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abstract: 'Active galactic Nuclei (AGNs) with their relativistic jets pointed towards the observer, form a subclass of luminous gamma-ray sources commonly known as blazars. The study of blazars is essential to improve our understanding on the AGNs emission mechanisms and evolution, as well as to map the extragalactic background light. To do so, however, one needs to correctly classify and measure a redshift for a large sample of these sources. The Third [*Fermi*]{}–LAT Catalog of High-Energy Sources (3FHL) contains $\approx1160$ blazars reported at energies greater than $10$GeV. However $\sim$25% of these sources are unclassified and $\sim$50% lack of redshift information. To increase the spectral completeness of the 3FHL catalog, we are working on an optical spectroscopic follow up campaign using 4–m and 8–m telescopes. In this paper, we present the results of the second part of this campaign, where we observed 23 blazars using the 4$m$ telescope at CTIO in Chile. We report all the 23 sources to be classified as BL Lacs, a confirmed redshift measurement for 3 sources, a redshift lower limit for 2 sources and a tentative redshift measurement for 3 sources.'
author:
- 'A.Desai'
- 'S. Marchesi'
- 'M. Rajagopal'
- 'M. Ajello'
title: 'Identifying the 3FHL catalog: III. Results of the CTIO-COSMOS optical spectroscopy campaign'
---
Introduction
============
Blazars are a peculiar class of active galactic nuclei (AGNs) which dominate the observable $\gamma$-ray Universe because of their extreme properties and abundant population. The blazar properties are a result of non-thermal emitting plasma traveling towards the observer causing relativistic amplification of flux. This leads to an amplification of low energy photons in the medium to intense levels via inverse Compton process, making blazars valuable sources to understand the physics of an AGN. The Third –LAT Catalog of High-Energy Sources [3FHL @ajello17], which encompasses seven years of observations made by the Large area telescope (LAT) aboard the [*Fermi Gamma-ray Space Telescope*]{} [@atwood09], contains more than 1500 sources detected at $>10$GeV, the vast majority of which ($\approx$ 1160) are blazars [@ajello17].
Innovative scientific results can be obtained using the blazar data collected by the LAT in the $\gamma$-ray regime, provided the redshift ($z$) of the observed blazar source is known. These are not only limited to blazar physics such as, understanding their basic emission processes [e.g. @ghisellini17] or their evolution with redshift [@ajello14], but also to other areas of study, like understanding the extragalactic background light (EBL), which encompasses all the radiation emitted by stars and galaxies and reprocessed radiation from interstellar dust, and its evolution with $z$ [@ackermann12; @dominguez13]. Out of the confirmed blazar sources reported in the 3FHL catalog a redshift measurement of only $\approx$50% sources is present [@ajello17]. To overcome this limitation, extensive optical spectroscopic campaigns, targeting those 3FHL objects still lacking redshift and classification, must be performed. Besides being used for redshift determination, optical spectroscopy campaigns of blazars are also essential to distinguish between blazar sub-classes, namely BL Lacs (BLL) and flat spectrum radio quasars (FSRQs). FSRQs are generally high redshift objects with average luminosity larger than that of the BLL [@padovani92; @paiano17]. As a result, the emission lines in the BLL spectra are weak or absent and the lines in FSRQs are extremely prominent. This is seen by the difference in the equivalent width (EW) of the lines where generally, FSRQ have lines with EW$>5$ and BLL have lines with EW$<5$ [@urry95; @ghisellini17]. The blazar sources not classified as FSRQ or BLL are listed as blazar candidates of uncertain type (BCU) in the 3FHL catalog, and constitute $\approx 25\,\%$ of the reported blazar sample [@ajello17]. Obtaining a spectroscopically complete classification of the blazars observed by LAT in the $\gamma$-ray regime is essential to validate claims of different cosmological evolution of the two classes [@ajello12; @ajello14].
The ground based telescopes used in the spectroscopy campaigns are generally of the 4–m,8–m and 10–m class type. While the 10–m and 8–m class telescopes are shown to be significantly more effective in obtaining redshift measurements for blazars [60–80% versus 25–40% success rate, see, e.g. @paiano17; @marchesi18], even 4–m class telescopes have proven to be useful for effectively distinguishing between the two different blazar subclasses [see @shaw13; @massaro14; @paggi14; @landoni15; @ricci15; @marchesini16; @alvarez16a; @alvarez16b].
This work is part of a larger spectroscopic follow-up campaign to classify the BCUs in the 3FHL catalog and measure their redshift. [The first part of the campaign took place in the second half of 2017, when we observed 28 sources in seven nights of observations at the 4–m telescope at Kitt Peak National Observatory (KPNO). The results of this work are reported in @marchesi18: we classified 27 out of 28 sources as BL Lacs, while the remaining object was found to be a FSRQ. Furthermore, we measured a redshift for 3 sources and set a lower limit on $z$ for other four objects; the farthest object in our KPNO sample has $z>$0.836. The spectroscopic campaign will then continue with seven nights of observations at the 4–m telescope at Cerro Tololo Interamerican Observatory (CTIO) in Chile and five nights of observations at the 8–m Gemini-N and Gemini-S telescopes (to be performed in 2019). In this work, we report the results]{} of the observations made during the first four nights at CTIO. Our source sample contains 23 BCUs in the 3FHL catalog without a redshift measurement. The paper is organized as follows: Section \[sample\_sel\] reports the criteria used in sample selection, Section \[obs\] describes the methodology used for the source observation and spectral extraction procedures, Section \[spectral\] lists the results of this work, both, for each individual source and also in general terms, while Section \[conclusion\] reports the conclusions inferred from this spectroscopic campaign.
Sample Selection {#sample_sel}
================
We selected the 23 objects in our sample among the BCUs in the 3FHL catalog, using the following three criteria.
- [**The object should have an measured optical magnitude measurement**]{}, and it should be V$\le$19.5. Based on previous works, sources with magnitude V$>$19.5 require more than two hours of observations to obtain an acceptable signal-to-noise ratio (S/N), therefore significantly reducing the number of sources that one can observe in a night.
- The 3FHL source should be bright in the hard $\gamma$-ray spectral regime ($f_{\rm 50-150 GeV}>$10$^{-12}$ erg s$^{-1}$ cm$^{-2}$). Selecting 3FHL objects bright in the 50–150GeV band ensures that the completeness of the 3FHL catalog evolves to lower fluxes as more optical observations are performed.
- The target should be observable from Cerro Tololo with an altitude above the horizon $\delta$$>$40(i.e., with airmass $<$1.5): this corresponds to a declination range -80$<$Dec$<$20. The target should also be observable in October, when the observations take place (i.e., it should have R.A.$\geq$09h0m00s and R.A.$\leq$0h30m00s).
A total of 77 3FHL sources satisfy all these criteria. Our 23 sources were selected among these 77 objects with the goal of covering a wide range of optical magnitudes (V=\[16–19.5\]) and, consequently, of potential redshifts and luminosities. [In Figure \[fig:hist\] we show the normalized V-band magnitude distribution of our sources, compared with the one of the overall population of 173 3FHL BCUs still lacking a redshift measurement and having available magnitude information. We also plot the magnitude distribution of the 28 sources studied in @marchesi18, where we sampled a larger number of bright sources (V$<$16) which all turned out to be featureless BL Lacs.]{} The sources used in our sample and their properties are listed in Table \[tab:sample\].
Observations and Data Analysis {#obs}
==============================
All the sources in our sample were observed using the 4$m$ Blanco telescope located at the Cerro Tololo Inter-American Observatory (CTIO) in Chile. The spectra were obtained using the COSMOS spectrograph with the Red grism and the 0.9$^{\prime\prime}$ slit. This experimental setup corresponds to a dispersion of $\sim$ 4Å pixel$^{-1}$, over a wavelength range $\lambda$=\[5000–8000\]Å, and a spectral resolution R$\sim$2100. The data were taken with the slit aligned along the parallactic angle. The seeing was 1.3$^{\prime\prime}$ during the first and third night, 1$^{\prime\prime}$ during the second night and 2.2$^{\prime\prime}$ in the last night, respectively; all four nights were photometric.
All spectra reported here are obtained by combining at least three individual observations of the source with varying exposure times. This allows us to reduce both instrumental effects and cosmic ray contribution. The data reduction is done following a standard procedure: the final spectra are all bias-subtracted, flat-normalized and corrected for bad pixels. [ We normalize the flat-field to remove any wavelength dependent variations that could be present in the flat-field source but not in the observed spectrum. This is done by fitting a cubic spline function on the calibration spectrum and taking a ratio of the flat-field to the derived fit [see response function in @iraf_doc]. We choose an order $>$5 for the cubic spline function fit with a $\chi^2$ value less than 1 to account for all variable features in the flat-field ]{}An additional visual inspection is also done on the combined spectra to remove any artificial features that may still be present. This data reduction and spectral extraction is done using the IRAF pipeline [@tody86].
The wavelength calibration for each source is done using the Hg-Ne lamp: we took a lamp spectrum after each observation of a source, to avoid potential shifts in the pixel-$\lambda$ calibration due to changes in the telescope position during the night. Finally, all spectra were flux-calibrated using a spectroscopic standard, which were observed using the same 0.9$^{\prime\prime}$ slit used in the rest of the analysis, and then corrected for the Galactic reddening using the extinction law by @cardelli89 and the $E(B-V)$ value based on the [@schlafly11] measurements, as reported in the NASA/IPAC Infrared Science Archive.[^1]
Spectral Analysis {#spectral}
=================
[ To visually enhance the spectral features of our sources, in Figure \[fig:spec\] we report the normalized spectra of the objects in our sample. These normalized spectra are obtained by dividing the flux-calibrated spectra using a continuum fit [an approach similar to the one reported in @landoni18]. The continuum is taken to be a power-law unless the optical shape is more complex, in which case the preferred fit is described in \[individual\_src\]. The S/N of the normalized spectrum is then measured in a minimum of five individual featureless regions in the spectrum with a width of $\Delta\lambda\approx40$.]{} The spectral analysis results for each source, including the computed S/N, are reported in Table \[tab:redshift\].
To find a redshift measurement, each spectrum was visually inspected for any absorption or emission feature. Any potential feature that matched known atmospheric lines[^2] was not taken into consideration. To test the reliability of any potential feature, its existence was verified in each of the individual spectral files used to obtain the final combined spectrum shown in Fig \[fig:spec\]. [ For example, the broad emission feature seen in the spectrum of 3FHL J0935.2-1735 around 5633 is not found in the individual files and is thus considered to be an artifact.]{} The verified features are then matched with common blazar lines, such as the Mg II doublet lines (2797 and 2803) or O III line (5007), to compute the redshift.
All the sources in our sample were classified as BLL based on their spectral properties. Out of the 23 sources, we were able to determine a redshift measurement for 3 sources, a lower limit on the redshift for 2 of them and a tentative redshift measurement for 3 of them. The remaining 15 sources in our sample were found to be featureless. [ Details for some of the sources for which a spectral feature or redshift is found are given in Sec \[individual\_src\]. These features are also listed in Table \[tab:redshift\] with the derived redshift measurement.]{}
Comments on Individual sources {#individual_src}
------------------------------
[**3FHL J0936.4-2109:**]{} This BCU is associated with the X-ray source 1RXS J093622.9-211031. The optical spectrum of this source shows the presence of two absorption features at 6176 and 6160. If they are associated with the Mg II doublet, a redshift measurement of 1.1974 and 1.1976 is obtained respectively. Corresponding to this $z$ value, other typical features observed in blazars, either in emission or in absorption (e.g., the Ca II doublet, the G-band, O II or O III features) will fall out of our observed wavelength range of $5000$$-8200$. We report a tentative lower limit of the redshift as $z>1.197$ for this BLL.
[**3FHL J1030.6-2029:**]{} This source is associated with the radio source NVSS J103040-203032. Its optical spectrum shows the presence of the Mg II doublet at 5579 and 5591 respectively. This gives a redshift lower limit of $z>0.995$.
[**3FHL J1042.8+0055:**]{} This source is associated with the X-ray source RBS 0895. A redshift value of 0.73 exists in the literature, [@mnras90], however the authors flagged it as an uncertain measurement. We were not able to detect any absorption or emission lines in our optical spectrum, so we classify this source as a BLL.
[**3FHL J1155.5-3418:**]{} This source is associated with the radio source NVSS J115520-341718. The Mg II doublet is identified in the optical spectrum of the source at 5174 and 5185 allowing us to measure the lower limit of the redshift as $z>0.849$.
[**3FHL J1212.1-2328:**]{} This source is associated with the radio source PMN J1212-2327. We obtain an optical spectrum with S/N of 102.8 and detect an emission feature at 8345 with an equivalent width of 0.8. If associated to the O III line, we derive a redshift $z$=0.666.
[**3FHL J1223.5-3033:**]{} This source is associated with the radio source NVSS J122337-303246. We see possible absorption features at 5245, 5256, 5577 and 6341. If 5245 and 5256 absorption features are associated with the Mg II line, a redshift of $0.875$ is measured. However we were not able to detect the presence of any other features and also identify the features at 5577 and 6341 to confirm the redshift measurement with certainty. This source is thus classified as a BLL and a tentative lower limit of $z$$>$0.875 is reported.
[**3FHL J1433.5-7304:**]{} This source is associated with the X-ray source 1RXS J143343.2-730433. One emission feature (H$_\alpha$) and four absorption features (G-band, Mg I,Na and Ca+Fe ) are detected in the spectrum. This gives us a redshift measurement of $z = 0.200$.
[**3FHL J1439.4-2524:**]{} This source is associated with the radio source NVSS J143934-252458. We detect two strong absorption lines at 6008 and 6115 and an absorption line at 6835 close to an atmospheric feature (6845) in its optical spectrum. If these lines are associated with the Mg I, Ca+Fe and NaD absorption features respectively, a redshift of $z=0.16$ is derived.
[**3FHL J1605.0-1140:**]{} The IR counterpart of this source is WISE J160517.53-113926.8. The optical spectrum shows the presence of an emission feature at 6801 with equivalent width of 7.044. This feature can be associated with the O II or O III line giving a redshift of 0.824 or 0.358 respectively, however due to no significant detection of any other emission or absorption features and a low S/N measurement, the redshift of this source cannot be measured with certainty.
Conclusion
==========
[ In this work, we present the results the optical spectroscopic campaign directed towards rendering the 3FHL a spectroscopically complete sample using the COSMOS spectrograph mounted on the 4$m$ Blanco telescope at CTIO in Chile.]{} We observed 23 extragalactic sources classified as BCU (blazars of uncertain classification) in the 3FHL catalog.
All the objects in our source sample are classified as BLL based on their observed optical spectrum. In the 3FHL catalog, out of the already classified 901 blazars $\approx 84.1\%$ sources are classified as BLL. Moreover out of the 28 sources observed by [@marchesi18], 27 are identified as BLL denoting that our results are not surprising.
Out of the 23 BLL in our sample we find a reliable [ redshift]{} measurement for 3 sources, a reliable [ redshift]{} constraint for 2 sources, a tentative [ redshift]{} constraint for 3 sources and a featureless spectrum with no [ redshift]{} measurement for the remaining 15 sources. Combining our results with the results of [@marchesi18], our optical spectroscopic campaign reports a redshift measurement for $\approx23.5\%$ of the observed BLL sources using 4$m$ telescopes. This measurement is in line with the expected consistency of $10-35\%$, obtained for redshift determination of pure BLL using using 4$m$ telescopes [@landoni15; @ricci15; @alvarez16a; @pena17]. Moreover, our work combined with [@marchesi18] also classifies, as either BLL or FSRQs, 51 blazars of previously uncertain classification.
The third and fourth part of our spectroscopic campaign will include observations from the 4$m$ CTIO telescope and 8$m$ Gemini-N and Gemini-S telescope respectively[^3]. Additionally we also aim to extend the campaign by inducing follow up observations[^4], similar to [@kaur18], using the Swift X-ray telescope. These follow up observations in the X-ray regime will help us confirm the classification of the blazar sources contributing to the spectral completion of the 3FHL catalog.
\[conclusion\]
Acknowledgements
================
A.D. acknowledges funding support from NSF through grant AST-1715256. S.M. acknowledges support from NASA contract 80NSSC17K0503. The authors thank Alberto Alvarez and Sean Points for the help provided during the observing nights at CTIO. This work made use of the TOPCAT software (Taylor 2005) for the analysis of data tables.
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[^1]: <http://irsa.ipac.caltech.edu/applications/DUST/>
[^2]: <https://www2.keck.hawaii.edu/inst/common/makeewww/Atmosphere/atmabs.txt>
[^3]: [*Fermi*]{} Guest Investigator Program Cycle 11, ID:111128, PI: S. Marchesi.
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|
---
abstract: 'We study the properties of the Green’s functions of the fermions in charged Gauss-Bonnet black hole. What we want to do is to investigate how the presence of Gauss-Bonnet coupling constant $\alpha$ affects the dispersion relation, which is a characteristic of Fermi or non-Fermi liquid, as well as what properties such a system has, for instance, the Particle-hole (a)symmetry. One important result of this research is that we find for $q=1$, the behavior of this system is different from that of the Landau Fermi liquid and so the system can be candidates for holographic dual of generalized non-Fermi liquids. More importantly, the behavior of this system increasingly similar to that of the Landau Fermi liquid when $\alpha$ is approaching its lower bound. Also we find that this system possesses the Particle-hole asymmetry when $q\neq 0$, another important characteristic of this system. In addition, we also investigate briefly the cases of the charge dependence.'
author:
- 'Jian-Pin Wu'
title: 'Holographic fermions in charged Gauss-Bonnet black hole'
---
Introduction
============
As is known to all, the success of the single electron picture of metals rests on Landau’s Fermi-liquid theory. The metal in such theory is treated as a gas of Fermi particles whose interactions are weak and not as significant as that of the original electrons. The reason is that particles of this model are not the original electrons but the electron-like quasi-particles that emerge from the interacting gas of electrons. However, recently some new materials, including the cuprate superconductors and other oxides, seem to lie outside this framework. For these new materials, we refer them to the non-Fermi liquid metals. They show lots of new physical properties which can not be understood in terms of weakly interacting electron-like objects. For the non-Fermi liquid, a sharp Fermi surface still exists. But quasi-particle picture breaks down generically. Although there have been many phenomenological models to describe the non-Fermi liquid, a general theoretical framework characterizing non-Fermi liquid metals remains a suspense. Therefore, it is necessary to develop a basic principle for non-Fermi liquid. Maybe the AdS/CFT correspondence can provide us with a possible clue to yield the basic principle of non-Fermi liquid. Indeed, by applying AdS/CFT correspondence [@Maldacena1997; @Gubser1998; @Witten1998], some breakthroughs have been achieved, that is, we find some new classes of non-Fermi liquids [@HongLiuNon-Fermi]. Varieties of holographic fermions models and their extensions are being explored. These can be seen from Refs. [@StringQEFL; @FermionsBTZBH; @HNFMagneticFieldBasu; @StrangeMetallicHartnoll; @SemiHFLPolchinski; @HNFLMagneticFGubankova; @HFLDynamicalGap; @HFLDipoleCoupling].
However, before string theory is fully understood, it is necessary to consider the higher curvature (or derivative) interactions in an effective gravitational theory. From the point of view of the AdS/CFT, the higher curvature interactions on the gravity side correspond to finite coupling corrections on the gauge theory side, thus broadening the class of field theories one can holographically study. The main motivation of considering these corrections derives from the fact that string theory contains higher curvature corrections arising from stringy effects. A simple and useful model with regard these corrections is Gauss-Bonnet (GB) gravity, which contains only the curvature-squared interaction. Several models of holographic superconductor have been exploited in this setting [@GBHS1; @GBHS2; @GBHS3; @GBHS4; @GBHS5; @GBHS6; @GBHS7]. Recently, a new higher derivative theory of gravity (quasi-topological gravity) is constructed, which contains not only the curvature-squared interaction but also a curvature-cubed interaction [@QT; @QTblackhole]. The corresponding holographic superconductor models have also been discussed in [@HSinQT1; @HSinQT2]. Another interesting extension comes from the coupling between the Maxwell field and the bulk Weyl tensor [@Weyl1; @Weyl2]. Also, in Refs. [@Weyl2; @WeylHS], they find that the Weyl corrections can describe the dual field theories with both the weak-coupling and the strong-coupling, which is a very interesting and significant holographic study on the dual field theories. Therefore, it is deserving to further exploit the effect of higher curvature interactions on fermions by using AdS/CFT correspondence step by step. In this paper, we consider the Gauss-Bonnet term as the first step to introduce the stringy correction into the gravitational action. And we will mainly focus on how the GB coupling constant affects the spectral function of the fermions.
Our paper is organized as follows. In section II, we briefly introduce the charged Gauss-Bonnet black hole. Following Ref. [@HongLiuNon-Fermi], we obtain the Dirac equation of the probe fermions in the charged Gauss-Bonnet black hole in section III. The numerical results are presented in section IV where we will mainly focus on the dispersion and the Particle-hole asymmetry in particular. Conclusions and discussions follow in section VI. Finally, in appendix A, we give a brief summary on the analytic treatment in the low frequency limit developed in Ref.[@HongLiuAdS2].
Charged Black Holes in Gauss-Bonnet gravity {#CBHinGBG}
===========================================
In this section, a brief review of black holes in Gauss-Bonnet gravity is given as follows. Conventionally, we begin with the following action coupled a vector field $A_{a}$ [^1]: $$\begin{aligned}
\label{GBaction}
S=\int d^{5}x \sqrt{-g}\left[R+\frac{12}{L^{2}}+\frac{\alpha}{2}\left(R^{abcd}R_{abcd}-4R^{ab}R_{ab}+R^{2}\right)-\frac{1}{4}F^{ab}F_{ab}\right],\end{aligned}$$ where $R_{abcd}$, $R_{ab}$ and $R$ are the Riemann curvature tensor, Ricci tensor, and the Ricci scalar, respectively. $L$ is the AdS radius and for convenience, we will set $L=1$. $\alpha$ is Gauss-Bonnet coupling constant and the constraints on it will also be discussed in the following parts of this paper. And as is commonly known, $F_{ab}=\partial_{a}A_{b}-\partial_{b}A_{a}$. By applying the principle of variation to (\[GBaction\]), we can easily obtain the equations of motion:
Maxwell¡¯s equations $$\begin{aligned}
\label{MaxwellE}
\nabla_{a}F^{ab}=0,\end{aligned}$$ and Einstein¡¯s equations $$\begin{aligned}
\label{EinsteinE}
R_{ab}-\frac{1}{2}Rg_{ab}+6g_{ab}-\alpha\left[H_{ab}-\frac{1}{4}Hg_{ab}\right]
=\frac{1}{2}F_{ac}F_{b}^{c}-\frac{1}{8}F_{cd}^{2}g_{ab},\end{aligned}$$ where $$\begin{aligned}
\label{TensorHab}
H_{ab}=R_{a}^{cde}R_{bcde}-2R_{ac}R_{b}^{c}-2R_{acbd}R^{cd}+RR_{ab}.\end{aligned}$$ and $$\begin{aligned}
\label{TensorHtrace}
H=H_{a}^{a}.\end{aligned}$$ Under a certain circumstance in which we take the following metric with planar symmetry $$\begin{aligned}
\label{MetricA}
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(dx^{2}+dy^{2}+dz^{2}),\end{aligned}$$ and the ansatz of the gauge fields as $$\begin{aligned}
\label{GaugeFA}
A_{a}=(A_{t}(r),0,0,0),\end{aligned}$$ the equations of motion (\[MaxwellE\]) and (\[EinsteinE\]) reduce to $$\begin{aligned}
\label{MaxwellESimplify}
A''_{t}+\frac{3}{r}A'_{t}=0,\end{aligned}$$ $$\begin{aligned}
\label{EinsteinESimplify}
\left(1-\frac{2\alpha f}{r^{2}}\right)f'+\frac{2}{r}f-4r
+\frac{r A'^{2}_{t}}{6}=0,\end{aligned}$$ where the prime represents derivative with respect to $r$. The solutions of the above equations are [@GBblackhole1; @GBblackhole2] $$\begin{aligned}
\label{metricf}
f(r)=\frac{r^{2}}{2\alpha}\left[1-\sqrt{1-4\alpha\left(1-\frac{r_{+}^{4}}{r^{4}}\right)+\frac{4\alpha \mu^{2}r_{+}^{2}}{3r^{4}}\left(1-\frac{r_{+}^{2}}{r^{2}}\right)}\right],\end{aligned}$$ and $$\begin{aligned}
\label{metricA}
A_{t}=\mu\left(1-\frac{r_{+}^{2}}{r^{2}}\right),\end{aligned}$$ where $r_{+}$ is the horizon radius determined by $f(r_{+})=0$[^2], and $\mu$ can be identified with the chemical potential of the dual field theory. In the Einstein limit $\alpha\rightarrow 0$ the formula above reduces to the Reissner-Nordstr$\ddot{o}$m AdS black hole $$\begin{aligned}
\label{metricfRN}
f(r)|_{\alpha\rightarrow 0}=r^{2}\left[\left(1-\frac{r_{+}^{4}}{r^{4}}\right)-\frac{\mu^{2}r_{+}^{2}}{3r^{4}}\left(1-\frac{r_{+}^{2}}{r^{2}}\right)\right].\end{aligned}$$ By using the standard approach of euclidean continuation near the black hole horizon, the Hawking temperature of the black hole in Gauss-Bonnet gravity is $$\begin{aligned}
\label{HawkingT}
T=\frac{f'(r_{+})}{4\pi}=\frac{1}{\pi}\left(1-\frac{\mu^{2}}{6}\right),\end{aligned}$$ which is also the temperature of the conformal field theory on the boundary of the AdS spacetime. It is independent of the Gauss-Bonnet coupling constant $\alpha$.
In the following, we will discuss the behaviors of places verging on the boundary ($r\rightarrow \infty$) and the horizon ($r\rightarrow 1$), respectively. Near the boundary of the bulk, the redshift factor $f(r)$ becomes $$\begin{aligned}
\label{metricfinfty}
f_{\infty}\equiv \lim_{r\rightarrow \infty}f(r)=\frac{r^{2}}{2\alpha}\left[1-\sqrt{1-4\alpha}\right]=r^{2}C_{\alpha},\end{aligned}$$ where we denote $C_{\alpha}\equiv(1-\sqrt{1-4\alpha})/2\alpha$. Then, the geometry of the black hole on the boundary can be reexpressed as $$\begin{aligned}
\label{MetricB}
ds^{2}=-r^{2}C_{\alpha}dt^{2}+\frac{dr^{2}}{r^{2}C_{\alpha}}+r^{2}(dx^{2}+dy^{2}+dz^{2}).\end{aligned}$$ It is also pure $AdS_{5}$, but depends on the GB coupling constant $\alpha$. In order to have a well-defined anti-de Sitter vacuum for the gravity theory, a certain condition $\alpha\leq 1/4$ should be required, whose upper bound $\alpha=1/4$ is known as the Chern-Simons limit. Furthermore, considering the causality of dual field theory on the boundary, there exists a stronger constraint on the GB coupling in five dimensions [@GBcouplingConstraint1; @GBcouplingConstraint2; @GBcouplingConstraint3; @GBcouplingConstraint4; @GBcouplingConstraint5; @GBcouplingConstraint6; @GBcouplingConstraint7; @GBcouplingConstraint8; @GBcouplingConstraint9; @GBcouplingConstraint10] $$\begin{aligned}
\label{GBcouplingConstraint}
-\frac{7}{36}\leq \alpha\leq \frac{9}{100}.\end{aligned}$$ As for the condition of the horizon, we in reality do a similar job, but for this part, we consider a special case, that is, the zero-temperature which means $\mu=\sqrt{6}$. Then, the redshift factor $f(r)$ becomes $$\begin{aligned}
\label{metricfT0}
f(r)|_{T=0}=\frac{r^{2}}{2\alpha}\left[1-\sqrt{1-4\alpha+\frac{12\alpha}{r^{4}}-\frac{8\alpha}{r^{6}}}\right].\end{aligned}$$ Obviously, when $r\rightarrow 1$, $f(r)\approx 12(r-1)^{2}$, which is also independent of the Gauss-Bonnet coupling constant $\alpha$. In light of this, near the horizon, the geometry is no longer the pure $AdS_{5}$, but the $AdS_{2}\times \mathbb{R}^{3}$ with the curvature radius of $AdS_{2}$, $L_{2}=\frac{1}{\sqrt{12}}L$.
Dirac equation
==============
Now, we consider probe fermions in the charged Gauss-Bonnet black hole. We have the bulk fermion action as following [@HongLiuNon-Fermi] $$\begin{aligned}
\label{actionspinor}
S_{D}=i\int d^{5}x \sqrt{-g}\overline{\zeta}\left(\Gamma^{a}\mathcal{D}_{a}-m_{\zeta}\right)\zeta,\end{aligned}$$ where $\mathcal{D}_{a}$ is the covariant derivative given by[^3]$^{,}$[^4] $$\begin{aligned}
\label{Dderivative}
\mathcal{D}_{a}=\partial_{a}+\frac{1}{4}(\omega_{\mu\nu})_{a}\Gamma^{\mu\nu}-iqA_{a},\end{aligned}$$ $(\omega_{\mu\nu})_{a}$ is the spin connection 1-forms given by $$\begin{aligned}
\label{spinconnectionD}
(\omega_{\mu\nu})_{a}=(e_{\mu})^{b}\nabla_{a}(e_{\nu})_{b},\end{aligned}$$ and $$\begin{aligned}
\label{spinconnection}
\Gamma^{\mu\nu}=\frac{1}{2}[\Gamma^{\mu},\Gamma^{\nu}],~~~~~~\Gamma^{a}=(e_{\mu})^{a}\Gamma^{\mu},\end{aligned}$$ where $(e_{\mu})^{a}$ form a set of orthogonal normal vector bases. The Dirac equation derived from the action $S_{D}$ is expressed as $$\begin{aligned}
\label{DiracEquation1}
\Gamma^{a}\mathcal{D}_{a}\zeta-m_{\zeta}\zeta=0.\end{aligned}$$ We should choose the following orthogonal normal vector bases $$\begin{aligned}
\label{VectorBases}
(e_{\mu})^{a}=\sqrt{g^{\mu\mu}}(\frac{\partial}{\partial
\mu})^{a},~~\mu=t,x,y,z,r.\end{aligned}$$ Using (\[spinconnectionD\]), one can calculate the non-vanishing components of spin connections as follows $$\begin{aligned}
\label{SpinConnections}
(\omega_{tr})_{a} &=& -(\omega_{rt})_{a}=-\sqrt{g^{rr}}\partial_{r}(\sqrt{g_{tt}})(dt)_{a},
\nonumber\\ \label{S1}(\omega_{ir})_{a} &=& -(\omega_{ri})_{a}=-\sqrt{g^{rr}}\partial_{r}(\sqrt{g_{ii}})(dx^{i})_{a},~~i=x,y,z.\end{aligned}$$ Following Ref.[@HongLiuNon-Fermi], we can make a transformation $\zeta=(-g g^{rr})^{-\frac{1}{4}}\mathcal{F}$ to remove the spin connection in Dirac equation. Then, the equation turns out to be $$\begin{aligned}
\label{DiracEquation2}
\sqrt{g^{rr}}\Gamma^{r}\partial_{r}\mathcal{F}
+\sqrt{g^{tt}}\Gamma^{t}(\partial_{t}-iq A_{t})\mathcal{F}
+\left(\sum_{i}\sqrt{g^{ii}}\Gamma^{i}\partial_{i}\right)\mathcal{F}
-m_{\zeta}\mathcal{F}=0.\end{aligned}$$ Next, we will work in Fourier space where we expand $\mathcal{F}$ as $\mathcal{F}=F e^{-i\omega t +ik_{i}x^{i}}$. Then, the Dirac equation can be rewritten as $$\begin{aligned}
\label{DiracEinFourier}
\sqrt{g^{rr}}\Gamma^{r}\partial_{r}F
-i(\omega+q)\sqrt{g^{tt}}\Gamma^{t}F
+i k \sqrt{g^{xx}}\Gamma^{x}F
-m_{\zeta}F=0.\end{aligned}$$ where due to rotational symmetry in $x-y-z$ directions, we set $k_{y}=k_{z}=0$ and $k_{x}=k$ without losing generality. We will choose the following basis for our gamma matrices as in[@HongLiuAdS2; @Photoemission] $$\begin{aligned}
\label{GammaMatrices}
&& \Gamma^{r} = \left( \begin{array}{cc}
-\sigma^3 & 0 \\
0 & -\sigma^3
\end{array} \right), \;\;
\Gamma^{t} = \left( \begin{array}{cc}
i \sigma^1 & 0 \\
0 & i \sigma^1
\end{array} \right), \;\;
\Gamma^{x} = \left( \begin{array}{cc}
-\sigma^2 & 0 \\
0 & \sigma^2
\end{array} \right),
\qquad \ldots\end{aligned}$$ Splitting the 4-component spinors into two 2-component spinors $F=(F_{1},F_{2})^{T}$, we have a new version of the Dirac equation as $$\begin{aligned}
\label{DiracEF}
\sqrt{g^{rr}}\partial_{r}\left( \begin{matrix} F_{1} \cr F_{2} \end{matrix}\right)
+m_{\zeta}\sigma^3\otimes\left( \begin{matrix} F_{1} \cr F_{2} \end{matrix}\right)
=\sqrt{g^{tt}}(\omega+qA_{t})i\sigma^2\otimes\left( \begin{matrix} F_{1} \cr F_{2} \end{matrix}\right)
\mp k \sqrt{g^{xx}}\sigma^1 \otimes \left( \begin{matrix} F_{1} \cr F_{2} \end{matrix}\right)
~,\end{aligned}$$ a decouple equation between $F_{1}$ and $F_{2}$. After achieving this equation, we will discuss it in a special case, that is, near the boundary ($r\rightarrow \infty$) in order to acquire some information about the Green function near the boundary. From the forgoing information, we know that in this special case, the geometry is an an asymptotic $AdS_{5}$, $g_{rr}\approx\frac{1}{r^{2}}$ and $g_{ii}\approx r^{2},~~~i=t,x,y,z$. Under such a condition, Eq. (\[DiracEF\]) becomes $$\begin{aligned}
\label{DiracEboundary}
(r\partial_{r}+m_{\zeta}\sigma^3)\otimes \left( \begin{matrix} F_{1} \cr F_{2} \end{matrix}\right)=0~,\end{aligned}$$ another decouple equation between $F_{1}$ and $F_{2}$ whose solution can be expressed as $$\begin{aligned}
\label{BoundaryBehaviour}
F_{\alpha} \buildrel{r \to \infty}\over {\approx} a_{\alpha}r^{m_{\zeta}}\left( \begin{matrix} 0 \cr 1 \end{matrix}\right)
+b_{\alpha}r^{-m_{\zeta}}\left( \begin{matrix} 1 \cr 0 \end{matrix}\right),
\qquad
\alpha = 1,2~.\end{aligned}$$
If $b_{\alpha}\left( \begin{matrix} 1 \cr 0 \end{matrix}\right)$ and $a_{\alpha}\left( \begin{matrix} 0 \cr 1 \end{matrix}\right)$ are related by $$\begin{aligned}
\label{EVEsourceRelation}
b_{\alpha}\left( \begin{matrix} 1 \cr 0 \end{matrix}\right)=\mathcal{S}a_{\alpha}\left( \begin{matrix} 0 \cr 1 \end{matrix}\right)~,\end{aligned}$$ then the boundary spinor Green functions $G$ is given by [@HongLiuSpinor] $$\begin{aligned}
\label{Grgamma}
G=-i \mathcal{S}\gamma^{0}~.\end{aligned}$$ where $\gamma^{0}$ is the gamma matrices of the boundary theory and $\gamma^{0}=i\sigma^1$. In order to find the matrix $\mathcal{S}$, we will make such a decomposition $F_{\pm}=\frac{1}{2}(1\pm \Gamma^{r})F$ according to eigenvalues of $\Gamma^{r}$. Then $$\begin{aligned}
\label{gammarDecompose}
F_{+}=\left( \begin{matrix} \mathcal{B}_{1} \cr \mathcal{B}_{2} \end{matrix}\right),~~~~~~~F_{-}=\left( \begin{matrix} \mathcal{A}_{1} \cr \mathcal{A}_{2} \end{matrix}\right),\end{aligned}$$ with $$\begin{aligned}
\label{gammarDecompose}
F_{\alpha} \equiv \left( \begin{matrix} \mathcal{A}_{\alpha} \cr \mathcal{B}_{\alpha} \end{matrix}\right).\end{aligned}$$
Now, by using (\[BoundaryBehaviour\]), (\[EVEsourceRelation\]) and (\[Grgamma\]), we can express the boundary Green functions as following $$\begin{aligned}
\label{GreenFBoundary}
G (\omega,k)= \lim_{r\rightarrow \infty} r^{2m_{\zeta}} \widetilde{G}(r,\omega,k),\end{aligned}$$ here we have defined the following matrices: $$\begin{aligned}
\label{AB}
\widetilde{G}(r,\omega,k)\equiv \left( \begin{array}{cc}
\widetilde{G}_{11} & 0 \\
0 & \widetilde{G}_{22} \end{array} \right) \ ,\end{aligned}$$ and $$\begin{aligned}
\label{GAB}
\widetilde{G}_{\alpha\alpha}(r,\omega,k)\equiv \frac{\mathcal{A}_{\alpha}}{\mathcal{B}_{\alpha}}~~~,\alpha=1,2.\end{aligned}$$
Moreover, under the same decomposition, the Dirac equation (\[DiracEF\]) can be rewritten as $$\begin{aligned}
\label{DiracEAB1}
(\sqrt{g^{rr}}\partial_{r}\pm m_{\zeta})\left( \begin{matrix} \mathcal{A}_{1} \cr \mathcal{B}_{1} \end{matrix}\right)
=\pm(\omega+qA_{t})\sqrt{g^{tt}}\left( \begin{matrix} \mathcal{B}_{1} \cr \mathcal{A}_{1} \end{matrix}\right)
-k \sqrt{g^{xx}} \left( \begin{matrix} \mathcal{B}_{1} \cr \mathcal{A}_{1} \end{matrix}\right)
~,\end{aligned}$$ $$\begin{aligned}
\label{DiracEAB2}
(\sqrt{g^{rr}}\partial_{r}\pm m_{\zeta})\left( \begin{matrix} \mathcal{A}_{2} \cr \mathcal{B}_{2} \end{matrix}\right)
=\pm(\omega+qA_{t})\sqrt{g^{tt}}\left( \begin{matrix} \mathcal{B}_{2} \cr \mathcal{A}_{2} \end{matrix}\right)
+k \sqrt{g^{xx}} \left( \begin{matrix} \mathcal{B}_{2} \cr \mathcal{A}_{2} \end{matrix}\right)
~.\end{aligned}$$
Using the method developed in [@HongLiuUniversality; @HongLiuSpinor; @HongLiuAdS2], one can package the Dirac equation (\[DiracEAB1\]) and (\[DiracEAB2\]) into the evolution equation of $\widetilde{G}(r,\omega,k)$, which will be more convenient to impose the initial conditions at the horizon and read off the boundary Green functions, $$\begin{aligned}
\label{DiracEF1}
(\sqrt{g^{rr}}\partial_{r}
+2m_{\zeta})\widetilde{G}
=\widetilde{G}\left(\sqrt{g^{tt}}(\omega+qA_{t})+k \sqrt{g^{xx}}\sigma^{3}\right)\widetilde{G}
+(\sqrt{g^{tt}}(\omega+qA_{t})-k \sqrt{g^{xx}}\sigma^{3})~.\end{aligned}$$
The boundary condition of the matrix $\widetilde{G}(r,\omega,k)$ in this new equation $$\begin{aligned}
\label{GatTip}
\widetilde{G}_{\alpha\alpha}(r,\omega,k)\buildrel{r \to 1}\over =i.\end{aligned}$$ can be derived from the requirement that the solutions of Eqs. (\[DiracEAB1\]) and (\[DiracEAB2\]) at the horizon, $r\rightarrow 1$, be in-falling.
However, we also note that for $T=0$ and $\omega=0$, the boundary condition has to be modified as follows $$\begin{aligned}
\label{GhorizonTw0}
\widetilde{G}_{\alpha\alpha}(r,\omega=0,k)\buildrel{r \to 1}\over =\frac{m_{\zeta}-\sqrt{m_{\zeta}^{2}+k^{2}-\frac{\mu_{q}^{2}}{12}-i\epsilon}}{k+\frac{\mu_{q}}{\sqrt{12}}}.\end{aligned}$$
Properties of spectral functions
================================
General behavior {#GeneralB}
----------------
As Ref.[@HongLiuNon-Fermi], we also have some symmetry properties of the Green function by direct inspection of the equation (\[DiracEF1\]). We itemize them as follows:
\(1) $G_{22}(\omega,k)=G_{11}(\omega,-k)$; (2) $G_{22}(\omega,k;-q)=G_{11}^{\ast}(-\omega,k;q)$;
For the case $m_{\zeta}=0$,
\(3) $G_{22}(\omega,k)=-\frac{1}{G_{11}(\omega,k)}$; (4) $G_{22}(\omega,k=0)=G_{11}(\omega,k=0)=i$.
When the background geometry is pure $AdS_{5}$ (Eq. (\[MetricB\])), for massless bulk fermion, the Dirac equation (\[DiracEF1\]) can be explicitly expressed as $$\begin{aligned}
\label{DiracEFpureAdS1}
r^{2}\partial_{r}\mathcal{G}_{11}=\left(\frac{\omega+\mu_{q}}{C_{\alpha}}+\frac{k}{C_{\alpha}^{1/2}}\right)\mathcal{G}_{11}^{2}
+\frac{\omega+\mu_{q}}{C_{\alpha}}-\frac{k}{C_{\alpha}^{1/2}},\end{aligned}$$ $$\begin{aligned}
\label{DiracEFpureAdS2}
r^{2}\partial_{r}\mathcal{G}_{22}=\left(\frac{\omega+\mu_{q}}{C_{\alpha}}-\frac{k}{C_{\alpha}^{1/2}}\right)\mathcal{G}_{22}^{2}
+\frac{\omega+\mu_{q}}{C_{\alpha}}+\frac{k}{C_{\alpha}^{1/2}},\end{aligned}$$ where we denote the Green function in pure $AdS_{5}$ background as $\mathcal{G}$. The solution of the above equations can be easily obtained as [^5] $$\begin{aligned}
\label{GreenFpureAdS}
\mathcal{G}_{11}=-\sqrt{\frac{(\omega+\mu_{q})/C_{\alpha}-k/C_{\alpha}^{1/2}+i\epsilon}{(\omega+\mu_{q})/C_{\alpha}+k/C_{\alpha}^{1/2}+i\epsilon}},~~~
\mathcal{G}_{22}=\sqrt{\frac{(\omega+\mu_{q})/C_{\alpha}+k/C_{\alpha}^{1/2}+i\epsilon}{(\omega+\mu_{q})/C_{\alpha}-k/C_{\alpha}^{1/2}+i\epsilon}},\end{aligned}$$ where $\epsilon\rightarrow 0$. For $q=0$, it is clear that the spectral function has a Particle-hole symmetry (symmetry under $(\omega,k)\rightarrow (-\omega,-k)$). However, when $q\neq 0$, the Particle-hole symmetry is broken (Particle-hole asymmetry). In addition, the spectral function $Im \mathcal{G}$ has also an edge-singularity along $\omega=\pm k$ and vanishs in the region $\omega\in (-k,k)$.
Now, we turn to the charged Gauss-Bonnet black hole background (Eqs. (\[MetricA\]) and (\[metricf\])). In this case, the Dirac equation (\[DiracEF1\]) becomes $$\begin{aligned}
\label{DiracEFCGB1}
f^{1/2}\partial_{r}\widetilde{G}_{11}+2m_{\zeta}\widetilde{G}_{11}
=\left[\frac{1}{f^{1/2}}(\omega+qA_{t})+k\frac{1}{r}\right]\widetilde{G}_{11}^{2}
+\frac{1}{f^{1/2}}(\omega+qA_{t})-k\frac{1}{r},\end{aligned}$$ $$\begin{aligned}
\label{DiracEFCGB2}
f^{1/2}\partial_{r}\widetilde{G}_{22}+2m_{\zeta}\widetilde{G}_{22}
=\left[\frac{1}{f^{1/2}}(\omega+qA_{t})-k\frac{1}{r}\right]\widetilde{G}_{22}^{2}
+\frac{1}{f^{1/2}}(\omega+qA_{t})+k\frac{1}{r}.\end{aligned}$$
We can solve the above equations numerically with the boundary conditions (\[GatTip\]) to investigate the properties of the spectral function. In this paper, we will only focus on the massless fermion ($m_{\zeta}=0$) and extremal charged GB black hole (zero temperature limit). The dependence of the spectral function on mass and temperature will be discussed in the future works.
Here we will do several quick checks on the consistency of our numerics. In FIG. \[GeneralB\], we show the spectral function $Im G_{22}$ at $k=2.2<\mu_{q}$ (left plot) and $k=4.0>\mu_{q}$ (right plot) for $m_{\zeta}=0$ and $q=1$ ($\mu_{q}=\sqrt{6}$). Firstly, the divergence in the vacuum turns out to be a peak of finite size at $\omega+\mu_{q}\approx \pm k$. In addition, independent of the parameter $\alpha$, for a fixed large $k\gg \mu_{q}$, $Im G_{22}$ is roughly zero in the region $\omega+\mu_{q}\in (-k,k)$ and asymptote to $1$ as $|\omega|\rightarrow \infty$, which recovers the behavior in the vacuum. It is consistent with the Green function $\mathcal{G}$ in pure $AdS_{5}$ background (Eq. (\[GreenFpureAdS\])). The height and width of the peak vary with the GB coupling constant $\alpha$. When $\alpha$ increases, the peak becomes sharper and narrower. We also note that for $k=2.2<\mu_{q}$ (left plot), the deviation from the vacuum behavior becomes significant. For some more concrete investigations, we will discuss some specific properties of the spectral function in the subsequent subsection.
Fermi surface and the dispersion relation
-----------------------------------------
In this subsection, we will focus on the dispersion relation between small $\tilde{k}=k-k_{F}$ and $\omega$, a characteristic of Fermi or non-Fermi liquid. In order to achieve this goal, firstly, we have to find the Fermi surface. As is known to all, the fermion is created near the Fermi surface and so it should have a long lifetime. Therefore, when the energy equals the Fermi energy and the momentum equals the Fermi momentum ($k_{F}$), the spectral function of this system should have a sharp quasi-particle peak. In this paper, we will adopt the conventions [@HongLiuNon-Fermi; @StringQEFL], where energy equal to the Fermi energy corresponds to the frequency vanishing. Therefore, in order to find the Fermi surface, we can solve the equations (\[DiracEFCGB1\]) and (\[DiracEFCGB2\]) numerically with the boundary conditions (\[GatTip\]). For definiteness, we firstly focus on the cases of $q=1$ (with $\mu_{q}=\sqrt{6}$) in this subsection and consider the cases of charge dependence in the next subsection. For $\alpha=-0.19$, we obtain a sharp quasi-particle-like peak near $k_{F}=2.071564$ (Fig.\[3D\] and Fig.\[fermiS\]). Similarly, we can also get different fermi momentums corresponding to different $\alpha$, such as $k_{F}\approx 1.7821$ for $\alpha=0.09$, $k_{F}\approx 1.8770$ for $\alpha=0.01$, $k_{F}\approx 1.8879$ for $\alpha=0.0001$ and $k_{F}\approx 1.8880$ for $\alpha=0$ (Fig.\[fermiS\]). We note that the fermi momentum is $\alpha$-dependent.
Now, we can move on to investigate the behavior of $Im G_{22}$ in the region of small $\tilde{k}=k-k_{F}$ and $\omega$. By fitting the data, we find that there exists a dispersion relation between $\tilde{k}$ ($\tilde{k}\rightarrow 0_{-}$) and $\tilde{\omega}(\tilde{k})$ (Fig.\[dispersion\]), $i.e.$, $$\begin{aligned}
\label{Ldispersion}
\tilde{\omega}(\tilde{k})\sim \tilde{k}^{\delta},~~~\end{aligned}$$ where $\delta\approx 1.13$ for $\alpha=-0.19$, $\delta\approx 1.35$ for $\alpha=0$ and $\delta\approx 1.55$ for $\alpha=0.09$.
In addition, we can also find that the scaling behavior of the height of $ImG_{22}$ at the maximum as follows: $$\begin{aligned}
\label{ScalingHeight}
Im G_{22}(\tilde{\omega},\tilde{k})\sim \tilde{k}^{-\beta},~~~\beta\approx 1,\end{aligned}$$ for all $\alpha$ (Fig.\[dispersion\]).
Here come some comments on the above two scaling behaviors. They are different from the landau Fermi liquid, which has exponents $\delta=\beta=1$. Therefore, the system can be candidates for holographic dual of generalized non-Fermi liquids. However, we also note that when the Gauss-Bonnet coupling constant $\alpha$ is approaching the lower bound, the parameter $\delta$ decreases, indicating the behavior of this system is more similar to that of the landau Fermi liquid. It seems that the GB term can also describe the dual field theories with both the weak-coupling and the strong-coupling.
In fact, when the Fermi momentum $k_{F}$ has been determined numerically, the scaling exponents $\delta$ in the dispersion relation (\[Ldispersion\]) can also be computed by the analytical method developing in the Ref.[@HongLiuAdS2][^6]. We should compare the numerical result with that obtained by the analytical method. The result summarized in Table \[Edelta\]. From Table \[Edelta\], we can see that the numerical result is agree well with the analytical that.
$~~\alpha~~$ $-0.19$ $0$ $0.09$
---------------------------------- --------------- --------------- --------------- -- -- --
$\delta$ (numerical result) $1.12612$ $1.35345$ $1.55103$
$\delta$ (analytical result) $1.14423$ $1.38474$ $1.59727$
: \[Edelta\] The scaling exponents $\delta$ for different GB parameter $\alpha$ numerically and analytically.
Charge dependence
-----------------
As observed in Ref.[@HongLiuNon-Fermi; @HongLiuAdS2], the Fermi momentum $k_{F}$ increases as we amplify the charge $q$. These features are still preserved in the charged Gauss-Bonnet black hole for fixed GB parameter $\alpha$. We list the values of the Fermi momentum $k_{F}$ for a few other values of charge $q$ for the primary Fermi surface in Table \[FermiM\][^7].
After the Fermi momentum $k_{F}$ is determined numerically, we can compute the scaling exponent $\delta$ by using Eqs. (\[LdispersionA\]) and (\[nuk\]). The results are showed in Table \[delta\]. For comparison, we also present the numerical fitting for $q=1.2$ and $\alpha=-0.19,~0.09$ in Fig.\[dispersionq12\][^8]. From Table \[delta\] and Fig.\[dispersionq12\], we can see that for fixed GB parameter $\alpha$, with the increase of charge $q$, the scaling exponent $\delta$ decreases rapidly and will asymptote to $1$ for larger values of $q$. It is in agreement with that found in Ref.[@HongLiuNon-Fermi]. When the charge $q$ is increased to certain values (for example, $q=1.2$), we find that the scaling exponent $\delta$ decreases with the decrease of the values of GB parameter $\alpha$ and asymptote to $1$ for smaller $\alpha$. While for larger charge $q$ (for example, $q=1.5$), the scaling exponent $\delta\approx 1$ independent of the $\alpha$. For furthermore exploration on how the charge $q$ and GB parameter $\alpha$ affect together the scaling exponent $\delta$, we leave it for future work.
$~~ ~~$ $q=0.5$ $q=1.2$ $q=1.5$
-------------------- ------------- ---------------- ----------------- -- -- --
$~~\alpha=-0.19~~$ $0.87$ $2.581414$ $3.3570706$
$~~\alpha=0~~$ $0.81$ $2.349913$ $3.0572678$
$~~\alpha=0.09~~$ $0.77$ $2.212214$ $2.8735578$
: \[FermiM\] The Fermi momentum $k_{F}$ for different charge $q$ and GB parameter $\alpha$.
$~~ ~~$ $q=0.5$ $q=1.2$ $q=1.5$
-------------------- --------------- --------------- ------------- -- -- --
$~~\alpha=-0.19~~$ $3.41726$ $1$ $1$
$~~\alpha=0~~$ $4.38389$ $1.06558$ $1$
$~~\alpha=0.09~~$ $5.68267$ $1.22051$ $1$
: \[delta\] The scaling exponent $\delta$ for different charge $q$ and GB parameter $\alpha$ (analytical result).
Particle-Hole (A)symmetry
-------------------------
As mentioned in subsection A, when the background geometry is pure $AdS_{5}$ and $q=0$, the spectral function has a Particle-hole symmetry, but when $q\neq 0$, the Particle-hole asymmetry presents. The similar case does occur in the background of charged Gauss-Bonnet black hole (Fig.\[ParticleHole\]). In Fig.\[ParticleHole\], the last two panels below show that the spectral function behavior is not symmetrical about the Fermi point (Particle-hole asymmetry) for $\alpha=-0.19$ and $q=1$ or $q=-1$. Moreover, we also find that when $q$ is restored to $0$, the system regains the Particle-hole symmetry and the asymmetry slowly becomes obvious with $q$ increasing (the first and second panel above in Fig.\[ParticleHole\]). Thus, we can reasonably conclude that whether the system has the Particle-hole symmetry or asymmetry is related to the different values of $q$. Some similar conclusions have also been pointed out in the investigations on fermions in charged BTZ black hole [@FermionsBTZBH]. In addition, we also notice that the Gauss-Bonnet coupling constant $\alpha$ has less effects on the (a)symmetry of the system. For comparison, we present the plot for $\alpha=0$ and $q=1$ or $q=-1$ (the last panel above and the first panel below in Fig.\[ParticleHole\]).
Conclusions and discussion
==========================
We have studied the main features of the fermions in charged Gauss-Bonnet black hole for zero temperature limit and massless fermions by AdS/CFT correspondence. The general behavior of the spectral function is similar with the case of RN black hole [@HongLiuNon-Fermi]. However, the Gauss-Bonnet coupling constant $\alpha$ changes the shape of the spectral function. Especially, near the quasi-particle like peak, the effect the $\alpha$ exerts on the shape of the spectral function is more significant. Therefore, it is interesting to further understand the behavior of Green¡¯s functions in the Quasi-topological gravity or the case with Weyl corrections, which has more physical contents.
Furthermore, we especially focus on the dispersion and the Particle-hole asymmetry. Their properties can be summarized as follows. Generally, at $q=1$, we find a dispersion relation $\tilde{\omega}(\tilde{k})\sim \tilde{k}^{\delta}$ and the scaling exponent $\delta\neq 1$, indicating this system is non-Fermi liquid. Therefore, the system can be candidates for holographic dual of generalized non-Fermi liquids. More importantly, the behavior of this system increasingly similar to that of the Landau Fermi liquid when $\alpha$ is approaching its lower bound. It seems that the GB term can also describe the dual field theories with both the weak-coupling and the strong-coupling. Also we discuss briefly the cases of the charge dependence. At larger values of $q$, new scaling behavior appears. For instance, for certain values of charge $q$ (for example, $q=1.2$), the scaling exponent $\delta$ decreases with the decrease of the values of GB parameter $\alpha$ and asymptote to $1$ for smaller $\alpha$. While for larger charge $q$ (for example, $q=1.5$), the scaling exponent $\delta\approx 1$ independent of the $\alpha$. Another important characteristic of this system is the Particle-hole asymmetry. From the density plot of the spectral function, we find that for zero temperature limit, the Particle-hole (a)symmetry is controlled by the parameter $q$ and the Gauss-Bonnet coupling constant $\alpha$ has less effects on it.
In addition, other features, such as the log period, studies in Ref. [@HongLiuNon-Fermi], are still preserved, which are determined by the geometry $AdS_{2}$ near the horizon. In the future works, we will address the particulars of such properties. In order to searching for the effect of the higher curvature corrections on the spectral function of the fermions, we can study the fermions in Quasi-topological gravity or the case with Weyl corrections in parallel.
I would like to thank my advisor, Prof. Yongge Ma, for his encouragement. And I am also grateful to Prof. Yi Ling, Dr. Hongbao Zhang, Dr. Wei-Jie Li, Dr. Xiangdong Zhang, Dr. Huaisong Zhao and Yue Cao for their useful discussions. In addition, I am also thank J. Shock for his pointing out some type mistakes and his comments. This work is partly supported by NSFC(No.10975017) and the Fundamental Research Funds for the central Universities.
Analytical treatment
====================
In this Appendix, we will give a brief summary on the analytic treatment in the low frequency limit developed in Ref.[@HongLiuAdS2]. As revealed in Ref.[@HongLiuAdS2], the dispersion relation can be given by[^9] $$\begin{aligned}
\label{LdispersionA}
\tilde{\omega}(\tilde{k})\propto \tilde{k}^{\delta}, \quad {\rm with} \quad \delta = \begin{cases} \frac{1}{2 \nu_{k_F}} & \nu_{k_F} < \frac{1}{2}\cr
1 & \nu_{k_F} > \frac{1}{2}
\end{cases}.\end{aligned}$$
While $\nu_{k_{F}}$ can be calculated by the equation (57) in Ref.[@HongLiuAdS2] $$\begin{aligned}
\label{nuk}
\nu_{k}=\frac{g_{F}q}{\sqrt{2d(d-1)}}\sqrt{\frac{2m_{\zeta}^{2}}{g_{F}^{2}q^{2}}
+\frac{d(d-1)}{(d-2)^{2}}\frac{k^{2}}{\mu_{q}^{2}}-1}~~.\end{aligned}$$
In our conventions, $g_{F}=2$. The above expressions was derived by the fact that the geometry near the horizon is $AdS_{2}\times \mathbb{R}^{d-1}$. As discussed in Section\[CBHinGBG\], the geometry near the horizon is also $AdS_{2}\times \mathbb{R}^{3}$ in Gauss-Bonnet gravity, independent of $\alpha$. Therefore, in order to obtain the dispersion relation, we only need to work out numerically the Fermi momentum $k_{F}$, which is controlled by UV physics.
However, we can also obtain the range of $k_{F}$ by WKB analysis[^10] [@HongLiuAdS2]. As observed in Ref.[@HongLiuAdS2], when $m_{\zeta}=0$, the range for allowed $k_{F}$ is[^11] $$\begin{aligned}
\label{RangekF}
\frac{d-2}{\sqrt{d(d-1)}}\leq \frac{k_{F}}{\mu_{q}}\leq 1.\end{aligned}$$ In order to test the robustness of our numerical result on the Fermi momentum. We list the values of the Fermi momentum $k_{F}$ determined numerically and the range of $k_{F}$ determined by Eq.(\[RangekF\]) in Table \[kF\] for $\alpha=0$ and different $q$. From Table \[kF\], we can see that our numerical result is well within the interval determined by Eq.(\[RangekF\]).
$~~q~~$ $0.5$ $1$ $1.2$ $1.5$
------------------------ --------------------- --------------------- --------------------- --------------------- -- --
$~~k_{F}$(numerical) $0.81$ $1.8880$ $2.349913$ $3.0572678$
$~~k_{F}$(WKB) $[0.707,1.225]$ $[1.414,2.450]$ $[1.697,2.939]$ $[2.121,3.674]$
: \[kF\] The Fermi momentum $k_{F}$ determined numerically and the range of $k_{F}$ obtained by WKB analysis.
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[^1]: Here we have set the the five dimensional gravitational constant $\kappa_{5}^{2}=1/2$ and the effective dimensionless gauge coupling $g_{F}=2$.
[^2]: In the following, for convenience, we will set $r_{+}=1$.
[^3]: Throughout the paper, we will use the conventions of [@Conventions1]. $a$, $b$ are the usual spacetime abstract index and $\mu$, $\nu$ are the tangent-space index.
[^4]: Note that the Dirac action (\[actionspinor\]), only through the effective chemical potential $\mu_{q}\equiv \mu q$, that is to say, through the combination of $g_{F}q$, depends on $q$. For convenience, the $g_{F}$ has been set as $2$ above and the $q$ is treated as a free parameter, but we should note that only the product of them is the relevant quantity. For more discussions, please refer to Ref.[@HongLiuNon-Fermi]. At the same time, we also want to remind readers to pay attention to the fact that in Ref.[@HongLiuNon-Fermi; @HongLiuAdS2], they set $g_{F}=1$, which is different from our conventions.
[^5]: We can also refer to Refs.[@GreenFpureAdS1; @GreenFpureAdS2].
[^6]: For details, please refer to the Ref.[@HongLiuAdS2]. We also give a brief summary in Appendix A.
[^7]: Since the peak of $G_{22}$ becomes sharper as the charge $q$ increases, we can pin down the Fermi surface to more digit for larger $q$.
[^8]: Another scaling exponent $\beta$ is still approximate to $1$ independent of $\alpha$ and $q$.
[^9]: The equation (93) in Ref.[@HongLiuAdS2].
[^10]: This analysis only applies to the Reissner-Nordstr$\ddot{o}$m AdS black hole but not Gauss-Bonnet AdS black hole.
[^11]: The equation (110) in Ref.[@HongLiuAdS2].
|
---
abstract: 'We present a new high-order finite volume reconstruction method for hyperbolic conservation laws. The method is based on a piecewise cubic polynomial which provides its solutions a fifth-order accuracy in space. The spatially reconstructed solutions are evolved in time with a fourth-order accuracy by tracing the characteristics of the cubic polynomials. As a result, our temporal update scheme provides a significantly simpler and computationally more efficient approach in achieving fourth order accuracy in time, relative to the comparable fourth-order Runge-Kutta method. We demonstrate that the solutions of PCM converges in fifth-order in solving 1D smooth flows described by hyperbolic conservation laws. We test the new scheme in a range of numerical experiments, including both gas dynamics and magnetohydrodynamics applications in multiple spatial dimensions.'
address:
- 'Applied Mathematics and Statistics, University of California, Santa Cruz, CA, U.S.A'
- 'Department of Physics, University of California, Santa Cruz, CA, U.S.A'
- 'Département de Physique, École Normale Supérieure, Paris, France'
author:
- Dongwook Lee
- Hugues Faller
- Adam Reyes
bibliography:
- 'mybibfile.bib'
title: 'The Piecewise Cubic Method (PCM) for Computational Fluid Dynamics'
---
High-order methods; piecewise cubic method; finite volume method; gas dynamics; magnetohydrodynamics; Godunov’s method.
Introduction
============
In this paper we are interested in solving multidimensional conservation laws of the Euler equations and the ideal MHD equations, written as \[Eq:cons\_law\] + () = 0, where $\bU$ is the vector of the conservative variables, and ()= \[F(), G(), H()\]\^T = \[, , \]\^T is the flux vector.
We present a new high-order piecewise cubic method (PCM) algorithm that is extended from the classical PPM and WENO schemes [@colella1984piecewise; @jiang1996efficient]. These two algorithms, by far, have been extremely successful in various scientific fields where there are challenging computational needs for [*[both]{}*]{} high-order accuracy in smooth flows and well-resolved solutions in shock/discontinuous flows. With the advent of high-performance computing (HPC) in recent years, such needs have been more and more desired, and have become a necessary requirement in conducting large scale, cutting edge simulations of gas dynamics and magnetohydrodynamics (MHD) [@Dongarra2010; @Dongarra2012; @Subcommittee2014; @Keyes2013].
As observed in the success stories of the PPM and WENO methods, discrete algorithms of data interpolation and reconstruction play a key role in numerical methods for PDE approximations [@LeVeque2002; @leveque2007finite; @Toro2009] within the broad framework of finite difference and finite volume discretization methods. In view of this, computational improvements of such interpolation and reconstruction schemes, particularly focused on the high-order property with great shock-capturing capability, take their positions at the center of HPC in modern computational fluid dynamics.
The properties of enhanced solution accuracy with lower numerical errors on a given grid resolution and faster convergence-to-solution rates are the key advantages in high-order schemes. The advantage of using high-order methods in HPC is therefore clear: one can obtain reproducible, admissible, and highly accurate numerical solutions in a faster computational time at the expense of increased rate of floating point operations, while at the same time, with the use of smaller size of grid resolutions. This is by no means exceedingly efficient in high-performance computing (HPC), in view of the fact that the increase of grid resolutions has a direct impact to an increase of memory footprints which are bounded in all modern computing architecture.
In this regards, our goal in this paper is to lay down a mathematical foundation in designing a new high-order method using piecewise cubic polynomials. We mainly focus on describing the detailed PCM algorithm in 1D finite volume framework for the scope of the current paper. For multidimensional problems, we adopt the classical “dimension-by-dimension” approach for simplicity. Although this approach has an advantage in its simplicity, it unfortunately fails to retain the high-order accurate property of the 1D baseline algorithm in multidimensional problems. Instead, it provides only a second-order accuracy in multidimensional nonlinear advection in finite volume method due to the lack of accuracy in approximating a face-averaged flux function as a result of mis-using an averaged quantity in place of a pointwise quantity, or vice versa [@shu2009high; @buchmuller2014improved; @zhang2011order; @mccorquodale2011high]. Although the baseline 1D PCM scheme can be extended to multiple spatial dimensions preserving its high-order accuracy by following more sophisticated treatments [@shu2009high; @buchmuller2014improved; @zhang2011order; @mccorquodale2011high], more careful work is needed to carry out the detailed design, and this will be considered in our future research. For a finite volume scheme in 1D we take the spatial average of Eq. (\[Eq:cons\_law\]) over the cell $I_i=[x_{i-1/2},x_{i+1/2}]$, yielding a semi-discrete form, $$\label{Eq:1D_avg}
\frac{\partial \avg{\bU}_i}{\partial t} = -\frac{1}{\Delta x}(\bF_{i+1/2}-\bF_{i-1/2})$$ to get an equation for the evolution of the volume averaged variables, $\avg{\bU}_i=\frac{1}{\dx}\int_{I_i}\bU(x,t)dx$. Typically to achieve high-order accuracy in time the temporal update is done using a TVD Runge-Kutta scheme in method-of-lines form [@shu1988tvd; @mccorquodale2011high]. In this approach the high-order accuracy comes from taking the [*[multiple Euler stages]{}*]{} of the RK time discretizations, which require repeated reconstructions in a single time step, increasing the computational cost.
Instead, as will be fully described in Section \[Sec:pcm\], one of the novel ideas in PCM is to employ the simple [*[single stage]{}*]{} predictor-corrector type temporal update formulation in which we take the time-average of Eq. (\[Eq:1D\_avg\]) $$\label{Eq:time_avg}
\avg{\bU}^{n+1}_i = \avg{\bU}^n_i -\frac{\Delta t}{\Delta x}(\bF ^{n+1/2}_{i+1/2}-\bF^{n+1/2}_{i-1/2}).$$ Here $\avg{\bU}^n_i = \avg{\bU}_i(t^n)$ is the volume averaged quantity at $t^n$, and $\bF^{n+1/2}_{i \pm 1/2} = \frac{1}{\dt}\int_{t^n}^{t^{n+1}}\bF_{i\pm 1/2}(t)dt$ is the time average of the interface flux from $t^n$ to $t^{n+1}$. In this way high-order in space and time is accomplished with a single reconstruction in contrast to the multiple Euler stages of the RK time discretizations, providing significant benefits in computational efficiency per solution accuracy. The organization of the paper is as follows: Section \[Sec:pcm\] describes the fifth-order accurate spatial reconstruction algorithm of PCM in 1D. We highlight several desirable properties of the PCM scheme in terms of computational efficiency and solution accuracy. Section \[Sec:chartracing\] introduces the fourth-order accurate temporal updating scheme of PCM using a predictor-corrector type characteristic tracing, which is much simpler than the typical high-order Runge-Kutta ODE updates. In Section \[Sec:pcmMultiD\] we discuss how to extend the 1D scheme in Section \[Sec:pcm\] to multiple spatial dimensions following the [*[dimension-by-dimension approach]{}*]{} [@buchmuller2014improved; @zhang2011order].
In Section \[Sec:results\] we test the PCM scheme on a wide spectrum of benchmark problems in 1D, 2D and 3D, both for hydrodynamics and magnetohydrodynamics (MHD) applications. We also compare the PCM solutions with PPM and WENO solutions in order to examine numerical accuracy, capability and efficiency in both smooth and shock flow regimes. We conclude our paper in Section \[Sec:conclusions\] with a brief summary.
The One-Dimensional Piecewise Cubic Method (PCM) Spatial Reconstruction {#Sec:pcm}
=======================================================================
In this section we describe a new PCM scheme in a one-dimensional finite volume formulation for solving hyperbolic conservation laws of hydrodynamics and magnetohydrodynamics. The new PCM scheme is a higher-order extension of Godunov’s method [@godunov1959difference], bearing its key components in the reconstruction algorithm on the relevant ideas of its high-order predecessors, the PPM scheme [@colella1984piecewise], the WENO schemes [@jiang1996efficient; @shi2002technique; @borges2008improved; @castro2011high], and Hermite-WENO schemes [@qiu2004hermite; @qiu2005hermite; @zhu2009hermite; @balsara2007sub].
For the purpose of this section, we take the $3 \times 3$ hyperbolic system of conservation laws of the 1D Euler equations + = 0. \[Eq:Euler1D\] The notations used are the vector of the conservative variables $\bU$ and fluxes $F(\bU)$, respectively, defined as = , F() = . Here $\rho$ is the fluid density, $u$ is the fluid velocity in $x$-direction, and $E$ is the total energy as the sum of the internal energy $\epsilon={p}/({\gamma-1})$ and the kinetic energy obeying the ideal gas law, E = + , where $p$ is the gas pressure, with the ratio of specific heats denoted as $\gamma$. We denote the cells in $x$-direction by $I_i = [x_{i-1/2}, x_{i+1/2}]$. We assume our grid is configured on an equidistant uniform grid for simplicity.
In addition to the system of the Euler equations in the conserved variables $\bU$ as given in Eq. (\[Eq:Euler1D\]), we often use the two other equivalent system of equations each of which can be written either in the primitive variables $\bV=[\rho,u,p]^T$ or in the characteristic variables $\bW$. The characteristic variable $\bW$ is readily obtained from $\bU$ or $\bV$ by multiplying the left eigenvectors corresponding to either $\bU$ or $\bV$, for instance, $\bW = \bL \bU$. In the latter $\bL \; (\equiv \bR^{-1}$) represents the $3\times 3$ matrix obtained from diagonalizing the coefficient matrix $\bA=\partial \mathcal{F}/\partial \bU = \bR \mathbf{\Lambda}\bR^{-1}$, whose rows are the $k$-th left eigenvectors $\bl^{(k)}$, $k=1, 2, 3$. The representation of the system in $\bW$ furnishes a completely linearly decoupled 1D system of equations, + = 0. \[Eq:Euler1D\_char\] The above system in the characteristic variables $\bW$ is therefore very handy for analyses, and also is a preferred choice of variable in order to furnish numerical solutions more accurate than third-order especially with better non-oscillatory controls, in particular when considering wave-by-wave propagations in a system of equations [@shu2009high]. In this reason the characteristic variable $\bW$ is taken as our default variable choice in the 1D PCM reconstruction steps via characteristic decompositions, albeit with an increased computational cost, among the other two choices of the primitive $\bV$ or the conservative variables $\bU$.
The methodology presented below can be similarly applied to the 1D ideal MHD equations (see for instance, [@brio1988upwind]).
Piecewise Cubic Profile
-----------------------
To begin with we first define a cubic polynomial $p_i(x)$ to approximate a $k$-th characteristic variable $q \in \bW$ on each interval $I_i$ by p\_i(x) = c\_0 + c\_1(x-x\_i) + c\_2(x-x\_i)\^2 + c\_3(x-x\_i)\^3. \[Eq:pcm\_poly\] The goal is now to determine the four coefficients $c_i$, $i \in \Z$, $0 \le i \le 3$, which can be achieved by imposing the following four conditions: \_[I\_i]{}p\_i(x) dx &=& \_i,\[Eq:pcm\_cond1\]\
p\_i(x\_[i-1/2]{}) &=& q\_[L,i]{},\[Eq:pcm\_cond2\]\
p\_i(x\_[i+1/2]{}) &=& q\_[R,i]{},\[Eq:pcm\_cond3\]\
p\_i’(x\_[i]{}) &=& q’\_[C,i]{},\[Eq:pcm\_cond4\] where \_i = \_[I\_i]{}q(x,t\^n) dx is the cell-averaged quantity at $t^n$ on $I_i$ which is given as an initial condition; q\_[L,i]{} = q(x\_[i-1/2]{},t\^n) + (\^p), q\_[R,i]{} = q(x\_[i+1/2]{},t\^n) + (\^p) are respectively the $p$-th order accurate pointwise left and the right Riemann states at $t^n$ on the cell $I_i$ that are unknown yet but are to be determined as described below; and lastly q’\_[C,i]{} = q’(x\_i,t\^n) + (\^r) is the $r$-th order accurate approximation to the slope of $q$ at $t^n$ evaluated at $x_i$, which is again unknown at this point but is to be determined as below.
For the moment let us assume that all four quantities $\overline{q}_i, q_{L,i}, q_{R,i}$ and $q'_{C,i}$ are known. It can be shown that the system of relations in Eqs. (\[Eq:pcm\_cond1\]) $\sim$ (\[Eq:pcm\_cond4\]) is equivalent to a system given as:
c\_0 + c\_2 &=& \_i, \[Eq:pcm\_cond1a\]\
c\_0 - c\_1 + c\_2 - c\_3 &=& q\_[L,i]{}, \[Eq:pcm\_cond2b\]\
c\_0 + c\_1 + c\_2 + c\_3 &=& q\_[R,i]{}, \[Eq:pcm\_cond3b\]\
c\_1 &=& q’\_[C,i]{},\[Eq:pcm\_cond4b\] which, in turn, can be solved for all four $c_i$, $i=1, \dots, 4$. The final expressions of the coefficients in terms of $\overline{q}_i, q_{L,i}, q_{R,i}$, and $q'_{C,i}$ are given as: c\_0 &=& ( - q\_[R,i]{} - q\_[L,i]{} + 6\_i ), \[Eq:pcm\_cond1c\]\
c\_1 &=& q’\_[C,i]{}, \[Eq:pcm\_cond2c\]\
c\_2 &=& (q\_[R,i]{} + q\_[L,i]{} -2\_i ),\[Eq:pcm\_cond3c\]\
c\_3 &=& ( q\_[R,i]{} - q\_[L,i]{} -q’\_[C,i]{} ).\[Eq:pcm\_cond4c\]
Therefore once we figure out the three unknowns, $q_{L,i}, q_{R,i}$, and $q'_{C,i}$, the cubic profile $p_i(x)$ in Eq. (\[Eq:pcm\_poly\]) can be completely determined and is ready to approximate $q$ on each $I_i$.
We now devote the following sections to describe how to determine $q_{L,i}, q_{R,i}$, and $q'_{C,i}$ so that the resulting PCM approximation to the variable $q$ lend its accuracy a fifth-order in space (Sections \[Sec:edges\] and \[Sec:qc\_prime\]) and a fourth-order in time (Section \[Sec:chartracing\]).
Reconstruction of the Riemann States $q_{L,i}$ and $q_{R,i}$ {#Sec:edges}
------------------------------------------------------------
We follow the fifth-order finite volume WENO approach, either of the classical WENO-JS [@jiang1996efficient] or WENO-Z [@borges2008improved; @castro2011high], in order to reconstruct the left and right Riemann states, $q_{L,i}$ and $q_{R,i}$, on each cell $I_i$. For the sake of providing a full self-contained description of the PCM scheme, we briefly present the two WENO Riemann state reconstruction strategies here.
The main idea in WENO is to employ its reconstruction procedure according to the nonlinear smoothness measurements on three ENO sub-stencils, $S_\ell$, $\ell=1,2,3$, each of which consisting three cells $I_i$, $i=i_1,i_2,i_3$. Let us first define $$\begin{aligned}
S_1&=&\{I_{i-2},I_{i-1},I_{i}\}, \\
S_2&=&\{I_{i-1},I_{i},I_{i+1}\}, \\
S_3&=&\{I_{i},I_{i+1},I_{i+2}\}.\end{aligned}$$ Formulating the WENO reconstruction consists of the following three steps:
#### Step 1: ENO-Build
We begin with building a second degree polynomial for each $\ell=1,2,3$, $$\label{Eq:WENO_p(x)}
p_\ell(x)=\sum_{k=0}^2a_{\ell,k}(x-x_i)^k,$$ each of which is defined on $S_\ell$, satisfying $$\label{Eq:WENO_p(x)_constraints}
\frac{1}{\Delta x}\int_{I_k}p_\ell(x)\mathop{dx}=\bar{q}_k,$$ for $k=i+\ell-3, \dots, i+\ell-1$. After a bit of algebra, we obtain the coefficients $a_{\ell,k}$ that determine $p_\ell(x)$ in Eq. (\[Eq:WENO\_p(x)\]).
For $\ell=1$, $$\begin{aligned}
\label{Eq:WENO_p(x)_coeffs_l=1}
%\frac{}{}\bar u_{i}
&&a_{1,0}=\left(-\frac{1}{24}\bar q_{i-2} + \frac{1}{12}\bar q_{i-1}+\frac{23}{24}\bar q_{i}\right),\\
&&a_{1,1}=\left( \frac{1}{2} \bar q_{i-2} -2 \bar q_{i-1} +\frac{3}{2} \bar q_{i}\right)\frac{1}{\Delta x},\\
&&a_{1,2}=\left( \frac{1}{2} \bar q_{i-2} -\bar q_{i-1}+\frac{1}{2} \bar q_{i}\right)\frac{1}{\Delta x^2},\end{aligned}$$
and for $\ell=2$, $$\begin{aligned}
\label{Eq:WENO_p(x)_coeffs_l=2}
%\frac{}{}\bar q_{i}
&&a_{2,0}=\left(-\frac{1}{24}\bar q_{i-1} + \frac{13}{12}\bar q_{i}-\frac{1}{24}\bar q_{i+1}\right),\\
&&a_{2,1}=\left(-\frac{1}{2} \bar q_{i-1} +\frac{1}{2} \bar q_{i+1}\right)\frac{1}{\Delta x},\\
&&a_{2,2}=\left( \frac{1}{2} \bar q_{i-1} -\bar q_{i}+\frac{1}{2} \bar q_{i+1}\right)\frac{1}{\Delta x^2}.\end{aligned}$$
Lastly, for $\ell=3$, we get $$\begin{aligned}
\label{Eq:WENO_p(x)_coeffs_l=3}
%\frac{}{}\bar q_{i}
&&a_{3,0}=\left( \frac{23}{24}\bar q_{i} + \frac{1}{12}\bar q_{i+1} -\frac{1}{24}\bar q_{i+2}\right),\\
&&a_{3,1}=\left(-\frac{3}{2} \bar q_{i} +2\bar q_{i+1} -\frac{1}{2} \bar q_{i+2}\right)\frac{1}{\Delta x},\\
&&a_{3,2}=\left( \frac{1}{2} \bar q_{i} -\bar q_{i+1} +\frac{1}{2} \bar q_{i+2}\right)\frac{1}{\Delta x^2}.\end{aligned}$$ Then the three sets of left and right states follow as $$\{p_1(x_{i-1/2}),p_2(x_{i-1/2}),p_3(x_{i-1/2})\},\mbox{ and } \{p_1(x_{i+1/2}),p_2(x_{i+1/2}),p_3(x_{i+1/2})\},$$ where each of $p_\ell(x_{i\pm 1/2})$ is the ENO approximation and is given by, first for $p_1$, $$\begin{aligned}
\label{Eq:WENO_p(x)_value_l=1}
%\frac{}{}\bar u_{i}
&&p_{1}(x_{i-1/2})=-\frac{1}{6}\bar q_{i-2} + \frac{5}{6}\bar q_{i-1} +\frac{1}{3}\bar q_{i},\\
&&p_{1}(x_{i+1/2})=\frac{1}{3}\bar q_{i-2} - \frac{7}{6}\bar q_{i-1} +\frac{11}{6}\bar q_{i},\end{aligned}$$ and for $p_2$, $$\begin{aligned}
\label{Eq:WENO_p(x)_value_l=2}
%\frac{}{}\bar q_{i}
&&p_{2}(x_{i-1/2})= \frac{1}{3}\bar q_{i-1} + \frac{5}{6}\bar q_{i} -\frac{1}{6}\bar q_{i+1},\\
&&p_{2}(x_{i+1/2})=-\frac{1}{6}\bar q_{i-1} + \frac{5}{6}\bar q_{i} +\frac{1}{3}\bar q_{i+1},\end{aligned}$$ and finally for $p_3$, $$\begin{aligned}
\label{Eq:WENO_p(x)_value_l=3}
%\frac{}{}\bar q_{i}
&&p_{3}(x_{i-1/2})=\frac{11}{6}\bar q_{i} - \frac{7}{6}\bar q_{i+1} +\frac{1}{3}\bar q_{i+2},\\
&&p_{3}(x_{i+1/2})=\frac{1}{3}\bar q_{i} + \frac{5}{6}\bar q_{i+1} -\frac{1}{6}\bar q_{i+2}.\end{aligned}$$
These left and right states respectively approximate the pointwise values at the interfaces $q(x_{i\pm 1/2})$ with third-order accuracy, i.e., $p_\ell(x_{i\pm 1/2})-q({x_{i\pm 1/2}})=O(\Delta x^3)$ (see [@jiang1996efficient]) by using the given cell-averaged quantities $\bar q_k$.
#### Step 2: Linear Constant Weights
The next step is to construct a fourth-degree polynomial $$\label{Eq:WENO_phi}
\phi(x) = \sum_{k=0}^4 b_k(x-x_i)^k$$ over the entire stencil $S=\cup_{\ell=1}^3 S_\ell$ which satisfies $$\label{Eq:WENO_phi(x)_constraints}
\frac{1}{\Delta x}\int_{I_k}\phi(x)dx=\bar{q}_k,$$ for $k=i-2,\dots,i+2$. We can show that the coefficients $b_k$ are given as $$\begin{aligned}
\label{Eq:WENO_phi_coeffs}
&&b_0=\frac{3}{640}\bar q_{i-2} -\frac{29}{480}\bar q_{i-1}+ \frac{1067}{960}\bar q_{i} -\frac{29}{480}\bar q_{i+1} +\frac{3}{640}\bar q_{i+2}, \label{Eq:b0}\\
%%\nonumber\\\\
&&b_1=\Big(\frac{5}{48}\bar q_{i-2} -\frac{17}{24}\bar q_{i-1}+ \frac{17}{24}\bar q_{i+1} -\frac{5}{48}\bar q_{i+2}\Big)\frac{1}{\dx}, \label{Eq:b1}\\
&&b_2=\Big(-\frac{1}{16}\bar q_{i-2} +\frac{3}{4}\bar q_{i-1} -\frac{11}{8}\bar q_{i} +\frac{3}{4}\bar q_{i+1}- \frac{1}{16}\bar q_{i+2}\Big)\frac{1}{\dx^2},\label{Eq:b2}\\
&&b_3=\Big(-\frac{1}{12}\bar q_{i-2} +\frac{1}{6}\bar q_{i-1} - \frac{1}{6}\bar q_{i+1} +\frac{1}{12}\bar q_{i+2}\Big)\frac{1}{\dx^3}, \label{Eq:b3}\\
&&b_4=\Big(\frac{1}{24}\bar q_{i-2} -\frac{1}{6}\bar q_{i-1}+ \frac{1}{4}\bar q_{i} -\frac{1}{6}\bar q_{i+1} +\frac{1}{24}\bar q_{i+2}\Big)\frac{1}{\dx^4}. \label{Eq:b4}\end{aligned}$$ WENO uses $\phi(x)$ to determine three linear constant weights $\gamma_{\ell}^{\pm}$, $\ell=1,2,3$, with $\sum_\ell \gamma_\ell^{\pm}=1$, such that $$\label{Eq:WENO_gammas}
\phi(x_{i\pm 1/2})=\sum_{\ell=1}^3 \gamma_\ell^{\pm} p_\ell(x_{i\pm 1/2}).$$ The values on the left-hand side become $$\label{Eq:WENO_phi_values_left}
\phi(x_{i-1/2})=-\frac{1}{20}\bar q_{i-2} +\frac{9}{20}\bar q_{i-1} + \frac{47}{60}\bar q_{i} -\frac{13}{60}\bar q_{i+1} +\frac{1}{30}\bar q_{i+2},$$ and $$\label{Eq:WENO_phi_values_right}
\phi(x_{i+1/2})=\frac{1}{30}\bar q_{i-2} -\frac{13}{60}\bar q_{i-1} + \frac{47}{60}\bar q_{i} +\frac{9}{20}\bar q_{i+1} -\frac{1}{20}\bar q_{i+2}.$$ Now, by inspection, we obtain a set of linear weights for the left state, $$\label{Eq:WENO_gamma_left}
\gamma_1^-=\frac{3}{10},\gamma_2^-=\frac{6}{10},\gamma_3^-=\frac{1}{10},$$ and for the right state, $$\label{Eq:WENO_gamma_right}
\gamma_1^+=\frac{1}{10},\gamma_2^+=\frac{6}{10},\gamma_3^+=\frac{3}{10}.$$
#### Step 3: Nonlinear Weights
The last step that imposes the non-oscillatory feature in the WENO approximations is to measure how smoothly the three polynomials $p_\ell(x)$ vary on $I_i$. This is done by determining non-constant, nonlinear weights $\omega_\ell^{\pm}$ (three of them for each $\pm$ state) that rely on the so-called smoothness indicator $\beta_\ell$, defined by $$\label{Eq:WENO_beta}
\beta_\ell=\sum_{s=1}^{2}\left(\Delta x^{2s-1}
%\int_{I_i} \Big[\frac{\mathop{d}^s }{\mathop{dx}^s} p_\ell(x) \Big]^2 dx \right),
\int_{I_i} \Big[\frac{d^s }{dx^s} p_\ell(x) \Big]^2 dx \right).$$ With this definition $\beta_\ell$ becomes small for smooth flows, and large for discontinuous flows.
For explicit expressions, we attain $$\begin{aligned}
\label{Eq:WENO_beta_explicit}
&&\beta_1=\frac{13}{12}\left(\bar q_{i-2}-2\bar q_{i-1}+\bar q_{i} \right)^2+\frac{1}{4}\left(\bar q_{i-2}-4\bar q_{i-1}+3\bar q_{i}\right)^2,\\
&&\beta_2=\frac{13}{12}\left(\bar q_{i-1}-2\bar q_{i}+\bar q_{i+1} \right)^2+\frac{1}{4}\left(\bar q_{i-1}-\bar q_{i+1}\right)^2,\\
&&\beta_3=\frac{13}{12}\left(\bar q_{i}-2\bar q_{i+1}+\bar q_{i+2} \right)^2+\frac{1}{4}\left(3\bar q_{i}-4\bar q_{i+1}+\bar q_{i+2}\right)^2.\end{aligned}$$ Equipped with these $\beta_\ell$, the nonlinear weights $\omega_\ell^{\pm}\ge 0$ are defined as:
- For WENO-JS: $$\label{Eq:WENO5_omega}
\omega_\ell^{\pm} = \frac{\tilde{\omega}_\ell^{\pm}}{ \sum_{s}\tilde{\omega}_s^{\pm}}, \mbox{ where }
\tilde{\omega}_\ell^{\pm} = \frac{\gamma_\ell^{\pm}}{(\epsilon + \beta_\ell)^m},$$
- For WENO-Z: $$\label{Eq:WENOZ_omega}
\omega_\ell^{\pm} = \frac{\tilde{\omega}_\ell^{\pm}}{ \sum_{s}\tilde{\omega}_s^{\pm}}, \mbox{ where }
\tilde{\omega}_\ell^{\pm} = {\gamma_\ell^{\pm}}\Biggl(1+\Bigl(\frac{|\beta_0-\beta_2|}{\epsilon + \beta_\ell}\Bigr)^m\Biggr).$$
Here $\epsilon$ is any arbitrarily small positive number that prevents division by zero, for which we choose $\epsilon=10^{-36}$. One of the classical choice of $\epsilon$ in many WENO literatures is found to be $\epsilon=10^{-6}$ [@jiang1996efficient; @shi2002technique]; however, it was suggested in [@borges2008improved] that $\epsilon$ should be chosen to be much smaller in order to force this parameter to play only its original role of avoiding division by zero in the definitions of the weights, Eqs. (\[Eq:WENO5\_omega\]) and (\[Eq:WENOZ\_omega\]). Another closely related point of discussion is with the value of $m$, the power in the denominators in Eqs. (\[Eq:WENO5\_omega\]) and (\[Eq:WENOZ\_omega\]). The parameter $m$ determines the rate of changes in $\beta_\ell$, and most of the WENO literatures use $m=2$. However, we observe that using $m=1$ resolves discontinuities sharper in most of our numerical simulations without exhibiting any numerical instability, so became the default value in our implementation. For more detailed discussions on the choices of $\epsilon$ and $m$, see [@borges2008improved; @castro2011high].
Using these nonlinear weights, we complete the WENO reconstruction procedure of producing the fifth-order spatially accurate, non-oscillatorily reconstructed values at each cell interface at each time step $t^n$ [@jiang1996efficient; @shu2009high], $$\label{Eq:WENO_final_states}
q_{L;R,i} = \sum_{\ell=1}^{3}\omega_\ell^{\pm} p_\ell(x_{i\pm 1/2}).$$
Reconstruction of the Derivative $q'_{C,i}$ {#Sec:qc_prime}
-------------------------------------------
The spatial reconstruction part of PCM proceeds to the next final step to obtain the derivative $q'_{C,i}$ in Eq. (\[Eq:pcm\_cond4\]). The approach is again to take the WENO-type reconstruction as before, but this time, to approximate a first derivative of a function [@shu2009high], i.e., $q'(x_i,t^n)$.
For this, we might consider using the same ENO-build strategy in Section \[Sec:edges\] in which the three second degree ENO polynomials in Eq. (\[Eq:WENO\_p(x)\]) are constructed over the five-point stencil $S=\cup_{\ell=1}^3 S_\ell$. However, this setup will provide only a third-order accurate approximation $q'_{C,i}$ to the exact derivative $q'(x_i)$. To see this, we first observe that the smoothness indicators $\beta_\ell$ with this setup will be including only a single term, $$\label{Eq:WENO_beta_1}
\beta_\ell=\Delta x^{3}
\int_{I_i} \Big[p''_\ell(x) \Big]^2 dx, \;\;\; \ell=1, 2, 3.$$ Through a Taylor expansion analysis on Eq. (\[Eq:WENO\_beta\_1\]) we see $$\label{Eq:WENO_beta_order}
\beta_\ell=D(1+\mcal{O}(\dx)),$$ where $D=(q''\dx^2)^2$ is a nonzero quantity independent of $\ell$ but may depend on $\dx$, assuming $q'' \ne 0$ on $S$. This results in a set of three nonlinear weights $\omega_\ell$, $\ell=1,2,3$, obtained either by Eq. (\[Eq:WENO5\_omega\]) or Eq. (\[Eq:WENOZ\_omega\]), satisfying \[Eq:WENO\_omega\_order\] \_= \_+ (), where the linear constant weights $\gamma_\ell$ are assumed to exist, when $q'(x,t^n)$ is smooth in $S$, such that q’\_[C,i]{} = \_[=1]{}\^3\_p\_’(x\_[i]{}) = q’(x\_i,t\^n) + (\^3). This finally implies the accuracy of $q'_{C,i}$ is found out to be third-order, $$\label{Eq:qC_prime_third_order}
q'_{C,i} = \sum_{\ell=1}^3\omega_\ell p_\ell'(x_{i}) = q'(x_i,t^n) + \mcal{O}(\dx^3),$$ because \[Eq:qC\_prime\_third\_order\_derivation\] \_[=1]{}\^3\_p\_’(x\_[i]{}) - \_[=1]{}\^3\_p\_’(x\_[i]{}) &=&\_[=1]{}\^3(\_- \_) (p\_’(x\_[i]{}) -q’(x\_i,t\^n))\
&=&\_[=1]{}\^3() (\^2) = (\^3). In the last equality, we used the fact that, for each $\ell$, $p'_\ell(x)$ is only a first degree polynomial which is accurate up to second-order when approximating $q'(x,t^n)$.
For this reason, we want a better strategy to obtain an approximation $q'_{C,i}$ at least fourth-order accurate in order that the overall nominal accuracy of the 1D PCM scheme achieves [*[at least]{}*]{} fourth-order accurate in both space and time.
#### Step 1: PPM-Build
An alternate strategy for this goal therefore would be to use a set of third degree polynomials instead. This can be designed using the two third degree polynomials, $\phi_{\pm}(x)$, from the PPM algorithm [@colella1984piecewise], \[Eq:PPM\_phi\_all\] \_(x)=\_[k=0]{}\^3 a\_k\^(x-x\_[i1/2]{})\^k. Following the description of PPM, we carry out to determine the coefficients $a_k^{\pm}$ by imposing the following constraints on $\phi_{\pm}(x)$ that are essential to keeping the volume averages on each cell $I_i$: $$\label{Eq:PPM_phi_minum}
\frac{1}{\Delta x}\int_{I_k} \phi_{-}(x)\mathop{dx}= \bar q_k^n, \mbox{ for } i-2\le k \le i+1,$$ and $$\label{Eq:PPM_phi_plus}
\frac{1}{\Delta x}\int_{I_k} \phi_{+}(x)\mathop{dx}= \bar q_k^n, \mbox{ for } i-1\le k \le i+2.$$ After a bit of algebra we obtain the coefficients $a_k^{\pm}$, with $s=1$ for $a_k^{+}$, while $s=0$ for $a_k^{-}$: $$\label{Eq:PPM_a0}
a_0^{\pm}=\frac{1}{12}\Big(-\bar q_{i-2+s} +7\bar q_{i-1+s} + 7\bar q_{i+s} -\bar q_{i+1+s} \Big),$$ $$\label{Eq:PPM_a1}
a_1^{\pm}=\frac{1}{12\Delta x}\Big(\bar q_{i-2+s} -15\bar q_{i-1+s} +15\bar q_{i+s} -\bar q_{i+1+s} \Big),$$ $$\label{Eq:PPM_a2}
a_2^{\pm}=\frac{1}{4\Delta x^2}\Big(\bar q_{i-2+s} -\bar q_{i-1+s} -\bar q_{i+s} +\bar q_{i+1+s} \Big),$$ $$\label{Eq:PPM_a3}
a_3^{\pm}=\frac{1}{6\Delta x^3}\Big(-\bar q_{i-2+s} +3\bar q_{i-1+s} -3\bar q_{i+s} +\bar q_{i+1+s} \Big).$$
#### Step 2: Linear Constant Weights
Now that the polynomials are determined over the stencil $S = \cup_{\ell=1}^3 S_\ell$, we use their first derivatives $\phi'_{\pm}$ to obtain a convex combination with two linear weights $\gamma_-$ and $\gamma_+$, \[Eq:PCM\_ucPrime\_linear\] q’\_[C,i]{} = \_- ’\_[-]{}(x\_i) + \_+ ’\_[+]{}(x\_i). The two linear weights can be determined by comparing Eq. (\[Eq:PCM\_ucPrime\_linear\]) with $\phi'(x_i)$ in Eq. (\[Eq:WENO\_phi\]), \_- ’\_[-]{}(x\_i) + \_+ ’\_[+]{}(x\_i) = ’(x\_i) This gives us \_- (a\_1\^- + a\_2\^- + 3 a\_3\^- ) + \_+ (a\_1\^+ - a\_2\^+ + 3 a\_3\^+ ) = b\_1 where $b_1$ is defined in Eq. (\[Eq:b1\]). By inspection, we obtain \_- = \_+ = .
#### Step 3: Nonlinear Weights
The smoothness indicators $\beta_\pm$ are now constructed using $\phi_{\pm}(x)$ as \[Eq:PCM\_beta\] \_=\_[s=2]{}\^[3]{}(x\^[2s-1]{} \_[I\_i]{} \^2 dx ). They can be written explicitly as \_- &=& 4 (a\_2\^-)\^2 \^4 + 12 (a\_2\^-) (a\_3\^-) \^5 + 48 (a\_3\^-)\^2 \^6\
&=& ( |q\_[i-2]{} -|q\_[i-1]{} -|q\_[i]{} +|q\_[i+1]{} )\^2\
&+& ( |q\_[i-2]{} -|q\_[i-1]{} -|q\_[i]{} +|q\_[i+1]{} )( -|q\_[i-2]{} +3|q\_[i-1]{} -3|q\_[i]{} +|q\_[i+1]{} )\
&+&( -|q\_[i-2]{} +3|q\_[i-1]{} -3|q\_[i]{} +|q\_[i+1]{} )\^2, and \_+ &=& 4 (a\_2\^+)\^2 \^4 - 12 (a\_2\^+) (a\_3\^+) \^5 + 48 (a\_3\^+)\^2 \^6\
&=& ( |q\_[i-1]{} -|q\_[i]{} -|q\_[i+1]{} +|q\_[i+2]{} )\^2\
&+& ( |q\_[i-1]{} -|q\_[i]{} -|q\_[i+1]{} +|q\_[i+2]{} )( -|q\_[i-1]{} +3|q\_[i]{} -3|q\_[i+1]{} +|q\_[i+2]{} )\
&+&( -|q\_[i-1]{} +3|q\_[i]{} -3|q\_[i+1]{} +|q\_[i+2]{} )\^2. Upon conducting Taylor series expansion analysis on $\beta_\pm$, we can see that \[Eq:beta\_pcm\] \_= D (1+()), where $D=(q''\dx^2)^2$ is a nonzero quantity independent of $\pm$ but might depend on $\dx$, assuming $q'' \ne 0$ on $S$.
The remaining procedure is to obtain the two nonlinear weights $\omega_\pm$ in the similar way done in the edge reconstructions in Eqs. (\[Eq:WENO5\_omega\]) – (\[Eq:WENOZ\_omega\]),
- For WENO-JS: $$\label{Eq:PCM_WENO5_omega}
\omega_{\pm} = \frac{\tilde{\omega}_{\pm}}{\tilde{\omega}_{-} + \tilde{\omega}_{+}}, \mbox{ where }
\tilde{\omega}_{\pm} = \frac{\gamma_{\pm}}{(\epsilon + \beta_\pm)^m},$$
- For WENO-Z: $$\label{Eq:PCM_WENOZ_omega}
\omega_{\pm} = \frac{\tilde{\omega}_{\pm}}{\tilde{\omega}_{-} + \tilde{\omega}_{+}}, \mbox{ where }
\tilde{\omega}_{\pm} = {\gamma_{\pm}}\Biggl(1+\Bigl(\frac{|\beta_+-\beta_-|}{\epsilon + \beta_\pm}\Bigr)^m\Biggr).$$
The final representation of the approximation $q'_{C,i}$ becomes q’\_[C,i]{} = \_- ’\_-(x\_i) + \_+ ’\_+(x\_i). Let us now verify that this approximation is fourth-order accurate after all, that is, \[Eq:PCM\_ucPrime\_final\] q’\_[C,i]{} = \_- \_-(x\_i) + \_+ \_+(x\_i) = q’(x\_i,t\^n) + (\^4). Similarly as before, using Eqs. (\[Eq:beta\_pcm\]), (\[Eq:PCM\_WENO5\_omega\]), and (\[Eq:PCM\_WENOZ\_omega\]), we can see that, with the help of the binomial series expansion, \[Eq:PCM\_omega\_order\] \_= \_+ (). Therefore, the desired accuracy claimed in Eq. (\[Eq:PCM\_ucPrime\_final\]) is readily verified by repeating the similar relationship in Eq. (\[Eq:qC\_prime\_third\_order\_derivation\]): \[Eq:qC\_prime\_fourth\_order\_derivation\] \_[=-,+]{}\_\_’(x\_[i]{}) - \_[=-,+]{}\_\_’(x\_[i]{}) &=&\_[=-,+]{}(\_- \_) (\_’(x\_[i]{}) -q’(x\_i,t\^n))\
&=&\_[=-,+]{}() (\^3) = (\^4). Comparing Eq. (\[Eq:qC\_prime\_fourth\_order\_derivation\]) with Eq. (\[Eq:qC\_prime\_third\_order\_derivation\]), we now see that it is fourth-order accurate due to the improved third-order accuracy in calculating $\phi_\ell'(x_{i}) -q'(x_i,t^n)$. This is a result of using the third degree PPM polynomials $\phi_\pm(x)$, with which $q'(x,t^n)$ can be accurately approximated by the second degree polynomials $\phi'_\pm(x)$ up to third-order.
We notice that there are some cases when $q'_{C,i}$ in Eq. (\[Eq:PCM\_ucPrime\_final\]) differs from $({q_{R,i}-q_{L,i}})/{\dx}$ by an order of magnitude. This may happen in two different cases: (i) they both are very small, approximating zero slopes, or (ii) one is larger (or smaller) than the other in regions where $q'(x,t^n)$ becomes singular at discontinuities or kinks at which the derivatives $q'(x_i,t^n)$ are not well defined. The first is simply due to the level of machine accuracy (e.g., one being $10^{-16}$ and the other being $10^{-15}$) and does not affect the overall spatial approximation of PCM. However, the latter needs to be taken with some spacial care because, at those singular points, any over/under predictions of $q'_{C,i}$ will result in undesirable oscillations, which can yield negative states in approximating density or pressure. To prevent this situation, we limit both $q'_{C,i}$ and $({q_{R,i}-q_{L,i}})/{\dx}$ using the MC slope limiter when it is detected there is an order of magnitude difference between the two, that is, \[Eq:qPrime\_flattening\] c\_1 = (q’\_[C,i]{}, ) if $\Big|\frac{\dx q'_{C,i}}{q_{R,i} - q_{L,i}}\Big|> 10$ or $\Big|\frac{\dx q'_{C,i}}{q_{R,i} - q_{L,i}}\Big| < 0.1$. This limiting does not get activated on smooth flows in general and does not affect the overall fifth-order accuracy of PCM (see Section \[Sec:convegence\_performance\]). However, in case the limiting is fully turned on and is activated on smooth flows, the accuracy is reduced to third-order because the solution accuracy is limited by the third-order dissipation of the MC limiter in smooth regions (see [@toth2008hall]). In what follows we call Eq. (\[Eq:qPrime\_flattening\]) the PCM flattening.
On a separate note, the PCM scheme reduces to a PPM-like algorithm when setting \[Eq:qPrime\_PPM\] q’\_[C,i]{} = , because, in this case, we have $c_3 = 0$ in Eq. (\[Eq:pcm\_cond4c\]) so that $p_i(x)$ in Eq. (\[Eq:pcm\_poly\]) loses its highest term, becoming a piecewise parabolic polynomial at most. The solution accuracy becomes third-order accurate, similar to the solution accuracy of PPM on 1D smooth flows.
This completes the PCM spatial reconstruction steps that provide the fifth-order accurate Riemann states $q_{L;R,i}$, and the fourth-order accurate derivative $q'_{C,i}$ in space.
The remaining task includes conducting a temporal updating step via tracing the characteristic lines using the piecewise cubic polynomials in Eq. (\[Eq:pcm\_poly\]). This step produces the Riemann states $(q_L,q_R)=(q_{R,i}^{n+1/2},q_{L,i+1}^{n+1/2})$ as predictor. We will show in the next section that these predictors are at least fourth-order accurate in time, and they are provided as the initial value problems for the Godunov fluxes at each interface $x_{i+1/2}$.
The PCM Characteristic Tracing for Temporal Updates {#Sec:chartracing}
===================================================
The PCM proceeds to the last step which advances the pointwise Riemann interface states at $t^n$ q\_[L;R,i]{} = p\_i(x\_[i1/2]{}), where $p_i(x)$ is the piecewise cubic polynomial in Eq. (\[Eq:pcm\_poly\]), to the half-time updated predictor states q\_[L;R,i]{}\^[n+1/2]{} by tracing characteristics. The idea is same as how the PPM characteristic tracing is performed [@colella1984piecewise], in which we seek a time averaged state. For instance, at the interface $x_{i+1/2}$, we consider q\^[n+1/2]{}\_[x+1/2]{}= \_[t\^n]{}\^[t\^[n+1]{}]{} q(x\_[i+1/2]{},t) dt. The initial condition at $t^n$ of a generalized Riemann problem is given as q(x\_[i+1/2]{},t\^n) = {
[ll]{} p\_i(x\_[i+1/2]{}), & x I\_i\
p\_[i+1]{}(x\_[i+1/2]{}), & x I\_[i+1]{}.
. Given a linear characteristic equation as in Eq. (\[Eq:Euler1D\_char\]), and for $t>t^n$ we then have \[Eq:char\_tracing\_IC\] q(x\_[i+1/2]{},t) = {
[ll]{} p\_i(x\_[i+1/2]{}-\_i (t-t\^n)), & x I\_i, \_i > 0,\
p\_[i+1]{}(x\_[i+1/2]{} - \_[i+1]{}(t-t\^n)), & x I\_[i+1]{}, \_[i+1]{} < 0.
. Here $\xi(t) = x_{i+1/2} - \lambda (t-t^n)$ is a characteristic line for an eigenvalue $\lambda$, assuming $t-t^n < \dt$.
We argue that the characteristically traced solution in Eq. (\[Eq:char\_tracing\_IC\]) is [*[almost]{}*]{} exact, provided the stability condition $t-t^n < \dt$ is satisfied (which is always true), inheriting all the desirable high-order accurate properties built in to the initial conditions which are, in this case, given by the piecewise cubic polynomial $p_i(x)$ (see [@leveque2007finite]). Therefore, the spatial accuracy designed in $p_i(x)$ naturally gets transferred to the evaluation of the time averaged state in Eq. (\[Eq:char\_tracing\_IC\]). In particular, for our case, the expected accuracy of the characteristic tracing using our cubic polynomial $p_i(x)$ for predicting a future state at $t>t^n$, satisfying $t-t^n < \dt$, is to be at least fourth-order accurate.
We now illustrate, for exposition purpose, the case with $x\in I_i$ with $\lambda_i > 0$ first. Using $\uparrow$ to denote the state from the left of $x_{i+1/2}$, q\^[n+1/2]{}\_[R,i]{} = q\^[n+1/2]{}\_[x+1/2,]{} &=& \_[t\^n]{}\^[t\^[n+1]{}]{} q(x\_[i+1/2]{},t) dt \[Eq:ERP\_pcm\_1\]\
&=& \_[x\_[i+1/2 - \_i ]{}]{}\^[x\_[i+1/2]{}]{} p\_i(x) dx. \[Eq:ERP\_pcm\_2\] Again, as seen in Eqs. (\[Eq:ERP\_pcm\_1\]) and (\[Eq:ERP\_pcm\_2\]), the half-time advancement of the spatially reconstructed state is given by the average of the reconstructed variable $p_i(x)$ over the domain of dependence $[x_{i+1/2 - \lambda_i \dt}, x_{i+1/2}]$ of the interface $x_{i+1/2}$. Therefore the accuracy of $q^{n+1/2}_{R,i}$ is inherited from that of the reconstruction algorithm of $p_i(x)$.
The outcome of the integration yields q\^[n+1/2]{}\_[x+1/2,]{} &=& c\_0 + (1- ) + (1-2 + ( )\^2) \^2\
&+&( 1-3 + 4()\^2 - 2()\^3 ) \^3. The case for $x\in I_{i+1}$ with $\lambda_{i+1} < 0$ can be obtained similarly, q\^[n+1/2]{}\_[L,i+1]{} &=& q\^[n+1/2]{}\_[x+1/2, ]{}\
&=& c\_0 + (-1- ) + (1+2 + ()\^2 ) \^2\
&+&( -1-3 - 4()\^2 - 2()\^3 ) \^3.
In the general case of a system of Euler equations, the above treatment is to be extended to include multiple characteristic waves correspondingly depending on the sign of each $k$-th eigenvalue $\lambda^{(k)}_i$. This gives us, for the two predictor states $\bV_{L;R,i}^{n+1/2}$ on each cell $I_i$ in primitive form, \[Eqn:PCM\_right\_state\_final\] &&\_[R,i]{}\^[n+1/2]{} = [C]{}\_0+ \_[k;\^[(k)]{}\_i>0]{}(1- )r\^[(k)]{} C\^[(k)]{}\_1\
&&+\_[k;\^[(k)]{}\_i>0]{}(1-2 +()\^2)r\^[(k)]{} C\^[(k)]{}\_2,\
&&+\_[k;\^[(k)]{}\_i>0]{}(1-3 +[4]{}()\^2 -[2]{}( )\^3)r\^[(k)]{} C\^[(k)]{}\_3,\
and \[Eqn:PCM\_left\_state\_final\] &&\_[L,i]{}\^[n+1/2]{} = [C]{}\_0+ \_[k;\^[(k)]{}\_i<0]{}(-1- )r\^[(k)]{} C\^[(k)]{}\_1\
&&+\_[k;\^[(k)]{}\_i<0]{}(1+2 +()\^2)r\^[(k)]{} C\^[(k)]{}\_2,\
&&+\_[k;\^[(k)]{}\_i<0]{}(-1-3 -[4]{}()\^2 -[2]{}( )\^3)r\^[(k)]{} C\^[(k)]{}\_3.\
Those new notations introduced in Eqs. (\[Eqn:PCM\_right\_state\_final\]) and (\[Eqn:PCM\_left\_state\_final\]) represent the $k$-th right eigenvector $r^{(k)}$, $k=1,2,3$, which is the $k$-th column vector of the $3\times 3$ matrix $\bR$ evaluated at $I_i$, = , and the $k$-th characteristic variable vector $\Delta \mbf C^{(k)}_m$ given as C\^[(k)]{}\_m = \^m \^[(k)]{} \_m, where, for $m=0,\dots,3$, \_m = \^T, in which $c^{(k)}_m$ is the $m$-th coefficient in Eqs. (\[Eq:pcm\_cond1c\]) $\sim$ (\[Eq:pcm\_cond4c\]) of the piecewise cubic polynomial in Eq. (\[Eq:pcm\_poly\]) applied to each of the $k$-th characteristic variable $\bar {q}_i$.
It is worth mentioning that, unlike the characteristic tracing of PPM (see [@colella1984piecewise]), the PCM scheme does not necessarily require any extra monotonicity enforcements on $\bV^{n+1/2}_{L;R,i}$. First of all, this is because the use of the WENO reconstruction algorithms provides $q_{L;R,i}$ and $q'_{C,i}$, all of which are, by design, non-oscillatory. Secondly, such monotonicity enforcements on the PPM’s parabolic polynomials are now redundant in PCM, since our building block polynomials are piecewise cubic. Compared to the parabolic polynomials, the cubic polynomials can easily adapt to fit $q_{L;R,i}$, $q'_{C,i}$, and $\bar{q}_i$ uniquely on each $I_i$, without needing to preserve such monotonicity constraints as in PPM, by readily varying its rate of change $p'_i(x)$ at an inflection point if needed, taking an advantage of an extra degree of freedom by being cubic.
Final Update Step in 1D
=======================
The only remaining task at this point is the final update to evolve $\avg{\bU}_i^n$ to $\avg{\bU}^{n+1}_i$. We proceed this using the high-order Godunov fluxes $\bF_{i+1/2}^{n+1/2}=\mathcal{RP}(\bU_L,\bU_R)=\mathcal{RP}(\bU_{R,i}^{n+1/2},\bU_{L,i+1}^{n+1/2})$ as corrector, where $\mathcal{RP}$ implies a solution of the Riemann problem. Note that the Riemann states in conservative variables $\bU_{R,i}^{n+1/2},\bU_{L,i+1}^{n+1/2}$ are obtained either by conversions from $\bV_{R,i}^{n+1/2},\bV_{L,i+1}^{n+1/2}$ in Eq. (\[Eqn:PCM\_right\_state\_final\]) and Eq. (\[Eqn:PCM\_left\_state\_final\]), or projecting the characteristic variables directly to the conservative variables in Eq. (\[Eqn:PCM\_right\_state\_final\]) and Eq. (\[Eqn:PCM\_left\_state\_final\]). We note that the first needs to be processed using high-order approximation [@mccorquodale2011high], in particular for multidimensonal problems, while such a high-order conversion is not required in 1D. In this regards the latter could be a better choice in multi spatial dimensions, because there is no need for any high-order conversion from the primitive Riemann states to the conservative Riemann states, knowing the fact that the [*[conservative]{}*]{} states variables are the type of inputs for the Riemann problems.
Multidimensional Extension of the 1D PCM Scheme {#Sec:pcmMultiD}
===============================================
Our primary purpose in the current paper is to focus on laying down the key algorithmic components of PCM in 1D. As described, the 1D PCM algorithm is formally fifth-order in space and fourth-order in time. Our test problems of one-dimensional smooth flows in Section \[Sec:convegence\_performance\] show that the algorithm delivers nominally a fifth-order accurate convergence rate, particularly with smaller $L_1$ errors than WENO-JS with RK4.
Although possible, extending such a high-order 1D algorithm to multiple spatial dimensions in a way to preserve the same order of convergence in 1D is an attentive task that requires some extra cares and attentions [@shu2009high; @buchmuller2014improved; @zhang2011order; @mccorquodale2011high] in the finite volume formulation. On the other hand, one of the simplest and easiest multidimensional extensions that has been widely adopted in many algorithmic choices [@mignone2007pluto; @mignone2011pluto; @mignone2010second; @stone2008athena; @fryxell2000flash; @dubey2009extensible; @lee2009unsplit; @lee2013solution; @bryan1995piecewise; @bryan2014enzo; @teyssier2002cosmological] is to use the dimension-by-dimension formalism in which the baseline 1D algorithm is extended in each normal sweep direction, requiring a very minimal effort for extension. However, the order of convergence from the resulting multidimensional extension is limited to be at most second-order due to the lack of accuracies that may arise in a couple of places in code implementations: (i) mis-using averaged quantities in place of pointwise quantities for Riemann states, (ii) using low-order approximations in converting between primitive and conservative variables, (iii) and applying low-order quadrature rules in flux function estimations [@shu2009high; @buchmuller2014improved; @zhang2011order; @mccorquodale2011high]. In our case, it takes more coding efforts, practically because the multidimensional PCM results we demonstrate in this paper have been obtained by integrating the PCM algorithm in the FLASH code framework [@fryxell2000flash; @dubey2009extensible; @dlee_flash]; hence carrying out the above-mentioned code changes in a large code such as FLASH requires extra efforts that are not the main points of the current paper. We leave such a high-order, multidimensional extension in our future work, and instead, we adopt the simple dimension-by-dimension formalism for our multidimensional extension of the 1D PCM algorithm.
Additionally, for our choice of multidimensional extension we use the computationally efficient unsplit corner transport upwind (CTU) formulation in FLASH [@lee2009unsplit; @lee2013solution], which requires smaller number of Riemann solves in both 2D and 3D than the conventional CTU approaches [@colella1990multidimensional; @gardiner2008unsplit; @saltzman1994unsplit], while achieving the maximum Courant condition of CFL $\approx$ 1 [@lee2009unsplit; @lee2013solution].
Results {#Sec:results}
=======
In this section we present numerical results of PCM in 1D, 2D and 3D for hydrodynamics and magnetohydrodynamics. The PCM results are compared with numerical solutions of other popular choices of reconstruction schemes including the second-order PLM [@colella1985direct], the third-order PPM [@colella1984piecewise] and the fifth-order WENO methods [@jiang1996efficient; @borges2008improved; @castro2011high]. As mentioned, the second-order accurate dimension-by-dimension approach has been adopted to extend all of the above baseline 1D reconstruction algorithms to multiple spatial dimensions. As this is the case, for multidimensional problems we have chosen the predictor-corrector type of characteristic tracing methods (charTr) for PLM, PPM, and WENO, not to mention PCM by design. In 1D problems, however, we treat WENO differently and integrate its spatial reconstruction with RK4 in consideration of fully demonstrating its orders of accuracy due from both space (i.e., $\mathcal{O}(\dx^5)$) and time (i.e., $\mathcal{O}(\dt^4)$). It should also be noted that the orders of WENO + RK4 in 1D are to be well comparable to those of PCM. Hence the choice provides a set of good informative comparisons between PCM and WENO + RK4 in particular, in which we will illuminate the advantages of PCM. In what follows, unless otherwise mentioned, we set the WENO-JS approach in Eq. (\[Eq:PCM\_WENO5\_omega\]) as the default choice for $q'_{C,i}$ in our PCM results. This default setting will be referred to as [*[PCM-JS]{}*]{} (or simply [*[PCM]{}*]{}), while the choice with the WENO-Z approach in Eq. (\[Eq:PCM\_WENOZ\_omega\]) will be referred to as [*[PCM-Z]{}*]{}.
1D Tests
--------
### 1D Convergence and Performance Tests {#Sec:convegence_performance}
####
In our first test we consider two configurations of 1D passive advection of smooth flows, involving initial density profiles of Gaussian and sinusoidal waves. We initialize the both problems on a computational box on \[0,1\] with periodic boundary conditions. The initial density profile of the Gaussian advection is defined by $\rho(x) = 1 + e^{-100(x-x_0)^2}$, with $x_0=0.5$, whereas for the sinusoidal advection the density is initialized by $\rho(x) = 1.5 - 0.5 \sin(2 \pi x)$. In both cases, we set constant velocity, $u=1$, and pressure, $P=1/\gamma$, and the specific heat ratio, $\gamma=5/3$.
The resulting profiles are propagated for one period through the boundaries, reaching $t=1$. At this point, both profiles return to its initial positions at which we conduct the $L_1$ error convergence tests compared with the initial conditions on the grid resolutions of $N_x=16, 32,64,128,256, 512$ and $1024$. Since the nature of the problem is a pure advection in both, any deformation of the initial profile is due to either phase errors or numerical diffusion. For stability we use a fixed Courant number, $C_{\mbox{cfl}}=0.8$ for both tests. We choose the HLLC Riemann solver [@toro1994restoration] in all cases.
[c]{}
The results of this study are shown in Fig. \[Fig:1DAdvections\]. From these numerical experiments, the PCM reconstruction shows the fifth-order convergence rates in both tests. Although both PCM and WENO-JS + RK4 demonstrate the same fifth-order of convergence rate, the $L_1$ errors of PCM are more than twice smaller than those of WENO-JS + RK4. The solutions of PPM converge with the rate of 2.5 which is the slowest among the three. Parameter choices for the PPM runs include the use of the MC slope limiter applied to characteristic variables, no flattening, no contact discontinuity steepening, and no artificial viscosity (this setting for PPM remains the same in what follows).
Scheme Speedup
--------------- ---------
PPM 0.65
PCM 1.00
WENO-JS + RK4 1.71
: Relative speedup of the PPM and WENO schemes compared to the PCM scheme for the 1D Gaussian and sine advection problems. The comparisons have been obtained from a serial calculation on a single CPU.[]{data-label="tab:performance"}
In Table \[tab:performance\] we compare the relative performance speedups of PCM, PPM and WENO-JS + RK4, all testing the Gaussian and sinusoidal advection problems. We can clearly see that there is a big performance advantage in PCM over WENO-JS + RK4 in delivering the target fifth-order accuracy. The major gain in PCM lies in its predictor-corrector type of characteristic tracing which affords not only the accuracy but also the computational efficiency. Such a relative computational efficiency of PCM in 1D is expected to grow much larger in multidimensional problems, considering that there have to be added algorithmic complexities in achieving high-order accurate solutions in multidimensional finite volume reconstruction [@shu2009high; @buchmuller2014improved; @zhang2011order; @mccorquodale2011high] from the perspectives of balancing optimal numerical stability and accuracy. We will report our strategies of multidimensional extension of PCM in our future work.
### 1D Discontinuous Tests
In this section we test PCM on a series of well-benchmarked shock-tube problems of one dimensional hydrodynamics and MHD that involve discontinuities and shocks. As all the tests here have already been well discussed in various literatures, we will describe their setups only briefly and put our emphasis more on discussing the code performance of PCM. Readers are encouraged to refer to the cited references in the texts for more detailed descriptions on each setup.
####
[cccc]{}\
\
The Sod’s problem [@sod1978survey] has been one of the most widely chosen popular tests in 1D to assess a code’s capability to handle shocks and contact discontinuities. The initial condition is consist of the left and the right states given as \[Eq:Sod\] (, u, p) = {
[ll]{} (1, 0, 1) & x < 0.5,\
(0.125, 0, 0.1) & x > 0.5,
. with the ratio of specific heats $\gamma = 1.4$ on the entire domain $[0,1]$. The outflow boundary conditions are imposed at $x=0$ and $x=1$. Shown in Fig. \[Fig:Sod\] include two numerical solutions of PCM, with and without the use of the PCM flattening given by Eq. (\[Eq:qPrime\_flattening\]); and two solutions of using PPM and WENO-JS + RK4. The Roe Riemann solver [@roe1981approximate] was used in all cases. The test cases (denoted in symbols) are resolved on the grid size of $N_x=128$, and are compared with the reference solutions (denoted in solid curves) computed using WENO-JS + RK4 on the grid resolution of $N_x=1024$. A fixed value of $C_{\mbox{cfl}}=0.8$ was used for all tests.
The result in Fig. \[Fig:Sod\](a) shows that the solutions of PCM without using the flattening well predict all nonlinear flow characteristics of the rarefaction wave, the contact discontinuity, and the shock. A notable thing in PCM is the number of points at the shock. We see in Fig. \[Fig:Sod\](a) that at the shock there is only one single point in all flow variables, whereas in all other cases, there are two points spread over the shock width.
We also tested the PCM flattening in in Fig. \[Fig:Sod\](b). We observe that the switch introduces some level of noisy oscillations, easily seen in the region between the rarefaction tail and the shock. As anticipated, the solution looks very similar to that of PPM in Fig. \[Fig:Sod\](c) because, in the limit of Eq. (\[Eq:qPrime\_PPM\]), the PCM flattening reduces the PCM scheme to a PPM-like algorithm.
####
[cccc]{}\
The second test is the Shu-Osher problem [@Shu1989]. In this problem we test PCM’s ability to resolve both small-scale smooth flow features and the shock. On \[-4.5, 4.5\], the initial condition launches a nominally Mach 3 shock wave at $x=-4.0$ propagating into a region ($x>-4.0$) of a constant density field with sinusoidal perturbations. As the shock advances, two sets of density features appear behind the shock. The first set has the same spatial frequency as the un-shocked perturbations, whereas and the second set behind the shock involves the frequency that is doubled. The important point of the test is to see how well a code can accurately resolve strengths of the oscillations behind the shock, as well as the shock itself.
The results of this test are shown for PLM, PPM, WENO-JS + RK4, and PCM in Fig. \[Fig:ShuOsher\]. The solutions are calculated at $t=1.8$ using a resolution of $N_x=256$ and are compared to a reference solution resolved on $N_x=1024$. All methods were solved using the Roe Riemann solver, with $C_{\mbox{cfl}}=0.8$. It is evident in Fig. \[Fig:ShuOsher\](b) that the PCM solution exhibits the least diffusive solution among the tested methods, producing a very-high order accurate solution that is more quickly approaching to the high resolution reference solution.
####
[cccc]{}\
\
First described by Einfeldt et al. [@einfeldt1991godunov] the main test point in this problem is to see how satisfactorily a code can compute physical variables, $p, u, \rho, \epsilon$, etc. in the low density region. Among the variables the internal energy $\epsilon=p/(\rho(\gamma-1))$, where $\gamma=1.4$, is the hardest to get it right due to the ratio of the pressure and density that are both close to zero. The ratio of the two small quantities will amplify any small errors in each, hence making the error in $\epsilon$ appear to be the largest in general [@Toro2009].
The large errors in $\epsilon$ are indeed observed in Fig. \[Fig:Rarefaction\](b) $\sim$ Fig. \[Fig:Rarefaction\](d) in that the error is the largest at or around $x=0.5$ in the presence of sudden increase of its peak values. On the contrary, the internal energy computed using PCM shown in Fig. \[Fig:Rarefaction\](a) behaves in a uniquely different way such that the value continues to drop when approaching $x=0.5$. From this viewpoint, and with the help of the exact solution available in [@Toro2009], it’s fair to say that the PCM solution in Fig. \[Fig:Rarefaction\](a) appears to predict the internal energy most accurately. It is seen that there are two slight bumps produced in Fig. \[Fig:Rarefaction\](a), at $x\approx 0.08$ and $x \approx 0.92$ in $\rho$, which disappear by turning on the PCM flattening as shown in Fig. \[Fig:Rarefaction\](b).
####
[cccc]{}\
This problem was introduced by Woodward and Colella [@woodward1984numerical] and was designed to test a code performance particularly on interactions of strong shocks and discontinuities. We follow the original setup to test PCM, and compare its solution with those of PPM and WENO-JS + RK4, using 128 grid points to resolve the domain $[0,1]$. In Fig. \[Fig:2blast\] the three density profiles at $t=0.038$ are plotted against the high-resolution solution of WENO-JS + RK4 on 1024 grid points. Overall, all methods we tested here produce an acceptable quality of solutions as illustrated in Fig. \[Fig:2blast\](a). Note however that, among the three methods, the PCM solution in Fig. \[Fig:2blast\](b) demonstrates the highest peak heights, following more closely the high-resolution solutions. As reported in [@stone2008athena] we see that all methods also smear out the contact discontinuity at $x\approx 0.6$ pretty much the same amount.
####
[cccc]{}\
\
An MHD version of the Sod’s shock tube problem was first studied by Brio and Wu [@brio1988upwind], and it has become a must-to-do test for MHD codes. Since then, the problem has revealed a couple of interesting findings including not only the discovery of the compound wave [@brio1988upwind], but also the existence of non-unique solutions [@torrilhon2003uniqueness; @torrilhon2003non]. More recently, Lee [@lee2011upwind] realized that there are unphysical numerical oscillations in using PPM and studied an approach to suppress the level of oscillations based on the upwind slope limiter. The presence of such oscillations in PPM has been also briefly reported in [@stone2008athena]. The study reported in [@lee2011upwind] shows that the origin of the oscillations arise from the numerical nature of a slowly moving shock as a function of the magnetic strength of tangential component. The slowly moving shock was first identified in [@woodward1984numerical], and the oscillatory behaviors have been studied by many researchers for more than 30 years, yet there is no ultimate resolution [@karni1997computations; @stiriba2003numerical; @arora1997postshock; @jin1996effects; @roberts1990behavior; @johnsen2008numerical].
In this test, as just mentioned, there are observable numerical oscillations found in all methods, PPM, PCM and WENO-JS + charTr. The results in Fig. \[Fig:BrioWu\] show that the oscillations are the largest in PPM, consistent with the findings in [@lee2011upwind], and there are less amount in PCM and WENO-JS + charTr. The PPM solutions are suffering from significant amount of spurious oscillations in all four variables, $\rho, u, p$ and $B_y$, as shown in Fig. \[Fig:BrioWu\](c). Such behaviors are less significant in PCM and WENO-JS + charTr, respectively illustrated in Fig. \[Fig:BrioWu\](a) and Fig. \[Fig:BrioWu\](d), in that, the oscillations in $\rho, p, B_y$ are much more controlled now, while the most outstanding oscillations are found in $u$ near $x\approx0.8$. Again, Fig. \[Fig:BrioWu\](b) shows that the PCM flattening makes PCM to perform very similar to PPM. It is worth mentioning that the oscillatory behaviors remain to be consistent regardless of the choice of Riemann solvers such as HLL [@harten1997upstream], HLLC [@li2005hllc], HLLD [@miyoshi2005multi], or Roe [@roe1981approximate] (tested here).
####
[cccc]{}\
\
Ryu et al. [@ryu1994numerical] studied a class of one dimensional MHD shock tube problems that are informative to run as a code verification test. We have chosen one of their setups, introduced in their figure 2a. In what follows the problem is referred to be as the RJ2a test. The viewpoint of this test is to monitor if all three dimensional MHD waves are successfully captured. We see that in Fig. \[Fig:RJ2a\] all structures of left- and right-going fast shocks, left- and right-going slow shocks, and a contact discontinuity are well captured in all methods tested, including PCM.
2D Tests
--------
We present two dimensional tests of hydrodynamics and MHD in this section. All test cases are computed using the second-order dimension-by-dimension extension of the baseline 1D algorithms, including PCM.
### 2D Convergence Test of the Isentropic Vortex Advection
![Convergence test of the 2D isentropic vortex advection problem. The errors in $\rho, u$, and $v$ are calculated in $L_1$ sense against the initial conditions. The tested PCM solutions are solved on $N_x \times N_y$, where $N_x=N_y= 32, 64, 128, 256$ and $512$. All runs reached to $t=10$ using the HLLC Riemann solver with $C_{\mbox{cfl}}=0.8$.[]{data-label="Fig:2dIsentropicVortex"}](./isen_vort_conv.png){width="3.4in"}
The first 2D test problem, considered in [@yee2000entropy], consists of the advection of an isentropic vortex along the diagonal of a cartesian computational box. The dynamics of the problem allows to quantify a code’s dissipative properties and the correct discretization balance of multidimensional terms through monitoring the preservation of the initial circular shape of the vortex. At $t=10$ the vortex finishes one periodic advection over the domain and returns to the initial position, where we can measure the solution accuracy against the initial condition. As such we have chosen this problem particularly to access the PCM’s order of convergence rate in 2D. We omit the details of the initial problem setup which can be found in [@yee2000entropy].
As expected, the results presented in Fig. \[Fig:2dIsentropicVortex\] clearly confirm that, when PCM is extended to 2D using the simple dimension-by-dimension formulation, the overall numerical solution accuracy converges in second-order, regardless of its inherent fifth-order property in 1D.
### 2D Discontinuous Tests
####
We consider the Sedov blast test [@sedov1993similarity] to check PCM’s ability to handle a spherical symmetry of the strong hydrodynamical shock explosion. The problem studies a self-similar evolution of a spherical shock wave propagation due to an initial point-source of a highly pressurized perturbation. The test has been used widely in various literatures, and we follow the same setup found in [@fryxell2000flash]. Panels in Fig. \[Fig:Sedov\_PCM\_a\] show the density field in linear scale at $t=0.05$ resolved on a grid size of $256\times 256$ for the domain $[0,1]\times [0,1]$. The HLLC Riemann solver was used in all runs with $C_{\mbox{cfl}}=0.8$. The range of the plotted densities in colors in all four panels is $0.01 \le \rho \le 4.9$, the same is also used for the 30 levels of density contour lines that are plotted in logarithmic scale.
As can be seen, the PCM solution in Fig. \[Fig:Sedov\_PCM\_a\](a) is superior not only in preserving a great deal of the spherical symmetry at the outermost shock front, but also in revealing more flow structures in the central low density region, again in the most spherical manner. This great ability of preserving the spherical symmetry in PCM is also found in Fig. \[Fig:Sedov\_PCM\_b\](a) where the two curves are the two section cuts of density fields along $y=0.5$ (black) and $y=x$ (cyan), respectively. We see that the two peak values of each section cut are matching each other very closely in terms of both their locations and their magnitudes. In the other schemes there are clearly much larger disagreements in the magnitude of the peak density values.
It is also interesting to note that the PCM flattening makes the symmetry worse, as illustrated in Fig. \[Fig:Sedov\_PCM\_a\](a) and Fig. \[Fig:Sedov\_PCM\_a\](b). For this reason, as well as for the observations we have collected in our 1D results, the default choice in PCM is to keep the flattening off, unless otherwise stated in what follows.
[cccc]{}\
[cccc]{}\
####
Next we test PCM for a family of well-known benchmarked Riemann problems whose mathematical classification was originally put forward by Zhang and Zheng [@zhang1990conjecture], in which the original 16 of admissible configurations were conjectured on polytropic gas. This claim was corrected by Schultz-Rinne [@schulz1993classification] that one of them was impossible, and the numerical testings for such 15 configurations were studied in [@schulz1993numerical]. Later, Lax and Liu showed that there are total of 19 genuinely different configurations available, providing numerical solutions of all 19 cases too [@lax1998solution]. See also [@chang19952]. Until today, this family of 2D Riemann problems has been chosen by many people to demonstrate that their numerical algorithms can predict these 19 configurations successfully in pursuance of code verification purposes [@buchmuller2014improved; @kurganov2002solution; @don2016hybrid; @balsara2010multidimensional].
We follow the setup as described in [@don2016hybrid] in the following two verification tests, Configuration 3 and Configuration 5. In both cases the calculations show numerical solutions on $[0,1]\times [0,1]$ using outflow boundary conditions.
#### (b) – Configuration 3
[cccc]{}\
\
Panels in Fig. \[Fig:2dRiemann\_conf3\] show numerical solutions of density at $t=0.8$ resolved on $400\times 400$ using the HLLC Riemann solvers with $C_{\mbox{cfl}}=0.8$. Also shown are the 40 contour lines of $\rho$. The range of $\rho$ is fixed as $0.1 \le \rho \le 1.8$ in both the pseudo-color figures and the contour lines. For the PCM method we employed both approaches of WENO-JS (see Eq. (\[Eq:PCM\_WENO5\_omega\])) and WENO-Z (see Eq. (\[Eq:PCM\_WENOZ\_omega\])) for the calculations of the smoothness stencils. The figures can be directly compared with Fig. 6 in [@don2016hybrid] where they used the same grid resolution for their hybrid compact-WENO scheme. First of all, including the two PCM solutions, we see that all calculations have produced their solutions successfully, in particular, without suffering unphysical oscillations near shocks and contact discontinuities. This test confirms that the PCM results are well comparable to the other solutions, except for some expected minor discrepancies.
One thing to notice is that the PPM solution in Fig. \[Fig:2dRiemann\_conf3\](b) has interestingly much more formations of Kelvin-Helmholtz instabilities, identified as vortical rollups along the slip lines (shown as the interface boundaries between the green triangular regions and the sky blue areas surrounding the mushroom-shaped jet). This feature is also often found in a test known as “Double Mach reflection" [@woodward1984numerical], where the similar pattern of rollups are detected along the slip line. As found in various studies [@zhang2011order; @qiu2005hermite; @mignone2007pluto] it is conventional to say that the amount of such vortical rollups at slip lines is one of the key factors to measure inherent numerical dissipations in a code. If we follow this approach, it then leads us to say that the PPM method is the least dissipative method among the six methods we tested. However, we find that this conclusion is somewhat arguable considering the nominal order of accuracy of PPM is lower than those of WENO-JS + CharTr, WENO-Z + CharTr and PCM. We think that there is to be more accurate assessment regarding this type of conclusion. Readers can find a very similar numerical comparison between PPM and WENO-JS in [@mignone2007pluto].
#### (b) – Configuration 5
[cccc]{}\
As a second 2D Riemann problem we consider Configuration 5 to test PCM and compare its solution with three other solutions of PLM, PPM and WENO-JS + CharTr. The WENO-JS approaches in Eqs. (\[Eq:WENO5\_omega\]) and (\[Eq:PCM\_WENO5\_omega\]) are adopted for PCM for the calculations of the smoothness stencils. The choice of grid resolution is $1024\times 1024$ in this test in order to directly compare our results with the results reported in Fig. 4 and Fig. 5 of [@buchmuller2014improved]. As obtained in Fig. \[Fig:2dRiemann\_conf5\], the PCM solution satisfactorily compares well with the solutions of PPM and WENO-JS + CharTr, as well as with the high-resolution results in [@buchmuller2014improved]. As also reported in [@buchmuller2014improved; @zhang2011order], when considering discontinuous flows in multiple space dimensions, the dimension-by-dimension approach works just as fine in terms of producing comparably accurate solutions. Likewise, the results in Fig. \[Fig:2dRiemann\_conf5\] show that we observe the same qualitative performances in all the methods we tested here, including PCM.
Although all the solutions are comparably admissible, there are few distinctive features in the PCM solution, displayed in Fig. \[Fig:2dRiemann\_conf5\] (d). We note that the minimum and maximum values of the computed density are respectively the smallest and the largest among the four results. This trend is consistent with the increasing order of accuracy in the four panels here. The same observation can be found also in Fig. \[Fig:2dRiemann\_conf3\] too. We consider this as an indication that the small scale features are better resolved in PCM with less amount of numerical diffusion.
####
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\
Next, we consider the MHD rotor problem [@Balsara1999; @Toth2000]. As the problem has been discussed by various people we rather focus on discussing the solution of PCM here. We use the same setup conditions as described in [@lee2009unsplit]. Exhibited in Fig. \[Fig:MHD\_rotor\](e) and Fig. \[Fig:MHD\_rotor\](f) are respectively the density and the 30 contour lines of the Mach number on $400\times 400$ cells, both at $t=0.15$. With minor discrepancies, we see that the PCM solution successfully demonstrates its ability to solve MHD flows in multiple space dimensions. To test PCM for multidimensional MHD flows, we integrated the PCM algorithm in the MHD scheme [@lee2009unsplit; @lee2013solution] of the FLASH code [@fryxell2000flash; @dubey2009extensible; @dlee_flash]. Of noteworthy point is that the contour lines of the Mach number in Fig. \[Fig:MHD\_rotor\](f) remain concentric in the central region without any distortion from the near-perfect symmetry. In all runs the HLLD Riemann solver [@miyoshi2005multi] was used with a fixed value $C_{\mbox{cfl}}=0.8$. The PPM run used the MC slope limiter for monotonicity.
3D Tests
--------
Lastly, for 3D cases, we have selected three test problems in MHD in such a way that we can fully quantify the performance of PCM both for assessing its convergence rate in 3D and for verifying its code capability in discontinuous flows.
### 3D Convergence Test
####
We solve the circularly polarized Alfvén Wave propagation problem [@lee2013solution; @gardiner2008unsplit] as our first 3D test problem to quantify the PCM’s order of accuracy in full 3D. The computational domain is resolved on $2N_x \times N_y \times N_z$ grid cells, where we choose $N_x = N_y = N_z = 8, 16, 32$ and $64$ for the grid convergence study. As in [@lee2013solution] we ran the same two configurations of the wave mode that are the standing wave mode and the traveling wave mode until $t=1$. In both we choose the Roe Riemann solver with $C_{\mbox{cfl}}=0.95$.
Respectively, Fig. \[Fig:3D\_conv\](a) and Fig. \[Fig:3D\_conv\](b) are the $L_1$ numerical errors on a logarithmic scale for the standing wave case and the traveling wave case. We observe that the rate of PCM convergence in 3D is second-order as expected, which agrees with the results reported in [@lee2013solution]. One difference is noted in the standing wave case in Fig. \[Fig:3D\_conv\](a) that PCM’s $L_1$ error in each grid resolution is much lower than those obtained with PPM + HLLD + F-CTU with $C_{\mbox{cfl}}=0.95$ in [@lee2013solution]. However, the magnitudes of the PCM error in the traveling wave case in Fig. \[Fig:3D\_conv\](b) look pretty much similar to the equivalent run in [@lee2013solution].
[ccc]{}
### 3D Discontinuous Tests
####
[ccc]{}\
We consider the 3D variant of the MHD blast problem by adopting the setup conditions in [@lee2013solution] to demonstrate the three-dimensional propagation of strong MHD shocks using the PCM algorithm. The original 2D version of the spherical blast wave problem was studied in [@koessl1990numerical], and later various people adopted the similar setup conditions [@stone2008athena; @lee2009unsplit; @lee2013solution; @Balsara1999; @zachary1994higher; @balsara2015divergence; @ziegler2011semi; @Mignone2010a; @li2008high; @kawai2013divergence; @londrillo2000high] for their code verifications in strongly magnetized shock flows.
We display four different fluid variables in each panel in Fig. \[Fig:3D\_blastBS\]. From the top right quadrant to the bottom right quadrant in counter clockwise direction, we show the gas pressure $p$, the density $\rho$, the total velocity $U=\sqrt{u^2+v^2+w^2}$, and the magnetic pressure $B_p$, all plotted at $t=0.01$. The grid resolution $128\times 128\times 128$ as well as all other parameters are chosen as same as in [@lee2013solution] in order to provide a direct comparison.
We tested PCM in three different plasma conditions defined by the three different strengths of $B_x=0, \frac{50}{\sqrt{4\pi}}$, and $\frac{100}{\sqrt{4\pi}}$, as displayed in Fig. \[Fig:3D\_blastBS\](a) $\sim$ Fig. \[Fig:3D\_blastBS\] (c). Of particular interest to note is with the initial low plasma $\beta$ conditions in the last two cases, $\beta=1\times 10^{-3}$ and $2.513\times 10^{-4}$, respectively. On the other hand, the first setup in Fig. \[Fig:3D\_blastBS\](a) produces the non-magnetized plasma flow, hence it allows us to test the PCM algorithm in the pure hydrodynamical limit in 3D. As clearly seen, all results have produced confidently accurate solutions. We also note that PCM has produced larger values of extrema in each variable than those reported in [@lee2013solution], without exhibiting any unphysical oscillations. This test demonstrate that the PCM algorithm is well-suited for simulating low-$\beta$ flows in full 3D.
####
[ccc]{}\
Since this problem was originally studied and reported in [@gardiner2005unsplit], the problem has become a popular benchmark case among various code developers to demonstrate their MHD algorithms’ capabilities in advecting the initial field loop which is weakly magnetized with a very high plasma $\beta=2\times 10^6$. The problem is known to be challenging [@gardiner2008unsplit; @gardiner2005unsplit], however, many have demonstrated that their codes can successfully produced comparable results [@lee2009unsplit; @lee2013solution; @balsara2010multidimensional; @li2008high; @balsara2013efficient; @kappeli2011fish; @stone2009simple]. In addition to the two original setups [@gardiner2008unsplit; @gardiner2005unsplit] where the initial field loops advect with the angle diagonal to the domain, Lee [@lee2013solution] recently reported that a small-angle advection is much more challenging. As an example, Lee adopted the advection angle $\theta \approx 0.573^\circ$ relative to the $x$-axis for the small-angle advection case in both 2D and 3D. The study found that a proper amount of multidimensional numerical dissipation plays a key role in maintaining the clean small-angle advection, and designed the algorithm called [*[upwind-MEC]{}*]{}. Here we repeat all three configurations (two large-angle advection cases and one small-angle advection case) by following the same setups in [@lee2013solution]. All the results in Fig. \[Fig:3D\_fieldloop\] were obtained using PCM and the Roe Riemann solver with $C_{\mbox{cfl}}=0.8$ on $64\times 64 \times 128$ cells.
First, Fig. \[Fig:3D\_fieldloop\](a) shows the small-angle advection with $\theta \approx 0.573^\circ$ relative to the $x$-axis with the velocity fields given by $\bU=(\cos\theta, \sin\theta, 2)^T$. Compared to this, in Fig. \[Fig:3D\_fieldloop\](b), we use $\bU=(1,1,2)^T$ which yields the large-angle advection. In both cases the tilt angle $\omega$ (see [@lee2013solution] for details) is set to be same as $\theta$. As manifested, both runs cleanly preserve the initial geometry of the field loop, convincing us that the PCM algorithm is robust and accurate in this challenging problem. As a final test we also perform the standard field loop advection setup of Gardiner and Stone [@gardiner2008unsplit]. The result is shown in Fig. \[Fig:3D\_fieldloop\](c). We see clearly that the PCM algorithm has produced well-behaving, accurate and confident solutions in this test. The results in Fig. \[Fig:3D\_fieldloop\] can be directly compared to the results reported in [@lee2013solution].
Conclusions {#Sec:conclusions}
===========
We summarize key features of the PCM algorithm studied in this paper.
- We have presented a new high-order finite volume scheme for the solutions of the compressible gas dynamics and ideal MHD equations in 1D. This baseline 1D algorithm uses piecewise cubic polynomials for spatial reconstruction by adopting the non-oscillatory approximations of the fifth order WENO schemes to determine the unique piecewise cubic polynomial on each cell. To provide the nominal fifth-order accuracy in space, we have developed a new non-oscillatory WENO-type reconstruction for $q'_{C,i}$ approximation. The new approach makes use of the two parabolic polynomials, termed as [*[PPM-Build]{}*]{}, to achieve fourth-order accuracy in establishing $q'_{C,i}$ approximation.
- We have formulated a new fourth-order temporal updating scheme, all integrated in PCM by design, based on the simple predictor-corrector type characteristic tracing approach. The overall solution accuracy of the baseline 1D PCM scheme, combining both spatially and temporally, seemingly converges with fifth-order. We show the PCM scheme compares greatly with the spatially fifth-order WENO integrated with RK4, demonstrating even smaller $L_1$ errors in PCM.
- A comparison of the computational expenses of PCM, PPM and WENO-JS + RK4 in 1D reveals that PCM has a superior advantage over the fifth-order counterpart WENO-JS + RK4 by a factor of 1.71.
- We have integrated the baseline 1D PCM algorithm for multidimensional cases by adopting the simple dimension-by-dimension approach. As anticipated, this approach yields at most second-order accurate solutions in multidimensional simulations of smooth flows. In the presence of flow discontinuities and shocks, however, the results obtained with the present simple multidimensional PCM extension shows a great level of confidence in predicting numerical solutions of hydrodynamics and MHD. An approach to extend the fifth-order property of the baseline 1D PCM to multiple spatial dimensions will be further investigated in our future work.
Acknowledgements
================
The software used in this work was in part developed by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago. D. Lee also gratefully acknowledges the FLASH group for supporting the current work.
References {#references .unnumbered}
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abstract: 'Despite the realizations of spin-orbit (SO) coupling and synthetic gauge fields in optical lattices, the associated time-reversal symmetry breaking, and 1D nature of the observed SO coupling pose challenges to obtain intrinsic $Z_2$ topological insulator. We propose here a model optical device for engineering intrinsic $Z_2$ topological insulator which can be easily set up with the existing tools. The device is made of a periodic lattice of quantum mechanically connected atomic wires (dubbed SO wires) in which the laser generated SO coupling ($\alpha_{\bf k}$, with ${\bf k}$ being the momentum) is reversed in every alternating wires as $\pm\alpha_{\bf k}$. The associated small Zeeman terms are also automatically reversed in any two adjacent SO wires, which allow to effectively restore the global time-reversal (TR) symmetry. Therefore, the two SO wires serve as the TR partner to each other which is an important ingredient for $Z_2$ topological insulators according to the Kane-Mele model. These properties ensure a non-trivial $Z_2$ invariant topological insulator phase with protected edge states. We also discuss that a non-local current measurement can be used to detect the chiral edge states.'
author:
- |
Sayonee Ray$^{1}$[^1], Kallol Sen$^{2}$[^2] and Tanmoy Das$^{1}$[^3]\
[$^{1}$Department of Physics, Indian Institute of Science, Bangalore-560012, India.\
$^{2}$Center for High Energy Physics, Indian Institute of Science, Bangalore-560012, India. ]{}
title: Assembling topological insulators with lasers
---
Optical lattice provides a model ‘breadboard’ to imprint diverse quantum and topological phases of ultracold fermionic and bosonic atoms.[@rev1; @rev2; @rev3; @rev4; @Gauge2; @Gauge_AB] The realization of the synthetic gauge field in optical lattices,[@Gauge; @Gauge2; @Gauge_AB] analogous to magnetic field in solid state systems, is one of the major triumph in this field. This discovery has opened up possibilities for devising new and exotic quantum and topological phases, some of which may have even no analog with the solid state counterparts. Many exotic properties such as geometric Berry phase, Harper-Hofstadter butterfly,[@HHB] spin-orbit coupling (SOC),[@lin] time-reversal (TR) symmetry breaking Haldane lattice,[@jotzu] quantum spin-Hall insulator (QSH) [@Goldman; @aidelsburger], non-trivial edge states,[@mancini; @Stuhl] are subsequently synthesized in these systems.
Chiral motion of particles, arising from either staggered hopping or SOC, is instrumental to various topological phases of matter[@TIreviewCK; @TIreviewSCZ; @TIreviewTD]. Diverse forms of TR invariant topological and QSH states have been realized in solid state frameworks. Furthermore, a recent realization of SOC with detuned lasers[@lin; @bliokh] has provided the opportunity that topological phases can also be obtained in optical lattices[@aidelsburger]. However a number of shortcomings, intrinsic to the optical lattice frameworks, hinders setting up $Z_2$ topological insulators (TIs) in this framework. For example, due to the inevitable presence of the Zeeman-like term, although often estimated to be small, the TR symmetry becomes inherently broken. Similarly, the SOC in optical devices can be synthesized easily in one-dimensional (1D) atomic chain,[@lin] while its generalization to higher dimensions is cumbersome.[@SOC2D; @SOC2D2] A similar difficulty arises in solids when Rashba- and Dresselhaus-type SOCs possess equal strengths, subsisting only a residual 1D SOC, or in quasi-1D quantum wires. Such 1D SOC prevents electrons to form closed orbit motion in the bulk. Since the localization of counter-propagating ‘chiral orbits’ without breaking the TR symmetry is the key for 2D $Z_2$ TIs, observation of them has proven challenging in optical devices.
We propose an optical device to assemble $Z_2$ TIs in 2D (extension to 3D follows the same principle), as illustrated in Fig. \[fig1\]. As two detuned lasers are directed perpendicular to each other, it generates a SOC, with equal Rashba and Dresselhaus strengths, at a 45$^o$ angle from both lasers, say $k_x$ direction[@lin]. We take two other lasers of the same configuration, but aligned anti-parallel to the above SO wire in such a way that a SOC commences along the $-k_x$ direction. These two counter-propagating SO wires are referred as ‘A’ and ‘B’ wires as shown in Fig. \[fig1\](a). For engineering advantage, layers of ‘A’ and ‘B’ wires do not need to be on the same plane, and can be assembled on a bi-layer framework.[@footnote_setup] If the spin-up atom is right-moving in ‘A’ wire, it becomes left moving in ‘B’ wire and vice versa. Furthermore, as shown in the supplementary material, the Zeeman term for two such wires consequently become completely reversed ($\pm\Omega$), giving opposite spin splittings as shown Fig. \[fig1\](c). Therefore, ‘A’ and ‘B’ wires possess positive and negative band gaps, as in the case of CdTe and HgTe systems, respectively[@bhz]. The byproduct of this setup is that as the two wires are brought closer, their combined structure creates an effective spin-degenerate band structure at all $k$-points. Therefore, a band gap can be opened at the TR invariant $k$-points without breaking the TR symmetry as shown in Fig. \[fig1\](d). The effective band gap at the $\Gamma$-point is a combination of the Zeeman term, and the inter-wire (spin conserving) hopping amplitude, $t$. We show below that as the SOC strength exceeds the atom’s kinetic energy terms, the valence band in Fig. \[fig1\](d) fails to cross the Fermi level ($E_F$) and an insulating state arises.
The associated emergence of $Z_2$ topological invariant without any further tuning can be understood from the combinations of band inversion phenomena as proposed in HgTe/CdTe heterostructure,[@bhz] and the ‘TR polarization’ as proposed by Kane and Mele.[@kane] According to the Kane-Mele formalism,[@kane] a $Z_2$ invariant arises if a fermion’s wavefunction switches to its TR conjugate odd number of times in traversing half of the Brillouin zone (BZ). This criterion is embedded automatically in our structure since the two SO wires serve as TR partners to each other. As the inter-wire hopping becomes comparable to the intra-wire hopping, the right-moving spin-up atom in ‘A’ wire hops to ‘B’ wire and becomes left-moving, and vice versa. Finally, as the spin up atom hops back to the original ‘A’ wire, it encircles a closed loop, as illustrated in Fig. \[fig1\](b). In this process opposite spin states form counter-helical orbits without breaking the TR symmetry and thus become localized in the bulk. In analogy with the HgTe/CdTe heterostucture,[@bhz] the inverted band gap between ‘A’ and ‘B’ wires, Fig. \[fig1\](c), ensures a band inversion at the $\Gamma$-point. We have computed the generalized $Z_2$ invariant for this setup, which holds in both 2D and 3D, and supplemented the results with calculations of edge states. The proposed setup is also applicable in solid state frameworks such as in quasi-1D quantum wires, where SOC and electronic properties are largely tunable.\
We begin with formulating the above-mentioned setup. For each SO wire, generation of SOC was demonstrated in the $\rm{Rb}^{87}$ atoms which has a ground state with total angular momentum $F=1$, and $m_F=1,0,-1$ multiplets. Lets us assume that the ultracold $\rm{Rb}^{87}$ atoms are optically trapped along the $x$-direction with inter-atomic distance being $a$. Each $\rm{Rb}^{87}$ atom is further regulated with two Raman lasers pointed in the $ \hat{x}+\hat{y}$ and $\hat{x}-\hat{y}$ directions, with slightly detuned frequencies by $\triangle\omega_{\rm L}$. The two electric fields are ${\bf E}_{1}=\sin(k_{\rm L}x)(\hat{x}+\hat{y})$, and ${\bf E}_{2}=\sin(k_{\rm L}x+\Delta\omega_{\rm L} t)(\hat{x}-\hat{y})$, where $k_{\rm L}$ is the wavevector of the lasers. For this setup, the lowest order light-matter interaction term extends upto the rank-1 (vector) terms, giving rise to light shift interaction with atoms as: $$H=\Omega^{(0)}{\bf E}_{1}\cdot {\bf E}_{2}+\Omega^{(1)}{\bf E}_{1}\times {\bf E}_{2}\cdot \textbf{F},$$ where $\Omega^{(i)}$ are the corresponding interaction strengths. Since ${\bf E}_{1}$ and ${\bf E}_{2}$ are orthogonal to each other, first term disappears (henceforth we drop the superscript in $\Omega^{(1)}$). In the second term, ${\bf E}_{1}\times {\bf E}_{2}$ appears as a magnetic field to the atoms and couples to the total angular momentum ${\bf F}$ of the atom (nucleus’ total moment + outermost electron’s spin momentum) thereby producing SOC. It is observed that the $|m_F=+1\rangle$ state in $\rm{Rb}^{87}$ atom lies at a much higher energy than the other two multiplets[@lin] and thus can be neglected. Following adiabatic elimination method (see supplementary materials), we can remove the $|m_F=+1\rangle$ state and obtain an effective 2-level model involving $m_{F}=0$ and $m_{F}=-1$ pseudospin states, defined as the $|\uparrow \rangle$ and $|\downarrow \rangle$ basis, as:[@lin] $$\label{eq: 2}
H_{\rm A}=\xi_k\ \mathbb{I}_2 +\frac{\Omega}{2}\sigma_z + \alpha_R k_x\sigma_y.$$ Here, the non-interacting dispersion for each SO wire is $\xi_{k}$. $\sigma_i$ are the usual $2\times 2$ Pauli matrices and $\mathbb{I}$ is the identity matrix. $\alpha_R=\hbar^2k_{\rm L}/m^*$ is the SOC strength, which is tunable both externally (by laser wavelength) and internally (by atom’s effective mass $m^*$).
The SOC is reversed in the adjacent ‘B’ SO wire by reversing the ${\bf E}_{1}$ laser, while keeping ${\bf E}_{2}$ laser the same,[@footnote_setup] as shown Fig. \[fig1\](a). This reverses the SOC as well as the intrinsic Zeeman like term to $-\Omega$ in the ‘B’ wire, resulting in the corresponding Hamiltonian as: $H_{\rm B}({\bf k},\Omega)=H_{\rm A}(-{\bf k},-\Omega)$ .
We find that for topological reasons, the $k$-dependent tunneling matrix element between ‘A’ and ‘B’ wires should carry a phase[@footnote_complexhopping]. Given the flexibility of the optical lattice, this can be achieved in multiple ways. A simple possible method would be to allow staggered hoppings between the upper- and lower-nearest neighbor wires, as in the case of the Su-Schrieffer-Heeger (SSH) lattice.[@ssh1] This is modeled by different hopping parameters from ‘B’ to the top ($t$) and bottom ($t^{\prime}$) ‘A’ wires as shown in Fig. \[fig1\](b). In the corresponding $k$-space, this leads to the net inter-wire hopping: $T(k)=-te^{ik_yb}-t^{\prime}e^{-ik_yb}$, where $b$ is the inter-wire distance.
For analytical solutions of the bulk energy states and the $Z_2$ invariant, we expand the Hamiltonian in the basis of Dirac matrices. The calculation becomes simpler if we set $t^{\prime}<<t$, and without any loss of generality, we set $t^{\prime}=0$. This does not change the overall band topology and the $Z_2$ invariant as subsequently confirmed with numerical simulation by inserting back the finite $t^{\prime}$ term. For the lattice generalization, we assume a nearest neighbor hopping for both dispersions and SOC which yield $\xi_k= -\gamma_1\cos{(k_xa)}-\gamma_2\cos{(k_yb)}-\mu$, and $\alpha_{k}=-i\alpha_R \sin{(k_xa)}$, where $\gamma_{1,2}$ are the usual tight-binding parameters between same spin species in the $x$ and $y$ directions, respectively, $\mu$ is the chemical potential, and $a$ and $b$ are the corresponding inter-atomic distances. Therefore, we can express the total Hamiltonian in a four-component spinor defined as $(|{\rm A}_\uparrow \rangle$, $|{\rm A}_\downarrow \rangle$, $|{\rm B}_\uparrow\rangle$, $|{\rm B}_\downarrow\rangle)$ (where ‘A’ and ‘B’ stand for atoms on ‘A’ and ‘B’ SO wires, respectively): $$\begin{aligned}
H(k)=\left(
\begin{array}{ c c c c }
\xi_k+\frac{\Omega}{2} & \alpha_k & -te^{ik_yb} & 0\\
-\alpha_k & \xi_k-\frac{\Omega} {2} & 0 & -te^{ik_yb} \\
-te^{-ik_yb} & 0 & \xi_k-\frac{\Omega}{2} & -\alpha_k\\
0 & -te^{-ik_yb} & \alpha_k &\xi_k+\frac{\Omega}{2}
\end{array} \right).
\label{Hamp}\end{aligned}$$ The corresponding eigenvalues are $E_{\pm}(k)=\xi_k\pm \frac{1}{2}\sqrt{4|\alpha_k|^2 + \Omega^2 + 4 t^{2}}$. The TR operator for the above spinor can be defined as $\tau=-i \mathbb{I}\otimes\sigma_y\mathbb{K}$[^4], where $\mathbb{I}$ is a 2$\times$2 identity matrix, and $\mathbb{K}$ is the complex conjugate. The system is TR invariant as $\tau H^*(\textbf{k}, \Omega) \tau^{-1}\ =\ H(-\textbf{k},-\Omega)$, where $\Omega\rightarrow -\Omega$ under TR operation since it represents spin splitting. Therefore, the full Hamiltonian is TR invariant despite it is broken in each $H_{\rm A,B}$ block. We have subsequently calculated the spin expectation values and shown that the total magnetic moment always vanishes in our setup, further supporting the TR invariance of the Hamiltonian (see Supplementary Material).
The TR invariance makes each band doubly degenerate, while the $\Gamma$-point is four-fold degenerate (two spins and two valleys) in the absence of $\Omega$ and $t$; see Fig. \[fig2\] and also the figure the supplementary material. Therefore, without breaking the TR symmetry, the $\Gamma$ point can be gapped out with a finite value of $\Omega$ \[see Fig. \[fig2\](b)\]. The valence band is gradually inverted at all $k$-points as SOC $\alpha_R$, and inter-wire hopping $t$ are increased above their corresponding kinetic energies (i.e. $\gamma_{1,2}$), see Fig. \[fig2\](c-d). As $t$ becomes comparable to the intra-wire kinetic energy, inter-wire hopping becomes more favorable. As a spin-up atom hops from one wire to the next one, its propagation direction becomes reversed, due to opposite SOC. Finally, by hopping back to the first wire, it forms a ‘chiral orbit’ (opposite chirality for the spin-down atom) in the bulk, with an associated $Z_2$ topological invariant.
Unlike the typical Hamiltonians of QSH insulators[@bhz; @kane], where two $2\times 2$ blocks for different spins are decoupled, in our case they are coupled by the SOC, mixing the spin states. Therefore, a simple Chern number for each spin cannot be deduced. Here the topological invariant can be calculated from the TR operator[@kane; @fu; @soluyanov], or from the Wilson loop.[@yu] We discuss the former procedure here. According to the Kane-Mele formalism,[@kane] the $Z_2$ invariance can be calculated by counting the number of pairs of zeros in the Pfaffian of the overlap matrix defined as: $P({\bf k})={\rm Pf}[\langle u_{i}({\bf k})|\tau|u_{j}({\bf k})\rangle]$, and $|u_i({\bf k})\rangle$ is the Bloch state for the $i^{\rm th}$-filled band. For the Hamiltonian in Eq. \[Hamp\], the valence band is twofold degenerate, so the Pfaffian is just the $i\ne j$ component of the overlap matrix. The exact evaluation of $P({\bf k})$ comes out to be $$P(k) = 2\ (1 + e^{-2 i k_yb}).
\label{Eq:Pk}$$ It is interesting to note that $P(k)$ only depends on the phase associated with the inter-wire hopping, and is parameter free. This justifies the inclusion of staggered inter-wire hopping allowing the survival of this complex momentum dependent phase. The loci of the nodes in $|P(k)|$ is $k_y^*=\pm\frac{\pi}{b}(n + \frac{1}{2})$ ($n$ is integer), for any value of $k_x$, except at the TR invariant point. Each node at $+k_y^*$ renders a positive winding number, also referred by ‘vorticity’ or ‘chiral orbit’, while that at $-k_y^*$ yields a negative winding number since $\pm k_y^*$ are related by TR invariance. The $Z_2$ invariance can therefore be evaluated by the winding number of $P(k)$ as [@kane; @fu]: $$\nu =\ \frac{1}{2 \pi i} \oint_{C_{1/2}} d \textbf{k}. \nabla_{\textbf{k}}\big( \log{[P(\textbf{k})+ i \delta]} \big)\ \rm{mod}\ 2,
\label{Eq:TI}$$ where $C_{1/2} $ denotes that the integral is over half of the BZ, $ k \in [0, \frac{\pi}{b}]$, enclosing either $k_y^*$ or $-k_y^*$ point. As the contour $C_{1/2}$ encloses a single Pfaffian node, the integral gives $\nu=1$, indicating the presence of non-trivial bulk topology with odd pair of edge states. To establish the robustness of the topological invariant of this setup, we have also evaluated the $Z_2$ invariant $\nu$ by inserting back the $t^{\prime}$ term in the Hamiltonian. We find that the inclusion of the $t^{\prime}$ term keeps the Pfaffian unchanged and we still obtain $\nu =1$ as long as $t^{\prime} \neq t$. We emphasize that the non-trivial $Z_2$ invariance is ensured by the geometry of our setup in which the adjacent ‘A’ and ‘B’ wires serve as TR conjugate to each other which enables odd number of TR partner inversion in half of the BZ, and therefore each half encloses a single node of $P(k)$.
The bulk-boundary correspondence of the TI dictates the presence of the conducting edge states, connecting the bulk conduction and valence bands.[@TIreviewTD; @TIreviewCK; @TIreviewSCZ] The properties of the edge states for the present setup are studied both analytically and numerically using the bulk Hamiltonian from Eq. (\[Hamp\]). For the edge states, the crystal symmetries play important roles. Terms involving $k_x$ and $k_y$ variables are decoupled into different off-diagonal terms in the Hamiltonian (Eq. \[Hamp\]) with different coupling constants, leading to a rotational ($C_4$) symmetry breaking. This makes the two edge states behave differently. As a byproduct, the system possess Mirror symmetry in the $x$-direction, which restricts that the eigenvalues should be even in $k_x$. Therefore, the leading term in the corresponding edge state becomes quadratic in momentum, rather than a quintessential linear dispersion. However, it remains helical owing to the SOC.
The edge parallel to the $x$-axis is made of just a single SO wire lying at $y=0$, coupled to the non-trivial bulk for $y>0$ and the vacuum at $y<0$. To make the analytical calculation manageable, we solve the bulk Hamiltonian in Eq. with periodic boundary condition along the $x$-direction, but open boundary condition along the $y$-direction with two SO wires. We solve the Schrödinger equation in the continuum limit ($\alpha_k\rightarrow i\alpha_R k_x$), obtain the helical low-energy edge states (up to the quartic term) as $$E_y (k_x)\ =\ \pm \bigg(t + \frac{\alpha_R^2}{2 t}\ k_x^2 -\frac{\alpha_R^4 }{8 t^3}\ k_x^4\bigg).
\label{edge}$$ We note that the two edge states are ‘apparently’ gapped by $t$, and the gap vanishes at some finite value of the momentum for a given parameter set ($k_x \approx \pm 0.015$ for the present case, where $t=0.1$ and $\alpha_R=15$). This is due to the finite size effect of the geometry. As the number of wire is increased, the gap decreases gradually and eventually vanishes at $k_x=0$. This result is confirmed by numerical simulation of a system with 50 pairs of SO wires, and the corresponding results are shown in Fig. \[fig3\]. The Fermi velocity of the edge state is proportional to the tunable parameters SOC strength $\alpha_R$, and the hopping amplitude $t$, see Eq. . We have also estimated the decay length scale of the edge state into the bulk, and find that it is directly proportional to $\Omega$, and thus can be monitored externally (see Supplementary Materials, section C. for further details).
For our choice of the staggered hopping along the $y$-direction, the corresponding edge state properties follow that of the SSH model. In the limit of $t^{\prime}\rightarrow 0$, the edge consists of a dimer of two atoms coming from the ‘A’ and ‘B’ wires. The dimer remains decoupled from the bulk, and therefore the edge state is gapped. As $t^{\prime}$ is increased, the corresponding gap decreases and gapless end states arise in the limit of $t^{\prime}>>t$ and $\alpha_R\rightarrow 0$. The present setup, however, makes the end states helical as SOC is turned on. In summary, for our proposed setup, conducting and helical edge states arise along the edges parallel to the direction of SOC wires, while the dispersionless end modes along the perpendicular direction can be tuned from gapped to gapless with SOC and inter-wire hopping. Therefore, these edge states promise multifunctional applications: while the conducting edge states can be used for transport related applications, the other edge can be used for optical switch and transistor related applications owing to the tunable bang gap.
In Fig. \[fig5\] we provide the relevant setup to detect the chiral edge states for the present case.[@Roth] We attach a current source and a ground to two different edge atoms. The voltage drop occurring between them is the expected local voltage ($V_l$). An additional current flow is expected to arise in the other direction (reaching to the ground from the other side of the edge), dictated by left moving arrow in Fig. \[fig5\]. This occurs due to the chiral property of the edge state in $Z_2$ TI. Measuring the corresponding non-local voltage drop ($V_{nl}$) with opposite sign gives us a concrete prove to the emergence of the TI in the bulk. The edge current is expected not to dissipate, in principle, with distance from the source, due to topological protection. This can be checked by measuring voltage $V_{nl}$ farther from the source atoms. The absence of bulk conductivity can be easily verified here by measuring the voltage drop in the inner atoms.
In condensed matter systems, quantum SO wires are routinely grown in Bi-, Pb- and related elements based atomic wires with tunable SOC.[@QW1; @QW2] Also as mentioned earlier, for the cases where the bulk (Dresselhaus) and surface (Rashba) SOC in non-centrosymmetric materials cancel each other, a similar 1D SOC arises. For such systems, the generation of tunable TI follows similarly. Generalization to a 3D TI can be obtained easily. We can seek to build a ‘strong’ 3D TI starting with the above-obtained 2D setup. For the second layer, we need to place another 2D layer, with ‘B’ wire sitting on top of the ‘A’ wire and vice versa. That means for each ‘A’ wire the surrounding nearest-neighbor lattice wire along all directions should be the ‘B’ wire with opposite SOC. Such a setup will lead to a TI with odd number of band inversion (at the $\Gamma$-point only) in the entire BZ, a criterion for the strong TI.[@TIreviewTD; @NCTD] Given that the engineered TIs in optical lattice harness remarkable flexibility and tunability, both bulk and boundary states can be easily manipulated. This can provide a versatile substitute for the topological solid state materials, seeking to overcome of materials challenges embedded in the present materials selections.
[**Acknowledgments**]{} We thank Aveek Bid for suggesting the experimental setup. We thank Srinivas Raghu, Benjamin Lev, Monika Schleier- Smith, Diptiman Sen and Jainendra Jain for useful discussions. SR would also like to thank CSIR for financial support during the work. The work is facilitated by the computer cluster facility at the Department of Physics at the Indian Institute of Science.
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Two SO wires are not required to be placed on the same plane. If they are placed on different planes (along the $z$-direction) to facilitate preparations, the inter-wire hopping in Hamiltonian (Eq. \[Hamp\]) would still remain the same as long as the distance ($b$) along the $y$ direction is kept to be the same.
In the real-space, the hoppings from ‘B’ to the top and the bottom ‘A’ wires are taken to be different ($t\ne t^{\prime}$) without any complex phase associated with it. This staggered hopping naturally leads to a complex Bloch in the momentum space, according to the SSH model,[@ssh1] and serves our purpose. This can be achieved by tuning the distance between the two adjacent wires to be slightly different, as illustrated in Fig. \[fig1\](b).
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Supplemental Material {#supplemental-material .unnumbered}
=====================
Setup and Hamiltonian
---------------------
The model setup for the generation of 1D SOC in optical lattice is given in Ref. [@lin] and the corresponding effective Hamiltonian is obtained in the same paper. Here we deduce the full Hamiltonian with the microscopic details, and show that the generation of their bilayer extension for our purpose is viable. The basic setup consists of a periodic array of ultracold bosons designed in optical lattice. We consider that the ultracold atoms are in the ground state with total angular momentum $F=1$, with three states $m_F$= +1, 0 and -1, which as denoted by $|+1\rangle$, $|0\rangle$, and $|-1\rangle$. $^{87}$Rb atom is such a bosonic atom with $F=1$ ground state which was utilized for the SOC generation[@lin]. We start with a three component spinor for the three states, and derive the Hamiltonian in the rank$-3$ Pauli matrices are labeled as $\sigma_{3,i}$: $$\begin{pmatrix}
0&1&0\\
1&0&1\\
0&1&0\\
\end{pmatrix},
\begin{pmatrix}
0& - i &0\\
i &0& - i\\
0& i &0\\
\end{pmatrix},
\begin{pmatrix}
1&0&0\\
0&0&0\\
0&0&-1\\
\end{pmatrix},$$ for $i=x,y$, and $z$, respectively. Finally, we will find in the following sections that the $|+1\rangle$ is pushed significantly up in energy compared to the other two states, allowing very weak hybridization with the other states. Therefore, we can write the Hamiltonian as an effective 2-level system.
An 1D SOC is created (say along $x$-direction) when two Raman lasers are aligned orthogonal to each other, along $\hat x\pm\hat{y}$ directions, and are detuned from each other by $\Delta\omega_{\rm L}$, respectively. In this case, the light-matter interaction basically has three terms: Interaction between the electric fields of the lasers and angular momentum of the Rb-atom, and two Zeeman-like (paramagnetic and diamagnetic) terms arising from the bias magnetic field coupled to the spin. We denote them as $H_{\rm R}$, $H_{\rm P}$, and $H_{\rm D}$. Considering also a generalized kinetic energy term of the bosons in a rectangular optical lattice as discussed in the main text, denoted by $H_{\rm K}$, we obtain the full Hamiltonian as $$H= H_{\rm K} + H_{\rm P} + H_{\rm D}+ H_{\rm R}.
\label{Ham}$$ The last term is crucial for the SOC. The total angular momentum ${\bf F}$ has contributions from both the nucleus and the last orbital electron. If the angular momentum is $\textbf{I}$ and the nuclear magnetic moment is $\textbf{J}$, then the total angular momentum is $\textbf{F=I+J}$. The atom-light shift interaction with two monochromatic light fields (${\bf E}_{1,2}$) detuned off resonance, have a generic Hamiltonian, $$H_{\rm R }=\alpha_{ij}E_{1 i} E_{2 j} \,.$$ Here $\alpha_{ij}$ denotes the rank of the interaction, i.e, $\alpha_{ij}=\delta_{ij}$ for scalar (rank 0) and $\alpha_{ij}=\epsilon_{ijl}F_l$ for vector (rank 1) component and so on. Restricting the Hamiltonian upto the rank-1 (vector) light shift interaction with the spin states of the atom, we get H\_[R]{} =\^[(0)]{}[**E**]{}\_[1]{}\_[2]{}+ \^[(1)]{}[**E**]{}\_[1]{}\_[2]{}**F**. \[HR1\] The strength of each interaction is denoted by $\Omega^{(i)}$. Since the two lasers are aligned orthogonal to each other and intersect at the origin, the first term in Eq. (\[HR1\]) vanishes. Henceforth, we drop the superscript ‘(1)’ from $\Omega^{(1)}$. The rank-1 light interaction acts as an anisotropic magnetic field which couples to the atom’s total angular momentum. We assume that both the laser wavelengths are denoted by $k_{\rm L}$, with frequencies $\omega_{\rm L}$, and $\omega_{\rm L}+\D\omega_{\rm L}$. We set $\omega_{\rm L}=0$ (which eventually drops out otherwise), we get $$\begin{aligned}
H_{\rm R}=&\Omega [\sin (k_L x)(\hat{x}-\hat{y})\times\sin (k_L x+\D\omega_L t)(\hat{x}+\hat{y})]\cdot \textbf{F}\nonumber\\
=& 2\Omega [\cos(2k_L x+\D\w_L t)-\cos (\D\w_L t)]\s_{3,z}\,.\end{aligned}$$ Furthermore, since the second term in the above equation gives only a constant energy shift to the ground state energy, we can also drop this term, yielding $H_{\rm R}=\Omega \cos[2k_L x+\D\w_L t] \s_{3,z}$.
The remaining terms in the Hamiltonian are usual. In the continuum limit, we take anisotropic effective mass for the intra-wire and inter-wire hoppings, which gives H\_[K]{}=\_[k]{}\_3= \_3, where $\mathbb{I}_3$ is the $3\times3$ identity matrix. $\xi_k$ is the non-interacting dispersion term for each spin states. $E_L$ is the energy shift due to the lasers. $\vec{B}\cdot \vec{\mu}$, is the paramagnetic term coming from the externally applied bias magnetic field $\vec{B}$. We choose the bias magnetic field along the $y$-direction which gives $H_{\rm P}=\delta\s_{3,y}$, where $\d=-\m_B B_y$. The diamagnetic term $H_{D}$ gives a quadratic Zeeman effect, denoted by $ \hbar \omega_q$, shifting the the $|+1\rangle$ state further from the $|0 \rangle$ and $|-1\rangle$.
With all the terms discussed above, we can now write down the 3-level Hamiltonian in the basis $|+1\rangle,|0\rangle$ and $|-1\rangle$ as follows,
H=&\_[k]{}\_3+
\_q&0&0\
0&0&0\
0&0&0\
\
&+\_[3,y]{} +\_[3,z]{}.
Since we choose the $\hat{y}$ axis as the natural quantization axis, we can perform a global rotation as $\s_{3,y}\rightarrow \s_{3,z},\s_{3,x}\rightarrow \s_{3,y}$ and $\s_{3,z}\rightarrow \s_{3,x}$. Then we add a constant term $ \frac{\d}{2}\ \mathbb{I}_3$ which shifts each of the levels by $\frac{\d}{2}$. Thereby second and third terms can now be combined to obtain\
\[Hm1\]
H=&\_[k]{}\_3+
+\_[q]{}&0&0\
0&&0\
0&0&-\
\
& +\_[3,x]{}.
$|+1\rangle$ state is thus separated by a relative large energy scale $\hbar(\delta+\w_q)$ from $|0\rangle$ and $|-1\rangle$ states. Next we perform another a rotation about the $\hat{z}$ axis to go to the co-moving frame rotating with frequency $\D\w_{\rm L}$. This helps eliminate the $\w_{\rm L}$ term from the Hamiltonian without changing anything else in it. This is done by using the so-called Rotating Wave Approximation (RWA).[@fujii; @sanchez] The RWA procedure is analogous to going from the Schrodinger picture (where the states are time evolving) to the Heisenberg picture (where the operators are time evolving and the states are not). In this case we want to go to a frame where this is static. A familiar choice of the transformation matrix is given by, U=. \[Op\] The Hamiltonian thus transform to $H'=U H U^\dagger\,$. The identity matrix remains invariant under this rotation. Therefore, the first and the second term, which can be decomposed into terms contained $\mathbb{I}$ and $\s_{3,z}$), also remain the unchanged under this $U$ transformation. $U$ gives a non-trivial effect for the SO term which can be seen as follows: $$\begin{aligned}
&&U \cos[2k_{\rm L} x+\D\w_{\rm L} t]\s_{3,x} U^\dagger \nonumber \\
&&\qquad\qquad\qquad=\cos(2k_{\rm L} x)U\cos(\D\w_{\rm L} t)\s_{3,x}U^\dagger\nonumber\\
&&\qquad\qquad\qquad~+\sin(2k_{\rm L} x)U\sin(\D\w_{\rm L} t)\s_{3,x}U^\dagger \,,\nonumber\\\end{aligned}$$ Let us first write down the $\s_{3,i}$ operators in a convenient fashion as, $\s_{3,\pm}=(\s_{3,x}\pm\s_{3,y})/2$ and further $e^{i\D\w_L t}=\cos \D\w_L t+i \sin \D\w_L t$. Thus we can write $$\begin{aligned}
\cos \D\w_L t \s_{3,x}&=\frac{1}{2}(e^{i\D\w_L t}+e^{-i\D\w_L t})(\s_{3,+}+\s_{3,-})\nonumber\\
&=\frac{1}{2}(e^{i\D\w_L t}\s_{3,+}+e^{-i\D\w_L t}\s_{3,-}\nonumber\\
& ~~~~+e^{i\D\w_L t}\s_{3,-}+e^{-i\D\w_L t}\s_{3,+})\,.\end{aligned}$$ Using the facts that, $U \s_{3,\pm}U^\dagger=e^{\mp i\D\w_L t}\s_{3,\pm}\,$ we can show that the first term becomes
U (\_[L]{} t)\_[3,x]{}U\^=&(\_[3,+]{}+\_[3,-]{}+e\^[2i\_[L]{} t]{}\_[3,+]{}\
&+e\^[-2i\_[L]{} t]{}\_[3,-]{}).
Neglecting the terms proportional to $e^{\pm i\D\w_L t}$, since they represent rapidly oscillating terms, we get, U (\_[L]{} t)\_[3,x]{}U\^=\_[3,x]{}. Similarly for the other term we get, U (\_[L]{} t) \_[3,x]{}U\^= -\_[3,y]{}. We thus get the resultant Hamiltonian in the rotated frame as, \[3lh\]
H=&\_[k]{}\_3+
+\_q&0&0\
0&&0\
0&0&-.
\
&+.
[**Effective two levels Hamiltonian.**]{} As discussed before, the $|1\rangle$ is shifted to much higher in energy compared to other two states due to the diamagnetic term. Therefore, we can neglect this terms and obtain an effective $2\times 2$ Hamiltonian described by usual Pauli matrices $\sigma_i$ \[The full derivation of $2\times2$ Hamiltonian using Adiabatic Elimination method is given in Sec. \[Sec:AEM\]\]. The corresponding Hamiltonian takes the form \[h2\] H=\_[k]{}\_2+\_z +.\
Now apply a pseudo spin rotation about the $\hat{z}$ axis with a phase of $\theta(x)=2k_{\rm L} x$, which gives $U_x=\exp[i k_{\rm L} x \s_z]\,$. As done previously, we find that, U\_xU\^=\_x. There is, however, an important different in this rotation compared to the rotation for operator in Eq. (\[Op\]). Here $U_x$ does not commute with the kinetic energy term $H_{\rm K}=\xi_k\mathbb{I}$. This is because the leading term in $\xi_k$ is $k_x^2$, which gives rise to an non trivial term linear in $k_x$ in the SOC Hamiltonian under $U_x$ rotation. This can be seen explicitly as U\_[k]{}U\^= E\_[L]{} + Uk\_x\^2U\^+k\_y\^2\
,. Terms containing $k_y$ do not change under $U_x$ rotation. We take an infinitesimal representation of the operator $U$ as, U=1+i\_z. Thus with $\theta=2k_L \hat{x}$ we have, $$\begin{aligned}
\label{kxtrans}
\frac{\hbar^2}{2m_1}e^{i\theta/2 \s_z}k_x^2e^{-i \theta/2 \s_z}&=\frac{\hbar^2}{2m_1}(1+i\frac{\theta}{2}\s_z)k_x^2(1-i\frac{\theta}{2}\s_z)\nonumber\\
&=\frac{\hbar^2k_x^2}{2m_1}+\frac{i\hbar^2}{4m_1}[\theta, k_x^2]\s_z\nonumber\\
&=\frac{\hbar^2k_x^2}{2m_1}+\frac{\hbar^2k_Lk_x}{m_1}\s_z\,.\end{aligned}$$ Thus the total 2-level Hamiltonian after the rotation becomes, H=\_[k]{}\_2+\_z+\_x+\_z. Finally, we employ a global rotation $\s_z\rightarrow\s_y$, $\s_y\rightarrow\s_x$ and $\s_x\rightarrow\s_z$, to get the final form of the Hamiltonian as, \[2lh\] H=\_[k]{}\_2+\_z+\_y+\_y.
In the final Hamiltonian, we can easily recognize that the last term gives a 1D SOC. We denote the SOC coupling strength as $\alpha_R=2\hbar^2k_{\rm L}/2m_1$. It is interesting to see that the SOC strength is inversely proportional to the band mass, which is opposite to the case for a simple SOC in solid state systems. Therefore, SOC can be tuned here by the effective mass of the electron propagating along the SOC wire, as well as by the wavevector of the incident laser. This constitute the Hamiltonian for the ‘A’ wire in the main text.
Recalling that $\delta$ arises from the external bias, and its value decreases with increasing laser frequency in the actual experiment[@lin], without losing generality, we can set it to be zero. The other term $\Omega$ is proportional to the direction of intrinsic magnetic field of the laser. For our bilayer setup, we set out to obtain an effective Hamiltonian which remains time-reversal invariant. This can be obtained by rotating one of the lasers into the opposite direction in the adjacent wire \[see Fig. 1 in the main text\]. This has two effects. It creates the SOC in the reverse direction ($-k_x$), as well as change the sign of $\Omega\rightarrow -\Omega$. The explicit form of the Hamiltonian for the ‘B’ wire is then \[2lh\] H\_[B]{}=\_[k]{}\_2-\_z-\_R k\_x \_y. Thereby we restore the time-reversal symmetry in the total Hamiltonian. We note that due to the presence of the quantum tunneling between the two layers, which is of the form $te^{ik_yb}$, we see that the block Hamiltonians cannot be separated, as was done for the quantum Spin Hamiltonian for the HgTe/CdTe quantum wells in Ref.[@bhz]. Therefore, we cannot compute the Chern number for each block. Calculation of the proper $Z_{2}$ invariant from the expectation value of the time-reversal operator,[@TIreviewCK; @TIreviewSCZ; @TIreviewTD; @kane] is thus necessary here.\
In the real-space, the hoppings from ‘B’ to the top and the bottom ‘A’ wires are taken to be different ($t\ne t^{\prime}$) without any complex phase associated with it. This staggered hopping naturally leads to a complex Bloch in the momentum space, according to the SSH model (Ref. 23), and serves our purpose. This can be achieved by tuning the distance between the two adjacent wires to be slightly different, as illustrated in Fig.1.(b).\
Two SO wires are not required to be placed on the same plane. If they are placed on different planes (along the $z$-direction) to facilitate preparations, the inter-wire hopping in Hamiltonian (Eq. 3) would still remain the same as long as the distance ($b$) along the $y$ direction is kept to be the same.
[**Generalization to Lattice Model.**]{} The above analysis can be generalized to a lattice model in which we replace $k_x\rightarrow \sin{(k_xa)}/a$, and $k_x^2\rightarrow 2(\cos{(k_xa)}-1)/a^2$, where $a$ is the lattice constant. If we assume the nearest neighbor hopping amplitudes as $\gamma_{1,2}$ along the $x$-, and $y$-directions, respectively, the non-interacting dispersion in 2D becomes $xi_{\vec{k}} = \hbar^2k_x^2/2m_1^*+\hbar^2k_y^2/2m_2^* \rightarrow \gamma_1\cos{k_x} +\gamma_2 \cos{k_y} -\mu$. Here $\gamma_{1}=\hbar^2/m_1^*a^2$, $\gamma_{2}=\hbar^2/m_2^*b^2$, and the chemical potential is $\mu=\hbar^2(1/m_2^*a^2+1/m_2^*b^2)$.
The SOC term arises by the same formalism from Eq. (\[kxtrans\]) in the following way: $$\begin{aligned}
\label{SOClattice}
& \bigg[ e^{i\theta/2 \s_z}(\xi_k)e^{-i \theta/2 \s_z} \bigg]\nonumber\\
& \approx (1+i\frac{\theta}{2}\s_z)(\gamma_1\cos{k_x}+\gamma_2 \cos{k_y}-\mu)(1-i\frac{\theta}{2}\s_z)\nonumber\\
&= \bigg[ (\xi_k)+ \frac{1}{4}[\theta, (\xi_k)]\s_z \bigg]\end{aligned}$$ By expanding $\cos$ term, we perform the commutation operation with $\theta$ with each power of $k_x$ and $k_y$. The commutator of $\theta$ with $k_y$ will naturally give zero. The commutator with $k_x$ will give rise to a $\sin{k_x}$ term, thus an effective 1D SOC, as follows: $$\begin{aligned}
& \bigg[ e^{i\theta/2 \s_z}(\gamma_1\cos{k_x} +\gamma_2 \cos{k_y} -\mu)e^{-i \theta/2 \s_z} \bigg] \nonumber \\
& = \xi_{\vec{k}}- (2\gamma_1 k_L) \sin{k_x} \s_z \nonumber \\
& = \xi_{\vec{k}}- \alpha_ R \sin{k_x} \s_z\,,\end{aligned}$$ where $\alpha_R=2\gamma_1k_L$.
Adiabatic Elimination method {#Sec:AEM}
----------------------------
Here we elaborate the derivation of the $2\times 2$ Hamiltonian in Eq. (\[h2\]) from the $3\times3$ Hamiltonian in Eq. (\[3lh\]). Here we use the Adiabatic Elimination method to eliminate the $|+1\rangle$ state. [@brion] This method primarily depends on the fact that one of the states (say $|+1\ra$) is so high in the energy compared to the states ($|0\ra$ and $|-1\ra$), eliminating the $ |+1 \rangle $ state will not alter the band structure . In other words, the time evolution of the excited state is not affected by the ground states. We begin by considering the Hamiltonian in given in the matrix form as, H=
\_k++\_q&e\^[2i k\_[L]{} x]{}&0\
e\^[-2i k\_[L]{} x]{}& \_k+&e\^[2i k\_[L]{} x]{}\
0&e\^[-2i k\_[L]{} x]{}&\_k-\
. Here we have neglected $E_{\rm L}$, the total energy due to the Raman lasers since this only gives a constant energy shift. We take a spinor for the three states $|\pm1\ra$ and $|0\ra$ as (t)=
(t)\
(t)\
(t)\
. where $\a(t)\equiv|+1\ra$, $\b(t)\equiv|0\ra$, and $\g(t)\equiv|-1\ra$, respectively. Substituting $\phi(t)$ in the Schr'’odinger equation $\pd_t\psi(t)=H\psi(t)\,$, gives a set of three coupled differential equations for $\a(t)$, $\b(t)$ and $\g(t)$. We are interested here to study only the effect on $\a(t)$ due to the other states. Thus, (t)=(\_k++\_q)(t)+e\^[2i k\_[L]{} x]{}(t). Setting $\dot{\a}(t)=0$ gives, (t)=(t). Putting this in the remaining set of the coupled differential equations, we immediately see that the spinor containing the two low lying states $\bar{\psi}(t)=(\b(t),\g(t))$ satisfy, \_t|(t)=H\_[2,eff]{}|(t), where, H\_[2,eff]{}=
\_k++&e\^[2i k\_[L]{} x]{}\
e\^[-2i k\_[L]{} x]{}&\_k-\
$H_{\rm 2,eff}$ can be put effectively in the basis as H\_[2,eff]{}=\_k\_2+(2k\_[L]{} x)\_x-(2k\_[L]{} x)\_y+\_z, where, =(-). Expanding $\r$ around $\w_q>\Omega\gg \d$ up to the first sub leading order we get, =-. Thus we can see that the coupling $\d$ is effectively modified by the additional term, \^[(2)]{}=-. Further expanding $\r$ upto one more order, we find that, \_1=+\^2. Note that the effective two level Hamiltonian now takes the form,
H\_[2,eff]{}=\_k(1+)\_2 & +(2k\_[L]{} x)\_x\
&-(2k\_[L]{} x)\_y+\_z,
for $k^2\gg \d$. Finally employing the RWA as before, we obtain
H’\_[2,eff]{} &\_k(1+)\_2 +(2k\_[L]{} x)\_x\
& -(2k\_[L]{} x)\_y+\_z+(+\^[(2)]{})\_z,
where $\a^{(2)}=\frac{\a}{8\w_q^2}\Omega^2\,$, and we can consider $\frac{\Omega^2}{8\w_q^2}\ll1$. Hence we obtain Eq. (\[h2\]).
Calculation of edge states
--------------------------
The edge state calculation follows the same procedure as used earlier,[@TIreviewCK; @TIreviewSCZ; @TIreviewTD; @shen] but unlike these models, where the two blocks of the Hamiltonian are decoupled, here they are coupled by either SOC or the inter-wire tunneling. Therefore, the edge state calculation requires special treatment. In our Hamiltonian, the coupling along the $x$- and $y$-directions are different, and the system does not possess the rotational $C_4$ symmetry. Therefore, both edge states have different characteristics, which can be evaluated from the bulk Hamiltonian, owing to the bulk-boundary correspondence of the topological insulator. Here we discuss the analytical results in the low-energy limits of the edge states. To make the problem manageable with exact diagonalization procedure, we consider the bulk Hamiltonian for two SO wires given in the main text, but relax the periodic boundary condition for the edge under consideration, while keep the periodic boundary condition in the perpendicular direction..
### Edge parallel to x-axis
We first consider the edge parallel to the $x$-axis, or parallel to the SO wire. The edge of this setup is a decoupled SOC wire, lying at, say $y=0$ position. Therefore, the edge state is made of two counter-propagating chiral spin states along the $x$-direction, while decaying exponentially along the $y$-direction. An important symmetry to recognize here is that the Hamiltonian has a Mirror symmetry along the $k_x$-direction, in addition to the time-reversal symmetry. Therefore, the eigenvalues obey the condition that $E_{k_x}=E_{-k_x}$, which restricts the lowest order term to be quadratic in $k_x$, which is indeed the full calculation suggests.
For this edge the $k_x$ remains a good quantum number of the eigenstate, while the $k_y \rightarrow i \frac{\partial}{\partial y}$. Given the condition that the wavefunction must die as $y \rightarrow \infty$, we take the trial wavefunction as \_[k\_x]{}(y)=
c\_[1k\_x]{}\
c\_[2k\_x]{}\
c\_[3k\_x]{}\
c\_[4k\_x]{}\
e\^[- y]{},\
\[edge\_wf\] where $c_{ik_x}$ and $\lambda$ are to be evaluated. We find that the calculation is dramatically simplified if we introduce an anisotropic term to the Dirac mass term as $\Omega_{k}=\frac{\Omega}{2} + B(k_x^2+k_y^2)$, and the final result is obtained with $B\rightarrow 0$. Since this additional term contains quadratic momentum dependence, this does not change the bulk topology. Since $y=0$ axis is set to be the edge, we assume the system for $y>0$ is a non-trivial insulator (having positive Dirac mass, i.e., $\Omega_{k}>0$), while that for $y<0$ is a trivial insulator ($\Omega_{k}<0$). Using the expression for our $4 \times 4$ Hamiltonian, we solve the Schr'’odinger’s equation for $y>0$, and find the following coupled equations. (The kinetic term only gives the plane wave solution, it is dropped out in the calculation of the edge states) : $$\label{eq: 1}
\begin{split}
\bigg \{\frac{\Omega}{2} + B (k_x^2 - \lambda^2) - E \bigg\}c_1 -i \alpha_R k_x c_2 - t e^{-\lambda b}c_3\ &=\ 0\\
i \alpha_R k_x c_1 + \bigg\{-\frac{\Omega}{2} - B (k_x^2 - \lambda^2)- E\bigg\}c_2 - t e^{-\lambda b}c_4\ &=\ 0\\
-t e^{\lambda b}c_1 + \bigg\{-\frac{\Omega}{2} - B (k_x^2 - \lambda^2)- E\bigg\}c_3 + i \alpha_R k_x c_4\ &=\ 0 \\
-t e^{\lambda b}c_2 -i \alpha_R k_x c_3 + \bigg\{\frac{\Omega}{2} + B (k_x^2 - \lambda^2 - E)\bigg\}c_4\ &=\ 0, \\
\end{split}$$ where $E$ is the corresponding eigenvalue. For $y<0$ we do the substitution on the mass term (along with the quadratic term) $M(p) \rightarrow -M(p)$ and the trial solutions are : $\begin{pmatrix}
c_1 \\
c_2 \\
c_3 \\
c_4 \\
\end{pmatrix} e^{ \lambda y}$\
For $y<0$ the derivatives are with respect to $-y$.\
We solve the four coupled equations for both cases of $y<0$ and $y>0$ separately and then match the wavefunction at $y=0$, to get the following dispersion and decay length: $$\begin{split}
E_y\ &=\ \pm \bigg(t + \frac{\alpha_R^2}{2 t}\ k_x^2 -\frac{\alpha_R^4 }{8 t^3}\ k_x^4\bigg) \\
\text{lim}\ & B \rightarrow 0\,, \ \ \lambda \rightarrow \sqrt{\frac{\Omega}{2 B}}
\end{split}$$
Thus, the edge state decays very fast in the limit $ B \rightarrow 0$. However, the dispersion is not affected. This result matches exactly with the numerical results when we see the dispersion for two 1D channels, where the gap is of the order $t$. The apparent gap in the edge state by $t$ is an artifact arising due to finite size effect, which disappears as the number of SO wires is increased, gradually reducing the hybridization between the two edges. This result is confirmed by numerical calculation as shown in the main text.
### Edge parallel to y-axis
The edge state behaves differently for the edge perpendicular to the SOC wires, i.e., parallel to the $y$-direction. Along this edge, the boundary state is topologically protected but gapped. We have chosen the hybridization between the ‘A’ and ‘B’ wires lying along the $+y$-direction to be finite, and the set the hybridization ($t^{\prime}$) along $-y$ to be zero (the idea is to have these two hybridization to be different so that the imaginary term in the net hybridization survives). This is the origin of a gap in the edge state along this edge. As $t^{\prime}$ is turned on slowly we find that the gap disappears.
When the edge is parallel to the $y$-axis, the eigenvalue and eigenstates are functions of $k_y$. However, we find that for the energy eigenvalues to be real, the decay length $1/\lambda$ has to be imaginary. This gives [*only*]{} standing wave solutions along the $x$-direction, and the corresponding wavefunction of have the form: \_[k\_y]{}(x)=
c\_[1k\_y]{}\
c\_[2k\_y]{}\
c\_[3k\_y]{}\
c\_[4k\_y]{}\
(e\^[i x]{}-e\^[-i x]{}).\
Following the aforementioned procedure used for the other edge, we arrive at the energy values: E\_[x]{}&=&\_R ,\
&& . The apparent divergence of $E_x$ and $\lambda$ as $B\rightarrow0$ is an artifact of the analytical computation for small system size and converges in the numerical calculations when system size is made large. Hence, the gap has leading order in $\alpha_R$, which satisfies the numerical results in the limit $ B\rightarrow 0$.
Time reversal invariance
------------------------
Under TR symmetry, not only the momentum and spin are reserved, but the magnetic field is also flipped. For this reason, the Zeeman term in our Hamiltonian is also reversed. In what follows, the TR invariance of our Hamiltonian dictates $$\label{TRS}
\mathcal{U} \mathcal{H}^*(\vec{k}, \Omega) \mathcal{U}^{-1}\ =\ \mathcal{H}(-\vec{k},-\Omega)$$ where $\mathcal{U}$ is the unitary operator acting on the spin basis in the TR operator $\tau$, where $\tau = \mathcal{U} \mathcal{K}$, and $\mathcal{K}$ is the complex conjugation operator. For our case, the basis states of our Hamiltonian (Eq. 3 in the main text) are $\{A_{\uparrow},A_{\downarrow},B_{\uparrow},B_{\downarrow}\}$, and hence, the unitary operator is: $$\label{Umatrix}
\mathcal{U} = \begin{pmatrix}
0 & -1&0&0\\
1&0&0&0\\
0& 0& 0& -1\\
0& 0& 1& 0
\end{pmatrix}$$ Using Eq. (\[Umatrix\]) in Eq. (\[TRS\]), we conclude that the $4 \times 4$ Hamiltonian Eq. 3 (in the main text) is TR symmetric, and hence each of the eigenvalues are doubly degenerate even in the presence of $\Omega$ and $t$.
Calculation of spin density
---------------------------
We also calculate the spin density for our Hamiltonian in Eq. 3 (in the main text) to further ascertain the TR invariance of the system. The three spin operators are $S_{x,y,z} = \mathcal{I}_{2\times 2}\otimes \sigma_{x,y,z}$. We indeed find that the expectation values of the spin operators in all three directions $\langle S_{x,y,z}\rangle = 0$. We explicitly discuss the $\langle S_{z}\rangle$ case here. Eigenstates $|1 \rangle$, $|3 \rangle$ and $|2 \rangle$, $|4 \rangle$ have equal and opposite $S_z$ expectation value: $ \langle 1| S_z| 1\rangle = -\langle 4| S_z| 4\rangle$ and $ \langle 2| S_z| 2\rangle = -\langle 3| S_z| 3\rangle$, giving $ \langle 1| S_z| 1\rangle + \langle 3| S_z| 3\rangle = -4(1 +\frac{\Omega^2}{t^2})$ and, $ \langle 2| S_z| 2\rangle + \langle 4| S_z| 4\rangle = 4(1 + \frac{\Omega^2}{t^2})$ . Thus, the total $S_z$ expectation value of all the eigenstates of the $4 \times 4$ lattice Hamiltonian vanishes, thus showing that TR invariance is preserved for the $4 \times 4$ Hamiltonian of the set up shown in Fig. (1).
Band progression
----------------
In Fig. \[supp\_band\] we present how the band structure evolves as we include different terms in the Hamiltonian separately. As ultra-cold atoms are placed in a 1D periodic array, the quantum tunneling between them produces a typical parabolic band. With Rasha-type SOC, the band splits in the momentum space, as illustrated in Fig. \[supp\_band\](b). In the ‘B’ wire, we reserve the direction of the SOC, which gives a same band splitting as the previous wire (‘A’ wire), but the spin expectation value of the bands is reversed. Therefore, the combined setup gives spin-degenerate SOC split bands (as in the case of inversion and TR symmetric systems). As we turn on the Zeeman coupling, but keep it reversed in the two adjacent wires, a band gap opens at the TR invariant $\Gamma$-point even without breaking the TR symmetry. The corresponding band structure is shown in Fig. \[supp\_band\](d). With tuning the SOC strength and the inter-wire hopping amplitude the valence band can be pulled back completely below the Fermi level. In this case, as an insulating gap forms, the non-trivial topology ensures a protected metallic surface state.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: Interchanging the basis to $\{A_\uparrow,B_\downarrow,B_\uparrow,A_\downarrow\}$ retrieves the familiar TR operator $\tau^\prime=-i \sigma_x\otimes \sigma_y$, $H\rightarrow H^\prime$ and hence $\tau^\prime H^{\prime*}(\textbf{k},\Omega)\tau^{\prime -1}\ =\ H^\prime (-\textbf{k},-\Omega)$.
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abstract: |
We discuss a mechanism for producing baryon density perturbations during inflationary stage and study the evolution of the baryon charge density distribution in the framework of the low temperature baryogenesis scenario. This mechanism may be important for the large scale structure formation of the Universe and particularly, may be essential for understanding the existence of a characteristic scale of $130h^{-1}$ Mpc (comoving size) in the distribution of the visible matter.
The detailed analysis showed that both the observed very large scale of the visible matter distribution in the Universe and the observed baryon asymmetry value could naturally appear as a result of the evolution of a complex scalar field condensate, formed at the inflationary stage.
Moreover, according to our model, the visible part of the Universe at present may consist of baryonic and antibaryonic regions, sufficiently separated, so that annihilation radiation is not observed.
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6.4in -1in by -1.5cm by -1cm 0.1in
[**Non-GUT Baryogenesis and Large Scale Structure of the Universe**]{}
D.P.Kirilova[^1] and M.V.Chizhov[^2]\
\
[*The Abdus Salam International Centre for Theoretical Physics,\
Strada Costiera 11, 34014 Trieste, Italy*]{}
keywords: cosmology – large-scale structure - baryogenesis
Introduction
============
The large scale texture of the Universe shows a great complexity and variety of observed structures, it shows a strange pattern of filaments, voids and sheets. Moreover, due to the increasing amount of different types of observational data and theoretical analysis the last years, it was realized, that there exists a characteristic very large scale of about $130h^{-1}$ Mpc in the large scale texture of the Universe. Namely, the galaxy deep pencil beam surveys (Broadhurst et al. 1988, 1990) found an intriguing periodicity in the very large scale distribution of the luminous matter. The data consisted of several hundred redshifts of galaxies, coming from four distinct surveys, in two narrow cylindrical volumes into the directions of the North and the South Galactic poles of our Galaxy, up to redshifts of more than $z\sim 0.3$, combined to produce a well sampled distribution of galaxies by redshift on a linear scale extending to $2000h^{-1}$ Mpc. The plot of the numbers of galaxies as a function of redshifts displays a remarkably regular redshift distribution, with most galaxies lying in discrete peaks, with a periodicity over a scale of about $130h^{-1}$ Mpc comoving size.
It was realized also that the density peaks in the regular space distribution of galaxies in the redshift survey of Broadhurst et al. (1990), correspond to the location of superclusters, as defined by rich clusters of galaxies in the given direction (Bahcall 1991). The survey of samples in other directions, located near the South Galactic pole also gave indications for a regular distribution on slightly different scales near $100h^{-1}$ Mpc (Ettori et al. 1995, see also Tully et al. 1992 and Guzzo et al. 1992, Willmer et al. 1994). This discovery of a large scale pattern at the galactic poles was confirmed in a wider angle survey of 21 new pencil beams distributed over 10 degree field at both galactic caps (Broadhurst et al. 1995) and also by the new pencil-beam galaxy redshift data around the South Galactic pole region (Ettori et al. 1997).
The analysis of other types of observations confirm the existence of this periodicity. Namely, such structure is consistent with the reported periodicity in the distribution of quasars and radio galaxies (Foltz et al. 1989, Komberg et al. 1996, Quashnock et al. 1996, Petitjeau 1996, Cristiani 1998) and Lyman-$\alpha$ forest (Chu & Zhu 1989); the studies on spatial distribution of galaxies (both optical and IRAS) and clusters of galaxies (Kopylov et al. 1984, de Lapparent et al. 1986, Geller & Hunchra 1989, Hunchra et al. 1990, Bertshinger et al. 1990, Rowan-Robinson et al. 1990, Buryak et al. 1994, Bahcall 1992, Fetisova et al. 1993a, Einasto et al. 1994, Cohen et al. 1996, Bellanger & de Lapparent 1995) as well as peculiar velocity information (Lynden-Bell et al. 1988, Lauer & Postman 1994, Hudson et al. 1999) suggest the existence of a large scale superclusters-voids network with a characteristic scale around $130h^{-1}$ Mpc.
An indication of the presence of this characteristic scale in the distribution of clusters has been found also from the studies of the correlation functions and power spectrum of clusters of galaxies (Kopylov et al. 1988, Bahcall 1991, Mo et al. 1992, Peacock & West 1992, Deckel et al. 1992, Saar et al. 1995, Einasto et al. 1993, Einasto & Gramann 1993, Fetisova et al. 1993b, Frisch et al. 1995, Einasto et al. 1997b, Retzlaff et al. 1998, Tadros et al. 1997). The galaxy correlation function of the Las Campanas redshift survey also showed the presence of a secondary maximum at the same scale and a strong peak in the 2-dimensional power spectrum corresponding to an excess power at about 100 Mpc (Landy et al. 1995, 1996, Shectman et al. 1996, Doroshkevich et al. 1996, Geller at al. 1997, Tucker et al. 1999). The supercluster distribution was shown also to be not random but rather described as some weakly correlated network of superclusters and voids with typical mean separation of $100-150h^{-1}$ Mpc. Many known superclusters were identified with the vertices of an octahedron superstructure network (Battaner 1998). The network was proven to resemble a cubical lattice, with a periodic distribution of the rich clusters along the main axis (coinciding with the supergalactic $Y$ axis) of the network, with a step $\sim 130 h^{-1}$ Mpc (Toomet et al. 1999). These results are consistent with the statistical analysis of the pencil beam surveys data (Kurki-Suonio et al. 1990, Amendola 1994), which advocates a regular structure.
Recently performed study of the whole-sky distribution of high density regions defined by very rich Abell and APM clusters of galaxies (Baugh 1996, Einasto et al. 1994, 1996, 1997a, Gaztanaga & Baugh 1997, Landy et al. 1996, Retzlaff et al. 1998, Tadros et al. 1997, Kerscher 1998) confirmed from 3-dimensional data the presence of the characteristic scale of about $130h^{-1}$ Mpc of the spatial inhomogeneity of the Universe, found by Broadhurst et al. (1988, 1990) from the one dimensional study. The combined evidence from cluster and CMB data (Baker et al. 1999, Scott et al. 1996) also favours the presence of a peak at $130h^{-1}$ Mpc and a subsequent break in the initial power spectrum (Atrio-Barandela et al. 1997, Broadhurst$\&$Jaffe 1999). For a recent review of the regularity of the Universe on large scales see Einasto (1997).
Concerning all these rather convenient data, pointing that different objects trace the same structure at large scales, we are forced to believe in the real existence of the $130h^{-1}$ Mpc as a typical scale for the matter distribution in the Universe (see also Einasto et al 1998). However, this periodicity points to the existence of a significantly larger scale in the observed today Universe structure than predicted by standard models of structure formation by gravitational instability (Davis 1990 , Szalay et al. 1991, Davis et al. 1992, Luo & Vishniac 1993, Bahcall 1994, Retzlaff et al. 1998, Atrio-Barandela et al. 1997, Lesgourgues et al. 1998, Meiksin et al. 1998, Eisenstein et al. 1998, Eisenstein & Hu 1997a, 1997b) and is rather to be regarded as a new feature appearing only when very large scales ($>100
h^{-1}$ Mpc) are probed.
The problem of the generation of the spatial periodicity in the density distribution of luminous matter at large scales was discussed in numerous publications (Lumsden et al. 1989, Ostriker & Strassler 1989, Davis 1990, Coles 1990, Kurki-Suonio et al. 1990, Trimble 1990, Kofman et al. 1990, Ikeuchi & Turner 1991, van de Weygaert 1991, Buchert & Mo 1991, SubbaRao & Szalay 1992, Coleman & Pietronero 1992, Hill et al. 1989, Tully et al. 1992, Chincarini 1992, Weis & Buchert 1993, Atrio-Barandela et al. 1997, Lesgourgues et al. 1998, Eisenstein & Hu 1997a, Meiksin et al. 1998, Eisenstein et al. 1998, etc.). It was shown that a random structure could not explain the observed distribution. Statistical analysis of the deviations from periodicity showed that even for a perfectly regular structure a somewhat favoured direction and/or location within the structure may be required. The presence of the observed periodicity up to a great distance and in different directions seams rather amazing. Having in mind this results and the difficulties that perturbative models encounter in explaining the very large scale structure formation (namely the existence of the very large characteristic scale and the periodicity of the visible matter distribution), we chose another way of exploration, namely, we assume these as a typical new feature characteristic only for very large scales ($>100
h^{-1}$ Mpc). I. e. we consider the possibility that density fluctuations required to explain the present cosmological largest scale structures of the universal texture may have arisen in a different from the standard way, they may be a result from a completely different mechanism not necessarily with gravitational origin.
Such a successful mechanism was already proposed (Chizhov & Dolgov 1992) and analyzed in the framework of high-temperature baryogenesis scenarios.[^3] According to the discussed mechanism an additional complex scalar field (besides inflaton) is assumed to be present at the inflationary epoch, and it yields the extra power at the very large scale discussed. Primordial baryonic fluctuations are produced during the inflationary period, due to the specific evolution of the space distribution of the complex scalar field, carrying the baryon charge.
In the present work we study the possibility of generating of periodic space distribution of primordial baryon density fluctuations at the end of inflationary stage, applying this mechanism for the case of low temperature baryogenesis with baryon charge condensate of Dolgov & Kirilova (1991). The preliminary analysis of this problem, provided in Chizhov & Kirilova (1994), proved its usefulness in that case. Here we provide detail analysis of the evolution of the baryon density perturbations from the inflationary epoch till the baryogenesis epoch and describe the evolution of the spatial distribution of the baryon density. The production of matter-antimatter asymmetry in this scenario proceeds generally at low energies ($\le 10^{9}$ GeV). This is of special importance having in mind that the low-temperature baryogenesis scenarios are the preferred ones, as far as for their realization in the postinflationary stage it is not necessary to provide considerable reheating temperature typical for GUT high temperature baryogenesis scenarios. Hence, the discussed model (Dolgov & Kirilova 1991) has several attractive features: (a) It is compatible with the inflationary models as far as it does not suffer from the problem of insufficient reheating. (b) Generally, this scenario evades the problem of washing out the previously produced baryon asymmetry at the electroweak transition. (c) And as it will be proved in the following it may solve the problem of large scale periodicity of the visible matter.
It was already discussed in (Dolgov 1992, Chizhov & Dolgov 1992) a periodic in space baryonic density distribution can be obtained provided that the following assumptions are realized:
\(a) There exists a complex scalar field $\phi$ with a mass small in comparison with the Hubble parameter during inflation.
\(b) Its potential contains nonharmonic terms.
\(c) A condensate of $\phi$ forms during the inflationary stage and it is a slowly varying function of space points.
All these requirements can be naturally fulfilled in our scenario of the scalar field condensate baryogenesis (Dolgov & Kirilova 1991) and in low temperature baryogenesis scenarios based on the Affleck and Dine mechanism (Affleck & Dine 1985).
In case when the potential of $\phi$ is not strictly harmonic the oscillation period depends on the amplitude $P(\phi_0(r))$, and it on its turn depends on $r$. Therefore, a monotonic initial space distribution will soon result into spatial oscillations of $\phi$ (Chizhov & Dolgov 1992). Correspondingly, the baryon charge, contained in $\phi$: $N_B=i\phi^* \stackrel {\leftrightarrow} {\partial}_0 \phi$, will have quasi-periodic behavior. During Universe expansion the characteristic scale of the variation of $N_B$ will be inflated up to a cosmologically interesting size. Then, if $\phi$ has not reached the equilibrium point till the baryogenesis epoch $t_B$, the baryogenesis would make a snapshot of the space distribution of $\phi(r,t_B)$ and $N_B(r,t_B)$, and thus the present periodic distribution of the visible matter may date from the spatial distribution of the baryon charge contained in the $\phi$ field at the advent of the $B$-conservation epoch.
Density fluctuations with a comoving size today of $130 h^{-1}$ Mpc reentered the horizon at late times at a redshift of about 10 000 and a mass of $10^{18}M_o$. After recombination the Jeans mass becomes less than the horizon and the fluctuations of this large mass begin to grow. We propose that these baryonic fluctuations, periodically spaced, lead to an enhanced formation of galaxy superclusters at the peaks of baryon overdensity. The concentration of baryons into periodic shells may have catalysed also the clustering of matter coming from the inflaton decays onto these “baryonic nuclei". After baryogenesis proceeded, superclusters may have formed at the high peaks of the background field (the baryon charge carrying scalar field, we discuss). (See the results of the statistical analysis (Plionis 1995), confirming the idea that clusters formed at the high peaks of background field, which is analogous to our assumption.) We imply that afterwards the self gravity mechanisms might have optimized the arrangement of this structure into the thin regularly spaced dense baryonic shells and voids in between with the characteristic size of $130 h^{-1} Mpc$ observed today.
The analysis showed that in the framework of our scenario both the generation of the baryon asymmetry and the periodic distribution of the baryon density can be explained simultaneously as due to the evolution of a complex scalar field.
Moreover, for a certain range of parameters the model predicts that the Universe may consist of sufficiently separated baryonic and antibaryonic shells. This possibility was discussed in more detail elsewhere (Kirilova 1998). This is an interesting possibility as far as the observational data of antiparticles in cosmic rays and the gamma rays data do not rule out the possibility for existence of superclusters of galaxies of antimatter in the Universe (Steigman 1976, Ahlen et al. 1982, 1988, Stecker 1985, 1989, Gao et al. 1990). The observations exclude the possibility of noticeable amount of antimatter in our Galaxy, however, they are not sensitive enough to test the existence of antimatter extragalactic regions. I.e. current experiments (Salamon et al. 1990, Ahlen et al. 1994, Golden et al. 1994, 1996, Yoshimura et al. 1995, Mitchell et al. 1996, Barbiellini & Zalateu 1997, Moiseev et al. 1997, Boesio et al. 1997, Orito et al. 1999, etc.) put only a lower limit on the distance to the nearest antimatter-rich region, namely $\sim 20$ Mpc. Future searches for antimatter among cosmic rays are expected to increase this lower bound by an order of magnitude. Namely, the reach of the AntiMatter Spectrometer is claimed to exceed 150 Mpc (Ahlen et al. 1982) and its sensitivity is three orders of magnitudes better than that of the previous experiments (Battiston 1997, Plyaskin et al. 1998). For a more detail discussion on the problem of existence of noticeable amounts of antimatter at considerable distances see Dolgov (1993), Cohen et al. (1998), Kinney et al. (1997).
The following section describes the baryogenesis model and the last section deals with the generation of the periodicity of the baryon density and discusses the results.
Description of the model. Main characteristics.
===============================================
Our analysis was performed in the framework of the low temperature non-GUT baryogenesis model described in (Dolgov & Kirilova 1991), based on the Affleck and Dine SUSY GUT motivated mechanism for generation of the baryon asymmetry (Affleck & Dine 1985). In this section we describe the main characteristics of the baryogenesis model, which are essential for the investigation of the periodicity in the next section. For more detail please see the original paper.
Generation of the baryon condensate.
------------------------------------
The essential ingredient of the model is a squark condensate $\phi$ with a nonzero baryon charge. It naturally appears in supersymmetric theories and is a scalar superpartner of quarks. The condensate $<\phi>\neq0$ is formed during the inflationary period as a result of the enhancement of quantum fluctuations of the $\phi$ field (Vilenkin & Ford 1982, Linde 1982, Bunch & Davies 1978, Starobinsky 1982): $<\phi^2>=H^3t/4\pi^2$. The baryon charge of the field is not conserved at large values of the field amplitude due to the presence of the B nonconserving self-interaction terms in the field’s potential. As a result, a condensate of a baryon charge (stored in $<\phi>$) is developed during inflation with a baryon charge density of the order of $H^3_I$, where $H_I$ is the Hubble parameter at the inflationary stage.
Generation of the baryon asymmetry.
-----------------------------------
After inflation $\phi$ starts to oscillate around its equilibrium point with a decreasing amplitude. This decrease is due to the Universe expansion and to the particle production by the oscillating scalar field (Dolgov & Kirilova 1990, 1991). Here we discuss the simple case of particle production when $\phi$ decays into fermions and there is no parametric resonance. We expect that the case of decays into bosons due to parametric resonance (Kofman et al. 1994, 1996, Shtanov et al. 1995, Boyanovski et al. 1995, Yoshimura 1995, Kaiser 1996), especially in the broad resonance case, will lead to an explosive decay of the condensate, and hence an insufficient baryon asymmetry. Therefore, we explore the more promising case of $\phi$ decaying into fermions.
In the expanding Universe $\phi$ satisfies the equation -a\^[-2]{}\^2\_i+3H+ [1 4]{}+U’\_=0, where $a(t)$ is the scale factor and $H=\dot{a}/a$.
The potential $U(\phi)$ is chosen in the form U()=[\_12]{}||\^4 +[\_24]{}(\^4+\^[\*4]{})+[\_34]{}||\^2 (\^2+\^[\*2]{}) The mass parameters of the potential are assumed small in comparison to the Hubble constant during inflation $m \ll H_I$. In supersymmetric theories the constants $\lambda_i$ are of the order of the gauge coupling constant $\alpha$. A natural value of $m$ is $10^2\div10^4$ GeV. The initial values for the field variables can be derived from the natural assumption that the energy density of $\phi$ at the inflationary stage is of the order $H^4_I$, then $\phi^{max}_o \sim H_I\lambda^{-1/4}$ and $\dot{\phi_o}=0$.
The term $\Gamma\dot{\phi}$ in the equations of motion explicitly accounts for the eventual damping of $\phi$ as a result of particle creation processes. The explicit account for the effect of particle creation processes in the equations of motion was first provided in (Chizhov & Kirilova 1994, Kirilova & Chizhov 1996). The production rate $\Gamma$ was calculated in (Dolgov & Kirilova 1990). For simplicity here we have used the perturbation theory approximation for the production rate $\Gamma = \alpha \Omega$, where $\Omega$ is the frequency of the scalar field.[^4] For $g<\lambda^{3/4}$, $\Gamma$ considerably exceeds the rate of the ordinary decay of the field $\Gamma_m=\alpha m$. Fast oscillations of $\phi$ after inflation result in particle creation due to the coupling of the scalar field to fermions $g\phi \bar{f}_1 f_2$, where $g^2/4\pi = \alpha_{SUSY}$. Therefore, the amplitude of $\phi$ is damped as $\phi \rightarrow
\phi \exp(-\Gamma t/4)$ and the baryon charge, contained in the $\phi$ condensate, is considerably reduced. It was discussed in detail in Dolgov & Kirilova (1991) that for a constant $\Gamma$ this reduction is exponential and generally, for a natural range of the model’s parameters, the baryon asymmetry is waved away till baryogenesis epoch as a result of the particle creation processes. Fortunately, in the case without flat directions of the potential, the production rate is a decreasing function of time, so that the damping process may be slow enough for a considerable range of acceptable model parameters values of $m$, $H$, $\alpha$, and $\lambda$, so that the baryon charge contained in $\phi$ may survive until the advent of the $B$-conservation epoch. Generally, in cases of more effective particle creation, like in the case with flat directions in the potential, or in the case when $\phi$ decays spontaneously into bosons due to parametric resonance, the discussed mechanism of the baryon asymmetry generation cannot be successful. Hence, it cannot be useful also for the generation of the matter periodicity.
Baryogenesis epoch $t_B$.
-------------------------
When inflation is over and $\phi$ relaxes to its equilibrium state, its coherent oscillations produce an excess of quarks over antiquarks (or vice versa) depending on the initial sign of the baryon charge condensate. This charge, diluted further by some entropy generating processes, dictates the observed baryon asymmetry. This epoch when $\phi$ decays to quarks with non-zero average baryon charge and thus induces baryon asymmetry we call baryogenesis epoch. The baryogenesis epoch $t_B$ for our model coincides with the advent of the baryon conservation epoch, i.e. the time after which the mass terms in the equations of motion cannot be neglected. In the original version (Affleck & Dine 1985) this epoch corresponds to energies $10^{2}-10^{4}$ GeV. However, as it was already explained, the amplitude of $\phi$ may be reduced much more quickly due to the particle creation processes and as a result, depending on the model’s parameters the advent of this epoch may be considerably earlier. For the correct estimation of $t_B$ and the value of the generated baryon asymmetry, it is essential to account for the eventual damping of the field’s amplitude due to particle production processes by an external time-dependent scalar field, which could lead to a strong reduction of the baryon charge contained in the condensate.
Generation of the baryon density periodicity.
=============================================
In order to explore the spatial distribution behavior of the scalar field and its evolution during Universe expansion it is necessary to analyze eq.(1). We have made the natural assumption that initially $\phi$ is a slowly varying function of the space coordinates $\phi(r,t)$. The space derivative term can be safely neglected because of the exponential rising of the scale factor $a(t)\sim\exp(H_It)$. Then the equations of motion for $\phi=x+iy$ read $$\begin{aligned}
&&\ddot{x}+3H\dot{x}+{1 \over 4} \Gamma_x \dot{x}+
(\lambda+\lambda_3)x^3+\lambda'xy^2=0
\nonumber
\\
&&\ddot{y}+3H\dot{y}+{1 \over 4} \Gamma_y \dot{y}+
(\lambda-\lambda_3)y^3+\lambda'yx^2=0\end{aligned}$$ where $\lambda=\lambda_1+\lambda_2$, $\lambda'=\lambda_1-3\lambda_2$.
In case when at the end of inflation the Universe is dominated by a coherent oscillations of the inflaton field $\psi=m_{PL}(3\pi)^{-1/2}\sin(m_{\psi}t)$, the Hubble parameter is $H=2/(3t)$. In this case it is convenient to make the substitutions $x=H_I(t_i/t)^{2/3}u(\eta)$, $y=H_I(t_i/t)^{2/3}v(\eta)$ where $\eta=2(t/t_i)^{1/3}$. The functions $u(\eta)$ and $v(\eta)$ satisfy the equations
[c]{} u”+ 0.75 \_u (u’-2u\^[-1]{})+ u\[(+\_3) u\^2+’v\^2-2\^[-2]{}\]=0\
v”+ 0.75 \_v (v’-2v\^[-1]{})+ v\[(-\_3) v\^2+’u\^2-2\^[-2]{}\]=0.
The baryon charge in the comoving volume $V=V_i(t/t_i)^2$ is $B=N_B \cdot V=2 (u'v-v'u)$. The numerical calculations were performed for $u_o,v_o \in [0, \lambda^{-1/4}]$, $u'_o,v'_o \in [0, 2/3 \lambda^{-1/4}]$. For simplicity we considered the case: $\lambda_1 > \lambda_2 \sim \lambda_3$, when the unharmonic oscillators $u$ and $v$ are weakly coupled. For each set of parameter values of the model $\lambda_i$ we have numerically calculated the baryon charge evolution $B(\eta)$ for different initial conditions of the field corresponding to the accepted initial monotonic space distribution of the field (see Figs. 1,2).
The numerical analysis confirmed the important role of particle creation processes for baryogenesis models and large scale structure periodicity (Chizhov & Kirilova 1994, 1996) which were obtained from an approximate analytical solution. In the present work we have accounted for particle creation processes explicitly. [^5]
The space distribution of the baryon charge is calculated for the moment $t_B$. It is obtained from the evolution analysis $B(\eta)$ for different initial values of the field, corresponding to its initial space distribution $\phi(t_i,r)$ (Fig. 3). As it was expected, in the case of nonharmonic field’s potential, the initially monotonic space behavior is quickly replaced by space oscillations of $\phi$, because of the dependence of the period on the amplitude, which on its turn is a function of $r$. As a result in different points different periods are observed and space behavior of $\phi$ becomes quasiperiodic. Correspondingly, the space distribution of the baryon charge contained in $\phi$ becomes quasiperiodic as well. Therefore, the space distribution of baryons at the moment of baryogenesis is found to be periodic.
The observed space distribution of the visible matter today is defined by the space distribution of the baryon charge of the field $\phi$ at the moment of baryogenesis $t_B$, $B(t_B,r)$. So, that at present the visible part of the Universe consists of baryonic shells, divided by vast underdense regions. [*For a wide range of parameters’ values the observed average distance of $130h^{-1}$ Mpc between matter shells in the Universe can be obtained. The parameters of the model ensuring the necessary observable size between the matter domains belong to the range of parameters for which the generation of the observed value of the baryon asymmetry may be possible in the model of scalar field condensate baryogenesis.*]{} This is an attractive feature of this model because both the baryogenesis and the large scale structure periodicity of the Universe can be explained simply through the evolution of a single scalar field.
Moreover, for some model’s variations the presence of vast antibaryonic regions in the Universe is predicted. This is an interesting possibility as far as the observational data do not rule out the possibility of antimatter superclusters in the Universe. The model proposes an elegant mechanism for achieving a sufficient separation between regions occupied by baryons and those occupied by antibaryons, necessary in order to inhibit the contact of matter and antimatter regions with considerable density.
It is interesting, having in mind the positive results of this investigation, to provide a more precise study of the question for different possibilities of particle creation and their relevance for the discussed scenario of baryogenesis and periodicity generation. In the case of narrow-band resonance decay the final state interactions regulate the decay rate, parametric amplification is effectively suppressed (Allahverdi & Campbell 1997) and does not drastically enhance the decay rate. Therefore, we expect that this case will be interesting to explore. Another interesting case may be the case of strong dissipative processes of the products of the parametric resonance. As far as the dissipation reduces the resonant decay rate (Kolb et al. 1996, Kasuya & Kawasaki 1996) it may be worthwhile to consider such a model as well.
Acknowledgments
===============
We are glad to thank A.D.Dolgov for stimulating our interest in this problem. We are thankful to ICTP, Trieste, where this work was finished, for the financial support and hospitality. We are grateful also to the referee for the useful remarks and suggestions.
This work was partially financially supported by Grant-in-Aid for Scientific Research F-553 from the Bulgarian Ministry of Education, Science and Culture.\
\
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[Captions]{}
\
[**Figure 1**]{}: The evolution of the baryon charge $B(\eta)$ contained in the condensate $<\phi>$ for $\lambda_1=5 \times 10^{-2}$, $\lambda_2=\lambda_3=\alpha=10^{-3}$, $H_I/m=10^7$, $\phi_o=H_I\lambda^{-1/4}$, and $\dot{\phi}_o=0$.
[**Figure 2**]{}: The evolution of the baryon charge $B(\eta)$ contained in the condensate $<\phi>$ for $\lambda_1= 5 \times 10^{-2}$, $\lambda_2=\lambda_3=\alpha=10^{-3}$, $H_I/m=10^7$, $\phi_o={1 \over 50}H_I\lambda^{-1/4}$, and $\dot{\phi}_o=0$.
[**Figure 3**]{}: The space distribution of baryon charge at the moment of baryogenesis for $\lambda_1=5 \times 10^{-2}$, $\lambda_2=\lambda_3=
\alpha=10^{-3}$, $H_I/m=10^7$.
[^1]: Permanent address: Institute of Astronomy at Bulgarian Academy of Sciences, Bul. Tsarigradsko Shosse 72, 1784 Sofia, Bulgaria
[^2]: Permanent address: Centre of Space Research and Technologies, Faculty of Physics, University of Sofia, 1164 Sofia, Bulgaria, E-mail: [email protected]
[^3]: By high-temperature baryogenesis scenarios we denote here those scenarios which proceed at very high energies of the order of the Grand Unification scale, and especially the GUT baryogenesis scenarios. In contrast, low temperature baryogenesis scenarios like Affleck and Dine scenario and electroweak baryogenesis, proceed at several orders of magnitude lower energies.
[^4]: For the toy model, we discuss here, we consider this approximation instructive enough.
[^5]: It was shown, that the damping effect due to the particle creation is proportional to the initial amplitudes of the field. As far as the particle creation rate is proportional to the field’s frequency, it can be concluded that the frequency depends on the initial amplitudes. This result confirms our analytical estimation provided in earlier works.
|
---
abstract: |
In dimensions $d\geq 4$, we prove that the Schrödinger map initial-value problem $$\begin{cases}
&\partial_ts=s\times\Delta s\,\text{ on }\,\mathbb{R}^d\times\mathbb{R};\\
&s(0)=s_0
\end{cases}$$ admits a unique solution $s:\mathbb{R}^d\times\mathbb{R}\to\mathbb{S}^2\hookrightarrow\mathbb{R}^3$, $s\in C(\mathbb{R}:H^{\infty}_Q)$, provided that $s_0\in H^{\infty}_Q$ and $\|s_0-Q\|_{\dot{H}^{d/2}}\ll 1$, where $Q\in\mathbb{S}^2$.
address:
- 'University of California – Los Angeles'
- 'University of Wisconsin – Madison'
- University of Chicago
author:
- 'I. Bejenaru'
- 'A. D. Ionescu'
- 'C. E. Kenig'
title: 'Global existence and uniqueness of Schrödinger maps in dimensions $d\geq 4$'
---
[^1]
Introduction {#intro}
============
In this paper we consider the Schrödinger map initial-value problem $$\label{Sch1}
\begin{cases}
&\partial_ts=s\times\Delta s\,\text{ on }\,\mathbb{R}^d\times\mathbb{R};\\
&s(0)=s_0,
\end{cases}$$ where $d\geq 4$ and $s:\mathbb{R}^d\times\mathbb{R}\to\mathbb{S}^2\hookrightarrow\mathbb{R}^3$ is a continuous function. The Schrödinger map equation has a rich geometric structure and arises naturally in a number of different ways; we refer the reader to [@NaStUh] or [@KePoStTo] for details.
For $\sigma\geq 0$ and $n\in\{1,2,\ldots\}$ let $H^{\sigma}=H^\sigma(\mathbb{R}^d;\mathbb{C}^n)$ denote the Banach spaces of $\mathbb{C}^n$-valued Sobolev functions on $\mathbb{R}^d$, i.e. $$H^{\sigma}=\{f:\mathbb{R}^d\to\mathbb{C}^n:\|f\|_{H^\sigma}=\big[\sum_{l=1}^n\|\mathcal{F}_{(d)}(f_l)\cdot ( |\xi|^2+1)^{\sigma/2}\|_{L^2}^2\big]^{1/2}<\infty\},$$ where ${\ensuremath{\mathcal{F}}}_{(d)}$ denotes the Fourier transform on $L^2({\ensuremath{\mathbb{R}}}^d)$. For $\sigma\geq 0$, $n\in\{1,2,\ldots\}$, and $f\in H^\sigma(\mathbb{R}^d;\mathbb{C}^n)$, we define $$\begin{split}
\|f\|_{\dot{H}^{\sigma}}=\big[\sum_{l=1}^n\|{\ensuremath{\mathcal{F}}}_{(d)}(f_l)(\xi)\cdot |\xi|^\sigma\|^2_{L^2}\big]^{1/2}.
\end{split}$$ For $\sigma\geq 0$ and $Q=(Q_1,Q_2,Q_3)\in\mathbb{S}^2$ we define the complete metric space $$\label{Sch2}
H^\sigma_Q=H^{\sigma}_Q(\mathbb{R}^d;\mathbb{S}^2\hookrightarrow\mathbb{R}^3)=\{f:\mathbb{R}^d\to\mathbb{R}^3:|f(x)|\equiv 1\text{ and }f-Q\in H^\sigma\},$$ with the induced distance $$\label{Ban2}
d^\sigma_Q(f,g)=\|f-g\|_{H^\sigma}.$$ For simplicity of notation, we let $\|f\|_{H^\sigma_Q}=d^\sigma_Q(f,Q)$ for $f\in H^\sigma_Q$. Let ${\ensuremath{\mathbb{Z}}}_+=\{0,1,\ldots\}$. For $n\in\{1,2,\ldots\}$ and $Q\in\mathbb{S}^2$ we define the complete metric spaces $$H^\infty=H^\infty({\ensuremath{\mathbb{R}}}^d;\mathbb{C}^n)=\bigcap_{\sigma\in{\ensuremath{\mathbb{Z}}}_+}H^\sigma\,\,\,\text{ and }\,\,\,H^\infty_Q=\bigcap_{\sigma\in{\ensuremath{\mathbb{Z}}}_+}H^\sigma_Q,$$ with the induced distances. Our main theorem concerns global existence and uniqueness of solutions of the initial-value problem for data $s_0\in H^{\infty}_Q$, with $\|s_0-Q\|_{\dot{H}^{d/2}}\ll1$.
\[section\]
\[Main1\] Assume $d\geq 4$ and $Q\in\mathbb{S}^2$. Then there is $\varepsilon_0=\varepsilon_0(d)>0$ such that for any $s_0\in H^{\infty}_Q$ with $\|s_0-Q\|_{\dot{H}^{d/2}}\leq \varepsilon_0$ there is a unique solution $$\label{amy1}
s=S_Q(s_0)\in
C({\ensuremath{\mathbb{R}}}:H^{\infty}_Q)$$ of the initial-value problem . Moreover $$\label{amy2}
\sup_{t\in{\ensuremath{\mathbb{R}}}}\|s(t)-Q\|_{\dot{H}^{d/2}}\leq C\|s_0-Q\|_{\dot{H}^{d/2}},$$ and $$\label{amy3}
\sup_{t\in[-T,T]}\|s(t)\|_{H^\sigma_Q}\leq C(\sigma,T,\|s_0\|_{H^\sigma_Q})$$ for any $T\in[0,\infty)$ and $\sigma\in{\ensuremath{\mathbb{Z}}}_+$.
[**[Remark:]{}**]{} We prove in fact a slightly stronger statement: there is $\sigma_0\in[d/2,\infty)\cap{\ensuremath{\mathbb{Z}}}$ sufficiently large such that for any $s_0\in H^{\sigma_0}_Q$ with $\|s_0-Q\|_{\dot{H}^{d/2}}\leq \varepsilon_0$ there is a unique solution $$s=S_Q(s_0)\in C({\ensuremath{\mathbb{R}}}:H^{\sigma_0-1}_Q)\cap L^\infty({\ensuremath{\mathbb{R}}}:H^{\sigma_0}_Q)$$ of the initial-value problem . Moreover, the bounds and (assuming $s_0\in H^\sigma_Q$, $\sigma\in{\ensuremath{\mathbb{Z}}}_+$) still hold.
The main point of Theorem \[Main1\] is the global (in time) existence of solutions. Its direct analogue in the setting of wave maps is the work of Tao [@Ta1] (see also [@KlMa], [@KlSe], [@Tat1], [@Tat2], [@Ta2], [@KlRo], [@ShSt], [@NaStUh3], and [@Tat3] for other local and global existence (or well-posedness) theorems for wave maps). However, our proof of Theorem \[Main1\] is closer to that of [@ShSt] and [@NaStUh3].
The initial-value problem has been studied extensively (also in the case in which the sphere $\mathbb{S}^2$ is replaced by more general targets). It is known that sufficiently smooth solutions exist locally in time, even for large data (see, for example, [@SuSuBa], [@ChShUh], [@DiWa2], [@Ga], [@KePoStTo] and the references therein). Such theorems for (local in time) smooth solutions are proved using delicate geometric variants of the energy method. For low-regularity data, the initial-value problem has been studied indirectly using the “modified Schrödinger map equations” (see, for example, [@ChShUh], [@NaStUh], [@NaStUh2], [@KeNa], [@Ka], and [@KaKo]) and certain enhanced energy methods.
In [@IoKe2], Ionescu–Kenig realized that the initial-value problem can be analyzed perturbatively using the stereographic model, in the case of “small data” (i.e. data that takes values in a small neighborhood of a point on the sphere), and proved local well-posedness for small data in $H^\sigma_Q$, $\sigma>(d+1)/2$, $d\geq 2$. The resolution spaces constructed in [@IoKe2] (see also [@IoKe] for the $1$-dimensional version of these spaces) are based on directional $L^{p,q}_{\ensuremath{\mathbf{e}}}$ physical spaces, which are related to local smoothing; in particular, the nonlinear analysis is based on local smoothing and the simple inclusion $$L^{\infty,2}_{\ensuremath{\mathbf{e}}}\cdot L^{2,\infty}_{\ensuremath{\mathbf{e}}}\cdot L^{2,\infty}_{\ensuremath{\mathbf{e}}}\subseteq L^{1,2}_{\ensuremath{\mathbf{e}}}.$$ We use the same resolution spaces and this simple inclusion in the perturbative analysis in section \[section3\] in this paper.
Slightly later and independently, Bejenaru [@Be] also realized that the stereographic model can be used for perturbative analysis, and proved local well-posedness for small data in $H^\sigma$, in the full subcritical range $\sigma>d/2$, $d\geq 2$. In the stereographic model Bejenaru observed, apparently for the first time in the setting of Schrödinger maps, that the gradient part of the nonlinearity has a certain null structure (similar to the null structure of wave maps, observed by S. Klainerman).[^2] The resolution spaces used in [@Be] for the perturbative argument are different from those of [@IoKe2]; these resolution spaces are based on the construction of suitably normalized wave packets, and had been previously used by Bejenaru in other subcritical problems (see [@Be2] and the references therein).
In [@IoKe3] Ionescu–Kenig proved the first global (in time) well-posedness theorem for small data in the critical Besov spaces $\dot{B}^{d/2}_Q$, in dimensions $d\geq 3$, using certain technical modifications of the resolution spaces of [@IoKe2] and the null structure observed in [@Be]. As explained in [@IoKe3], the main difficulty in proving this result in dimension $d=2$ is the logarithmic failure of the scale-invariant $L^{2,\infty}_{\ensuremath{\mathbf{e}}}$ estimate.
Unlike its Besov analogue, the condition $\|s_0-Q\|_{\dot{H}^{d/2}}\ll 1$ in Theorem \[Main1\] does not guarantee that the data $s_0$ takes values in a small neighborhood of $Q$. Because of this, the stereographic model used in [@IoKe2], [@Be], and [@IoKe3] is not relevant, and it does not appear possible to prove Theorem \[Main1\] using a direct perturbative construction. We construct the solution $s$ indirectly, using [*[ a priori ]{}*]{} estimates: we start with a solution $s\in C([-T,T]:H^{\infty}_Q)$ of , where $T=T( \|s_0\|_{H^{\sigma_0}_Q})>0$, $\sigma_0$ sufficiently large, and transfer the quantitative bounds on the function $s$ at time $0$ to suitable quantitative bounds on the functions $\psi_m$ at time $0$ (the functions $\psi_m$ are solutions of the modified Schrödinger map equations, see section \[gauge\]). Then we study the modified Schrödinger map equations perturbatively, and prove uniform quantitative bounds on the functions $\psi_m$ at all times $t\in[-T,T]$. Finally, we transfer these bounds back to the solution $s$; this gives uniform quantitative bounds on $s$ at all times $t\in[-T,T]$, which allow us to extend the solution $s$ up to time $T=1$. By scaling, we can construct a global solution.
The rest of the paper is organized as follows: in section \[gauge\] we explain how to derive the modified Schrödinger map equations (MSM)[^3], and prove quantitative bounds on the solutions $\psi_m$ of the MSM at time $t=0$. In section \[section3\] we use a perturbative argument and the resolution spaces defined in [@IoKe2] (and some of their properties) to prove bounds on the solutions $\psi_m$ of the MSM on the time interval $[-T,T]$. The proofs of some of the technical nonlinear bounds are deferred to section \[section5\]. In section \[section4\] we transfer the bounds on $\psi_m$ to a priori bounds on solution $s$ of , and use a local existence theorem to close the argument.
We will always assume in the rest of the paper that $d\geq 3$ (we have not constructed yet suitable resolution spaces in dimension $d=2$). In subsection \[section3.3\] and sections \[section4\] and \[section5\] we assume the stronger restriction $d\geq 4$; the reason for this restriction is mostly technical, as it leads to simple proofs of the nonlinear estimates in Lemma \[Lemmaq5\]. In many estimates, we will use the letter $C$ to denote constants that may depend only on the dimension $d$.
We would like to thank S. Klainerman, I. Rodnianski, J. Shatah, and T. Tao for several useful discussions.
The modified Schrödinger map {#gauge}
============================
In this section we give a self-contained derivation of the modified Schrödinger map equations, using orthonormal frames[^4]. In the context of wave maps, orthonormal frames have been used in [@ChTa], [@ShSt], [@KlRo], [@NaStUh3] etc. In the context of Schrödinger maps, orthonormal frames (on the pullback of $T^\ast M$ under the solution $s$) have been used for the first time in [@ChShUh] to construct the modified Schrödinger map equations. See also [@Ga]. Complete expositions of this construction have been presented by J. Shatah on several occasions.
In this section we assume $d\geq 3$ (some technical changes are needed in dimension $d=2$, but we will not discuss them here).
A topological construction {#section2.1}
--------------------------
Assume $n\in[1,\infty)\cap{\ensuremath{\mathbb{Z}}}$, $a_1,\ldots,a_n\in[0,\infty)$, and let $$\mathcal{D}^n=[-a_1,a_1]\times\ldots\times[-a_n,a_n].$$ For $n=0$ let $\mathcal{D}^0=\{0\}$.
\[section\]
\[Lemmag1\] Assume $n\geq 0$ and $s:\mathcal{D}^n\to\mathbb{S}^2$ is a continuous function. Then there is a continuous function $v:\mathcal{D}^n\to\mathbb{S}^2$ with the property that $$s(x)\cdot v(x)=0\text{ for any }x\in\mathcal{D}^n.$$
We argue by induction over $n$ (the case $n=0$ is trivial). Since $s$ is continuous, there is $\epsilon>0$ with the property that $$\label{t1}
|s(x)-s(y)|\leq 2^{-10}\text{ for any }x,y\in\mathcal{D}^n\text{ with }|x-y|\leq\epsilon.$$ For $x\in\mathcal{D}^n$ we write $x=(x',x_n)\in\mathcal{D}^{n-1}\times[-a_n,a_n]$. For any $b\in[-a_n,a_n]$ let $\mathcal{D}^n_b=\mathcal{D}^{n-1}\times[-a_n,b]=\{x=(x',x_n)\in\mathcal{D}^n:x_n\in[-a_n,b]\}$. By the induction hypothesis, we can define $v:\mathcal{D}^n_{-a_n}\to\mathbb{S}^2$ continuous such that $$s(x)\cdot v(x)=0\text{ for any }x\in\mathcal{D}^n_{-a_n}.$$ We extend now the function $v$ to $\mathcal{D}^n$. With $\epsilon$ as in , it suffices to prove that if $b,b'\in[-a_n,a_n]$, $0\leq b'-b\leq\epsilon$, $v:\mathcal{D}^n_b\to\mathbb{S}^2$ is continuous, and $s(x)\cdot v(x)=0$ for any $x\in\mathcal{D}_b^n$, then $v$ can be extended to a continuous function $\widetilde{v}:\mathcal{D}^n_{b'}\to\mathbb{S}^2$ such that $s(x)\cdot \widetilde{v}(x)=0$ for any $x\in\mathcal{D}^n_{b'}$.
Let $$\label{t4}
\mathcal{R}=\{(u_1,u_2)\in\mathbb{R}^3\times\mathbb{R}^3:\,|u_1|,|u_2|\in(1/2,2)\text{ and }|u_1\cdot u_2|< 2^{-5}\},$$ and let $N:\mathcal{R}\to\mathbb{S}^2$ denote the smooth function $$\label{t5}
N[u_1,u_2]=\frac{u_1-((u_1\cdot u_2)/|u_2|^2)\,u_2}{|u_1-((u_1\cdot u_2)/|u_2|^2)\,u_2|}.$$ So $N[u_1,u_2]$ is a unit vector orthogonal to $u_2$ in the plane generated by the vectors $u_1$ and $u_2$. We construct now the extension $\widetilde{v}:\mathcal{D}^n_{b'}\to\mathbb{S}^2$. For $x'\in\mathcal{D}^{n-1}$ and $x_n\in[-a_n,b']$ let $$\widetilde{v}(x',x_n)=
\begin{cases}
N[v(x',b),s(x',x_n)]&\text{ if }x_n\in[b,b'];\\
v(x',x_n)&\text{ if }x_n\in[-a_n,b].
\end{cases}$$ In view of , the function $\widetilde{v}:\mathcal{D}^n_{b'}\to\mathbb{S}^2$ is well-defined, continuous, and $s(x)\cdot \widetilde{v}(x)=0$ for any $x\in\mathcal{D}_{b'}^n$. This completes the proof of Lemma \[Lemmag1\].
\[Lemmag1\][Lemma]{}
\[Lemmag2\] Assume $T\in[0,2]$, $Q,Q'\in\mathbb{S}^2$, $Q\cdot Q'=0$, and $s:\mathbb{R}^d\times[-T,T]\to\mathbb{S}^2$ is a continuous function with the property that $$\lim_{x\to\infty}s(x,t)=Q\text{ uniformly in }t\in[-T,T].$$ Then there is a continuous function $v:\mathbb{R}^d\times[-T,T]\to\mathbb{S}^2$ with the property that $$\begin{cases}
&s(x,t)\cdot v(x,t)=0\text{ for any }(x,t)\in\mathbb{R}^d\times[-T,T];\\
&\lim\limits_{x\to\infty}v(x,t)=Q'\text{ uniformly in }t\in[-T,T].
\end{cases}$$
We fix $R>0$ such that $$|s(x,t)-Q|\leq 2^{-10}\text{ if }|x|\geq R \text{ and }t\in[-T,T].$$ Using Lemma \[Lemmag1\], we can define a continuous function $v_0:B_R\times[-T,T]\to\mathbb{S}^2$ such that $s(x,t)\cdot v_0(x,t)=0$ for $(x,t)\in B_R\times[-T,T]$, where $B_R=\{x\in\mathbb{R}^d:|x|\leq R\}$. Let $S_R=\{x\in\mathbb{R}^d:|y|=R\}$ and $\mathbb{S}^1_Q=\{x\in\mathbb{S}^2:x\cdot Q=0\}$. We define the continuous function $$w:S_R\times[-T,T]\to\mathbb{S}^1_Q,\,\,\,w(y,t)=\frac{(s(y,t)\cdot Q)v_0(y,t)-(v_0(y,t)\cdot Q)s(y,t)}{|(s(y,t)\cdot Q)v_0(y,t)-(v_0(y,t)\cdot Q)s(y,t)|},$$ so $w(y,t)$ is a vector in $\mathbb{S}^1_Q$ and in the plane generated by $s(y,t)$ and $v_0(y,t)$. Since $d\geq 3$, the space $S_R\times[-T,T]$ is simply connected (and compact), thus the function $w$ is homotopic to a constant function. Thus there is a continuous function $$\widetilde{w}:S_R\times[-T,T]\times[1,2]\to\mathbb{S}^1_Q\,\text{ such that }\,\widetilde{w}(y,t,1)=w(y,t)\text{ and }\widetilde{w}(y,t,2)\equiv Q'.$$
With $N$ is as in , we define $$v_1(x,t)=N[\widetilde{w}(Rx/|x|,t,|x|/R),s(x,t)]$$ for $|x|\in[R,2R]$, and $$v_2(x,t)=N[Q',s(x,t)]$$ for $|x|\geq 2R$. The function $v$ in Lemma \[Lemmag2\] is obtained by gluing the functions $v_0$, $v_1$, and $v_2$.
Derivation of the modified Schrödinger map equations {#section2.2}
----------------------------------------------------
Assume now that $T\in[0,1]$, $Q,Q'\in\mathbb{S}^2$, and $Q\cdot Q'=0$. Assume that $$\label{t10}
\begin{cases}
&s\in C([-T,T]:H_Q^{\infty});\\
&\partial_ts\in C([-T,T]:H^{\infty}).
\end{cases}$$ We extend the function $s$ to a function $\widetilde{s}\in C([-T-1,T+1]:H_Q^{\infty})$ by setting $\widetilde{s}(.,t)=s(.,T)$ if $t\in[T,T+1]$ and $\widetilde{s}(.,t)=s(.,-T)$ if $t\in[-T-1,-T]$. Clearly, the function $\widetilde{s}:\mathbb{R}^d\times[-T-1,T+1]\to\mathbb{S}^2$ is continuous and $\lim_{x\to\infty}\widetilde{s}(x,t)=Q$ uniformly in $t$. We apply Lemma \[Lemmag2\] to construct a continuous function $\widetilde{v}:\mathbb{R}^d\times[-T-1,T+1]\to\mathbb{S}^2$ such that $\widetilde{s}\cdot\widetilde{v}\equiv 0$ and $\lim_{x\to\infty}\widetilde{v}(x,t)=Q'$ uniformly in $t$.
We regularize now the function $\widetilde{v}$. Let $\varphi:\mathbb{R}^d\times\mathbb{R}\to[0,\infty)$ denote a smooth function supported in the ball $\{(x,t):|x|^2+t^2\leq 1\}$ with $\int_{\mathbb{R}^d\times\mathbb{R}}\varphi\,dxdt=1$. Since $\widetilde{v}$ is a uniformly continuous function, there is $\epsilon=\epsilon(\widetilde{v})$ with the property that $$|\widetilde{v}(x,t)-(\widetilde{v}\ast\varphi_\epsilon)(x,t)|\leq 2^{-20}\text{ for any }(x,t)\in\mathbb{R}^d\times[-T-1/2,T+1/2],$$ where $\varphi_\epsilon(x,t)=\epsilon^{-d-1}\varphi(x/ \epsilon,t/ \epsilon)$. Using a partition of $1$, we replace smoothly $(\widetilde{v}\ast\varphi_\epsilon)(x,t)$ with $Q'$ for $|x|$ large enough. Thus we have constructed a smooth function $v':\mathbb{R}^d\times(-T-1/2,T+1/2)\to\mathbb{R}^3$ with the properties $$\label{t11}
\begin{cases}
&|v'(x,t)|\in[1-2^{-10},1+2^{-10}]\text{ for any }(x,t)\in\mathbb{R}^d\times[-T,T];\\
&|v'(x,t)\cdot s(x,t)|\leq 2^{-10} \text{ for any }(x,t)\in\mathbb{R}^d\times[-T,T];\\
&v'(x,t)=Q'\text{ for }|x|\text{ large enough and }t\in[-T,T].
\end{cases}$$
With $N$ as in , we define $$v(x,t)=N[v'(x,t),s(x,t)].$$ In view of , the continuous function $v:\mathbb{R}^d\times[-T,T]\to\mathbb{S}^2$ is well-defined, $s(x,t)\cdot v(x,t)\equiv 0$, and $$\label{t12}
\begin{cases}
&\partial_mv\in C([-T,T]:H^{\infty})\text{ for }m=1,\ldots,d;\\
&\partial_tv\in C([-T,T]:H^{\infty}).
\end{cases}$$
Given $s$ as in and $v$ as in , we define $$w(x,t)=s(x,t)\times v(x,t).$$ Since $H^\sigma$ is an algebra for $\sigma>d/2$, we have $$\label{t13}
\begin{cases}
&\partial_mw\in C([-T,T]:H^{\infty})\text{ for }m=1,\ldots,d;\\
&\partial_tw\in C([-T,T]:H^{\infty }).
\end{cases}$$ To summarize, given a function $s$ as in we have constructed continuous functions $v,w:\mathbb{R}^d\times[-T,T]\to\mathbb{S}^2$ such that $s\cdot v=s\cdot w=v\cdot w\equiv 0$, and and hold.
We use now the functions $v$ and $w$ to construct a suitable Coulomb gauge. Let $$A_m=(\partial_mv)\cdot w=-(\partial_mw)\cdot v\text{ for }m=1,\ldots,d.$$ Clearly, the functions $A_m$ are real-valued, $$\label{t20}
A_m\in C([-T,T]:H^{\infty})\text{ and }\partial_tA_m\in C([-T,T]:H^{\infty}).$$ We would like to modify the functions $v$ and $w$ such that $\sum_{m=1}^d\partial_mA_m\equiv 0$. Let $$\begin{cases}
&v'=(\cos \chi)v+(\sin\chi)w;\\
&w'=(-\sin\chi)v+(\cos\chi)w,
\end{cases}$$ for some function $\chi:\mathbb{R}^d\times[-T,T]\to\mathbb{R}$ to be determined. Then, using the orthonormality of $v$ and $w$ (which gives $\partial_mv\cdot v=\partial_mw\cdot w\equiv 0$), $$A'_m=(\partial_mv')\cdot w'=A_m+\partial_m\chi.$$ The condition $\sum_{m=1}^d\partial_mA'_m\equiv 0$ gives $$\Delta\chi=-\sum_{m=1}^d\partial_mA_m.$$ Thus we define $\chi$ by the formula $$\chi(x,t)=c\int_{\mathbb{R}^d}e^{ix\cdot \xi}|\xi|^{-2}\sum_{m=1}^d(i\xi_m)\,\mathcal{F}_{(d)}(A_m)(\xi,t)\,d\xi.$$ The integral defining the function $\chi$ converges absolutely since $A_m\in C([-T,T]:H^{\infty})$ and $d\geq 3$. Using , it follows that $\chi:\mathbb{R}^d\times[-T,T]\to\mathbb{R}$ is a bounded, continuous function, $\partial_m\chi\in C([-T,T]:H^{\infty})$ and $\partial_t\chi\in C([-T,T]:H^{\infty })$. To summarize, we proved the following proposition:
\[Lemmag1\][Proposition]{}
\[Lemmag3\] Assume $T\in[0,1]$, $Q\in\mathbb{S}^2$, and $$\label{t45}
\begin{cases}
&s\in C([-T,T]:H_Q^{\infty});\\
&\partial_ts\in C([-T,T]:H^{\infty}).
\end{cases}$$ Then there are continuous functions $v,w:\mathbb{R}^d\times[-T,T]\to\mathbb{S}^2$, $s\cdot v\equiv 0$, $w=s\times v$, such that $$\label{t30}
\partial_mv,\partial_mw\in C([-T,T]:H^{\infty})\text{ for }m=0,1,\ldots,d,\\$$ where $\partial_0=\partial_t$. In addition, $$\label{t40}
\text{ if }A_m=(\partial_mv)\cdot w\text{ for }m=1,\ldots,d,\text{ then }\sum_{j=1}^d\partial_mA_m\equiv 0.$$
Assume now that $s,v,w$ are as in Proposition \[Lemmag3\]. In addition to the functions $A_m$, we define the continuous functions $\psi_m:\mathbb{R}^d\times[-T,T]\to\mathbb{C}$, $m=1,\ldots,d$, $$\label{t41}
\psi_m=(\partial_ms)\cdot v+i(\partial_ms)\cdot w.$$ Let $\partial_0=\partial_t$. We also define the continuous functions $A_0:\mathbb{R}^d\times[-T,T]\to\mathbb{R}$ and $\psi_0:\mathbb{R}^d\times[-T,T]\to\mathbb{C}$, $$\label{t42}
\begin{cases}
&\psi_0=(\partial_0s)\cdot v+i(\partial_0s)\cdot w;\\
&A_0=(\partial_0v)\cdot w=-(\partial_0w)\cdot v.
\end{cases}$$ Clearly, $\psi_m,A_m\in C([-T,T]:H^\infty)$ for $m=0,1,\ldots,d$, and $\partial_t\psi_m,\partial_tA_m\in C([-T,T]:H^\infty)$ for $m=1,\ldots,d$. In view of the orthonormality of $s,v,w$, for $m=0,1,\ldots,d$ $$\label{t49}
\begin{cases}
&\partial_ms=\Re(\psi_m)v+\Im(\psi_m)w;\\
&\partial_mv=-\Re(\psi_m)s+A_mw;\\
&\partial_mw=-\Im(\psi_m)s-A_mv.
\end{cases}$$
A direct computation using the orthonormality of $s,v,w$ gives $$\label{t43}
(\partial_l+iA_l)\psi_m=(\partial_m+iA_m)\psi_l\text{ for any }m,l=0,1,\ldots,d.$$ A direct computation also shows that $$\label{t51}
\partial_lA_m-\partial_mA_l=\Im(\psi_l\,\overline{\psi}_m)\text{ for any }m,l=0,1,\ldots,d.$$ We combine these identities with the Coulomb gauge condition $\sum_{m=1}^d\partial_mA_m\equiv 0$ and solve the div-curl system for each $t$ fixed. The result is $$\label{t50}
\Delta A_m=-\sum_{l=1}^d\partial_l[\Im(\psi_m\,\overline{\psi}_l)]\text{ for }m=1,\ldots,d.$$ Thus, using , for $m=1,\ldots,d$, $$\label{t52}
A_m=\nabla^{-1}\big[\sum_{l=1}^dR_l[\Im(\psi_m\,\overline{\psi}_l)]\big],$$ where $R_l$ denotes the Riesz transform defined by the Fourier multiplier $\xi\to i\xi_l/|\xi|$ and $\nabla^{-1}$ is the operator defined by the Fourier multiplier $\xi\to|\xi|^{-1}$.
Assume now that the function $s$ satisfies the identity $$\label{t60}
\partial_ts=s\times\Delta s\,\text{ on }\,\mathbb{R}^d\times[-T,T],$$ in addition to . For $m=0,1,\ldots,d$ we define the covariant derivatives $D_m=\partial_m+iA_m$. Using the definition, $$\psi_0=(s\times\Delta s)\cdot v+i(s\times\Delta s)\cdot w.$$ In addition, using , $$\partial_m^2s=\big(\partial_m\Re(\psi_m)-A_m\cdot \Im(\psi_m)\big)v+\big(\partial_m\Im(\psi_m)+A_m\cdot \Re(\psi_m)\big)w-|\psi_m|^2s.$$ Thus, using $s\times v=w$, $s\times w=-v$, $$\label{t61}
\begin{split}
\psi_0&=-\sum_{m=1}^d\big(\partial_m\Im(\psi_m)+A_m\cdot \Re(\psi_m)\big)+i\sum_{m=1}^d\big(\partial_m\Re(\psi_m)-A_m\cdot \Im(\psi_m)\big)\\
&=i\sum_{m=1}^dD_m\psi_m.
\end{split}$$
We use now and to convert into a nonlinear Schrödinger equation. We rewrite the identities and in the form $$\begin{cases}
&D_l\psi_m=D_m\psi_l\,\text{ for any }m,l=0,1,\ldots,d;\\
&D_lD_mf-D_mD_lf=i\Im(\psi_l\overline{\psi}_m)f\,\text{ for any }m,l=0,1,\ldots,d.
\end{cases}$$ Thus, using , for $m=1,\ldots,d$, $$\begin{split}
D_0\psi_m&=D_m\psi_0=i\sum_{l=1}^dD_mD_l\psi_l=i\sum_{l=1}^dD_lD_m\psi_l-\sum_{l=1}^d\Im(\psi_m\overline{\psi}_l)\psi_l\\
&=i\sum_{l=1}^dD_lD_l\psi_m-\sum_{l=1}^d\Im(\psi_m\overline{\psi}_l)\psi_l.
\end{split}$$ Thus, using again , for $m=1,\ldots,d$, $$\label{t62}
(i\partial_t+\Delta_x)\psi_m=-2i\sum_{l=1}^dA_l\cdot \partial_l\psi_m+\big(A_0+\sum_{l=1}^dA_l^2\big)\psi_m-i\sum_{l=1}^d\Im(\psi_m\overline{\psi}_l)\psi_l.$$
We find now the coefficient $A_0$. Using and , $$\label{t70}
\Delta A_0=\sum_{l=1}^d\partial_l(\partial_0A_l+\Im(\psi_l\overline{\psi}_0))=\sum_{l=1}^d\partial_l\,\Im(\psi_l\overline{\psi}_0).$$ Using , and the identity $\overline{\psi}_l\cdot D_m\psi_m=\partial_m(\overline{\psi}_l\psi_m)-\psi_m\cdot\overline{D_m\psi_l}$, $$\begin{split}
\Im(\psi_l\overline{\psi}_0)&=-\sum_{m=1}^d\Re(\overline{\psi}_l\cdot D_m\psi_m)=-\sum_{m=1}^d\partial_m\Re(\overline{\psi}_l\psi_m)+\sum_{m=1}^d\Re(\psi_m\cdot\overline{D_m\psi_l})\\
&=-\sum_{m=1}^d\partial_m\Re(\overline{\psi}_l\psi_m)+\frac{1}{2}\partial_l\big(\sum_{m=1}^d\psi_m\overline{\psi}_m\big).
\end{split}$$ It follows from that $$\Delta A_0=-\sum_{m,l=1}^d\partial_l\partial_m\Re(\overline{\psi}_l\psi_m)+\frac{1}{2}\Delta\big(\sum_{m=1}^d\psi_m\overline{\psi}_m\big).$$ Thus $$\label{t75}
A_0=\sum_{m,l=1}^dR_lR_m\big(\Re(\overline{\psi}_l\psi_m)\big)+\frac{1}{2}\sum_{m=1}^d\psi_m\overline{\psi}_m.$$
\[Lemmag1\][Proposition]{}
\[Lemmag4\] Assume $s,v,w$, and $A_m$, $m=1,\ldots,d$ are as in Proposition \[Lemmag3\]. Assume in addition that the function $s$ satisfies the identity $$\partial_ts=s\times\Delta s\,\text{ on }\,\mathbb{R}^d\times[-T,T].$$ For $m=1,\ldots,d$ let $$\label{t80}
\psi_m=(\partial_ms)\cdot v+i(\partial_ms)\cdot w\text{ on }\mathbb{R}^d\times[-T,T].$$ Then $\psi_m,A_m,\partial_t\psi_m,\partial_tA_m\in C([-T,T]:H^{\infty})$ and $$\label{t81}
\begin{cases}
&(\partial_l+iA_l)\psi_m=(\partial_m+iA_m)\psi_l\text{ for any }m,l=1,\ldots,d;\\
&A_m=\nabla^{-1}\big[\sum_{l=1}^dR_l[\Im(\psi_m\,\overline{\psi}_l)]\big]\text{ for any }m=1,\ldots,d,
\end{cases}$$ where $R_l$ denotes the Riesz transform defined by the Fourier multiplier $\xi\to i\xi_l/|\xi|$ and $\nabla^{-1}$ is the operator defined by the Fourier multiplier $\xi\to |\xi|^{-1}$. In addition, the functions $\psi_m$ satisfy the system of nonlinear Schrödinger equations $$\label{t82}
(i\partial_t+\Delta_x)\psi_m=-2i\sum_{l=1}^dA_l\cdot \partial_l\psi_m+\big(A_0+\sum_{l=1}^dA_l^2\big)\psi_m+i\sum_{l=1}^d\Im(\psi_l\overline{\psi}_m)\psi_l,$$ for $m=1,\ldots,d$, where $$\label{t83}
A_0=\sum_{l,l'=1}^dR_lR_{l'}\big(\Re(\overline{\psi}_l\psi_{l'})\big)+\frac{1}{2}\sum_{l=1}^d\psi_l\overline{\psi}_l.$$
A quantitative estimate {#section2.3}
-----------------------
We prove now quantitative estimates for the functions $\psi_m$.
\[Lemmag1\][Lemma]{}
\[Lemmag5\] With the notation in Propositions \[Lemmag3\] and \[Lemmag4\], if the function $s_0(x)=s(x,0)$ has the additional property $\|s_0-Q\|_{\dot{H}^{d/2}}\leq1$ and $\sigma_0=d+10$ then, for $m=1,\ldots,d$, $$\label{t90}
\begin{cases}
&\|\psi_m(.,0)\|_{\dot{H}^{(d-2)/2}}\leq C\cdot \|s_0-Q\|_{\dot{H}^{d/2}};\\
&\|\psi_m(.,0)\|_{H^{\sigma'-1}}\leq C( \|s_0\|_{H^{\sigma'}_Q})\text{ for any }\sigma'\in[1,\sigma_0]\cap{\ensuremath{\mathbb{Z}}}.
\end{cases}$$
The main difficulty is that our construction does not give effective control of the Sobolev norms of $v$ and $w$ in terms of the norms of $s$. We argue indirectly, using a bootstrap argument and the identities , , and . For $\sigma\in [-1,\infty)$ let $\nabla^\sigma$ denote the operator (acting on functions in $H^{\infty}$) defined by the Fourier multiplier $\xi\to|\xi|^\sigma$. For $\sigma\in[-1/2,d/2]$ let $p_\sigma=d/(\sigma+1)$. Then, in view of the Sobolev imbedding theorem (recall $d\geq 3$), $$\label{t91}
\|\nabla^\sigma f\|_{L^{p_\sigma}}\leq C\|\nabla^{\sigma'}f\|_{L^{p_{\sigma'}}}\text{ if }-1/2\leq \sigma\leq\sigma'\leq d/2\text{ and }f\in H^{\infty}.$$
Let $s_0(x)=s(x,0)$, $v_0(x)=v(x,0)$, $w_0(x)=w(x,0)$, $\psi_{m,0}(x)=\psi_m(x,0)$, and $A_{m,0}(x)=A_m(x,0)$, and let $\epsilon_0=\|s_0-Q\|_{\dot{H}^{d/2}}\leq 1$. To start our bootstrap argument, we use , and the fact that $|v_0|=|w_0|=1$ to obtain $$\|\psi_{m,0}\|_{L^{p_0}}\leq C\epsilon_0\text{ for }m=1,\ldots,d.$$ Then, using , $$\|\nabla^1A_{m,0}\|_{L^{p_1}}\leq C\epsilon_0\text{ for }m=1,\ldots,d.$$ Thus, using , $\|A_{m,0}\|_{L^{p_0}}\leq C\epsilon_0$ for $m=1,\ldots,d$. We use now the identity and the fact that for $f\in H^{\infty }$ $$\label{t92}
\|\nabla^nf\|_{L^p}\approx \sum_{n_1+\ldots+n_d=n}\|\partial_1^{n_1}\ldots\partial_d^{n_d}f\|_{L^p}\text{ if }n\in\mathbb{Z}_+\text{ and }p\in[p_{d/2},p_{-1/2}].$$ Thus $$\|\nabla^1v_0\|_{L^{p_0}}+\|\nabla^1w_0\|_{L^{p_0}}\leq C\epsilon_0.$$ Therefore $$\label{t94}
\sum_{m=1}^d\|\psi_{m,0}\|_{L^{p_0}}+\sum_{m=1}^d\|\nabla^1A_{m,0}\|_{L^{p_1}}+\|\nabla^1v_0\|_{L^{p_0}}+\|\nabla^1w_0\|_{L^{p_0}}\leq C\epsilon_0.$$
We prove now that $$\label{t96}
\sum_{m=1}^d\|\nabla^n\psi_{m,0}\|_{L^{p_n}}+\sum_{m=1}^d\|\nabla^{n+1}A_{m,0}\|_{L^{p_{n+1}}}+\|\nabla^{n+1}v_0\|_{L^{p_n}}+\|\nabla^{n+1}w_0\|_{L^{p_n}}\leq C\epsilon_0,$$ for any $n\in\mathbb{Z}\cap[0,(d-2)/2]$. We argue by induction over $n$. The case $n=0$ was already proved in . Assume $n\geq 1$ and holds for any $n'\in[0,n-1]\cap\mathbb{Z}$. Using , , and the induction hypothesis $$\begin{split}
\|\nabla^n\psi_{m,0}\|_{L^{p_n}}&\leq C\|\nabla^{n+1}s_0\|_{L^{p_n}}\cdot \|v_0\|_{L^\infty}\\
&+C\sum_{n'=0}^{n-1}\|\nabla^{n-n'}s_0\|_{L^{p_{n-n'-1}}}\cdot \|\nabla^{n'+1}v_0\|_{L^{p_{n'}}},
\end{split}$$ which suffices to control the first term in the left-hand side of . For the second term, using and , $$\|\nabla^{n+1}A_{m,0}\|_{L^{p_{n+1}}}\leq C\sum_{l,l'=1}^d\sum_{n'=0}^{n}\|\nabla^{n'}\psi_{l,0}\|_{L^{p_{n'}}}\cdot \|\nabla^{n-n'}\psi_{l',0}\|_{L^{p_{n-n'}}},$$ which suffices in view of the induction hypothesis and the bound on the first term proved before. The bound on the last two terms in the left-hand side of follows in a similar way, using , , and the bound on the first two terms.
If $d$ is even then suffices to prove the first inequality in , simply by taking $n=(d-2)/2$. If $d$ is odd, the bounds with $n=(d-3)/2$ and give $$\label{t98}
\|\nabla^{\sigma+1}v_0\|_{L^{p_\sigma}}+\|\nabla^{\sigma+1}w_0\|_{L^{p_\sigma}}\leq C\epsilon_0\text{ for }\sigma\in[-1/2,(d-3)/2].$$ In view of the hypothesis and , we also have the bound $$\label{t99}
\|\nabla^{\sigma+1}s_0\|_{L^{p_\sigma}}\leq C\epsilon_0\text{ for }\sigma\in[-1/2,(d-2)/2].$$ We need the following Leibniz rule (a particular case of [@KePoVe Theorem A.8]): $$\label{Le}
\|\nabla^{1/2}(fg)-g\nabla^{1/2}f\|_{L^2}\leq C\|\nabla^{1/2}g\|_{L^{q_1}}\cdot \|f\|_{L^{q_2}}$$ if $1/q_1+1/q_2=1/2$ and $q_1,q_2\in[p_{d/2},p_{-1/2}]$. Then, using and $$\|\nabla^{(d-2)/2}\psi_{m,0}\|_{L^2}\leq C\sum_{u_0\in\{v_0,w_0\}}\sum_{n=0}^{(d-3)/2}\|\nabla^{1/2}(\partial_mD^ns_0\cdot D^{(d-3)/2-n}u_0)\|_{L^2},$$ where $D^n$ denotes any derivative of the form $\partial_1^{n_1}\ldots\partial_d^{n_d}$, with $n_1+\ldots+n_d=n$. The first inequality in then follows from , , and the fact that $|u_0|\equiv 1$.
For the second inequality in , we notice first that $\|\psi_{m,0}\|_{H^0}\leq C\cdot \|s_0\|_{H^{1}_Q}$, since $|v_0|=|w_0|\equiv 1$. In view of the first inequality in , we may assume $\sigma'\geq (d+1)/2$. We use a similar argument as before: the bootstrap inequality that replaces is $$\label{t96.1}
\begin{split}
\sum_{m=1}^d\|\nabla^n&\psi_{m,0}\|_{L^2\cap L^{p_{n-\sigma'+d/2}}}+\sum_{m=1}^d\|\nabla^{n}A_{m,0}\|_{L^2\cap L^{p_{n-\sigma'+d/2}}}\\
&+\sum_{u_0\in \{v_0,w_0\}}\|\nabla^{n+1}u_0\|_{L^2\cap L^{p_{n-\sigma'+d/2}}}\leq C( \|s_0\|_{H^{\sigma'}_Q}),
\end{split}$$ for any $n\in[0,\sigma'-1]\cap{\ensuremath{\mathbb{Z}}}$, where $p_\sigma=p_{-1/2}=2d$ if $\sigma\leq-1/2$. As before, the bound follows by induction over $n$, using the identities , , and , and the inequalities , , and $$\sum_{n_1+\ldots+n_d\leq \sigma'-(d+1)/2}||\partial_1^{n_1}\ldots\partial_d^{n_d}s_0||_{L^\infty}\leq C( \|s_0\|_{H^{\sigma'}_Q}).$$ The second inequality in follows from the bound with $n=\sigma'-1$.
Perturbative analysis of the modified Schrödinger map {#section3}
=====================================================
In this section we analyze the Schrödinger map system derived in Propositions \[Lemmag3\] and \[Lemmag4\]. In the rest of this section we assume $d\geq 3$; this restriction is used implicitly in many estimates.
The resolution spaces and their properties {#section3.1}
------------------------------------------
In this subsection we define our main normed spaces and summarize some of their basic properties. These resolution spaces have been used in [@IoKe2] and, with slight modifications, in [@IoKe3], and we will refer to these papers for most of the proofs.
Let $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier transform and the inverse Fourier transform operators on $L^2(\mathbb{R}^{d+1})$. For $l=1,\ldots,d$ let $\mathcal{F}_{(l)}$ and $\mathcal{F}_{(l)}^{-1}$ denote the Fourier transform and the inverse Fourier transform operators on $L^2(\mathbb{R}^l)$. We fix $\eta_0:\mathbb{R}\to[0,1]$ a smooth even function supported in the set $\{\mu\in\mathbb{R}:|\mu|\leq 8/5\}$ and equal to $1$ in the set $\{\mu\in\mathbb{R}:|\mu|\leq 5/4\}$. Then we define $\eta_j:\mathbb{R}\to[0,1]$, $j=1,2,\ldots$, $$\label{gu1}
\eta_j(\mu)=\eta_0(\mu/2^j)-\eta_0(\mu/2^{j-1}),$$ and $\eta_k^{(d)}:\mathbb{R}^d\to[0,1]$, $k\in{\ensuremath{\mathbb{Z}}}$, $$\label{gu1.1}
\eta_k^{(d)}(\xi )=\eta_0( |\xi|/2^k)-\eta_0( |\xi|/2^{k-1}).$$ For $j\in{\ensuremath{\mathbb{Z}}}_+$, we also define $\eta_{\leq j}=\eta_0+\ldots+\eta_j$.
For $k\in\mathbb{Z}$ let $I_k^{(d)}=\{\xi\in{\ensuremath{\mathbb{R}}}^d:|\xi|\in[2^{k-1},2^{k+1}]\}$; for $j\in{\ensuremath{\mathbb{Z}}}_+$ let $I_j=\{\mu\in{\ensuremath{\mathbb{R}}}:|\mu|\in[2^{j-1},2^{j+1}]\}$ if $j\geq 1$ and $I_j=[-2,2]$ if $j=0$. For $k\in{\ensuremath{\mathbb{Z}}}$ and $j\in{\ensuremath{\mathbb{Z}}}_+$ let $$D_{k,j}=\{(\xi,\tau)\in\mathbb{R}^d\times\mathbb{R}:\xi \in I_k^{(d)}\text{ and }|\tau+|\xi|^2|\in I_j\}\text{ and }D_{k,\leq j}=\bigcup\limits_{0\leq j'\leq j}D_{k,j'}.$$
For $k\in{\ensuremath{\mathbb{Z}}}$ we define first the normed spaces $$\label{v1}
\begin{split}
X_k=\{f\in L^2(\mathbb{R}^d\times&\mathbb{R}):f\text{ supported in }I_k^{(d)}\times\mathbb{R}\text { and } \\
&\|f\|_{X_k}=\sum_{j=0}^\infty 2^{j/2}\|\eta_j(\tau+|\xi|^2)\cdot f\|_{L^2}<\infty\}.
\end{split}$$ The spaces $X_k$ are not sufficient for our estimates, due to various logarithmic divergences. For any vector $\mathbf{e}\in\mathbb{S}^{d-1}$ let $$P_{\mathbf{e}}=\{\xi\in\mathbb{R}^d:\xi\cdot\mathbf{e}=0\}$$ with the induced Euclidean measure. For $p,q\in[1,\infty]$ we define the normed spaces $L^{p,q}_{\mathbf{e}}=L^{p,q}_{\mathbf{e}}(\mathbb{R}^d\times\mathbb{R})$, $$\label{vv1}
\begin{split}
L^{p,q}_{\mathbf{e}}&=\{f\in L^2(\mathbb{R}^d\times \mathbb{R}):\\
&\|f\|_{L^{p,q}_{\mathbf{e}}}=\Big[\int_{\mathbb{R}}\Big[\int_{P_\mathbf{e}\times \mathbb{R}}|f(r\mathbf{e}+v,t)|^q\,dvdt\Big]^{p/q}\,dr\Big]^{1/p}<\infty\}.
\end{split}$$ For $k\in{\ensuremath{\mathbb{Z}}}$ and $j\in \mathbb{Z}_+$ let $$D_{k,j}^{\mathbf{e}}=\{(\xi,\tau)\in D_{k,j}:\xi\cdot\mathbf{e}\geq 2^{k-20}\}\text{ and }D_{k,\leq j}^{\mathbf{e}}=\bigcup_{0\leq j'\leq j}D_{k,j}^{\mathbf{e}}.$$ For $k\geq 100$ and $\mathbf{e}\in\mathbb{S}^{d-1}$, we define the normed spaces $$\label{v2}
\begin{split}
Y_k^{\mathbf{e}}&=\{f\in L^2(\mathbb{R}^d\times\mathbb{R}):f\text{ supported in }D_{k,\leq 2k+10}^{\mathbf{e}}\text { and } \\
&\|f\|_{Y_k^{\mathbf{e}}}=2^{-k/2}\|\mathcal{F}^{-1}[(\tau+|\xi|^2+i)\cdot f]\|_{L^{1,2}_{\mathbf{e}}}<\infty\}.
\end{split}$$ For simplicity of notation, we also define $Y_k^{\mathbf{e}}=\{0\}$ for $k\leq 99$.
We fix $L=L(d)$ large and $\mathbf{e}_1,\ldots,\mathbf{e}_L\in\mathbb{S}^{d-1}$, $\mathbf{e}_l\neq \mathbf{e}_{l'}$ if $l\neq l'$, such that $$\label{vm4}
\text{ for any }\mathbf{e}\in\mathbb{S}^{d-1}\text{ there is }l\in\{1,\ldots,L\}\text{ such that }|\mathbf{e}-\mathbf{e}_l|\leq 2^{-100}.$$ We assume in addition that if $\mathbf{e}\in\{\mathbf{e}_1,\ldots,\mathbf{e}_L\}$ then $-\mathbf{e}\in\{\mathbf{e}_1,\ldots,\mathbf{e}_L\}$. For $k\in\mathbb{Z}$ we define the normed spaces $$\label{v3'}
Z_k=X_k+Y_{k}^{\mathbf{e}_1}+\ldots+Y_k^{\mathbf{e}_L}.$$ The spaces $Z_k$ are our main normed spaces.
For $k\in {\ensuremath{\mathbb{Z}}}_+$ let $\Xi_k=2^k\cdot{\ensuremath{\mathbb{Z}}}^d$. Let $\chi^{(1)}:\mathbb{R}\to[0,1]$ denote an even smooth function supported in the interval $[-2/3,2/3]$ with the property that $$\sum_{n\in{\ensuremath{\mathbb{Z}}}}\chi^{(1)}(\xi-n)\equiv 1\text{ on }{\ensuremath{\mathbb{R}}}.$$ Let $\chi:\mathbb{R}^d\to[0,1]$, $\chi(\xi)=\chi^{(1)}(\xi_1)\cdot\ldots\cdot \chi^{(1)}(\xi_d)$. For $k\in {\ensuremath{\mathbb{Z}}}_+$ and $n\in\Xi_k$ let $$\chi_{k,n}(\xi)=\chi((\xi-n)/2^k).$$ Clearly, $\sum_{n\in\Xi_k}\chi_{k,n}\equiv 1$ on ${\ensuremath{\mathbb{R}}}^d$.
We summarize now some of the main properties of the spaces $Z_k$.
\[section\]
\[Lemmas1\] (a) If $k\in{\ensuremath{\mathbb{Z}}}$, $m\in L^\infty(\mathbb{R}^d)$, $\mathcal{F}_{(d)}^{-1}(m)\in L^1(\mathbb{R}^d)$, and $f\in Z_k$, then $m(\xi)\cdot f\in Z_k$ and $$\label{im1}
||m(\xi)\cdot f||_{Z_k}\leq C||\mathcal{F}_{(d)}^{-1}(m)||_{L^1(\mathbb{R}^d)}\cdot ||f||_{Z_k}.$$
\(b) If $k\in{\ensuremath{\mathbb{Z}}}$, $j\in\mathbb{Z}_+$ and $f\in Z_k$ then $$\label{im2}
\|f\cdot \eta_j(\tau+|\xi|^2)\|_{X_k}\leq C\|f\|_{Z_k}.$$
\(c) If $k\in{\ensuremath{\mathbb{Z}}}$, $j\in{\ensuremath{\mathbb{Z}}}_+$, and $f\in Z_k$ then $$\label{im3}
||\eta_{\leq j}(\tau+|\xi|^2)\cdot f||_{Z_k}\leq C||f||_{Z_k}.$$
\(d) If $k\in{\ensuremath{\mathbb{Z}}}$ and $f$ is supported in $D^{{\ensuremath{\mathbf{e}}}}_{k,\leq \infty}$ for some ${\ensuremath{\mathbf{e}}}\in\{{\ensuremath{\mathbf{e}}}_1,\ldots,{\ensuremath{\mathbf{e}}}_L\}$ then $$\label{im4}
||f||_{Z_k}\leq C 2^{-k/2}||\mathcal{F}^{-1}
[(\tau+|\xi|^2+i)\cdot f]||_{L^{1,2}_{\ensuremath{\mathbf{e}}}}.$$
\(e) (Energy estimate) If $k\in{\ensuremath{\mathbb{Z}}}$ and $f\in Z_k$ then $$\label{im5}
\sup_{t\in\mathbb{R}}\|\mathcal{F}^{-1}(f)(.,t)\|_{L^2_x}
\leq C\|f\|_{Z_k}.$$
\(f) (Localized maximal function estimate) If $k\in{\ensuremath{\mathbb{Z}}}$, $k'\in(-\infty,k+10d]\cap{\ensuremath{\mathbb{Z}}}$, $f\in Z_k$, and $\mathbf{e}'\in\mathbb{S}^{d-1}$ then $$\label{im7}
\big[\sum_{n\in\Xi_{k'}}||\mathcal{F}^{-1}(\chi_{k',n}(\xi)\cdot \widetilde{f})||_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}^2\big]^{1/2}\leq C2^{(d-1)k/2}\cdot 2^{-(d-2)(k-k')/2}(1+|k-k'| )\cdot \|f\|_{Z_k},$$ where $\mathcal{F}^{-1}(\widetilde{f})\in\{\mathcal{F}^{-1}(f),\overline{\mathcal{F}^{-1}(f)}\}$ .
\(g) (Local smoothing estimate) If $k\in {\ensuremath{\mathbb{Z}}}$, $\mathbf{e}'\in\mathbb{S}^{d-1}$, $l\in[-1,40]\cap{\ensuremath{\mathbb{Z}}}$, and $f\in Z_k$ then $$\label{im8}
\|\mathcal{F}^{-1}[\widetilde{f}\cdot
\eta_{1}(\xi\cdot\mathbf{e}'/2^{k-l})]\|_{L^{\infty,2}_{\mathbf{e}'}}\leq
C2^{-k/2}\|f\|_{Z_k},$$ where $\mathcal{F}^{-1}(\widetilde{f})\in\{\mathcal{F}^{-1}(f),\overline{\mathcal{F}^{-1}(f)}\}$.
The bound follows directly from the definitions. The bound is proved in [@IoKe3 Lemma 2.1]. The bound is proved in [@IoKe3 Lemma 2.3]. The bound follows from the estimate (2.15) in [@IoKe3]. The energy estimate is proved in [@IoKe3 Lemma 2.2]. The localized maximal function estimate follows from [@IoKe3 Lemma 4.1] and . Finally, the local smoothing estimate is proved in [@IoKe3 Lemma 4.2].
The estimate in part (f) with $k'=k$ will often be referred to as the “global ”. For $k'\leq k-C$ we refer to this estimate as the “localized ”.
Linear estimates {#section3.2}
----------------
We fix a large constant $\sigma_0$, say $$\label{me1}
\sigma_0=d+10.$$ For $\sigma\in[(d-2)/2,\sigma_0-1]$ we define the normed space $$\label{no5}
\begin{split}
\dot{F}^\sigma&=\{u\in C(\mathbb{R}:H^{\infty}):\\
&\|u\|_{\dot{F}^\sigma}=\big[\sum_{k\in{\ensuremath{\mathbb{Z}}}}(2^{2\sigma k}+2^{(d-2)k})\,\|\eta_k^{(d)}(\xi)\cdot \mathcal{F}(u)\|^2_{Z_k}\big]^{1/2}<\infty\}.
\end{split}$$ For $\sigma\in[(d-2)/2,\sigma_0-1]$, $T\in[0,1]$, $u\in C([-T,T]:H^{\infty})$, and $T'\in[0,T]$ we define $$\label{no6.1}
E_{T'}(u)(t)=
\begin{cases}
u(t)&\text{ if }|t|\leq T';\\
0&\text{ if }|t|>T',
\end{cases}$$ and $$\label{no6}
\begin{split}
\|u\|_{\dot{N}^\sigma[-T',T']}=\big[\sum_{k\in{\ensuremath{\mathbb{Z}}}}(2^{2\sigma k}+2^{(d-2)k})\,\|\eta_k^{(d)}(\xi)\cdot (\tau+|\xi|^2+i)^{-1}\cdot \mathcal{F}(E_{T'}u)\|_{Z_k}^2\big]^{1/2}.
\end{split}$$ The definition shows that if $k\in{\ensuremath{\mathbb{Z}}}$ and $f$ is supported in $I_k^{(d)}\times{\ensuremath{\mathbb{R}}}$ then $$\|(\tau+|\xi|^2+i)^{-1}\cdot f\|_{Z_k}\leq C\|f\|_{L^2},$$ thus, for $\sigma\in[(d-2)/2,\sigma_0-1]$ and $T_1,T_2\in[0,T]$ $$\label{no6.2}
\big|\|u\|_{\dot{N}^\sigma[-T_1,T_1]}-\|u\|_{\dot{N}^\sigma[-T_2,T_2]}\big|\leq C|T_1-T_2|^{1/2}\cdot \sup_{t\in[-T,T]}\|u(.,t)\|_{H^\sigma}.$$
For $\phi\in H^{\sigma}$ let $W(t)(\phi)\in
C(\mathbb{R}:H^{\sigma})$ denote the solution of the free Schrödinger evolution.
\[Lemmas1\][Proposition]{}
\[Lemmaqq1\] If $\sigma\in [(d-2)/2,\sigma_0-1]$ and $\phi\in H^{\infty}$ then $$\|\eta_0(t)\cdot W(t)(\phi)\|_{\dot{F}^{\sigma}}\leq C(\|\phi\|_{\dot{H}^\sigma}+\|\phi\|_{\dot{H}^{(d-2)/2}}).$$
See [@IoKe3 Lemma 3.1] for the proof.
\[Lemmas1\][Proposition]{}
\[Lemmaq3\] If $\sigma\in[(d-2)/2,\sigma_0-1]$, $T\in[0,1]$, and $u\in C([-T,T]:H^{\infty})$ then $$\Big|\Big|\eta_0(t)\cdot \int_0^tW(t-s)(E_T(u)(s))\,ds\Big|\Big|_{\dot{F}^{\sigma}}\leq C||u||_{\dot{N}^{\sigma}[-T,T]},$$ where $E_T(u)$ is defined in .
See [@IoKe3 Lemma 3.2] for the proof.
Nonlinear estimates {#section3.3}
-------------------
In this subsection we assume that $d\geq 4$. Assume that $T\in[0,1]$ and $\psi_m\in C([-T,T]:H^{\infty })$, $m=1,\ldots,d$. Let $\Psi=(\psi_1,\ldots,\psi_d)$ and define $$\label{bh1}
\begin{cases}
&A_0=\sum_{l,l'=1}^dR_lR_{l'}\big(\Re(\overline{\psi}_l\psi_{l'})\big)+\frac{1}{2}\sum_{l=1}^d\psi_l\overline{\psi}_l;\\
&A_m=\nabla^{-1}\big[\sum_{l=1}^dR_l[\Im(\psi_m\,\overline{\psi}_l)]\big]\text{ for any }m=1,\ldots,d,
\end{cases}$$ and $$\label{bh2}
\mathcal{N}_m(\Psi)=-2i\sum_{l=1}^dA_l\cdot \partial_l\psi_m+\big(A_0+\sum_{l=1}^dA_l^2\big)\psi_m+i\sum_{l=1}^d\Im(\psi_l\overline{\psi}_m)\psi_l.$$ Clearly, $A_m,\mathcal{N}_m(\Psi)\in C([-T,T]:H^{\infty})$ (recall that $d\geq 3$). We assume also that on $\mathbb{R}^d\times[-T,T]$ we have the integral equation $$\label{inte}
\psi_m(t)=W(t)(\psi_{m,0})+\int_0^tW(t-s)(\mathcal{N}_m(\Psi)(s))\,ds,$$ where $\psi_{m,0}=\psi_m(0)$. In dimensions $d\geq 4$ we will not need the compatibility conditions $$(\partial_l+iA_l)\psi_m=(\partial_m+iA_m)\psi_l\text{ for any }m,l=1,\ldots,d.$$
We define the extensions $\widetilde{E}_T(\psi_m)\in C({\ensuremath{\mathbb{R}}}:H^{\infty})$, $m=1,\ldots,d$, $$\label{ty4}
\widetilde{E}_T(\psi_m)(t)=\eta_0(t)\cdot W(t)(\psi_{m,0})+\eta_0(t)\cdot \int_0^tW(t-s)(E_T(\mathcal{N}_m(\Psi))(s))\,ds.$$ Using Propositions \[Lemmaqq1\] and \[Lemmaq3\], for $\sigma\in[(d-2)/2,\sigma_0-1]$ $$\|\widetilde{E}_T(\psi_m)\|_{\dot{F}^\sigma}\leq C\cdot ( \|\psi_{m,0}\|_{\dot{H}^\sigma\cap\dot{H}^{(d-2)/2}}+\|\mathcal{N}_m(\Psi)\|_{\dot{N}^\sigma[-T,T]}).$$ Let $\widetilde{E}_T(\Psi)=(\widetilde{E}_T(\psi_1),\ldots,\widetilde{E}_T(\psi_d))$. For $\sigma\in[(d-2)/2,\sigma_0-1]$ let $$\label{bh2.1}
\|\widetilde{E}_T(\Psi)\|_{\dot{F}^\sigma}=\sum_{m=1}^d\|\widetilde{E}_T(\psi_m)\|_{\dot{F}^\sigma}.$$ The main result of this subsection is the following proposition.
\[Lemmas1\][Proposition]{}
\[Lemmaq7\] Assume $d\geq 4$. Then, for any $\sigma\in[(d-2)/2,\sigma_0-1]$ and $m=1,\ldots,d$, $$\label{vu20}
\|\mathcal{N}_m(\Psi)\|_{\dot{N}^\sigma[-T,T]}\leq C\|\widetilde{E}_T(\Psi)\|_{\dot{F}^\sigma}( \|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}^2+\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}^4 ).$$
The rest of this subsection is concerned with the proof of Proposition \[Lemmaq7\]. For $\sigma\in[(d-2)/2,\sigma_0-1]$ and $k\in{\ensuremath{\mathbb{Z}}}$ let $$\label{go1}
\beta_k(\sigma)=\sum_{m=1}^d\sum_{k'\in{\ensuremath{\mathbb{Z}}}}2^{-|k-k'|/10}\cdot (2^{\sigma k'}+2^{(d-2)k'/2})\|\eta_{k'}^{(d)}(\xi)\cdot \mathcal{F}(\widetilde{E}_T(\psi_m))\|_{Z_{k'}}.$$ Clearly, $\beta_{k_1}(\sigma)\leq C2^{|k_1-k_2|/10}\beta_{k_2}(\sigma)$ for any $k_1,k_2\in{\ensuremath{\mathbb{Z}}}$, and $$[\sum_{k\in{\ensuremath{\mathbb{Z}}}}\beta_k(\sigma)^2]^{1/2}\leq C\|\widetilde{E}_T(\Psi)\|_{\dot{F}^\sigma}\text{ for any }\sigma\in[(d-2)/2,\sigma_0-1].$$
For $k\in {\ensuremath{\mathbb{Z}}}$ let $P_{k}$ denote the operator defined by the Fourier multiplier $(\xi,\tau)\to\eta_k^{(d)}(\xi)$, and let $P_{\leq k}=\sum_{k'\leq k}P_{k'}$. For $k\in{\ensuremath{\mathbb{Z}}}$ and $n\in \Xi_k$ let $\widetilde{P}_{k,n}$ denote the operator defined by the Fourier multiplier $(\xi,\tau)\to\chi_{k,n}(\xi)$.
\[Lemmas1\][Lemma]{}
\[Lemmaq5\] If $d\geq 4$, $k\in{\ensuremath{\mathbb{Z}}}$, ${\ensuremath{\mathbf{e}}}'\in\mathbb{S}^{d-1}$, $\sigma\in[(d-2)/2,\sigma_0-1]$, and $$\label{ty1}
F\in\{E_T(A_0),E_T(A_m^2),E_T(\widetilde{\psi}_m\cdot\widetilde{\psi}_l):m,l=1,\ldots,d,\widetilde{\psi}\in\{\psi,\overline{\psi}\}\}$$ then $$\label{vu2}
(2^{\sigma k}+2^{(d-2)k/2})\|P_{k}(F)\|_{L^2}\leq C\beta_k(\sigma)\cdot (||\widetilde{E}_T(\Psi)||_{\dot{F}^{(d-2)/2}}+\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}^3),$$ and $$\label{vu1}
\|P_{\leq k}(F)\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}\leq C2^k(||\widetilde{E}_T(\Psi)||^2_{\dot{F}^{(d-2)/2}}+\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}^4).$$ In addition, for $m=1,\ldots,d$, $$\label{vu3}
(2^{\sigma k}+2^{(d-2)k/2})\|P_{k}(E_T(A_m))\|_{L^2}\leq C2^{-k}\beta_k(\sigma)\cdot \|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}},$$ and $$\label{vu10}
\|P_{\leq k}(E_T(A_m))\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}\leq C\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}^2.$$
The main reason we assume $d\geq 4$ (rather than $d\geq 3$) is to have a simple proof of . We defer the proof of Lemma \[Lemmaq5\] to section \[section5\], and complete now the proof of Proposition \[Lemmaq7\]. For it suffices to prove that $$\label{vu21}
\begin{split}
(2^{\sigma k}+2^{(d-2)k/2})&\|(\tau+|\xi|^2+i)^{-1}\cdot \mathcal{F}(P_k(E_T(\mathcal{N}_m(\Psi))))\|_{Z_k}\\
&\leq C\beta_k(\sigma)\cdot ( \|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}^2+\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}^4)
\end{split}$$ for any $k\in{\ensuremath{\mathbb{Z}}}$. Since $E_T(\mathcal{N}_m(\Psi))$ is a sum of terms of the form $F\cdot \widetilde{E}_T(\psi_m)$ and $E_T(A_l)\cdot\partial_l\widetilde{E}_T(\psi_m)$, where $F$ is as in , it suffices to prove that $$\label{ty2}
\begin{split}
&(2^{\sigma k}+2^{(d-2)k/2})\|(\tau+|\xi|^2+i)^{-1}\cdot \mathcal{F}(P_k(F\cdot \widetilde{E}_T(\psi_m)))\|_{Z_k}\\
&+(2^{\sigma k}+2^{(d-2)k/2})\|(\tau+|\xi|^2+i)^{-1}\cdot \mathcal{F}(P_k(E_T(A_l)\cdot \partial_l\widetilde{E}_T(\psi_m)))\|_{Z_k}
\end{split}$$ is dominated by the right-hand side of for any $m,l=1,\ldots,d$. We always estimate the expressions in using .
We consider first the term $F\cdot \widetilde{E}_T(\psi_m)$, and write $P_k(F\cdot \widetilde{E}_T(\psi_m))$ as $$\label{vu40}
\sum_{|k_1-k|\leq 2}P_k[P_{\leq k-10}(F)\cdot P_{k_1}(\widetilde{E}_T(\psi_m))]+\sum_{k_1\geq k-9}P_k[P_{k_1}(F)\cdot P_{\leq k_1+20}(\widetilde{E}_T(\psi_m))].$$ Let $c_\sigma(k)=2^{\sigma k}+2^{(d-2)k/2}$. To control the term in the first line of it suffices to prove that for any $v\in I_k^{(d)}$, the quantities $$\label{vu41}
\sum_{|k_1-k|\leq 2}c_\sigma(k)\|\eta_0( |\xi-v|/2^{k-50})(\tau+|\xi|^2+i)^{-1}\mathcal{F}(P_k[P_{\leq k-10}(F)\cdot P_{k_1}(\widetilde{E}_T(\psi_m))])\|_{Z_k}$$ and $$\label{vu41.1}
\sum_{k_1\geq k-9}c_\sigma(k)\|\eta_0( |\xi-v|/2^{k-50})(\tau+|\xi|^2+i)^{-1}\mathcal{F}(P_k[P_{k_1}(F)\cdot P_{\leq k_1+20}(\widetilde{E}_T(\psi_m))])\|_{Z_k}$$ are dominated by the right-hand side of .
To bound the expression in , we may assume that $\mathcal{F}(P_{k_1}(\widetilde{E}_T(\psi_m)))$ is supported in $I_{k_1}^{(d)}\times{\ensuremath{\mathbb{R}}}\cap
\{(\xi,\tau):|\xi-w| \leq 2^{k_1-50}\}$ for some $w\in
I_{k_1}^{(d)}$. We use the following simple geometric observation (cf. [@IoKe3 Section 8]): if $\widehat{v},\widehat{w}\in\mathbb{S}^{d-1}$ then there is $\mathbf{e}\in\{\mathbf{e}_1,\ldots,\mathbf{e}_L\}$ such that $$\label{ma2}
\mathbf{e}\cdot\widehat{v}\geq 2^{-5}\text{ and }|\mathbf{e}\cdot\widehat{w}|\geq 2^{-5}.$$ We fix ${\ensuremath{\mathbf{e}}}$ as in (with $\widehat{v}=v/|v|$ and $\widehat{w}=w/|w|$). Using , the expression in is dominated by $$\begin{split}
&Cc_\sigma(k)\sum_{|k_1-k|\leq 2}2^{-k/2}\|P_{\leq k-10}(F)\cdot P_{k_1}(\widetilde{E}_T(\psi_m))\|_{L^{1,2}_{\ensuremath{\mathbf{e}}}}\\
&\leq Cc_\sigma(k)\sum_{|k_1-k|\leq 2}2^{-k/2}\|P_{\leq k-10}(F)\|_{L^{1,\infty}_{\ensuremath{\mathbf{e}}}}\cdot\| P_{k_1}(\widetilde{E}_T(\psi_m))\|_{L^{\infty,2}_{\ensuremath{\mathbf{e}}}},
\end{split}$$ which suffices, in view of and .
To bound the expression in , we fix ${\ensuremath{\mathbf{e}}}\in\{{\ensuremath{\mathbf{e}}}_1,\ldots,{\ensuremath{\mathbf{e}}}_l\}$ such that $|{\ensuremath{\mathbf{e}}}-v/|v||\leq 2^{-100}$ and use . The second sum in is dominated by $$\begin{split}
&Cc_\sigma(k)\sum_{k_1\geq k-9}2^{-k/2}\|P_k[P_{k_1}(F)\cdot P_{\leq k_1+20}(\widetilde{E}_T(\psi_m))]\|_{L^{1,2}_{\ensuremath{\mathbf{e}}}}\\
&\leq Cc_\sigma(k)\sum_{k_1\geq k-9}2^{-k/2}\negmedspace\negmedspace\negmedspace\sum_{n,n'\in\Xi_k\text{ and }|n-n'|\leq C2^k}\|\widetilde{P}_{k,n}P_{k_1}(F)\cdot \widetilde{P}_{k,n'}P_{\leq k_1+20}(\widetilde{E}_T(\psi_m))]\|_{L^{1,2}_{\ensuremath{\mathbf{e}}}}\\
&\leq Cc_\sigma(k)\sum_{k_1\geq k-9}2^{-k/2}\|P_{k_1}(F)\|_{L^2}\big[\sum_{n'\in\Xi_k}\|\widetilde{P}_{k,n'}P_{\leq k_1+20}(\widetilde{E}_T(\psi_m))]\|_{L^{2,\infty}_{\ensuremath{\mathbf{e}}}}^2\big]^{1/2}\\
&\leq Cc_\sigma(k)\sum_{k_1\geq k-9}2^{-k/2}\frac{\beta_{k_1}(\sigma)\cdot M}{c_\sigma(k_1)}\cdot 2^{k_1/2}2^{-|k_1-k|/4}\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}},
\end{split}$$ where $M=(||\widetilde{E}_T(\Psi)||_{\dot{F}^{(d-2)/2}}+\|\widetilde{E}_T(\Psi)\|^3_{\dot{F}^{(d-2)/2}})$, and we used the localized and in the last estimate. This suffices since $\beta_{k_1}(\sigma)\leq C2^{|k_1-k|/10}\beta_{k}(\sigma)$ and $d\geq 4$.
We consider now $E_T(A_l)\cdot \partial_l\widetilde{E}_T(\psi_m)$. We write $P_k(E_T(A_l)\cdot\partial_l\widetilde{E}_T(\psi_m))$ as $$\begin{split}
&\sum_{|k_1-k|\leq 2}P_k[P_{\leq k-10}(E_T(A_l))\cdot P_{k_1}(\partial_l\widetilde{E}_T(\psi_m))]\\
&+\sum_{k_1\geq k-9}P_k[P_{k_1}(E_T(A_l))\cdot P_{\leq k_1+20}(\partial_l\widetilde{E}_T(\psi_m))],
\end{split}$$ and argue as before, using and instead of and .
Proof of Theorem \[Main1\] {#section4}
==========================
In this section we assume $d\geq 4$.
A priori estimates {#section4.1}
------------------
In this subsection we prove the following:
\[section\]
\[Lemmad1\] Assume that $\sigma_0=d+10$ is as in , $T\in[0,1]$ and $s\in C([-T,T]:H^{\infty }_Q)$ is a solution of the initial-value problem $$\label{fi0}
\begin{cases}
&\partial_ts=s\times\Delta s\,\,\text{ on }\,\,\mathbb{R}^d\times[-T,T];\\
&s(0)=s_0.
\end{cases}$$ If $\|s_0-Q\|_{\dot{H}^{d/2}}\leq\varepsilon_0\ll 1$ then $$\label{fi20}
\begin{cases}
\sup\limits_{t\in[-T,T]}\|s(t)-Q\|_{\dot{H}^{d/2}}\leq C\|s_0-Q\|_{\dot{H}^{d/2}};\\
\sup\limits_{t\in[-T,T]}\|s(t)\|_{H^{\sigma'}_Q}\leq C( \|s_0\|_{H^{\sigma'}_Q})\text{ for any }\sigma'\in[0,\sigma_0]\cap{\ensuremath{\mathbb{Z}}}.
\end{cases}$$
We construct $\psi_m,A_m\in C([-T,T]:H^{\infty})$ as in Proposition \[Lemmag4\]. In view of Lemma \[Lemmag5\], $$\label{fi8}
||\psi_{m,0}||_{\dot{H}^{(d-2)/2}}\leq C||s_0-Q||_{\dot{H}^{d/2}}\leq C\varepsilon_0.$$ For any $T'\in[0,T]$ we define the functions $E_{T'}(\mathcal{N}_m(\Psi))$ and $\widetilde{E}_{T'}(\psi_m)$ as in and . Using Propositions \[Lemmaqq1\] and \[Lemmaq3\], for $\sigma\in[(d-2)/2,\sigma_0-1]$ and $T'\in[0,T]$, $$\label{fi1}
\|\widetilde{E}_{T'}(\Psi)\|_{\dot{F}^\sigma}\leq C\cdot ( \sum_{m=1}^d\|\psi_{m,0}\|_{\dot{H}^\sigma\cap\dot{H}^{(d-2)/2}}+\sum_{m=1}^d\|\mathcal{N}_m(\Psi)\|_{\dot{N}^\sigma[-T',T']}).$$ In addition, using Lemma \[Lemmaq7\], for $\sigma\in[(d-2)/2,\sigma_0-1]$ and $T'\in[0,T]$, $$\label{fi2}
\sum_{m=1}^d\|\mathcal{N}_m(\Psi)\|_{\dot{N}^\sigma[-T',T']}\leq C\|\widetilde{E}_{T'}(\Psi)\|_{\dot{F}^\sigma}( \|\widetilde{E}_{T'}(\Psi)\|_{\dot{F}^{(d-2)/2}}^2+\|\widetilde{E}_{T'}(\Psi)\|_{\dot{F}^{(d-2)/2}}^4 ).$$ The inequality shows that the function $L(T')=\sum_{m=1}^d\|\mathcal{N}_m(\Psi)\|_{\dot{N}^\sigma[-T',T']}$ is continuous on the interval $[0,T]$. Also, $L(0)=0$. Thus we can combine and (with $\sigma=(d-2)/2$), together with the smallness of $\|\psi_{m,0}\|_{\dot{H}^{(d-2)/2}}$, to conclude that $$\sum_{m=1}^d\|\mathcal{N}_m(\Psi)\|_{\dot{N}^\sigma[-T',T']}\leq C\sum_{m=1}^d||\psi_{m,0}||_{\dot{H}^{(d-2)/2}}\text{ for any }T'\in[0,T].$$ Using again, it follows that $$\label{fi3}
\|\widetilde{E}_{T}(\Psi)\|_{\dot{F}^{(d-2)/2}}\leq C\sum_{m=1}^d||\psi_{m,0}||_{\dot{H}^{(d-2)/2}}\ll 1.$$ We combine and again; using , for any $\sigma\in[(d-2)/2,\sigma_0-1]$ $$\label{fi4}
\|\widetilde{E}_{T}(\Psi)\|_{\dot{F}^{\sigma}}\leq C\sum_{m=1}^d||\psi_{m,0}||_{\dot{H}^\sigma\cap\dot{H}^{(d-2)/2}}.$$ Using , it follows that for any $\sigma\in[(d-2)/2,\sigma_0-1]$ $$\label{fi5}
\sum_{m=1}^d\sup_{t\in[-T,T]}\|\psi_m(t)\|_{\dot{H}^\sigma\cap\dot{H}^{(d-2)/2}}\leq C\sum_{m=1}^d||\psi_{m,0}||_{\dot{H}^\sigma\cap\dot{H}^{(d-2)/2}}.$$
We use to get a priori estimates on the solution $s$. Using and , $$\label{fi6}
\sum_{m=1}^d\sup_{t\in[-T,T]}\|\psi_m(t)\|_{\dot{H}^{(d-2)/2}}\leq C\|s_0-Q\|_{\dot{H}^{d/2}}.$$ We define the operators $\nabla^\sigma$, $\sigma\in [-1/2,d/2]$, as in the proof of Lemma \[Lemmag5\]. Let $p_\sigma=d/(\sigma+1)$. Then, in view of the Sobolev imbedding theorem (recall $d\geq 3$), $$\label{fi7}
\|\nabla^\sigma f\|_{L^{p_\sigma}}\leq C\|\nabla^{\sigma'}f\|_{L^{p_{\sigma'}}}\text{ if }-1/2\leq \sigma\leq\sigma'\leq d/2\text{ and }f\in H^{\sigma_0-1}.$$ Let $n_0$ denote the smallest integer $\geq (d-2)/2$. Using , , and the definition of the coefficients $A_m$, $$\label{fi9}
\|A_m(t)\|_{\dot{H}^{(d-2)/2}}\leq \|\nabla^{n_0}(A_m(t))\|_{L^{p_{n_0}}}\leq C\|s_0-Q\|_{\dot{H}^{d/2}},$$ for any $t\in[-T,T]$ and $m=1,\ldots,d$.
To prove estimates on the solution $s$, recall the identity , $$\label{fi12}
\begin{cases}
&\partial_ms=\Re(\psi_m)v+\Im(\psi_m)w;\\
&\partial_mv=-\Re(\psi_m)s+A_mw;\\
&\partial_mw=-\Im(\psi_m)s-A_mv.
\end{cases}$$ Since $|s|=|v|=|w|\equiv 1$, we use , , and to see that $$\sum_{m=1}^d\big[\|\partial_m(s(t))\|_{L^{p_0}}+\|\partial_m(v(t))\|_{L^{p_0}}+\|\partial_m(w(t))\|_{L^{p_0}}\big]\leq C\|s_0-Q\|_{\dot{H}^{d/2}},$$ for any $t\in[-T,T]$. As in the proof of Lemma \[Lemmag5\], a simple inductive argument using , , , and shows that $$\label{fi14}
\sum_{m=1}^d\big[\|\nabla^{n}\partial_m(s(t))\|_{L^{p_n}}+\|\nabla^{n}\partial_m(v(t))\|_{L^{p_n}}+\|\nabla^{n}\partial_m(w(t))\|_{L^{p_n}}\big]\leq C\|s_0-Q\|_{\dot{H}^{d/2}},$$ for any $n\in{\ensuremath{\mathbb{Z}}}\cap[0,(d-2)/2]$ and $t\in[-T,T]$. If $d$ is even, this gives $$\label{fi15}
\|s(t)-Q\|_{\dot{H}^{d/2}}\leq C\|s_0-Q\|_{\dot{H}^{d/2}}\text{ for any }t\in[-T,T].$$ If $d$ is odd then, using with $n=(d-3)/2$ and , we have $$\sum_{m=1}^d\big[\|\nabla^{\sigma}\partial_m(s(t))\|_{L^{p_\sigma}}+\|\nabla^{\sigma}\partial_m(v(t))\|_{L^{p_\sigma}}+\|\nabla^{\sigma}\partial_m(w(t))\|_{L^{p_\sigma}}\big]\leq C\|s_0-Q\|_{\dot{H}^{d/2}}$$ for any $\sigma\in[-1/2,(d-3)/2]$. The bound follows in this case as well, using the Leibniz rule .
We show now that for $\sigma'\in[0,\sigma_0]\cap{\ensuremath{\mathbb{Z}}}$ $$\label{fi30}
\sup_{t\in[-T,T]}\|s(t)\|_{H^{\sigma'}_Q}\leq C( \|s_0\|_{H^{\sigma'}_Q}).$$ For this we observe first that we have the conservation law $$\label{conserve}
\|s(t)\|_{H^0_Q}=\|s_0\|_{H^0_Q}\text{ for any }t\in[-T,T],$$ which follows by integration by parts from the initial-value problem . Thus, we need to estimate $\|s(t)-Q\|_{\dot{H}^{\sigma'}}$ for $t\in[-T,T]$. Using the first inequality in , we may assume $\sigma'\geq (d+1)/2$. In view of and $$\sum_{m=1}^d\sup_{t\in[-T,T]}\|\psi_m(t)\|_{\dot{H}^{\sigma'-1}}\leq C( \|s_0\|_{H^{\sigma'}_Q}).$$ In addition, due to the energy conservation law $$\sum_{l=1}^d||\partial_ls(t)||_{L^2}^2=\sum_{l=1}^d||\partial_ls(0)||_{L^2}^2,$$ and the definition $\psi_m=(\partial_ms)\cdot v+i(\partial_ms)\cdot w$, we control $\sup_{t\in[-T,T]}\|\psi_m(t)\|_{L^2}\leq C( \|s_0\|_{H^{\sigma'}_Q})$. Thus $$\sum_{m=1}^d\sup_{t\in[-T,T]}\|\psi_m(t)\|_{H^{\sigma'-1}}\leq C( \|s_0\|_{H^{\sigma'}_Q}).$$ Using the definition of the coefficients $A_m$, it follows easily that $$\sum_{m=1}^d\sup_{t\in[-T,T]}\|A_m(t)\|_{H^{\sigma'-1}}\leq C( \|s_0\|_{H^{\sigma'}_Q}).$$ We combine the last two inequalities, , and the fact that $|s|=|v|=|w|$; a simple inductive argument gives $\sup_{t\in[-T,T]}||\partial_ms||_{H^{\sigma'-1}}\leq C( \|s_0\|_{H^{\sigma'}_Q})$, which completes the proof of .
Existence and uniqueness of solutions {#section4.2}
-------------------------------------
The uniqueness statement in part (a) is proved in [@IoKe2 section 2]: assume $s, s'\in C([T_1,T_2]:H^{\sigma_0}_Q)$ solve the equation $\partial_ts=s\times\Delta_x s$ on $\mathbb{R}^d\times[T_1,T_2]$, and $s(T_1)=s'(T_1)$. Let $q=s'-s$, so $$\label{Sch9}
\begin{cases}
&\partial_tq=(s+q)\times\Delta_x (s+q)-s\times\Delta_x s\,\text{ on }\,\mathbb{R}^d\times[T_1,T_2];\\
&q(T_1)=0.
\end{cases}$$ We multiply by $q(t)$ and integrate by parts over $\mathbb{R}^d$ to obtain $$\label{Sch91}
\begin{split}
\frac{1}{2}\partial_t[\|q(t)\|_{L^2}^2]&=\int_{\mathbb{R}^d}[s(t)\times\Delta_x
q(t)]\cdot q(t)\,dx\\
&\leq C_s(||q(t)||_{L^2}^2+\sum_{l=1}^d||\partial_lq(t)||_{L^2}^2).
\end{split}$$ Then we apply $\partial_l$ to , multiply by $\partial_lq(t)$, add up over $l=1,\ldots,d$, and integrate by parts over $\mathbb{R}^d$. The result is $$\label{Sch92}
\begin{split}
\frac{1}{2}\partial_t[\sum_{l=1}^d\|\partial_lq(t)\|_{L^2}^2]&=-\int_{\mathbb{R}^d}[q(t)\times\Delta_x s(t)]\cdot \Delta_xq(t)\,dx\\
&\leq C_s(||q(t)||_{L^2}^2+\sum_{l=1}^d||\partial_lq(t)||_{L^2}^2).
\end{split}$$ Using and , $q\equiv 0$ on ${\ensuremath{\mathbb{R}}}^d\times[T_1,T_2]$, as desired.
To construct the global solution, we need the following local existence result:
\[Lemmad1\][Proposition]{}
\[Lemmad2\] Assume $s_0\in H_Q^{\infty}$. Then there is $T_{\sigma_0}=T( \|s_0\|_{H^{\sigma_0}_Q})>0$ and a solution $s\in C([-T_{\sigma_0},T_{\sigma_0}]:H^{\infty}_Q)$ of the initial-value problem $$\begin{cases}
&\partial_ts=s\times\Delta s\,\text{ on }\,\mathbb{R}^d\times[-T_{\sigma_0},T_{\sigma_0}];\\
&s(0)=s_0.
\end{cases}$$ In addition, the time $T_{\sigma_0}$ can be chosen such that $$\label{co1}
\begin{cases}
&\sup\limits_{t\in[-T_{\sigma_0},T_{\sigma_0}]}\|s(t)\|_{H^{\sigma_0}_Q}\leq C(\|s_0\|_{H^{\sigma_0}_Q});\\
&\sup\limits_{t\in[-T_{\sigma_0},T_{\sigma_0}]}\|s(t)\|_{H^\sigma_Q}\leq C(\sigma,\|s_0\|_{H^\sigma_Q})\text{ if }\sigma\in[\sigma_0,\infty)\cap{\ensuremath{\mathbb{Z}}}.
\end{cases}$$
The local existence Proposition \[Lemmad2\] is proved, for example, in [@KePoStTo]. The bound is not stated in this paper, but follows from the key estimate (5.32) in [@KePoStTo]. Assuming Proposition \[Lemmad2\], by scale invariance, it suffices to construct the solution $s$ in Theorem \[Main1\] on the time interval $[-1,1]$. In view of Proposition \[Lemmad2\], there is $T_{\sigma_0}>0$ and a solution $s$ on the time interval $[-T_{\sigma_0},T_{\sigma_0}]$. Assume the solution $s\in C([T,T]:H^{\infty}_Q)$ is constructed on some time interval $[-T,T]$, $T\leq 1$. In view of Proposition \[Lemmad1\], $$\sup_{t\in[-T,T]}\|s(t)\|_{H^{\sigma_0}_Q}\leq C( \|s_0\|_{H^{\sigma_0}_Q}),$$ uniformly in $T$. Using Proposition \[Lemmad2\], the solution $s$ can be extended to the time interval $[-T-T',T+T']$ for some $T'=T'( \|s_0\|_{H^{\sigma_0}})>0$ (which does not depend on $T$). The theorem follows.
Proof of Lemma \[Lemmaq5\] {#section5}
==========================
We use the notation in section \[section3\] and assume in this section that $d\geq 4$. For simplicity of notation, we let $\psi$ denote any of the functions $\widetilde{E}_T(\psi_m)$ or $\overline{\widetilde{E}_T(\psi_m)}$, $m=1,\ldots,d$, $A$ denote any of the functions $A_m$, $m=1,\ldots,d$, and $R$ denote any operator of the form $R_lR_{l'}$, $l,l'=0,1,\ldots,d$, $R_0=I$. With this convention, we show first that for any $k\in{\ensuremath{\mathbb{Z}}}$ and $\sigma\in[(d-2)/2,\sigma_0-1]$ $$\label{vg1}
(2^{\sigma k}+2^{(d-2)k/2})\|P_k(R(\psi\cdot\psi))\|_{L^2}\leq C\beta_k(\sigma)\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}.$$ The left-hand side of is dominated by $$\begin{split}
&C(2^{\sigma k}+2^{(d-2)k/2})\sum_{|k_1-k|\leq 2}\sum_{k_2\leq k-4}\|P_{k_1}(\psi)\cdot P_{k_2}(\psi)\|_{L^2}\\
&+C(2^{\sigma k}+2^{(d-2)k/2})\sum_{k_1,k_2\geq k-4,|k_1-k_2|\leq 10}\|P_k(P_{k_1}(\psi)\cdot P_{k_2}(\psi))\|_{L^2}
\end{split}$$ Using , we estimate $\|P_{k_1}\psi\|$ in $L^{\infty,2}_{\ensuremath{\mathbf{e}}}$ (after suitable localization), and, using the global , we estimate $\|P_{k_2}\psi\|$ in $L^{2,\infty}_{\ensuremath{\mathbf{e}}}$. The bound follows since $\beta_{k_1}(\sigma)\leq C2^{|k_1-k|/10}\beta_k(\sigma)$. The bounds for $F\in\{E_T(A_0),E_T(\widetilde{\psi}_m\cdot\widetilde{\psi}_l):m,l=1,\ldots,d,\widetilde{\psi}\in\{\psi,\overline{\psi}\}\}$, and clearly follow from . Also, it follows from that $$\label{vg2}
(2^{\sigma k}+2^{(d-2)k/2})\cdot \|P_{k}(A)\|_{L^{\infty,2}_{{\ensuremath{\mathbf{e}}}'}}\leq C2^{-k/2}\cdot \beta_k(\sigma)\cdot \|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}},$$ for any ${\ensuremath{\mathbf{e}}}'\in\mathbb{S}^{d-1}$.
We prove now that for any ${\ensuremath{\mathbf{e}}}'\in\mathbb{S}^{d-1}$ $$\label{vg3}
\sum_{k\in{\ensuremath{\mathbb{Z}}}}2^{-k}\|P_{k}(R(\psi\cdot\psi))\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}\leq C\|\widetilde{E}_T(\Psi)\|^2_{\dot{F}^{(d-2)/2}}.$$ For any $k\in{\ensuremath{\mathbb{Z}}}$ $$\label{vg5}
\begin{split}
\|P_{k}(R(\psi\cdot\psi))\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}&\leq C\sum_{|k_1-k|\leq 2}\sum_{k_2\leq k-4}\|P_{k_1}(\psi)\cdot P_{k_2}(\psi)\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}\\
&+C\sum_{k_1,k_2\geq k-4,|k_1-k_2|\leq 10}\|P_k(P_{k_1}(\psi)\cdot P_{k_2}(\psi))\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}.
\end{split}$$ For the first sum in , we use the global : $$\begin{split}
\sum_{|k_1-k|\leq 2}&\sum_{k_2\leq k-4}\|P_{k_1}(\psi)\cdot P_{k_2}(\psi)\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}\\
&\leq C\sum_{|k_1-k|\leq 2}\sum_{k_2\leq k-4}(2^{(d-1)k_1/2}\|P_{k_1}(\psi)\|_{Z_{k_1}})\cdot (2^{(d-1)k_2/2}\|P_{k_2}(\psi)\|_{Z_{k_2}})\\
&\leq C2^k\beta_k((d-2)/2)^2.
\end{split}$$ For the second sum, we use the localized and the assumption $d\geq 4$: $$\begin{split}
\|P_k(P_{k_1}\psi\cdot &P_{k_2}\psi)\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}\leq\,C\sum_{n,n'\in\Xi_k\text{ and }|n-n'|\leq C2^k}\|\widetilde{P}_{k,n}P_{k_1}(\psi)\cdot \widetilde{P}_{k,n'}P_{k_2}(\psi)\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}\\
&\leq C\big[\sum_{n\in\Xi_k}\|\widetilde{P}_{k,n}P_{k_1}(\psi)\|^2_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}\big]^{1/2}\cdot\big[\sum_{n'\in\Xi_k}\|\widetilde{P}_{k,n}P_{k_2}(\psi)\|^2_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}\big]^{1/2}\\
&\leq C2^{-3|k_1-k|/2}\cdot (2^{(d-1)k_1/2}\|P_{k_1}(\psi)\|_{Z_{k_1}})\cdot (2^{(d-1)k_2/2}\|P_{k_2}(\psi)\|_{Z_{k_2}})\\
&\leq C2^k2^{-|k_1-k|/4}\beta_k((d-2)/2)^2.
\end{split}$$ The bound follows from and the last two estimates. The bounds for $F\in\{E_T(A_0),E_T(\widetilde{\psi}_m\cdot\widetilde{\psi}_l):m,l=1,\ldots,d,\widetilde{\psi}\in\{\psi,\overline{\psi}\}\}$, and clearly follow from .
It remains to prove the bounds and for $F=E_T(A_m^2)$. We will need the following technical lemma:
\[section\]
\[Lemmaz5\] If $k\in{\ensuremath{\mathbb{Z}}}$, $k'\in(-\infty, k+10d]\cap{\ensuremath{\mathbb{Z}}}$, and ${\ensuremath{\mathbf{e}}}'\in\mathbb{S}^{d-1}$ then $$\label{jj1}
\big[\sum_{n\in\Xi_{k'}}\|\widetilde{P}_{k',n}P_{k}(A)\|^2_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}\big]^{1/2}\leq C2^{k/2}2^{-3|k-k'|/4}\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}^2.$$
Assuming Lemma \[Lemmaz5\], for it suffices to prove that $$\label{vg9}
(2^{\sigma k}+2^{(d-2)k/2})\|P_k(A\cdot A)\|_{L^2}\leq C\beta_k(\sigma)\|\widetilde{E}_T(\Psi)\|^3_{\dot{F}^{(d-2)/2}}.$$ The proof of is similar to the proof of , using the $L^{\infty,2}_{{\ensuremath{\mathbf{e}}}'}$ estimate in and the global (that is $k'=k$) $L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}$ estimate in . For it suffices to prove that $$\label{vg10}
\|P_{k}(A\cdot A)\|_{L^{1,\infty}_{{\ensuremath{\mathbf{e}}}'}}\leq C2^k\|\widetilde{E}_T(\Psi)\|^4_{\dot{F}^{(d-2)/2}},$$ for any $k\in{\ensuremath{\mathbb{Z}}}$ and ${\ensuremath{\mathbf{e}}}'\in\mathbb{S}^{d-1}$. The proof of is similar to the proof of , using the localized $L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}$ estimate in .
In view of the definitions, we may assume $k'\leq k-10d$ and it suffices to prove that $$\label{jj2}
\big[\sum_{n\in\Xi_{k'}}\|\widetilde{P}_{k',n}P_{k}(\psi\cdot \psi)\|^2_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}\big]^{1/2}\leq C2^{3k/2}2^{-3|k-k'|/4}\|\widetilde{E}_T(\Psi)\|^2_{\dot{F}^{(d-2)/2}}.$$ We will use the following bound: if $k\in{\ensuremath{\mathbb{Z}}}$, $k'\in(-\infty,k+10d]\cap{\ensuremath{\mathbb{Z}}}$, and $f\in Z_k$ then $$\label{im6}
\big[\|\sum_{n\in\Xi_{k'}}\mathcal{F}^{-1}(\chi_{k',n}(\xi)\cdot \widetilde{f})\|_{L^\infty_{x,t}}^2\big]^{1/2}
\leq C2^{dk/2}\cdot 2^{-d|k-k'|/2}(1+|k-k'| )\cdot\|f\|_{Z_k},$$ where $\mathcal{F}^{-1}(\widetilde{f})\in\{\mathcal{F}^{-1}(f),\overline{\mathcal{F}^{-1}(f)}\}$. For $k-k'\leq C$ this follows directly from and the Sobolev imbedding theorem. For $k-k'\geq C$, the bound follows by analyzing the cases $f\in X_k$ and $f\in Y_k^{\ensuremath{\mathbf{e}}}$ (see Lemma 4.1 in [@IoKe3] for a similar proof).
The left-hand side of is dominated by $$\label{jj3}
\begin{split}
&C\sum_{|k_1-k|\leq 2}\sum_{k_2\leq k'}\big[\sum_{n\in\Xi_{k'}}\|\widetilde{P}_{k',n}P_{k}(P_{k_1}(\psi)\cdot P_{k_2}(\psi))\|^2_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}\big]^{1/2}\\
&+C\sum_{|k_1-k|\leq 2}\sum_{k'\leq k_2\leq k-4}\big[\sum_{n\in\Xi_{k'}}\|\widetilde{P}_{k',n}P_{k}(P_{k_1}(\psi)\cdot P_{k_2}(\psi))\|^2_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}\big]^{1/2}\\
&+C\sum_{k_1,k_2\geq k-4,\,|k_1-k_2|\leq 10}\big[\sum_{n\in\Xi_{k'}}\|\widetilde{P}_{k',n}P_{k}(P_{k_1}(\psi)\cdot P_{k_2}(\psi))\|^2_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}\big]^{1/2}.
\end{split}$$ We use the $L^\infty_{x,t}$ estimate on the lower frequency term and the localized $L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}$ estimate on the higher frequency term. The first sum in is dominated by $$C\sum_{|k_1-k|\leq 2}\sum_{k_2\leq k'}(2^{k_1/2}2^{-3|k-k'|/4}\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}})\cdot (2^{k_2}\|\widetilde{E}_T(\Psi)\|^2_{\dot{F}^{(d-2)/2}}),$$ which suffices for . The second sum in is dominated by $$\begin{split}
&C\sum_{|k_1-k|\leq 2}\sum_{k'\leq k_2\leq k-4}\big[\sum_{n\in\Xi_{k'}}\|\widetilde{P}_{k',n}P_{k_1}(\psi)\|^2_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}\big]^{1/2}\cdot \big[\sum_{n\in\Xi_{k'}}\|\widetilde{P}_{k',n}P_{k_2}(\psi)\|_{L^\infty}\big]\\
&\leq C\sum_{|k_1-k|\leq 2}\sum_{k'\leq k_2\leq k-4}(2^{k/2}2^{-7|k-k'|/8}\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}})\negmedspace\cdot\negmedspace (2^{k_2}|k-k'|\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}})
\end{split}$$ which suffices for . The third sum in is dominated by $$\begin{split}
&C2^{d|k-k'|/2}\negmedspace\negmedspace\negmedspace\sum_{k_1,k_2\geq k-4,\,|k_1-k_2|\leq 10}\negmedspace\big[\sum_{n\in\Xi_{k'}}\|\widetilde{P}_{k',n}P_{k_1}(\psi)\|^2_{L^{2,\infty}_{{\ensuremath{\mathbf{e}}}'}}\big]^{1/2}\negmedspace\cdot \negmedspace\big[\sum_{n\in\Xi_{k'}}\|\widetilde{P}_{k',n}P_{k_2}(\psi)\|_{L^\infty}^2\big]^{1/2}\\
&\leq C2^{d|k-k'|/2}\negmedspace\negmedspace\sum_{k_1,k_2\geq k-4,\,|k_1-k_2|\leq 10}\negmedspace2^{3k_1/2}\|\widetilde{E}_T(\Psi)\|_{\dot{F}^{(d-2)/2}}^2\cdot 2^{-(d-1)|k_1-k'|}(1+|k_1-k'|)^2,
\end{split}$$ which suffices for since $d\geq 4$. This completes the proof of Lemma \[Lemmaz5\].
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[^1]: The second author was supported in part by an NSF grant and a Packard fellowship. The third author was supported in part by an NSF grant.
[^2]: This null structure was not observed in the earlier paper of Ionescu–Kenig [@IoKe2]; without this null structure the restriction $\sigma>(d+1)/2$ in [@IoKe2] is necessary for the perturbative argument.
[^3]: The MSM were first derived in [@ChShUh], using orthonormal frames, and [@NaStUh], using the stereographic projection.
[^4]: This elementary construction was suggested to us by T. Tao.
|
---
abstract: 'Three experimental concepts investigating possible anisotropy of the speed of light are presented. They are based on i) beam deflection in a 180$^\circ$ magnetic arc, ii) narrow resonance production in an electron-positron collider, and iii) the ratio of magnetic moments of an electron and a positron moving in opposite directions.'
author:
- 'B. Wojtsekhowski'
title: On measurement of the isotropy of the speed of light
---
There are several well known experiments which investigate the one-way speed of light $c$; see the review and analysis in Refs. [@CW1992; @CW2014]. Here we present related experimental schemes based on a high energy electron (positron) beam. The experiments test isotropy of the maximum speed, but we refer to the speed of light assuming the photon to be massless. A recent experiment at the storage ring ESRF (Grenoble) established a stringent constraint on the speed of light anisotropy of $1.6 \times 10^{-14}$ using the Compton back-scattering method [@GM1996; @GR2010].
In this letter we discuss three other methods which could be used in a search for directional variation of the speed of light and/or related effects. Their common feature is a large value of the Lorentz factor of the electron (positron) beam, $\gamma_{-(+)}$, of a few $10^4$. They are based on various techniques for the beam momentum measurement.
Momenta in opposite directions
==============================
Deflection of the beam in a magnetic field is the simplest method for a momentum measurement. Such a measurement could provide very accurate monitoring of slight changes of momentum assuming stability of the magnetic field and the beam position detectors. Two beam momentum monitors at opposite sides of a 180$^\circ$ bending magnet allow one to measure the ratio of the particle momenta $p_+, \, p_-$ moving in opposite directions, $R = p_+ / p_-$ (see Fig. \[fig:arc\]). The sensitivity of the method, $\delta c/c \sim (\delta R/R) /\gamma^2$, is defined by the beam Lorentz factor and relative accuracy of the momentum monitors.
![Diagram for measurement of the beam momenta with a 180$^\circ$ arc magnet.[]{data-label="fig:arc"}](Arc){width="25.00000%"}
Assuming that there are no acceleration elements between those two momentum measurements and a small correction (calculable) on the radiative energy loss, such a ratio should be stable even when the actual beam energy varies. The value of $R$ will be close to 1 with any deviation mainly due to the systematics uncertainties of the momentum monitors. The configuration could have a large portion of the arc used as a momentum monitor (the area A (B) shown in Fig. \[fig:arc\]) because it allows higher dispersion and better sensitivity in spite of the angle’s being smaller than 180$^\circ$ between the two momentum monitors. The systematics of the $R$ measurement are strongly suppressed when the measurements in both monitors perform synchronously. The proposed measurement of the ratio $R$ is sensitive to the directional variation of the speed of light as formulated, for example, in the Mansouri-Sexl test theory [@TestT]. The results could also be interpreted in terms of the Standard Model Extension [@SME], whose odd-parity parameter $\tilde{k}_{o \,+}$ would be constrained. The time dependence of $R$ could reveal, for example, the directional variation of the speed of light due to changes in the beam direction because of the accelerator rotation together with the Earth and variation of the speed of motion relative to the Cosmic Microwave Background dipole [@CMB].
A number of accelerators with beam energies in the several GeV range have (or could build) beam momentum monitors with a level of $10^{-6}$ relative precision over a period of one millisecond. Indeed, the accuracy of the momentum monitor is defined by the precision of the beam position monitors, BPMs, and the deflection of the beam trajectory in the dipole magnet. The RF-based BPMs have an accuracy on the level of 1 $\mu$m for a 0.1 nC beam bunch charge [@KEK-BPM]. Measurement of the magnet temperature with 0.01$^\circ$K accuracy and the magnetic field by means of NMR should allow sufficient short term stability of the magnetic field integral. The BMPs located next to the dipole section with dispersion of 2-3 meters should provide the required accuracy of the momentum measurement. When data are averaged over a few hours, the potential sensitivity of $R$ should be on the level of $10^{-9}$ (limited by the long term stability of the beam position monitors, which could be determined by means of the optical interferometer to sufficient precision as shown e.g. in Ref. [@ILC-BPM]). The relatively small size of each momentum monitor of 20 meters will help to minimize known distortions such as tide and ground instability. The combined effect of these uncertainties on the precision of the $R$ measurement we estimate to be less than $10^{-7}$.
The stability of measurement could be greatly improved by means of two beams (electron and positron) moving in the same magnetic system, which would allow a full compensation of drift of the momentum monitor characteristics with normalization of the electron ratio $R^e = p_+^e / p_-^e$ to the positron ratio $R^p = p_+^p / p_-^p$. The double ratio $R_{e/p} = R^e/R^p$ would be immune to most instabilities. This double ratio has also doubled sensitivity to the directional variation of the speed of light. Such a measurement could be performed with a single storage ring collider, e.g. CESR or VEPP-4.
The sidereal periodicity of the possible signal when used in the Fourier analysis of the data will allow suppression of the systematics by a factor of 10-20 depending on the duration of the experiment. By combining the resulting sensitivity of the momenta ratio measurement $R$ of $10^{-8}$ with the factor of the beam ($\gamma^{-2} \sim 10^{-8}$), one can find that the signal sensitivity is $10^{-16}$ or better for $\delta c/c$, which would be a significant improvement for the limit on the odd-parity anisotropy, whose current value is $10^{-14}$ [@CAV].
The JLab CEBAF accelerator [@CEBAF] has the most advantageous parameters for the proposed experiment because its beam has a high current of 100 $\mu$A, a relative energy spread of a few 10$^{-5}$, and geometrical emittance of 10$^{-9}$ m$\times$rad (in spite of significant broadening due to radiation losses in the last few turns). The accelerator sections are separated by 180$^\circ$ arcs, and the beam recently reached $\gamma \sim 2 \times 10^4$, the highest Lorentz factor among the currently operating accelerators. The absolute value of the beam energy in a linear accelerator could vary, but this does not impact the proposed investigation because only the ratio of the beam momenta at opposite sides of the arc needs to be stable. In addition, the value of the anisotropy could be constrained at 10 different arcs with increasing beam energy.
Resonance production
====================
The masses of the narrow resonances ($\varphi, J/\Psi, \Upsilon $, and Z) have been measured with high precision in electron-positron colliders. For example, the mass of the $J/\Psi$ is known with a relative accuracy of $3.5 \times 10^{-6}$ and the mass of the $\Upsilon$ with $1 \times 10^{-5}$ [@PDG]. These collider results were obtained using a precision measurement of the beam energies via the spin resonance method, which allows accurate determination of the absolute value of the beam Lorentz factor [@POL]. Variation of the speed of light leads to a change of the Lorentz factor along the beam orbit in a storage ring. The resonance could be used as the second method of beam energy measurement needed in a search for a sidereal time variation of the beam $\gamma$.
Because the beam directions at the location of the energy meter and the $e^+e^-$ collision point are different (in some experiments), the average observed Lorentz factor of the beams is not necessarily equal to the ratio of the meson and electron masses. For a symmetric-energy collider (equal energies of a positron and an electron beam) to first order the effect is canceled out. The B-factories at SLAC and KEK [@BFAC] could access the non-isotropic component of the speed of light due to the large difference in the Lorentz factor of the electron and positron beams. The variation in the ratio $\beta = v/c$ could be expressed as ${\delta \beta}/{\beta} \,=\, {\delta m}/{m} /(\gamma_+^2 - \gamma_-^2)$. The full observed width of the $\Upsilon$ resonance is of 10 MeV (mainly due to the beam energy spread). This would lead to an estimate of a potential constraint on $\delta c/c$ of $3 \times 10^{-12}$. Finally, very large statistics of accumulated $\Upsilon$ events in the KEK experiment [@Belle], $1 \times 10^8$, boosts potential sensitivity to $3 \times 10^{-16}$. The sidereal variation of the observed resonance mass (at the fixed beams energies) would be a signal of the anisotropy of the speed of light.
Leptons’ anomalous magnetic moments
===================================
The anomalous magnetic moments $a_{-(+)}$ of an electron and positron were measured to a $2 \times 10^{-12}$ relative accuracy in single particle traps [@g-2t], which puts a stringent constraint on the CPT violating parameter of SME [@SME]. Independent measurement on the level of $ 10^{-8}$ was performed at the VEPP2m storage ring with 650 MeV beams of electrons and positrons [@g-2s]. These beams of electrons and positrons move in the storage ring in opposite directions and their Lorentz factors are obtained by the spin resonance method. Assuming that $a_+ = a_-$ (constraint [@g-2t]) and equal masses of an electron and a positron, we can find the difference of the Lorentz factors of these two beams and use such an experiment for a search of the anisotropy of the speed of light. Sensitivity of the experiment [@g-2s] corresponds to .
There is the potential to reach a higher precision of $(a_+ - a_-)/a_{avg}$ of $10^{-10}$ with the beam method [@POL]; alternatively, one can perform such a measurement using the VEPP-4 storage ring, where an almost ten times higher beam $\gamma$ factor leads to higher sensitivity. The resulting estimate for $(\gamma_+ - \gamma_-)/\gamma_{avg}$ of $10^{-10}$ means that the search sensitivity for is about $10^{-16}$ from multiple measurements with different sidereal phases.
The author takes pleasure in acknowledging helpful discussions with V. G. Gurzadyan of the GRAAL experiment and with C. Keppel, B. Vlahovic, and V. Zelevinsky of the current concepts. He would like to extend thanks to P. Evtushenko and Y. Roblin for information on CEBAF parameters. This work was supported by the U.S. Department of Energy. Jefferson Science Associates, LLC, operates Jefferson Lab for the U.S. DOE under U.S. DOE contract DE-AC05-060R23177.
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|
---
abstract: 'We derive generalized Kronig identities expressing quadratic fermionic terms including momentum transfer to bosonic operators and use them to obtain the exact solution for one-dimensional fermionic models with linear dispersion in the presence of position-dependent interactions and scattering potential. In these Luttinger droplets, which correspond to Luttinger liquids with spatial variations or constraints, the position dependences of the couplings break the translational invariance of correlation functions and modify the Luttinger-liquid interrelations between excitation velocities.'
author:
- Sebastian Huber
- Marcus Kollar
bibliography:
- 'bosopapbib\_short.bib'
nocite: '[@pereira_dynamical_2007; @enciso_fermion_2006; @karabali_exact_2014; @von_delft_bosonization_1998]'
title: 'From Luttinger liquids to Luttinger droplets via higher-order bosonization identities'
---
Introduction {#sec:introduction}
============
An important goal of condensed matter theory is a reliable description of the correlated behavior of electrons which is rooted in the Coulomb interaction between them. In one-dimensional geometries they exhibit a special coherence at low energies:[@senechal_introduction_1999; @giamarchi_quantum_2003] the dispersion can be approximately linearized in the vicinity of the Fermi points $\pm k_{\text{F}}$ as $\epsilon_k$ $\simeq$ $\pm {v_{\text{F}}}(k\pm k_\text{F})$, so that the energy $\delta\epsilon$ $=$ ${v_{\text{F}}}\delta k$ of a particle-hole excitation from $k_1$ to $k_2$ is a function only of the momentum transfer $\delta k$ $=$ $k_2-k_1$. By contrast, in higher dimensions the magnitude and relative orientation of the two momenta usually enter into $\delta\epsilon$, leading to a continuum of excitation energies for a given momentum transfer. This coherence in one dimension is prominently featured in the Tomonaga-Luttinger model,[@tomonaga_remarks_1950; @luttinger_exactly_1963] which is based on the approximation that one can regard a physical electron field operator $\Psi(x)$ for a wire of length $L$ at low energies as a sum of two independent fields,
\[eq:psiphys\] $$\begin{aligned}
\Psi(x) &=
\sum_{k}
\frac{e^{{{\text{i}}}kx}}{\sqrt{L}}
{{\cal C}}{{\vphantom{{+}}}}_k
=
\sum_{\substack{
k>-k_\text{F}\\
\lambda=\pm
}}
\frac{e^{{{\text{i}}}\lambda(k_\text{F}+k)x}}{\sqrt{L}}
{{\cal C}}{{\vphantom{{+}}}}_{\lambda(k_\text{F}+k)}
\label{eq:psiphysdef}
\\&
\simeq
\frac{
e^{{{\text{i}}}k_\text{F}x}
{\psi_{\text{R}}^{{\vphantom{{+}}}}(x)}
+
e^{-{{\text{i}}}k_\text{F}x}
{\psi_{\text{L}}^{{\vphantom{{+}}}}(x)}
}{\sqrt{2\pi}}
\label{eq:psiphysapprox}
\,,
\end{aligned}$$
![Wire of length $L$ with position-dependent interaction potential $V(x)$ in , with $V(x)$ $=$ $V(-x)$, sketched here for the repulsive case with larger $V(x)$ near $x$ $=$ $0$ so that particles tend to keep a larger distance from each other there. An additional single-particle potential $W(x)$ $=$ $W(-x)$ may also be present. (a) A general smooth interaction potential. (b) A piecewise constant interaction potential, i.e., with piecewise constant value $V(0)$ inside and $V(L/2)$ outside a central region of width $2R$, as solved explicitly in Sec. \[subsubsec:piecewise\] for $L$ $\to$ $\infty$ and finite $R$.\[fig:sketch\]](fig-sketch.png){width="\columnwidth"}
where the lower summations limits $-k_{\text{F}}$ in were replaced by $-\infty$ . Then ${\psi_{\text{R,L}}^{{\vphantom{{+}}}}(x)}$ $=$ ${\psi_{1,2}^{{\vphantom{{+}}}}(\mp x)}$ and ${\psi_{\eta}^{{\vphantom{{+}}}}(x)}$ $=$ $(2\pi/L)^{\frac12}$ $\sum_{k} {{\text{e}}}^{-{{\text{i}}}kx} {c^{{{\vphantom{{+}}}}}_{k\eta}}$ are defined in terms of canonical fermions ${c^{{{\vphantom{{+}}}}}_{k\eta}}$ $=$ ${{\cal C}}{{\vphantom{{+}}}}_{\pm(k_\text{F}+k)}$ which correspond to the physical fermions ${{\cal C}}_k$ near the two Fermi points for $\eta$ $=$ $1,2$. In the Tomonaga-Luttinger model the dispersion is linearized near the Fermi points and only forward-scattering density interactions between left- and right-moving fermions are kept. The Tomonaga-Luttinger model can be solved by bosonization,[@tomonaga_remarks_1950; @luttinger_exactly_1963; @mattis_exact_1965; @schick_flux_1968; @schotte_tomonagas_1969; @mattis_new_1974; @luther_single-particle_1974; @coleman_quantum_1975; @mandelstam_soliton_1975; @heidenreich_sine-gordon_1975; @haldane_coupling_1979; @*haldane_luttinger_1981] which expresses the above-mentioned coherence of excitations into an exact mapping to bosonic degrees of freedom (at the operator level[@luther_single-particle_1974; @emery_theory_1979; @voit_one-dimensional_1995; @kotliar_toulouse_1996; @von_delft_bosonization_1998; @von_delft_finite-size_1998; @*zarand_analytical_2000] or in a path-integral formulation;[@lee_functional_1988; @yurkevich_bosonisation_2002; @grishin_functional_2004; @galda_impurity_2011; @filippone_tunneling_2016] throughout we use Ref. ’s constructive finite-size bosonization approach, which is recapped below). Bosonization has led to such remarkable results and concepts as spin-charge separation of elementary excitations,[@tomonaga_remarks_1950; @luttinger_exactly_1963] interaction-dependent exponents of correlation functions,[@luther_single-particle_1974; @meden_spectral_1992; @*schonhammer_nonuniversal_1993; @*schonhammer_erratum:_1993; @meden_nonuniversality_1999; @markhof_spectral_2016] and the Luttinger-liquid paradigm[@haldane_luttinger_1981; @haldane_general_1980; @*haldane_demonstration_1981; @*haldane_effective_1981] which states that the relations between excitation velocities and correlation exponents of the Tomonaga-Luttinger model remain valid even for weakly nonlinear dispersion. These topics are nowadays presented in many reviews[@senechal_introduction_1999; @emery_theory_1979; @solyom_fermi_1979; @voit_one-dimensional_1995; @schonhammer_interacting_1997; @*schonhammer_luttinger_2004; @*schonhammer_physics_2013; @von_delft_bosonization_1998; @miranda_introduction_2003; @cazalilla_one_2011] and textbooks.[@kopietz_bosonization_1997; @giamarchi_quantum_2003; @gogolin_bosonization_2004; @bruus_many-body_2004; @giuliani_quantum_2008; @phillips_advanced_2012; @mastropietro_luttinger_2013] Characteristic signatures of one-dimensional electron liquids have been observed in a variety of experiments.[@milliken_indications_1996; @maasilta_line_1997; @chang_chiral_2003; @bockrath_luttinger-liquid_1999; @ishii_direct_2003; @aleshin_one-dimensional_2004; @boninsegni_luttinger_2007; @*del_maestro_4he_2011; @*duc_critical_2015; @jompol_probing_2009; @barak_interacting_2010; @blumenstein_atomically_2011; @mebrahtu_quantum_2012; @*mebrahtu_observation_2013; @yang_quantum_2017; @cedergren_insulating_2017; @stuhler_tomonaga-luttinger_2019] The theory of nonlinear dispersion terms has been of particular further interest,[@schick_flux_1968; @haldane_luttinger_1981; @busche_how_2000; @*pirooznia_dynamic_2008; @teber_bosonization_2007; @karrasch_low_2015] including refermionization techniques which use bosonization identities in reverse to map diagonalized bosonic systems back to free fermions.[@rozhkov_variational_2003; @*rozhkov_fermionic_2005; @*rozhkov_class_2006; @*rozhkov_density-density_2008; @*rozhkov_one-dimensional_2014; @imambekov_universal_2009; @*imambekov_phenomenology_2009; @*imambekov_one-dimensional_2012; @teber_bosonization_2007; @maebashi_structural_2014; @essler_spin-charge-separated_2015; @markhof_investigating_2019] For Luttinger liquids out of equilibrium[@cazalilla_effect_2006; @*iucci_quantum_2009; @*nessi_quantum_2013; @uhrig_interaction_2009; @foster_quantum_2010; @perfetto_thermalization_2011; @dziarmaga_excitation_2011; @dora_crossover_2011; @*dora_generalized_2012; @*dora_absence_2015; @*dora_momentum-space_2016; @karrasch_luttinger-liquid_2012; @*rentrop_quench_2012; @coira_quantum_2013; @ngo_dinh_interaction_2013; @sabetta_nonequilibrium_2013; @kennes_luttinger_2013; @*kennes_spectral_2014; @sachdeva_finite-time_2014; @schiro_transport_2015; @mastropietro_quantum_2015] nonlinear dispersion effects are also essential.[@gutman_bosonization_2010; @*protopopov_many-particle_2011; @*protopopov_relaxation_2014; @*protopopov_dissipationless_2014; @*protopopov_equilibration_2015; @lin_thermalization_2013; @buchhold_nonequilibrium_2015; @*buchhold_kinetic_2015; @*buchhold_prethermalization_2016; @*huber_thermalization_2018] The technical hallmarks of bosonization are the following. On the one hand, a two-body density interaction term for fermions ${c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}$ becomes quadratic in terms of canonical bosons, defined for $q$ $>$ $0$ as ${b^{{{\vphantom{{+}}}}}_{q{{\eta}}}}$ $=$ $-{{\text{i}}}\sum_k{c^{{+}}_{k-q{{\eta}}}}{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}/\sqrt{\smash[b]{n_q}}$. Here the momentum sum runs over $k$ $=$ $\frac{2\pi}{L} (n_k-\tfrac12\delta_\text{b})$ with integer $n_k$, and the parameter $0$ $\leq$ $\delta_\text{b}$ $<$ $2$ fixes the boundary conditions, ${\psi_{{{\eta}}}^{{\vphantom{{+}}}}(x+L/2)}$ $=$ ${{\text{e}}}^{i\pi\delta_\text{b}}$ ${\psi_{{{\eta}}}^{{\vphantom{{+}}}}(x-L/2)}$. On the other hand the fermionic kinetic energy also translates into free bosons by means of the so-called Kronig identity,[@kronig_zur_1935; @dover_properties_1968; @schonhammer_interacting_1997]
\[eq:H01\] $$\begin{aligned}
H_{0{{\eta}}}^{(1)}
&=
\sum_k
k
\,
{{{}^\ast_\ast}{
{c^{{+}}_{k{{\eta}}}}
{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}
}{{}^\ast_\ast}}
\label{eq:H01-fermionic}
\\&
=
\sum_{q>0}
q
\,
{b^{{+}}_{q{{\eta}}}}
{b^{{{\vphantom{{+}}}}}_{q{{\eta}}}}
+
\frac{\pi}{L}
({\hat{N}}_{{{\eta}}}+1-\delta_\text{b}){\hat{N}}_{{{\eta}}}
\,,
\label{eq:H01-bosonic}
\end{aligned}$$
where the fermionic number operator is given by[^1] ${\hat{N}}_{{{\eta}}}$ $=$ $\sum_k{{{}^\ast_\ast}{{c^{{+}}_{k{{\eta}}}}{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}}{{}^\ast_\ast}}$, which commutes with ${b^{{{\vphantom{{+}}}}}_{q{{\eta}}}}$. The normal ordering ${{{}^\ast_\ast}{\cdots}{{}^\ast_\ast}}$ is defined with respect to the state ${|{\bm{0}}\rangle_0}$, where ${|{\bm{N}}\rangle_0}$ is an eigenstate of all ${c^{{+}}_{k{{\eta}}}}{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}$ (with eigenvalue 1 if $n_k$ $\leq$ $N_{{\eta}}$ and 0 otherwise). Furthermore, real-space fermionic and bosonic fields are related by the celebrated bosonization identity,[@schotte_tomonagas_1969; @mattis_new_1974; @luther_single-particle_1974; @von_delft_bosonization_1998] $$\begin{aligned}
{\psi_{{{\eta}}}^{{\vphantom{{+}}}}(x)}
&=
\big(\tfrac{2\pi}{L}\big)^{\frac12}
{F_{{{\eta}}}^{{\vphantom{{+}}}}}
\,
{{\text{e}}}^{-{{\text{i}}}\frac{2\pi}{L}({\hat{N}}_{{{\eta}}}-\frac12\delta_\text{b})x}
\,
{{\text{e}}}^{-{{\text{i}}}{\varphi_{{{\eta}}}^{{+}\!}(x)}}
\,
{{\text{e}}}^{-{{\text{i}}}{\varphi_{{{\eta}}}^{{{\vphantom{{+}}}}}(x)}}
, \label{eq:intro:bosonizationidentity}
\end{aligned}$$ which allows the calculation of fermionic in terms of bosonic correlation functions.[@luther_single-particle_1974] Here the fermionic Klein factor ${F_{{{\eta}}}^{{\vphantom{{+}}}}}$ decreases the fermionic particle number ${\hat{N}}_{{{\eta}}}$ by one, and ${\varphi_{{{\eta}}}^{{{\vphantom{{+}}}}}(x)}$ $=$ $-\sum_{q>0} {b^{{{\vphantom{{+}}}}}_{q{{\eta}}}}\,{{\text{e}}}^{-{{\text{i}}}qx-aq}/\sqrt{\smash[b]{n_q}}$. The regularization parameter $a$ $\to$ $0^+$ is needed to obtain a finite commutator,[@von_delft_bosonization_1998] $[{\varphi_{{{\eta}}}^{{{\vphantom{{+}}}}}(x)},{\varphi_{{{\eta}}}^{{+}\!}(x{^{\prime\!}})}]$ $=$ $-\ln\!\big( 1-{{\text{e}}}^{-\frac{2\pi{{\text{i}}}}{L}(x-x{^{\prime\!}}-{{\text{i}}}a)}
\big)$. A final ingredient to the solution of the Tomonaga-Luttinger model is a Boguljubov transformation, which absorbs the interaction between left- and right-moving fermions into the free bosonic theory.[@tomonaga_remarks_1950]
In the present work we will study Luttinger liquids with additional spatial constraints, which we term *Luttinger droplets*. Namely, we consider a (spinless) fermionic Hamiltonian with linear dispersion, position-dependent interactions $V(x)$ and $U(x)$, and scattering potential $W(x)$ (all assumed to be real symmetric functions of $x$),
\[eq:dropletHfermionicLR-intro\] $$\begin{aligned}
\label{eq:dropletHfermionicLR-intro-Hamiltonian}
H &=
\int \!
\frac{{\ensuremath{\text{d}x}}}{L}\,
{{}^\ast_\ast}{v_{\text{F}}}[
{\psi_{\text{R}}^{+}(x)}
{{\text{i}}}\partial_x
{\psi_{\text{R}}^{{\vphantom{{+}}}}(x)}
-
{\psi_{\text{L}}^{+}(x)}
{{\text{i}}}\partial_x
{\psi_{\text{L}}^{{\vphantom{{+}}}}(x)}
]
\nonumber\\&\,\hphantom{=}\,
+
W(x)[n_{\text{L}}(x)+n_{\text{R}}(x)]
+
U(x)
n_{\text{L}}(x)n_{\text{R}}(x)
\nonumber\\&\,\hphantom{=}\,
+
\frac12V(x)[n_{\text{L}}(x)^2+n_{\text{R}}(x)^2]
{{}^\ast_\ast}\,,
\end{aligned}$$ in terms of densities $n_{\text{R,L}}(x)$ $=$ ${\psi_{\text{R,L}}^{+}(x)}{\psi_{\text{R,L}}^{{\vphantom{{+}}}}(x)}/(2\pi)$. Without $W(x)$ and with constant $V(x)$ and $U(x)$, $H$ reduces to the usual translationally invariant Tomonaga-Luttinger model (with contact interactions). Below we will diagonalize exactly for the special case $$\begin{aligned}
U(x)
&=
\gamma\,[2\pi {v_{\text{F}}}+V(x)]
\,,~~~~
-1<\gamma<1
\,,
\label{eq:dropletHfermionicLR-intro-parameters}
\end{aligned}$$
for otherwise arbitrary $V(x)$ $>$ $-2\pi{v_{\text{F}}}$ and a constant $\gamma$. This means that real Fourier components $V_q$ $=$ $V_{-q}$ as well as $U_{q=0}$ can be chosen freely; then $\gamma$ $=$ $U_0/[2\pi {v_{\text{F}}}+V_0]$ and $U_{q\neq0}$ $=$ $\gamma
V_q$. Thus $\gamma$ characterizes the relative strength of interbranch interactions. Below we obtain the single-particle Green function for the ground state of this model, the exponents of which will reflect the spatial dependence of the couplings. We first derive generalized Kronig-type identities in Sec. \[sec:kronig\], which we then use to solve a single-flavor chiral version of in Sec. \[sec:chiral\]. We then proceed to the two-flavor case in Sec. \[sec:droplet\], with a discussion of the similiarities and differences of the spectrum and Green function compared to the translationally invariant case. One representative choice of $V(x)$ to be discussed below involves a central region with stronger repulsion than at the edges of the system, as shown in Fig. \[fig:sketch\]a. An explicit evaluation is provided for a piecewise constant $V(x)$ as shown in Fig. \[fig:sketch\]b in Sec. \[subsubsec:piecewise\]. Relations between the excitation velocities and Green function exponents are discussed in Sec. \[sec:dropletparadigm\], followed by a summary in Sec. \[sec:conclusion\].
Many results for inhomogeneous Luttinger liquids are of course known, e.g., with barriers,[@kane_transport_1992; @*kane_transmission_1992; @*kane_resonant_1992; @rylands_quantum_2016; @*rylands_quantum_2017; @*rylands_quantum_2018] impurities,[@meden_single_2002; @hattori_quantum_2014] boundaries,[@meden_luttinger_2000; @schneider_recursive_2010; @rylands_exact_2019] leads,[@eggert_scanning_2000; @filippone_tunneling_2016] confinements[@wonneberger_luttinger-model_2001] and so on. Models with (effective) position-dependent Luttinger liquid parameters or interaction potentials have also been investigated.[@maslov_landauer_1995; @safi_transport_1995; @ponomarenko_renormalization_1995; @rech_electronic_2008; @*rech_resistivity_2008; @grishin_functional_2004; @*galda_impurity_2011] Our goal is to provide a complementary perspective on these setups with the exact solution of the rather flexible model , i.e., the Hamiltonian with parameters from the manifold , and to possibly enable new applications, e.g., to ultradilute quantum droplets held together by weak cohesive forces.[@ferrier-barbut_ultradilute_2019]
Kronig-type identities with arbitrary momentum transfer {#sec:kronig}
=======================================================
Bosonic forms of bilinear fermionic terms
-----------------------------------------
Consider a general bilinear fermionic term,
\[eq:intro:Hqm\] $$\begin{aligned}
\label{eq:intro:Hqm-k}
H_{q{{\eta}}}^{(m)}
&=
\sum_k
k^m
\,
{{{}^\ast_\ast}{
{c^{{+}}_{k-q{{\eta}}}}
{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}
}{{}^\ast_\ast}}
\\&
\label{eq:intro:Hqm-x}
=
\int \!
\frac{{\ensuremath{\text{d}x}}}{2\pi}\,
{{\text{e}}}^{{{\text{i}}}qx}
\,
{{{}^\ast_\ast}{
{\psi_{{{\eta}}}^{+}(x)}
({{\text{i}}}\partial_x)^m
{\psi_{{{\eta}}}^{{\vphantom{{+}}}}(x)}
}{{}^\ast_\ast}}
\,,
\end{aligned}$$
for integer exponents $m$ $\geq$ $0$ and momentum transfer $q$ $=$ $\frac{2\pi}{L}$ $n_q$ with integer $n_q$; here and throughout real-space integrals without indicated endpoints extend over the interval $[-L/2,L/2]$. Arbitrary dispersion terms are included in for $q$ $=$ $0$, such as for $m$ $=$ $1$. Forming the product of with its hermitian conjugate at different positions $x$ and $x+\ell$, canceling the Klein factors (${F_{{{\eta}}}^{+}}{F_{{{\eta}}}^{{\vphantom{{+}}}}}$ $=$ $1$), commuting the bosonic fields, taking $a$ to zero, and combining exponentials, we obtain $$\begin{aligned}
&\frac{L}{2\pi}
\,
{\psi_{{{\eta}}}^{+}(x)}
{\psi_{{{\eta}}}^{{\vphantom{{+}}}}(x+\ell)}
\label{eq:intro:psidagpsi-x}
\\&
=
{{\text{e}}}^{\pi{{\text{i}}}(\delta_\text{b}-2{\hat{N}}_{{{\eta}}})\ell/L}
\,
{{\text{e}}}^{{{\text{i}}}{\varphi_{{{\eta}}}^{{+}\!}(x)}}
\,
{{\text{e}}}^{{{\text{i}}}{\varphi_{{{\eta}}}^{{{\vphantom{{+}}}}}(x)}}
\,
{{\text{e}}}^{-{{\text{i}}}{\varphi_{{{\eta}}}^{{+}\!}(x+\ell)}}
\,
{{\text{e}}}^{-{{\text{i}}}{\varphi_{{{\eta}}}^{{{\vphantom{{+}}}}}(x+\ell)}}
\nonumber\\&
=
\frac{
{{\text{e}}}^{\pi{{\text{i}}}(\delta_\text{b}-2{\hat{N}}_{{{\eta}}})\ell/L}
}{1-{{\text{e}}}^{2\pi{{\text{i}}}\ell/L}}
\,
{{\text{e}}}^{{{\text{i}}}({\varphi_{{{\eta}}}^{{+}\!}(x)}-{\varphi_{{{\eta}}}^{{+}\!}(x+\ell)})}
\,
{{\text{e}}}^{{{\text{i}}}({\varphi_{{{\eta}}}^{{{\vphantom{{+}}}}}(x)}-{\varphi_{{{\eta}}}^{{{\vphantom{{+}}}}}(x+\ell)})}
\,,\nonumber
\end{aligned}$$ A generating function of the terms in then reads $$\begin{aligned}
\label{eq:intro:Hq}
&\sum_{m=0}^{\infty}
\frac{(-{{\text{i}}}\ell)^{m}}{m!}
\,
H_{q{{\eta}}}^{(m)}
=
\sum_k
{{\text{e}}}^{-{{\text{i}}}k\ell}
\,
{{{}^\ast_\ast}{
{c^{{+}}_{k-q{{\eta}}}}
{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}
}{{}^\ast_\ast}}
\\&
=
\int \!\frac{{\ensuremath{\text{d}x}}}{2\pi}\,
{{\text{e}}}^{{{\text{i}}}qx}
\,
{{{}^\ast_\ast}{
{\psi_{{{\eta}}}^{+}(x)}
{\psi_{{{\eta}}}^{{\vphantom{{+}}}}(x+\ell)}
}{{}^\ast_\ast}}
=
\int \!\frac{{\ensuremath{\text{d}x}}}{L}\,
\frac{
{{\text{e}}}^{\pi{{\text{i}}}\delta_\text{b}\ell/L}
\,
{{\text{e}}}^{{{\text{i}}}qx}
}{1-{{\text{e}}}^{2\pi{{\text{i}}}\ell/L}}
\nonumber\\&~~~\times
\,
\big(
{{\text{e}}}^{-2\pi{{\text{i}}}{\hat{N}}_{{{\eta}}}\ell/L}
\,
{{\text{e}}}^{{{\text{i}}}{\varphi_{{{\eta}}}^{{+}\!}(x)}-{{\text{i}}}{\varphi_{{{\eta}}}^{{+}\!}(x+\ell)}}
\,
{{\text{e}}}^{{{\text{i}}}{\varphi_{{{\eta}}}^{{{\vphantom{{+}}}}}(x)}-{{\text{i}}}{\varphi_{{{\eta}}}^{{{\vphantom{{+}}}}}(x+\ell)}}-1\big)
\,,\nonumber
\end{aligned}$$ where we summed the Taylor series of the terms , inserted relation , and performed the normal ordering. Taylor expanding the exponentials and taking coefficients of $\ell^m$ on both sides of now yields $H_{q{{\eta}}}^{(m)}$ in terms of bosonic operators, as discussed below. Relation thus provides explicit bosonic representations of general bilinear fermionic operators, including .[^2] We also introduce operators which use the more convenient powers of the integer $n_k$ instead of momentum $k$,
\[eq:K-def\] $$\begin{aligned}
\label{eq:Kq}
\!\!\!\!\!\!
\!\!\!\!\!\!
\!\!\!\!\!\!
\!\!\!\!\!\!
{K^{\vphantom{(}}}_{q{{\eta}}}(\lambda)
&=
\sum_k
{{\text{e}}}^{\lambda n_k}
\;
{{{}^\ast_\ast}{
{c^{{+}}_{k-q{{\eta}}}}
{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}
}{{}^\ast_\ast}}
=
\sum_{m=0}^\infty
\frac{\lambda^m}{m!}
\,
K_{q{{\eta}}}^{(m)}
\,,
\\
\label{eq:Kqm}
K_{q{{\eta}}}^{(m)}
&=
\sum_k
n_{k}^{m}
\;
{{{}^\ast_\ast}{
{c^{{+}}_{k-q{{\eta}}}}
{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}
}{{}^\ast_\ast}}
\,,
\\
\label{eq:intro:Kq0}
K_{q{{\eta}}}^{(0)}
&=
\sum_k
{{{}^\ast_\ast}{
{c^{{+}}_{k-q{{\eta}}}}
{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}
}{{}^\ast_\ast}}
=
\begin{cases}
\phantom{-}{\hat{N}}_{{{\eta}}}
&
\text{\!\!\!\!if\,}
n_q\!=\!0,
\\
\phantom{-}{{\text{i}}}\sqrt{\smash[b]{n_q}}\phantom{{}_{-}}\,{b^{{{\vphantom{{+}}}}}_{q{{\eta}}}}
&\text{\!\!\!\!if\,}
n_q\!>\!0,
\\
-{{\text{i}}}\sqrt{n_{-q}}\,{b^{{+}}_{-q{{\eta}}}}
&\text{\!\!\!\!if\,}
n_{q}\!<\!0,
\end{cases}
\!\!\!\!\!\!
\end{aligned}$$
so that the terms are then given by $H_{q{{\eta}}}^{(m)}$ $=$ $(\pi/L)^m$ $\sum_{n=0}^{m}$ $\binom{m}{n}$ $(-\delta_\text{b})^{m-n}$ $2^n$ $K_{q{{\eta}}}^{(n)}$ and the bosonic commutation relations become $[K_{-q{{\eta}}}^{(0)},K_{q{^{\prime\!}}{{\eta'}}}^{(0)}]$ $=$ $-n_q\delta_{qq{^{\prime\!}}}\delta_{{{\eta}}{{\eta'}}}$. The operators ${K^{\vphantom{(}}}_{q{{\eta}}}(\lambda)$, which are operator-valued formal power series in the (complex) indeterminate $\lambda$ with coefficients $K_{q{{\eta}}}^{(m)}$, obey the intriguing operator algebra $$\begin{aligned}
\label{eq:intro:comm1}
\!\!\Big[{K^{\vphantom{(}}}_{-q{{\eta}}}(\lambda),{K^{\vphantom{(}}}_{q{^{\prime\!}}{{\eta'}}}(\lambda{^{\prime\!}})\Big]
&=
{{\delta}}_{{{\eta}}{{\eta'}}}\bigg[
\delta_{qq{^{\prime\!}}}\frac{{{\text{e}}}^{-\lambda n_{q}}-{{\text{e}}}^{\lambda{^{\prime\!}}n_{q}}}{1-{{\text{e}}}^{-\lambda-\lambda{^{\prime\!}}}}
\\&~~~
+
({{\text{e}}}^{-\lambda n_{q{^{\prime\!}}}}-{{\text{e}}}^{\lambda{^{\prime\!}}n_{q}}){K^{\vphantom{(}}}_{q{^{\prime\!}}-q{{\eta}}}(\lambda+\lambda{^{\prime\!}})
\bigg],\nonumber
\end{aligned}$$ which is reminiscient of affine Lie algebras,[@francesco_conformal_1997] but not immediately recognizable. From , or alternatively from , the generating function becomes
\[eq:intro:K-lambda-result\] $$\begin{aligned}
\!{K^{\vphantom{(}}}_{q{{\eta}}}(\lambda)
&=
\frac{
{{\text{e}}}^{\lambda{\hat{N}}_{{{\eta}}}}
Y_{q{{\eta}}}(\lambda)
-
\delta_{q0}
}{1-{{\text{e}}}^{-\lambda}}
\,,
\label{eq:intro:Kq-lambda-result}
\\
\!Y_{q{{\eta}}}(\lambda)
&=
\sum_{n,r=0}^{\infty}
\frac{1}{n!r!}
\sum_{\substack{p_1,\ldots,p_n>0\\p_1{^{\prime\!}},\ldots,p_r{^{\prime\!}}>0}}
\delta_{p_1+\cdots p_n+q,p_1{^{\prime\!}}+\cdots p_r{^{\prime\!}}}
\nonumber
\\
\nonumber
&\;\times\!
\Big(
\prod_{i=1}^n
\frac{1-{{\text{e}}}^{-\lambda n_{p_i}}}{n_{p_i}}
K_{-p_i{{\eta}}}^{(0)}
\Big)
\! \Big(
\prod_{j=1}^r
\frac{{{\text{e}}}^{\lambda n_{p_j{^{\prime\!}}}}-1}{n_{p_j{^{\prime\!}}}}
K_{p_j{^{\prime\!}}{{\eta}}}^{(0)}
\Big)
\\
&=
\sum_{m=0}^\infty
\frac{\lambda^m}{m!}
Y_{q{{\eta}}}^{(m)}
\,.
\end{aligned}$$
The coefficients $K_{q{{\eta}}}^{(m)}$ and $Y_{q{{\eta}}}^{(m)}$ of $\lambda^m$ in these expression are given by
\[eq:intro:Kqm:all\] $$\begin{aligned}
K_{q{{\eta}}}^{(m)}
&=
\sum_{m=0}^\infty
\frac{\lambda^m}{m!}
\,
K_{q{{\eta}}}^{(m)}
=
\frac{
B_{m+1}({\hat{N}}_{{{\eta}}}+1)-B_{m+1}(1)
}{m+1}
\delta_{q0}
\nonumber
\\&\label{eq:intro:Kqm}
+
\sum_{n=0}^{m}
\binom{m}{n}
\frac{
(-1)^n
}{
m+1-n
}
B_n(-{\hat{N}}_{{{\eta}}})
\,
Y_{q{{\eta}}}^{(m+1-n)}
\,,
\end{aligned}$$ $$\begin{aligned}
Y_{q{{\eta}}}^{(m)}
&=
\int \!\frac{{\ensuremath{\text{d}x}}}{L}
\,
{{\text{e}}}^{{{\text{i}}}qx}
\sum_{n=0}^{m}
\binom{m}{n}
{\mathfrak{B}}_{m}\big(K_{+{{,\eta}}}^{(1)\!}(x),...,K_{+{{,\eta}}}^{(m)\!}(x)\big)
\nonumber\\&\times
{\mathfrak{B}}_{m}\big(K_{-{{,\eta}}}^{(1)\!}(x),...,K_{-{{,\eta}}}^{(m)\!}(x)\big)
\,,
\end{aligned}$$
Here $K_{\pm{{,\eta}}}^{(m)\!}(x)$ $=$ $\sum_{\pm p>0} n_{-p}^{m-1} K_{-p{{\eta}}}^{(0)} {{\text{e}}}^{{{\text{i}}}px}$ and $B_n(x)$ and ${\mathfrak{B}}_m(x_1,\ldots,x_m)$ are the Bernoulli and complete Bell polynomials, respectively, defined by[@comtet_advanced_1974]
$$\begin{aligned}
\frac{
\lambda\,{{\text{e}}}^{\lambda x}
}{
{{\text{e}}}^{\lambda}-1
}
&=
\sum_{m=0}^\infty
\frac{\lambda^m}{m!}B_m(x)
\,,\\
\exp\!\bigg(
\sum_{m=1}^\infty
\frac{\lambda^m}{m!}\,x_m
\bigg)
&=
\sum_{m=0}^\infty
\frac{\lambda^m}{m!}
{\mathfrak{B}}_m(x_1,\ldots,x_m)
\,.
\end{aligned}$$
A detailed derivation of - will be presented elsewhere.
Bosonic representation of a fermionic scattering term
-----------------------------------------------------
Generalized Kronig identities for arbitrary order $m$ follow from the equivalence of and , with the latter involving only fermionic number operators and normal-ordered bosonic operators. As a special case, we obtain for $m$ $=$ $1$ and $q$ $\neq$ $0$ the finite-$q$ generalization of , $$\begin{aligned}
\!\!\!
K_{q{{\eta}}}^{(1)}
&=
\sum_k
n_k
\,
{{{}^\ast_\ast}{
{c^{{+}}_{k-q{{\eta}}}}
{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}
}{{}^\ast_\ast}}
\nonumber\\&
=
\Big(
\frac{n_q+1}{2}
+
{\hat{N}}_{{{\eta}}}
\Big)
K_{q{{\eta}}}^{(0)}
+
\frac12
\sum_{p(\neq0,q)}\!\!
K_{q-p{{\eta}}}^{(0)}
K_{p{{\eta}}}^{(0)}
\,,
\label{eq:intro:Kq1-bosonic}
\end{aligned}$$ which can also be expressed as
\[eq:Hq1\] $$\begin{aligned}
H_{q{{\eta}}}^{(1)}
&=
\sum_k
k
\,
{{{}^\ast_\ast}{
{c^{{+}}_{k-q{{\eta}}}}
{c^{{{\vphantom{{+}}}}}_{k{{\eta}}}}
}{{}^\ast_\ast}}
=\frac{2\pi}{L}K_{q{{\eta}}}^{(1)}-\frac{\pi\delta_{\text{b}}}{L}{\hat{N}}_{{{\eta}}}
\label{eq:Hq1-fermionic}
\\&
=
\Big(\frac{q}{2}+\frac{\pi}{L}(2{\hat{N}}_{{{\eta}}}+1-\delta_{\text{b}})\Big)
\,
{{\text{i}}}\sqrt{\smash[b]{n_q}}\,
{b^{{{\vphantom{{+}}}}}_{q{{\eta}}}}
\nonumber\\&~~~~
-
\frac12\!
\sum_{q>p>0}
\!
\sqrt{(q-p)p}\,
{b^{{{\vphantom{{+}}}}}_{q-p{{\eta}}}}
{b^{{{\vphantom{{+}}}}}_{p{{\eta}}}}
\nonumber\\&~~~~
+
\sum_{p>0}
\sqrt{p(q+p)}\,
{b^{{+}}_{p{{\eta}}}}
{b^{{{\vphantom{{+}}}}}_{q+p{{\eta}}}}
\,,
~~~~
(q>0)
\label{eq:Hq1-bosonic}
\end{aligned}$$
so as to make the modification of the momentum-diagonal identity more apparent.
Chiral Luttinger droplets {#sec:chiral}
=========================
Droplet model with only right movers
------------------------------------
As a simple application of and for later reference we first consider a single species of spinless fermions with density $$\begin{aligned}
n_{{{}}}(x) =
\frac{1}{2\pi}\,{\psi_{{{}}}^{+}(x)}{\psi_{{{}}}^{{\vphantom{{+}}}}(x)} =
\frac{1}{L}\,\sum_qK_{q{{}}}^{(0)}e^{-iqx}\,,
\end{aligned}$$ subjected to a single-particle potential $w(x)$ $=$ $w(-x)$ and a position-dependent interaction $g(x)$ $=$ $g(-x)$, with Fourier transforms $w_q$ $=$ $\int\!w(x)\,{{\text{e}}}^{-{{\text{i}}}qx}{\ensuremath{\text{d}x}}/L$ $=$ $w_{-q}$ and so on. For simplicity we choose antiperiodic boundary conditions ($\delta_{\text{b}}$ $=$ $1$). For a linear dispersion the Hamiltonian of such a ‘chiral Luttinger droplet’ is given by $$\begin{aligned}
\label{eq:chiralH}
H_{\text{chiral}}
&=
{v_{\text{F}}}\sum_{k}
k
\,
{{{}^\ast_\ast}{
{c^{{+}}_{k{{}}}}
{c^{{{\vphantom{{+}}}}}_{k{{}}}}
}{{}^\ast_\ast}}
+
\int \!\frac{{\ensuremath{\text{d}x}}}{L}\,
w(x)\,
{{{}^\ast_\ast}{
n_{{{}}}(x)
}{{}^\ast_\ast}}
\\\nonumber&\,\hphantom{=}\,
+
\frac12
\int \!
\frac{{\ensuremath{\text{d}x}}}{L}\,
g(x)\,
{{{}^\ast_\ast}{
n_{{{}}}(x)^2
}{{}^\ast_\ast}}
\,.
\end{aligned}$$
Diagonalization of the chiral model
-----------------------------------
On the one hand, we can now express the fermionic Hamiltonian $H_{\text{chiral}}$ in terms of bosonic operators. We define $$\begin{aligned}
H_{[\bm{\tilde{g}},\hat{\bm{\tilde{w}}};\bm{K}]}^{\text{bosonic}}
&=
\frac{\tilde{g}_0}{L}
\sum_{q>0}
K_{-q{{}}}^{(0)}
K_{q{{}}}^{(0)}
\label{eq:chiralHbosonic} \\&\,\hphantom{=}\,
+
\frac{1}{L}
\sum_{q\neq0}
\bigg[
\hat{\tilde{w}}_q
K_{q{{}}}^{(0)}
+
\frac{\tilde{g}_q}{2}
\!\sum_{p(\neq0,q)}\!
K_{p{{}}}^{(0)}
K_{q-p{{}}}^{(0)}
\bigg]
\,,
\nonumber
\end{aligned}$$ with symmetric parameters $\hat{\tilde{w}}_q$ (that may contain ${\hat{N}}$) and $\tilde{g}_q$. For $\tilde{g}_q$ $=$ $2\pi {v_{\text{F}}}\delta_{q0}$ $+$ $g_q$ and $\hat{\tilde{w}}_q$ $=$ $w_qL$ $+$ $g_q{\hat{N}}_{{{}}}$ we find that $$\begin{aligned}
H_{\text{chiral}}
&=
H_{[\bm{\tilde{g}},\hat{\bm{\tilde{w}}};\bm{K}]}^{\text{bosonic}}
+
\frac{\tilde{g}_0}{2L}{\hat{N}}_{{{}}}^2
+
w_0{\hat{N}}_{{{}}}
\,.
\end{aligned}$$ On the other hand, the fermionic basis permits a full diagonalization as follows. Using to eliminate the last term in we arrive at a fermionic scattering Hamiltonian, $$\begin{aligned}
\label{eq:chiralHfermionic} H_{\text{chiral}}
&=
\sum_{kk{^{\prime\!}}}
T_{kk{^{\prime\!}}}
\,
{{{}^\ast_\ast}{
{c^{{+}}_{k{{}}}}
{c^{{{\vphantom{{+}}}}}_{k{^{\prime\!}}{{}}}}
}{{}^\ast_\ast}}
\,,
\\
T_{kk{^{\prime\!}}}
&=
{v_{\text{F}}}k\,\delta_{kk{^{\prime\!}}}
+
w_{k{^{\prime\!}}-k}
+
(k+k{^{\prime\!}}\, )\frac{g_{k{^{\prime\!}}-k}}{4\pi}
\,.\nonumber
\end{aligned}$$ We conclude that the four-fermion interaction terms in cancel, as they do in the Kronig identity . In terms of field operators we obtain
\[eq:chiralHfermionicrealspace\] $$\begin{aligned}
H_{\text{chiral}}
&=
\int \!{\ensuremath{\text{d}x}}\,
{{{}^\ast_\ast}{
\frac{
{\psi_{{{}}}^{+}(x)}
h(x)
{\psi_{{{}}}^{{\vphantom{{+}}}}(x)}
}{2\pi}
}{{}^\ast_\ast}}
\,,
\\
h(x)
&=
\tilde{g}(x)(-{{\text{i}}}s\partial_x)-\frac12{{\text{i}}}s\tilde{g}{^{\prime\!}}(x)+w(x)
\,,\label{eq:linearkinetic}
\end{aligned}$$
where $\tilde{g}(x)$ $=$ $2\pi {v_{\text{F}}}$ $+$ $g(x)$ as above, and $s$ $=$ $-1/(2\pi)$.
Next we use the spectrum of the first-quantized Hamiltonian in , $h$ $=$ $s[\tilde{g}(X)P+P\tilde{g}(X)]/2+w(X)$ with $[X,P]$ $=$ ${{\text{i}}}$. The eigenvalue equation $h(x)\xi_k(x)$ $=$ $E_k\xi_k(x)$ is separable because $h$ is linear in $P$. For a constant real scale $s$ and real functions $\tilde{g}(x)$, $w(x)$ on an interval $[x_1,x_2]$ with $\tilde{g}(x)$ $>$ $0$ and $\tilde{g}(x_1)$ $=$ $\tilde{g}(x_2)$, and demanding $\xi_k(x_2)$ $=$ $\xi_k(x_1){{\text{e}}}^{\pi{{\text{i}}}\delta_\text{b}}$, we find $E_k$ $=$ $(S_1$ $-$ $sLk)/S_0$, $\xi_k(x)$ $=$ $[\tilde{g}(x)S_0]^{-\frac12}\exp({{\text{i}}}[s_0(x,0)E_k$ $-$ $s_1(x,0)]/s)$, where the momentum $k$ takes on the same discrete values $k_n$ as before. Here $S_j$ $=$ $s_j(x_2,x_1)$ with $s_j(x,x')$ $=$ $\int_{x'}^x{\ensuremath{\text{d}y}}\,(\delta_{j0}+\delta_{j1}w(y))/\tilde{g}(y)$. These eigenstates correspond to plane waves subject to a local scale transformation induced by the interaction potential, reminiscient of eikonal wave equations or semiclassical Schrödinger equations. We note the eigenstate expectation values ${\langle{w(X)}\rangle}$ $=$ $S_1/S_0$ $=$ $E_k-s{\langle{P}\rangle}$.
Setting $x_1$ $=$ $-x_2$ $=L/2$ and $\delta_\text{b}$ $=$ $1$ and requiring $g_{q=0}$ $>$ $-2\pi {v_{\text{F}}}$, we thus diagonalize , , in terms of new canonical fermions, $\{{\Xi_{k{{}}}^{{\vphantom{{+}}}}},{\Xi_{k{^{\prime\!}}{{}}}^{+}}\}$ $=$ $\delta_{kk{^{\prime\!}}}$, as $$\begin{aligned}
\label{eq:chiralH_diagonal} H_{\text{chiral}}
&=
\sum_kE_k\,{{{}^\ast_\ast}{{\Xi_{k{{}}}^{+}}{\Xi_{k{{}}}^{{\vphantom{{+}}}}}}{{}^\ast_\ast}}
\equiv
H_{[\bm{\tilde{g}},\bm{w};\bm{{\Xi_{{{}}}^{{\vphantom{{+}}}}}}]}^{\text{diagonal}}
\,,
\\
E_k
&=
\tilde{v}(k-\tilde{k})
\,,
\,~~
{\hat{N}}=
\sum_k
{{{}^\ast_\ast}{
{\Xi_{k{{}}}^{+}}
{\Xi_{k{{}}}^{{\vphantom{{+}}}}}
}{{}^\ast_\ast}}
=
\sum_k
{{{}^\ast_\ast}{
{c^{{+}}_{k{{}}}}
{c^{{{\vphantom{{+}}}}}_{k{{}}}}
}{{}^\ast_\ast}}
\,,\nonumber
\\
{\Xi_{k{{}}}^{{\vphantom{{+}}}}}
&=
\int\!\frac{{\ensuremath{\text{d}x}}}{\sqrt{2\pi}}\,\xi_k(x)\,
{\psi_{{{}}}^{{\vphantom{{+}}}}(x)}
\,,~~
\tilde{k}
=
-\int\!\frac{{\ensuremath{\text{d}x}}}{L}\,\frac{2\pi w(x)}{\tilde{g}(x)}
\,,\nonumber
\\
\xi_k(x)
&=
\frac{
\sqrt{2\pi\tilde{v}}\;
{{\text{e}}}^{-{{\text{i}}}[\tilde{r}_0(x)k-\tilde{r}_1(x)]}
}{
\sqrt{L\,\tilde{g}(x)}
}
\,,~
\tilde{r}_0(x)
=
\int_0^x\!{\ensuremath{\text{d}y}}\,\frac{2\pi\tilde{v}}{\tilde{g}(y)}
\,,
\nonumber\\
\tilde{r}_1(x)
&=
\tilde{k}\,\tilde{r}_0(x)
+ \int_0^x\!{\ensuremath{\text{d}y}}\,\frac{2\pi w(y)}{\tilde{g}(y)}
,~
\tilde{v}
=
\bigg[\int\!\frac{{\ensuremath{\text{d}x}}}{L}\,\frac{2\pi}{\tilde{g}(x)}\bigg]^{-1}
\!.\nonumber
\end{aligned}$$ Note that the renormalized dressed Fermi velocity $\tilde{v}$ is given by the spatial harmonic average of the renormalized ‘local’ Fermi velocity ${v_{\text{F}}}$ $+$ $g(x)/(2\pi)$ $=$ $\tilde{g}(x)/(2\pi)$.
Green function for the chiral model
-----------------------------------
From the above solution it is straightforward to obtain the time-ordered Green function for the Heisenberg operators of the chiral field, $$\begin{aligned}
G(x,x{^{\prime\!}};t)
&=
\theta(t){G^>}(x,x{^{\prime\!}};t)
-
\theta(-t){G^<}(x,x{^{\prime\!}};t)
\,,
\\
{G^{\gtrless}}(x,x{^{\prime\!}};t)
&=
\begin{cases}
-{{\text{i}}}{\langle{{\psi_{{{}}}^{{\vphantom{{+}}}}(x,t)}\,{\psi_{{{}}}^{+}(x{^{\prime\!}},0)}}\rangle}
\,,\\
-{{\text{i}}}{\langle{{\psi_{{{}}}^{+}(x{^{\prime\!}},0)}\,{\psi_{{{}}}^{{\vphantom{{+}}}}(x,t)}}\rangle}
\,,
\end{cases}
\label{eq:green_chiral}
\end{aligned}$$ with $\theta(\pm t)$ $=$ $(1\pm\text{sgn}(t))/2$. At zero temperature in a state with fixed particle number $N$ we find $$\begin{aligned}
\label{eq:Gchiral_zerotemp}
{{\text{i}}}G(x,x{^{\prime\!}};t)
&=
\frac{
\tilde{v}
}{
\sqrt{
\tilde{g}(x)\tilde{g}(x{^{\prime\!}})
}
}
\frac{
{{\text{e}}}^{
{{\text{i}}}S(x,x',t)
}
}{
\frac{L}{\pi}{\sinh}\frac{\pi}{L}({{\text{i}}}R(x,x',t)+a\,\text{sgn}\,t)
}
\,,\nonumber
\\
R(x,x',t)
&=
\tilde{r}_0(x)-\tilde{r}_0(x')-\tilde{v}t
\,,
\\
S(x,x',t)
&=
\tilde{r}_1(x)
- \tilde{r}_1(x')
+\tilde{v}\tilde{k}t
-\frac{2\pi N}{L} R(x,x',t)
\,,\nonumber
\end{aligned}$$ where $a$ $\to$ $0^+$ stems from a convergence factor that was included in the momentum sum. For constant $g(x)$ and $w(x)$ we recover the translationally invariant case, $G(x,x{^{\prime\!}};t)$ $\propto$ $1/(x-x'+\tilde{v}t+a\,\text{sgn}\, t)$, with renormalized Fermi velocity. Position-dependent couplings, on the other hand, may lead to a substantial redistribution of spectral weight. The critical behavior however remains unaffected, in the sense that the exponent of the denominator involving $R(x,x',t)$ remains unity for the chiral model.
Luttinger droplets {#sec:droplet}
==================
Droplet model with with right and left movers
---------------------------------------------
We now study a generalization of the two-flavor Tomonaga-Luttinger model to position-dependent interactions and scattering potentials. Such a ‘Luttinger droplet’ involves right- and left-moving fermions, ${\psi_{\text{R}}^{{\vphantom{{+}}}}(x)}$ $=$ ${\psi_{1}^{{\vphantom{{+}}}}(-x)}$ and ${\psi_{\text{L}}^{{\vphantom{{+}}}}(x)}$ $=$ ${\psi_{2}^{{\vphantom{{+}}}}(x)}$ (see introduction) with linear dispersion in opposite directions, subject to the one-particle potential $W(x)$, as well as intrabranch and interbranch density interactions $V(x)$ and $U(x)$, respectively, as given in . In terms of fermions with flavor $\eta$ $=$ $1,2$ we have $$\begin{aligned}
&H =
{{}^\ast_\ast}\sum_{\eta}
\Bigg[
{v_{\text{F}}}\sum_{k}
k \,
{c^{{+}}_{k\eta}}
{c^{{{\vphantom{{+}}}}}_{k\eta}}
+
\int \!
\frac{{\ensuremath{\text{d}x}}}{L}\,
W(x)
n_{\eta}(x)
\nonumber\\&~~
+
\frac12V(x)n_{\eta}(x)^2
\Bigg]
+
\int \!\frac{{\ensuremath{\text{d}x}}}{L}\,
U(x)
n_{1}(-x)n_{2}(x)
{{}^\ast_\ast}\,,
\end{aligned}$$ i.e., compared to the couplings $g(x)$ and $w(x)$ were relabeled as $V(x)$ and $W(x)$, indices $\eta$ were put on operators, and the interaction term with $U(x)$ was included.
Diagonalization of the Luttinger droplet model
----------------------------------------------
### Bosonic form of the Hamiltonian
Rewritten with bosonic operators this becomes $$\begin{aligned}
\label{eq:dropletHbosonic}
H &=
H_\text{TL}+H'+H''
\,,
\\
H_\text{TL}
&=
\sum_\eta
\bigg[
\frac{2\pi {v_{\text{F}}}+V_0}{L}
\bigg(
\frac{{\hat{N}}_{\eta}^2}{2}
+
\sum_{q>0}
K_{-q\eta}^{(0)}
K_{q\eta}^{(0)}
\bigg)
\bigg]
\nonumber\\&\,\hphantom{=}\,
+
\frac{U_0}{L}
\bigg[
{\hat{N}}_{1}{\hat{N}}_{2}
+
\sum_{q>0}
\bigg(
K_{-q1}^{(0)}
K_{-q2}^{(0)}
+
K_{q1}^{(0)}
K_{q2}^{(0)}
\bigg)
\bigg]
,\nonumber
\\
H'
&=
\sum_{\eta}
\bigg[
W_0
{\hat{N}}_{\eta}
+
\sum_{q\neq0}
\bigg(
W_q
+
\frac{V_q}{L}
{\hat{N}}_{\eta}
+
\frac{U_q}{2L}
{\hat{N}}_{\bar{\eta}}
\bigg)
K_{q\eta}^{(0)}
\bigg]
,\nonumber
\\
H''
&=
\sum_{\eta}
\sum_{q\neq0}
\sum_{p(\neq0,q)}\!\!
K_{p\eta}^{(0)}
\bigg[
\frac{V_q}{2L}
K_{q-p\eta}^{(0)}
+
\frac{U_q}{2L}
K_{p-q\bar{\eta}}^{(0)}
\bigg]
\,.\nonumber
\end{aligned}$$ $H$ contains a standard (i.e., translationally invariant) Tomonaga-Luttinger model $H_\text{TL}$ involving only the zero-momentum (space-averaged) couplings, which by itself can be diagonalized by a Bogoljubov transformation. For position-dependent couplings, on the other hand, also $H'$ (linear in bosons) and $H''$ (quadratic in bosons with momentum transfer) are present.
### Specialization to common spatial dependence
For simplicity we set from now on $$\begin{aligned}
\label{eq:dropletspatialdependence} \binom{V(x)}{U(x)}
&=
\binom{V_0}{U_0}
+
\binom{V}{U}
\sum_{q\neq0}f_q\cos(qx)
\,,
\end{aligned}$$ with constant prefactors $V$ and $U$ and $f_q$ $=$ $V_q/V$ $=$ $U_q/U$ $=$ $f_{-q}$ for $q$ $\neq$ $0$. We can then simplify the momentum-offdiagonal term $H''$ by a Bogoljubov transformation to $K_{q\sigma}^{(0)}$ (for $q$ $\neq$ $0$, $\sigma$ $=$ $-\bar{\sigma}$ $=$ $\pm$, letting $\eta_{\sigma}$ $=$ $(3$$-$$\sigma)/2$, $\sigma_\eta$ $=$ $3$$-$$2\eta$ for $\eta$ $=$ $3$$-$$\bar{\eta}$ $=$ $1,2$),
\[eq:main:bogoljubov\] $$\begin{aligned}
K_{q\sigma}^{(0)}
&=
{{u}}\,
K_{q\eta_{\sigma}}^{(0)}
+
{{v}}\,
K_{-q\bar{\eta}_{\sigma}}^{(0)}
\,,
\\
K_{q\eta}^{(0)}
&=
{{u}}\,
K_{q\sigma_\eta}^{(0)}
-
{{v}}\,
K_{-q\bar{\sigma}_\eta}^{(0)}
\,,
\end{aligned}$$
$u$ $=$ ${\cosh}\!\theta$, $v$ $=$ ${\sinh}\!\theta$, which preserves the bosonic algebra, $[K_{-q\sigma}^{(0)},K_{q{^{\prime\!}}\sigma{^{\prime\!}}}^{(0)}]$ $=$ $-n_q\delta_{qq{^{\prime\!}}}\delta_{\sigma\sigma{^{\prime\!}}}$. The choice $U/V$ $=$ ${\mathop{\text{tanh}}}2\theta$, assuming $|U|$ $<$ $V$, yields $$\begin{aligned}
\label{eq:dropletHbosonicsigma}
H &=
\sum_{\sigma=\pm}\big(H_{\sigma}^{(0)}+H_{\sigma}^{(1)}\big)+H^{(2)}+{\hat{H}}_\text{N}+E_0
\,,
\\
\!\!\!\!
H_{\sigma}^{(0)}
&=
\frac{\bar{V}}{L}
\sum_{q>0}
K_{-q\sigma}^{(0)}
K_{q\sigma}^{(0)}
+
\sum_{q\neq0}
\frac{\bar{U}\!f_q}{2L}
\sum_{p(\neq0,q)}\!\!
K_{p\sigma}^{(0)}
K_{q-p\sigma}^{(0)}
\,,\nonumber
\\
\!\!\!\!
H_{\sigma}^{(1)}
&=
\frac{1}{L}
\sum_{q\neq0}
\hat{\bar{w}}_{q\sigma}
K_{q\sigma}^{(0)}
\,,~~~~
H^{(2)}
=
\frac{\bar{V}'}{L}
\sum_{q\neq0}
K_{q+}^{(0)}
K_{q-}^{(0)}
\,,\nonumber
\\
{\hat{H}}_{\text{N}}
&=
\frac{2\pi {v_{\text{F}}}+V_0}{2L}\sum_\eta{\hat{N}}_\eta^2+\frac{U_0}{L}{\hat{N}}_1{\hat{N}}_2
+W_0\sum_\eta{\hat{N}}_\eta
\end{aligned}$$ where $E_0$ is a constant energy shift, omitted from now on, which diverges due to the contact interactions in $H$. Here and below we use the following abbreviations and relations, $$\begin{aligned}
\label{eq:ham_par_bos}
\bar{V}
&=
\frac{(2\pi {v_{\text{F}}}+V_0)V-U_0U}{\bar{U}}
\,,~~
\bar{U}
=
\bar{\gamma}V
,~
\\
\bar{V}'
&=
\frac{U_0V-(2\pi {v_{\text{F}}}+V_0)U}{\bar{U}}
\,,\nonumber
\\
\hat{\bar{w}}_{q\sigma}
&=
LW_q{{\text{e}}}^{-\theta}
+
\bar{\gamma}\,V_q
\big[
{{u}}^3{\hat{N}}_1\delta_{\sigma+}
-
{{v}}^3{\hat{N}}_2\delta_{\sigma-}
\big]
,~(q\neq0)\!\!
\nonumber
\\
\gamma
&=
\frac{U}{V}={\mathop{\text{tanh}}}\!2\theta
\,,~
\bar{\gamma}
=
\sqrt{1-\gamma^2}
=
\text{sech}{2\theta}
\,,\nonumber
\\
\gamma_3
&=
u^3-v^3
=(1+\tfrac{1}{2}\gamma)(1-\gamma)^{-\frac14}(1+\gamma)^{-\frac34}
\,,\nonumber
\\
{{\text{e}}}^{-\theta}
&=
(V-U)^{\frac14}(V+U)^{-\frac14}
=
(1-\gamma)^{\frac14}(1+\gamma)^{-\frac14}
\,,\nonumber
\\
2v^2
&=
2\sinh^2\!\theta=(1-\gamma^2)^{-\frac12}-1
\,.\nonumber
\end{aligned}$$ The Hamiltonian $H$ has thus become diagonal in the new flavors $\sigma$ except for the term $H^{(2)}$ in .
### Specialization to interrelated interaction strengths
For simplicity we now assume that $\bar{V}'$ $=$ $0$, i.e., that the bare Fermi velocity ${v_{\text{F}}}$ and the strengths of the position-averaged ($V_0$ and $U_0$) and position-dependent interactions ($V$ and $U$) combine so that $H^{(2)}$ is absent. This corresponds to the special case $$\begin{aligned}
\gamma
&=
\frac{U}{V}=\frac{U_0}{2\pi {v_{\text{F}}}+V_0}
\,,\label{eq:gamma-specialcase}
\end{aligned}$$ which together with is equivalent to . From now on we will thus consider ${v_{\text{F}}}$, $V_q$, $\gamma$ to be chosen freely (with $V_0$ $>$ $-2\pi{v_{\text{F}}}$), with the other parameters in $H$ then being given by
$$\begin{aligned}
U_q&=\gamma
\,
(2\pi {v_{\text{F}}}\delta_{q0}+V_q)
\,,
\\
\bar{V}
&=
\bar{\gamma}
\,
(2\pi {v_{\text{F}}}+V_0)
\,,~
\bar{U}
=
\bar{\gamma}
\,
V
\,,~
\bar{V}'
=
0
\,,~
\end{aligned}$$
i.e., $\bar{U} f_q$ $=$ $\bar{\gamma} V_q$ for $q$ $\neq$ $0$. Then for $\sigma$ $=$ $\pm1$ each decoupled Hamiltonian has precisely the form of the bosonic Hamiltonian encountered in the chiral model, $$\begin{aligned}
\label{eq:dropletHbosonic-sigmas} H
&=
{\hat{H}}_\text{N}
+
\sum_{\sigma=\pm}
{H}_{\sigma}
\,,\\
H_{\sigma}
&
=
H_{\sigma}^{(0)}+H_{\sigma}^{(1)}
=
H_{[\bar{\bm{g}},\hat{\bar{\bm{w}}}_\sigma;\bm{K}_\sigma]}^{\text{bosonic}}
\,,\nonumber
\end{aligned}$$ with effective interaction $\bar{g}_{q}$ $=$ $\bar{V}\delta_{q0} + (1-\delta_{q0})\bar{U}f_q$, i.e.,
$$\begin{aligned}
\bar{g}_{q}
&=
\bar{\gamma}
\,
[2\pi {v_{\text{F}}}\delta_{q0}+V_q]
\,,
~~
\bar{g}(x)
=
\bar{\gamma}
\,
[2\pi {v_{\text{F}}}+V(x)]
\,,
\\
\bar{v}
&=
\bigg[\int\!\frac{{\ensuremath{\text{d}x}}}{L}\,\frac{2\pi}{\bar{g}(x)}\bigg]^{-1}
\,,\label{eq:vbar+Wbar-def}
~~
\bar{W}
=
\int\!\frac{{\ensuremath{\text{d}x}}\,2\pi\bar{v}}{L}\,\frac{W(x)}{\bar{g}(x)}
\,,
\end{aligned}$$
where we also introduced the renormalized Fermi velocity $\bar{v}$ and averaged one-particle potential $\bar{W}$ which will emerge below.
### Refermionization as separately diagonalizable chiral models
We thus refermionize each $H_\sigma$, first in terms of new fermions ${\psi_{\sigma}^{{\vphantom{{+}}}}(x)}$, with bosonic fields $\phi_\sigma(x)$ $=$ ${\varphi_{\sigma}^{{+}\!}(x)}$ $+$ ${\varphi_{\sigma}^{{{\vphantom{{+}}}}}(x)}$ built from the $K_{q\sigma}^{(0)}$ analogously to , $$\begin{aligned}
{\psi_{\sigma}^{{\vphantom{{+}}}}(x)}
&=
\sqrt{\frac{2\pi}{L}}\;
\sum_{k} {{\text{e}}}^{-{{\text{i}}}kx} {c^{{{\vphantom{{+}}}}}_{k\sigma}}
\nonumber\\
&=
\frac{{F_{\sigma}^{{\vphantom{{+}}}}}}{\sqrt{a}}
\,
{{\text{e}}}^{-{{\text{i}}}\frac{2\pi}{L}({\hat{N}}_{\sigma}-\frac12)x}
\,
{{\text{e}}}^{-{{\text{i}}}\phi_\sigma(x)}
\,.
\label{eq:refermionization:sigma}
\end{aligned}$$ Below we will fix the connection between the fermionic number operators ${\hat{N}}_\sigma$ and their associated Klein factors ${F_{\sigma}^{{\vphantom{{+}}}}}$ to the original fermions ${c^{{{\vphantom{{+}}}}}_{k\eta}}$, which is not determined by the purely bosonic Bogoljubov transformation .
Next each chiral-type Hamiltonian $H_\sigma$ is diagonalized with fermions $\Xi_{k\sigma}$ according to , $$\begin{aligned}
\label{eq:X-refermionize}
H_\sigma
=
H_{[\bar{\bm{g}},\hat{\bar{\bm{w}}}_\sigma;\bm{K}_\sigma]}^{\text{bosonic}}
&=
H_{[\bar{\bm{g}},\hat{\bar{\bm{w}}}_\sigma;\bm{\Xi}_{\sigma}]}^{\text{diagonal}}
-
\frac{\bar{g}_0}{2L}{\hat{N}}_{\sigma}^2
\,,
\\
H_{[\bar{\bm{g}},\hat{\bar{\bm{w}}}_\sigma;\bm{\Xi}_{\sigma}]}^{\text{diagonal}}
&=
\bar{v}\sum_k(k-\hat{\bar{k}}_{\sigma}){{{}^\ast_\ast}{{\Xi_{k\sigma}^{+}}{\Xi_{k\sigma}^{{\vphantom{{+}}}}}}{{}^\ast_\ast}}
\,.
\nonumber
\end{aligned}$$ with the two types of fermions $\psi_\sigma$ and $\Xi_\sigma$ related by
$$\begin{aligned}
{\Xi_{k\sigma}^{{\vphantom{{+}}}}}
&=
\int\!\frac{{\ensuremath{\text{d}x}}}{\sqrt{2\pi}}\,\xi_{k\sigma}(x)\,
{\psi_{\sigma}^{{\vphantom{{+}}}}(x)}
\,,
\label{eq:Xi-k-sigma-definition}
\\
{\hat{N}}_\sigma
&=
\sum_k{{{}^\ast_\ast}{{\Xi_{k\sigma}^{+}}{\Xi_{k\sigma}^{{\vphantom{{+}}}}}}{{}^\ast_\ast}}
= \sum_k {{{}^\ast_\ast}{{c^{{+}}_{k\sigma}} {c^{{{\vphantom{{+}}}}}_{k\sigma}}}{{}^\ast_\ast}}
\label{eq:Nsigma-densities}
\,,
\end{aligned}$$
in terms of the following functions and parameters
\[eq:xi+r0\] $$\begin{aligned}
\xi_{k\sigma}(x)
&=
\frac{
\sqrt{2\pi\bar{v}}\;
{{\text{e}}}^{-{{\text{i}}}[r_0(x)k-\hat{r}_{1\sigma}
(x)]}
}{
\sqrt{L\,\bar{g}(x)}
}
\,,
\\
r_0(x)
&=
\int_0^x\!{\ensuremath{\text{d}y}}\,\frac{2\pi\bar{v}}{\bar{g}(y)}
=
-r_0(-x)
\,,
\\
\hat{r}_{1\sigma}(x)
&=
\hat{\bar{k}}_{\sigma}\,r_0(x)
+ \int_0^x\!{\ensuremath{\text{d}y}}\,\frac{2\pi \hat{\bar{w}}_{\sigma}(y)}{\bar{g}(y)}
\,,
\\
\hat{\bar{k}}_{\sigma}
&=
-\int\!\frac{{\ensuremath{\text{d}x}}}{L}\,\frac{2\pi}{\bar{g}(x)}\sum_{q\neq0}\frac{\hat{\bar{w}}_{q\sigma}}{L}{{\text{e}}}^{-{{\text{i}}}qx}
\,.
\end{aligned}$$
### Rebosonization into canonical form with quadratic number operator terms
Due to the linear dispersion we can rebosonize the $\Xi_{k\sigma}$ in terms of new canonical bosons ${B^{{{\vphantom{{+}}}}}_{q\sigma\!}}$, which will also be needed for the calculation of Green functions below. The corresponding (re-)bosonization identity reads $$\begin{aligned}
\!\!\!
\Xi_{\sigma}(x)
&=
\frac{{{\cal F}_{\sigma}^{{\vphantom{{+}}}}}}{\sqrt{a}}
{{\text{e}}}^{-\frac{2\pi{{\text{i}}}}{L}({\hat{N}}_\sigma-\frac12)x +\sum\limits_{q>0}\!\!
\frac{{{\text{i}}}{{\text{e}}}^{-a|q|}}{\sqrt{n_q}}
\![
{B^{{{\vphantom{{+}}}}}_{q\sigma}}{{\text{e}}}^{-{{\text{i}}}qx}
+
{B^{{+}}_{q\sigma}}{{\text{e}}}^{{{\text{i}}}qx}
]\!
},\!\!\!\label{eq:Xi_rebosonization}
\end{aligned}$$ where ${{\cal F}_{\sigma}^{{\vphantom{{+}}}}}$ is another Klein factor which lowers ${\hat{N}}_\sigma$ by one. We note that once we fix ${F_{\sigma}^{{\vphantom{{+}}}}}$, then ${{\cal F}_{\sigma}^{{\vphantom{{+}}}}}$ is determined by , , , although its explicit form is not needed in the following. The transformation yields $$\begin{aligned}
\label{eq:droplet-but-numberoperators}
H
&=
H_{\Xi}+H_\text{N}
-
\sum_\sigma
\bigg(
\frac{\bar{g}_0}{2L}{\hat{N}}_\sigma^2
+
\bar{v}
\bar{k}_{\sigma}{\hat{N}}_{\sigma}
\bigg)
,\!\!
\\
H_{\Xi}
&=
\sum_{\sigma;k}\bar{v}k\,{{{}^\ast_\ast}{{\Xi_{k\sigma}^{+}}{\Xi_{k\sigma}^{{\vphantom{{+}}}}}}{{}^\ast_\ast}}
=
\sum_{\sigma;q>0}\bar{v}q{B^{{+}}_{q\sigma}}{B^{{{\vphantom{{+}}}}}_{q\sigma}}
+
\frac{\pi}{2L}\sum_\sigma{\hat{N}}_\sigma^2
,\nonumber
\end{aligned}$$ We observe that even for position-dependent interactions, collective bosonic excitations with linear dispersion emerge.
To complete the diagonalization of $H$ in , we must still define the new number operators ${\hat{N}}_\sigma$ (with integer eigenvalues) and Klein factors ${F_{\sigma}^{{\vphantom{{+}}}}}$ in terms of the original ${\hat{N}}_{\eta}$ and ${F_{\eta}^{{\vphantom{{+}}}}}$ (which also appear in $H_\text{N}$). We set $$\begin{aligned}
{\hat{N}}_\sigma
&=
{\hat{N}}_1\delta_{\sigma-}
+
{\hat{N}}_2\delta_{\sigma+}
\,,
\end{aligned}$$ which ensures that the ground state (without bosonic excitations ${B^{{+}}_{q\sigma\!}}$) remains in a sector with finite ${\hat{N}}_1$ $=$ ${\hat{N}}_2$, because then only the density terms $({\hat{N}}_1^2+{\hat{N}}_2^2)$ and ${\hat{N}}_1{\hat{N}}_2$ appear in the Hamiltonian. We note that no other form of ${\hat{N}}_\sigma$ that is linear in ${\hat{N}}_1$ and ${\hat{N}}_2$ has this feature The corresponding Klein factors are then given by $$\begin{aligned}
{F_{\sigma}^{{\vphantom{{+}}}}}
&=
{F_{\mathrm{1}}^{{\vphantom{{+}}}}}\delta_{\sigma-}
+
{F_{\mathrm{2}}^{{\vphantom{{+}}}}}\delta_{\sigma+}
\,.
\end{aligned}$$ Collecting terms, the diagonalization of the Luttinger droplet Hamiltonian is then finally complete, $$\begin{aligned}
\!\!H
&=
\sum_{\sigma;q>0}\bar{v}q{B^{{+}}_{q\sigma}}{B^{{{\vphantom{{+}}}}}_{q\sigma}}
+
\frac{\pi}{2L}\big[v_{{{\cal N}}}{\hat{\cal N}}^2+v_{{{\cal J}}}{\hat{\cal J}}^2]
+
\epsilon{\hat{\cal N}}\,,\!\!\!\!
\,,\label{eq:luttingerdroplet-diagonalized}
\\
{\hat{\cal N}}&=
{\hat{N}}_1+{\hat{N}}_2
\,,~~
{\hat{\cal J}}=
{\hat{N}}_1-{\hat{N}}_2
\,,\nonumber
\end{aligned}$$ in which the following parameters appear, $$\begin{aligned}
v_{{{\cal N}}}
&=
v_1+v_2
\,,
&
v_{{{\cal J}}}
&=v_1-v_2
\,,\nonumber
\\
v_1
&=
\tilde{v}_\text{F}+\Delta v
\,,
&
\tilde{v}_\text{F}
&={v_{\text{F}}}+\tfrac{1}{2\pi}V_0
\,,\nonumber
\\
v_2
&=
\gamma\tilde{v}_\text{F}+\gamma_3\Delta v\,,
&
\Delta v&=\bar{v}-\bar{\gamma}\tilde{v}_\text{F}
\,,\nonumber
\\
\epsilon
&=\bar{W}{{\text{e}}}^{-\theta}
+W_0(1-{{\text{e}}}^{-\theta})
\,,\label{eq:luttingerdroplet-velocities}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\end{aligned}$$ and $\bar{v}$ and $\bar{W}$ were defined in . Here the total and relative fermionic number operators, ${\hat{\cal N}}$ and ${\hat{\cal J}}$, take on integer values and commute with the two flavors of bosonic operators. We note the ground-state value of ${\hat{\cal N}}$ may shift due to the one-particle potential $W(x)$ according to the value $\epsilon$, which also depends on the interaction via $\bar{W}$.
We consider to be the canonical form of the diagonalized Luttinger droplet Hamiltonian, as it is essentially the same as that of the bosonized translationally invariant Tomonaga-Luttinger model. Namely, both are characterized by the renormalized Fermi velocity $\bar{v}$ for collective bosonic particle-hole excitations with linear dispersion, as well as $v_{{{{\cal N}}},{{{\cal J}}}}$ for total and relative particle number changes. For the Luttinger droplet, however, spatial dependencies enter into the diagonalization and lead to qualitatively different behavior for the fermionic degrees of freedom, as discussed below.
Spectrum of the Luttinger droplet model
---------------------------------------
### Recovery of the translationally invariant case
For position-independent potentials, the translationally invariant case is fully recovered by setting $f_{q\neq0}$ $=$ $0$, so that $\bar{v}$ $=$ $\bar{\gamma}\tilde{v}_\text{F}$ and $\Delta v$ $=$ $0$. We thus find that
$$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!
W(x)&=W_0\,,~
V(x)=V_0\,,~
U(x)=U_0
\nonumber
\\[1ex]
\!\!\!\!\!\!\!\!\!\!
\Rightarrow~~
H&=H_\text{TL}+W_0{\hat{\cal N}}\,,~
\gamma= \frac{U_0}{2\pi{v_{\text{F}}}+V_0}
\,,
\\
v_{{{{\cal N}}},{{{\cal J}}}}
&=
{v_{\text{F}}}+\frac{V_0\pm U_0}{2\pi}
\nonumber\\&
=
\bar{v}
\bigg[
\frac{1+\gamma}{1-\gamma}
\bigg]^{\!\pm\frac12}
\,,
\\
\bar{v}&=
\sqrt{\Big({v_{\text{F}}}+\frac{V_0}{2\pi}\Big)^2-\Big(\frac{U_0}{2\pi}\Big)^2}
\nonumber\\&
=
\bar{\gamma}\Big({v_{\text{F}}}+\frac{V_0}{2\pi}\Big)
=
\bar{\gamma}\tilde{v}_\text{F}
\,,
\end{aligned}$$
\[eq:translationallyinvariantcase\]
i.e., the parameter $\gamma$ of only relates ${v_{\text{F}}}$, $V_0$, $U_0$ to one another, as the interactions $V$ and $U$ are absent for the translationally invariant case. As before, $\gamma$ characterizes the relative strength of (translationally invariant) interbranch interactions. It is one of the characteristic properties of a Luttinger liquid[@haldane_luttinger_1981] that the relations $$\begin{aligned}
\bar{v}
&=
\sqrt{v_{{{\cal N}}}v_{{{\cal J}}}^{\vphantom{x}}}
\,,
\label{eq:luttingerliquid-vrelation}
&
\gamma
&=
\frac{v_{{{\cal N}}}-v_{{{\cal J}}}}{v_{{{\cal N}}}+v_{{{\cal J}}}}
\,,
\end{aligned}$$ remain valid even if the dispersion in $H_\text{TL}$ is weakly nonlinear. This connects the excitation velocities $\bar{v}$, $v_{{{\cal N}}}$, $v_{{{\cal J}}}$ as well as the power-law exponents in the single-particle Green function, which contain the parameter $\gamma$, as discussed below.
### Excitation velocities for position-dependent interactions {#subsubsec:velocities}
By contrast, for the Luttinger droplet with position-dependent interactions, the renormalized Fermi velocity $\bar{v}$ depends on $V(x)$ according to , so that $\bar{v}$ can be varied independently from the average interaction potential $V_0$. Namely if $\bar{v}$ $\neq$ $\bar{\gamma}\tilde{v}_\text{F}$ in , i.e., if $$\begin{aligned}
\int\!\frac{{\ensuremath{\text{d}x}}}{2\pi{v_{\text{F}}}+V(x)}
&\neq
\int\!\frac{{\ensuremath{\text{d}x}}}{2\pi{v_{\text{F}}}+V_0}
\,,\label{eq:vbar-indep-condition}
\end{aligned}$$ the three velocities $\bar{v}$, $v_{{{\cal N}}}$, $v_{{{\cal J}}}$ are independent of each other (but together determine $\gamma$).
In the following, however, we will adopt a different perspective. We regard $\gamma$ as given by the interactions as in , $$\begin{aligned}
\gamma&= \frac{U_0}{2\pi{v_{\text{F}}}+V_0}= \frac{U(x)}{2\pi{v_{\text{F}}}+V(x)}
\,.\label{eq:luttingerdroplet-gamma}
\end{aligned}$$ Then it follows from that the velocities are related by
$$\begin{aligned}
\!\!\!
\bar{v}
=
\frac{\gamma-\bar{\gamma}\gamma_3-(1-\bar{\gamma})}{2(\gamma-\gamma_3)}
&v_{{{{\cal N}}}}
+
\frac{\gamma-\bar{\gamma}\gamma_3+(1-\bar{\gamma})}{2(\gamma-\gamma_3)}
v_{{{{\cal J}}}}
\,,\!\!
\\
\!\!\!
\tilde{v}_{\text{F}}
=
\frac{1-\gamma_3}{2(\gamma-\gamma_3)}
&v_{{{{\cal N}}}}
-
\frac{1+\gamma_3}{2(\gamma-\gamma_3)}
v_{{{{\cal J}}}}
\,,
\end{aligned}$$
\[eq:luttingerdroplet-vrelation\]
which replaces .
Hence we may already conclude that the Luttinger droplet is strictly speaking *not* a Luttinger liquid, in the sense that $\bar{v}$ $\neq$ $\sqrt{v_{{{\cal N}}}v_{{{\cal J}}}}$ if holds, so that the Luttinger liquid relation is violated and the linear relations between the velocities $\bar{v}$, $v_{{{{\cal N}}}}$, $v_{{{{\cal J}}}}$, $\tilde{v}_{\text{F}}$ hold instead.
Note also that while the canonical form of the Hamiltonian and its eigenvalues are very similar to the translationally invariant case, their relation to the original fermions is more complex since it was obtained from a position-dependent canonical transformation. As a result, the position dependence of the interaction appears in the Green function, which we calculate next.
Green function for Luttinger droplet model {#sub:dropletgreen}
------------------------------------------
### Rebosonization route to the Green function
As in the translationally invariant case, the Green function is obtained from the bosonization identity and the Bogoljubov transformation , but also makes use of the refermionization and the rebosonization . Using $\phi_\eta(x)$ $=$ ${\varphi_{\eta}^{{+}\!}(x)}$ $+$ ${\varphi_{\eta}^{{{\vphantom{{+}}}}}(x)}$ $=$ $u\phi_{\sigma_\eta}(x)$ $+$ $\phi_{\bar{\sigma}_\eta}(-x)$, we have $$\begin{aligned}
{\psi_{\eta}^{{\vphantom{{+}}}}(x)}
&=
\frac{1}{\sqrt{a}}\,{F_{\eta}^{{\vphantom{{+}}}}}
\,
{{\text{e}}}^{-{{\text{i}}}\frac{2\pi}{L}({\hat{N}}_{\eta}-\frac12)x}
\,
{{\text{e}}}^{-{{\text{i}}}[
{{u}}\phi_{\sigma_\eta\!}(x)
+
{{v}}\phi_{\bar{\sigma}_\eta\!}(-x)
]}
\,.
\label{eq:bosonizationidentity-eta-sigma}
\end{aligned}$$ To evaluate correlation functions of this field, we need to express it in the diagonalizing fermionic basis . We define the auxiliary functions $$\begin{aligned}
\lambda_{q}(x)
&=
{{\text{i}}}\frac{{{\text{e}}}^{-{{\text{i}}}q x -a |q|/2}}{n_q}
\,,
~
\tilde{\lambda}(x-x')
=
\sum_{q\neq0}\lambda_{q}(x){{\text{e}}}^{{{\text{i}}}q x'}
\,,
\nonumber\\
\tilde{\lambda}(x)
&=
{{\text{i}}}\sum_{q\neq0}
\frac{{{\text{e}}}^{-{{\text{i}}}q x -a |q|/2}}{n_q}
=2\sum_{n=1}^\infty\frac{1}{n}\sin\frac{2\pi n x}{L}
\\&
=
\pi\,\text{sgn}(x)-\frac{2\pi x}{L}
\,,
~~~~(-L<x<L)
\nonumber
\end{aligned}$$ in terms of which we can express the bosonic fields as $$\begin{aligned}
\phi_{\sigma}(x)
&=
\sum_{q\neq0}
\lambda_{q}(x)
K_{q\sigma}^{(0)}
=
\sum_{q \neq 0,k}
\lambda_{q}(x)
{c^{{+}}_{k-q\sigma}}{c^{{{\vphantom{{+}}}}}_{k\sigma}}
\nonumber\\&
=
\int\!\frac{{\ensuremath{\text{d}x}}'}{2\pi}\,\tilde{\lambda}(x-x')\,{{{}^\ast_\ast}{{\psi_{\sigma}^{+}(x')}{\psi_{\sigma}^{{\vphantom{{+}}}}(x')}}{{}^\ast_\ast}}
\\&
=
\sum_{k,k'}
\chi_{k-k'}(x)
{{{}^\ast_\ast}{{\Xi_{k\sigma}^{+}}{\Xi_{k'\sigma}^{{\vphantom{{+}}}}}}{{}^\ast_\ast}}
\,.
\nonumber
\end{aligned}$$ Here further auxiliary functions were introduced, $$\begin{aligned}
\chi_{q}(x)
&=
2\pi\bar{v}
\int\!\frac{{\ensuremath{\text{d}y}}}{L}\,
\frac{\tilde{\lambda}(x-y)}{\bar{g}(y)}
\,
{{\text{e}}}^{-{{\text{i}}}qr_0(y)}
\nonumber\\&
=
\frac{2\pi {{\text{i}}}}{qL}
\big({{\text{e}}}^{-{{\text{i}}}qr_0(x)}
-\bar{R}_q\big)
\,,
~~
\chi_{0}(x)
=
\frac{2\pi}{L}r_0(x)
\,,
\label{eq:chiq-def}
\\
\bar{R}_q
&=
\frac{2}{L}\int_0^{L/2}\!{\ensuremath{\text{d}x}}\,\cos(qr_0(x))
\nonumber\\&
=
\frac{2}{L}\int_0^{L/2}\!{\ensuremath{\text{d}r}}\,x_0'(r)\cos(qr)
\,,~\bar{R}_0=1\,,
\label{eq:Rq-def}
\\
x_0(r)
&=
r+2\sum_{q>0}\bar{R}_q\frac{\sin qr}{q}
\,\label{eq:x0-from-Rq}
\,,
\end{aligned}$$ where $x_0(r)$ is the unique inverse function of $r_0(x)$, which was substituted in the integral in and expressed in terms of $\bar{R}_q$ via Fourier transform in for later reference. The rebosonization relation then yields $$\begin{aligned}
\phi_{\sigma}(x)
&=
\sum_{q}
\chi_{-q}(x)
\sum_{k'}
{{{}^\ast_\ast}{{\Xi_{k'-q\sigma}^{+}}{\Xi_{k'\sigma}^{{\vphantom{{+}}}}}}{{}^\ast_\ast}}
\nonumber\\
&=
\chi_{0}(x){\hat{N}}_{\sigma}
+ {{\text{i}}}A_\sigma(x)
\,,\\
A_\sigma(x)
&=
\sum_{q>0}\chi_{-q}(x)\sqrt{n_q}{B^{{{\vphantom{{+}}}}}_{q\sigma}}
-\text{h.c.}
\,,
\end{aligned}$$ finally expressing the fermionic field in the diagonal bosonic basis . For the Green function we also need the time dependence of the Klein factors, which originates from $H_\text{N}+H'$ in and . This leads to a sum over $K_{q\sigma_\eta}^{(0)}$ which we calculate from the inversion $K_{q\sigma}^{(0)}$ $=$ $\int\!\frac{{\ensuremath{\text{d}x}}}{2\pi}\, {{\text{e}}}^{-{{\text{i}}}q x}
\partial_x\phi_{\sigma}(x)$ ($q$ $\neq$ $0$), namely $\sum_{q\neq0} V_q K_{q\sigma}^{(0)}$ $=$ $\bar{\kappa}_{0}{\hat{N}}_{\sigma}/L$ $+$ ${{\text{i}}}\bar{A}_\sigma$, where $$\begin{aligned}
\bar{A}_\sigma
&=
\sum_{q>0}
\bar{\kappa}_{-q}\sqrt{n_q}{B^{{{\vphantom{{+}}}}}_{q\sigma}}
-
\text{h.c.}
\,,
\\
\bar{\kappa}_{q}
&=
\int\!\frac{{\ensuremath{\text{d}x}}}{2\pi L}\,
(V(x)-V_0)
\chi_{q}'(x)
=
-\frac{2\pi\tilde{v}_\text{F}}{L}\bar{R}_q
\,.
\end{aligned}$$ Using the hyperbolic relation ${{\text{e}}}^{\mp\theta}(1\pm\gamma/2)$ $=$ $\bar{\gamma}(u^3\mp v^3)$ and eliminating $U_0$ with , the time-dependent Klein factor then becomes $$\begin{aligned}
{F_{\eta}^{{\vphantom{{+}}}}}(t)
&=
{{\text{e}}}^{{{\text{i}}}(H_\text{N}+H')t}
{F_{\eta}^{{\vphantom{{+}}}}}\,
{{\text{e}}}^{-{{\text{i}}}(H_\text{N}+H')t}
\nonumber\\&
=
{F_{\eta}^{{\vphantom{{+}}}}}
{{\text{e}}}^{
-{{\text{i}}}t
[
2\pi\tilde{v}_\text{F}({\hat{N}}_\eta+\gamma{\hat{N}}_{\bar{\eta}}-\frac12)/L+W_0
+
\bar{\gamma}\bar{\kappa}_0(
u^3{\hat{N}}_{\bar{\eta}}
-
v^3{\hat{N}}_{\eta})]
}\nonumber\\&~~~~~~~~\times{{\text{e}}}^{
{{\text{i}}}\bar{\gamma}(
u^3
\bar{A}_{\sigma_\eta}
-
v^3\bar{A}_{\bar{\sigma}_\eta}
)
}
\,.
\end{aligned}$$ We evaluate the Green function in the ground state with ${\hat{N}}_\eta$ $=$ ${{{\cal N}}}/2$ $=$ ${\hat{N}}_\sigma$ and ${B^{{+}}_{q\sigma}}{B^{{{\vphantom{{+}}}}}_{q\sigma}}$ $=$ $0$ for all $q$ $>$ $0$, where ${{\cal N}}$ is the integer closest to $-\epsilon/(2v_{\cal N})$, $$\begin{aligned}
G_\eta(x,x{^{\prime\!}};t)
&=
\theta(t){G^>}_{\eta\eta}(x,x{^{\prime\!}};t)
-
\theta(-t){G^<}_{\eta\eta}(x,x{^{\prime\!}};t)
\,,
\end{aligned}$$ with $\theta(\pm t)$ $=$ $(1\pm\text{sgn}(t))/2$. The greater and lesser Green functions, $$\begin{aligned}
{G^{\gtrless}}_{\eta\eta'}(x,x';t)
&=
\begin{cases}
-{{\text{i}}}{\langle{{\psi_{\eta}^{{\vphantom{{+}}}}(x,t)}\,{\psi_{\eta'}^{+}(x',0)}}\rangle}
\,,\\
-{{\text{i}}}{\langle{{\psi_{\eta'}^{+}(x',0)}\,{\psi_{\eta}^{{\vphantom{{+}}}}(x,t)}}\rangle}
\,,
\end{cases}
\nonumber\\&
=
\delta_{\eta\eta'}
{G^{\gtrless}}_{\eta\eta}(x,x';t)
\,,
\end{aligned}$$ are then flavor-diagonal. They are evaluated by first clearing the Klein factors, inserting the Bogoljubov-transformed bosonic fields, separate them according to the index $\sigma$, and then express them with ${\hat{N}}_\sigma$, ${B^{{{\vphantom{{+}}}}}_{q\sigma}}$, ${B^{{+}}_{q\sigma}}$. This leads to $$\begin{aligned}
&{{\text{i}}}a{G^{\gtrless}}_{\eta}(x,x{^{\prime\!}};t)
=
M^{\gtrless}_{x,x',t}
\,
M_{\sigma_\eta}(t\bar{\gamma}u^3,\pm u,x_{\gtrless},t_{\gtrless},x_{\lessgtr},t_{\lessgtr})
\nonumber\\&~~~~~~\times
M_{\bar{\sigma}_\eta}(-t\bar{\gamma}v^3,\pm v,-x_{\gtrless},t_{\gtrless},-x_{\lessgtr},t_{\lessgtr})
\,, \label{eq:Gdroplet-result}
\end{aligned}$$ with a phase factor and $\sigma$-diagonal exponential bosonic expectation values $$\begin{aligned}
&M^{\gtrless}_{x,x',t}
=
{{\text{e}}}^{-\frac{{{\text{i}}}\pi}{L}[({{\cal N}}\pm1)(x-x')+v_{\gtrless}t]-{{\text{i}}}{{\text{e}}}^{-\theta}[\chi_0(x)-\chi_0(x')]\frac{{{\cal N}}}{2}}
\!,
\nonumber\\
&M_{\sigma}(\tau{},\nu,x,t,x',t')
=
\langle
{{\text{e}}}^{\tau{}\bar{A}_{\sigma}}
{{\text{e}}}^{\nu A_{\sigma}(x,t)}
{{\text{e}}}^{-\nu A_{\sigma}(x',t')}
\rangle_{\sigma}
,
\label{eq:Msigma-def}
\end{aligned}$$ with $x_>$ $=$ $x$, $x_<$ $=$ $x'$, $t_>$ $=$ $t$, $x_<$ $=$ $0$, and a velocity parameter given by $v_{\gtrless}$ $=$ $(\tilde{v}_\text{F}$ $-$ $\bar{\gamma}\kappa_0v^3/(2\pi))({{\cal N}}$ $+$ $1$ $\pm$ $1)$ $+$ $(\tilde{v}_\text{F}\gamma$ $+$ $\bar{\gamma}\kappa_0u^3/(2\pi)){{\cal N}}$ $+$ $LW_0/\pi$ $-$ $\tilde{v}_\text{F}$. To evaluate the remaining expectation value, we use the identity[@von_delft_bosonization_1998] $$\begin{aligned}
{\langle{{{\text{e}}}^{A_1}{{\text{e}}}^{A_2}{{\text{e}}}^{A_3}}\rangle} = {{\text{e}}}^{{\langle{A_1 A_2+A_2 A_3 + A_1 A_3 + \tfrac{1}{2}(A_1^2+A_2^2+A_3^2)}\rangle}}\,,
\end{aligned}$$ valid for linear bosonic operators $A_1$, $A_2$, $A_3$ and eigenstates of the bosonic particle numbers. We obtain $$\begin{aligned}
\label{eq:Msigma-result}
&M (\tau{},\nu,x,t,x',t')
=
{{\text{e}}}^{
-\tfrac{1}{2}\tau^2\bar{S}_0^{[a]}
-
\tau\nu
\big(\bar{S}_1^{[\bar{v}t,a]}(x)-\bar{S}_1^{[\bar{v}t'\!,a]}(x')\big)
}\nonumber
\\&~~\times{{\text{e}}}^{
\frac{1}{2}\nu^2\big(
2\bar{S}_2^{[\bar{v}(t'-t),a]}(x,x')
-\bar{S}_2^{[0,a]}(x,x)
-\bar{S}_2^{[0,a]}(x',x')
\big)
}
\,,
\end{aligned}$$ where the index $\sigma$ was omitted because $M_\sigma$ is independent of it, and we used the abbreviations
\[eq:Sfunc-def\] $$\begin{aligned}
\bar{S}_0
&=
\sum_{q>0}
n_q|\bar{\kappa}_{q}|^2{{\text{e}}}^{iqs}{{\text{e}}}^{-aq}
\,,
\\
\bar{S}_1^{[s,a]}(y)
&=
\sum_{q>0}
n_q\bar{\kappa}_{-q}\chi_{q}(y){{\text{e}}}^{iqs}{{\text{e}}}^{-aq}
\,,
\\
\bar{S}_2^{[s,a]}(x,y)
&=
\sum_{q>0}
n_q\chi_{-q}(x)\chi_{q}(y){{\text{e}}}^{iqs}{{\text{e}}}^{-aq}
\,.
\end{aligned}$$
Using the explicit wave functions and the definition , they evaluate to
\[eq:Sfunc-result\] $$\begin{aligned}
\bar{S}_0
&=
\Big(\frac{2\pi\tilde{v}_\text{F}}{L}\Big)^2
\bar{R}_{1,2}^{[0,a]}
\,,
\\
\bar{S}_1^{[s,a]}(y)
&=
\frac{2\pi\tilde{v}_\text{F}}{L}
{{\text{i}}}\Big(
\bar{R}_{0,2}^{[s,a]}
-
\bar{R}_{0,1}^{[s-r_0(y),a]}
\Big)
\,,
\\
\bar{S}_2^{[s,a]}(x,y)
&=
\bar{R}_{-1,0}^{[s+r_0(x)-r_0(y),a]}
+
\bar{R}_{-1,2}^{[s,a]}
\nonumber\\&~~~~~~
-
\bar{R}_{-1,1}^{[s+r_0(x),a]}
-
\bar{R}_{-1,1}^{[s-r_0(y),a]}
\,.
\end{aligned}$$
Here we introduced the functions $$\begin{aligned}
\bar{R}_{m,n}^{[s,a]}
&=
\sum_{q>0}
n_q^m\bar{R}_q^n{{\text{e}}}^{{{\text{i}}}qs}{{\text{e}}}^{-aq}
\,,\label{eq:Rfunc-def} \end{aligned}$$ which for $n$ $\neq$ $0$ depend on the position dependence of $V(x)$ through $\bar{R}_q$ of . Putting , , into , the calculation of the Green function is complete, and can be summarized as
$$\begin{aligned}
&{G^>}_{\eta}(x,x{^{\prime\!}};t)=M^{>}_{x,x',t}
\\&~~\times
M(t\bar{\gamma}u^3,+u,x,t,x',0)
M(-t\bar{\gamma}v^3,+v,-x,t,-x',0)
\,,\nonumber
\\
&{G^<}_{\eta}(x,x{^{\prime\!}};t)=M^{<}_{x,x',t}
\\&~~\times
M(t\bar{\gamma}u^3,-u,x',0,x,t)
M(-t\bar{\gamma}v^3,-v,-x',0,-x,t)
\,,\nonumber
\end{aligned}$$
with the factors given by and . We now discuss this result for different settings, referring for simplicity only to ${G^>}_{\eta}(x,x{^{\prime\!}};t)$.
### Recovery of the translationally invariant case
In the translationally invariant case we have $r_0(x)$ $=$ $x$, due to the constant function $r_0'(x)$ $=$ $\bar{v}/(\bar{\gamma}\tilde{v}_\text{F})$ $=$ $1$, cf. . Also $\bar{R}_q$ $=$ $\delta_{q0}$, so that all sums over $\bar{R}_q$ (with $q$ $>$ $0$) vanish. In $\bar{S}_2^{[\bar{v}(t'-t),a]}(x,x')$ only the usual logarithmic sum $$\begin{aligned}
\bar{R}_{-1,0}^{[s,a]}
&=
\sum_{q>0}
\frac{{{\text{e}}}^{{{\text{i}}}qs}{{\text{e}}}^{-aq}}{n_q}
\\&
=
-\ln\Big(1-{{\text{e}}}^{\frac{2\pi}{L}({{\text{i}}}s-a)}\Big)
\stackrel{L\to\infty}{\longrightarrow}
-\ln\Big(\frac{2\pi}{L}(a-{{\text{i}}}s)\Big)
\nonumber
\end{aligned}$$ survives, so that the contributions to the Green function for $L$ $\to$ $\infty$ become $$\begin{aligned}
M(\tau,\nu,x,t,x',t')
&=
\bigg[\frac{a}{{{\text{i}}}[x-x'-\bar{v}(t-t')]+a}\bigg]^{\nu^2}
\,.
\end{aligned}$$ The Green function then takes the familiar power-law form $$\begin{aligned}
\label{eq:G-result-translinv}
&{G^>}_{\eta}(x,x{^{\prime\!}};t)
=M^{>}_{x,x',t}
\\&~~~~\times
\bigg[\frac{-{{\text{i}}}a}{x-x'-\bar{v}t-{{\text{i}}}a}\bigg]^{1+v^2}
\bigg[\frac{{{\text{i}}}a}{x-x'+\bar{v}t+{{\text{i}}}a}\bigg]^{v^2}
\,,\nonumber
\end{aligned}$$ with dependence on only $x-x'\pm{{\text{i}}}\bar{v}t$. The interaction-dependent exponent, $v^2$ $=$ $(\sqrt{v_{{{\cal N}}}/v_{{{\cal J}}}}$ $-$ $\sqrt{v_{{{\cal J}}}/v_{{{\cal N}}}})^2/4$, depends only on the velocity ratio of $v_{{{\cal N}}}/v_{{{\cal J}}}$, which is a characteristic feature of the Luttinger liquid that remains valid even for weakly nonlinear dispersions.[@haldane_luttinger_1981] Furthermore, in the translationally invariant case without interaction we have $\gamma$ $=$ $0$ and hence $v$ $=$ $0$, so that only the first factor with unit exponent correctly remains in .
### Weak quadratic position dependence of the interactions
Next we consider position-dependent potentials that are regular at the origin, i.e., $V(x)$ $=$ $V(0)$ $+$ $V''(0)x^2/2$ $+$ $O(x^4)$, which is sketched in Fig. \[fig:sketch\]a for the repulsive case. From we find for the function $r_0(x)$ that $$\begin{aligned}
r_0(x)
&=
r_0'(0)\,x+\frac16r_0'''(0)\,x^3 + O(x^5)
\,,
\\
r_0'(0)
&=
\frac{2\pi\bar{v}}{\bar{\gamma}(2\pi{v_{\text{F}}}+V(0))}\equiv\alpha
\,,~
r_0''(0)
=
0
\,,\nonumber
\\
r_0'''(0)
&=
\frac{-2\pi\bar{v}V''(0)}{\bar{\gamma}(2\pi{v_{\text{F}}}+V(0))^2}\equiv6\beta
\,.\nonumber
\end{aligned}$$ We will be interested in the asymptotic behavior of Green functions (rather than their periodicity in $L$) and thus will eventually take the limit $L$ $\to$ $\infty$. We therefore consider a weak correction to the linear behavior $r'(0)$, i.e., $$\begin{aligned}
r_0(x)
&=
\alpha x
+
\beta x^3 + O(\beta^2 x^5)\,,
&
\beta
&=
\frac{\text{const}}{L^2}
\,,\label{eq:cubic-r0-approx-withbeta}
\\
x_0(r)
&=
\bar{\alpha} r
-
\bar{\beta} r^3 + O(\bar{\beta}^2 r^5)\,,
&
\bar{\alpha}
&=
\frac{1}{\alpha}
\,,
~
\bar{\beta}
=
\frac{\beta}{\alpha^4}
\,.\nonumber
\end{aligned}$$ For the potential this means $$\begin{aligned}
\!\!\!\!
V(x)
&
=
V(0)-\frac{6\pi\tilde{v}_{\text{F}}\beta}{\alpha}x^2
+
O\Big(\frac{x^4V_0}{L^4}\Big)
\,.\!\!\!\!
\end{aligned}$$ The following choice of coefficients $\bar{R}_q$ turn out to produce this behavior, $$\begin{aligned}
\bar{R}_q
&=
e^{-{c}|q|L/\pi}
\,,\label{eq:Rq-exponential}
\end{aligned}$$ where ${c}$ is positive dimensionless parameter, because from we find $$\begin{aligned}
x_0(r)
&=
r+
\frac{L}{\pi}
\arctan\frac{\sin\frac{2\pi r}{L}}{{{\text{e}}}^{2{c}}-\cos\frac{2\pi r}{L}}
\,,
\end{aligned}$$ which for small $|x/L|$ corresponds to with
$$\begin{aligned}
\bar{\alpha}
&=
{\mathop{\text{coth}}}{c}\,,
&
\bar{\beta}
&=
\frac{\pi^2}{3}
\frac{{\cosh}{c}}{{\sinh}^3{c}}
\Big(\frac{2}{L}\Big)^2
\,,
\\
\alpha
&=
{\mathop{\text{tanh}}}{c}\,,
&
\beta
&=
\frac{\pi^2}{3}
\frac{{\sinh}{c}}{{\cosh}^3{c}}
\Big(\frac{2}{L}\Big)^2
\,.
\end{aligned}$$
The functions are evaluated from as
$$\begin{aligned}
\bar{R}_{-1,n}^{[s,a]}
&=
-\ln\Big(1-{{\text{e}}}^{\frac{2\pi}{L}({{\text{i}}}s-a-n{c}L/\pi)}\Big)
\,,
\\
\bar{R}_{m,n}^{[s,a]}
&=
\frac{{{\text{e}}}^{(-\frac{2\pi}{L}({{\text{i}}}s-a)+2n{c})m}}{[{{\text{e}}}^{-\frac{2\pi}{L}({{\text{i}}}s-a)+2n{c}}-1]^{m+1}}
\,.~~(m=0,1)
\end{aligned}$$
For large $L$, the last logarithmic term in the exponent of then dominates, containing $$\begin{aligned}
\bar{S}_2^{[s,a]}(x,y)
&=
-\ln\bigg[
\frac{
{\sinh}\frac{\pi}{L}({{\text{i}}}[s+r_0(x)-r_0(y)]-a)
}{
{\sinh}\frac{\pi}{L}({{\text{i}}}[s+r_0(x)]-a-{c}L/\pi)
}\nonumber
\\&~~~~\times
\frac{
{\sinh}\frac{\pi}{L}({{\text{i}}}s-a-{c}L/\pi)
}{
{\sinh}\frac{\pi}{L}({{\text{i}}}[s-r_0(y)]-a-{c}L/\pi)
}
\bigg]
\,.
\end{aligned}$$ To leading order in $x/L$, $x'/L$, the Green function then becomes $$\begin{aligned}
&{G^>}_{\eta}(x,x{^{\prime\!}};t)
= M^{>}_{x,x',t}
\label{eq:G-result-weakpositiondependence}
\\&\times
\bigg[\frac{-{{\text{i}}}a}{\alpha(x-x')-\bar{v}t-{{\text{i}}}a}\bigg]^{1+v^2}
\bigg[\frac{{{\text{i}}}a}{\alpha(x-x')+\bar{v}t+{{\text{i}}}a}\bigg]^{v^2}
\,,\nonumber
\end{aligned}$$ i.e., translational invariance is only broken in finite-size corrections.
Note that according to a fermionic single-particle perturbation near $x$ $=$ $0$, as measured by the Green function, propagates with velocity $\bar{v}/\alpha$ $=$ $\bar{v}/r_0'(0)$ $=$ $\bar{\gamma}({v_{\text{F}}}+V(0)/(2\pi)$. This which differs from the translationally invariant case with corresponding velocity $\bar{\gamma}({v_{\text{F}}}+V_0/(2\pi)$ for which only the position-averaged interaction $V_0$ matters. For the Luttinger droplet, the position dependence of $V(x)$ is thus observable in the propagation velocity described by the Green function. This can be observed in more detail for a stronger position dependence of $V(x)$, as discussed in the next subsection.
We also note that the exponent $v^2$ (expressed in terms of $\gamma$ in ) is no longer related only to the velocity ratio of $v_{{{\cal N}}}/v_{{{\cal J}}}$, hence this feature of the Luttinger liquid is also no longer present.
### Piecewise constant interaction potential {#subsubsec:piecewise}
As a minimal example which explicitly breaks the translational invariance of the Green function, we consider an interaction potential that is piecewise constant, $$\begin{aligned}
\label{eq:piecewiseV}
V(x)
&=
\begin{cases}
V(0)&\text{~if~}|x|<R\,,
\\
V(\tfrac{L}{2})&\text{~if~}|x|>R\,,
\end{cases}
\end{aligned}$$ i.e., the particles interact differently inside a central region and outside of it, as depicted in Fig. \[fig:sketch\]b for the repulsive case. The average of this function is given by $$\begin{aligned}
V_0
&=
r\,V(0)+
(1-r)\,V(\tfrac{L}{2})
\,,~~
r=\frac{2R}{L}
\,.
\end{aligned}$$ Here $r$ is the fraction of the central region with interaction $V(0)$, which tends to zero if we consider a fixed finite central interval of width $2R$ but let $L$ tend to infinity, see below. For the potential we find
$$\begin{aligned}
\bar{v}
&=
\frac{\bar{\gamma}}{2\pi}\frac{1}{rs+(1-r)\tilde{s}}
\,,
\\
r_0(x)
&=
\begin{cases}
\alpha x
&\text{~if~}|x|\leq R\,,
\\
\tilde{\alpha}x+\text{sgn}(x)(\alpha-\tilde{\alpha})R
&\text{~if~}|x|\geq R\,,
\end{cases}
\label{eq:r0-result-box}
\\
\bar{R}_q
&=
r
\,
\frac{V(0)-V(\tfrac{L}{2})}{2\pi{v_{\text{F}}}+V(0)}
\,
\frac{\sin n_q\pi r \alpha}{n_q\pi r \alpha}
\,,~~(q>0)
\,,\label{eq:Rq-result-box}
\end{aligned}$$
with the abbreviations $$\begin{aligned}
s
&=
\frac{1}{2\pi{v_{\text{F}}}+V(0)}
\,,&
\tilde{s}
&=
\frac{1}{2\pi{v_{\text{F}}}+V(\tfrac{L}{2})}
\,,
\\
\alpha
&=
r_0'(0)=\frac{s}{\tilde{s}+(s-\tilde{s})r}
\,,&
\tilde{\alpha}
&=
r_0'(\tfrac{L}{2})=\frac{\tilde{s}}{\tilde{s}+(s-\tilde{s})r}
\,.\nonumber
\end{aligned}$$ From now on we consider only fixed finite $R$ and let $L$ $\to$ $\infty$, i.e., $r$ $\to$ $0$. The second fraction in involving the sine function can then be replaced by unity. In this limit the summations evaluate to $$\begin{aligned}
\bar{R}_{m,n}^{[s,a]}
&=\Bigg[r
\,
\frac{V(0)-V(\tfrac{L}{2})}{2\pi{v_{\text{F}}}+V(0)}
\Bigg]^n
\bar{R}_{m,0}^{[s,a]}
\,.
\end{aligned}$$ The logarithmic term in $\bar{S}_2^{[s,a]}(x,y)$ then again provides the leading term in for $L$ $\to$ $\infty$, $$\begin{aligned}
&M(\tau,\nu,x,t,x',t')\nonumber
\\&~~=
\bigg[\frac{a}{{{\text{i}}}[r_0(x)-r_0(x')-\bar{v}(t-t')]+a}\bigg]^{\nu^2}
\,.
\end{aligned}$$ The Green function then takes a power-law form with piecewise linear argument $$\begin{aligned}
\label{eq:resultG_piecewise}
{G^>}_{\eta}(x,x{^{\prime\!}};t)
&=M^{>}_{x,x',t}
\\\nonumber&~~~~\times
\bigg[\frac{-{{\text{i}}}a}{r_0(x)-r_0(x')-\bar{v}t-{{\text{i}}}a}\bigg]^{1+v^2}
\\\nonumber&~~~~\times
\bigg[\frac{{{\text{i}}}a}{r_0(x)-r_0(x')+\bar{v}t+{{\text{i}}}a}\bigg]^{v^2}
\!,
\end{aligned}$$ with the exponent $v^2$ given in terms of $\gamma$ in . As listed in , in the present case $r_0(x)$ is piecewise linear in $x$ with a change in slope at $|x|$ $=$ $R$. Hence if $x$ and $x'$ lie both inside or both outside the central region, the Green function is essentially the same as in the case of weak position dependence or the translationally invariant case , respectively. However, if only one of $x$ and $x'$ is inside the central region, the two coordinates enter with different prefactors into the Green function, breaking its translational invariance. The Green function and velocity relation indicate that for the interaction potential the Luttinger droplet is distinguishable from the Luttinger liquid.
Moreover, the Green function shows that a fermionic single-particle perturbation created at position $x$ will initially propagate with velocity $\bar{v}/r_0'(x)$ $=$ $\bar{\gamma}({v_{\text{F}}}+V(x)/(2\pi)$, which is piecewise constant in the present case. As might have been expected, the position dependence of $V(x)$ thus translates into a position-dependent ‘local’ propagation velocity. Its relation to the other excitation velocitues of the Luttinger droplet model will be discussed the next section.
Towards a Luttinger droplet paradigm {#sec:dropletparadigm}
====================================
The translationally invariant Tomonaga-Luttinger model obeys the relations between excitation velocities and Green function exponents, i.e., in our notation between $\bar{v}$, $v_{{{\cal N}}}$, $v_{{{\cal J}}}$, and $\gamma$. In particular, the dressed Fermi velocity $\bar{v}$ appears in the Green function as the velocity with which a fermion ${\psi_{\eta}^{+}(x)}$ propagates when added to the Luttinger liquid ground state. For the Luttinger droplet model (with linear dispersion) we found different relations between the excitation velocities and $\gamma$, as given in . Furthermore, the Green functions of Sec. \[sub:dropletgreen\] show that a fermion ${\psi_{\eta}^{+}(x)}$, inserted into the Luttinger droplet ground state at position $x$, initially propagates with velocity $\bar{v}/r_0'(x)$. This behavior was observed explicitly for a weak and piecewise constant position dependence of the interaction potential $V(x)$ in and , respectively. It can be traced to , where a phase $r_0(x)k$ appears in the exponent of the eigenfunctions $\xi_k(x)$ of the refermionized model . We can therefore expect that a ‘local’ propagation velocity of fermionic perturbations, $$\begin{aligned}
v^{\text{loc}}(x)
&=
\frac{\bar{v}}{r_0'(x)}
=
\bar{\gamma}\bigg({v_{\text{F}}}+\frac{V(x)}{2\pi}\bigg)
\,,
\end{aligned}$$ will appear in the Green function also for more general $V(x)$. Compared to the translationally invariant case this is a new range of velocities, which we will now relate to the other excitation velocities of the Luttinger droplet.
For this purpose we first seek to characterize the scales of $v^{\text{loc}}(x)$. One way to do this uses its arithmetic and harmonic averages over the entire system. For these we find
$$\begin{aligned}
\!\!
\overline{v^{\text{loc}}}
\equiv
\langle\langle v^{\text{loc}}(x) \rangle\rangle_{\text{arith}}
&\equiv
\hphantom{\bigg[}\,
\int\!\frac{{\ensuremath{\text{d}x}}}{L}\,v^{\text{loc}}(x)\,
\hphantom{\bigg]^{-1}}
=
\bar{\gamma}\tilde{v}_{\text{F}}
\,,\!\!
\\
\!\!
\langle\langle v^{\text{loc}}(x) \rangle\rangle_{\text{harm}}
&\equiv
\bigg[\int\!\frac{{\ensuremath{\text{d}x}}}{L}\,\frac{1}{v^{\text{loc}}(x)}\bigg]^{-1}
=
\bar{v}
\,,
\end{aligned}$$
\[eq:dropletaveragecelocities\]
where, as above, $\tilde{v}_{\text{F}}$ $=$ ${v_{\text{F}}}$ $+$ $V_0/(2\pi)$. For general $V(x)$ these two averages are different, but coincide in the translationally invariant case. With the excitations of the Luttinger droplet characterized by the velocities $\bar{v}$, $\overline{v^{\text{loc}}}$, $v_{{{{\cal N}}}}$, $v_{{{{\cal J}}}}$, we then obtain their interrelation from ,
\[eq:dropletrelations\] $$\begin{aligned}
\bar{v}
&=
c_{{{{\cal N}}}}(\gamma)\;v_{{{{\cal N}}}}
+
c_{{{{\cal J}}}}(\gamma)\;v_{{{{\cal J}}}}
\,,
\\
\overline{v^{\text{loc}}}
&=
c^{\text{loc}}_{{{{\cal N}}}}(\gamma)\,v_{{{{\cal N}}}}
+
c^{\text{loc}}_{{{{\cal J}}}}(\gamma)\,v_{{{{\cal J}}}}
\,,
\end{aligned}$$
where the prefactors are given by
\[eq:coefficients\] $$\begin{aligned}
c_{{{{\cal N}}},{{{\cal J}}}}(\gamma)
&=
\frac{(\gamma-\bar{\gamma}\gamma_3)\mp(1-\bar{\gamma})}{2(\gamma-\gamma_3)}
\,,
\\
c^{\text{loc}}_{{{{\cal N}}},{{{\cal J}}}}(\gamma)
&=
\bar{\gamma}\,
\frac{(\pm1-\gamma_3)}{2(\gamma-\gamma_3)}
\,,
\end{aligned}$$
Furthermore $\gamma$, which characterizes the relative strength of interbranch interactions, determines the Green function exponent $v^2$ according to . The dependence of the coefficients on $\gamma$ is shown in
![Coefficients in the linear relation between excitation velocities in the Luttinger droplet model as a function of the interaction parameter $\gamma$ given by , .\[fig:coeffs\]](fig-coeffs.pdf){width="\columnwidth"}
Fig. \[fig:coeffs\]. We note that for $\gamma$ $=$ $0$, the two branches in the Hamiltonian do not mix; in this case $v_{{{{\cal N}}}}$ and $v_{{{{\cal J}}}}$ contribute equally to $\bar{v}$ and $\overline{v^{\text{loc}}}$ equals $v_{{{{\cal N}}}}$. On the other hand, for only interbranch interactions ($\gamma$ $\to$ $\pm1$), $\bar{\gamma}$ vanishes and hence so do $v_{{{{\cal N}}},{{{\cal J}}}}$.
A preliminary physical interpretation of the velocities might be that $\bar{v}$ plays the role of group velocity, as $\bar{v}q$ is the energy of a bosonic excitation in which involves a nonlocal and mixed-flavor superposition of original fermions. On the other hand, since $v^{\text{loc}}(x)$ plays the role of a local phase velocity, its scale is presumably captured by the arithmetic average $\overline{v^{\text{loc}}}$. Note that for the translationally invariant case $\bar{v}$ $=$ $\bar{\gamma}\tilde{v}_{\text{F}}$, and indeed the group velocity and (position-independent) phase velocity are both given by $\bar{v}$, cf. , , .
We conclude that for the Luttinger droplet model the quantities $\bar{v}$, $\overline{v^{\text{loc}}}$, $v_{{{{\cal N}}}}$, $v_{{{{\cal J}}}}$, and $\gamma$ are related, extending the Luttinger liquid relations between $\bar{v}$, $v_{{{\cal N}}}$, $v_{{{\cal J}}}$, $\gamma$ to the position-dependent case. However, it remains to clarify how the relations evolve away from the special case . Furthermore, in order to be regarded as a paradigm for one-dimensional electronic systems with position-dependent interactions, these relations would have to remain valid also for weak nonlinearites in the dispersion. Both of these questions would therefore be worthwhile to address, e.g., by perturbative methods.
Conclusion {#sec:conclusion}
==========
Using higher-order bosonization identities, i.e., Kronig-type relations with finite momentum transfer, we solved the Luttinger droplet model for a large class of position-dependent interactions and arbitrary one-particle potentials. While the diagonalized Hamiltonian has the same operator expression as for the Luttinger liquid, the relation between its velocity parameters is not fulfilled in general, as the bosonic excitations and particle number changes involve different averages of the interaction potential over all positions. Similarly the Green functions retain their power-law form for weak position-dependence of the interaction potential, but their exponents also no longer depend only on the ratio of excitation velocities for particle-number changes. For weak position-dependent interactions the Luttinger-liquid characteristics are rather robust regarding their functional form, although the interrelation of the dressed scales and exponents is somewhat different. On the other hand, for an interaction potential with different (e.g., constant) values inside or outside a central region of finite width, not only are the Luttinger-liquid velocity relations modified, but also the Green function is no longer translationally invariant and exhibits a position-dependent propagation velocity of single-particle excitations. This may mean that the group velocity of such an excitation differs from its (position-dependent) phase velocity, in contrast to the Luttinger liquid. We conclude that the Luttinger droplet model has a ground state with different characteristics than the Luttinger liquid. It remains to be seen how the velocity relations obtained for evolve for more general one-dimensional models with position-dependent interactions, and whether a Luttinger droplet paradigm emerges for them.
The authors would like to thank Matthias Punk and Jan von Delft for valuable discussions. M.K. would also like to thank Sebastian Diehl, Erik Koch, Volker Meden, Lisa Markhof, Aditi Mitra, Herbert Schoeller, and Eva Pavarini for useful discussions. S.H. gratefully acknowledges support by the German Excellence Cluster Nanosystems Initiative Munich (NIM) and by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC-2111-390814868. M.K. was supported in part by Deutsche Forschungsgemeinschaft under Projektnummer 107745057 (TRR 80) and performed part of this work at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.
[^1]: Throughout, hats appear only on those operators which involve fermionic number operators.
[^2]: We note that our derivation of is similar in spirit to the procedure in Sec. 3.2 of Ref. , and also bears some resemblance to the analysis of higher-order dispersion terms in Ref. . However our approach is more general since we also allow finite momentum transfer $q$ and work with exact operator identities. Note also that one may view as a fermionic representation of certain properties of vertex operators,[@von_delft_bosonization_1998] i.e., exponentiated bosonic fields.
|
---
abstract: 'Recent experiments[@kim2016YPtBiSCj=3/2] have revealed the evidence of nodal-line superconductivity in half-Heusler superconductors, e.g. YPtBi. Theories have suggested the topological nature of such nodal-line superconductivity and proposed the existence of surface Majorana flat bands on the (111) surface of half-Heusler superconductors. Due to the divergent density of states of the surface Majorana flat bands, the surface order parameter and the surface impurity play essential roles in determining the surface properties. In this work, we studied the effect of the surface order parameter and the surface impurity on the surface Majorana flat bands of half-Heusler superconductors based on the Luttinger model. To be specific, we consider the topological nodal-line superconducting phase induced by the singlet-quintet pairing mixing, classify all the possible translationally invariant order parameters for the surface states according to irreducible representations of $C_{3v}$ point group, and demonstrate that any energetically favorable order parameter needs to break time-reversal symmetry. We further discuss the energy splitting in the energy spectrum of surface Majorana flat bands induced by different order parameters and non-magnetic or magnetic impurities. We proposed that the splitting in the energy spectrum can serve as the fingerprint of the pairing symmetry and mean-field order parameters. Our theoretical prediction can be examined in the future scanning tunneling microscopy experiments.'
author:
- Jiabin Yu
- 'Chao-Xing Liu'
title: 'Surface Majorana Flat Bands in $j=\frac{3}{2}$ Superconductors with Singlet-Quintet Mixing'
---
Introduction
============
Recent years have witnessed increasing research interests in half-Heusler compounds ($R$PdBi or $R$PtBi with $R$ a rare-earth element)[@Graf2011heusler] due to their non-trivial band topology[@Lin2010Half; @Chadov2010tunable; @Xiao2010Half; @AlSawai2010Half; @Yan2014half; @Liu2016halfSS; @Logan2016SShalf; @Cano2016chiral; @Ruan2016WSM; @Hirschberger2016WSM; @Shekhar2016observationCMT; @Suzuki2016largeAHE; @Yang2017HHTP; @Liu2018TI], magnetism[@Pan2013ErPdBiSC; @Gofryk2011Mag; @Muller2014Mag; @Nikitin2015HoPdBiSC; @Nakajima2015RPdBiSC; @Pavlosiuk2016AFMSCHH; @Pavlosiuk2016MagHH; @Yu2017ModelAFMHH; @Pavlosiuk2018MagHH] and unconventional superconductivity[@Goll2008LaBiPtSC; @Butch2011SCYPtBi; @Bay2012SCYPtBi; @kim2016YPtBiSCj=3/2; @Tafti2013LuPtBiSC; @Pan2013ErPdBiSC; @Nakajima2015RPdBiSC; @Xu2014LuPdBiSC; @Pavlosiuk2015LuPdBiSC; @Nikitin2015HoPdBiSC; @Meinert2015UnconverntialSCYPtBi; @Pavlosiuk2016AFMSCHH; @TbPdBi2018SC]. Half-Heusler superconductors (SCs) are of particular interest because of the low carrier density ($10^{18}\sim 10^{19} cm^{-3}$), the power-law temperature dependence of London penetration depth, and the large upper critical field. Furthermore, it was theoretically proposed that electrons near Fermi level in half-Heusler SCs possess total angular momentum $j=\frac{3}{2}$ as a result of the addition of the $\frac{1}{2}$ spin and the angular momentum of p atomic orbitals ($l=1$).[@kim2016YPtBiSCj=3/2; @Brydon2016j=3/2SC] Therefore, half-Heusler SCs provide a great platform to study the superconductivity of $j=\frac{3}{2}$ fermions. Such $j=\frac{3}{2}$ fermions were also studied in anti-perovskite materials[@Kawakami2018j=3/2electrons] and the cold atom system[@Wu2006spin3/2CAS; @Kuzmenko2018F=3/2CFG]. Due to the $j=\frac{3}{2}$ nature, the spin of Cooper pairs can take four values: $S=0$ (singlet), 1 (triplet), 2 (quintet) and 3 (septet), among which quintet and septet Cooper pairs cannot appear for spin-$\frac{1}{2}$ electrons.
In order to understand the unconventional superconductivity, various pairing states were proposed, including mixed singlet-septet pairing[@Brydon2016j=3/2SC; @kim2016YPtBiSCj=3/2; @Yang2017Majoranaj=3/2SC; @Timm2017nodalj=3/2SC], mixed singlet-quintet pairing[@yu2017Singlet-Quintetj=3/2SC; @Wang2018j=3/2SCSurface; @Yu2018SSUCFDE], s-wave quintet pairing [@Brydon2016j=3/2SC; @Roy2017j=3/2SC; @Timm2017nodalj=3/2SC; @Boettcher2018j=3/2SC] , d-wave quintet pairing[@Yang2016j=3/2Fermions; @Venderbos2018j=3/2SC] , odd-parity (triplet and septet) parings[@Yang2016j=3/2Fermions; @Venderbos2018j=3/2SC; @Savary2017j=3/2SC; @Ghorashi2017j=3/2SCdisorder], [*et al*]{}[@Venderbos2018j=3/2SC; @Brydon2018BFS]. In particular, [Ref.\[\]]{} proposed that the power-law temperature dependence of London penetration depth can be explained by topological nodal-line superconductivity (TNLS) generated by the pairing mixing between different spin channels. In particular, it has been shown that two types of pairing mixing states, the singlet-quintet mixing and singlet-septet mixing, can both give rise to nodal lines in certain parameter regimes.
In this work, we focus on the singlet-quintet mixing, which was proposed in [Ref.\[\]]{}. As a consequence of TNLS, the Majorana flat bands (MFBs) are expected to exist on the surface perpendicular to certain directions. Such surface MFBs (SMFBs) are expected to show divergent quasi-particle density of states (DOS) at the Fermi energy and thus can be directly probed through experimental techniques, such as scanning tunneling microscopy (STM). [@Yada2011SDOSTSC] Due to the divergent DOS, certain types of interaction [@Li2013MZMJC; @Potter2014EdgeMZM; @Timm2015SurfInsNodalSC; @Hofmann2016EdgeMZMIns] and surface impurities[@Ikegaya2015APE; @Ikegaya2017MZMDirtyNSC; @Ikegaya2018SymABSDirty] are expected to have a strong influence on SMFBs. This motivates us to study the effect of the interaction-induced surface order parameter and the surface impurity on the SMFBs of the superconducting Luttinger model with the singlet-quintet mixing. Specifically, we classify all the mean-field translationally invariant order parameters of the SMFBs according to the irreducible representations (IRs) of $C_{3v}$ group, identify their possible physical origins, and show their energy spectrum by calculating the corresponding DOS. We find that the order parameter needs to break the time-reversal (TR) symmetry in order to either gap out the SMFBs or convert the SMFBs to nodal-lines or nodal points. We also study the quasi-particle local DOS (LDOS) of SMFBs with a surface charge impurity or a surface magnetic impurity (whose magnetic moment is perpendicular to the surface), and show that the peak splitting induced by different types of impurities can help to distinguish the pairing symmetries and surface order parameters.
The rest of the paper is organized as the following. In [Sec.\[sec:model\_H\]]{} and \[sec:surf\_MFB\], we briefly review the superconducting Luttinger model with singlet-quintet mixing and illustrate the symmetry properties of SMFBs. In [Sec.\[sec:MF\_order\_MFB\]]{}, we classify all the mean-field translationally invariant order parameters according to the IRs of $C_{3v}$ and identify their physical origin. We also calculate the energy spectrum and DOS of SMFBs with different order parameters. In [Sec.\[sec:imp\_MFB\]]{}, the impurity effect on the LDOS of MFBs with/without the surface order parameter is discussed. Finally, our work is concluded in [Sec.\[sec:conclusion\]]{}
Model Hamiltonian {#sec:model_H}
=================
The model that we used to generate MFBs in this work is the same as that studied in [Ref.\[\]]{}, which describes the superconductivity in the Luttinger model with mixed s-wave singlet and isotropic d-wave quintet channels. The Bogoliubov-de-Gennes (BdG) Hamiltonian in the continuous limit reads $$\label{eq:H_BdG}
H=\frac{1}{2}\sum_{{\boldsymbol{k}}}\Psi_{{\boldsymbol{k}}}^{\dagger}h_{BdG}({\boldsymbol{k}})\Psi_{{\boldsymbol{k}}}+const.\ ,$$ where $\Psi^{\dagger}_{{\boldsymbol{k}}}=(c_{{\boldsymbol{k}}}^{\dagger},c_{-{\boldsymbol{k}}}^{T})$ is the Nambu spinor and $c_{{\boldsymbol{k}}}^{\dagger}=(c_{{\boldsymbol{k}},\frac{3}{2}}^{\dagger},c_{{\boldsymbol{k}},\frac{1}{2}}^{\dagger},c_{{\boldsymbol{k}},-\frac{1}{2}}^{\dagger},c_{{\boldsymbol{k}},-\frac{3}{2}}^{\dagger})$ are creation operators of $j=\frac{3}{2}$ fermionic excitations. The term $$h_{BdG}({\boldsymbol{k}})=
\left(
\begin{matrix}
h({\boldsymbol{k}})& \Delta({\boldsymbol{k}})\\
\Delta^{\dagger}({\boldsymbol{k}})& -h^T(-{\boldsymbol{k}})\\
\end{matrix}
\right)$$ consists of the normal part $h({\boldsymbol{k}})$ that is the Luttinger model[@Luttinger1956LuttingerModel; @Chadov2010tunable; @Winkler2003SOC; @yu2017Singlet-Quintetj=3/2SC] $$\begin{aligned}
\label{Eqn:h}
h({\boldsymbol{k}})
=(\frac{k^2}{2m}-\mu)\Gamma^0+
c_1 \sum_{i=1}^{3}g_{{\boldsymbol{k}},i}\Gamma^i+c_2 \sum_{i=4}^{5}g_{{\boldsymbol{k}},i}\Gamma^i\end{aligned}$$ and the paring part $\Delta({\boldsymbol{k}})$ that contains s-wave singlet and isotropic d-wave quintet channels $$\label{eq:pairing}
\Delta({\boldsymbol{k}})
=\Delta_0\frac{\Gamma^0}{2}\gamma
+
\Delta_1\sum_{i=1}^5 \frac{a^2 g_{{\boldsymbol{k}},i}\Gamma^i}{2}\gamma,$$ where $\mu$ is the chemical potential, $c_1,c_2$ indicate the strength of the centrosymmetric spin orbital coupling (SOC) which is the coupling between the orbital and the 3/2-“spin”, d-wave cubic harmonics $g_{{\boldsymbol{k}},i}$ and five $\Gamma$ matrices are shown in Appendix.\[app:conv\_expn\], $\Delta_{0,1}$ are order parameters of singlet and quintet channels, respectively, $a$ is the lattice constant of the material, and $\gamma=-\Gamma^1\Gamma^3$ is the TR matrix. The coexistence of the two order parameters is allowed by their same symmetry properties [@yu2017Singlet-Quintetj=3/2SC; @Blount1985SC; @Ueda1985SC; @Volovik1985SC; @Sigrist1991SC; @Annett1990SC; @Annett1991SC; @Annett1996SC].
Before demonstrating the SMFB generated by [Eq.]{}, we first discuss the symmetry properties of the Hamiltonian $H$. As discussed in [Ref.\[\]]{}, $H$ has TR symmetry, and its point group is $O(3)$ or $O_h$ for $c_1=c_2$ or $c_1\neq c_2$, respectively. Due to the coexistence of TR and inversion symmetries, the Luttinger model $h({\boldsymbol{k}})$ has two doubly degenerate bands $\xi_{\pm}({\boldsymbol{k}})=k^2/(2m_{\pm})-\mu$, where $m_{\pm}=m \widetilde{m}_{\pm}$ are effective masses of two bands, $\widetilde{m}_{\pm}=1/(1\pm 2mQ_c)$, $Q_c=\sqrt{c_1^2 Q_1^2+c_2^2 Q_2^2}$, $Q_1=\sqrt{\hat{g}^2_{1}+\hat{g}^2_{2}+\hat{g}^2_{3}}$, $Q_2=\sqrt{\hat{g}^2_{4}+\hat{g}^2_{5}}$, and $\hat{g}_i=g_i/k^2$. In addition, particle-hole (PH) symmetry can be defined as $-\mathcal{C} h^*_{BdG}(-{\boldsymbol{k}})\mathcal{C}^{\dagger}=h_{BdG}({\boldsymbol{k}}) $ and $\Psi_{{\boldsymbol{k}}}^{\dagger} \mathcal{C}=\Psi_{-{\boldsymbol{k}}}^T$ for the BdG Hamiltonian, where $\mathcal{C}=\tau_x$ with $\tau_x$ the Pauli matrix for the PH index. Combining the PH and TR symmetries, we have the chiral symmetry $-\chi h_{BdG}({\boldsymbol{k}}) \chi^{\dagger}=h_{BdG}({\boldsymbol{k}})$, where $\chi=i \mathcal{T C^*}$ and $\mathcal{T}=\text{diag}(\gamma,\gamma^*)$ is the TR matrix on the Nambu bases. The representations of other symmetry operators are shown in Appendix.\[app:rep\_sym\].
![\[fig:surf\_MFB\] This is the distribution of SMFBs for $|2 m| c_1=0.8$, $|2 m| c_2=0.5$, $\widetilde{\Delta}_0/|\mu|=1$ and $\widetilde{\Delta}_1/|\mu|=1.6$, where $\widetilde{\Delta}_0=\text{sgn}(c_1)\Delta_0$, $\widetilde{\Delta}_1=2m\mu a^2 \Delta_1$ and $\tilde{k}_{1,2}=k_{1,2}/\sqrt{2m\mu}$. The surface zero modes in red(orange) regions have $1$($-1$) chiral eigenvalue, and $A^{l_c,l_{\chi}}$’s are labeled according to the convention. The dashed lines are given by $k_{{\shortparallel},1}=0$ and $k_{{\shortparallel},2}=\pm k_{{\shortparallel},1}/2$, where the surface zero modes cannot exist. ](Interacting_MZMs_surf_MZM.pdf){width="\columnwidth"}
Surface Majorana Flat Bands {#sec:surf_MFB}
===========================
In this work, we choose $\mu<0$, $m<0$ and $c_1 c_2>0$, and focus on the case where $c_1\neq c_2$, $m_{\pm}<0$, and SMFBs exist on the $(111)$ surface. [@yu2017Singlet-Quintetj=3/2SC] To solve for SMFBs, we consider a semi-infinite configuration ($x_{\perp}<0$) of [Eq.]{} along the $(111)$ direction with an open boundary condition at the $x_{\perp}=0$ surface, where $x_{\perp}$ labels the position along $(111)$. In this case, the point group is reduced from $O_h$ to $C_{3v}$, which is generated by three-fold rotation $\hat{C}_3$ along the $(111)$ direction and the mirror operation $\hat{\Pi}$ perpendicular to the $(\bar{1}10)$ direction. Although the translational invariance along $(111)$ is broken, the momentum ${\boldsymbol{k}}_{{\shortparallel}}$ that lies inside the $(111)$ plane is still a good quantum number, and we define $k_{{\shortparallel},1}$ and $k_{{\shortparallel},2}$ along the $(11\bar{2})$ and $(\bar{1}10)$ directions, respectively.
Following [Ref.\[\]]{}, we find that SMFBs can exist in certain regions of the surface Brillouin zone, denoted as $A$ in [Fig.\[fig:surf\_MFB\]]{}, and originate from the non-trivial one-dimensional AIII bulk topological invariant $(N_{w}=\pm 2)$. At each ${\boldsymbol{k}}_{{\shortparallel}}\in A$, the semi-infinite model has two orthonormal solutions of zero energy that are localized near the $x_{\perp}=0$ surface and have the same chrial eigenvalues, coinciding with the bulk topological invariant $N_{w}=\pm 2$. We label the creation operators for the two zero-energy solutions at ${\boldsymbol{k}}_{{\shortparallel}}\in A$ as $b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}$ with $i=1,2$, and they satisfy the anti-commutation relation $$\{ b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}, b_{j,{\boldsymbol{k}}'_{{\shortparallel}}}\}=\delta_{ij}\delta_{{\boldsymbol{k}}_{{\shortparallel}} {\boldsymbol{k}}'_{{\shortparallel}}}\ .$$ The subscript $i=1,2$ of $b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}$ can be regarded as the pseudospin index, since $b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}$ can furnish the same representation of TR, $\hat{C}_3$ and $\hat{\Pi}$ operators as a two dimensional $j=1/2$ fermion by choosing the convention $$\label{eq:rep_TR_C3v_surf}
\left\{
\begin{array}{l}
\hat{\mathcal{T}}b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}\hat{\mathcal{T}}^{-1}=
b^{\dagger}_{-{\boldsymbol{k}}_{{\shortparallel}}}\mathcal{T}_b \\
\hat{C}_3 b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}\hat{C}_3^{-1}=
b^{\dagger}_{C_3{\boldsymbol{k}}_{{\shortparallel}}}C_{3,b} \\
\hat{\Pi} b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}\hat{\Pi}^{-1}=
b^{\dagger}_{\Pi{\boldsymbol{k}}_{{\shortparallel}}}\Pi_{b}
\end{array}
\right. \ ,$$ where $\mathcal{T}_b=i\sigma_2$, $C_{3,b}=e^{-i \sigma_3 \frac{\pi}{3}}$, $\Pi_b=-e^{-i \sigma_2 \frac{\pi}{2}}$, and $\sigma_{1,2,3}$ are Pauli matrices for the pseudospin of SMFBs. Since the chiral matrix $\chi$ commutes with any operation in $C_{3v}$ and anti-commutes with TR operation, the chiral eigenvalue of $b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}$ is the same as $b^{\dagger}_{i,R{\boldsymbol{k}}_{{\shortparallel}}}$, but opposite to $b^{\dagger}_{i,-{\boldsymbol{k}}_{{\shortparallel}}}$, where $R\in C_{3v}$. As a result, the surface zero-energy modes cannot exist on three lines parametrized by $k_{{\shortparallel},1}=0$ and $k_{{\shortparallel},2}=\pm k_{{\shortparallel},1}/2$, dividing the region $A$ into six patches as shown in [Fig.\[fig:surf\_MFB\]]{}. Since the chiral eigenvalues of the zero-energy modes in one patch are the same, we can label each patch as $A_{l_{\chi}, \l_c}$ with $l_{\chi}=\pm$ for the chiral eigenvalues $\pm 1$ and $l_c=1,2,3$ marking three patches related by $\hat{C}_{3}$ rotation. Furthermore, we choose $A_{l_{\chi}, 3}$ to be symmetric under $k_{{\shortparallel},2}\rightarrow -k_{{\shortparallel},2}$, i.e. the mirror operation perpendicular to $(\bar{1}10)$. Due to the PH symmetry, the surface zero modes at $\pm {\boldsymbol{k}}_{{\shortparallel}}$ are related by $$\label{eq:b_PH}
b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}(-\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_{2})=b^T_{-{\boldsymbol{k}}_{{\shortparallel}}}\ ,$$ where $\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}$ is the chiral eigenvalue of the zero modes at ${\boldsymbol{k}}_{\shortparallel}$, i.e. $\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}=\pm 1$ for ${\boldsymbol{k}}_{{\shortparallel}}\in A_{\pm}$ with $A_{l_\chi}=\cup_{l_c} A_{l_\chi,l_c}$. TR and $C_{3v}$ symmetries imply $\delta^{\chi}_{-{\boldsymbol{k}}_{{\shortparallel}}}=-\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}$ and $\delta^{\chi}_{R{\boldsymbol{k}}_{{\shortparallel}}}=\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}$ with $R\in C_{3v}$. (See Appendix.\[app:surf\_modes\] for details.)
Mean-field Order Parameters of Surface Majorana Flat Bands {#sec:MF_order_MFB}
==========================================================
Due to the divergent DOS, the interaction may result in the nonvanishing order parameters at the surface and give rise to a gap of SMFBs. In this section, we study the possible mean-field order parameters on the $(111)$ surface that preserve the in-plane translation symmetry. We find that the order parameters must break the TR symmetry in order to gap out the SMFB; all the TR-breaking surface order parameters are classified based on the IRs of $C_{3v}$ and their physical origins are identified. Then, to the leading order approximation where the surface order parameters are independent of ${\boldsymbol{k}}_{{\shortparallel}}$ in each of the surface mode regions, we find the SMFBs can be generally gapped out by these order parameters, and the gapless modes are only possible for certain IRs with certain finely tuned values of parameters. We further study the LDOS structure of SMFBs in the presence of various order parameters and find the splitting patterns of the LDOS peak can be used to distinguish different order parameters as summarized in [Fig.\[fig:LDOS\_no\_imp\]]{} and \[fig:LDOS\_peak\_split\].
Symmetry Classification and Physical Origin
-------------------------------------------
The general form of translationally invariant fermion-bilinear terms for SMFBs can be constructed as $$\label{eq:H_mf}
H_{mf}=\frac{1}{2}\sum_{{\boldsymbol{k}}_{{\shortparallel}}\in A} b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}} m({\boldsymbol{k}}_{{\shortparallel}}) b_{{\boldsymbol{k}}_{{\shortparallel}}}+const.\ ,$$ where $m({\boldsymbol{k}}_{{\shortparallel}})$ is a $2 \times 2$ Hermitian matrix. The PH symmetry makes $m({\boldsymbol{k}}_{{\shortparallel}})$ satisfy $m({\boldsymbol{k}}_{{\shortparallel}})=-\sigma_2\ m^T(-{\boldsymbol{k}}_{{\shortparallel}})\sigma_2$ up to a shift of ground state energy based on [Eq.]{}, while TR symmetry requires $\mathcal{T}_b m^*(-{\boldsymbol{k}}_{{\shortparallel}})\mathcal{T}_b^{\dagger}= m({\boldsymbol{k}}_{{\shortparallel}})$ according to [Eq.]{}. As a result, the combination of PH and TR symmetries, which is equivalent to the chiral symmetry, leads to $m({\boldsymbol{k}}_{{\shortparallel}})=0$, indicating that the existence of a non-vanishing fermion bilinear term $m({\boldsymbol{k}}_{{\shortparallel}})$ for the SMFBs requires the breaking of TR symmetry, i.e. [$$\mathcal{T}_b m^*(-{\boldsymbol{k}}_{{\shortparallel}})\mathcal{T}_b^{\dagger}=-m({\boldsymbol{k}}_{{\shortparallel}})\ .$$]{} As the $C_{3v}$ point group symmetry can also be spontaneously broken by these fermion-bilinear terms, we can further classify these TR-breaking order parameters according to the IR of $C_{3v}$, of which the character table ([Tab.\[tab:cha\_C3v\]]{}) is shown in Appendix.\[app:conv\_expn\]. Since $C_{3v}$ has three IRs $A_1$, $A_2$ and $E$, [Eq.]{} can be expressed as the linear combination of the three corresponding parts $$\label{eq:H_mf_IR}
m({\boldsymbol{k}}_{{\shortparallel}})=m_{A_1}({\boldsymbol{k}}_{{\shortparallel}})+m_{A_2}({\boldsymbol{k}}_{{\shortparallel}})+m_{E}({\boldsymbol{k}}_{{\shortparallel}})\ .$$ Here the $A_1$ term $m_{A_1}({\boldsymbol{k}}_{{\shortparallel}})$ preserves $C_{3v}$ symmetry, and the $A_2$ term $m_{A_2}({\boldsymbol{k}}_{{\shortparallel}})$ preserves $\hat{C}_{3}$ symmetry but has odd mirror parity. The $E$ term has the expression $m_{E}({\boldsymbol{k}}_{{\shortparallel}})=a_1 m_{E,1}({\boldsymbol{k}}_{{\shortparallel}})+a_2 m_{E,2}({\boldsymbol{k}}_{{\shortparallel}})$ with $(m_{E,1}({\boldsymbol{k}}_{{\shortparallel}}), m_{E,2}({\boldsymbol{k}}_{{\shortparallel}}))$ a two-component vector that can furnish a $E$ IR; it breaks the entire $C_{3v}$ symmetry except for some special values of $(a_1,a_2)$, e.g. one of the three mirrors is preserved but the $\hat{C}_3$ is broken for $(a_1,a_2)\propto (1,0), (1,\sqrt{3})$ or $(1,-\sqrt{3})$.
Next we illustrate the physical origin of each term in [Eq.]{} by considering the following on-site mean-field Hamiltonian that are independent of ${\boldsymbol{k}}_{{\shortparallel}}$ $$\begin{aligned}
\label{eq:Ht_mf}
&&\widetilde{H}_{mf}=\sum_{{\boldsymbol{k}}_{{\shortparallel}}}^{A}\int_{-\infty}^{0} d x_{\perp} [c^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}} \widetilde{M}(x_{\perp}) c_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}+\\
&& \frac{1}{2} c^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}} \widetilde{D}(x_{\perp}) (c^{\dagger}_{-{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}})^T + \frac{1}{2} c^T_{-{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}} \widetilde{D}^{\dagger}(x_{\perp}) c_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}]\ ,\nonumber\end{aligned}$$ where $\widetilde{M}^{\dagger}(x_{\perp})=\widetilde{M}(x_{\perp})$ and $-\widetilde{D}^T(x_{\perp})=\widetilde{D}(x_{\perp})$. [Eq.]{} can be obtained by projecting the above Hamiltonian onto the surface, and such projection does not change the symmetry properties. Since $m({\boldsymbol{k}}_{{\shortparallel}})$ must be TR odd in order to be non-vanishing, it requires $\widetilde{M}(x_{\perp})$ and $\widetilde{D}(x_{\perp})$ to be TR odd. Then, the TR-breaking $\widetilde{M}$ and $\widetilde{D}$ can be classified into different IRs of $C_{3v}$: $$\widetilde{M}(x_{\perp})=\widetilde{M}_{A_1}(x_{\perp})+\widetilde{M}_{A_2}(x_{\perp})+\widetilde{M}_E(x_{\perp})\ ,$$ and $$\widetilde{D}(x_{\perp})=\widetilde{D}_{A_1}(x_{\perp})+\widetilde{D}_{A_2}(x_{\perp})+\widetilde{D}_E(x_{\perp})\ ,$$ where $\widetilde{M}_{\beta}(x_{\perp})$ and $\widetilde{D}_{\beta}(x_{\perp})$ can only give rise to $m_{\beta}({\boldsymbol{k}}_{{\shortparallel}})$ in [Eq.]{} with $\beta=A_1, A_2, E$. (See Appendix.\[app:H\_mf\_c2b\] for details.) Concretely, we have $$\label{eq:Mt_Dt}
\left\{
\begin{array}{l}
\widetilde{M}_{A_1}(x_{\perp})=\zeta_2(x_{\perp}) n_2\\
\widetilde{M}_{A_2}(x_{\perp})=\sum_{j=3}^5 \zeta_j(x_{\perp}) n_j\\
\widetilde{M}_{E}(x_{\perp})=\sum_{j=8}^{10} {\boldsymbol{\zeta}}_j(x_{\perp})\cdot {\boldsymbol{n}}_j\\
\widetilde{D}_{A_1}(x_{\perp})=\sum_{j=0}^1 i \zeta_j(x_{\perp}) n_j \gamma\\
\widetilde{D}_{A_2}(x_{\perp})=0\\
\widetilde{D}_{E}(x_{\perp})=\sum_{j=6}^7 i {\boldsymbol{\zeta}}_j(x_{\perp})\cdot {\boldsymbol{n}}_j \gamma
\end{array}
\right.\ ,$$ where $n_i$’s are listed in [Tab.\[tab:n\_class\]]{} of Appendix.\[app:conv\_expn\], and $\zeta_j(x_{\perp})$’s are real. Physically, $n_0\gamma$ corresponds to the singlet pairing, $n_1\gamma$, ${\boldsymbol{n}}_6\gamma$ and ${\boldsymbol{n}}_7\gamma$ generate quintet pairings, and $n_4, n_{8,1}, n_{8,2}$ give FM in $(111)$, $(1\bar{1}0)$ and $(11\bar{2})$ directions, respectively. Since $n_2, n_3, n_5, {\boldsymbol{n}}_9$ and ${\boldsymbol{n}}_{10}$ can be represented by the linear combinations of $c^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}S^{3m} c_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ with the septet spin tensor $S^{3m}$($m=-3,-2,...,3$), we dub these terms the spin-septet order parameters. As a summary, $m_{A_1}({\boldsymbol{k}}_{{\shortparallel}})$ can be generated by the singlet pairing, the quintet pairing, and the spin-septet order parameter; $m_{A_2}({\boldsymbol{k}}_{{\shortparallel}})$ can be generated by $(111)$-directional ferromagnetism (FM) and the spin-septet order parameter; $m_{E}({\boldsymbol{k}}_{{\shortparallel}})$ can be generated by the quintet pairing, the FM perpendicular to the $(111)$ direction, and the spin-septet order parameter.
[c|m[5 cm]{}|c|c|c]{} $C_{3v}$ & Bases & TR & PH & $\chi$\
$A_1$ & $\sum_{l_{\chi},l_c} \delta^{l_{\chi}l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}=1$ for ${\boldsymbol{k}}_{{\shortparallel}}\in A$ & + & $+$ & $+$\
$A_1$ & $\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}=\sum_{l_{\chi},l_c} l_{\chi} \delta^{l_{\chi}l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}$ & $-$ & $-$ & $+$\
$E$ & $(\delta^{E_1,+}_{{\boldsymbol{k}}_{{\shortparallel}}},\delta^{E_2,+}_{{\boldsymbol{k}}_{{\shortparallel}}})$ & + & $+$ & $+$\
$E$ & $(\delta^{E_1,-}_{{\boldsymbol{k}}_{{\shortparallel}}},\delta^{E_2,-}_{{\boldsymbol{k}}_{{\shortparallel}}})$ & $-$& $-$ & $+$\
$A_1$ & $\sigma_0$ & $+$ & $+$ & $+$\
$A_2$ & $\sigma_3$ & $-$ & $-$ & $+$\
$E$ & $(-\sigma_2,\sigma_1)$ & $-$& $-$ & $+$\
$A_1$ & $\rho_0$ & $+$ & $+$ & $+$\
$A_1$ & $\rho_1$ & $+$ & $-$ & $-$\
$A_1$ & $\rho_2$ & $+$ & $-$ & $-$\
$A_1$ & $\rho_3$ & $-$ & $-$ & $+$\
$A_1$ & $\Lambda_1=\lambda_0$ & $+$ & $+$ & $+$\
$A_1$ & $\Lambda_2=\frac{1}{\sqrt{2}}(\lambda_1+\lambda_4+\lambda_6)$ & $+$ & $+$ & $+$\
$A_2$ & $\Lambda_3=\frac{1}{\sqrt{2}}(\lambda_2-\lambda_5+\lambda_7)$ & $-$ & $-$ & $+$\
$E$ & ${\boldsymbol{\Lambda}}_4=\frac{\sqrt{3}}{2}(\lambda_8,-\lambda_3)$ & $+$& $+$ & $+$\
$E$ & ${\boldsymbol{\Lambda}}_5=\sqrt{\frac{3}{8}}(\lambda_5+\lambda_7,\frac{-2\lambda_2-\lambda_5+\lambda_7}{\sqrt{3}})$ & $-$& $-$ & $+$\
$E$ & ${\boldsymbol{\Lambda}}_6=\sqrt{\frac{3}{8}}(\frac{-2\lambda_1+\lambda_4+\lambda_6}{\sqrt{3}},\lambda_4-\lambda_6)$ & $+$& $+$ & $+$\
Surface Local Density of States
-------------------------------
In the following, we focus on the order parameters that are independent of ${\boldsymbol{k}}_{{\shortparallel}}$ in every one of six surface mode regions $A_{l_{\chi}, l_c}$’s. In this case, [Eq.]{} can be expanded as $$\label{eq:m_unif_gen}
m({\boldsymbol{k}}_{{\shortparallel}})=\sum_{l=0}^4\sum_{l_{\chi}=\pm}\sum_{l_c=1}^3 f_{l}^{l_{\chi} l_c}\sigma_l \delta_{{\boldsymbol{k}}_{{\shortparallel}}}^{l_{\chi} l_c}\ ,$$ where $f_{l}^{l_{\chi} l_c}$ is real, $\delta^{l_{\chi}l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}=1$ if ${\boldsymbol{k}}_{{\shortparallel}}\in A_{l_{\chi},l_c}$ and $\delta^{l_{\chi}l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}=0$ otherwise, and $\sigma_l$ labels the Pauli matrix for pseudospin. Then, for any symmetry transformation of $m({\boldsymbol{k}})$, we can convert the transformation of pseudospin index and ${\boldsymbol{k}}_{{\shortparallel}}$ dependence of $m({\boldsymbol{k}})$ to the transformation of $\sigma_l$ and $\delta^{l_{\chi}l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}$, respectively. Based on the symmetry transformation, we can classify $\delta^{l_{\chi}l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}$ and $\sigma_l$ according to the IRs of $C_{3v}$ and the parities under TR, PH and $\chi$, as shown in the top and second top parts of [Tab.\[tab:IR\_C3v\_TR\_PH\_chi\]]{}, respectively. The symmetry classification of TR-odd terms in $m({\boldsymbol{k}}_{{\shortparallel}})$ can be obtained by the tensor product of $\sigma_l$ and $\delta^{l_{\chi}l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}$, as shown in [Tab.\[tab:N\]]{} of Appendix.\[app:conv\_expn\] with various terms labeled by $N_i$’s. As a result, we have the following general expressions of the order parameters in different IRs of $C_{3v}$: $$\label{eq:H_mf_A1}
m_{A_1}({\boldsymbol{k}}_{{\shortparallel}})=\sum_{j=1}^2 m_j N_{j}({\boldsymbol{k}}_{{\shortparallel}})\ ,$$ $$\label{eq:H_mf_A2}
m_{A_2}({\boldsymbol{k}}_{{\shortparallel}})=\sum_{j=3}^4 m_j N_{j}({\boldsymbol{k}}_{{\shortparallel}})\ ,$$ and $$\label{eq:H_mf_E}
m_{E}({\boldsymbol{k}}_{{\shortparallel}})=\sum_{j=5}^8 {\boldsymbol{m}}_j\cdot {\boldsymbol{N}}_{j}({\boldsymbol{k}}_{{\shortparallel}})\ .$$ Here all $m_j$’s are real.
With [Eq.]{}-, we next discuss the energy spectrum and LDOS of SMFBs after including these order parameters. Due to the PH symmetry, only half of the energy spectrum (non-negative energy part) gives the quasi-particle LDOS of SMFBs. However, it is more convenient to study the full spectrum, since the LDOS, which is probed by the tunneling conductance of STM, must symmetrically distribute with respect to the zero energy in experiments [@tinkham1996introductionSC]. Since the order parameters in each patch are ${\boldsymbol{k}}_{{\shortparallel}}$-independent, we choose the mode at the geometric center ${\boldsymbol{K}}^{l_{\chi}, l_c}_{{\shortparallel}}$ of each patch $A_{l_{\chi},l_c}$ as the representative mode. In the following, we only consider the representative modes and use the term “degeneracy" to refer to the [*extra*]{} degeneracy determined by the symmetry, excluding the large degeneracy given by the flatness of the dispersion in each patch. For convenience, we define the creation operator $b^{\dagger}_{i,l_{\chi},l_{c}}=b^{\dagger}_{i,{\boldsymbol{K}}^{l_{\chi}, l_c}_{{\shortparallel}}}$ to label the representative mode in the patch $A_{l_{\chi},l_c}$ with the pseudo-spin index $i$. Since only the uniform order parameters are considered, $l_{\chi}$ and $l_c$ are good quantum numbers, while different pseudo-spin components (the $\sigma_l$ part) are typically coupled by the order parameter $m({\boldsymbol{k}}_{{\shortparallel}})$. Thus, we introduce the band index $s=\pm$ and label the eigen-mode as $\widetilde{b}^{\dagger}_{s,l_\chi,l_c}=\sum_{i}X^{s,l_\chi,l_c}_i b^{\dagger}_{i,l_{\chi},l_{c}}$ with [$$\sum_{l_\chi,l_c}m({\boldsymbol{K}}^{l_{\chi}, l_c}_{{\shortparallel}})X^{s,l_\chi,l_c}=\sum_{l_\chi,l_c} E^{s,l_\chi,l_c} X^{s,l_\chi,l_c}$$]{} the eigen-equation. Without any order parameters, all these 12 modes, including 6 patches and 2 pseudospin components, are degenerate and thus the SMFBs has a zero-bias peak for LDOS, as shown in [Fig.\[fig:LDOS\_no\_imp\]]{}a. For the $A_1$ order $m_{A_1}({\boldsymbol{k}}_{{\shortparallel}})$, the eigen-energies are given by $m_1 \delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\pm |m_2|$, and once $|m_1|\neq|m_2|$, all the zero energy peaks will be split for SMFBs. As a result, the LDOS of the $A_1$ order parameter typically has 4 peaks shown in [Fig.\[fig:LDOS\_no\_imp\]]{}b. This peak structure of LDOS can be understood from symmetry consideration. Due to the breaking of TR symmetry, as well as the chiral symmetry, we only need to consider the the point group symmetry $C_{3v}$. As mentioned before, any operation in $C_{3v}$ does not change the $l_{\chi}$ index, and since $A_{1}$ order parameter is $C_{3v}$ invariant, the band index $s$ cannot be changed either. The $C_3$ rotation only transforms the $l_c={1,2,3}$ index counter-clockwise, resulting in the three-fold degeneracy among the eigen-modes $\widetilde{b}^{\dagger}_{s,l_{\chi},l_c}$ with the same $s$ and $l_{\chi}$. On the other hand, $\Pi$ interchanges $l_c=1,2$ and makes sure $\widetilde{b}^{\dagger}_{s,l_{\chi},1}$ has the same energy as $\widetilde{b}^{\dagger}_{s,l_{\chi},2}$, meaning that $\Pi$ does not give extra constraints compared with $C_3$. Thus, there are $12/3=4$ peaks in the LDOS of the $A_1$ order parameter with each peak of 3-fold degeneracy. For the $A_2$ order parameter $m_{A_2}({\boldsymbol{k}}_{{\shortparallel}})$, the eigen-energies are given by $\pm \sqrt{m_3^2+m_4^2}$, leading to 2 peaks in the LDOS ([Fig.\[fig:LDOS\_no\_imp\]]{}c), resulted from the six-fold degeneracy of each eigen-energy due to the symmetry. Among the six-fold degeneracy, three-fold degeneracy is due to translational invariance and $C_3$ symmetry as the $A_1$ order parameter, meaning that $\widetilde{b}^{\dagger}_{s,l_\chi,1}$, $\widetilde{b}^{\dagger}_{s,l_\chi,2}$ and $\widetilde{b}^{\dagger}_{s,l_\chi,3}$ have the same energy. The remaining double degeneracy originates from the combination of the odd mirror parity of the $A_2$ order parameter and the PH symmetry, i.e. $\Pi_b \sigma_2 m_{A_2}^* (-\Pi^{-1}{\boldsymbol{k}}_{{\shortparallel}}) (\Pi_b \sigma_2)^{\dagger}=m_{A_2}({\boldsymbol{k}}_{{\shortparallel}})$. This combined symmetry does not change the band index $s$, but transforms $l_\chi$ as $+\leftrightarrow -$ and $l_c$ as $1\leftrightarrow 2$. As a result, $\widetilde{b}^{\dagger}_{s,\pm,l_c}$ with fixed $s$ and $l_c$ also have the same energy, giving the extra double degeneracy. For the $E$ order parameter $m_{E}({\boldsymbol{k}}_{{\shortparallel}})$, the eigen-energies are $\sum_{l_\chi,l_c} (l_\chi \bar{m}_{l_c}\pm \bar{m}'_{l_c})\delta^{l_{\chi},l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}$, where [$$\begin{aligned}
\begin{split}
&\bar{m}_{1}=\frac{m_{5,1}}{2}-\frac{\sqrt{3}}{2}m_{5,2}\\
&\bar{m}_{2}=\frac{m_{5,1}}{2}+\frac{\sqrt{3}}{2}m_{5,2}\ ,\ \bar{m}_{3}=-m_{5,1}\\
&\bar{m}'_{1}=[(\frac{\sqrt{3}}{2}m_{6,1}+\frac{m_{6,2}}{2})^2+(-m_{7,1}+\frac{m_{8,1}}{2}+\frac{\sqrt{3}}{2}m_{8,2})^2\\
&+(m_{7,2}-\frac{\sqrt{3}}{2}m_{8,1}+\frac{m_{8,2}}{2})^2]^{1/2}\\
&\bar{m}'_{2}=[(-\frac{\sqrt{3}}{2}m_{6,1}+\frac{m_{6,2}}{2})^2+(-m_{7,1}+\frac{m_{8,1}}{2}-\frac{\sqrt{3}}{2}m_{8,2})^2\\
&+(m_{7,2}+\frac{\sqrt{3}}{2}m_{8,1}+\frac{m_{8,2}}{2})^2]^{1/2}\\
&\bar{m}'_{3}=[m^2_{6,2}+(m_{7,1}+m_{8,1})^2+(m_{7,2}-m_{8,2})^2]^{1/2}\ .
\end{split}\end{aligned}$$]{} Therefore, all the modes are typically split for the $E$ order and the corresponding LDOS generally has 12 peaks shown in [Fig.\[fig:LDOS\_no\_imp\]]{}d. We would like to mention that if including the momentum dependence of the surface order parameter in each surface-mode region, it can broaden the LDOS peaks in [Fig.\[fig:LDOS\_no\_imp\]]{}. In addition, the momentum dependence may also lead to the existence of arcs of surface zero modes in certain small parameter regions as discussed Appendix.\[app:MZM\_arc\].
![\[fig:LDOS\_no\_imp\] (a), (b), (c) and (d) show the LDOS on the $(111)$ surface as a function of the energy ($E/|\mu|$) without any order parameters, with the $A_1$ order parameter, with the $A_2$ order parameter and with the $E$ order parameter, respectively. Due to PH symmetry, only non-negative-energy half of the LDOS is physical. The broadening of each peak is plotted via Gaussian distribution with standard deviation being $10^{-3}$.The parameters choices for each order if exist are $m_1/|\mu|=0.05$ and $m_2/|\mu|=0.1$ for the $A_1$ order parameter (\[eq:H\_mf\_A1\]), $m_3/|\mu|=0.05$ and $m_4/|\mu|=-0.1$ for the $A_2$ order parameter (\[eq:H\_mf\_A2\]), and ${\boldsymbol{m}}_5/|\mu|=(0.01,0.02)$,${\boldsymbol{m}}_6/|\mu|=(0.03,0.04)$,${\boldsymbol{m}}_7/|\mu|=(0.05,0.06)$ and ${\boldsymbol{m}}_{8}/|\mu|=(0.07, 0.08)$ for the $E$ order parameter (\[eq:H\_mf\_E\]). Here we don’t show the numbers on the vertical axis [@Bi2019TBG] since only the position of LDOS peak can be probed in the STM experiments. ](LDOS_no_imp.pdf){width="\columnwidth"}
Impurity Effect {#sec:imp_MFB}
===============
In this section, we will study the effect of surface non-magnetic and magnetic impurities. The effect of non-magnetic impurity on SMFBs in the absence of the mean-field order parameters has been studied in [Ref.\[\]]{}, showing that any non-magnetic impurity can generally induce a local gap for the SMFBs of DIII TNLS. Our work here aims to present a systematic study on how the LDOS of SMFBs is split around a single non-magnetic or magnetic impurity in the absence/presence of the mean-field order parameters.
Preliminaries
-------------
To consider the local potential, we first need to transform SMFBs to the real space with $$\label{eq:b_r_k}
b^{\dagger}_{l_{\chi},l_c,i,{\boldsymbol{r}}_{{\shortparallel}}}=\frac{1}{\sqrt{\mathcal{S}_{{\shortparallel}}}}\sum_{{\boldsymbol{k}}_{{\shortparallel}}}^{A_{l_{\chi},l_c}}e^{-i{\boldsymbol{k}}_{{\shortparallel}}\cdot {\boldsymbol{r}}_{{\shortparallel}}}b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}\ ,$$ where the momentum summation is limited into the surface mode region $A_{l_{\chi},l_c}$. Under the symmetry operations, the indexes $i,l_\chi,l_c$ of $b^{\dagger}_{l_{\chi},l_c,i,{\boldsymbol{r}}_{{\shortparallel}}}$ defined here are transformed in the same way as those of $b^{\dagger}_{i,l_{\chi},l_c}$ defined in [Sec.\[sec:MF\_order\_MFB\]]{}. In the following, we adopt the following approximation $$\label{eq:appro_local_d}
\frac{1}{S_{{\shortparallel}}}\sum_{{\boldsymbol{k}}_{{\shortparallel}}}^{A_{l_{\chi},l_{c}}}e^{i({\boldsymbol{k}}_{{\shortparallel}}-{\boldsymbol{K}}^{l_{\chi},l_{c}}_{{\shortparallel}})\cdot {\boldsymbol{r}}_{{\shortparallel}}}\approx \delta^{(2)}({\boldsymbol{r}}_{{\shortparallel}})\ ,$$ resulting in $$\{ b^{\dagger}_{l_{\chi},l_c,i,{\boldsymbol{r}}_{{\shortparallel}}}, b_{l_{\chi}',l_c',i',{\boldsymbol{r}}_{{\shortparallel}}'}\}=\delta_{l_{\chi}l_{\chi}'}\delta_{l_c l_c'}\delta_{i i'}\delta^{(2)}({\boldsymbol{r}}_{{\shortparallel}}-{\boldsymbol{r}}_{{\shortparallel}}')\ .$$ Further, we define [$$d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}}=(b^{\dagger}_{{+,1},{\boldsymbol{r}}_{{\shortparallel}}}, b^{\dagger}_{{+,2},{\boldsymbol{r}}_{{\shortparallel}}}, b^{\dagger}_{{+,3},{\boldsymbol{r}}_{{\shortparallel}}}, b^{\dagger}_{{-,1},{\boldsymbol{r}}_{{\shortparallel}}}, b^{\dagger}_{{-,2},{\boldsymbol{r}}_{{\shortparallel}}}, b^{\dagger}_{{-,3},{\boldsymbol{r}}_{{\shortparallel}}})$$]{} for convenience.
The behavior of $d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}}$ under the symmetry transformation is crucial for the understanding of LDOS. In general, the relation required by the PH symmetry has the form $d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}}\mathcal{C}_d=d_{{\boldsymbol{r}}_{{\shortparallel}}}^T$, and the transformation under TR, $\hat{C}_3$, and $\hat{\Pi}$ operations reads $\hat{\mathcal{T}}d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}}\hat{\mathcal{T}}^{-1}=
d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}}\mathcal{T}_d$, $\hat{C}_3 d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}}\hat{C}_3^{-1}=
d^{\dagger}_{C_3{\boldsymbol{r}}_{{\shortparallel}}}C_{3,d}$, and $\hat{\Pi} d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}}\hat{\Pi}^{-1}=
d^{\dagger}_{\Pi{\boldsymbol{r}}_{{\shortparallel}}}\Pi_{d}$, respectively. As $d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}}$, besides ${\boldsymbol{r}}_{\shortparallel}$, carries three indexes $l_\chi,l_c,i$ that transform independently under the symmetry operation, the transformation matrices presented above should be in the tensor product form as [$$\begin{aligned}
\begin{split}
\label{eq:sym_d}
&\mathcal{C}_d=\mathcal{C}_{\chi}\otimes \mathcal{C}_c \otimes \sigma_2\ \text{with}\ \mathcal{C}_{\chi}=-i\rho_2\ \text{and}\ \mathcal{C}_c=\lambda_0\ ,\\
&\mathcal{T}_d=\mathcal{T}_{\chi}\otimes \mathcal{T}_{c}\otimes \mathcal{T}_b\ \text{with}\ \mathcal{T}_{\chi}=\rho_1\ \text{and}\ \mathcal{T}_{c}=\lambda_0\ ,\\
&C_{3,d}=C_{3,\chi}\otimes C_{3,c} \otimes C_{3,b}\ \text{with}\ C_{3,\chi}=\rho_0\ ,\\
&\Pi_d=\Pi_{\chi}\otimes \Pi_{c}\otimes \Pi_{b}\ \text{with}\ \Pi_{\chi}=\rho_0\ ,
\end{split}\end{aligned}$$]{} where $C_{3,c}=\exp(-i \frac{\lambda_2-\lambda_5+\lambda_7}{\sqrt{3}} \frac{2\pi}{3})$, $\Pi_c=-\exp(i \frac{\lambda_5+\lambda_7}{\sqrt{2}} \pi)$, $\rho_i$’s are Pauli matrices for $l_\chi=\pm$ index, $\sigma_i$’s are for the pseudo-spin of the surface modes as before, and $\lambda_i$’s are Gell-Mann matrices (Appendix.\[app:conv\_expn\]) for $l_c=1,2,3$ index with $\lambda_0$ the $3\times 3$ identity matrix. In addition, the representation of the translation operator perpendicular to $(111)$ direction is $\hat{T}_{{\boldsymbol{x}}_{{\shortparallel}}}d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}} \hat{T}^{-1}_{{\boldsymbol{x}}_{{\shortparallel}}}=d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}+{\boldsymbol{x}}_{{\shortparallel}}}$.
With the above definition of $d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}}$ operator, we next consider the Hamiltonian that describes the effect of a surface impurity on the SMFBs, given by $$\label{eq:H_imp_d}
H_V=\int d^2 r_{{\shortparallel}} d_{{\boldsymbol{r}}_{{\shortparallel}}}^{\dagger}M_V({\boldsymbol{r}}_{{\shortparallel}})d_{{\boldsymbol{r}}_{{\shortparallel}}}+const.\ ,$$ where $M_{V}({\boldsymbol{r}}_{{\shortparallel}})$ is Hermitian, the PH symmetry requires $\mathcal{C}_d M_V^*({\boldsymbol{r}}_{{\shortparallel}}) \mathcal{C}_d^{\dagger}=-M_V({\boldsymbol{r}}_{{\shortparallel}})$, and the impurity is chosen to be at ${\boldsymbol{r}}_{{\shortparallel}}=0$ without the loss of generality. Such form of impurity Hamiltonian is justified in Appendix.\[app:H\_d\_bases\]. $M_V({\boldsymbol{r}}_{{\shortparallel}})$ in general is the linear combination of $\rho_j\otimes \lambda_k \otimes \sigma_l$ with coefficients depending on ${\boldsymbol{r}}_{{\shortparallel}}$. In this case, we can convert the symmetry transformation of $l_\chi$ and $l_c$ indexes of $M_V({\boldsymbol{r}}_{{\shortparallel}})$ to the transformations of $\rho_j$’s and $\lambda_k$’s, respectively. Based on [Eq.]{}, $\rho_j$’s and $\lambda_k$’s can be classified according to the IRs of $C_{3v}$ and parities of TR, PH and $\chi$, as shown in the second lowest and lowest parts of [Tab.\[tab:IR\_C3v\_TR\_PH\_chi\]]{}. Then, the terms in $M_{V}({\boldsymbol{r}}_{{\shortparallel}})$ with certain symmetry properties can be constructed via the tensor product of the classified $\rho_j$’s, $\lambda_k$’s and $\sigma_l$’s listed in [Tab.\[tab:IR\_C3v\_TR\_PH\_chi\]]{}, which can further determine the number of LDOS peaks. Similar as [Sec.\[sec:MF\_order\_MFB\]]{}, the LDOS discussed here is based on the full spectrum of $M_V({\boldsymbol{r}}_{{\shortparallel}})$, of which only the half with non-negative energy is physical. In the following, we study the LDOS at the impurity position ${\boldsymbol{r}}_{{\shortparallel}}=0$ with the focus on two types of impurities: (i) non-magnetic charge impurity, and (ii) magnetic impurity with magnetization along the $(111)$ direction.
{width="\textwidth"}
{width="90.00000%"}
Non-magnetic Charge Impurity
----------------------------
For a charge impurity, the potential term $M_V({\boldsymbol{r}}_{{\shortparallel}}=0)=M_c$ possesses the TR symmetry $\mathcal{T}_d M^*_{c}\mathcal{T}_d^{\dagger}=M_c$, the $C_{3v}$ symmetries centered at the impurity $R_d M_c R_d^{\dagger}=M_c$ with $R\in C_{3v}$, and the chiral symmetry $\chi_d M_c \chi_d^{\dagger}=-M_c$. (See Appendix.\[app:H\_d\_bases\] for details.) According to its symmetry properties and [Tab.\[tab:IR\_C3v\_TR\_PH\_chi\]]{}, the generic form of $M_c$ reads $$\begin{aligned}
\label{eq:c_imp}
&&M_c=(\eta_1\rho_1+\eta_2\rho_2)\otimes \Lambda_1\otimes \sigma_0\\
&&+(\eta_3\rho_1+\eta_4\rho_2)\otimes \Lambda_2\otimes \sigma_0+(\eta_5\rho_1+\eta_6\rho_2)\otimes \Lambda_3\otimes \sigma_3\nonumber\\
&&+(\eta_7\rho_1+\eta_8\rho_2)\otimes (-\Lambda_{5,1}\otimes \sigma_2+\Lambda_{5,2}\otimes \sigma_1)\nonumber\ ,\end{aligned}$$ where $\eta_{1,...,8}$ are real. Below we examine the LDOS on a single charge impurity for SMFBs and compare the case without any order parameter to the cases with $A_1$ (\[eq:H\_mf\_A1\]), $A_2$ (\[eq:H\_mf\_A2\]), and $E$ (\[eq:H\_mf\_E\]) order parameters. The LDOS around the charge impurity is shown in Figs.\[fig:LDOS\]a-d, which reveal the following features. (1) Since PH symmetry exists in all the cases, the LDOS is always symmetric with respect to zero energy. (2) If no order parameters exist, there are six peaks ([Fig.\[fig:LDOS\]]{}a), given by the TR protected double degeneracy of each eigenvalue of $M_c$ according to the Kramer’s degeneracy. (3) In the presence of the $A_1$ order parameter, 8 peaks exist at the impurity ([Fig.\[fig:LDOS\]]{}b). The reason is the following. Since the translational invariance is absent, the modes with different $l_\chi$ or $l_c$ are coupled by the charge impurity, and the three-fold degeneracy for the pure $A_1$ order parameter case is lifted. Moreover, the appearance of the order parameter breaks the TR symmetry, leaving only the $C_{3v}$ symmetries to protect the degeneracy. For convenience, we choose the eigenstates of $\hat{C}_3$ rotation as the bases to make the representation $C_{3,d}$ diagonal as $$\label{eq:C3t_d}
\widetilde{C}_{3,d}
=
\left(
\begin{array}{ccc}
e^{-i \frac{\pi}{3}}\mathds{1}_4 & & \\
& -\mathds{1}_4 & \\
& & e^{i \frac{\pi}{3}}\mathds{1}_4\\
\end{array}
\right)\ ,$$ where $\mathds{1}_n$ is the $n\times n$ identity matrix. Due to the presence of the $A_1$ order order parameter, the Hamiltonian at the charge impurity becomes $M_c+M_{A_1}$ with $M_{A_1}$ given by transforming [Eq.]{} to the $d$ bases. (See Appendix.\[app:H\_d\_bases\].) With the eigen-bases of $\hat{C}_3$ rotation, $M_c+M_{A_1}$ can be block diagonalized as $\text{diag}(h_1,h_2,h_3)$, where $h_1$, $h_2$ and $h_3$ are $4\times 4$ Hermitian matrices. With the same bases, the mirror matrix $\Pi_d$ has the form $$\label{eq:Pit_d}
\widetilde{\Pi}_{d}
=
\left(
\begin{array}{ccc}
& & U_\Pi\\
& U_\Pi & \\
U_\Pi & & \\
\end{array}
\right)$$ with $$U_\Pi=
\left(
\begin{array}{cccc}
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 \\
\end{array}
\right)\ .$$ The mirror symmetry gives $U_{\Pi} h_3 U_{\Pi}^{\dagger}=h_1$ and $U_{\Pi} h_2 U_{\Pi}^{\dagger}=h_2$, which means the eigenvalues of $h_1$ are the same as those of $h_3$. In fact, the representations of symmetry operations show that the bases of $h_1$ and $h_3$ belong to two dimensional IRs of $C_{3v}$ while those of $h_2$ belong to one dimensional IRs of $C_{3v}$. Therefore, $M_c+M_{A_1}$ has four doubly degenerate and four single eigenvalues, resulting in the 8 LDOS peaks. (4) The 12 LDOS peaks exist at the impurity in the presence of the $A_2$ order parameter ([Fig.\[fig:LDOS\]]{}c) since the translational invariance and the odd mirror parity of the $A_2$ order parameter are broken by impurity, and there are no symmetries ensuring any degeneracy. (5) The 12 LDOS peaks at the impurity for the $E$ order parameter ([Fig.\[fig:LDOS\]]{}d) are because no new symmetries are brought by the impurity. Besides the above five features, the sign change of the charge does not affect the LDOS peaks since the order parameters are all chiral anti-symmetric while the charge impurity is chiral symmetric.
Magnetic Impurity
-----------------
$M_V({\boldsymbol{r}}_{{\shortparallel}}=0)=M_m$ is still Hermitian and PH symmetric at a magnetic impurity with magnetic momentum along (111) direction. Moreover, it is TR-odd $\mathcal{T}_d M_m^*\mathcal{T}_d^{\dagger}=-M_m$, $\hat{C}_{3}$-symmetric $C_{3,d} M_m C_{3,d}^{\dagger}=M_m$, and $\hat{\Pi}$-odd $\Pi_{d} M_m\Pi_{d}^{\dagger}=-M_m$. (Appendix.\[app:H\_d\_bases\].) According to the symmetry properties and [Tab.\[tab:IR\_C3v\_TR\_PH\_chi\]]{}, the generic form of $M_m$ reads $$\begin{aligned}
\label{eq:m_imp_111}
&&M_m=\eta_9\rho_0\otimes \Lambda_1\otimes \sigma_3\nonumber\\
&&+\eta_{10}\rho_0\otimes \Lambda_2\otimes \sigma_3+\eta_{11}\rho_0\otimes \Lambda_3\otimes \sigma_0\nonumber\\
&&+\eta_{12}\rho_0\otimes (\Lambda_{4,2}\otimes \sigma_2+\Lambda_{4,1}\otimes \sigma_1)\nonumber\\
&&+\eta_{13}\rho_3\otimes (\Lambda_{5,2}\otimes \sigma_2+\Lambda_{5,1}\otimes \sigma_1)\nonumber\\
&&+\eta_{14}\rho_0\otimes (\Lambda_{6,2}\otimes \sigma_2+\Lambda_{6,1}\otimes \sigma_1)\ ,\end{aligned}$$ where $\eta_{9,...,14}$ are real. Figs.\[fig:LDOS\]e-h show the LDOS around the magnetic impurity and reveal the following features. (1) PH symmetry again ensures that the LDOS is always symmetric with respect to zero energy and the $E$ order parameter still has 12 LDOS peaks at the magnetic impurity since no new symmetries appear as shown in [Fig.\[fig:LDOS\]]{}h. (2) If no order parameters exist, there are six peaks (Figs.\[fig:LDOS\]e), resulted from the double degeneracy given by the combination of the PH symmetry and odd $\hat{\Pi}$ parity. It is because the combination of the PH symmetry and odd $\Pi$ parity gives $\Pi_{d} \mathcal{C}_d M_m \mathcal{C}_d^{\dagger} \Pi_{d}^{\dagger} = M_m$, and since $\Pi_{d} \mathcal{C}_d (\Pi_{d} \mathcal{C}_d)^*=-1$, each eigenvalue of $M_m$ must be doubly degenerate (similar to Kramer’s theorem). (3) The original 4 peaks of the $A_1$ order are splitted into 12 peaks since the magnetic impurity breaks the translational invariance and $\hat{\Pi}$ symmetry ([Fig.\[fig:LDOS\]]{}f). (4) As shown in [Fig.\[fig:LDOS\]]{}g, the 6 LDOS peaks of the magnetic impurity remain in the presence of the $A_2$ order since the PH symmetry and odd $\hat{\Pi}$ parity are not broken. Besides the above four features, flipping the direction of the magnetic moment, i.e. $M_m\rightarrow -M_m$, does not affect the LDOS distribution in presence of the $A_1$ order parameter, since the $A_{1}$ order parameter has $\hat{\Pi}$ symmetry while $M_m$ has odd $\hat{\Pi}$ parity.
Summary for Impurity Effect
---------------------------
To sum up, the number of LDOS peaks at a charge impurity or a magnetic impurity with magnetic moment in $(111)$ direction is 6 or 6 for no order parameters, 8 or 12 for the $A_1$ order parameter, 12 or 6 for the $A_2$ order parameter, and 12 or 12 for the $E$ order parameter, respectively, as summarized in [Fig.\[fig:LDOS\_peak\_split\]]{}. Combining the above results with the LDOS peaks without impurity given in [Sec.\[sec:MF\_order\_MFB\]]{}, it is more than enough to identify the order parameters in our system. In the above analysis, we adopt the approximation (\[eq:appro\_local\_d\]), only consider translationally invariant order parameters that are ${\boldsymbol{k}}_{\shortparallel}$-independent in each surface mode region, and assume the surface mode wavefunctions are ${\boldsymbol{k}}_{\shortparallel}$-independent in each surface mode region to deal with the impurity. Those approximations neglect high-order effects which typically can only broaden the LDOS peaks without affecting the qualitative result.
Conclusion and Discussion {#sec:conclusion}
=========================
In this work, we studied the energy spectrum (or LDOS) of the SMFBs localized on (111) surface of the half-Heusler SCs with translationally invariant order parameters or magnetic/non-magnetic impurities based on the Luttinger model with singlet-quintet mixing. Our work demonstrates that the zero-bias peak of SMFBs can be split to reveal a rich peak structure when different types of order parameters induced by interaction or magnetic/non-magnetic impurities are introduced. Such peak structure can be viewed as a fingerprint to distinguish different types of order parameters in the standard STM experiments. In addition, we notice that the SMFBs induced by singlet-septet mixing proposed in [Ref.\[\]]{} possess six patches without any additional pseudospin degeneracy in the surface Brillouin zone (see Fig.5a and the discussion in [Ref.\[\]]{}). Due to the different number of degeneracy, we expect the peak structures given by the order parameters and magnetic/non-magnetic impurities will be different in two cases, which thereby may help distinguish the singlet-quintet mixing from the singlet-septet mixing in experiments.
Acknowledgement
===============
We acknowledge the helpful discussion with C.Wu. J.Y thanks Yang Ge, Rui-Xing Zhang, Jian-Xiao Zhang and Tongzhou Zhao for helpful discussion. We acknowledge the support of the Office of Naval Research (Grant No. N00014-18-1-2793), Kaufman New Initiative research grant KA2018-98553 of the Pittsburgh Foundation and the U.S. Department of Energy (Grant No. DESC0019064).
Convention and Expressions {#app:conv_expn}
==========================
The Fourier transformation of creation operators in the continuous limit reads $$c^{\dagger}_{{\boldsymbol{r}}}=\frac{1}{\sqrt{\mathcal{V}}}\sum_{{\boldsymbol{k}}}e^{-i{\boldsymbol{k}}\cdot{\boldsymbol{r}}}c^{\dagger}_{{\boldsymbol{k}}}\ ,$$ where $\mathcal{V}$ is the total volume of the entire space.
The five d-orbital cubic harmonics read [@Murakami2004SU2] $$\left\{
\begin{array}{l}
g_{{\boldsymbol{k}},1}=\sqrt{3} k_y k_z\\
g_{{\boldsymbol{k}},2}=\sqrt{3} k_z k_x\\
g_{{\boldsymbol{k}},3}=\sqrt{3} k_x k_y\\
g_{{\boldsymbol{k}},4}=\frac{\sqrt{3}}{2} (k_x^2-k_y^2)\\
g_{{\boldsymbol{k}},5}=\frac{1}{2}(2 k_z^2-k_x^2-k_y^2)\\
\end{array}
\right..$$
The $j=\frac{3}{2}$ angular momentum matrices are [@Winkler2003SOC] $$J_x=\left(
\begin{array}{cccc}
0 & \frac{\sqrt{3}}{2} & 0 & 0 \\
\frac{\sqrt{3}}{2} & 0 & 1 & 0 \\
0 & 1 & 0 & \frac{\sqrt{3}}{2} \\
0 & 0 & \frac{\sqrt{3}}{2} & 0 \\
\end{array}
\right)$$ $$J_y=\left(
\begin{array}{cccc}
0 & -\frac{i \sqrt{3}}{2} & 0 & 0 \\
\frac{i \sqrt{3}}{2} & 0 & -i & 0 \\
0 & i & 0 & -\frac{i \sqrt{3}}{2} \\
0 & 0 & \frac{i \sqrt{3}}{2} & 0 \\
\end{array}
\right)$$ $$J_z=\left(
\begin{array}{cccc}
\frac{3}{2} & 0 & 0 & 0 \\
0 & \frac{1}{2} & 0 & 0 \\
0 & 0 & -\frac{1}{2} & 0 \\
0 & 0 & 0 & -\frac{3}{2} \\
\end{array}
\right).$$
The five Gamma matrices are [@Murakami2004SU2] $$\left\{
\begin{array}{l}
\Gamma^1=\frac{1}{\sqrt{3}} (J_y J_z+J_z J_y)\\
\Gamma^2=\frac{1}{\sqrt{3}} (J_z J_x+J_x J_z)\\
\Gamma^3=\frac{1}{\sqrt{3}} (J_x J_y+J_y J_x)\\
\Gamma^4=\frac{1}{\sqrt{3}} (J_x^2-J_y^2)\\
\Gamma^5=\frac{1}{3} (2 J_z^2-J_x^2-J_y^2)\\
\end{array}
\right..$$ Clearly, $\{\Gamma^a,\Gamma^b\}=2\delta_{ab}\Gamma^0$ where $\Gamma^0$ is the 4 by 4 identity matrix.
$C_{3v}$ $\mathds{1}$ $C_3$ $\Pi$
---------- -------------- ------- -------
$A_1$ 1 1 1
$A_2$ 1 1 -1
$E$ 2 -1 0
: \[tab:cha\_C3v\] Character table of $C_{3v}$. Here $\mathds{1}$ means identity operation.[@Aroyo2006BilbaoIR]
[|c|m[6 cm]{}|c|]{} $C_{3v}$ & & TR\
$A_1$ & $n_0=\Gamma_0 $ & $+$\
$A_1$ & $n_1=\frac{1}{\sqrt{3}}(\Gamma_1+\Gamma_2+\Gamma_3) $ & $+$\
$A_1$ & $n_2=\frac{1}{\sqrt{3}}(V_x+V_y+V_z) $ & $-$\
$A_2$ & $n_3= J_{xyz} $ & $-$\
$A_2$ & $n_4=\frac{1}{\sqrt{3}}(J_x+J_y+J_z)$ & $-$\
$A_2$ & $n_5=\frac{1}{\sqrt{3}}(P_x+P_y+P_z)$ & $-$\
$E$ & ${\boldsymbol{n}}_{6}=$ $( \frac{1}{\sqrt{6}}( \Gamma_1+\Gamma_2-2\Gamma_3 ), \frac{1}{\sqrt{2}}( -\Gamma_1+\Gamma_2 ))$ & $+$\
$E$ & ${\boldsymbol{n}}_{7}=$ $(\Gamma_5, \Gamma_4)$ & $+$\
$E$ & ${\boldsymbol{n}}_{8}=$ $(\frac{1}{\sqrt{2}}(J_x - J_y ), \frac{1}{\sqrt{6}}(J_x+J_y -2 J_z ) )$ & $-$\
$E$ & ${\boldsymbol{n}}_{9}=$ $(\frac{1}{\sqrt{2}}( P_x-P_y ), \frac{1}{\sqrt{6}}(P_x+P_y -2 P_z) )$ & $-$\
$E$ & ${\boldsymbol{n}}_{10}=$ $( \frac{1}{\sqrt{6}}( V_x+V_y-2V_z ), \frac{1}{\sqrt{2}}( -V_x+V_y ) )$ & $-$\
[|c|m[6 cm]{}|c|]{} $C_{3v}$ & & TR\
$A_1$ & $N_{1}({\boldsymbol{k}}_{{\shortparallel}})=\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_0 $ & $-$\
$A_1$ & $N_{2}({\boldsymbol{k}}_{{\shortparallel}})=\delta^{E_1,+}_{{\boldsymbol{k}}_{{\shortparallel}}}(-\sigma_2) +\delta^{E_2,+}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_1 $ & $-$\
$A_2$ & $N_{3}({\boldsymbol{k}}_{{\shortparallel}})=\sigma_3 $ & $-$\
$A_2$ & $N_{4}({\boldsymbol{k}}_{{\shortparallel}})=-\delta^{E_2,+}_{{\boldsymbol{k}}_{{\shortparallel}}}(-\sigma_2) +\delta^{E_1,+}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_1 $ & $-$\
$E$ & ${\boldsymbol{N}}_{5}({\boldsymbol{k}}_{{\shortparallel}})=$ $(\delta^{E_1,-}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_0 , \delta^{E_2,-}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_0 )$ & $-$\
$E$ & ${\boldsymbol{N}}_{6}({\boldsymbol{k}}_{{\shortparallel}})=$ $(-\delta^{E_2,+}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_3 , \delta^{E_1,+}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_3 )$ & $-$\
$E$ & ${\boldsymbol{N}}_{7}({\boldsymbol{k}}_{{\shortparallel}})=$ $(-\sigma_2 ,\sigma_1 )$ & $-$\
$E$ & ${\boldsymbol{N}}_{8}({\boldsymbol{k}}_{{\shortparallel}})=$ $(-\delta^{E_1,+}_{{\boldsymbol{k}}_{{\shortparallel}}}(-\sigma_2) +\delta^{E_2,+}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_1 $, $\delta^{E_1,+}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_1 + \delta^{E_2,+}_{{\boldsymbol{k}}_{{\shortparallel}}}(-\sigma_2) )$ & $-$\
The list of Gell-Mann matrices[@GellMannMatrices1962] $$\begin{array}{cc}
\lambda_1 =
\left(
\begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0 \\
\end{array}
\right) &
\lambda_2=\left(
\begin{array}{ccc}
0 & -i & 0 \\
i & 0 & 0 \\
0 & 0 & 0 \\
\end{array}
\right) \\
\lambda_3=\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0 \\
\end{array}
\right)&
\lambda_4=\left(
\begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0 \\
\end{array}
\right)
\\
\lambda_5=\left(
\begin{array}{ccc}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0 \\
\end{array}
\right)
&
\lambda_6=\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0 \\
\end{array}
\right)\\
\lambda_7=\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & -i \\
0 & i & 0 \\
\end{array}
\right)&
\lambda_8=\frac{1}{\sqrt{3}}\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -2 \\
\end{array}
\right)
\end{array}\ .$$ And $\lambda_0$ is defined as the $3\times 3$ identity matrix.
Representations of Symmetry Operators {#app:rep_sym}
=====================================
In this section, we show the representation of symmetry operators on the $c^{\dagger}_{{\boldsymbol{k}}}$ bases and the Nambu bases. Before showing the representation, we define the following notations: $\hat{P}_F$ is the fermion parity operator, $\hat{T}_{{\boldsymbol{x}}}$ with ${\boldsymbol{x}}\in \mathds{R}^3$ is a generic translation operator, the generators of $O_h$ group $\hat{C}_{3}$, $\hat{P}$, $\hat{C}_{4}$ and $\hat{\Pi}$ are 3-fold rotations along $(111)$, inversion, 4-fold rotation along $(001)$ and mirror perpendicular to $(1\bar{1}0)$, respectively, and $\hat{\mathcal{T}}$ is the time-reversal operator. Representations of $O(3)$ are not shown here since we only care about the $c_1\neq c_2$ case.
### The $c^{\dagger}_{\mathbf{k}}$ Bases
$$\hat{P}_F c^{\dagger}_{{\boldsymbol{k}}} \hat{P}_F^{-1}=-c^{\dagger}_{{\boldsymbol{k}}}\ ,\ \hat{P}_F c_{{\boldsymbol{k}}} \hat{P}_F^{-1}=-c_{{\boldsymbol{k}}}\ ,$$
$$\hat{T}_{{\boldsymbol{x}}} c^{\dagger}_{{\boldsymbol{k}}} \hat{T}_{{\boldsymbol{x}}}^{-1}=e^{-i {\boldsymbol{k}}\cdot{\boldsymbol{x}}}c^{\dagger}_{{\boldsymbol{k}}}\ ,\
\hat{T}_{{\boldsymbol{x}}} c_{{\boldsymbol{k}}} \hat{T}_{{\boldsymbol{x}}}^{-1}=e^{i {\boldsymbol{k}}\cdot{\boldsymbol{x}}}c_{{\boldsymbol{k}}}\ ,$$
$$\hat{C}_{3} c^{\dagger}_{{\boldsymbol{k}}} \hat{C}_{3}^{-1}=c^{\dagger}_{C_{3}{\boldsymbol{k}}}C_3\ ,\
\hat{C}_{3} c_{{\boldsymbol{k}}} \hat{C}_{3}^{-1}=C_3^{\dagger}c_{C_{3}{\boldsymbol{k}}}\ ,$$
$$\hat{P} c^{\dagger}_{{\boldsymbol{k}}} \hat{P}^{-1}=-c^{\dagger}_{-{\boldsymbol{k}}}\ ,\
\hat{P} c_{{\boldsymbol{k}}} \hat{P}^{-1}=-c_{-{\boldsymbol{k}}}\ ,$$
$$\hat{C}_{4} c^{\dagger}_{{\boldsymbol{k}}} \hat{C}_{4}^{-1}=c^{\dagger}_{C_{4}{\boldsymbol{k}}}C_4\ ,\
\hat{C}_{4} c_{{\boldsymbol{k}}} \hat{C}_{4}^{-1}=C_4^{\dagger} c_{C_{4}{\boldsymbol{k}}}\ ,$$
$$\hat{\Pi} c^{\dagger}_{{\boldsymbol{k}}} \hat{\Pi}^{-1}=c^{\dagger}_{\Pi{\boldsymbol{k}}}\Pi\ ,\
\hat{\Pi} c_{{\boldsymbol{k}}} \hat{\Pi}^{-1}=\Pi^{\dagger} c_{\Pi {\boldsymbol{k}}}\ ,$$
$$\hat{\mathcal{T}} c^{\dagger}_{{\boldsymbol{k}}} \hat{\mathcal{T}}^{-1}=c^{\dagger}_{-{\boldsymbol{k}}}\gamma\ ,\
\hat{\mathcal{T}} c_{{\boldsymbol{k}}} \hat{\mathcal{T}}^{-1}=\gamma^{\dagger} c_{-{\boldsymbol{k}}}\ ,$$
where $C_3= \exp(-i\frac{J_x+J_y+J_z}{\sqrt{3}}\frac{2\pi}{3})$, $C_3{\boldsymbol{k}}=(k_z,k_x,k_y)$, $C_4= \exp(-i J_z \frac{2\pi}{4})$, $C_4{\boldsymbol{k}}=(-k_y,k_x,k_z)$, $\Pi=-\exp(-i\frac{J_x-J_y}{\sqrt{2}}\frac{2\pi}{2})$ and $\Pi {\boldsymbol{k}}=(k_y,k_x,k_z)$.
### The Nambu Bases
$$\hat{P}_F \Psi^{\dagger}_{{\boldsymbol{k}}} \hat{P}_F^{-1}=-\Psi^{\dagger}_{{\boldsymbol{k}}}\ ,\ \hat{P}_F \Psi_{{\boldsymbol{k}}} \hat{P}_F^{-1}=-\Psi_{{\boldsymbol{k}}}\ ,$$
$$\hat{T}_{{\boldsymbol{x}}} \Psi^{\dagger}_{{\boldsymbol{k}}} \hat{T}_{{\boldsymbol{x}}}^{-1}=e^{-i {\boldsymbol{k}}\cdot{\boldsymbol{x}}}\Psi^{\dagger}_{{\boldsymbol{k}}}\ ,\
\hat{T}_{{\boldsymbol{x}}} \Psi_{{\boldsymbol{k}}} \hat{T}_{{\boldsymbol{x}}}^{-1}=e^{i {\boldsymbol{k}}\cdot{\boldsymbol{x}}}\Psi_{{\boldsymbol{k}}}\ ,$$
$$\hat{C}_{3} \Psi^{\dagger}_{{\boldsymbol{k}}} \hat{C}_{3}^{-1}=\Psi^{\dagger}_{C_{3}{\boldsymbol{k}}}\widetilde{C}_3\ ,\
\hat{C}_{3} \Psi_{{\boldsymbol{k}}} \hat{C}_{3}^{-1}=\widetilde{C}_3^{\dagger}\Psi_{C_{3}{\boldsymbol{k}}}\ ,$$
$$\hat{P} \Psi^{\dagger}_{{\boldsymbol{k}}} \hat{P}^{-1}=-\Psi^{\dagger}_{-{\boldsymbol{k}}}\ ,\
\hat{P} \Psi_{{\boldsymbol{k}}} \hat{P}^{-1}=-\Psi_{-{\boldsymbol{k}}}\ ,$$
$$\hat{C}_{4} \Psi^{\dagger}_{{\boldsymbol{k}}} \hat{C}_{4}^{-1}=\Psi^{\dagger}_{C_{4}{\boldsymbol{k}}}\widetilde{C}_4\ ,\
\hat{C}_{4} \Psi_{{\boldsymbol{k}}} \hat{C}_{4}^{-1}=\widetilde{C}_4^{\dagger} \Psi_{C_{4}{\boldsymbol{k}}}\ ,$$
$$\hat{\Pi} \Psi^{\dagger}_{{\boldsymbol{k}}} \hat{\Pi}^{-1}=\Psi^{\dagger}_{\Pi{\boldsymbol{k}}}\widetilde{\Pi}\ ,\
\hat{\Pi} \Psi_{{\boldsymbol{k}}} \hat{\Pi}^{-1}=\widetilde{\Pi}^{\dagger} \Psi_{\Pi{\boldsymbol{k}}}\ ,$$
$$\hat{\mathcal{T}} \Psi^{\dagger}_{{\boldsymbol{k}}} \hat{\mathcal{T}}^{-1}=\Psi^{\dagger}_{-{\boldsymbol{k}}}\mathcal{T}\ ,\
\hat{\mathcal{T}} \Psi_{{\boldsymbol{k}}} \hat{\mathcal{T}}^{-1}=\mathcal{T}^{\dagger} \Psi_{-{\boldsymbol{k}}}\ ,$$
where $\widetilde{C}_3=\text{diag}(C_3,C_3^*)$, $\widetilde{C}_4= \text{diag}(C_4,C_4^*)$, $\widetilde{\Pi}= \text{diag}(\Pi,\Pi^*)$ and $\mathcal{T}=\text{diag}(\gamma, \gamma^*)$. $\widetilde{C}_3$, $\widetilde{\Pi}$, $\mathcal{T}K$ and $\mathcal{C}K$ commute with each other, where $K$ is the complex conjugate operation. $\chi$ anti-commutes with $\mathcal{T}K$ and $\mathcal{C}K$ and commutes with $\widetilde{C}_3$ and $\widetilde{\Pi}$.
Surface Majorana Flat Bands {#app:surf_modes}
===========================
### Existence of Surface Zero Modes
Due to the topological invariant $N_{w}=\pm 2$ at each non-trivial ${\boldsymbol{k}}_{{\shortparallel}}$, we expect two boundary modes at each non-trivial ${\boldsymbol{k}}_{{\shortparallel}}$ on one surface of our model. [@yu2017Singlet-Quintetj=3/2SC] Therefore, we consider a semi-infinite version of [Eq.]{} ($x_{\perp}<0$) with open boundary condition at $x_{\perp}=0$, where $x_{\perp}$ is the position on $(111)$ axis. The corresponding Hamiltonian reads $$\begin{aligned}
\label{eq:H_BdG_perp}
&&H_{\perp}=\frac{1}{2}\sum_{{\boldsymbol{k}}_{{\shortparallel}}}\int_{-\infty}^0 dx_{\perp}\Psi_{{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp}}^{\dagger}h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})\Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}\nonumber\\
&& +\sum_{{\boldsymbol{k}}_{{\shortparallel}}}\int^{+\infty}_0 dx_{\perp} E_{\infty} c^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp}} c_{{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp}} +const.\ ,\end{aligned}$$ where $c^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp}}=\frac{1}{\sqrt{L_{\perp}}}\sum_{k_{\perp}}e^{-i k_{\perp} x_{\perp}}c^{\dagger}_{{\boldsymbol{k}}}$ with $L_{\perp}$ the length along the $(111)$ direction of the entire space, $h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})$ is obtained by replacing $k_{\perp}$ in $h_{BdG}({\boldsymbol{k}})$ by $-i\partial_{x_{\perp}}$, $\Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp}}=(c_{{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp}}^{\dagger},c_{-{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp}}^{T})$, and $E_{\infty}\rightarrow +\infty$ is for the open boundary condition. For such a semi-infinite system, the translation symmetry in the $(111)$ direction, the inversion symmetry and the 4-fold rotational symmetry along $(001)$ are broken. The Hamiltonian $h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})$ still has PH, TR, chiral and $C_{3v}$ symmetries $-\mathcal{C}[h_{BdG}(-{\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})]^*\mathcal{C}^{\dagger}=h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})$, $\mathcal{T}[h_{BdG}(-{\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})]^*\mathcal{T}^{\dagger}=h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})$, $-\chi h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})\chi^{\dagger}=h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})$ and $\widetilde{R} h_{BdG}(R^{-1}{\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})\widetilde{R}^{\dagger}=h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})$, respectively, where $R=C_3, \Pi$. In addition, the PH symmetry requires $\mathcal{C}(\Psi^{\dagger}_{-{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp}})^T=\Psi_{{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp}}$ and the commutation relation is $$\begin{aligned}
&&\{\Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp},\alpha,s},\Psi_{{\boldsymbol{k}}_{{\shortparallel}}', x_{\perp}',\alpha',s'} \}=\delta_{{\boldsymbol{k}}_{{\shortparallel}},{\boldsymbol{k}}_{{\shortparallel}}'}\delta(x_{\perp}-x_{\perp}')\delta_{\alpha\alpha'}\delta_{s s'}\\
&&\{\Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}, x_{\perp},\alpha,s},\Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}', x_{\perp}',\alpha',s'} \}=\delta_{{\boldsymbol{k}}_{{\shortparallel}},-{\boldsymbol{k}}_{{\shortparallel}}'}\delta(x_{\perp}-x_{\perp}')(\tau_x)_{\alpha\alpha'}\delta_{s s'}\nonumber\ ,\end{aligned}$$ where $\alpha,\alpha'=1,2$ stand for the particle-hole index and $s,s'$ are spin index of the $j=3/2$ fermion.
The surface mode with zero energy $b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}$ of $H_{\perp}$ in [Eq.]{} is defined as $$b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}=\int_{-\infty}^0 d x_{\perp} \Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}^{\dagger} v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}},$$ which satisfies $[H_{\perp},b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}]=0$ and $v_{{\boldsymbol{k}}_{{\shortparallel}},0}=v_{{\boldsymbol{k}}_{{\shortparallel}},-\infty}=0$. With the PH symmetry and the commutation relation, the equation $[H_{\perp},b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}]=0$ can be simplified as $$\label{eq:surf_modes_ori}
h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}}) v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=0\ .$$ Now we try to figure out the properties of the solution. First, transform the above equation to chiral eigen-bases: $$\label{eq:surf_modes}
U^{\dagger}_{\chi} h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}}) U_{\chi} U^{\dagger}_{\chi}v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=0\ ,$$ where $$U_{\chi}
=
\frac{1}{\sqrt{2}}
\left(
\begin{matrix}
\mathds{1}_4 & \mathds{1}_4\\
i\gamma & -i\gamma\\
\end{matrix}
\right)$$ is the unitary matrix that diagonalizes $\chi$: $$U^{\dagger}_{\chi} \chi U_{\chi}=
\left(
\begin{matrix}
\mathds{1}_4 & \\
& -\mathds{1}_4\\
\end{matrix}
\right)$$ , $$U^{\dagger}_{\chi} h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}}) U_{\chi}=
\left(
\begin{matrix}
& q({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})\\
[q({\boldsymbol{k}}_{{\shortparallel}},i\partial_{x_{\perp}})]^{\dagger} & \\
\end{matrix}
\right)\ ,$$ and $$q({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})=h({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})-i \Delta({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})\gamma\ .$$ The TR and PH matrices in the chiral representation read $$U^{\dagger}_{\chi}\mathcal{T}U^{*}_{\chi}=\left(
\begin{matrix}
& \gamma\\
\gamma & \\
\end{matrix}
\right)$$ and $$U^{\dagger}_{\chi}\mathcal{C}U^{*}_{\chi}=\left(
\begin{matrix}
& i \gamma\\
-i \gamma & \\
\end{matrix}
\right)\ .$$ In the chiral representation, both TR and PH symmetries give the same condition on $q$: $$\gamma [q(-{\boldsymbol{k}}_{{\shortparallel}},i\partial_{x_{\perp}})]^{T} \gamma^{\dagger}=q({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})\ .$$ By defining $U^{\dagger}_{\chi}v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=(u^T_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}},w^T_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}})^T$ with $u$($w$) corresponding to chiral eigen-wavefunction with chiral eigenvalues $1$($-1$), [Eq.]{} can be expressed as $$\label{eq:surf_modes_q}
\left\{
\begin{array}{l}
q({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})w_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=0\\
q^{\dagger}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})u_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=0
\end{array}
\right. \ .$$ Since $h_{BdG}(-{\boldsymbol{k}}_{{\shortparallel}},i\partial_{x_{\perp}})=h_{BdG}({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})$ originated from the bulk inversion symmetry, we have $q({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})=q(-{\boldsymbol{k}}_{{\shortparallel}},i\partial_{x_{\perp}})$. Combined with TR, the equation of $u$ in [Eq.]{} can be transformed to $$q({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})\gamma^T u^*_{{\boldsymbol{k}}_{{\shortparallel}},-x_{\perp}}=0\ .$$ Since $u_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=0$ for $x_{\perp}=0,-\infty$ which means $\gamma^T u^*_{{\boldsymbol{k}}_{{\shortparallel}},-x_{\perp}}=0$ for $x_{\perp}=0,+\infty$, the above equation is the same as the equation of $w$ except that the open boundary conditions are at $x_{\perp}=0,+\infty$. Therefore, we can solve the equation of $w$ in [Eq.]{}, i.e. $$\label{eq:surf_modes_q_w}
q({\boldsymbol{k}}_{{\shortparallel}},-i\partial_{x_{\perp}})w_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=0 \ ,$$ with $w_{{\boldsymbol{k}}_{{\shortparallel}},0}=w_{{\boldsymbol{k}}_{{\shortparallel}},-\infty}=0$ to have the solutions of $w$ and with $w_{{\boldsymbol{k}}_{{\shortparallel}},0}=w_{{\boldsymbol{k}}_{{\shortparallel}},\infty}=0$ to have the solutions of $u$ by $u_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=\gamma w^*_{{\boldsymbol{k}}_{{\shortparallel}},-x_{\perp}}$.
With the ansatz $w_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=e^{\lambda x_{\perp}}\bar{w}_{{\boldsymbol{k}}_{{\shortparallel}}}$, the [Eq.]{} becomes $$\label{eq:surf_modes_q_w_ansatz}
q({\boldsymbol{k}}_{{\shortparallel}},-i\lambda)\bar{w}_{{\boldsymbol{k}}_{{\shortparallel}}}=0$$ with the solution determined by the octic equation $\text{det}[q({\boldsymbol{k}}_{{\shortparallel}},-i\lambda)]=0$ for $\lambda$. The equation has 4 double roots $\lambda_{1,2,3,4}$ since $\text{det}[q({\boldsymbol{k}}_{{\shortparallel}},-i\lambda)]$ can be written in the form of the square of certain function, $\text{det}[q({\boldsymbol{k}}_{{\shortparallel}},-i\lambda)]=[\widetilde{q}({\boldsymbol{k}}_{{\shortparallel}},-i\lambda)]^2$.[@yu2017Singlet-Quintetj=3/2SC] In addition, since $\widetilde{q}({\boldsymbol{k}}_{{\shortparallel}},-i\lambda)$ does not have $\lambda^3$ term, the sum of $\lambda_{1,2,3,4}$ is zero. Each double root $\lambda_i$ can give two orthogonal solutions $\bar{w}_{{\boldsymbol{k}}_{{\shortparallel}},i,j}$ of [Eq.]{} with $i=1,2,3,4$ and $j=1,2$. Then the general solution of [Eq.]{} without boundary condition reads $$w_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=\sum_{i=1}^4\sum_{j=1}^2 b_{ij} e^{\lambda_i x_{\perp}}\bar{w}_{{\boldsymbol{k}}_{{\shortparallel}},i,j}\ .$$ Now let us impose the boundary condition. $w_{{\boldsymbol{k}}_{{\shortparallel}},\infty}=0$ or $w_{{\boldsymbol{k}}_{{\shortparallel}},-\infty}=0$ requires $Re[\lambda_i]<0$ or $Re[\lambda_i]>0$, respectively, and $w_{{\boldsymbol{k}}_{{\shortparallel}},0}=0$ requires $\sum_{i,j} b_{ij} \bar{w}_{{\boldsymbol{k}}_{{\shortparallel}},i,j}=0$. Since the sum of the four $\lambda_i$’s is zero, it is impossible to have four $Re[\lambda_i]$’s with the same sign. If only two $Re[\lambda_i]$’s have the same sign, there will be typically no solutions, since the corresponding four four-component $\bar{w}_{{\boldsymbol{k}}_{{\shortparallel}},i,j}$’s typically can not be linearly dependent. If three $\lambda_i$’s satisfy $Re[\lambda_i]>0$($Re[\lambda_i]<0$), there are six corresponding four-component $\bar{w}_{{\boldsymbol{k}}_{{\shortparallel}},i,j}$’s, resulting in two solutions to $w$($u$) corresponding to two surface zero modes $v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=U_{\chi}(0,w^T_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}})^T$($v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=U_{\chi}(u^T_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}},0$)) with chiral eigenvalue $-1$($1$). Therefore, the generic number of surface zero modes at a fixed ${\boldsymbol{k}}_{{\shortparallel}}$ on one surface, if exist, is two and those two modes are chiral eigenstates of the same chiral eigenvalues.
### Symmetries of Surface Zero Modes
Now we will show the symmetry properties of the surface zero modes. We take $v_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ with $i=1,2$ as the two orthonormal surface wavefunctions that satisfies [Eq.]{} at ${\boldsymbol{k}}_{{\shortparallel}}$ with the boundary conditions. Orthonormality requires $$\int_{-\infty}^0 d x_{\perp} v^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}v_{j,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=\delta_{ij}\ .$$ The creation operators of surface modes read $$b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}=\int_{-\infty}^0 d x_{\perp} \Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}^{\dagger} v_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}\ ,$$ and the orthonormal condition of $v_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ leads to the anti-commutation relations $$\left\{b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}, b_{j,{\boldsymbol{k}}'_{{\shortparallel}}}\right\}=\delta_{ij}\delta_{{\boldsymbol{k}}_{{\shortparallel}} {\boldsymbol{k}}'_{{\shortparallel}}}\ .$$ The effective Hamiltonian for the surface zero modes can thus be expressed as $$\label{eq:H_surf}
H_{surf}=E_{surf} \sum_{{\boldsymbol{k}}_{{\shortparallel}}\in A}b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}} b_{{\boldsymbol{k}}_{{\shortparallel}}}\ ,$$ where $A$ stands for the entire surface mode regions in the surface Brillouin zone, $E_{surf}=0$ and $b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}=(b^{\dagger}_{1,{\boldsymbol{k}}_{{\shortparallel}}},b^{\dagger}_{2,{\boldsymbol{k}}_{{\shortparallel}}})$. Fermion parity operator will transform the $
b_{{\boldsymbol{k}}_{{\shortparallel}}}$ operators as $
b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}\rightarrow -b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}
$ and $
b_{{\boldsymbol{k}}_{{\shortparallel}}}\rightarrow -b_{{\boldsymbol{k}}_{{\shortparallel}}}
$. The 2D translation read $
\hat{T}_{{\boldsymbol{x}}_{{\shortparallel}}}b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}\hat{T}^{-1}_{{\boldsymbol{x}}_{{\shortparallel}}}=e^{-i{\boldsymbol{k}}_{{\shortparallel}}\cdot {\boldsymbol{x}}_{{\shortparallel}}}b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}
$ and $
\hat{T}_{{\boldsymbol{x}}_{{\shortparallel}}}b_{{\boldsymbol{k}}_{{\shortparallel}}}\hat{T}^{-1}_{{\boldsymbol{x}}_{{\shortparallel}}}=e^{i{\boldsymbol{k}}_{{\shortparallel}}\cdot {\boldsymbol{x}}_{{\shortparallel}}}b_{{\boldsymbol{k}}_{{\shortparallel}}}
$ . Due to the TR symmetry, two orthonormal surface wavefunctions $v_{i,-{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ at $-{\boldsymbol{k}}_{{\shortparallel}}$ can be given by the linear combinations of $\mathcal{T}v^*_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$. Due to $\{ \mathcal{T} K, \chi \}=0$, $v_{i,-{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ and $v_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ have opposite chiral eigenvalues. It means that $A_{\pm}$ can be related by ${\boldsymbol{k}}_{{\shortparallel}}\rightarrow -{\boldsymbol{k}}_{{\shortparallel}}$, where $A_{\pm}$ are the surface mode regions in the ${\boldsymbol{k}}_{{\shortparallel}}$ space that are filled with the momenta of surface zero modes with chiral eigenvalue $\pm 1$, respectively. Based on the same logic, $C_{3v}$ symmetries gives that $ v_{i, C_3 {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ are linear combinations of $\widetilde{C}_3 v_{i, {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ and $ v_{i, \Pi {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ are linear combinations of $\widetilde{\Pi} v_{i, {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$. Furthermore, since $\chi$ commutes with any operation in $C_{3v}$, $v_{i, C_3 {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$’s and $ v_{i, \Pi {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$’s have the same chiral eigenvalue as $v_{i, {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$, meaning that both $A_+$ and $A_-$ are $C_{3v}$ symmetric. The representations of $\hat{\mathcal{T}}$, $\hat{C}_3$ and $\hat{\Pi}$ rely on the convention that we choose for $v_{i, {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$’s. For convenience, we choose a special convention such that $$\label{eq:v_conv}
\left\{
\begin{array}{l}
\mathcal{T}v_{i, {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}^*=\sum_j v_{j, -{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}(i\sigma_2)_{ji} \\
\widetilde{C}_3 v_{i, {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=
\sum_{j}v_{j, C_3{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}(e^{-i \sigma_3 \frac{\pi}{3}})_{ji} \\
\widetilde{\Pi} v_{i, {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=
\sum_{j}v_{j, \Pi{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}(-e^{-i \sigma_2 \frac{\pi}{2}})_{ji}
\end{array}
\right. \ .$$ As a result, $b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}$ imitates a $j=1/2$ fermion: $$\left\{
\begin{array}{l}
\hat{\mathcal{T}}b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}\hat{\mathcal{T}}^{-1}=
b^{\dagger}_{-{\boldsymbol{k}}_{{\shortparallel}}}i\sigma_2 \\
\hat{C}_3 b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}\hat{C}_3^{-1}=
b^{\dagger}_{C_3{\boldsymbol{k}}_{{\shortparallel}}}e^{-i \sigma_3 \frac{\pi}{3}} \\
\hat{\Pi} b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}\hat{\Pi}^{-1}=
b^{\dagger}_{\Pi {\boldsymbol{k}}_{{\shortparallel}}}(-e^{-i \sigma_2 \frac{\pi}{2}})
\end{array}
\right. \ ,$$ where $\sigma_{1,2,3}$ are Pauli matrices for the double degeneracy of the surface modes. And we can treat the double degeneracy of the surface modes as the pseudospin of the surface modes. Since the PH symmetry is related with TR and chiral symmetries by $\chi=i \mathcal{T C^*}$, we have $$\label{eq:PH_surf_v}
v_{i,-{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=\sum_{j=1}^2 \mathcal{C} v_{j,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}^*(\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_{2})_{ji}\ ,$$ where $\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}=\pm 1$ for ${\boldsymbol{k}}_{{\shortparallel}}\in A_{\pm}$, $\chi v_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}= \delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}} v_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$, $\delta^{\chi}_{-{\boldsymbol{k}}_{{\shortparallel}}}=-\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}$ since $v_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ and $v_{i,-{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ have opposite chiral eigenvalues, and $\delta^{\chi}_{R{\boldsymbol{k}}_{{\shortparallel}}}=\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}$ with $R\in C_{3v}$ since $v_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ and $v_{i,R{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ have the same chiral eigenvalue. Furthermore, using $\Psi_{-{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}^{\dagger}= \Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}^{T} \mathcal{C}$, we can get $$b^{\dagger}_{-{\boldsymbol{k}}_{{\shortparallel}}}=b^T_{{\boldsymbol{k}}_{{\shortparallel}}}(\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_{2})\Leftrightarrow b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}(-\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_{2})=b^T_{-{\boldsymbol{k}}_{{\shortparallel}}}\ .$$ Thus, the PH symmetry gives rise to the following relation $$\begin{aligned}
&&\left\{b_{i,{\boldsymbol{k}}_{{\shortparallel}}}^{\dagger}, b_{j,{\boldsymbol{k}}'_{{\shortparallel}}}^{\dagger}\right\}=
\left\{b_{i,{\boldsymbol{k}}_{{\shortparallel}}}^{\dagger}, b_{i',-{\boldsymbol{k}}'_{{\shortparallel}}} (\delta^{\chi}_{-{\boldsymbol{k}}'_{{\shortparallel}}}\sigma_{2})_{i' j}\right\}\\
&&=
(\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_{2})_{ij}\delta_{{\boldsymbol{k}}_{{\shortparallel}},- {\boldsymbol{k}}'_{{\shortparallel}}}\ ,\end{aligned}$$ which implies that only half the surface modes are actually physical due to the double counting of the BdG Hamiltonian. In this case, we can treat the surfaces modes as two Majorana zero modes(MZMs) at each ${\boldsymbol{k}}_{{\shortparallel}}$ as described below. In general, the fermionic creation operator $b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}$ can be expressed as the linear combination of two Majorana operators: $
b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}=\frac{1}{2}(\gamma_{i,{\boldsymbol{k}}_{{\shortparallel}}}+i\widetilde{\gamma}_{i,{\boldsymbol{k}}_{{\shortparallel}}})\ ,
$ where $$\label{eq:exp_MFB}
\gamma_{i,{\boldsymbol{k}}_{{\shortparallel}}}=b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}+b_{i,{\boldsymbol{k}}_{{\shortparallel}}}\ ,$$ and $
\widetilde{\gamma}_{i,{\boldsymbol{k}}_{{\shortparallel}}}=\frac{1}{i}(b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}-b_{i,{\boldsymbol{k}}_{{\shortparallel}}})
$. Due to [Eq.]{}, $\gamma_{i,{\boldsymbol{k}}_{{\shortparallel}}}$ and $\widetilde{\gamma}_{i,{\boldsymbol{k}}_{{\shortparallel}}}$ depend on each other by the relation $
\gamma_{i,-{\boldsymbol{k}}_{{\shortparallel}}}=-\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sum_{j}\widetilde{\gamma}_{j,{\boldsymbol{k}}_{{\shortparallel}}}(i\sigma_2)_{ji}\ .
$ Therefore, $\widetilde{\gamma}_{i,{\boldsymbol{k}}_{{\shortparallel}}}$’s can be chosen to be redundant and we can treat the physical degrees of freedom as two MZMs at each ${\boldsymbol{k}}_{{\shortparallel}}$, of which the Majorana operators are $\gamma_{i,{\boldsymbol{k}}_{{\shortparallel}}}$. And the $\gamma_{i,{\boldsymbol{k}}_{{\shortparallel}}}$ operators satisfy the following anti-commutation relation: $$\begin{aligned}
&&\{\gamma_{i,{\boldsymbol{k}}_{{\shortparallel}}},\gamma_{j,{\boldsymbol{k}}'_{{\shortparallel}}}\}=\nonumber\\
&&\{ b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}},b^{\dagger}_{j,{\boldsymbol{k}}_{{\shortparallel}}'}\}+\{ b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}},b_{j,{\boldsymbol{k}}_{{\shortparallel}}'}\}+\{ b_{i,{\boldsymbol{k}}_{{\shortparallel}}},b^{\dagger}_{j,{\boldsymbol{k}}_{{\shortparallel}}'}\}+\{ b_{i,{\boldsymbol{k}}_{{\shortparallel}}},b_{j,{\boldsymbol{k}}_{{\shortparallel}}'}\}\nonumber\\
&&=2\delta_{ij}\delta_{{\boldsymbol{k}}_{{\shortparallel}},{\boldsymbol{k}}'_{{\shortparallel}}}+(\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_{2})_{ij}\delta_{{\boldsymbol{k}}_{{\shortparallel}},- {\boldsymbol{k}}'_{{\shortparallel}}}+(\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_{2})_{ij}^*\delta_{{\boldsymbol{k}}_{{\shortparallel}},- {\boldsymbol{k}}'_{{\shortparallel}}}\nonumber\\
&& =2\delta_{ij}\delta_{{\boldsymbol{k}}_{{\shortparallel}},{\boldsymbol{k}}'_{{\shortparallel}}}\ .\end{aligned}$$ Although the actual physical degrees of freedom are MZMs, we still use $b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}}$ and $b_{{\boldsymbol{k}}_{{\shortparallel}}}$ in the following for convenience.
Projecting [Eq.]{} onto the surface to get [Eq.]{} {#app:H_mf_c2b}
==================================================
In this part, we will derive [Eq.]{} by projecting [Eq.]{} onto the surface. First, we show the relation between the surface modes $b^{\dagger}$ and the Nambu bases $\Psi^{\dagger}$. Due to the completeness of eigenstates of Hermitian operator, $\Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp},\alpha,s}$ and $\Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp},\alpha,s}$ can be expressed in terms of eigenstates of [Eq.]{} for $x_{\perp}<0$ and ${\boldsymbol{k}}_{{\shortparallel}}\in A$: $$\left\{
\begin{array}{l}
\Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp},\alpha,s}=\sum_{i} v^*_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp},\alpha,s} b^{\dagger}_{i,{\boldsymbol{k}}_{{\shortparallel}}}+\text{bulk modes}\\
\Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp},\alpha,s}=\sum_{i} v_{i,{\boldsymbol{k}}_{{\shortparallel}},x_{\perp},\alpha,s} b_{i,{\boldsymbol{k}}_{{\shortparallel}}}+\text{bulk modes}
\end{array}
\right. \ ,$$ where $\alpha=e,h$ is the particle-hole index and $s=\pm\frac{3}{2},\pm\frac{1}{2}$. Let us define $v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ as a $8\times 2$ matrix with $(\alpha,s)$ labeling the row and $i$ being the column index, and then the above relations can be expressed in the matrix version: $$\label{eq:rel_b_psi}
\left\{
\begin{array}{l}
\Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}= b^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}}} v^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}+\text{bulk modes}\\
\Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}= v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}} b_{{\boldsymbol{k}}_{{\shortparallel}}}+\text{bulk modes}
\end{array}
\right. \ .$$ In the matrix version, the symmetries of the surface eigenvectors become $$\label{eq:v_conv_mat}
\left\{
\begin{array}{l}
\mathcal{T}v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}^*= v_{ -{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}} \mathcal{T}_b \\
\widetilde{C}_3 v_{ {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=
v_{ C_3{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}C_{3,b}\\
\widetilde{\Pi} v_{ {\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}=
v_{\Pi{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}\Pi_b\\
v_{-{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}= \mathcal{C} v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}^*\delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}\sigma_{2}\\
\chi v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}= \delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}}v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}\\
\end{array}
\right. \ .$$ If ${\boldsymbol{k}}_{{\shortparallel}}$ is outside the surface mode regions, $\Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp},\alpha,s}$ and $\Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp},\alpha,s}$ only contain bulk modes.
In the Nambu bases, [Eq.]{} reads $$\label{eq:Ht_mf_Psi}
\widetilde{H}_{mf}=\frac{1}{2}\sum_{{\boldsymbol{k}}_{{\shortparallel}}}^{A}\int_{-\infty}^{0} d x_{\perp} \Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}
\widetilde{h}(x_{\perp}) \Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}+const.\ ,$$ where $$\label{eq:ht}
\widetilde{h}(x_{\perp})
=
\left(
\begin{array}{cc}
\widetilde{M}(x_{\perp}) & \widetilde{D}(x_{\perp}) \\
\widetilde{D}^{\dagger}(x_{\perp}) & -\widetilde{M}^T(x_{\perp})
\end{array}
\right)\ .$$ Using [Eq.]{} and neglecting terms involving bulk modes, we can obtain [Eq.]{} with $m({\boldsymbol{k}}_{{\shortparallel}})=\int_{-\infty}^0 dx_{\perp} v^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}} \widetilde{h}(x_{\perp}) v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ being Hermitian. Due to the PH symmetry of $\widetilde{h}(x_{\perp})
$, i.e. $-\mathcal{C}\widetilde{h}^T(x_{\perp})\mathcal{C}^{\dagger}=\widetilde{h}(x_{\perp})
$, and $v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ in Eq. (\[eq:v\_conv\_mat\]), the obtained $m({\boldsymbol{k}}_{{\shortparallel}})$ is PH symmetric. Only the TR odd part of $m({\boldsymbol{k}}_{{\shortparallel}})$, as well as $\widetilde{h}(x_{\perp})$, is allowed for the surface orders and thereby we only need to consider $\widetilde{h}(x_{\perp})$ satisfying $\mathcal{T}\widetilde{h}^*(x_{\perp})\mathcal{T}^{\dagger}=-\widetilde{h}(x_{\perp})$, which is equivalent to $\gamma \widetilde{M}^*(x_{\perp}) \gamma^{\dagger}=-\widetilde{M}(x_{\perp})$ and $\gamma \widetilde{D}^*(x_{\perp}) \gamma^{T}=-\widetilde{D}(x_{\perp})$. Suppose $\widetilde{h}(x_{\perp})$ is the linear combination of $\widetilde{h}_i(x_{\perp})$ and $\widetilde{R} \widetilde{h}_i(x_{\perp})\widetilde{R}^{\dagger}=\sum_{j} f_{ij} \widetilde{h}_j(x_{\perp})$ with $f_{ij}\in \mathds{R}$, where the latter is equivalent to $R \widetilde{M}_i(x_{\perp}) R^{\dagger}=f_{ij} \widetilde{M}_j(x_{\perp})$ and $R \widetilde{D}_i(x_{\perp}) R^{T}=f_{ij} \widetilde{D}_j(x_{\perp})$, and $R\in C_{3v}$. According to the transformation of $v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}$ under $C_{3v}$ (\[eq:v\_conv\_mat\]), we have $R_b \widetilde{m}_i (R^{-1}{\boldsymbol{k}}_{{\shortparallel}}) R_b^{\dagger}=\sum_{j} f_{ij} \widetilde{m}_j ({\boldsymbol{k}}_{{\shortparallel}})$, where $ \widetilde{m}_i ({\boldsymbol{k}}_{{\shortparallel}})$ is the surface projection of $\widetilde{h}_i(x_{\perp})$. Therefore, if $\widetilde{h}_i(x_{\perp})$, or equivalently $\widetilde{M}_i(x_{\perp})$ and $\widetilde{D}_i(x_{\perp})$, belongs to a certain IR of $C_{3v}$, the corresponding surface projection belongs to the same IR.
Arcs of Majorana Zero Modes {#app:MZM_arc}
===========================
![\[fig:surf\_MZM\_arc\] (a),(b) and (c) show the distribution of surface MZMs in the presence of $A_2$ surface translationally invariant order parameter without the $E$ order parameters, with the $\Pi$ anti-symmetric component of $E$ order parameters and with the $\Pi$ symmetric component of $E$ order parameters, respectively. Blue lines are the boundaries of surface mode regions shown in [Fig.\[fig:surf\_MFB\]]{} and one MZM exists on each point of orange lines. $m_3/|\mu|=0.05$, $m_4/|\mu|=0.04$, $B_0\sqrt{2m/\mu}=0.8$, $B_1\sqrt{2m/\mu}=B_3\sqrt{2m/\mu}=1$ and $B_2\sqrt{2m/\mu}=-0.5$ are chosen for (a),(b) and (c), while $(m_{7,1}/|\mu|,m_{7,2}/|\mu|)=(0,0)$ for (a), $(m_{7,1}/|\mu|,m_{7,2}/|\mu|)=(0,0.05)$ for (b) and $(m_{7,1}/|\mu|,m_{7,2}/|\mu|)=(0.05,0)$ for (c). The non-zero values of $(m_{7,1}/|\mu|,m_{7,2}/|\mu|)$ indicate the existence of $E$ order. The values of all other parameters are the same as [Fig.\[fig:surf\_MFB\]]{}.](Interacting_MZMs_surf_MZM_arcs.pdf){width="\columnwidth"}
In this section, we will discuss the condition for the arcs of MZMs in the ${\boldsymbol{k}}_{\shortparallel}$-space induced by order parameters. The analysis in [Sec.\[sec:MF\_order\_MFB\]]{} only included orders that are uniform in each $A_{l_{\chi} l_c}$, and thereby the surface zero modes either exist or disappear at all ${\boldsymbol{k}}_{{\shortparallel}}$ points in one $A_{l_{\chi} l_c}$ simultaneously. If the momentum dependence of the orders within each $A_{l_{\chi} l_c}$ is considered, it is possible that MZMs exist at lines in the surface mode regions. To illustrate that, we consider the $A_2$ order parameter to the linear order of momentum, which has no MZMs according to the analysis in [Sec.\[sec:MF\_order\_MFB\]]{}. To take into account the momentum dependence inside $A_{l_{\chi}, l_c}$, we define ${\boldsymbol{K}}^{l_{\chi}, l_c}_{{\shortparallel}}$ to be the geometric center of $A_{l_{\chi}, l_c}$, and define $h_{A_2}^{l_{\chi}, l_c}({\boldsymbol{q}}_{{\shortparallel}})\equiv m_{A_2}({\boldsymbol{q}}_{{\shortparallel}}+{\boldsymbol{K}}^{l_{\chi}, l_c}_{{\shortparallel}})$ with ${\boldsymbol{q}}_{{\shortparallel}}\equiv {\boldsymbol{k}}_{{\shortparallel}}-{\boldsymbol{K}}^{l_{\chi}, l_c}_{{\shortparallel}}$. Due to the odd mirror parity of $A_2$ order parameter and the $\Pi$ symmetry of $A_{+, 3}$, $h_{A_2}^{+,3}({\boldsymbol{q}}_{{\shortparallel}})$ to the first order of ${\boldsymbol{q}}_{{\shortparallel}}$ is $$\begin{aligned}
\label{eq:h_RP3_A_2_k1}
&&h_{A_2}^{+,3}({\boldsymbol{q}}_{{\shortparallel}})=B_0 q_{{\shortparallel},2}\sigma_0 + (-m_4 + B_1 q_{{\shortparallel},1})\sigma_1+(- B_2 q_{{\shortparallel},2})\sigma_2 \nonumber\\
&& + (m_3 +B_3 q_{{\shortparallel},1})\sigma_3\ ,\end{aligned}$$ where $K^{+,3}_{{\shortparallel},2}=0$ is used. In the following, we assume $B_{1,2,3,4} \neq 0$. Using $C_{3v}$ and PH symmetries, we have $h_{A_2}^{+,1}({\boldsymbol{q}}_{{\shortparallel}})=C_{3,b} h_{A_2}^{+,3}(C_3^{-1}{\boldsymbol{q}}_{{\shortparallel}})C_{3,b}^{\dagger}$, $h_{A_2}^{+,2}({\boldsymbol{q}}_{{\shortparallel}})=C_{3,b}^{\dagger} h_{A_2}^{+,3}(C_3{\boldsymbol{q}}_{{\shortparallel}})C_{3,b}$, and $h_{A_2}^{-,l_c}({\boldsymbol{q}}_{{\shortparallel}})=-\sigma_2 [h_{A_2}^{+,l_c}(-{\boldsymbol{q}}_{{\shortparallel}})]^T \sigma_2$. As a result, the number of MZMs at ${\boldsymbol{k}}_{{\shortparallel}}$ is the same as that at $C_3{\boldsymbol{k}}_{{\shortparallel}}$, $\Pi {\boldsymbol{k}}_{{\shortparallel}}$ and $-{\boldsymbol{k}}_{{\shortparallel}}$, and thereby we only need to study the existence of MZMs in $A_{+, 3}$. The eigenvalues of $h_{A_2}^{+,3}({\boldsymbol{q}}_{{\shortparallel}})$ are $$B_0 q_{{\shortparallel},2}\pm \sqrt{(m_4 -B_1 q_{{\shortparallel},1})^2 + (B_2 q_{{\shortparallel},2})^2 + (m_3+ B_3 q_{{\shortparallel},1})^2}\ .$$ In the case where $-m_3/B_3=m_4/B_1$, two MZMs exist at ${\boldsymbol{q}}_{{\shortparallel}}=(m_4/B_1, 0)$ if $(m_4/B_1, 0)\in A_{+, 3}$, and one MZM exists at every other point(in $A_{+, 3}$) on the straight line $(m_4/B_1, q_{{\shortparallel},2})$ if $B_0^2- B_2 ^2=0$ or on the straight lines $(q_{{\shortparallel},1}, \pm \sqrt{\frac{B_3^2+B_1^2}{B_0^2-B_2^2}} (m_4/B_1 - q_{{\shortparallel},1}))$ if $B_0^2- B_2 ^2>0$. In the case where $-m_3/B_3\neq m_4/B_1$, one MZM exists at every point on the part of the hyperbolas $(q_{{\shortparallel},1}, \pm \sqrt{\frac{(m_4 - B_1 q_{{\shortparallel},1})^2+(m_3 + B_3 q_{{\shortparallel},1})^2}{B_0^2-B_2^2})}$ that is in $A_{+, 3}$ if $B_0^2- B_2 ^2>0$. If none of the conditions listed above are satisfied, no MZMs exist. As an example, [Fig.\[fig:surf\_MZM\_arc\]]{}a shows the surface Majorana arcs for $B_0^2- B_2 ^2>0$ and $-m_3/B_3\neq m_4/B_1$, where only one MZM exists at each point of the arcs and the distribution of MZMs has $C_{3v}$ and PH symmetries as mentioned before. In the plot, we assume only surface order is formed and the bulk nodal lines as well as the boundaries of surface mode regions do not change. Such distribution of Majorana arcs is possible to be generated by surface FM along the $(111)$ direction since it is an $A_2$ order parameter.
Next we consider how the $E$ order parameter changes the distribution of Majorana arcs. Suppose the surface Majorana arcs exist for the $A_2$ order which is given by surface FM in the $(111)$ direction. In this case, the presence of the small $E$ order parameter can be achieved by tuning the surface magnetic moment slightly away from the $(111)$ direction with a weak external magnetic field, which can change the distribution of the surface Majorana arcs. To illustrate that, we add only the momentum independent $E$ order parameter ${\boldsymbol{m}}_7\cdot{\boldsymbol{N}}_7$ to the $A_2$ order $h_{A_2}^{l_\chi,l_c}({\boldsymbol{q}}_{{\shortparallel}})$ for simplicity. If the magnetic moment is tilted to $(11\bar{2})$ direction, then the system still has odd $\Pi$ parity, meaning that $m_{7,1}=0$. In this case, the $C_3$ symmetry of the distribution of surface Majorana arc is broken while its $\Pi$ symmetry is preserved, which is exactly shown in [Fig.\[fig:surf\_MZM\_arc\]]{}b. If the magnetic moment is tilted to $(\bar{1}10)$ direction, then the extra term should be $\Pi$ symmetric, meaning that $m_{7,2}=0$. As a result, the entire $C_{3v}$ symmetry of the surface Majorana arc distribution is broken, which matches [Fig.\[fig:surf\_MZM\_arc\]]{}c.
More Details on Impurity Effect {#app:H_d_bases}
===============================
In this section, we will provide more details on the impurity effect of SMFBs.
### Order Parameters in $\mathbf{r}_{{\shortparallel}}$ space
In this part, we will discuss the transformation of order parameters from the ${\boldsymbol{k}}_{{\shortparallel}}$ space to the ${\boldsymbol{r}}_{{\shortparallel}}$ space. Let us consider the general order parameters that are independent of ${\boldsymbol{k}}_{{\shortparallel}}$ in each $A_{l_\chi, l_c}$, i.e. [Eq.]{} with $m({\boldsymbol{k}}_{{\shortparallel}})$ having the form [Eq.]{}. Using [Eq.]{} and [Eq.]{}, we have $$H_{mf}=\frac{1}{2}\int dr^2_{{\shortparallel}} d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}} M d_{{\boldsymbol{r}}_{{\shortparallel}}}\ ,$$ with $M_{l_\chi l_\chi',l_c l_c',i i'}=\sum_{l=0}^3 f^{l_\chi,l_c}_{l}(\sigma_l)_{ii'}\delta_{l_\chi l_\chi'} \delta_{l_c l_c'}$. $f^{l_\chi,l_c}_{l}$’s for different $l_\chi,l_c$ are given by $1$ or $\delta^{\alpha}_{{\boldsymbol{k}}_{{\shortparallel}}}$ with $\alpha=\chi, (E_1,\pm), (E_2,\pm)$. Specifically, we have $$\begin{aligned}
&& 1 =\sum_{l_\chi, l_c} (\rho_0)_{l_\chi l_\chi} (\Lambda_{1})_{l_c l_c}\delta^{l_\chi,l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}\nonumber\\
&& \delta^{\chi}_{{\boldsymbol{k}}_{{\shortparallel}}} =\sum_{l_\chi, l_c} (\rho_3)_{l_\chi l_\chi} (\Lambda_{1})_{l_c l_c}\delta^{l_\chi,l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}\nonumber\\
&& \delta^{E_1,+}_{{\boldsymbol{k}}_{{\shortparallel}}} =\sum_{l_\chi, l_c} (\rho_0)_{l_\chi l_\chi} (\Lambda_{4,1})_{l_c l_c}\delta^{l_\chi,l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}\nonumber\\
&& \delta^{E_1,-}_{{\boldsymbol{k}}_{{\shortparallel}}} =\sum_{l_\chi, l_c} (\rho_3)_{l_\chi l_\chi} (\Lambda_{4,1})_{l_c l_c}\delta^{l_\chi,l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}\nonumber\\
&& \delta^{E_2,+}_{{\boldsymbol{k}}_{{\shortparallel}}} =\sum_{l_\chi, l_c} (\rho_0)_{l_\chi l_\chi} (\Lambda_{4,2})_{l_c l_c}\delta^{l_\chi,l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}\nonumber\\
&& \delta^{E_2,-}_{{\boldsymbol{k}}_{{\shortparallel}}} =\sum_{l_\chi, l_c} (\rho_3)_{l_\chi l_\chi} (\Lambda_{4,2})_{l_c l_c}\delta^{l_\chi,l_c}_{{\boldsymbol{k}}_{{\shortparallel}}}\ ,\end{aligned}$$ where that all matrices involved are diagonal due to translation symmetry. Using the above correspondence, [Tab.\[tab:N\]]{} and [Eq.]{}-\[eq:H\_mf\_E\], we can get $$H_{mf}^{\alpha}=\frac{1}{2}\int d^2 {\boldsymbol{r}}_{{\shortparallel}} d^{\dagger}_{{\boldsymbol{r}}_{{\shortparallel}}} M_\alpha d_{{\boldsymbol{r}}_{{\shortparallel}}}+const.\ ,$$ where $\alpha=A_1,A_2,E$, $$\label{eq:M_A1}
M_{A_1}=m_1\rho_3\otimes \Lambda_1 \otimes \sigma_0+m_2(-\rho_0\otimes \Lambda_{4,1} \otimes \sigma_2+\rho_0\otimes \Lambda_{4,2} \otimes \sigma_1)\ ,$$ $$\label{eq:M_A2}
M_{A_2}=m_3 \rho_0\otimes \Lambda_1 \otimes \sigma_3+m_4 (\rho_0\otimes\Lambda_{4,2}\otimes\sigma_2+\rho_0\otimes\Lambda_{4,1}\otimes\sigma_1)\ ,$$ and $$\begin{aligned}
\label{eq:M_E}
&&M_E=m_{5,1} \rho_3\otimes\Lambda_{4,1}\otimes\sigma_0+m_{5,2}\rho_3\otimes\Lambda_{4,2}\otimes\sigma_0\nonumber\\
&&+m_{6,1}(-\rho_0\otimes\Lambda_{4,2}\otimes\sigma_3)+m_{6,2} (\rho_0\otimes\Lambda_{4,1}\otimes \sigma_3)\nonumber\\
&&+m_{7,1}(-\rho_0\otimes\Lambda_1\otimes\sigma_2)+m_{7,2}(\rho_0\otimes\Lambda_1\otimes\sigma_1)\nonumber\\
&&+m_{8,1}(\rho_0\otimes\Lambda_{4,1}\otimes\sigma_2+\rho_0\otimes\Lambda_{4,2}\otimes\sigma_1)\nonumber\\
&&+m_{8,2}(\rho_0\otimes\Lambda_{4,1}\otimes\sigma_1-\rho_0\otimes\Lambda_{4,2}\otimes\sigma_2)\ .\end{aligned}$$ According to [Tab.\[tab:IR\_C3v\_TR\_PH\_chi\]]{}, [Eq.]{}, [Eq.]{} and [Eq.]{} are the most general PH symmetric uniform order parameters for the $A_1$, $A_2$ and $E$ IRs.
### Verification of LDOS Peaks for Translational Invariant Order Parameters with $d$ Bases
The purpose for this section is to re-derive the distribution of LDOS peaks from the symmetry aspect of the order parameters in [Eq.]{}-\[eq:M\_E\] with the $d$ bases and establish the formalism that can be generalized to the case with charge/magnetic impurities. Since the position ${\boldsymbol{r}}_{{\shortparallel}}$ is now approximately a good quantum number, the number of LDOS peaks is directly determined by the number of different eigenvalues of $M_{\alpha}$. It means that the numbers of LDOS peaks far away from impurities should be typically 1,4,2 and 12 for no order parameters, the $A_1$ order parameter, the $A_2$ order parameter and the $E$ order parameter, respectively, as indicated in [Sec.\[sec:MF\_order\_MFB\]]{}. 12 LDOS peaks for the $E$ order parameter are justified by the fact that $M_{\alpha}$’s are all $12\times 12 $ matrices with 12 eigenvalues and the $E$ order parameter typically has no symmetries to ensure any degeneracy. To discuss $A_1$ and $A_2$ order parameters, we again transform all the symmetry operators to the eigenbases of $C_{3,d}$ as discussed in the main text. By choosing the same convention (\[eq:C3t\_d\],\[eq:Pit\_d\]) in the main text, the representations of the symmetry operations other than $\hat{C}_{3}$ and $\hat{\Pi}$ are $$\widetilde{U}_{T}
=
\left(
\begin{array}{ccc}
& \mathds{1}_4 & \\
& & \mathds{1}_4\\
\mathds{1}_4 & & \\
\end{array}
\right)\ ,$$ $$\widetilde{\mathcal{C}}_{d}
=
\left(
\begin{array}{ccc}
& & U_c\\
& U_c & \\
U_c & & \\
\end{array}
\right)$$ with $$U_c=
\left(
\begin{array}{cccc}
0 & 0 & 0 & i \\
0 & 0 & -i & 0 \\
0 & -i & 0 & 0 \\
i & 0 & 0 & 0 \\
\end{array}
\right)\ ,$$ and $$\widetilde{\chi}_{d}
=
\left(
\begin{array}{ccc}
U_\chi & & \\
& U_\chi & \\
& & U_\chi\\
\end{array}
\right)$$ with $$U_\chi=
\left(
\begin{array}{cccc}
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)\ ,$$ where $\widetilde{R}$ means the matrix form of $R$ in the $C_{3,d}$ eigenbases and $U_T$ is defined such that $M$ is diagonal for $l_c$ index if and only if $[M,U_T]=0$. The $A_1$ order parameter satisfies $[M_{A_1},C_{3,d}]=[M_{A_1},U_T]=0$. Due to the commutation relation with $C_{3,d}$, $\widetilde{M}_{A_1}$ should be block-diagonal and written as $\widetilde{M}_{A_1}=\text{diag}(h_1,h_2,h_3)$, where $h_{1,2,3}$ are Hermitian $4\times 4$ matrices. Furthermore, due to the commutation relation with $U_T$, we requires $h_1=h_2=h_3$, which leads to the three-fold degeneracy of each eigenvalues. As a result, $M_{A_1}$ has typically 4 LDOS peaks. The $A_2$ order parameter satisfies not only $[M_{A_2},C_{3,d}]=[M_{A_2},U_T]=0$ but also $[M_{A_2}, \Pi_d \mathcal{C}_d K]=0$, in which we have $(\Pi_d \mathcal{C}_d K)^2=-1$. The former leads to $\widetilde{M}_{A_2}=\text{diag}(h_1,h_1,h_1)$ as mentioned above, while $\Pi_d \mathcal{C}_d M^* \mathcal{C}_d^{\dagger} \Pi_d^{\dagger}=M$ results in $U_{\Pi} U_c h_1^* U_c^{\dagger} U_\Pi^{\dagger}=h_1$. Thereby, each eigenvalues of $h_1$ have double degeneracy due to $U_{\Pi} U_c (U_{\Pi} U_c)^*=-1$. As a result, all eigenvalues of $M_{A_2}$ have six-fold degeneracy and the $A_2$ order parameter typically has 2 peaks. In addition, $M_{\alpha}$’s are PH symmetric, which guarantees that LDOS peaks are symmetric with respect to zero energy.
### Derivation of [Eq.]{} and the Symmetry Properties
In this part, we will derive [Eq.]{} and discuss the corresponding symmetry properties. The surface impurity Hamiltonian that we consider has the general form $$\label{eq:H_imp_c}
H_V=\int d^3 r c^{\dagger}_{{\boldsymbol{r}}} V({\boldsymbol{r}}) c_{{\boldsymbol{r}}}\ ,$$ where the position of the impurity is at ${\boldsymbol{r}}=0$ (certainly on the $x_{\perp}=0$ surface) and $V({\boldsymbol{r}})^{\dagger}=V({\boldsymbol{r}})$ decays fast away from ${\boldsymbol{r}}=0$. First we express [Eq.]{} in the Nambu bases as $$\begin{aligned}
&&H_V=\frac{1}{2}\int d^2 r_{{\shortparallel}} \int d x_{\perp} \frac{1}{S_{{\shortparallel}}}\sum_{{\boldsymbol{k}}_{{\shortparallel}},{\boldsymbol{k}}_{{\shortparallel}}'}e^{-i {\boldsymbol{k}}_{{\shortparallel}}\cdot {\boldsymbol{r}}_{{\shortparallel}}+i {\boldsymbol{k}}_{{\shortparallel}}'\cdot {\boldsymbol{r}}_{{\shortparallel}}}\nonumber\\
&&\Psi^{\dagger}_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}
\widetilde{V}({\boldsymbol{r}})
\Psi_{{\boldsymbol{k}}_{{\shortparallel}}',x_{\perp}}
+ const.\ ,\end{aligned}$$ where $$\widetilde{V}({\boldsymbol{r}})
=\left(
\begin{array}{cc}
V({\boldsymbol{r}}) & \\
& -V^*({\boldsymbol{r}})\\
\end{array}
\right)\ ,$$ and $
\Psi_{{\boldsymbol{r}}}^{\dagger}=\frac{1}{\sqrt{S_{{\shortparallel}}}}\sum_{{\boldsymbol{k}}_{{\shortparallel}}}e^{-i {\boldsymbol{k}}_{{\shortparallel}}\cdot {\boldsymbol{r}}_{{\shortparallel}}}\Psi_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}^{\dagger}
$ is used. Using [Eq.]{}, we only keep terms that involve surface modes and assume $v_{{\boldsymbol{k}}_{{\shortparallel}},x_{\perp}}\approx v_{{\boldsymbol{K}}_{{\shortparallel}}^{l_\chi, l_c},x_{\perp}}$ for all ${\boldsymbol{k}}_{{\shortparallel}}\in A_{l_\chi, l_c}$ and all ${l_\chi, l_c}$. This leads to [Eq.]{} with $$[M_V({\boldsymbol{r}}_{{\shortparallel}})]_{l_\chi l_\chi',l_c l_c', i i'}=\int_{-\infty}^0 d x_{\perp} v^{\dagger}_{i,{\boldsymbol{K}}^{l_\chi,l_c}_{{\shortparallel}},x_{\perp}} \widetilde{V}({\boldsymbol{r}}) v_{i',{\boldsymbol{K}}^{l_\chi',l_c'}_{{\shortparallel}},x_{\perp}}\ .$$ Since $V^{\dagger}({\boldsymbol{r}})=V({\boldsymbol{r}})$, we have $M_V^{\dagger}({\boldsymbol{r}}_{{\shortparallel}})=M_V({\boldsymbol{r}}_{{\shortparallel}})$. Due to $$\sum_{l_\chi',l_c',i'}[\mathcal{C}_d]_{l_{\chi} l_\chi',l_c l_c',i i'} v_{i',{\boldsymbol{K}}_{{\shortparallel}}^{l_\chi', l_c'},x_{\perp}}=\mathcal{C} v^*_{i,{\boldsymbol{K}}_{{\shortparallel}}^{l_\chi, l_c},x_{\perp}}\ ,$$ $M_V({\boldsymbol{r}}_{{\shortparallel}})$ is PH symmetric, written as $$-\mathcal{C}_d M_V^T({\boldsymbol{r}}_{{\shortparallel}}) \mathcal{C}_d^{\dagger}=M_V({\boldsymbol{r}}_{{\shortparallel}})\ .$$ Due to $$\sum_{l_\chi',l_c',i'}[\mathcal{T}_d]_{l_{\chi} l_\chi',l_c l_c',i i'} v_{i',{\boldsymbol{K}}_{{\shortparallel}}^{l_\chi', l_c'},x_{\perp}}=\mathcal{T}^T v^*_{i,{\boldsymbol{K}}_{{\shortparallel}}^{l_\chi, l_c},x_{\perp}}\ ,$$ $M_V({\boldsymbol{r}}_{{\shortparallel}})$ has the same TR properties as $\widetilde{V}({\boldsymbol{r}})$: $$\begin{aligned}
&&[\mathcal{T}_d M_V^*({\boldsymbol{r}}_{{\shortparallel}}) \mathcal{T}_d^{\dagger}]_{l_{\chi} l_\chi',l_c l_c',i i'}=\nonumber\\
&&\int_{-\infty}^0 d x_{\perp} v^{\dagger}_{i,{\boldsymbol{K}}^{l_\chi,l_c}_{{\shortparallel}},x_{\perp}} \mathcal{T}\widetilde{V}^*({\boldsymbol{r}}) \mathcal{T}^{\dagger}v_{i',{\boldsymbol{K}}^{l_\chi',l_c'}_{{\shortparallel}},x_{\perp}}\ .\end{aligned}$$ Similarly, due to $$\sum_{l_\chi',l_c',i'}[R_d]_{l_{\chi} l_\chi',l_c l_c',i i'} v^{\dagger}_{i',{\boldsymbol{K}}_{{\shortparallel}}^{l_\chi', l_c'},x_{\perp}}= v^{\dagger}_{i,{\boldsymbol{K}}_{{\shortparallel}}^{l_\chi, l_c},x_{\perp}}\widetilde{R}\ ,$$ $M_V({\boldsymbol{r}}_{{\shortparallel}})$ has the same $C_{3v}$ properties as $\widetilde{V}({\boldsymbol{r}})$: $$\begin{aligned}
&&[\mathcal{R}_d M_V({\boldsymbol{r}}_{{\shortparallel}}) \mathcal{R}_d^{\dagger}]_{l_{\chi} l_\chi',l_c l_c',i i'}=\nonumber\\
&&\int_{-\infty}^0 d x_{\perp} v^{\dagger}_{i,{\boldsymbol{K}}^{l_\chi,l_c}_{{\shortparallel}},x_{\perp}} \widetilde{R}\widetilde{V}({\boldsymbol{r}})\widetilde{R}^{\dagger}v_{i',{\boldsymbol{K}}^{l_\chi',l_c'}_{{\shortparallel}},x_{\perp}}\ ,\end{aligned}$$ where $R\in C_{3v}$. Furthermore, since $\widetilde{V}({\boldsymbol{r}})$ behaves the same as $V({\boldsymbol{r}})$, the TR and $C_{3v}$ properties of $M_{V}({\boldsymbol{r}}_{{\shortparallel}})$ are the same as those of $V({\boldsymbol{r}})$.
For a charge impurity, $V({\boldsymbol{r}})=V_c({\boldsymbol{r}})\mathds{1}_{4\times 4}$ with $V_c({\boldsymbol{r}})$ a real scalar function. In this case, $V_c({\boldsymbol{r}})\mathds{1}_{4\times 4}$ has TR symmetry $\gamma (V_c({\boldsymbol{r}})\mathds{1}_{4\times 4})^* \gamma^{\dagger}=V_c({\boldsymbol{r}})\mathds{1}_{4\times 4}$ and satisfies $R (V_c({\boldsymbol{r}})\mathds{1}_{4\times 4}) R^{\dagger}=V_c({\boldsymbol{r}})\mathds{1}_{4\times 4}$ with $R\in C_{3v}$. As a result, Hermitian and PH symmetric $M_V({\boldsymbol{r}})$ has TR symmetry $\mathcal{T}_d M^*_{V}({\boldsymbol{r}}_{{\shortparallel}})\mathcal{T}_d^{\dagger}=M_V({\boldsymbol{r}}_{{\shortparallel}})$ and satisfies $R_d M_V({\boldsymbol{r}}_{{\shortparallel}}) R_d^{\dagger}=M_V({\boldsymbol{r}}_{{\shortparallel}})$ with $R\in C_{3v}$. Combining TR and PH symmetries, we have chiral symmetry for $M_V({\boldsymbol{r}}_{{\shortparallel}})$, i.e. $\chi_d M_V({\boldsymbol{r}}_{{\shortparallel}}) \chi_d^{\dagger}=-M_V({\boldsymbol{r}}_{{\shortparallel}})$. By defining $M_c=M_V({\boldsymbol{r}}_{\shortparallel}=0)$, the symmetry properties of $M_c$ can be directly obtained.
For a magnetic impurity, we choose the magnetic moment of the impurity to be perpendicular to the surface and couple to the electron spin locally, i.e. choosing $V({\boldsymbol{r}})=V_m({\boldsymbol{r}}) {\boldsymbol{e}}_{\perp}\cdot {\boldsymbol{J}}$ with $V_m({\boldsymbol{r}})$ a real scalar function and ${\boldsymbol{e}}_{\perp}= (1,1,1)/\sqrt{3}$. In this case, $V_m({\boldsymbol{r}}) {\boldsymbol{e}}_{\perp}\cdot {\boldsymbol{J}}$ is TR odd $\gamma (V_m({\boldsymbol{r}}) {\boldsymbol{e}}_{\perp}\cdot {\boldsymbol{J}})^* \gamma^{\dagger}=-V_m({\boldsymbol{r}}) {\boldsymbol{e}}_{\perp}\cdot {\boldsymbol{J}}$, and satisfies $C_3 (V_m({\boldsymbol{r}}) {\boldsymbol{e}}_{\perp}\cdot {\boldsymbol{J}}) C_3^{\dagger}=V_m({\boldsymbol{r}}) {\boldsymbol{e}}_{\perp}\cdot {\boldsymbol{J}}$ and $\Pi (V_m({\boldsymbol{r}}) {\boldsymbol{e}}_{\perp}\cdot {\boldsymbol{J}}) \Pi^{\dagger}=-V_m({\boldsymbol{r}}) {\boldsymbol{e}}_{\perp}\cdot {\boldsymbol{J}}$. As a result, the Hermitian and PH symmetric $M_V({\boldsymbol{r}}_{{\shortparallel}})$ has TR antisymmetry $\mathcal{T}_d M_V^*({\boldsymbol{r}}_{{\shortparallel}})\mathcal{T}_d^{\dagger}=-M_V({\boldsymbol{r}}_{{\shortparallel}})$, and satisfies $C_{3,d} M_V({\boldsymbol{r}}_{{\shortparallel}})C_{3,d}^{\dagger}=M_V({\boldsymbol{r}}_{{\shortparallel}})$ and $\Pi_{d} M_V({\boldsymbol{r}}_{{\shortparallel}})\Pi_{d}^{\dagger}=-M_V({\boldsymbol{r}}_{{\shortparallel}})$. By defining $M_m=M_V({\boldsymbol{r}}_{\shortparallel}=0)$, the symmetry properties of $M_m$ can be obtained.
In [Fig.\[fig:LDOS\]]{}, $V_c({\boldsymbol{r}})/|\mu|=2/(|{\boldsymbol{r}}|\sqrt{2m\mu}+0.02)^2$ if the charge impurity is considered, and $V_m({\boldsymbol{r}})/|\mu|=5 e^{x_{\perp} \sqrt{2m\mu}/2}\theta(|{\boldsymbol{r}}_{{\shortparallel},0}|-|{\boldsymbol{r}}_{{\shortparallel}}|)$ with $|{\boldsymbol{r}}_{{\shortparallel}}|<|{\boldsymbol{r}}_{{\shortparallel},0}|$ if the magnetic impurity is considered.
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|
---
abstract: 'If one restricts an irreducible representation $V_{\lambda}$ of $Gl_{2n}$ to the orthogonal group (respectively the symplectic group), the trivial representation appears with multiplicity one if and only if all parts of $\lambda$ are even (resp. the conjugate partition $\lambda''$ is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered $q,t$-generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the $q=0$ limit (Hall-Littlewood), and provided direct combinatorial arguments for these identities; this approach led to various generalizations and a finite-dimensional analog of a recent summation identity of Warnaar. In this paper, we reformulate some of these results using $p$-adic representation theory; this parallels the representation-theoretic interpretation in the Schur case. The nonzero values of the identities are interpreted as certain $p$-adic measure counts. This approach provides a $p$-adic interpretation of these identities (and a new identity), as well as independent proofs. As an application, we obtain a new Littlewood summation identity that generalizes a classical result due to Littlewood and Macdonald. Finally, our $p$-adic method also leads to a generalized integral identity in terms of Littlewood-Richardson coefficients and Hall polynomials.'
address: 'Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139'
author:
- Vidya Venkateswaran
title: 'A $p$-adic interpretation of some integral identities for Hall-Littlewood polynomials'
---
Introduction
============
A crucial problem in representation theory can be described in the following way: let $G$ and $H$ be complex algebraic groups, with an embedding $H \hookrightarrow G$. Also let $V$ be a completely reducible representation of $G$, and $W$ an irreducible representation of $H$. What information can one obtain about $[V,W] := \text{dim } \text{Hom}_{H}(W,V)$, the multiplicity of $W$ in $V$? Here $V$ is viewed as a representation of $H$ by restriction. Such branching rules have important connections to physics as well as other areas of mathematics. There are often beautiful combinatorial objects describing these multiplicities. One prototypical example is that of the symmetric groups $G = S_{n}$ and $H= S_{n-1}$: the resulting rule has a particularly nice description in terms of Young tableaux.
Two particularly interesting examples involving matrix groups are the restriction of $Gl(2n)$ to $Sp(2n)$ (the symplectic group) and $Gl(n)$ to $O(n)$ (the orthogonal group); the combinatorics of these branching rules was first developed by D. Littlewood and continues to be a well-studied and active area at the forefront of algebraic combinatorics and invariant theory. These pairs are also important because they give examples of symmetric spaces. That is, $G$ is a reductive algebraic group and $H$ is the fixed point set of an involution on $G$; $S = G/H$ is the resulting symmetric space. The multiplicities in these branching rules are given in terms of Littlewood-Richardson coefficients, another important entity described in terms of tableaux and and lattice permutations. In fact, since Schur functions are characters of irreducible polynomial representations of $Gl(2n)$, one may rephrase these rules in terms of Schur functions and symplectic characters (respectively, orthogonal characters). This gives the following integral identities
\[symplecticgp\] [@H; @Mac] (1) For any even integer $n \geq 0$, we have $$\begin{aligned}
\int_{S \in Sp(n)} s_{\lambda}(S) dS
=\begin{cases} 1, &\text{if all parts of $\lambda$ have even multiplicity} \\
0, & \text{otherwise}
\end{cases}\end{aligned}$$ (where the integral is with respect to Haar measure on the symplectic group).
\(2) For any integer $n \geq 0$ and partition $\lambda$ with at most $n$ parts, we have $$\begin{aligned}
\int_{O \in O(n)} s_{\lambda}(O) dO
= \begin{cases} 1, &\text{if all parts of $\lambda$ are even } \\
0, & \text{otherwise}
\end{cases}\end{aligned}$$ (where the integral is with respect to Haar measure on the orthogonal group).
Proofs of these identities may be found in [@Mac]; they involve structure results for the two Gelfand pairs $(GL_{n}(\mathbb{H}), U_{n}(\mathbb{H}))$ and $(GL_{n}(\mathbb{R}), O_{n}(\mathbb{R}))$. Using the Weyl integration formula, we may rephrase the above identities in terms of the eigenvalue densities for the orthogonal and symplectic groups. For example, the left hand side of the symplectic integral above can be rewritten as $$\label{schursymplectic}
\frac{1}{2^{n}n!}\int_{T} s_{\lambda}(z_{1}, z_{1}^{-1}, z_{2}, z_{2}^{-1}, \dots, z_{n}, z_{n}^{-1})
\prod_{1 \leq i \leq n} |z_{i} - z_{i}^{-1}|^{2} \prod_{1 \leq i<j \leq n} |z_{i} + z_{i}^{-1} - z_{j} - z_{j}^{-1}|^{2} dT,$$ where $$\begin{aligned}
T &= \{ (z_{1}, \dots, z_{n}) : |z_{1}| = \dots = |z_{n}| = 1 \} \\
dT &= \prod_{j} \frac{dz_{j}}{2 \pi \sqrt{-1} z_{j}}\end{aligned}$$ are the $n$-torus and Haar measure, respectively.
Macdonald polynomials, $P_{\lambda}(x;q,t)$, are an important family of symmetric polynomials generalizing the Schur polynomials ($P_{\lambda}(x;t,t) = s_{\lambda}(x)$) (see [@Mac]). In [@RV], Rains and Vazirani provided $(q,t)$-generalizations of the restriction identities for Schur functions. That is, they exhibited densities such that when one integrates a Macdonald polynomial against it (over the $n$-torus), the integral vanishes unless the indexing partition satisfies an explicit condition. Moreover, when $q=t$, one obtains a known Schur identity. In fact, they were also able to find Macdonald identities with interesting vanishing conditions, but whose significance is unknown at the Schur level. To prove these results, Rains and Vazirani used techniques involving affine Hecke algebras; however, this method does not work directly at $q=0$ (another important special case of Macdonald polynomials: the Hall-Littlewood polynomials, $P_{\lambda}(x;t)$), although one can obtain the results as a limit.
In previous work [@VV], we provided a combinatorial approach for proving the results of Rains and Vazirani at $q=0$; this method allowed for several generalizations, one of which provided a connection with a summation identity of Warnaar [@W]. We were also able to use this approach to settle some conjectures of Rains at the $q=0$ level. In some cases, the affine Hecke algebra technique of Rains and Vazirani allowed them to determine when a given integral vanishes, but did not yield the nonzero values in the case that the integral is non-vanishing. Using our method, we were able to compute these values explicitly.
This paper provides an interpretation of the results of [@VV] in terms of $p$-adic representation theory. The motivation for this connection stems from the appearance of Hall-Littlewood polynomials in the representation theory of $p$-adic groups [@MacP], [@Mac Ch. V]. In particular, let $G = Gl_{n}(\mathbb{Q}_{p})$, and let $K = Gl_{n}(\mathbb{Z}_{p})$ be its maximal compact subgroup. Then $G/K$ is the affine Grassmannian and the spherical Hecke algebra $\mathcal{H}(G,K)$ is the convolution algebra of compactly supported, $K$-bi-invariant, complex valued functions on $G$; it has a basis given by $\{c_{\lambda}\}_{l(\lambda)\leq n}$, where $c_{\lambda}$ is the characteristic function of the double coset $Kp^{\lambda}K$ and $p^{\lambda} = diag(p^{\lambda_{1}}, \dotsc, p^{\lambda_{n}})$. Macdonald provides a Plancherel theorem in this context, where the zonal spherical functions are given in terms of Hall-Littlewood polynomials with $t=p^{-1}$. One consequence of this is another interpretation of the statement of Hall-Littlewood orthogonality:
[@MacP], [@Mac Ch. V] For partitions $\lambda, \mu$ of length at most $n$, we have $$\int_{T} P_{\lambda}(z_{1}, \dots, z_{n};p^{-1})P_{\mu}(z_{1}^{-1}, \dots, z_{n}^{-1}; p^{-1}) \tilde \Delta_{S}^{(n)}(z;p^{-1}) dT
= \frac{n!}{v_{n}(p^{-1})} p^{-\langle \lambda, \rho\rangle - \langle \mu, \rho \rangle}\int_{G} c_{\lambda}(g) c_{\mu}(g) dg,$$ where $\rho = \frac{1}{2}(n-1, n-3, \dots, 1-n)$ and $v_{n}(p^{-1}) = \big(\prod_{i=1}^{n} (1-p^{-i})\big)/(1-p^{-1})^{n}$.
Here $\tilde \Delta_{S}$ is the symmetric $q=0$ Macdonald density [@RV]. Since the double cosets $Kp^{\lambda}K$ and $Kp^{\mu}K$ do not intersect unless $\lambda = \mu$, the right hand side vanishes unless $\lambda = \mu$. In the case $\lambda = \mu$, one may also compute the $p$-adic measure of $Kp^{\lambda}K$ using [@MacP]; in particular, this provides an alternate approach for computing the left-hand side of the integral. Given the structural similarity between orthogonality and the vanishing results of [@VV], we were lead to search for $p$-adic interpretations of the latter results.
In this paper, we show that the vanishing results for Hall-Littlewood polynomials have a $p$-adic interpretation analogous to that of the Schur branching rules. We also consider some evaluation identities, and show that they, too, may be proved using $p$-adic representation theory. More precisely, let $F$ be a non-archimedean local field with residual field of odd characteristic. Let $E$ be an unramified quadratic extension of $F$. We set up the following cases:
Cases $G$ $H$
------------ -------------- ------------------------------
**Case 1** $Gl_{2n}(F)$ $Gl_{n}(E)$
**Case 2** $Gl_{2n}(E)$ $Gl_{2n}(F)$
**Case 3** $Gl_{2n}(F)$ $Sp_{2n}(F)$
**Case 4** $Gl_{2n}(F)$ $Gl_{n}(F) \times Gl_{n}(F)$
For simplicity, we will assume $F = \mathbb{Q}_{p}$ and $E = \mathbb{Q}_{p}(\sqrt{a})$, for $p$ an odd prime and $a$ prime to $p$ and without a square root. However, our arguments used in this paper apply to *any* $F,E$ satisfying the above conditions. Let $K$ denote the maximal compact subgroup of $G$ and $K'$ the maximal compact subgroup of $H$. In all four cases, there is an embedding of $H$ inside $G$, and an involution on $G$ that has $H$ as its set of fixed points; $S := G/H$ is the resulting $p$-adic symmetric space (see Background subsection 2 for more details). For these symmetric spaces, relative zonal spherical functions and a Plancherel theorem were computed by Offen in [@Offen] and Hironaka-Sato in [@HS]. The method used is that of Casselman and Shalika [@C; @CS], who provide another derivation of Macdonald’s formula for zonal spherical functions (see [@MacP] for the general reductive group case) using the theory of admissible representations of $p$-adic reductive groups. We use these works to prove the following integral identities in the third section of the paper:
\[genthm\] Let $l(\lambda) \leq 2n$, $l(\mu) \leq n$, and $c_{\lambda} \in \mathcal{H}(G,K)$ be the characteristic function of the double coset $Kp^{\lambda}K$. Then we have the following integral evaluations
1. (Case 1) $$\begin{gathered}
\frac{1}{Z}\int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; \pm p^{-1/2},0,0) \tilde \Delta_{K}^{(n)}(x;p^{-1}; \pm p^{-1/2},0,0) dT \\
= p^{\langle\mu,\rho_{1}\rangle - \langle\lambda, \rho_{2}\rangle} \frac{V_{0}}{V_{\mu}}\int_{H} c_{\lambda}(g_{\mu}h) dh\end{gathered}$$
2. (Case 2) $$\begin{gathered}
\frac{1}{Z}\int_{T}P_{\lambda}^{(2n)}(x_{i}^{\pm 1};p^{-2}) K_{\mu}^{BC_{n}}(x; p^{-2}; 1, p^{-1},0,0) \tilde \Delta_{K}^{(n)}(x;p^{-2};1,p^{-1},0,0)dT \\
= p^{2\langle \mu,\rho_{1} \rangle - 2 \langle \lambda, \rho_{2} \rangle} \frac{V_{0}}{V_{\mu}}\int_{H} c_{\lambda}(g_{\mu}k_{0}h) dh\end{gathered}$$
3. (Case 3) $$\begin{gathered}
\frac{1}{Z}\int_{T} P_{\lambda}^{(2n)}(p^{\pm 1/2}x_{i};p^{-1})P_{\mu}^{(n)}(x^{-1};p^{-2}) \tilde \Delta_{S}^{(n)}(x;p^{-2})dT \\
= p^{\langle \mu, \rho_{3}\rangle - \langle \lambda, \rho_{2}\rangle} \frac{V_{0}}{V_{\mu}}\int_{H} c_{\lambda}(g_{-\mu}h) dh\end{gathered}$$
4. (Case 4) $$\begin{gathered}
\frac{1}{Z}\int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; p^{-1/2}, p^{-1/2},0,0) \tilde \Delta_{K}^{(n)}(x;p^{-1}; p^{-1/2}, p^{-1/2},0,0) dT \\
= p^{\langle\mu,\rho_{1}\rangle - \langle\lambda, \rho_{2}\rangle} \frac{V_{0}}{V_{\mu}}\int_{H} c_{\lambda}(g_{\mu}k_{0}h) dh\end{gathered}$$
where $\rho_{1} = (n - 1/2, n-3/2, \dots, 1/2) \in \mathbb{C}^{n}$, $\rho_{2} = (n-1/2, n-3/2, \dots, 1/2 - n) \in \mathbb{C}^{2n}$, $\rho_{3} = (n-1, n-3, \dots, 1-n) \in \mathbb{C}^{n}$ and the normalization $Z$ is the evaluation of the integral at $\lambda = \mu = 0$. Thus, when $\mu = 0$, up to a scaling factor it is equal to $$\int_{H} c_{\lambda}(h) dh.$$
Explicit formulas for $V_{\mu}$, $g_{\mu} \in G$, and $k_{0} \in K$ for each case may be found in Sections 2 and 3, respectively. Also, $K_{\mu}^{BC_{n}}(x;t;a,b,c,d)$ are the Koornwinder polynomials at $q=0$; they are generalizations of symplectic and orthogonal group characters and multivariate generalizations of Askey-Wilson polynomials, see [@K]. These Laurent polynomials are invariant under permutations of variables and inverting variables. We note the symmetric function theory interpretation of the above identities: in Case 1 for example, if one expands the specialized Hall-Littlewood polynomial $P_{\lambda}^{(2n)}(x_{i}^{\pm 1};t)$ in terms of the corresponding Koornwinder basis, the integral gives the coefficient on $K_{\mu}^{BC_{n}}(x;t; \pm \sqrt{t},0,0)$ (with $t = p^{-1}$).
In the first three cases, specializing $\mu = 0$ in the above theorem provides an interpretation of Corollary 14, Corollary 15 and Theorem 22 of [@VV] using $p$-adic representation theory. It also provides a $p$-adic proof of these identities: we will evaluate the right hand side by using the Cartan decompositions for $G$ and $H$, along with some measure computations. As a consequence, we will show that the $\mu = 0$ version of the integrals above in Case (1) and (3) vanish unless $\lambda = \nu^{2}$ for some $\nu$ (resp. $\lambda = \nu \bar{\nu}$), and if this is satisfied it is a certain $p$-adic measure count. In Case (2), we obtain an explicit integral evaluation at $\mu = 0$, again in terms of a rational function arising from a certain $p$-adic quantity. The last case, Case (4), provides a new integral evaluation at $\mu = 0$, which is a $t$-generalization of Theorem \[symplecticgp\] part (1) (in a different direction than Case (1), which is also a $t$-generalization of the same result). One can refer to Theorem \[new\] within the paper for details.
We remark that the above integrals at $\mu = 0$ have a direct application to Littlewood summation identities. Indeed, as demonstrated in [@VV], one can start with the integral identities and use a procedure of [@R] to show that in the $n \rightarrow \infty$ limit, one obtains a Littlewood summation identity. In particular, we do this for Case (4) in this paper and obtain a new Littlewood summation identity:
\[Littlewood\] The following formal identity holds: $$\sum_{\mu, \nu} P_{\mu \cup \nu}(x;t) t^{\langle \mu + \nu, \rho \rangle - \frac{1}{2}\langle \mu \cup \nu, \rho \rangle} \frac{b_{\mu \cup \nu}(t)}{b_{\mu}(t) b_{\nu}(t)} = \prod_{j<k} \frac{1-tx_{j}x_{k}}{1-x_{j}x_{k}} \prod_{j} \frac{1+\sqrt{t}x_{j}}{1-\sqrt{t}x_{j}},$$ where $\rho = (1,3,5, \dots)$.
One can refer to Section 2 for the definition of $b_{\lambda}(t), \mu \cup \nu$ etc. This is a $t$-generalization of the classical Littlewood identity for Schur functions (see [@L], [@Mac]): $$\sum_{\substack{\lambda \\ \lambda' \text{ even }}} s_{\lambda}(x) = \prod_{i<j} \frac{1}{1-x_{i}x_{j}}.$$
Our method of using integration over $p$-adic groups supports a generalization of the usual vanishing identities at the Hall-Littlewood level; this is provided in the second part of the paper, and we briefly detail it here. Note that Theorem \[symplecticgp\] part (1) provides the coefficient on the trivial character when one expands the restricted Schur function in the basis of symplectic characters. A natural question, then, is whether there are interesting vanishing conditions for the other coefficients in this expansion. This is addressed by the following classical result:
(Littlewood and Weyl) \[WL\] If $l(\lambda) \leq n$, we have the branching rule $$s_{\lambda}^{(2n)}(x^{\pm 1}) = \sum_{l(\mu) \leq n} sp_{\mu}(x_{1}, \dots, x_{n}) \Bigg( \sum_{\substack{\beta \in \Lambda_{2n}^{+} \\ \beta = \nu^{2}}} c^{\lambda}_{\mu, \beta} \Bigg),$$ where $c^{\lambda}_{\mu, \beta}$ are the Littlewood-Richardson coefficients and $sp_{\mu}$ is an irreducible symplectic character.
One may rephrase this in terms of an integral identity as follows: for $l(\lambda) \leq n$, the integral $$\int_{S \in Sp(2n)} s_{\lambda}(S) sp_{\mu}(S) dS$$ vanishes if and only if $c^{\lambda}_{\mu, \beta} = 0$ for all $\beta = \nu^{2} := (\nu_{1}, \nu_{1}, \nu_{2}, \nu_{2}, \dots) \in \Lambda_{2n}^{+}$. We prove the following $t$-analog of this result:
\[genvan\] Let $\lambda, \mu \in \Lambda_{n}^{+}$. Then the following three statements are equivalent:
1. The generalized integral $$\frac{1}{\int_{T} \tilde \Delta_{K}^{(n)}(x; \pm \sqrt{t},0,0;t) dT} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1};t) K_{\mu}^{BC_{n}}(x;t; \pm \sqrt{t},0,0) \tilde \Delta_{K}^{(n)}(x; \pm \sqrt{t},0,0;t)dT$$ vanishes as a rational function of $t$.
2. The Hall polynomials $$g^{\lambda}_{\mu, \beta}(t)$$ vanish as a function of $t$, for all $\beta \in \Lambda_{2n}^{+}$ with $\beta = \nu^{2}$ for some $\nu$.
3. The Littlewood-Richardson coefficients $$c^{\lambda}_{\mu, \beta}$$ are equal to $0$ for all $\beta \in \Lambda_{2n}^{+}$ with $\beta = \nu^{2}$ for some $\nu$.
The proof of this relies on $p$-adic arguments similar to those used to prove Theorem \[genthm\], as well as some technical arguments involving Hall polynomials. We also provide some examples following the proof of Theorem \[genvan\] that makes use of known information about the vanishing of the Littlewood-Richardson coefficients (or Hall polynomials).
In the first section of the paper, we’ll review some relevant background and notation. In the second section, we’ll give an interpretation of some identities in [@VV] using $p$-adic representation theory that may be viewed as an analog of the Schur identities. Finally, in the last section, we’ll use this approach to provide a generalization of the integral identities considered in this paper.
**Acknowledgements:** The author would like to thank E. Rains for initially suggesting this problem and for many helpful conversations along the way. She would also like to thank A. Borodin and M. Vazirani for useful discussions. She also thanks O. Warnaar for pointing out a typo in the density computation in Case 4 in an earlier version of this article.
Background
==========
Symmetric function theory
-------------------------
We first review the relevant notations that we will be using in this paper.
Let $$\Lambda_{n}^{+} = \{ (\lambda_{1}, \dots, \lambda_{n}) \in \mathbb{Z}^{n}| \lambda_{1} \geq \cdots \geq \lambda_{n} \geq 0 \}$$ be the set of *partitions*. We will refer to the $\lambda_{i}$ as the *parts* of $\lambda$. The *length* of a partition $\lambda$ is the number of nonzero $\lambda_{i}$. Also let $m_{i}(\lambda)$ be the number of $\lambda_{j}$ equal to $i$ for each $i \geq 0$; this is the *multiplicity* of the part $i$ in $\lambda$. We will write $\lambda = \mu^{2}$ if $\lambda = (\mu_{1}, \mu_{1}, \mu_{2}, \mu_{2}, \dots)$ and we will say $\lambda$ has all parts occuring with even multiplicity. Note that this is the same as the conjugate partition $\lambda'$ having all even parts. For example, if $\lambda = (3,3,2,2,1,1,1,1)$ then $\lambda = \mu^{2}$ with $\mu = (3,2,1,1)$.
Following [@Mac], we define $\lambda \cup \mu$ to be the partition whose parts are those of $\lambda$ and $\mu$, arranged in descending order. For example, if $\lambda = (3,2,1)$ and $\mu = (2,2)$ then $\lambda \cup \mu = (3,2,2,2,1)$. Also $\langle \lambda, \mu \rangle = \lambda \cdot \mu = \lambda_{1}\mu_{1} + \cdots + \lambda_{n}\mu_{n}$.
We define $$\rho_{1} = (n - 1/2, n-3/2, \dots, 1/2) \in \mathbb{C}^{n}$$ $$\rho_{2} = (n-1/2, n-3/2, \dots, 1/2 - n) \in \mathbb{C}^{2n}$$ $$\rho_{3} = (n-1, n-3, \dots, 1-n) \in \mathbb{C}^{n};$$ which will be used throughout the paper.
The symmetric $q=0$ Macdonald density [@RV] is $$\begin{aligned}
\label{Sd}
\tilde \Delta_{S}^{(n)}(x;t) &= \frac{1}{n!}\prod_{1 \leq i \neq j \leq n} \frac{1-x_{i}x_{j}^{-1}}{1-tx_{i}x_{j}^{-1}}\end{aligned}$$ and the symmetric $q=0$ Koornwinder density [@K] is $$\label{Kd}
\tilde \Delta_{K}^{(n)}(x;t;a,b,c,d) = \frac{1}{2^{n}n!} \prod_{1 \leq i \leq n} \frac{1-x_{i}^{\pm 2}}{(1-ax_{i}^{\pm 1})(1-bx_{i}^{\pm 1})(1-cx_{i}^{\pm 1})(1-dx_{i}^{\pm 1})} \prod_{1 \leq i<j \leq n} \frac{1-x_{i}^{\pm 1}x_{j}^{\pm 1}}{1-tx_{i}^{\pm 1}x_{j}^{\pm 1}},$$ where we write $1-x_{i}^{\pm 2}$ for the product $(1-x_{i}^{2})(1-x_{i}^{-2})$ and $1-x_{i}^{\pm 1}x_{j}^{\pm 1}$ for $(1-x_{i}x_{j})(1-x_{i}^{-1}x_{j}^{-1})(1-x_{i}^{-1}x_{j})(1-x_{i}x_{j}^{-1})$ etc.
Hall-Littlewood polynomials $P_{\lambda}^{(n)}(x; t)$ indexed by partitions $\lambda$ with length at most $n$ form an orthogonal basis with respect to (\[Sd\]). Similarly, Koornwinder polynomials $K_{\mu}^{(n)}(x;t;a,b,c,d)$ indexed by partitions $\mu$ with length at most $n$ form an orthogonal basis with respect to (\[Kd\]). An explicit formula for $K_{\mu}^{(n)}(x;t;a,b,c,d)$ was provided in [@VV2].
We also define $$v_{m}(t) = \prod_{i=1}^{m} \frac{1-t^{i}}{1-t} = \frac{\phi_{m}(t)}{(1-t)^{m}}.$$ We use this to define $$\begin{aligned}
v_{\lambda}(t) &= \prod_{i \geq 0} v_{m_{i}(\lambda)}(t);\end{aligned}$$ this is the factor that makes the Hall-Littlewood polynomials of type A monic. Also let $$b_{\lambda}(t) = \prod_{i \geq 1} \phi_{m_{i}(\lambda)}(t).$$
Finally, for ease of notation, we set-up some shorthand notation for some specific densities that will be used throughout the paper. Let $p$ be a prime, and define $$\label{den1}
\Delta_{1} = \tilde \Delta_{K}^{(n)}(x; p^{-1}; \pm p^{-1/2},0,0)$$ $$\label{den2}
\Delta_{2} = \tilde \Delta_{K}^{(n)}(x;p^{-2};1,p^{-1},0,0)$$ $$\label{den3}
\Delta_{3} = \tilde \Delta_{S}^{(n)}(x;p^{-2})$$ $$\label{den4}
\Delta_{4} = \tilde \Delta_{K}^{(n)}(x;p^{-1}; p^{-1/2}, p^{-1/2},0,0).$$
$P$-adic represention theory
----------------------------
Let $F$ be a non-archimedean local field with residual field of odd characteristic. Let $E$ be an unramified quadratic extension of $F$. We set up the following cases corresponding to the above theorems:
**Case 1:** $G = Gl_{2n}(F), H = Gl_{n}(E)$
**Case 2:** $G = Gl_{2n}(E), H = Gl_{2n}(F)$
**Case 3:** $G = Gl_{2n}(F), H = Sp_{2n}(F)$
**Case 4:** $G = Gl_{2n}(F), H = Gl_{n}(F) \times Gl_{n}(F)$.
For simplicity, from now on we will assume $F = \mathbb{Q}_{p}$ and $E = \mathbb{Q}_{p}(\sqrt{a})$, for $p$ an odd prime and $a$ prime to $p$ and without a square root. However, the argument applies to *any* $F,E$ satisfying the above conditions. The number of elements in the residual field of $F$ is $p$, and for $E$ it is $p^{2}$. Throughout, we will use $K$ to denote the maximal compact subgroup of $G$ (so $K=Gl_{2n}(\mathbb{Z}_{p})$ in Cases 1 and 3 and $K = Gl_{2n}(\mathbb{Z}_{p}(\sqrt{a}))$ in Case 2) and $K'$ the maximal compact subgroup of $H$.
Define $$\Lambda_{n}^{+} = \{ \lambda = (\lambda_{1}, \dots, \lambda_{n}) \in \mathbb{Z}^{n} | \lambda_{1} \geq \cdots \geq \lambda_{n} \geq 0 \}$$ and $$\Lambda_{n} = \{ \lambda = (\lambda_{1}, \dots, \lambda_{n}) \in \mathbb{Z}^{n} | \lambda_{1} \geq \cdots \geq \lambda_{n} \},$$ so that $\Lambda_{n}^{+}$ is the set of partitions with length at most $n$, while $\lambda \in \Lambda_{n}$ is allowed to have negative parts.
We set-up some notation following [@Offen], [@HS]. Let $g \mapsto g^*$ denote the involution on $G$ given by:
**Case 1:** $g^* = g^{-1}$.
**Case 2:** $g^* = \bar g^{-1}$.
**Case 3:** $g^* = g^t$.
**Case 4:** $g^* = \epsilon g^{-1} \epsilon$, where $$\epsilon = \left( \begin{array}{cc}
I_{n} & 0 \\
0 & -I_{n} \\ \end{array} \right) \in G$$
Fix the element $s_{0} \in G$ to be $$s_0 = \begin{cases}
\left( \begin{array}{cc}
0 & w_{n} \\
aw_{n} & 0 \\ \end{array} \right), & \textrm{Case 1}\\
I_{2n}, & \textrm{Case 2}\\
J_n, & \textrm{Case 3} \\
I_{2n}, & \textrm{Case 4}
\end{cases},$$ where $J_n = (\begin{smallmatrix} 0 & I_n\\-I_n & 0\end{smallmatrix})$ and $w_{n}$ is the $n \times n$ matrix with ones on the anti-diagonal and zeroes everywhere else. Define $S = G \cdot s_{0}$, where the action is $g \cdot s_{0} = gs_{0}g^{*}$. In all cases, $H$ may be identified with the stabilizer of $s_{0}$ in $G$:
In **Case 1**: $$Stab(s_{0}) = \Bigg\{ \left( \begin{array}{cc}
i & j \\
aw_{n}jw_{n} & w_{n}iw_{n} \\ \end{array} \right) \in G | i,j \in Gl_{n}(\mathbb{Q}_{p}) \Bigg \} \cong Gl_{n}(\mathbb{Q}_{p}(\sqrt{a})).$$ In **Case 2**: $$Stab(s_{0}) = GL_{2n}(F).$$ In **Case 3**: $$Stab(s_{0}) = Sp_{2n}(F)$$ In **Case 4**: $$Stab(s_{0}) = \Bigg\{ \left( \begin{array}{cc}
g_{1} & 0 \\
0 & g_{2} \\ \end{array} \right) \in G | g_{1},g_{2} \in Gl_{n}(\mathbb{Q}_{p}) \Bigg \} \cong Gl_{n}(\mathbb{Q}_{p}) \times Gl_{n}(\mathbb{Q}_{p}).$$
In Case 1, we identify the maximal compact subgroup $K' = Gl_{n}(\mathbb{Z}_{p}(\sqrt{a})) \subset Gl_{n}(\mathbb{Q}_{p}(\sqrt{a}))$ with $K \cap H \subset Gl_{2n}(\mathbb{Q}_{p})$. In the other two cases, $K' = Gl_{2n}(\mathbb{Z}_{p})$ and $Sp_{2n}(\mathbb{Z}_{p})$, respectively. (In all cases, we identify $K'$ (the maximal compact subgroup of $H$) with its image in $G$ under the corresponding embedding.) The map $\theta: G \rightarrow S$ defined by $\theta(g) = gs_{0}g^{*} = g \cdot s_{0}$ induces a bijection between $G/H$ and $S$.
Let $\mathcal{H}(G,K)$ be the Hecke algebra of $G$ with respect to $K$; it is the convolution algebra of compactly supported, $K$-bi-invariant, complex valued functions on $G$. Let $C^{\infty}(K\setminus S)$ be the space of $K$-invariant complex valued functions on $S$. Also define $\mathcal{S}(K \setminus S)$ to be the $\mathcal{H}(G,K)$-submodule of $K$-invariant functions on $S$ with compact support. Define a $\mathcal{H}(G,K)$-module structure on $C^{\infty}(K \setminus S)$ via the convolution operation $$\begin{aligned}
f \star \phi(s) = \int_{G} f(g) \phi(g^{-1} \cdot s) dg\end{aligned}$$ where $f \in \mathcal{H}(G,K)$ and $\phi \in C^{\infty}(K \setminus S)$ and $dg$ is the Haar measure on $G$ normalized so $\int_{K} dg = 1$.
[@Offen; @HS] A **relative spherical function** on $S$ is an eigenfunction $\Omega \in C^{\infty}(K \setminus S)$ of $\mathcal{H}(G,K)$ under this convolution, normalized so that $\Omega(s_{0}) = 1$.
Define the elements $d_\lambda$ in $G$ as follows:
**Case 1**: $$d_{\lambda} = \text{antidiag}.(p^{\lambda_{1}}, \dots, p^{\lambda_{n}}, ap^{-\lambda_{n}}, \dots, ap^{-\lambda_{1}}).$$ **Case 2**: $$d_{\lambda} = \text{antidiag}.(p^{\lambda_1},\dotsc,p^{\lambda_n},p^{-\lambda_n},\dotsc,p^{-\lambda_1}).$$ **Case 3**: $$d_\lambda = \text{antidiag}.(p^{\lambda_1},\dotsc,p^{\lambda_n},-p^{\lambda_n},\dotsc,-p^{\lambda_1}).$$ **Case 4**: $$d_{\lambda} = \text{antidiag}.(p^{\lambda_1},\dotsc,p^{\lambda_n},-p^{-\lambda_n},\dotsc,-p^{-\lambda_1}).$$
Note that $d_{0} = s_{0}$ in Cases 1 and 3.
[@Offen Proposition 3.1], [@HS] The $K$-orbits of $S$ are given by the disjoint union $$S = \cup K \cdot d_{\lambda},$$ varying over $\lambda \in \Lambda_{n}^{+}$.
Let $ch_{\lambda}$ denote the characteristic function for the $K$-orbit $K \cdot d_{\lambda}$, then the space $\mathcal{S}(K \setminus S)$ is spanned by the functions $\{ ch_{\lambda} | \lambda \in \Lambda_{n}^{+} \}$.
(Cartan decomposition for $G$, see [@Mac] for example) We have the disjoint union $$G = \cup Kp^{\lambda}K,$$ varying over $\lambda \in \Lambda_{2n}$, and $p^{\lambda}$ denotes the diagonal matrix $\text{diag}.(p^{\lambda_{1}}, p^{\lambda_{2}}, \dots, p^{\lambda_{2n}})$.
Let $c_{\lambda}$, with $\lambda \in \Lambda_{2n}$, be the characteristic function for the double coset $Kp^{\lambda}K$ inside $G$. These functions form a basis for $\mathcal{H}(G,K)$.
Let the constant $V_{\lambda}$ (for any $l(\lambda) \leq n$) be given via the following integral evaluations (see Equations \[den1\], \[den2\], \[den3\], \[den4\] for notation):
In **Case 1:** $$\frac{1}{V_{\lambda}} = \int_{T} K_{\lambda}^{BC_{n}}(x;p^{-1}; \pm p^{-1/2},0,0)^{2} \Delta_{1} dT$$
In **Case 2:** $$\frac{1}{V_{\lambda}} = \int_{T} K_{\lambda}^{BC_{n}}(x;p^{-2};1,p^{-1},0,0)^{2} \Delta_{2} dT$$
In **Case 3:** $$\frac{1}{V_{\lambda}} = \int_{T} P_{\lambda}^{(n)}(x;p^{-2}) P_{\lambda}^{(n)}(x^{-1};p^{-2}) \Delta_{3} dT$$
In **Case 4:** $$\frac{1}{V_{\lambda}} = \int_{T} K_{\lambda}^{BC_{n}}(x;p^{-1}; p^{-1/2}, p^{-1/2},0,0)^{2} \Delta_{4} dT$$
In particular, $V_{0}$ is the reciprocal of the integral of the density function in each of the cases. Note that for Cases 1, 2, and 4, the $V_{\lambda}$ are determined explicitly in [@VV2] for general parameters $t_{0}, \dots, t_{3}$ of the Koornwinder $q=0$ polynomials and in [@MacP] for these choices of parameters. In the third case, the norm is computed in [@VV], for example.
Finally, for $z = (z_{1}, \dots, z_{n}) \in \mathbb{C}^{n}$ and $f \in \mathcal{H}(G,K)$, define (for **Case 1**, **Case 2**, and **Case 4**) $$\tilde f(z) = \hat{f}(z_{1}, \dots, z_{n}, -z_{1}, \dots, -z_{n})$$ where $\hat{}$ denotes the Satake transform on $\mathcal{H}(G,K)$. For **Case 3**, define $$\tilde f(z) = \hat{f}(z_{1} + 1/2, z_{1} - 1/2, \dots, z_{n}+1/2, z_{n} -1/2)$$
In fact by [@Offen Lemma 4.2] and [@HS Lemma 2.1], $f \rightarrow \tilde f(z)$ is the eigenvalue map, that is $$(f \star \Omega_{z})(s) = \tilde f(z) \Omega_{z}(s),$$ where $\Omega_{z}(s)$, $z \in \mathbb{C}^{n}$ are the relative spherical functions, as determined in [@Offen] and [@HS].
Also for $f \in \mathcal{H}(G,K), g \in G$ put $\check f(g) := f(g^{-1})$.
Main Results
============
In this section, we will prove Theorem \[genthm\] and discuss the related corollaries. The technique is to use the results of [@Offen; @HS] as well as additional $p$-adic arguments.
\[prop1\] Let $l(\lambda) \leq 2n$ and $l(\mu) \leq n$. Then we have $$\begin{gathered}
\int_{S} ( c_{\lambda} \star ch_{0})(s) ch_{\mu}(s) ds \\
=\begin{cases} \frac{ p^{\langle\mu,\rho_{1}\rangle + \langle\lambda, \rho_{2}\rangle}}{Z} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; \pm p^{-1/2},0,0) \Delta_{1} dT, & \text{in \textbf{Case 1}} \\
\frac{p^{2\langle \mu,\rho_{1} \rangle + 2 \langle \lambda, \rho_{2} \rangle}}{Z} \int_{T}P_{\lambda}^{(2n)}(x_{i}^{\pm 1};p^{-2}) K_{\mu}^{BC_{n}}(x; p^{-2}; 1, p^{-1},0,0) \Delta_{2} dT ,& \text{in \textbf{Case 2}}\\
\frac{p^{\langle \mu, \rho_{3}\rangle + \langle \lambda, \rho_{2}\rangle}}{Z} \int_{T} P_{\lambda}^{(2n)}(p^{\pm 1/2}x_{i};p^{-1})P_{\mu}^{(n)}(x^{-1};p^{-2}) \Delta_{3} dT,& \text{in \textbf{Case 3}}\\
\frac{p^{\langle \mu, \rho_{1} \rangle + \langle \lambda, \rho_{2} \rangle}}{Z}\int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; p^{-1/2},p^{-1/2},0,0) \Delta_{4} dT, & \text{in \textbf{Case 4}},
\end{cases}\end{gathered}$$ where $\rho_{1} = (n - 1/2, n-3/2, \dots, 1/2) \in \mathbb{C}^{n}$, $\rho_{2} = (n-1/2, n-3/2, \dots, 1/2 - n) \in \mathbb{C}^{2n}$, $\rho_{3} = (n-1, n-3, \dots, 1-n) \in \mathbb{C}^{n}$ and the normalization $Z$ is the evaluation of the integral at $\lambda = \mu = 0$.
**Case 1.** We use the spherical Fourier transform on $\mathcal{S}(K \setminus S)$: $$\int_{S} f_{1}(s) \overline{f_{2}(s)} ds = \int_{T} \hat{f_{1}}(z) \overline{\hat{f_{2}}(z)} d_{\mu}(z);$$ here $d_{\mu}(z)$ is the Plancherel measure on $\mathcal{S}(K \setminus S)$. We apply this to $$\int_{S} (c_{\lambda} \star ch_{0})(s) ch_{\mu}(s) ds.$$
Note that the spherical Fourier transform satisfies (by Lemma 4.4 [@Offen]) $$(c_{\lambda} \star ch_{0}) \hat (z) = \tilde c_{\lambda}(z) \hat{ch}_{0}(z) = \tilde c_{\lambda}(z),$$ since $\hat{ch}_{0}(s) = 1$. Here $$\tilde c_{\lambda}(z) = \hat{c_{\lambda}}(z_{1}, \dots, z_{n}, -z_{1}, \dots, -z_{n})$$ where $\hat{c_{\lambda}}$ denotes here the usual Satake transform on $\mathcal{H}(Gl_{2n}(\mathbb{Q}_{p}), Gl_{2n}(\mathbb{Z}_{p}))$. But [@Mac Ch. V] , this is equal to $$p^{\langle\lambda, \rho_{2}\rangle} P_{\lambda}^{(2n)}(p^{-z_{1}}, \dots, p^{-z_{n}}, p^{z_{1}}, \dots, p^{z_{n}}; p^{-1}).$$
Also, using [@Offen] Theorem 1.2 and Proposition 5.15, we have $$\begin{gathered}
\hat{ch_{\mu}}(z) = \Big\{ \int_{K \cdot d_{\mu}} ds \Big\} \Omega_{z}(d_{\mu}) = \Big \{ p^{2\langle\mu, \rho_{1}\rangle} \frac{V_{0}}{V_{\mu}} \Big \} p^{-\langle\mu, \rho_{1}\rangle} \frac{V_{\mu}}{V_{0}} K_{\mu}^{BC_{n}}(p^{z_{i}}; p^{-1}; \pm p^{-1/2},0,0) \\
= p^{\langle\mu, \rho_{1}\rangle} K_{\mu}^{BC_{n}}(p^{z_{i}}; p^{-1}; \pm p^{-1/2},0,0).\end{gathered}$$
Finally, by [@Offen] Theorem 1.3 the Plancherel density is $$\frac{\tilde \Delta_{K}^{(n)}(p^{z_{i}};p^{-1}; \pm p^{-1/2},0,0 )}{\int_{T} \tilde \Delta_{K}^{(n)}(p^{z_{i}};p^{-1}; \pm p^{-1/2},0,0)dT}.$$ Combining these, using Equation \[den1\], and putting $x_{i} = p^{z_{i}}$ gives the result.
**Case 2.** The argument is the same as Case 1, but the Plancherel measure and zonal spherical functions are different. We indicate the differences (see the above references of [@Offen] but for Case 2, and [@Mac Ch. V] for the group $Gl_{2n}(E)$): $$\tilde c_{\lambda}(z) = \hat{c_{\lambda}}(z_{1}, \dots, z_{n}, -z_{1}, \dots, -z_{n}) = p^{2\langle \lambda, \rho_{2} \rangle} P_{\lambda}^{(2n)} (p^{-2z_{1}}, \dots, p^{-2z_{n}}, p^{2z_{1}}, \dots, p^{2z_{n}}; p^{-2}).$$ We also have $$\begin{gathered}
\hat{ch_{\mu}}(z) = \Big\{ \int_{K \cdot d_{\mu}} ds \Big\} \Omega_{z}(d_{\mu}) = \Big \{ p^{4 \langle \mu, \rho_{1} \rangle} \frac{V_{0}}{V_{\mu}} \Big \} p^{-2\langle \mu, \rho_{1} \rangle} \frac{V_{\mu}}{V_{0}} K_{\mu}^{BC_{n}}(p^{2z_{i}}; p^{-2}; 1,p^{-1},0,0) \\
= p^{2 \langle \mu, \rho_{1} \rangle}K_{\mu}^{BC_{n}}(p^{2z_{i}}; p^{-2}; 1,p^{-1},0,0).\end{gathered}$$ Finally, the Plancherel density is $$\frac{\tilde \Delta_{K}^{(n)}(p^{2z_{i}}; p^{-2}; 1, p^{-1},0,0)}{\int_{T}\tilde \Delta_{K}^{(n)}(p^{2z_{i}}; p^{-2}; 1, p^{-1},0,0) dT}.$$ Combining these, and putting $x_{i} = p^{2z_{i}}$ gives the result.
**Case 3.** The argument is the same as in the above cases, but the Plancherel measure and zonal spherical functions are different. We indicate the differences (see [@HS]): $$\begin{gathered}
\tilde c_{\lambda}(z) = \hat{c_{\lambda}}(z_{1} + 1/2, z_{1} - 1/2, \dots, z_{n} + 1/2, z_{n} - 1/2) \\
= p^{\langle \lambda, \rho_{2} \rangle} P_{\lambda}^{(2n)}(p^{-z_{1} - 1/2}, p^{-z_{1} + 1/2}, \dots, p^{-z_{n} - 1/2}, p^{-z_{n} + 1/2}; p^{-1}).\end{gathered}$$ We also have $$\begin{gathered}
\hat{ch_{\mu}}(z) = \Big\{ \int_{K \cdot d_{\mu}} ds \Big\} \Omega_{z}(d_{\mu}) = \Big \{ p^{2\langle \mu, \rho_{3} \rangle} \frac{V_{0}}{V_{\mu}} \Big\} p^{-<\mu, \rho_{3}>} \frac{V_{\mu}}{V_{0}} P_{\mu}(p^{z_{1}}, \dots, p^{z_{n}}; p^{-2}) \\
= p^{\langle \mu, \rho_{3} \rangle} P_{\mu}(p^{z_{1}}, \dots, p^{z_{n}}; p^{-2}). \end{gathered}$$ Finally, the Plancherel density is $$\frac{\tilde \Delta_{S}^{(n)}(p^{z_{i}};p^{-2})}{\int_{T}\tilde \Delta_{S}^{(n)}(p^{z_{i}};p^{-2}) dT }.$$ Combining these, and putting $x_{i} = p^{z_{i}}$ gives the result.
**Case 4.** The argument is the same as Case 1, but the Plancherel measure and zonal spherical functions are different. We indicate the differences (see the above references of [@Offen] but for Case 1, as well as [@Mac]): $$\tilde c_{\lambda}(z) = p^{\langle\lambda, \rho_{2}\rangle} P_{\lambda}^{(2n)}(p^{-z_{1}}, \dots, p^{-z_{n}}, p^{z_{1}}, \dots, p^{z_{n}}; p^{-1}).$$ We also have $$\begin{gathered}
\hat{ch_{\mu}}(z) = \Big\{ \int_{K \cdot d_{\mu}} ds \Big\} \Omega_{z}(d_{\mu}) = \Big \{ p^{2\langle\mu, \rho_{1}\rangle} \frac{V_{0}}{V_{\mu}} \Big \} p^{-\langle \mu, \rho_{1} \rangle} \frac{V_{\mu}}{V_{0}} K_{\mu}^{BC_{n}}(p^{z_{i}}; p^{-1}; p^{-1/2},p^{-1/2},0,0) \\
= p^{ \langle \mu, \rho_{1} \rangle}K_{\mu}^{BC_{n}}(p^{z_{i}}; p^{-1}; p^{-1/2},p^{-1/2},0,0).\end{gathered}$$ Finally, the Plancherel density is $$\frac{\tilde \Delta_{K}^{(n)}(p^{z_{i}}; p^{-1}; p^{-1/2}, p^{-1/2},0,0)}{\int_{T}\tilde \Delta_{K}^{(n)}(p^{z_{i}}; p^{-1}; p^{-1/2}, p^{-1/2},0,0) dT}.$$ Combining these, and putting $x_{i} = p^{z_{i}}$ gives the result.
Let $g_{\mu} = \text{diag}(1, \dots, 1, p^{-\mu_{n}}, \dots, p^{-\mu_{1}}) \in Gl_{2n}(\mathbb{Q}_{p})$ for $\mu = (\mu_{1}, \dots, \mu_{n})$.
\[prop2\] We have $$\int_{S} (\check c_{\lambda} \star ch_{0})(s) ch_{\mu}(s) ds = \begin{cases} p^{2\langle\mu, \rho_{1}\rangle} \frac{V_{0}}{V_{\mu}}\int_{H} c_{\lambda}(g_{\mu}h) dh, & \text{in \textbf{Case 1}} \\
p^{4\langle \mu, \rho_{1} \rangle} \frac{V_{0}}{V_{\mu}}\int_{H} c_{\lambda}(g_{\mu}k_{0}h) dh, & \text{in \textbf{Case 2}} \\
p^{2\langle \mu, \rho_{3}\rangle}\frac{V_{0}}{V_{\mu}}\int_{H} c_{\lambda}(g_{-\mu}h) dh, & \text{in \textbf{Case 3}} \\
p^{2\langle\mu, \rho_{1}\rangle} \frac{V_{0}}{V_{\mu}}\int_{H} c_{\lambda}(g_{\mu}k_{0}h) dh, & \text{in \textbf{Case 4}},
\end{cases}$$ where $k_{0} \in K$ is a specific element in Cases 2,4. In particular, when $\mu = 0$, the RHS is $\int_{H} c_{\lambda}(g) dg$.
**Case 1:** We first note that $d_{0} = s_{0}$ in this case. We have $$\int_{S} ( c_{\lambda} \star ch_{0})(s) ch_{\mu}(s) ds = \int_{K \cdot d_{\mu}} ({c_{\lambda}} \star ch_{0})(s) = meas.(K \cdot d_{\mu}) ({c_{\lambda}} \star ch_{0})(d_{\mu}),$$ where the first equality follows since $ch_{\mu}(s)$ vanishes off of $K \cdot d_{\mu}$, and the second follows since $({c_{\lambda}} \star ch_{0})$ is $K$-invariant. Now by definition of the convolution action, we have $$({c_{\lambda}} \star ch_{0})(d_{\mu}) = \int_{G} c_{\lambda}(g^{-1}) ch_{0}(g \cdot d_{\mu})dg.$$ Letting $H_{\mu} = \{g \in G | g \cdot d_{0} = d_{\mu}\}$, we have $$g \cdot d_{\mu} \in K \cdot d_{0} \Leftrightarrow (kg) \cdot d_{\mu} = d_{0} \text{ for some $k \in K$} \Leftrightarrow g \in K H_{\mu}^{-1}.$$ Now one can check that $g_{\mu} \cdot d_{0} = d_{\mu}$, so that $H_{\mu} = g_{\mu} H$ (clearly $g_{\mu}H \subset H_{\mu}$, for the other direction let $g \in H_{\mu}$ then $g \cdot d_{0} = d_{\mu} = g_{\mu} \cdot d_{0}$, so $g_{\mu}^{-1}g \in H$) and so $KH_{\mu}^{-1} = KHg_{\mu}^{-1}$. Thus, the above integral can be rewritten as $$\int_{KHg_{\mu}^{-1}} c_{\lambda}(g^{-1})dg = \int_{KH} c_{-\lambda}(gg_{\mu}^{-1}) dg.$$ Finally, write $$KH = \cup K x_{i},$$ a disjoint union and $x_{i} \in H$. Then we claim $H = \cup K' x_{i}$, again a disjoint union. That the union is contained inside $H$ is clear, suppose next that $h \in H$. But then $h = kx_{i}$ for some $k \in K$ and $x_{i}$. But since $h, x_{i} \in H$ we have $k \in H$, i.e., $k \in K'$. Clearly the union is disjoint, since $K'x_{i} \subset Kx_{i}$ for all $i$. Thus, $$\begin{gathered}
\int_{KH} c_{-\lambda}(gg_{\mu}^{-1}) dg = \sum_{x_{i}} \int_{Kx_{i}} c_{-\lambda}(gg_{\mu}^{-1})dg = \sum_{x_{i}} c_{-\lambda}(x_{i}g_{\mu}^{-1})dg \\ = \sum_{x_{i}} \int_{K'x_{i}} c_{-\lambda}(x_{i}g_{\mu}^{-1})dg = \int_{H} c_{-\lambda}(hg_{\mu}^{-1})dh.\end{gathered}$$ Finally, we have to multiply this by $meas.(K \cdot d_{\mu})$, see the previous proof for these values in each Case.
**Case 3:** Analogous to Case 1, except that $g_{-\mu} \cdot d_{0} = d_{\mu}$, so one uses $g_{-\mu}$ instead of $g_{\mu}$.
**Case 2 and Case 4:** We have $s_{0} = I_{2n} \neq d_{0}$. In both cases, $s_{0} \in K \cdot d_{0}$, so $s_{0} = k_{0} \cdot d_{0}$ for some $k_{0} \in K$. Then one can check that $\text{Stab}(d_{0}) = k_{0}^{-1}Hk_{0}$, so that $H_{\mu} = g_{\mu} k_{0}^{-1} H k_{0}$ in the proof above. Using this, we have $KH_{\mu}^{-1} = Kk_{0}^{-1}Hk_{0}g_{\mu}^{-1} = KHk_{0}g_{\mu}^{-1}$, so repeating the arguments of Case 1, we have $$\int_{KHk_{0}g_{\mu}^{-1}} c_{\lambda}(g^{-1}) dg = \int_{KH} c_{-\lambda}(gk_{0}g_{\mu}^{-1})dg.$$ Exactly as argued in Case 1, we have $$\int_{KH} c_{-\lambda}(gk_{0}g_{\mu}^{-1})dg = \int_{H} c_{-\lambda}(hk_{0}g_{\mu}^{-1}) dh.$$ Finally, we have to multiply this by $meas.(K \cdot d_{\mu})$, see the previous proof for these values in each Case.
The last part follows since $$g_{0} = \text{diag}(1, \dots, 1) = \text{Id}_{2n},$$ and $c_{\lambda}(k_{0}h) = c_{\lambda}(h)$ as the characteristic functions are $K$-bi-invariant.
Putting the previous two propositions together proves the theorem in the introduction.
We will now focus on the special case $\mu = 0$, where we can obtain some more explicit results. First, we will use measure-theoretic arguments to compute the RHS of the integral identity in Theorem \[genthm\].
\[lemma\] We have the following:
**Case 1:** $$\int_{H} c_{\lambda}(h) dh = \begin{cases} 0 & \text{if $\lambda \neq \mu^{2}$ for any $\mu$,}
\\
p^{2\langle\mu, \rho_{3}\rangle} \frac{v_{n}(p^{-2})}{v_{\mu}(p^{-2})} &\text{if $\lambda = \mu^{2}$ for some $\mu$.}
\end{cases}$$
**Case 2:** $$\int_{H} c_{\lambda}(h) dh = p^{2\langle \lambda, \rho_{2} \rangle} \frac{v_{2n}(p^{-1})}{v_{\lambda}(p^{-1})}.$$
**Case 3:** $$\int_{H} c_{\lambda}(h) dh = \begin{cases} 0 & \text{if $\lambda \neq \mu \bar{\mu}$ for any $\mu$,} \\
p^{2<\mu, \rho_{1}>} \frac{\phi_{n}(p^{-2})}{\phi_{n-l(\mu)}(p^{-2})(1-p^{-1})^{l(\mu)}v_{\mu^{+}}(p^{-1})} &\text{if $\lambda = \mu \bar{\mu}$ for some $\mu$.}
\end{cases}$$
**Case 4:** $$\int_{H} c_{\lambda}(h) dh = \sum_{\mu \cup \nu = \lambda} p^{\langle \mu, \rho_{3} \rangle + \langle \nu, \rho_{3} \rangle} \frac{v_{n}(p^{-1})^{2}}{v_{\mu}(p^{-1})v_{\nu}(p^{-1})}$$
**Case 1:** Note first that the integral of the LHS is the measure of the intersection $H \cap Kp^{\lambda}K$. We recall the Cartan decomposition of $G = Gl_{2n}(\mathbb{Q}_{p})$: $$Gl_{2n}(\mathbb{Q}_{p})= \cup K p^{\lambda} K, \text{ (disjoint union)}$$ where $p^{\lambda}$ is the element $\text{diag}.(p^{\lambda_{1}}, \dots, p^{\lambda_{2n}})$ in $G$. Similarly, we have the Cartan decomposition for $Gl_{n}(\mathbb{Q}_{p}(\sqrt{a}))$: $$Gl_{n}(\mathbb{Q}_{p}(\sqrt{a})) = \cup K' p^{\mu} K', \text{ (disjoint union)}$$ where $p^{\mu}$ is the element $\text{diag}.(p^{\mu_{1}}, \dots, p^{\mu_{n}})$ in $Gl_{n}(\mathbb{Q}_{p}(\sqrt{a}))$ and $K' = Gl_{n}(\mathbb{Z}_{p}(\sqrt{a}))$. Note that under the isomorphism $Gl_{n}(\mathbb{Q}_{p}(\sqrt{a})) \rightarrow H$, $K'$ is mapped to $K \cap H$, which is contained in $K$. Also the element $\text{diag}.(p^{\mu_{1}}, \dots, p^{\mu_{n}}) \in Gl_{n}(\mathbb{Q}_{p}(\sqrt{a})) $ is mapped to the diagonal matrix $\text{diag}.(p^{\mu_{1}}, \dots, p^{\mu_{n}}, p^{\mu_{n}}, \dots, p^{\mu_{1}})$, which is an element of $Kp^{\lambda}K$, where $\lambda = \mu_{1} \mu_{1} \mu_{2} \mu_{2} \dots \mu_{n} \mu_{n}$. Thus, $H$ may be realized inside $G$ as the disjoint union of the double cosets $\{(K \cap H) p^{(\mu_{1}, \dots, \mu_{n}, \mu_{n}, \dots, \mu_{1})} (K \cap H)\}$, where $\mu$ is a partition of length at most $n$.
This implies $H \cap K p^{\lambda} K$ is empty unless $\lambda = \mu^{2}$ for some partition $\mu$, which gives the vanishing part of the claim. If $\lambda = \mu^{2}$, the integral is equal to $meas.((K \cap H) p^{(\mu_{1}, \dots, \mu_{n}, \mu_{n}, \dots, \mu_{1})} (K \cap H))$, which is equivalent to $meas.(K'p^{\mu}K')$ inside $Gl_{n}(\mathbb{Q}_{p}(\sqrt{a}))$. We can compute this last quantity using [@Mac Ch. V]. Applying that result to the group $Gl_{n}(\mathbb{Q}_{p}(\sqrt{a}))$, and noting that $p^{2}$ is the size of the residue field of $\mathbb{Q}_{p}(\sqrt{a})$ gives $$|K'p^{\mu}K'| = (p^{2})^{\langle\mu, \rho_{3}\rangle} \frac{\Big(\prod_{i=1}^{n} (1-p^{-2i})\Big)/ (1-p^{-2})^{n}}{\Big(\prod_{j \geq 0} \prod_{i=1}^{m_{j}(\mu)} (1-p^{-2i})\Big)/(1-p^{-2})^{n}} = p^{2\langle\mu, \rho_{3}\rangle} \frac{v_{n}(p^{-2})}{v_{\mu}(p^{-2})}.$$
**Case 2:** Note that we have the following Cartan decompositions: $$G = Gl_{2n}(\mathbb{Q}_{p}(\sqrt{a})) = \bigcup_{\lambda \in \Lambda_{2n}} \Big(Gl_{2n}(\mathbb{Z}_{p}(\sqrt{a})) p^{\lambda} Gl_{2n}(\mathbb{Z}_{p}(\sqrt{a})) \Big)$$ and $$H = Gl_{2n}(\mathbb{Q}_{p}) = \bigcup_{\lambda \in \Lambda_{2n}} \Big(Gl_{2n}(\mathbb{Z}_{p}) p^{\lambda} Gl_{2n}(\mathbb{Z}_{p})\Big),$$ where in both cases the unions are disjoint. Note also that $Gl_{2n}(\mathbb{Z}_{p}) = K' \subset K = Gl_{2n}(\mathbb{Z}_{p}(\sqrt{a}))$. Thus, the intersection $Kp^{\lambda}K \cap H$ is exactly $K'p^{\lambda}K'$. Finally, from [@Mac Ch. V (2.9)], we have $$\text{measure of } K'p^{\lambda}K' = p^{2\langle \lambda, \rho_{2} \rangle} \frac{v_{2n}(p^{-1})}{v_{\lambda}(p^{-1})}=p^{2\langle \lambda, \rho_{2} \rangle} \frac{\phi_{2n}(p^{-1})}{\prod_{i \geq 0} \phi_{m_{i}(\lambda)}(p^{-1})},$$ as desired.
**Case 3:** Note that we have the following Cartan decompositions: $$G= Gl_{2n}(\mathbb{Q}_{p}) = \bigcup_{\lambda \in \Lambda_{2n}} \Big(Gl_{2n}(\mathbb{Z}_{p}) p^{\lambda} Gl_{2n}(\mathbb{Z}_{p})\Big)$$ and $$H= Sp_{2n}(\mathbb{Q}_{p}) = \bigcup_{\substack{\lambda = \mu \bar{\mu}\\ \text{in } \Lambda_{2n}}} \Big( Sp_{2n}(\mathbb{Z}_{p}) p^{\lambda} Sp_{2n}(\mathbb{Z}_{p}) \Big)$$ where in both cases the unions are disjoint. This implies that the intersection $Kp^{\lambda}K \cap H$ is zero if $\lambda \neq \mu \bar{\mu}$ for some $\mu$ giving the vanishing part of the result. If $\lambda = \mu \bar{\mu}$, the intersection is $K'p^{\lambda}K'$. We use [@MacP] (which deals with the general reductive $p$-adic group case) to compute $$\text{measure of } K'p^{\mu{\bar{\mu}}}K' = p^{2\langle \mu, \rho_{1}\rangle } \frac{\phi_{n}(p^{-2})}{\phi_{n-l(\mu)}(p^{-2})(1-p^{-1})^{l(\mu)}v_{\mu^{+}}(p^{-1})},$$ as desired.
**Case 4:** Analogous to the arguments of the previous cases. We will indicate the differences. The double coset $Kp^{\lambda}K$ in $G = Gl_{2n}(\mathbb{Q}_{p})$ contains $Gl_{n}(\mathbb{Z}_{p})p^{\mu}Gl_{n}(\mathbb{Z}_{p}) \times Gl_{n}(\mathbb{Z}_{p})p^{\nu}Gl_{n}(\mathbb{Z}_{p})$, for any $\mu, \nu$ such that $\mu \cup \nu = \lambda$ (identifying this with its image in $G$ under the embedding of Case 4), since the element $$\left( \begin{array}{cc}
p^{\mu} & 0 \\
0 & p^{\nu} \\ \end{array} \right)$$ is in $Kp^{\lambda}K$ and $(k_{1}, k_{2}) \in Gl_{2n}(\mathbb{Z}_{p})$ for any $k_{1}, k_{2} \in Gl_{n}(\mathbb{Z}_{p})$. One can also easily show that $Kp^{\lambda}K$ only contains double cosets of $H$ of this form. Finally, one computes the measures of $Gl_{n}(\mathbb{Z}_{p})p^{\mu}Gl_{n}(\mathbb{Z}_{p})$ and $Gl_{n}(\mathbb{Z}_{p})p^{\nu}Gl_{n}(\mathbb{Z}_{p})$ using [@Mac Ch. V], for example, to obtain the result.
We now discuss the related integral identities corresponding to the special case $\mu = 0$.
\[symplectic\] Symplectic identity (Case 1), [@VV Corollary 14]. Let $\lambda$ be a partition of length at most $2n$. Then $$\frac{1}{Z} \int_{T} P_{\lambda}^{(2n)}(x_{1}^{\pm 1}, \dots, x_{n}^{\pm 1} ;t) \tilde \Delta_{K}^{(n)}(x; t;\pm \sqrt{t},0,0) dT = 0,$$ unless $\lambda = \mu^{2}$. In this case, the nonzero value is $$\frac{v_{n}(t^{2})}{v_{\mu}(t^{2})}$$ (here the normalization factor $Z = \int_{T} \tilde \Delta_{K}^{(n)}(x;t;\pm \sqrt{t},0,0) dT$).
\[Kawanaka\] Finite-dimensional version of Kawanaka’s identity (Case 2), [@VV Corollary 15]. Let $\lambda$ be a partition of length at most $2n$. Then $$\begin{aligned}
\frac{1}{Z} \int_{T} P_{\lambda}(x_{1}^{\pm 1}, \dots, x_{n}^{\pm 1};t) \tilde \Delta_{K}^{(n)}(x;t;1,\sqrt{t},0,0) &= \frac{v_{2n}(\sqrt{t})}{v_{\lambda}(\sqrt{t})} \end{aligned}$$ (here the normalization factor $Z = \int_{T} \tilde \Delta_{K}^{(n)}(x;t;1,\sqrt{t},0,0) dT$).
\[other\] (Case 3), [@VV Theorem 22]. Let $\lambda$ be a weight of the double cover of $GL_{2n}$, i.e., a half-integer vector such that $\lambda_{i} - \lambda_{j} \in \mathbb{Z}$ for all $i,j$. Then $$\begin{aligned}
\frac{1}{Z}\int_{T} P_{\lambda}^{(2n)}( \cdots t^{\pm 1/2}z_{i} \cdots ;t) \tilde \Delta_{S}^{(n)}(x; t^{2}) dT &=0,\end{aligned}$$ unless $\lambda = \mu \bar{\mu}$. In this case, the nonzero value is $$\begin{aligned}
\frac{\phi_{n}(t^{2})}{(1-t)^{n}v_{\mu}(t) (1+t)(1+t^{2}) \cdots (1+t^{n-l(\mu)})}\end{aligned}$$ (here the normalization factor $Z = \int_{T} \tilde \Delta_{S}(z;t^{2})dT$).
\[new\] (Case 4). Let $\lambda$ be a partition of length at most $2n$. Then $$\frac{1}{Z} \int_{T} P_{\lambda}^{(2n)}(x_{1}^{\pm 1}, \dots, x_{n}^{\pm 1};t) \tilde \Delta_{K}^{(n)}(x;t; \sqrt{t}, \sqrt{t},0,0) dT = \sum_{\mu \cup \nu = \lambda} t^{-\langle \mu, \rho_{3} \rangle - \langle \nu, \rho_{3} \rangle + \langle \lambda, \rho_{2} \rangle} \frac{v_{n}(t)^{2}}{v_{\mu}(t)v_{\nu}(t)}$$
Note that the second identity is *not* a vanishing identity. This identity is a finite-dimensional analog of a result of Kawanaka (see [@Ka1], [@Ka2]). Kawanaka’s identity has an interesting representation-theoretic significance for general linear groups over finite fields: it encodes the fact that the symmetric space $Gl_{n}(\mathbb{F}_{p^{2}})/Gl_{n}(\mathbb{F}_{p})$ is multiplicity free.
We also note that the fourth identity is new (as far as we know); also if we set $t=0$, we recover Theorem \[symplecticgp\], part (1), so it is a generalization of that result.
We are now prepared to provide $p$-adic proofs of Theorems \[symplectic\], \[Kawanaka\], \[other\], and \[new\].
The vanishing part of the statement is automatic from **Case 1** of Theorem \[genthm\] and Lemma \[lemma\]. To obtain the nonzero value, let $\lambda = \mu^{2}$. Then we can compute $$\begin{gathered}
2\langle\mu, \rho_{3}\rangle = 2 \Big( (n-1)\mu_{1} + (n-3)\mu_{2} + \cdots + (1-n)\mu_{n} \Big)
= (n-1)(\mu_{1} + \mu_{1}) + (n-3)(\mu_{2} + \mu_{2})+ \cdots + (1-n)(\mu_{n} + \mu_{n}) \\
= (n-1)(\lambda_{1} + \lambda_{2}) + (n-3)(\lambda_{3} + \lambda_{4}) + \cdots + (1-n)(\lambda_{2n-1} + \lambda_{2n}) \\
= \lambda_{1}(n-1/2) + \lambda_{2}(n- 3/2) + \cdots + \lambda_{2n}(1/2 - n) = \langle\lambda, \rho_{2}\rangle.\end{gathered}$$ Thus, we obtain $$\frac{1}{Z} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1};p^{-1}) \tilde \Delta_{K}^{(n)}(x; \pm p^{-1/2},0,0;p^{-1}) dT = \begin{cases} 0 & \text{if $\lambda \neq \mu^{2}$ for any $\mu$,}
\\
\frac{v_{n}(p^{-2})}{v_{\mu}(p^{-2})} &\text{if $\lambda = \mu^{2}$ for some $\mu$.}
\end{cases}$$ Thus the equation in the statement of Theorem \[symplectic\] holds for all $t = p^{-1}$, for $p$ an odd prime, and the left hand side of the equation is a rational function in $t$. This provides an infinite sequence of values for $t$ for which the equation holds, so in particular it holds for all values of $t$ as desired.
The identity follows from **Case 2** of Theorem \[genthm\] and Lemma \[lemma\], as in the proof of Theorem \[symplectic\] above. Note that these arguments show that the theorem holds for all $t = p^{-2}$. This provides an infinite sequence of values for $t$ for which the equation holds, so in particular it holds for all values of $t$ as desired.
The identity follows from **Case 3** of Theorem \[genthm\] and Lemma \[lemma\], as in the proof of Theorem \[symplectic\] above. If $\lambda = \mu\bar{\mu}$ for some $\mu$, the integral is non-vanishing. The evaluation follows by noting that $2\langle \mu, \rho_{1} \rangle = \langle \lambda, \rho_{2} \rangle$. Note that these arguments show that the theorem holds for all $t = p^{-1}$. This provides an infinite sequence of values for $t$ for which the equation holds, so in particular it holds for all values of $t$ as desired.
The identity follows from **Case 4** of Theorem \[genthm\] and Lemma \[lemma\], as in the proof of Theorem \[symplectic\] above. Note that these arguments show that the theorem holds for all $t = p^{-1}$. This provides an infinite sequence of values for $t$ for which the equation holds, so in particular it holds for all values of $t$ as desired.
In **Case 1**, the involution is $g \rightarrow g^{\star} = g^{-1}$ and the action is $g \cdot x = gxg^{*}$. Then $H$ is the stabilizer in $G$ of $s_{0}$ under this action. But $H = \{ g \in G | gs_{0}g^{*} = s_{0} \} = \{g \in G | g = s_{0}g^{*^{-1}}s_{0}^{-1} \}$. So $H$ is the set of fixed points of the order $2$ homomorphism $g \rightarrow s_{0}g^{*^{-1}}s_{0}^{-1}$. This provides an analog of Theorem (\[symplecticgp\]), where one restricts $s_{\lambda}$ to the subgroup of fixed points of a suitable involution. The other cases are analogous.
We now demonstrate how one can use Theorem \[new\] to obtain a new Littlewood summation identity, namely that of Theorem \[Littlewood\].
From [@R Lemma 7.18], we obtain the following integral identity $$\label{inf}
\frac{1}{Z} \int_{T} \prod_{j,k} \frac{1-tx_{j}y_{k}^{\pm 1}}{1-x_{j}y_{k}^{\pm 1}} \tilde \Delta_{K}(y;t;\sqrt{t}, \sqrt{t}, 0, 0) dT = \prod_{j<k} \frac{1-tx_{j}x_{k}}{1-x_{j}x_{k}} \prod_{j} \frac{1-tx_{j}^{2}}{(1-\sqrt{t}x_{j})(1-\sqrt{t}x_{j})},$$ where the integral in the LHS is with respect to the $y$-variables. Recall that the Cauchy identity for Hall-Littlewood functions is $$\sum_{\lambda} P_{\lambda}(x;t) Q_{\lambda}(y;t) = \prod_{i,j \geq 1} \frac{1-tx_{i}y_{j}}{1-x_{i}y_{j}},$$ where $$Q_{\lambda}(x;t) = b_{\lambda}(t)P_{\lambda}(x;t).$$ Using this on the product in the LHS of (\[inf\]) allows us to rewrite that equation as $$\begin{gathered}
\label{inf2}
\sum_{\lambda} P_{\lambda}(x;t) \lim_{n \rightarrow \infty} \frac{1}{Z} \Bigg[ \int_{T} b_{\lambda}(t) P_{\lambda}(y_{1}^{\pm 1}, \dots, y_{n}^{\pm 1};t) \tilde \Delta_{K}^{(n)}(y;t;\sqrt{t}, \sqrt{t},0,0) dT \Bigg] \\
= \prod_{j<k} \frac{1-tx_{j}x_{k}}{1-x_{j}x_{k}} \prod_{j} \frac{1-tx_{j}^{2}}{(1-\sqrt{t}x_{j})^{2}} = \prod_{j<k} \frac{1-tx_{j}x_{k}}{1-x_{j}x_{k}} \prod_{j} \frac{1+\sqrt{t}x_{j}}{1-\sqrt{t}x_{j}}.\end{gathered}$$ Recall that by Theorem \[new\], we have $$\begin{gathered}
\frac{1}{Z} \Bigg[ \int_{T} b_{\lambda}(t) P_{\lambda}(y_{1}^{\pm 1}, \dots, y_{n}^{\pm 1};t) \tilde \Delta_{K}^{(n)}(y;t;\sqrt{t}, \sqrt{t},0,0) dT \Bigg] \\= \sum_{\substack{\mu \cup \nu = \lambda \\ l(\mu), l(\nu) \leq n}} t^{-\langle \mu, \rho_{3} \rangle - \langle \nu, \rho_{3} \rangle + \langle \lambda, \rho_{2}\rangle} \frac{v_{n}(t)^{2} b_{\lambda}(t)}{v_{\mu}(t) v_{\nu}(t)}.\end{gathered}$$
So we want to compute $$\label{limit}
\lim_{n \rightarrow \infty} \sum_{\substack{\mu \cup \nu = \lambda \\ l(\mu), l(\nu) \leq n}} t^{-\langle \mu, \rho_{3} \rangle - \langle \nu, \rho_{3} \rangle + \langle \lambda, \rho_{2}\rangle} \frac{v_{n}(t)^{2} b_{\lambda}(t)}{v_{\mu}(t) v_{\nu}(t)}.$$ First, note that $$\frac{v_{n}(t)^{2}b_{\lambda}(t)}{v_{\mu}(t)v_{\nu}(t)} = \frac{\phi_{n}(t)^{2} \prod_{i \geq 1} \phi_{m_{i}(\lambda)}(t)}{\prod_{i \geq 0} \phi_{m_{i}(\mu)}(t) \prod_{i \geq 0} \phi_{m_{i}(\nu)}(t)}.$$ Also $$\lim_{n \rightarrow \infty} \frac{\phi_{n}(t)^{2}}{\phi_{m_{0}(\mu)}(t) \phi_{m_{0}(\nu)}(t)} = \lim_{n \rightarrow \infty} (1-t^{m_{0}(\mu) + 1}) \cdots (1-t^{n})(1-t^{m_{0}(\nu) + 1}) \cdots (1-t^{n}) = 1,$$ since as $n \rightarrow \infty$, we have $m_{0}(\nu), m_{0}(\mu) \rightarrow \infty$ with $n-m_{0}(\mu), n - m_{0}(\nu)$ fixed. Note that $$\rho_{2} = n^{2n}- \Big(\frac{1}{2}, \frac{3}{2}, \dots, \big(2n - \frac{1}{2}\big) \Big),$$ and $$\rho_{3} = n^{n} - (1,3,5, \dots, (2n-1)).$$ So for $\mu, \nu$ such that $\mu \cup \nu = \lambda$ we have $$-\langle \mu + \nu, n^{n} \rangle + \langle \lambda, n^{2n} \rangle = 0.$$
Thus, (\[limit\]) is equal to $$\sum_{\substack{\mu \cup \nu = \lambda }} t^{\langle \mu + \nu, (1,3,5, \dots) \rangle - \langle \lambda, (\frac{1}{2}, \frac{3}{2}, \dots) \rangle} \frac{b_{\lambda}(t)}{b_{\mu}(t) b_{\nu}(t)}.$$ Putting this back into (\[inf2\]), the resulting Littlewood identity is $$\sum_{\mu, \nu} P_{\mu \cup \nu}(x;t) t^{\langle \mu + \nu, \rho \rangle - \frac{1}{2}\langle \mu \cup \nu, \rho \rangle} \frac{b_{\mu \cup \nu}(t)}{b_{\mu}(t) b_{\nu}(t)} = \prod_{j<k} \frac{1-tx_{j}x_{k}}{1-x_{j}x_{k}} \prod_{j} \frac{1+\sqrt{t}x_{j}}{1-\sqrt{t}x_{j}}.$$
Generalized integral identity
=============================
In this section, we deal only with the symplectic case, **Case 1**; the notation is as in that case. We will prove some stronger results by extending the methods above.
Throughout this section, we fix $l(\lambda) \leq 2n$ and $l(\mu) \leq n$. Then, by Theorem \[genthm\] (Case 1), we have $$\begin{gathered}
\frac{1}{Z} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; \pm p^{-1/2},0,0) \tilde \Delta_{K}^{(n)}(x; \pm p^{-1/2},0,0; p^{-1}) dT \\
= p^{\langle\mu, \rho_{1}\rangle - \langle\lambda, \rho_{2}\rangle} \frac{V_{0}}{V_{\mu}} \int_{H} c_{\lambda}(hg_{\mu}^{-1}) dh.\end{gathered}$$ Using the Cartan decomposition for $(Gl_{n}(\mathbb{Q}_{p}(\sqrt{a})), Gl_{n}(\mathbb{Z}_{p}(\sqrt{a})))$ and the embedding into $Gl_{2n}(\mathbb{Q}_{p})$, we have $$\int_{H} c_{\lambda}(hg_{\mu}^{-1}) dh = \sum_{\substack{\beta \in \Lambda_{2n} \\ \beta = \nu_{1} \dots \nu_{n} \nu_{n} \dots \nu_{1} \\ \text{for some $\nu$}}} \int_{K' p^{\beta} K'} c_{\lambda}(hg_{\mu}^{-1}) dh;$$ also note that $$\int_{K'p^{\beta}K'} c_{\lambda}(hg_{\mu}^{-1}) dh = meas.(Kp^{\lambda}Kg_{\mu} \cap K'p^{\beta}K'),$$ where the measure is with respect to the measure on $H$. Thus, $$\begin{gathered}
\label{negpts}
\frac{1}{Z} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; \pm p^{-1/2},0,0) \tilde \Delta_{K}^{(n)}(x; \pm p^{-1/2},0,0; p^{-1}) dT \\
= p^{\langle\mu, \rho_{1}\rangle - \langle\lambda, \rho_{2}\rangle} \frac{V_{0}}{V_{\mu}} \sum_{\substack{\beta \in \Lambda_{2n} \\ \beta = \nu_{1} \dots \nu_{n} \nu_{n} \dots \nu_{1} \\ \text{for some $\nu$}}} meas.(Kp^{\lambda}Kg_{\mu} \cap K'p^{\beta}K').\end{gathered}$$
Let $\beta = \nu_{1} \dots \nu_{n} \nu_{n} \dots \nu_{1} \in \Lambda_{2n}$ have at least one negative part. Then $meas.(Kp^{\lambda}Kg_{\mu} \cap K'p^{\beta}K') = 0$.
Note that if $Kp^{\lambda}K \cap K'p^{\beta}K'g_{\mu}^{-1} \neq \emptyset $, then $Kp^{\lambda}K \cap p^{\beta}K' g_{\mu}^{-1} \neq \emptyset$. We will show that $Kp^{\lambda}K \cap p^{\beta}K' g_{\mu}^{-1} = \emptyset$, which proves the claim.
Note first that $g_{\mu}^{-1} = \text{diag}.(1, \dots, 1,p^{\mu_{n}}, \dots, p^{\mu_{1}})$. We will write $\bar{\mu} = (\mu_{n}, \dots, \mu_{1})$. Suppose for contradiction that $$k'= \left( \begin{array}{cc}
i & j \\
aw_{n}jw_{n} & w_{n}iw_{n} \\ \end{array} \right)$$ is an element in $K'$ such that $p^{\beta}k'g_{\mu}^{-1} \in Kp^{\lambda}K$. By a direct computation we have $$p^{\beta}k'g_{\mu}^{-1} = \left( \begin{array}{cc}
p^{\nu} & 0 \\
0 & p^{\bar{\nu}} \\ \end{array} \right)
\left( \begin{array}{cc}
i & j \\
aw_{n}jw_{n} & w_{n}iw_{n} \\ \end{array} \right)
\left( \begin{array}{cc}
1 & 0 \\
0 & p^{\bar{\mu}} \\ \end{array} \right)
=
\left( \begin{array}{cc}
p^{\nu}i & p^{\nu}jp^{\bar{\mu}} \\
p^{\bar{\nu}}aw_{n}jw_{n} & p^{\bar{\nu}}w_{n}iw_{n}p^{\bar{\mu}} \\ \end{array} \right).$$ Now noting that $p^{\bar{\nu}}w_{n} = w_{n}p^{\nu}$, the above becomes $$\left( \begin{array}{cc}
p^{\nu}i & p^{\nu}jp^{\bar{\mu}} \\
aw_{n}p^{\nu}jw_{n} & w_{n}p^{\nu}ip^{\mu}w_{n} \\ \end{array} \right).$$ Since $p^{\beta}k'g_{\mu}^{-1} \in Kp^{\lambda}K \subset M_{2n}(\mathbb{Z}_{p})$, it follows that $p^{\nu}i$ and $p^{\nu}j$ are in $M_{n}(\mathbb{Z}_{p})$. Since $\nu_{n} < 0$, it follows that the $n$-th row of $k'$ has entries all of which are divisible by $p$ in $\mathbb{Z}_{p}$. Let $B$ be the matrix obtained from $k'$ by dividing the $n$-th row by $p$; note that $B \in M_{2n}(\mathbb{Z}_{p})$. Then $$\det(k') = p \det(B) \in p \cdot \mathbb{Z}_{p},$$ which contradicts $|\det(k')| = 1$.
Thus, using the previous lemma, (\[negpts\]) now becomes $$\begin{gathered}
\label{pospts}
\frac{1}{Z} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; \pm p^{-1/2},0,0) \tilde \Delta_{K}^{(n)}(x; \pm p^{-1/2},0,0; p^{-1}) dT \\
= p^{\langle\mu, \rho_{1}\rangle - \langle\lambda, \rho_{2}\rangle} \frac{V_{0}}{V_{\mu}} \sum_{\substack{\beta \in \Lambda_{2n} ^{+}\\ \beta = \nu_{1} \dots \nu_{n} \nu_{n} \dots \nu_{1} \\ \text{for some $\nu$}}} meas.(Kp^{\lambda}Kg_{\mu} \cap K'p^{\beta}K').\end{gathered}$$
**Littlewood-Richardson coefficients and Hall polynomials:**
Recall that the Littlewood-Richardson coefficient $c^{\lambda}_{\mu \nu}$ is equal to the number of tableaux $T$ of shape $\lambda - \mu$ and weight $\nu$ such that $w(T)$, the word of $T$, is a lattice permutation. We have $$s_{\mu}s_{\nu} = \sum_{\lambda} c^{\lambda}_{\mu \nu} s_{\lambda},$$ where $s_{\mu}$ is the Schur function (see [@Mac] for more details).
We briefly recall the Hall polynomials $g_{\mu \nu}^{\lambda}(q)$ [@Mac Chs. II and V]. Let $\mathcal{O}$ be a complete (commutative) discrete valuation ring, $\mathcal{P}$ its maximal ideal and $k = \mathcal{O}/\mathcal{P}$ the residue field. We assume $k$ is a finite field. Let $q$ be the number of elements in $k$. Let $M$ be a finite $\mathcal{O}$-module of type $\lambda$. Then the number of submodules of $N$ of $M$ with type $\nu$ and cotype $\mu$ is a polynomial in $q$, called the Hall polynomial, denoted $g_{\mu \nu}^{\lambda}(q)$. One can consider our motivating case of $\mathbb{Q}_{p}$ and its ring of integers $\mathcal{O} = \mathbb{Z}_{p}$ and $G = Gl_{n}(\mathbb{Q}_{p})$, so that $q=p$. Then they are also the structure constants for the ring $\mathcal{H}(G^{+},K)$. That is, for $\mu, \nu \in \Lambda_{2n}^{+}$, we have $$c_{\mu} \star c_{\nu} = \sum_{\lambda \in \Lambda_{2n}^{+}} g_{\mu \nu}^{\lambda}(p) c_{\lambda}.$$ Note that, in particular, $$g_{\mu \nu}^{\lambda}(p) = (c_{\mu} \star c_{\nu})(p^{\lambda}) = \int_{G} c_{\mu}(p^{\lambda}y^{-1})c_{\nu}(y)dy = meas.(p^{\lambda}Kp^{-\nu}K \cap Kp^{\mu}K).$$
Several important facts are known (see [@Mac Ch. II]):
1. If $c^{\lambda}_{\mu \nu} = 0$, then $g^{\lambda}_{\mu \nu}(t) = 0$ as a function of $t$.
2. If $c^{\lambda}_{\mu \nu} \neq 0$, then $g^{\lambda}_{\mu \nu}(t)$ has degree $n(\lambda) - n(\mu) - n(\nu)$ and leading coefficient $c^{\lambda}_{\mu \nu}$, where the notation $n(\lambda) = \sum (i-1) \lambda_{i}$.
3. We have $g^{\lambda}_{\mu \nu}(t) = g^{\lambda}_{\nu \mu}(t)$.
Also if one multiplies two Hall-Littlewood polynomials, and expands the result in the Hall-Littlewood basis, one has $$P_{\mu}(x;t) P_{\nu}(x;t) = \sum_{\lambda} f^{\lambda}_{\mu \nu}(t) P_{\lambda}(x;t),$$ with $f^{\lambda}_{\mu \nu}(t) = t^{n(\lambda) - n(\mu) - n(\nu)} g^{\lambda}_{\mu \nu}(t^{-1})$.
\[2d\] Let $\lambda, \mu, \beta \in \Lambda_{2n}$. Then we have $$\int_{G} c_{-\mu}(g') \int_{G} c_{\beta}(g) c_{\lambda}(gg') dg dg' = meas.(Kp^{-\mu}K) \int_{G} c_{-\lambda}(p^{\mu}g^{-1})c_{\beta}(g)dg$$
Write $Kp^{-\mu}K$ as the disjoint union $\cup k_{i}p^{-\mu}K$, where $k_{i} \in K$. Then $$\begin{gathered}
\int_{G} c_{-\mu}(g') \int_{G} c_{\beta}(g) c_{\lambda}(gg') dg dg' = \int_{Kp^{-\mu}K} \int_{G} c_{\beta}(g) c_{\lambda}(gg')dg dg' = \sum_{k_{i}p^{-\mu}} \int_{K} \int_{G} c_{\beta}(g)c_{\lambda}(gk_{i}p^{-\mu}k) dg dk \\
= \sum_{k_{i}p^{-\mu}} \int_{G} c_{\beta}(g) c_{\lambda}(gk_{i}p^{-\mu}) dg = \sum_{k_{i}p^{-\mu}}\int_{G} c_{\beta}(yk_{i}^{-1})c_{\lambda}(yp^{-\mu})dy = \sum_{k_{i}p^{-\mu}} \int_{G} c_{\beta}(y)c_{\lambda}(yp^{-\mu})dy \\
= meas.(Kp^{-\mu}K) \int_{G} c_{\beta}(g)c_{\lambda}(gp^{-\mu})dg = meas.(Kp^{-\mu}K) \int_{G} c_{-\lambda}(p^{\mu}g^{-1})c_{\beta}(g)dg.\end{gathered}$$
Let $\lambda \in \Lambda_{2n}^{+}$ and $\mu \in \Lambda_{n}^{+}$ and fix a prime $p \neq 2$. Suppose $g^{\lambda}_{\mu, \beta}(p) = 0$ for all $\beta \in \Lambda_{2n}^{+}$ with all parts occurring with even multiplicity. Then the integral $$\frac{1}{\int_{T} \tilde \Delta_{K}^{(n)}(x; \pm p^{-1/2},0,0; p^{-1}) dT} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; \pm p^{-1/2},0,0) \tilde \Delta_{K}^{(n)}(x; \pm p^{-1/2},0,0; p^{-1}) dT$$ vanishes.
The starting point is (\[pospts\]) from the discussion above, recall that we have $$\begin{gathered}
\frac{1}{Z} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; \pm p^{-1/2},0,0) \tilde \Delta_{K}^{(n)}(x; \pm p^{-1/2},0,0; p^{-1}) dT \\
= p^{\langle\mu, \rho_{1}\rangle - \langle\lambda, \rho_{2}\rangle} \frac{V_{0}}{V_{\mu}} \sum_{\substack{\beta \in \Lambda_{2n} ^{+}\\ \beta = \nu_{1} \dots \nu_{n} \nu_{n} \dots \nu_{1} \\ \text{for some $\nu$}}} meas.(Kp^{\lambda}Kg_{\mu} \cap K'p^{\beta}K').\end{gathered}$$ Now if we write $$(Kp^{\lambda}Kg_{\mu} \cap K'p^{\beta}K') = \cup K' x_{i},$$ a disjoint union and $x_{i} \in p^{\beta}K'$, then the above measure is the number of $x_{i}$’s. But we also have $$\cup K x_{i} \subset (Kp^{\lambda}Kg_{\mu} \cap Kp^{\beta}K),$$ and the union is disjoint ($k_{1}x_{i} = k_{2}x_{j}$ implies $k_{2}^{-1}k_{1}x_{i} = x_{j}$, but $x_{i},x_{j} \in H$ so $k_{2}^{-1}k_{1} \in K'$, a contradiction to the definition of the $x_{j}$’s). Thus, $$meas.(Kp^{\lambda}Kg_{\mu} \cap K'p^{\beta}K') = \# \{x_{i} \} = meas.(\cup Kx_{i}) \leq meas.(Kp^{\lambda}Kg_{\mu} \cap Kp^{\beta}K),$$ so that $$\int_{K'p^{\beta}K'}c_{\lambda}(hg_{\mu}^{-1}) dh \leq \int_{Kp^{\beta}K}c_{\lambda}(gg_{\mu}^{-1})dg = \int_{G} c_{\beta}(g) c_{\lambda}(gg_{\mu}^{-1})dg = \int_{G} c_{-\lambda}(g_{\mu}g^{-1})c_{\beta}(g)dg.$$ Recall that $g_{\mu} = p^{(0^{n},-\mu_{n}, \dots, -\mu_{1})}$. By Lemma (\[2d\]), we have $$\int_{G} c_{-\lambda}(g_{\mu}g^{-1})c_{\beta}(g)dg = \frac{1}{meas.(Kp^{\mu 0^{n}}K) } \int_{G} c_{\mu 0^{n}}(g') \int_{G} c_{\beta}(g) c_{\lambda}(gg')dgdg'.$$ But, using a change of variables, we have $$\begin{gathered}
\int_{G} c_{\mu 0^{n}}(g') \int_{G} c_{\beta}(g) c_{\lambda}(gg')dgdg' = \int_{G} c_{\mu0^{n}}(g') \int_{G} c_{\beta}(yg'^{-1})c_{\lambda}(y) dy dg' \\
= \int_{G} c_{\mu0^{n}}(g') \int_{G} c_{\beta}(y^{-1}g'^{-1})c_{-\lambda}(y) dy dg'
= \int_{G} c_{-\lambda}(y)\int_{G} c_{\mu 0^{n}}(g') c_{-\beta}(g'y) dg'dy \\
= meas.(Kp^{-\lambda}K) \int_{G} c_{\beta}(p^{\lambda}g^{-1})c_{\mu0^{n}}(g) dg = meas.(Kp^{-\lambda}K) g_{\beta, \mu0^{n}}^{\lambda}(p).
\end{gathered}$$ Thus, $$\int_{G} c_{-\lambda}(g_{\mu}g^{-1})c_{\beta}(g)dg = \frac{meas.(Kp^{-\lambda}K)}{meas.(Kp^{\mu0^{n}}K)} g_{\beta, \mu0^{n}}^{\lambda}(p) = \frac{meas.(Kp^{-\lambda}K)}{meas.(Kp^{\mu0^{n}}K)} g_{ \mu0^{n}, \beta}^{\lambda}(p),$$ where the last equality follows from Fact 3 about Hall polynomials above.
Thus, we have $$\int_{H} c_{\lambda}(hg_{\mu}^{-1})dh = \sum_{\substack{\beta \in \Lambda_{2n} ^{+}\\ \beta = \nu_{1} \dots \nu_{n} \nu_{n} \dots \nu_{1} \\ \text{for some $\nu$}}} meas.(Kp^{\lambda}Kg_{\mu} \cap K'p^{\beta}K') \leq \sum_{\substack{\beta \in \Lambda_{2n}^{+} \\ \beta = \nu^{2} \\ \text{for some $\nu$}}} \frac{meas.(Kp^{-\lambda}K)}{meas.(Kp^{\mu0^{n}}K)} g_{ \mu0^{n}, \beta}^{\lambda}(p).$$
Since by assumption $g^{\lambda}_{\mu, \beta}(p) = 0$ for all $\beta= \nu^{2} \in \Lambda_{2n}^{+}$, the result follows.
\[HLSchurvan\] Let $\lambda, \mu \in \Lambda_{n}^{+}$. Then the integral $$\frac{1}{\int_{T} \tilde \Delta_{K}^{(n)}(x;0,0,0,0;t) dT}\int_{T} s_{\lambda}^{(2n)}(x_{i}^{\pm 1}) sp_{\mu}(x_{1}, \dots, x_{n}) \tilde \Delta_{K}^{(n)}(x;0,0,0,0;t) dT$$ vanishes if and only if the integral $$\frac{1}{\int_{T} \tilde \Delta_{K}^{(n)}(x; \pm \sqrt{t},0,0;t) dT} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1};t) K_{\mu}^{BC_{n}}(x;t; \pm \sqrt{t},0,0) \tilde \Delta_{K}^{(n)}(x; \pm \sqrt{t},0,0;t)dT$$ vanishes as a rational function of $t$.
The “if" direction follows by setting $t=0$ in the Hall-Littlewood polynomial integral to obtain the Schur case. We consider the other direction: i.e., suppose the integral involving Schur polynomials vanishes. We will show the integral involving the Hall-polynomial vanishes.
Fix an odd prime $p$. By Theorem \[WL\], since the above Schur integral vanishes, we must have $c^{\lambda}_{\mu, \beta} = 0$ for all $\beta \in \Lambda_{2n}^{+}$ with all parts occurring with even multiplicity.
By Fact 1 about Hall polynomials above, this implies $g^{\lambda}_{\mu, \beta}(p)=0$ for all $\beta \in \Lambda_{2n}^{+}$ with all parts occurring with even multiplicity. Thus, by the previous proposition, we have $$\frac{1}{\int_{T} \tilde \Delta_{K}^{(n)}(x; \pm p^{-1/2},0,0; p^{-1}) dT} \int_{T} P_{\lambda}^{(2n)}(x_{i}^{\pm 1}; p^{-1}) K_{\mu}^{BC_{n}}(x; p^{-1}; \pm p^{-1/2},0,0) \tilde \Delta_{K}^{(n)}(x; \pm p^{-1/2},0,0; p^{-1}) dT = 0.$$ This shows that the integral in question vanishes for all values $t = p^{-1}$, $p$ an odd prime. Thus it vanishes for all values of $t$.
We are now ready to provide a proof of Theorem \[genvan\], mentioned in the Introduction.
Follows from Proposition \[HLSchurvan\] and Theorem \[WL\].
Let $\lambda$ have all parts occurring with even multiplicity, and $\mu = (r)$ only one part (assume $r \neq 0)$. Let $\beta$ have all parts occurring with even multiplicity. We have $g^{\lambda}_{\beta, (r)}(t) = 0$ unless $\lambda - \beta$ is a horizontal $r$-strip [@Mac]. But $\lambda - \beta$ is a horizontal-strip if and only if $\lambda_{1} \geq \beta_{1} \geq \lambda_{2} \geq \beta_{2} \cdots$ (interlaced), so $\lambda = \beta$. Thus $g^{\lambda}_{(r), \beta}(t) = 0$ for all $\beta$ with all parts occurring with even multiplicity. So for these conditions on $\lambda$, $\mu$, the integral of Theorem \[genvan\] part (1) vanishes.
Let $\mu = 0$. Then by [@Mac], $c^{\lambda}_{\mu, \beta} = 0$ for all $\beta \neq \lambda$. Thus, the integral of Theorem \[genvan\] part (1) vanishes unless $\lambda = \beta$, where $\beta$ has all parts occuring with even multiplicity.
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author:
- Alexey Vladimirov
bibliography:
- 'RtRd.bib'
title: Structure of rapidity divergences in soft factors
---
Introduction
============
Soft factors are the inherent part of the modern factorization theorems, and have the common structure of vacuum matrix elements of a product of Wilson lines. The geometrical configuration of a soft factor reflects the classical picture of the scattering. In this way, the massless initial- and final- state partons represent themselves as a half-infinite lightlike Wilson lines, and give rise to the variety of divergences. Apart of usual collinear and ultraviolet (UV) divergences, they produce rapidity divergences. The latter is a subject of special treatment and factorization procedure. Many aspects of rapidity divergences are still unstudied. In this paper, we present the study of the rapidity divergences, and their connection to the geometry of soft factor. We demonstrate that the rapidity divergences are related to particular spatial configurations, and formulate the requirements on the structure of soft factors that guaranty the factorization of rapidity divergences. It allows us to formulate and prove the renormalization theorem for rapidity divergences.
To make an introduction to the problem, and demonstrate its practical importance, let us recall the transverse momentum dependent (TMD) factorization theorem, where the rapidity divergences and their factorization play one of the central roles. The TMD factorization theorem describes such processes as Drell-Yan (DY) and semi-inclusive-deep-inelastic-scattering (SIDIS) in the regime of low transverse momentum $q_T$. Within the TMD factorization the expression for the hadron tensor takes the form (see e.g. [@Becher:2007ty; @Becher:2010tm; @Collins:2011zzd; @Collins:2011ca; @GarciaEchevarria:2011rb]) $$\begin{aligned}
\label{intro:W=HFSF}
W_{\text{TMD}}=H\otimes {\left[}\bar F(\delta^-) S(\delta^-,\delta^+) F(\delta^+){\right]}+\mathcal{O}{\left(}\frac{q_T}{Q}{\right)},\end{aligned}$$ where $H$ is the Wilson coefficient for the hard-collinear matching, $F$ and $\bar F$ are hadron matrix elements of collinear and anti-collinear fields, and $S$ is the TMD soft factor. Here, the argument $\delta$ represents a regulator for rapidity divergences associated with particular hadron. The rapidity divergences cancel in the product of factors, so the expression (\[intro:W=HFSF\]) is finite. However, the factorization formula (\[intro:W=HFSF\]) is not practical, because it does not define a measurable parton density. The main difficulty is caused by the soft factor which mixes the rapidity divergences of both hadrons. To finalize the factorization and to define universal TMD parton distribution one has to perform the factorization of rapidity divergence in the soft factor.
There are several approaches to formulate the factorization of rapidity divergences for the TMD soft factor. The differences among approaches are originated from the differences in regularization schemes. It appears to be difficult to find a commonly convenient regularization for the rapidity divergences, since they are insensitive to the dimensional regularization [@Collins:1992tv]. Nowadays, there are three most popular approaches to the factorization of rapidity divergences. *(i)* Explicit evaluation of the soft factor and the manual split of divergent contributions [@Collins:2011zzd; @Collins:2011ca; @GarciaEchevarria:2011rb; @Echevarria:2012js]. In this case, the soft factor takes the form of the product of divergent terms, like ${S(\delta^-,\delta^+)=\sqrt{S(\delta^-;\zeta)}\sqrt{S(\delta^+;\zeta)}}$. *(ii)* Formulation of a scaleless regularization for rapidity divergences in which the soft factor is unity at all orders of perturbation theory (e.g. the analytical regularization [@Smirnov:1997gx]). The effect of factorization arises via an anomalous-like contribution, aka collinear anomaly [@Becher:2010tm; @Becher:2012yn]. *(iii)* Subtraction of the rapidity divergences at the symmetric point by the renormalization procedure similar to the UV renormalization [@Chiu:2011qc; @Chiu:2012ir]. All three schemes have been checked by explicit next-to-next-to-leading order (NNLO) calculations (see [@Echevarria:2015byo; @Echevarria:2016scs] for *(i)*, [@Gehrmann:2012ze; @Gehrmann:2014yya] for *(ii)* and [@Luebbert:2016itl; @Li:2016axz] for *(iii)*). The results agree with each other.
In fact, all these schemes imply that the logarithm of soft factor is linear in the rapidity divergence, i.e. $\ln S(\delta^-,\delta^+)\sim \ln(\delta^+\delta^-)$ (here, the divergences are represented by $\ln \delta$). This statement automatically leads to the factorization of rapidity divergences and to the equivalence of all approaches. The linearity in the rapidity divergences seems natural. Indeed, the structure of the exponentiated diagrams for the TMD soft factor is rather simple and gives some intuition how the cancellation of higher-order divergences takes place. This intuition is also supported by NNLO calculation. However, the factorization procedure is not proven, to our best knowledge. The absence of any proof conceptually prevents the extension of factorization to more difficult processes such as multi-parton scattering [@Diehl:2011yj; @Manohar:2012jr; @Diehl:2015bca], or processes with a richer final state that involve complicated soft factors (see e.g.[@Stewart:2010tn; @Jouttenus:2011wh]).
To access the problem of rapidity divergences on a more general level, we study the multi-parton scattering (MPS) factorization and its soft factor. Structurally, the factorized MPS hadron tensor repeats the DY hadron tensor (\[intro:W=HFSF\]), but obtains a non-trivial color structure, $$\begin{aligned}
\label{intro:W=HFSF2}
W_{\text{MPS}}=\sum_{i=1}^N H_i\otimes {\left[}\bar F_{a_1...a_N}(\delta^-) S^{a_1...a_N,b_1...b_N}(\delta^-,\delta^+) F_{b_1...b_N}(\delta^+){\right]}+\mathcal{O}{\left(}\frac{q_T}{Q}{\right)},\end{aligned}$$ where $N/2$ is the number of partons involved in the MPS. The MPS factorization is a direct generalization of TMD ($N=2$), and double-parton scattering ($N=4$) cases. Practically, the MPS is not that important, since it is only the one of many channels contributing to the multi-particle production reaction. However, theoretically, it is very interesting, and allows to look at the problem of rapidity divergences from a new side. In particular, it clearly shows that the rapidity divergences are associated with planes rather than with vectors, which is typical assumption. Therefore, the MPS soft factor has only two rapidity divergences, although it is a composition of many Wilson lines. To our best knowledge, the MPS soft factor has not been studied. Therefore, we start the paper from the presentation of details on the structure of MPS soft factor in sec.\[sec:MPS\_SF\]. It includes the presentation of the all-order color-structure and NNLO expression in sec.\[sec:color\].
The association of the rapidity divergences with the planes has far going consequences. First, it gives the simple and intuitive geometrical criterion of non-overlapping rapidity divergences, namely, the corresponding planes should not intersect. Second, it allows the conversion of rapidity divergences to UV divergences by a conformal transformation. In sec.\[sec:Cnn\] we construct the conformal transformation which maps the distant transverse plane to a point, and demonstrate the transition of divergences. The equivalence of rapidity divergences and UV divergences leads to the renormalization theorem for rapidity divergences (RTRD).
The relation between rapidity and UV divergences and RTRD have multiple consequences. The most important one is the factorization of rapidity divergences for soft factors. In the case of TMD factorization this statement is known, and thus, RTRD brings a little new, apart of some formality. However, it is novel for the double-parton scattering and MPS. The relation between different kind of divergences allows to relate the corresponding anomalous dimensions. In our case, it gives the correspondence between the soft anomalous dimension (SAD) and the rapidity anomalous dimension (RAD), which has been discovered in [@Vladimirov:2016dll]. The RTRD formulated in this article has a number of limitations. In particular, it is formulated for the soft factors that could be presented as a single T-ordered operator. Such soft factors arise in the processes with DY kinematics or annihilation kinematics. The status of factorization for processes with timelike separation is not clear. However, we show that the SIDIS TMD soft factor is equal to DY TMD soft factor, which was expected for a long time.
The structure of the paper is following. In sec.\[sec:notation\] we collect all necessary notation. In sec.\[sec:MPS\_SF\] we present the MPS soft factor, which is the main object of discussion. In particular, its all-order color structure and the explicit expression up to the three-loop order are given in sec.\[sec:color\]. The derivation of this result is given in the appendix \[app:color\]. The factorization of rapidity divergences at the fixed order (two-loop) is presented in sec.\[sec:N=2\] and sec.\[sec:N=4\]. In sec.\[sec:rap\_div\] we explore the origin of rapidity divergences on the level of Feynman diagrams. We start from the classification of divergences in the one-loop example in the position space in sec.\[sec:1loop\]. In sec.\[sec:geom\_rap\_div\] we discuss a general case and associate the rapidity divergent parts with a particular spatial configuration. Namely, we show that the gluons radiated to/by the transverse (to a given lightlike direction) plane positioned at the infinity, produce rapidity divergences. In sec.\[sec:2loop\] we illustrate the general statement by two-loop examples and present the graphical counting rules for rapidity divergences, which appears to be topologically similar to counting rules for UV divergences. Section \[sec:RTRD\_gen\] is devoted to the formulation and the proof of RTRD. In particular, in sec.\[sec:Cnn\] we introduce the transformation $C_{n\bar n}$ which distinguishes the rapidity divergences, and in sec.\[sec:RTRD\] we prove the theorem in a conformal theory and QCD. In sec.\[sec:consiquences\], some consequences, and applications of the theorem are presented. We discuss the definition of multi-parton distributions (which include the TMD distributions and double-parton distributions as particular cases) in sec.\[sec:MPS\_fac\]. The universality of TMD soft factor for DY and SIDIS process is proven in sec. \[sec:TMD\_UNIVERSAL\]. Finally, we discuss the correspondence between the soft anomalous dimension and the rapidity anomalous dimension and derive the three-loop rapidity anomalous dimension for TMD and MPS cases in sec.\[sec:corespondance\]. Some additional materials are collected in the set of appendices.
Notation and definitions {#sec:notation}
========================
In the most part of the paper, we discuss the kinematics with two selected lightlike directions. Conventionally, we denote these directions as $n$ and $\bar n$, with $n^2=\bar n^2=0,\qquad (n\cdot \bar n)=1$. The decomposition of a vector over light-cone components is defined as $$\begin{aligned}
x^\mu=\bar n^\mu x^++ n^\mu x^-+x_\perp^\mu.\end{aligned}$$ Consequently, the components of the vector $x$ are $$x^+=(n \cdot x),\qquad x^-=(\bar n \cdot x),\qquad (n \cdot x_\perp)=(\bar n\cdot x_\perp)=0,$$ and the scalar product is $$\begin{aligned}
(x\cdot y)=x^+ y^-+x^-y^++(x_\perp\cdot y_\perp),\end{aligned}$$ i.e. subscript $\perp$ denotes the transverse part in the Minkowski space ($x_\perp^2<0$).
Throughout the text, we use the color matrix notation, see e.g.[@Catani:1996vz; @Beneke:2009rj; @Gardi:2009qi]. Namely, we use the bold font for the color-matrices, and multi-matrices, i.e. for objects with two sets of color indices. The color vectors, i.e. the objects with one set of color indices are written in a usual font. The convolution between such objects is denoted by $\times$-symbol, e.g. $A_{a_1 a_2}B^{a_1a_2,b_1b_2}=A\times \mathbf{B}$. The generators of the color gauge group are denoted by $\mathbf{T}_i^A$, where $i$ labels the gauge-group representation. If some representation sub-space is not specified, this part of a matrix is unity.
The main objects of the discussion are soft factors. By a soft factor we widely understand a vacuum matrix element of any product of Wilson lines. The Wilson line from the point $x$ to the point $y$ reads $$\begin{aligned}
\pmb{[y,x]}=P\exp{\left(}ig \int_x^y dz^\mu A_\mu^A(z)\mathbf{T}^A{\right)},\end{aligned}$$ where the path of integration is the straight line from $x$ to $y$. Under the gauge transformation Wilson lines transforms as $$\begin{aligned}
\pmb{[y,x]}\to \mathbf{U}(y)\pmb{[y,x]}\mathbf{U}^\dagger (x).\end{aligned}$$ The group representation of the Wilson line is carried solely by the generator. For example, it implies that quark and anti-quark Wilson lines differ only by the color representation (fundamental and anti-fundamental), but not by the path.
A typical soft factor that arises in the factorization theorems, is build of half-infinite Wilson lines, which are specified by the direction and the initial point. The half-infinite Wilson line that is rooted at the position $x$ and points in the direction $v$ is denoted as $$\begin{aligned}
\mathbf{\Phi}_v(x)=\pmb{[v\infty+x,x]}=P\exp{\left(}ig \int_0^\infty d\sigma v^\mu A_\mu^A(v\sigma+x)\mathbf{T}^A{\right)}.\end{aligned}$$ The half-infinite Wilson line pointing in the opposite direction is $$\begin{aligned}
\mathbf{\Phi}_{-v}(x)=\pmb{[-v\infty+x,x]}=P\exp{\left(}ig \int_0^{-\infty} d\sigma v^\mu A_\mu^A(v\sigma+x)\mathbf{T}^A{\right)}.\end{aligned}$$
In the most part of the article, the discussion is not restricted to any rapidity regularization. However, for the demonstrations of particular expressions we use the $\delta$-regularization. The synopsis of $\delta$-regularization and some of its properties are given in appendix \[app:delta-reg\].
MPS soft factor {#sec:MPS_SF}
===============
The starting and the main object of our analysis is the soft factor of multi-Drell-Yan (multi-DY) process. Such a soft factor would appear in the description of hadron-hadron collision with the inclusive production of multiple heavy electro-weak bosons, e.g. $h_1+h_2\to Z_1+...+Z_N+X$ with the momenta of $Z$-bosons $Q_i\gg \Lambda_{QCD}$. The factorization theorem for multi-DY process contains many terms and various kinds of contributions. In particular, we are interested in the contribution which corresponds to the so-called multi-parton-scattering (MPS) subprocess. The MPS is characterized by the vector boson production by uncorrelated pairs of partons. The detailed discussion on these processes and possibilities to study them practically can be found in refs.[@Diehl:2011yj; @Manohar:2012jr; @Diehl:2015bca]. For our discussion, the multi-DY process is interesting as a generalization of the DY TMD factorization. It preserves the general structure of the factorization theorem and suffers from the same problem, namely the mix of rapidity divergences. The factorization of the MPS contribution of factorization theorem is discussed in sec.\[sec:MPS\_fac\].
The soft factor for multi-DY process reads [@Diehl:2011yj] (in the following we call it MPS soft factor for shortness) $$\begin{aligned}
\label{SF:MPS_Tordered}
\mathbf{\Sigma}(\{b\})=\langle 0|\bar T\{[\mathbf{\Phi}_{-n}\mathbf{\Phi}^\dagger_{-\bar n}](b_N)\dots\}T\{\dots[\mathbf{\Phi}_{-n}\mathbf{\Phi}^\dagger_{-\bar n}](b_1)\} |0\rangle,\end{aligned}$$ where Wilson lines inside square brackets belong to the same color-representation and thus contracted by the internal index, and vectors $b$ have only transverse components, i.e. $b^+_i=b^-_i=0$. We stress that the MPS soft factor is a multi-matrix in the color space. To clarify the notation we write this expression with all color indices explicit $$\begin{aligned}
\label{def:MPS_SF_openI}
\Sigma^{\{a_N...a_1\},\{d_N...d_1\}}(\{b\})=\langle 0|\bar T\{[\Phi^{a_Nc_N}_{-n}\Phi^{\dagger c_N d_N}_{-\bar n}](b_N)\dots\}T\{\dots[\Phi^{a_1c_1}_{-n}\Phi^{\dagger c_1d_1}_{-\bar n}](b_1)\} |0\rangle.\end{aligned}$$ The MPS soft factor can be visualized as a set of lightlike cusps located at the transverse plane, as it is shown in fig.\[fig:MPS+TMD\_SF\].
Only the color-singlet components of the soft factor matrix contribute to the factorization theorem. Generally, one can build the vector $C_K$ that selects the $K$’th singlet component. The complete set of vectors $C_K$ can be normalized and orthogonalized: $C^T_M\times C_N=\delta_{MN}$. The physically relevant part of the MPS soft factor reads $$\begin{aligned}
\Sigma_{MN}(\{b\})=C^T_M\times \mathbf{\Sigma}(\{b\})\times C_N.\end{aligned}$$ Only these components of the MPS soft factor are gauge-invariant, and IR-finite. Within the color-matrix notation the singlet components can be selected out by requiring (for the origin of this equation see e.g. [@Catani:1996vz; @Beneke:2009rj]) $$\begin{aligned}
\label{def:colorless}
\sum_{i=1}^N\mathbf{T}^A_i=0.\end{aligned}$$ In the following, we consider only physical components, which pick out by the relation (\[def:colorless\]).
![\[fig:MPS+TMD\_SF\] Visualization of the expression for the MPS soft factor (\[def:MPS\_SF\_openI\]) (left) and the TMD soft factor (right). The lines with arrows represent the Wilson lines with the color-flow in the direction of the arrow. The purple letters show the color indices, the black letters show the transverse positions and directions.](Figures/MDY_SF.pdf "fig:"){width="50.00000%"} ![\[fig:MPS+TMD\_SF\] Visualization of the expression for the MPS soft factor (\[def:MPS\_SF\_openI\]) (left) and the TMD soft factor (right). The lines with arrows represent the Wilson lines with the color-flow in the direction of the arrow. The purple letters show the color indices, the black letters show the transverse positions and directions.](Figures/TMD_SF.pdf "fig:"){width="41.00000%"}
To our best knowledge the MPS soft factor has been never considered in the literature (some discussion can be found in ref.[@Diehl:2011yj]). In the following subsections, we discuss some important cases and properties of MPS soft factor, including the all-order color structure and the explicit NNLO expression, which are presented here for the first time.
T-ordering
----------
The soft factors for the DY-like kinematics have Wilson lines settled on the past-light-cone. It has an important consequence, which makes possible the following analysis. Namely, all Wilson lines within the soft factor operator can be set under the single T-ordering. I.e. the expression (\[SF:MPS\_Tordered\]) can be written as $$\begin{aligned}
\label{SF:MPS}
\mathbf{\Sigma}(\{b\})=\langle 0|T\{[\mathbf{\Phi}_{-n}\mathbf{\Phi}^\dagger_{-\bar n}](b_N)\dots[\mathbf{\Phi}_{-n}\mathbf{\Phi}^\dagger_{-\bar n}](b_1)\} |0\rangle.\end{aligned}$$ It can be demonstrated as follows: (i) The distances between points of any two Wilson lines are spacelike. And hence, the T- and anti-T-orderings can be ignored due to the causality condition. (ii) The path ordering of a Wilson line overrides[^1] the anti-T ordering. Therefore, all Wilson lines can be T-ordered. (iii) Finally, using the causality we collect all Wilson lines under the single T-ordering.
The overall T-ordering of the operator is important for future discussion. It is not the general property for soft factors. For example, in the SIDIS kinematics, soft factors are built from $[\mathbf{\Phi}_{-n}\mathbf{\Phi}^\dagger_{\bar n}]$-cusps. In this case, not all distances are spacelike. Hence, the T-ordering cannot be eliminated. Another important example is the $e^+e^-$-annihilation, where soft factors are composed from $[\mathbf{\Phi}_{n}\mathbf{\Phi}^\dagger_{\bar n}]$-cusps, and can be presented as a single T-product.
Particular case $N=2$: the TMD soft factor {#sec:N=2}
------------------------------------------
The MPS soft factor at $N=2$ reduces to the TMD soft factor for the DY process, see e.g.[@Bauer:2001yt; @Becher:2010tm; @GarciaEchevarria:2011rb; @Collins:2011zzd; @Echevarria:2015byo]. In $N=2$ case, the color-neutrality condition (\[def:colorless\]) relates the generators of the first and the second Wilson lines as $\mathbf{T}_1^A=-\mathbf{T}_2^A$. The matrix $\mathbf{\Sigma}$ has only single colorless entry $\sim\delta^{a_1a_2}/\text{dim}_1=I_{\mathbf{1}}$. Projecting the singlet contribution we obtain $$\begin{aligned}
\label{SF:TMD}
\Sigma_{\text{TMD}}(b)=I_{\mathbf{1}}\times \mathbf{\Sigma}_{N=2}(b)\times I_{\mathbf{1}} = \frac{1}{\text{dim}_1}
\langle 0|\bar T\{\Phi^{dc_2}_{-n}(b)\Phi^{\dagger c_2a}_{-\bar n}(b)\}T\{\Phi^{ac_1}_{-n}(0)\Phi^{\dagger c_1d}_{-\bar n}(0)\} |0\rangle,\end{aligned}$$ where one of the vectors $b$ is eliminated by the translation invariance. To derive this relation we have used the relation $$\begin{aligned}
\mathbf{\Phi}_v(x)[-\mathbf{T}]=\mathbf{\Phi}^*_v(x)={\left(}\mathbf{\Phi}^\dagger_v(x){\right)}^T.\end{aligned}$$ The visualization of the expression (\[SF:TMD\]) is given in fig.\[fig:MPS+TMD\_SF\].
The TMD soft factor is a Wilson loop. Therefore, the non-Abelian exponentiation theorem [@Gatheral:1983cz; @Frenkel:1984pz] can be applied, and the soft factor takes the form $$\begin{aligned}
\label{SF:TMD->sigma}
\Sigma_{\text{TMD}}(b)=\exp{\left(}C_1a_s \sigma(b){\right)},\end{aligned}$$ where $C_1$ is the eigenvalue of the quadratic Casimir for the representation $1$, and $\sigma$ is given by the sum of the web-diagrams. The LO expression for $\sigma$ in the $\delta$-regularization reads $$\begin{aligned}
\label{SF:1loop}
\sigma^{[0]}=-4\Gamma(-\epsilon)(\mu^2 B)^\epsilon(L_\delta-\psi(-\epsilon)-\gamma_E),\end{aligned}$$ where $a_s=g^2/(4\pi)^2$, $B=b^2/4 e^{-2\gamma_E},$ and $ L_\delta=\ln(\delta^+\delta^- B)$. The parameters $\delta^+$ and $\delta^-$ regularize the rapidity divergences which arise due to the interaction with Wilson lines $\Phi_{-n}$ and $\Phi_{-\bar n}$ correspondingly. Obviously, the rapidity divergences belonging to different sectors can be split into separate functions by presentation of $\ln(\delta^+\delta^-)$ as $\ln\delta^++\ln\delta^-$.
The explicit calculations performed in different regularizations [@Echevarria:2015byo; @Luebbert:2016itl; @Li:2016axz], demonstrate that the linearity of the TMD soft factor in $L_\delta$ (or corresponding rapidity divergent function) holds at NLO as well. I.e. $$\begin{aligned}
\label{TMD:structure}
\sigma(b)=A(b,\epsilon)L_\delta+B(b,\epsilon),\end{aligned}$$ where $A$ and $B$ are known up to $a_s^2$-order. As it is discussed in the introduction, the status of this formula at higher orders is not clear. However, the expression (\[TMD:structure\]) is expected to hold at all orders of the perturbation theory. In particular, it holds at the leading order of large-$N_f$ expansion [@Scimemi:2016ffw]. Using the representation (\[TMD:structure\]), the TMD soft factor can be written in the form $$\begin{aligned}
\label{TMD:FAC}
\Sigma_{\text{TMD}}(b,\delta^+,\delta^-)=\sqrt{\Sigma_{\text{TMD}}(b,\delta^+,\delta^+)}\sqrt{\Sigma_{\text{TMD}}(b,\delta^-,\delta^-)}.\end{aligned}$$ This relation is the foundation for the TMD factorization and the definition of TMD distributions.
Particular case $N=4$: DPS soft factor {#sec:N=4}
--------------------------------------
The $N=4$ case describes the double-patron-scattering (DPS) process. The details on the factorization theorems for this process can be found in [@Diehl:2011yj; @Manohar:2012jr; @Diehl:2015bca; @Vladimirov:2016qkd]. There are many possible configurations for $N=4$ soft factors, which correspond to different parton content of the scattering subprocess. For the demonstration purpose, we present the simplest case of two quarks and two anti-quarks, which already gives six possibilities for the color flow. For definiteness, we present here the combination of {fundamental, anti-fundamental, fundamental, anti-fundamental} Wilson lines, that corresponds to {quark,anti-quark,quark,anti-quark} scattering or the contribution of double-parton distributions $F_{q\bar q}$. Such a composition is projected to the singlets by two vectors $$\begin{aligned}
\label{DPD:proj}
I_{\bf{1}}=\frac{\delta_{a_1a_4}\delta_{a_2a_3}}{N_c^2},\qquad I_{\bf{8}}=\frac{2t_{a_1a_4}^{A}t^A_{a_3a_2}}{N_c\sqrt{N_c^2-1}}.\end{aligned}$$ Therefore, the DPD soft factor is the two-by-two matrix. Practically, it is convenient to present it in the following form [@Vladimirov:2016qkd] $$\begin{aligned}
\label{DPD:SFmatrix}
\Sigma_{\text{DPD}}(\{b\})&=&{\left(}\begin{array}{cc}
\Sigma^{\mathbf{11}}_{\text{DPD}}(\{b\})& \Sigma^{\mathbf{18}}_{\text{DPD}}(\{b\})
\\
\Sigma^{\mathbf{81}}_{\text{DPD}}(\{b\})& \Sigma^{\mathbf{88}}_{\text{DPD}}(\{b\})
\end{array}
{\right)}\\{\nonumber}&=&
\frac{1}{N_c^2}{\left(}\begin{array}{cc}
\Sigma^{[2]}_{\text{DPD}}(b_{1,3,4,2})&
\frac{N_c\Sigma^{[1]}_{\text{DPD}}(b_{1,2,3,4})-\Sigma^{[2]}_{\text{DPD}}(b_{1,2,3,4})}{\sqrt{N_c^2-1}}
\\
\frac{N_c\Sigma^{[1]}_{\text{DPD}}(b_{1,2,3,4})-\Sigma^{[2]}_{\text{DPD}}(b_{1,2,3,4})}{\sqrt{N_c^2-1}} &
\frac{N_c^2\Sigma^{[2]}_{\text{DPD}}(b_{1,4,3,2})-2N_c\Sigma^{[1]}_{\text{DPD}}(b_{1,2,3,4})+\Sigma^{[2]}_{\text{DPD}}(b_{1,2,3,4})}{N_c^2-1}
\end{array}{\right)},\end{aligned}$$ where arguments $b_{i,j,k,l}$ are short notation for $(b_i,b_j,b_k,b_l)$, and $\Sigma^{\mathbf{ij}}_{\text{DPD}}(\{b\})=I_{\mathbf{i}}\times\mathbf{\Sigma}_{N=2}(\{b\})\times I_{\mathbf{i}}$. The soft factors $\Sigma^{[1]}_{\text{DPD}}$ and $\Sigma^{[2]}_{\text{DPD}}$ are soft factors with Wilson lines connected into single and double color loop, see fig.\[fig:DPD\_SF\].
![\[fig:DPD\_SF\] Visualization of $\Sigma^{[1]}_{\text{DPD}}$ (left), which has a topology of the single Wilson loop, and $\Sigma^{[2]}_{\text{DPD}}$(right), which has topology of the double Wilson loop.](Figures/DDY_SF.pdf "fig:"){width="45.00000%"} ![\[fig:DPD\_SF\] Visualization of $\Sigma^{[1]}_{\text{DPD}}$ (left), which has a topology of the single Wilson loop, and $\Sigma^{[2]}_{\text{DPD}}$(right), which has topology of the double Wilson loop.](Figures/DDY_SF_doubleLoop.pdf "fig:"){width="45.00000%"}
The explicit calculation of the DPS soft factor at NNLO has been made in [@Vladimirov:2016qkd]. It has been shown that DPS soft factor has a number of peculiarities. The most important one is the exact cancellation of the three-Wilson line interactions. Due to this cancellation the NNLO soft factor can be expresed via the TMD soft factor only. The result is not entirely trivial. The single-loop and double-loop components are $$\begin{aligned}
\label{DPD:NNLO1}
&&\ln\Sigma^{[1]}_{\text{DPD}}(\{b\})=a_sC_F{\left(}\sigma(b_{12})-\sigma(b_{13})+\sigma(b_{14})+\sigma(b_{23})-\sigma(b_{24})+\sigma(b_{34}){\right)}\\{\nonumber}&&\qquad\quad+a_s^2\frac{C_A C_F}{4}{\left(}\sigma(b_{13})-\sigma(b_{14})-\sigma(b_{23})+\sigma(b_{24}){\right)}{\left(}\sigma(b_{12})-\sigma(b_{13})-\sigma(b_{24})+\sigma(b_{34}){\right)}+\mathcal{O}(a_s^3),
\\\label{DPD:NNLO2}
&& \ln\Sigma_{\text{DPD}}^{[2]}(\{b\})=a_sC_F{\left(}\sigma(b_{14})+\sigma(b_{23}){\right)}\\{\nonumber}&&\qquad\quad-a_s^2\frac{C_F}{2}{\left(}C_F-\frac{C_A}{2}{\right)}{\left(}\sigma(b_{12})-\sigma(b_{13})-\sigma(b_{24})+\sigma(b_{34}){\right)}^2+\mathcal{O}(a_s^3),\end{aligned}$$ where $\sigma(b)$ is the logarithm of the TMD soft factor (\[SF:TMD->sigma\]), and $b_{ij}=b_i-b_j$. One can see that these components have double rapidity logarithms which do not cancel.
Combining the expression for components (\[DPD:NNLO1\]-\[DPD:NNLO2\]) into the matrix of the DPS soft factor (\[DPD:SFmatrix\]) one obtains a complicated expression. However, this expression can be presented in the form $$\begin{aligned}
\label{DPD:FAC}
\Sigma_{\text{DPD}}(\{b\},\delta^+,\delta^-)=s^T(\{b\},\delta^+)s(\{b\},\delta^-),\end{aligned}$$ where $s^T$ is the transposed matrix $s$. The expression for matrices $s$ is cumbersome (see [@Vladimirov:2016qkd]) but it has a general form $$\begin{aligned}
\label{DPD:small_s}
s(\{b\},\delta)=\exp{\left(}\mathcal{A}(\{b\})L_{\delta}+\mathcal{B}(\{b\}){\right)},\end{aligned}$$ where the $2\times 2$ matrices $\mathcal{A}$ and $\mathcal{B}$ are composed from functions $A$ and $B$ defined in (\[TMD:structure\]).
The decomposition (\[DPD:FAC\]) is the matrix generalization of the TMD decomposition formula (\[TMD:FAC\]). It defines the finite double parton distribution (DPD) in the very same manner as the decomposition (\[TMD:FAC\]) defines TMD distributions (see details in sec.\[sec:MPS\_fac\]). The main difference between (\[TMD:FAC\]) and (\[DPD:FAC\]) is the matrix structure, which leads to the matrix rapidity evolution equation for DPDs. In the next section, we demonstrate the generalization of this expression for the MPS soft factor. However, we also demonstrate that at the three-loop level the new types of terms appear, that do not reduce to the TMD soft factors. The analysis of these terms is difficult, and their rapidity divergences structure is unknown.
Color structure {#sec:color}
---------------
The MPS soft factor has reach color structure. Its expression is greatly simplified in the color matrix notation. In the appendix \[app:color\] we present the detailed evaluation of the MPS soft factor in the terms of the generating functions for web-diagrams [@Vladimirov:2015fea; @Vladimirov:2014wga]. Such decomposition extracts the color structure explicitly, and reveals the common structure of diagrams. In this section, we present the final result of the decomposition.
The first and the most important observation on the color structure of MPS soft factor follows from the rotation invariance. Performing the rotation that interchange $n\leftrightarrow \bar n$, we obtain $$\begin{aligned}
\mathbf{\Sigma}(\{b\})=\mathbf{\Sigma}^\dagger(\{b\}).\end{aligned}$$ For the generator of the color group this transformation acts as $\mathbf{T}_i\to -\mathbf{T}_i$. Therefore, the terms with the odd-number of color-generators vanish. This statement also holds for the exponentiated expression (see (\[app:odd=0\]) and the discussion around). This observation describes the absence of the three-Wilson lines interaction terms in the DPS soft factor (\[DPD:NNLO1\]-\[DPD:NNLO2\]), which has been shown on the level of diagrams in [@Vladimirov:2016qkd]. The general all-order structure of MPS soft factor reads $$\begin{aligned}
\label{MPS:all_order_color}
\mathbf{\Sigma}(\{b\})=\exp{\left(}\sum_{\substack{n=2\\n\in\text{even}}}^{\infty}a_s^{n/2}\sum_{i_1,...,i_n=1}^N\{\mathbf{T}^{A_1}_{i_1}...\mathbf{T}^{A_n}_{i_n}\}\sigma^{n;i_1...i_n}_{A_1...A_n}(\{b\}){\right)},\end{aligned}$$ where curly brackets denote the symmetrization over the color generators belonging to the same Wilson lines. The functions $\sigma^n\sim \mathcal{O}(a^0_s)$ obey the same symmetry pattern under permutation of labels $i$ and $A$ as the color structure. Note, that all matrices belonging to the same Wilson lines appear in the symmetric combinations. The anti-symmetric combinations are absent due to the algebra. The $n=0$ term is rewritten as $n=2$ contribution using the color-conservation.
The number of independent color components grows rapidly with order. However, their number is finite at given $N$, thanks to color-algebra and color-conservation condition (\[def:colorless\]). In particular, at the three-loop level there are three independent structures $$\begin{aligned}
\label{SF:color_dec}
\mathbf{\Sigma}&=&\exp\Bigg[-a_s \sum_{[i,j]}\mathbf{T}_i^A\mathbf{T}_j^A \sigma(b_{ij})
\\{\nonumber}&&
+a_s^3 \Big(\sum_{[i,j,k]}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^C\mathbf{T}_k^D if^{AC;BD}Y_{4}^{ijk}
+\sum_{[i,j,k,l]}\mathbf{T}_i^{A}\mathbf{T}_j^B\mathbf{T}_k^C\mathbf{T}_l^D if^{AC;BD}X_{4}^{ijkl}\Big)+\mathcal{O}(a_s^4)\Bigg],\end{aligned}$$ where $if^{AC;BD}=if^{AC\alpha}if^{\alpha BD}$, with $f^{ABC}$ being the structure constant, $b_{ij}=b_i-b_j$, and $\mathbf{T}_i^{\{AB\}}=\{\mathbf{T}_i^A,\mathbf{T}_i^B\}/2$. Here, the summation runs from $1$ to $N$ for each summation label with no label equals to any other label, which is denoted by the square brackets. Functions $\sigma$, $Y_4$, and $X_4$ contain all orders of perturbation series starting from the LO. Their explicit form in the terms of generating function is given in section \[app:results\]. The expression (\[SF:color\_dec\]) is simpler then the expressions (\[DPD:SFmatrix\]-\[DPD:NNLO2\]), to which they turn after application of projectors (\[DPD:proj\]).
The color-dipole term in the decomposition (\[SF:color\_dec\]) is proportional to the TMD soft factor, which can be checked by setting $N=2$. Assuming the linearity of $\sigma$ in $\ln(\delta^+\delta^-)$ at all-orders of the perturbation theory and also the linearity of $X_4$ and $Y_4$ in $\ln(\delta^+\delta^-)$ we can present this expression in the factorized form (\[DPD:FAC\]) as well (up to terms $\sim a_s^4$).
Divergences of soft factors {#sec:rap_div}
===========================
The soft factors with lightlike Wilson lines are utterly singular objects. Diagram-by-diagram there are UV-, IR-, and rapidity divergences. To define the soft factor completely, a sufficient set of regulators should be introduced. Typically, it includes the dimensional regularization for UV- and IR-divergences, and an extra regulator for the rapidity divergences. In this section we discuss the diagrammatic origin of the rapidity divergences, and show their similarity to the UV divergences.
Divergences of soft factor at one loop {#sec:1loop}
--------------------------------------
To begin with let us consider the LO contribution to the interaction of Wilson lines. At LO there could be many diagrams (depending on the structure of the soft factor, the gauge conditions and calculation technique). However, there is a single loop-integral that appears at this order. This integral describes the single-gluon exchange between Wilson lines $\Phi_{v_i}(b_i)$ and $\Phi_{v_j}(b_j)$ (for the demonstration purpose we keep vectors $v$ and $b$ unrestricted). In the coordinate representation, it reads $$\begin{aligned}
\label{1loop:1}
I_{ij}&=&
a_s 2^{2-2\epsilon}\Gamma(1-\epsilon)\int_0^\infty d\sigma_1 d\sigma_2\frac{(v_i\cdot v_j)}{(-(v_i\sigma_1+b_i-v_j\sigma_2-b_j)^2+i0)^{1-\epsilon}},\end{aligned}$$ where $a_s=g^2/(4\pi)^{2-\epsilon}$. We use the dimensional regularization with $d=4-2\epsilon$, and do not specify the rapidity divergence regulator.
In the expression (\[1loop:1\]) the variables $\sigma$ represent the distances of gluon radiation/absorption along Wilson line. It is convenient to change the variables as $\sigma_1=\alpha L$ and $\sigma_2=\alpha^{-1} L$. In these terms, the variable $L$ represents the general “size” of the loop, and the variable $\alpha$ represents the $n/\bar n$-asymmetry of the gluon positioning. The integral (\[1loop:1\]) takes the form $$\begin{aligned}
&&I_{ij}=
a_s 2^{2-2\epsilon}\Gamma(1-\epsilon)\int_0^\infty dL \int_0^\infty d\alpha\frac{2L}{\alpha}
\\{\nonumber}&&\quad\frac{(v_i\cdot v_j)}{(-(v_i^2\alpha^2-2(v_i\cdot v_j)+v_j^2\alpha^{-2})L^2-2(v_i \alpha -v_j \alpha^{-1})\cdot (b_i-b_j)L-(b_i-b_j)^2+i0)^{1-\epsilon}}.\end{aligned}$$ Let us sort the singularities of this integral, and depict the corresponding space configurations. Starting from the obvious:
- *UV divergence*. In the case $b_i=b_j$, there is UV singularity at $L\to 0$. The integral behaves as $I\sim L^{-1+2\epsilon}$, and is regularized by $\epsilon>0$. The UV divergence is a subject of the usual renormalization procedure.
- *IR divergence*. For any configuration one has IR singularity at $L\to \infty$. The integral behaves as $I\sim L^{-1+2\epsilon}$, and is regularized by $\epsilon<0$. For color singlet configurations the IR singularities cancel in the sum of diagrams. At LO the cancellation of IR-singularities is evident. Indeed, in the limit $L\to\infty$ vectors $b$ drop from the integral, and thus all IR-divergent integrals are equivalent. The proof of the cancellation at arbitrary perturbative order is given in the appendix \[sec:delta-structure\].
- *Rapidity divergence*. In the special case, $v_i^2=v_j^2=0$ and $v_i\cdot (b_i-b_j)=v_j\cdot (b_i-b_j)=0$ the integral over $\alpha$ decouples from the integral over $L$, $$\begin{aligned}
\label{rapdiv:1-loop}
I_{ij}&=&a_s 2^{2-2\epsilon}\Gamma(1-\epsilon) \int_0^\infty d L
\frac{2L(v_i\cdot v_j)}{(2(v_i\cdot v_j)L^2-(b_i-b_j)^2+i0)^{1-\epsilon}}\int_0^\infty \frac{d\alpha}{\alpha}.\end{aligned}$$ The integral over $\alpha$ is logarithmically divergent at both limits $\alpha \to0$ and $\alpha\to \infty$. Such singularity is called the rapidity divergence.
The visual representation of the divergent configurations for the case of TMD soft factor is shown in fig.\[fig:divergences\].
The rapidity divergences are present even if a single vector $v_i$ is lightlike and orthogonal to the rooting plane, i.e. $v_i^2=0$ and $v_i\cdot (b_i-b_j)=0$ (and the second vector $v_j$ is arbitrary). In this case, the integral is regular at $\alpha\to0 $, but divergent at $\alpha\to \infty$. Moreover, the coefficient of this divergence is just the same as in (\[rapdiv:1-loop\]).
![\[fig:divergences\] Divergent configurations of the TMD soft factor at one-loop. The arrows indicate the direction in which the position of the particle should be limited.](Figures/div_example.pdf){width="50.00000%"}
Spatial structure of rapidity divergences {#sec:geom_rap_div}
-----------------------------------------
In the one-loop example, the rapidity divergence arises from the integration over the half-infinite path of lightlike Wilson lines. Let us demonstrate that it is a general feature, and the rapidity divergence can arise for each coupling of the gluon to $\mathbf{\Phi}$. Note, that it is difficult to present the strict definition of rapidity divergences, because they are related to a particular component of gluon fields, and therefore, depend on the gauge fixation condition. In the following, we use the Feynman gauge for simplicity.
A general diagram with a single gluon radiated by $\mathbf{\Phi}_{v}(b)$ ($v^2=0$) has the following form in the coordinate representation $$\begin{aligned}
I^{[1]}&=&\int_0^\infty d\sigma \int d^d y \frac{1}{(-(v\sigma+b-y)^2+i0)^{p}}F(y),\end{aligned}$$ where $p$ is the power of propagator that connects the Wilson line with the rest of the diagram which is denoted by $F(y)$. The function $F(y)$ can have its own divergences which are not interesting in the current context. For a given lightlike vector $v$, we introduce the decomposition $$\begin{aligned}
\label{rapdiv:y_decomp}
y^\mu=v^\mu y_s+s^\mu y_v+y_\perp^\mu,\end{aligned}$$ where $(v\cdot y_\perp)=0$. Without loss of generality we can set $(v\cdot s)=1$. The components $y_v$, $y_s$, and $y_\perp$ are independent, and $d^dy=dy_vdy_sd^{d-2}y_\perp$. Rescaling variables $$y_v\to \frac{y_v}{\sigma}+(v\cdot b),$$ we obtain $$\begin{aligned}
\label{rapdiv:ex1}
I^{[1]}&=&\int_0^\infty \frac{d\sigma}{\sigma}\int d y_sdy_v d^{d-2}y_\perp
\\{\nonumber}&&\frac{F(y_s,y_v/\sigma+(v\cdot b),y_\perp)}{{\left[}2 y_v-
{\left(}b-y_\perp-(v\cdot b){\right)}^2-\frac{2y_v}{\sigma}{\left(}y_s+(s\cdot y_\perp)+s^2(v\cdot b)-(s\cdot b){\right)}-\frac{y_v^2 s^2}{\sigma^2}+i0{\right]}^{p}},\end{aligned}$$ Here, the rapidity divergence appears in the limit $\sigma\to \infty$, where the expression (\[rapdiv:ex1\]) takes the form $$\begin{aligned}
I^{[1]}_{\text{rap.div.}}&=&\int^\infty \frac{d\sigma}{\sigma}\int d y_sdy_v d^{d-2}y_\perp
\frac{F(y_s,(v\cdot b),y_\perp)}{{\left[}2 y_v-
{\left(}b-y_\perp-(v\cdot b){\right)}^2+i0{\right]}^{p}}.\end{aligned}$$ Note, that the divergent factor decouples from the rest of the diagram. Such configuration corresponds to the radiation of a gluon from the *transverse* to $v^\mu$ plane to the far end of the Wilson line $\mathbf{\Phi}_v$.
If there are several gluons coupled to the Wilson line $\mathbf{\Phi}_v$ we can perform the rescaling for each coupled coordinate $y_i$ and obtain the rapidity divergent configurations. The power of rapidity divergence is at most equal to the number of gluons coupled to $\mathbf{\Phi}$’s. We should also take into account that the coupling of gluons to a Wilson line is ordered, e.g. for three coupled gluons we have the integral $\int^\infty d\sigma_1\int^{\sigma_1} d\sigma_2\int^{\sigma_2} d\sigma_3$. Thus the limits $\sigma_i\to \infty$ should be taken in the same order, which however could be impossible due to the internal structure of the function $F$. In particular, such situation appears if the coordinate $y$ coupled to another Wilson lines (see the examples given in the next section).
To summarize the geometry of rapidity divergent configuration, we introduce special notation. Let us denote by $(v)_\perp^y$ the two-dimensional (or $(d-2)$-dimensional) plane which is transverse to $v$ and intersects the axis $v$ at the coordinate $y$. The rapidity divergences arise in the configuration with the gluon is radiated within the plane $(v)_\perp^y$ and absorbed within the plane $\lim_{\sigma\to \infty}(v)_\perp^\sigma=(v)_\perp^\infty$. In other words, the rapidity divergences associated with the gluons that are localized in the space between $(v)_\perp^y$ and $(v)_\perp^\infty$. Since the particular value of $y$ has no sense, we can relate rapidity divergences to the plane $(v)_\perp^\infty$ for simplicity.
If there are several Wilson lines pointing in the same lightlike direction, which is the typical situation, then they share $(v)_\perp^\infty$. The rapidity divergences of this configuration are shared. They can be regularized by a single regularization parameter, and should not be distinguished. If there are several sets of Wilson lines with directions $v_i$, then there are also several planes $(v_i)_\perp^\infty$. If these planes do not intersect then the associated rapidity divergences do not overlap. In this case, they can be regularized separately (and as we show later separately renormalized). If the planes $(v_i)_\perp^\infty$ intersect then the rapidity divergences overlap and could not be separated. Fortunately, soft factors with such geometry do not appear practically. Important to note, that the definition of the transverse plane is not unique, since the vector $s^\mu$ which specifies the plane, has not unique definition.
Two loop examples and counting of rapidity divergences {#sec:2loop}
------------------------------------------------------
In this section, we give some two-loop examples of rapidity divergences, and specify their counting.
![\[fig:divergences2\] Examples of two-loop diagrams studied in the text. The diagrams $A$, $C$ and $D$ has the second power of rapidity divergence which appears if the positions of vertices are limited according to arrows. The diagram $B$ has the first power of rapidity divergence, since the positions of vertices cannot be limited according to arrows successively. The other combinations of divergent limits possible.](Figures/div_example_2loop.pdf){width="83.00000%"}
As it was shown in the previous section, the overall power of the rapidity divergence in a diagram could not exceed the number of gluons attached to $\mathbf{\Phi}$’s. The maximum power of the divergences is achieved if all limits $\sigma_i\to \infty$, can be taken successively and decoupled from each other. This however is limited by the structure of the rest of the diagram. For example, it cannot be done if the divergent gluon is coupled to another Wilson line.
Let us give an example of similar diagrams which produce different power of rapidity divergences due to ordering of limits. These diagrams are shown in fig.\[fig:divergences2\] A and B, and given by similar expressions (we omit the prefactors of loop-integrals for brevity) $$\begin{aligned}
I_A&=&\int_0^\infty d\sigma_1\int_0^{\sigma_1} d\sigma_2 \int_0^\infty d\tau_1\int_0^{\tau_1} d\tau_2 \frac{1}{(2\sigma_1 \tau_2-b_{12}^2)^{1-\epsilon}
(2\sigma_2 \tau_1-b_{12}^2)^{1-\epsilon}},
\\{\nonumber}I_B&=&\int_0^\infty d\sigma_1\int_0^{\sigma_1} d\sigma_2 \int_0^\infty d\tau_1\int_0^{\tau_1} d\tau_2 \frac{1}{(2\sigma_1 \tau_1-b_{12}^2)^{1-\epsilon}
(2\sigma_2 \tau_2-b_{12}^2)^{1-\epsilon}},\end{aligned}$$ where $b_{12}$ is the transverse distance between lines. To extract the divergences associated with $\sigma\to \infty$, we rescale $\tau$ and obtain $$\begin{aligned}
I_A&=&\int_0^\infty d\tau'_1\int_0^{\frac{\sigma_1}{\sigma_2}\tau'_1} d\tau'_2 \frac{1}{(2\tau'_2-b_{12}^2)^{1-\epsilon}
(2\tau'_1-b_{12}^2)^{1-\epsilon}}\int_0^\infty \frac{d\sigma_1}{\sigma_1}\int_0^{\sigma_1} \frac{d\sigma_2}{\sigma_2},
\\{\nonumber}I_B&=& \int_0^\infty d\tau'_1\int_0^{\frac{\sigma_2}{\sigma_1}\tau'_1} d\tau'_2 \frac{1}{(2\tau'_1-b_{12}^2)^{1-\epsilon}
(2 \tau'_2-b_{12}^2)^{1-\epsilon}}\int_0^\infty \frac{d\sigma_1}{\sigma_1}\int_0^{\sigma_1} \frac{d\sigma_2}{\sigma_2}.\end{aligned}$$ In the integral $I_A$ the limit $\sigma_1\to \infty$ decouples from the limit $\sigma_2\to \infty$ and we obtain the second power of rapidity divergence, as $(\int^\infty d\sigma/\sigma)^2$. In the integral $I_B$, the limit $\sigma_1\to \infty$ neglects the expression, and thus there is only single rapidity divergence which appears if both $\sigma$’s are sent to infinity simultaneously. The visual representation of the rapidity divergent configurations is given in fig.\[fig:divergences2\]A and B.
The diagram $A$ has the overlap of rapidity divergences associated with different directions. It appears in the limit $\sigma_1\to \infty $ and $\tau_2\to \infty$, which can be taken independently. It gives the rapidity divergences in both direction, $\big(\int^\infty d\sigma/\sigma\big) \big(\int^\infty d\tau/\tau\big)$. The corresponded geometrical configuration is shown in fig.\[fig:divergences2\] A$^*$. In the $\delta$-regularization these substructures of diagram combine together into the Lorentz invariant expression $\sim \ln^2\delta^+\delta^-$ (here, $\ln^2 \delta^+$ corresponds to the double divergence in the $n$-direction, $\ln(\delta^+)\ln(\delta^-)$ to the mixed divergences and so on.)
The diagram $A$ does not contribute to the TMD soft factor. It is not a web diagram and thus, it is eliminated by the exponentiation procedure. However, in the case of the TMD soft factor, there are two other diagram topologies that give the double rapidity divergences. These diagrams are shown in fig.\[fig:divergences2\] C and D. The explicit expression for these diagrams can be found e.g. in [@Echevarria:2015byo]. These diagrams have the same leading rapidity divergent structure proportional to $B^{2\epsilon}\Gamma^2(-\epsilon)\ln^2(\delta^+\delta^-)$, in the $\delta$-regularization. These double divergences cancel in the soft factor due to the different sign of the color coefficients. Note, that the diagram $C$ is simply a square of one-loop diagrams. The diagram $D$ has a more complicated expression, which can be reduced to the product of one-loop integrals in the rapidity divergent limit. Let us mention, that to obtain the rapidity divergent configuration in the diagram $D$ one of the vertices on the Wilson lines should be sent to the origin, while another two to infinity. We do not present the derivation here and refer the reader to ref.[@Erdogan:2011yc] where a similar evaluation (with the only absence of vector $b$) is performed.
The examples that are given here confirm the general conclusion made in the previous section: The rapidity divergences are associated with gluons localized at $(v)_\perp^\infty$. To count the maximum power of rapidity divergence for a given diagram, one should draw the diagram and move the end point of a gluon attached to $\mathbf{\Phi}$ towards infinity, while the opposite side of this gluon is to be moved toward the origin (here we expect the “two-dimensional” TMD-like configuration of Wilson lines). If a gluon (or a subgraph) can be moved to the rapidity divergent limit without affecting the rest of the diagram, it decouples. The number of the vertices sent to infinities corresponds to the power of rapidity divergence. It is straightforward to show that the absolute maximum power of rapidity divergence does not exceed the number of coupling to Wilson lines, or the number of loops, whatever is smaller.
The graph-topological structure of rapidity divergences reminds the graph-topological structure of UV divergences. The only difference is that positions of gluon couplings for UV divergent subgraphs should be limited to the same point, while for rapidity divergent subgraphs they should be limited to separate transverse planes. As we discuss in the next section it is not accidental, but the result of the fundamental relation between rapidity and UV divergences. Since the rapidity divergences have the same structure of the sub-divergences, we expect that they can be iterated by the Ward identities in a similar manner as the cusp UV divergence (or UV divergence of multi-cusp for the case of the MPS soft factor). Here we again refer to the detailed calculation made in ref.[@Erdogan:2011yc], which can be nearly one-to-one repeated for the TMD soft factor. In fact, we expect that the renormalization theorem for rapidity divergences presented in the following sections can be proved in much the same way as the UV renormalization of the Wilson line cusp [@Dotsenko:1979wb; @Brandt:1981kf], i.e. by solving the chain of Ward identities.
Renormalization theorem for rapidity divergences {#sec:RTRD_gen}
================================================
Conformal transformations of soft factor {#sec:Cnn}
----------------------------------------
The rapidity divergences in many aspects resemble the UV divergences. The main difference is that the rapidity divergences are associated with the localization of gluons at the distant transverse plane $(-n)_\perp^{\infty}$, while the UV divergences are associated with the localization at a point. Let us build the conformal transformation which relates the plane $(-n)_\perp^{\infty}$ to a point (for simplicity we take the origin). It can be made by the following chain of transformations: (i) translation by $\{\frac{\lambda-1}{2 a},0^-,0_\perp\}$, (ii) special conformal transformation along the light-cone direction $n$ with the vector $\{0^+,a,0_\perp\}$ (iii) translation by $\{-(2 a)^{-1},0^-,0_\perp\}$. The resulting transformation reads $$\begin{aligned}
\mathcal{C}_{\bar n}: \{x^+,x^-,x_\perp\}\to\{\frac{-1}{2 a}\frac{1}{\lambda+2 a x^+},x^-+\frac{a x_\perp^2}{\lambda+2 a x^+},\frac{x_\perp}{\lambda+2 a x^+}\}.\end{aligned}$$ In the same manner we can build the transformation that relates the $(-\bar n)_\perp^\infty$ to the origin, $$\begin{aligned}
\mathcal{C}_{n}: \{x^+,x^-,x_\perp\}\to\{x^++\frac{\bar a x_\perp^2}{\bar \lambda+2 \bar a x^-},\frac{-1}{2 \bar a}\frac{1}{\bar \lambda+2 \bar a x^-},\frac{x_\perp}{\bar \lambda+2 \bar a x^-}\}.\end{aligned}$$ The parameters $a$ and $\lambda$ are free real parameters.
The combined transformation $$\begin{aligned}
C_{n\bar n}=\mathcal{C}_n\mathcal{C}_{\bar n}=\mathcal{C}_{\bar n}\mathcal{C}_{n},\end{aligned}$$ has a number of useful properties. The main geometric elements of the soft factor transform as $$\begin{aligned}
{\nonumber}&&C_{n\bar n}(-\bar n)_\perp^\infty = \{0^+,\frac{-1}{2\bar a \bar \lambda},0_\perp\},
\\
&&C_{n\bar n}(-\bar n)_\perp^0 = C_{n\bar n}(-\bar n)_\perp^0 = S,
\\{\nonumber}&&C_{n\bar n}(-n)_\perp^\infty = \{\frac{-1}{2 a \lambda},0^-,0_\perp\},\end{aligned}$$ where $S$ is the two-dimensional surface $$S(y)=\frac{1}{\lambda\bar \lambda-2a\bar a y_T^2}\{\frac{-\bar\lambda}{2 a},\frac{-\lambda}{2 \bar a},y_T\},$$ with $y_T$ being arbitrary two-dimensional (Euclidean) vector.
One can see that the plane $S$ is made by the intersection of two light-cones that are set at points $\{0^+,\frac{-1}{2\bar a \bar \lambda},0_\perp\}$ and $\{\frac{-1}{2 a \lambda},0^-,0_\perp\}$. The light-cones intersect by upper and lower branches, which form two disconnected branches of the surface $S$, parametrized by a single vector $y_T$. The boundary of the branch is determined by the equation $\lambda\bar \lambda=2a\bar a y_T^2$. Depending on the values of parameters $a$ and $\lambda$ the transformation realizes various configurations.
To apply the transformation to the soft factor geometry, we make the following restriction on the parameters $$\begin{aligned}
\label{Cnn:restriction}
a\lambda<0,\qquad\bar a\bar \lambda<0,\qquad (a\bar a)^2<\frac{1}{2 \rho^2_T},\end{aligned}$$ where $\rho_T$ is the traverse position of the most distant (from the origin) Wilson line, i.e. $\rho^2_T=\max\{-b_i^2\}$. Then the part of the transverse plane that contains the points $b_i$, transforms into the upper branch of the surface $S$. The paths of Wilson lines transform as $$\begin{aligned}
\label{Cnn:contour_tranform}
-\bar n \sigma+b_\perp\quad &\to& \quad \bar r+\omega \,\bar v(b_\perp),
\\{\nonumber}-n \sigma+b_\perp\quad &\to& \quad r+\omega \,v(b_\perp),\end{aligned}$$ where $0<\omega<1$. The end-points and the directions vectors are $$\begin{aligned}
\label{Cnn:vectors}
\bar v(b)&=&\frac{1}{\lambda\bar \lambda+2 a \bar a b^2}\{-\frac{\bar \lambda}{2 a},\frac{a b^2}{\bar \lambda},b\},\qquad
\bar r=\{0^+,\frac{-1}{2\bar a\bar \lambda},0_\perp\}
\\{\nonumber}v(b)&=&\frac{1}{\lambda\bar \lambda+2 a \bar a b^2}\{\frac{\bar a b^2}{\lambda},-\frac{\lambda}{2 \bar a},b\},\qquad
r=\{\frac{-1}{2 a \lambda},0^-,0_\perp\}.\end{aligned}$$ The vectors $v$ and $\bar v$ are lightlike, $v^2\bar v^2=0$. The end-points of the original $\mathbf{\Phi}$ at $\sigma\to \infty (0)$ correspond to the end-points of the new Wilson line at $\omega \to 0(1)$. Therefore, the transformation $C_{n\bar n}$ transforms straight half-infinite Wilson lines $\mathbf{\Phi}_{-n}$ and $\mathbf{\Phi}_{-\bar n}$ into the straight finite Wilson lines, $$\begin{aligned}
C_{n\bar n}\mathbf{\Phi}_{-\bar n}(b)&=&\pmb{[\bar r,S(b)]},
\\
C_{n\bar n}\mathbf{\Phi}_{-n}(b)&=&\pmb{[r,S(b)]}.\end{aligned}$$ Correspondingly, the MPS soft factor under the action of $C_{n\bar n}$ turns into the soft factor localized in the compact domain of the space-time, $$\begin{aligned}
C_{n\bar n}\mathbf{\Sigma}(\{b\})=\mathbf{\Omega}(\{v(b),\bar v(b)\}),\end{aligned}$$ where $$\begin{aligned}
\Omega^{\{a_N...a_1\},\{d_N...d_1\}}(\{v,\bar v\})=\langle 0|
T\{
{\left(}[r,S(b_N)][S(b_N),\bar r]{\right)}^{a_Nd_N}...
{\left(}[r,S(b_1)][S(b_1), \bar r]{\right)}^{a_1d_1}
\}
|0\rangle.\end{aligned}$$ The graphical representation of the transformed soft factor is given in fig.\[fig:Cnn\].
![\[fig:Cnn\] The shape of the MPS soft factor before (left) and after (right) the transformation $C_{n\bar n}$ (with restrictions (\[Cnn:restriction\])). The transverse planes at light-cone infinities transforms to the points (the correspondence is shown by the same color). The traverse plane at the light-cone origin transforms into the plane $S$ formed by intersection of light-cones that are set in the green and red points.](Figures/Cnn.pdf){width="80.00000%"}
The soft factor $\mathbf{\Omega}$ has only UV divergences. So, we conclude that the rapidity divergences of the original soft factor $\mathbf{\Sigma}$ turn into the UV divergences of $\mathbf{\Omega}$. Such transmutation of divergences is a known feature of conformal transformation, and it can be used to relate different aspects of the theory. Probably the most known example is the relation of the BK/JIMWLK kernel to the BMS kernel [@Hatta:2008st] at LO. Another example is the correspondence between rapidity and soft anomalous dimensions (which is discussed in sec.\[sec:corespondance\] in details) shown in ref.[@Vladimirov:2016dll]. In both references the used transformation is ${\mathcal{C}_{\bar n}(\lambda=0,a=2^{-1/2})}$. This transformation moves the transverse plane $(n)_\perp^0$ to the light-cone infinity. It precisely corresponds the relation between BK and BMS geometries, but rather disadvantageous for the TMD (and similar) soft factors because it locates a part of $\mathbf{\Sigma}$ at the light-cone infinity.
RTRD for Drell-Yan-like soft factors {#sec:RTRD}
------------------------------------
The renormalization theorem for rapidity divergences (RTRD) for the (color-singlet singlet entries of) DY-like MPS soft factor $\mathbf{\Sigma}(\{b\})$ reads:
*The rapidity divergences associated with different directions in the MPS soft factor can be factorized from each other. At any finite order of the perturbation theory there exist the rapidity divergence renormalization factor $\mathbf{R}_n$, which contains the rapidity singularities associated with the $(-n)_\perp^\infty$, such that the rapidity renormalized soft factor $$\begin{aligned}
\label{RTRD}
\mathbf{\Sigma}^R(\{b\},\nu^+,\nu^-)=\mathbf{R}_{n}(\{b\},\nu^+)\mathbf{\Sigma}(\{b\})\mathbf{R}^\dagger_{\bar n}(\{b\},\nu^-),\end{aligned}$$ is free from rapidity divergences.*
The variables $\nu^\pm$ in (\[RTRD\]) are the scales of the rapidity renormalization. The proof of RTRD is split into two parts. The first part is to prove RTRD in a conformal field theory. The second part is to extend it to QCD.
To prove RTRD in the conformal field theory we are going to use the relation between soft factors $\mathbf{\Omega}$ and $\mathbf{\Sigma}$. These soft factors are related by the conformal transformation $C_{n\bar n}$ and hence their color-singlet parts (or in other words gauge invariant parts) equal each other in the conformal field theory. The soft factor $\mathbf{\Omega}$ has only UV divergences at cusps and multi-cusps which can be renormalized individually. Therefore, to proof RTRD in conformal field theory, it is enough to find the correspondence between divergences of soft factors, and proof that they do not mix under the transformation $C_{n\bar n}$. Then the statement of the theorem is equivalent to the statement on the existence of the renormalization of Wilson lines [@Dotsenko:1979wb; @Brandt:1981kf].
To associate the divergences of $\mathbf{\Sigma}$ to the divergences $\mathbf{\Omega}$ we make a geometrical deformation of $\mathbf{\Sigma}$. The deformation parameter that regularizes a particular divergence in one soft factor regularizes its analog in another soft factor. Clearly, it could be cumbersome to trace the transformation of divergences on the level of the diagrams, since the conformal transformation also affects the gauge-fixation condition.
There are UV and rapidity divergences in $\mathbf{\Sigma}$. To start with, we consider the UV divergence of $\mathbf{\Sigma}$, that appears at the cusp located at $b_i$. To regularize it we perform a tiny displacement (in the transverse direction) of the end point of $\mathbf{\Phi}_{-n}(b_i)\to\mathbf{\Phi}_{-n}(b_i+\delta b)$, but leave $\mathbf{\Phi}_{-\bar n}(b_i)$ unchanged. The parameter $\delta b$ regularizes only the UV divergence at the cusp located at $b_i$, and does not affect any other divergences. In the soft factor $\mathbf{\Omega}$ it leads to the displacement of the end-point for Wilson line $\pmb{[r,S(b_i)]}\to \pmb{[r,S(b_i+\delta b)]}$, and thus regularizes the UV divergence of the cusp located at $S(b_i)$. Therefore, each cusp UV singularity of $\mathbf{\Sigma}$ maps to the cusp UV singularity of $\mathbf{\Omega}$.
To regularize the rapidity divergences, the Wilson lines $\mathbf{\Phi}_{-n}$ should be deformed[^2] away from the plane $(-n)_\perp^\infty$. There are three alternative ways to do so. For clarity we present all of them.
- *(i)* The half-infinite Wilson lines $\mathbf{\Phi}$ could be cut at a large distance $L$, preventing their intersection with $(-n)_\perp^\infty$. It corresponds to the restriction $0<\sigma<L$ in the parameterization of contours. In the transformed soft factor, this deformation turns into the restriction $cL^{-1}<\omega<1$ on the contour parameterization (\[Cnn:contour\_tranform\]), where $c$ is a constant. Therefore, the Wilson lines do not reach the point $r$ but stop at the sphere with radius $\sim L^{-1}$ which surround this point.
- *(ii)* The directions of Wilson lines can be tilt from the light-cone infinitesimally [@Collins:2011zzd]. E.g. for the Wilson lines $\mathbf{\Phi}_{-\bar n}$ we change $\bar n\to\{1^+,-\alpha,0_\perp\}$, where $\alpha\to 0$. Then the vector $\bar r$ gains the infinitesimal[^3] addition $\alpha \,\delta \bar r(b)$. Thus the Wilson lines do not intersect at the point $\bar r$.
- *(iii)* The end-points of Wilson lines can be pushed away from $(-n)_\perp^\infty$ by shifting rooting positions outside of the transverse plane ${\mathbf{\Phi}_{-n}\mathbf{\Phi}^\dagger_{-\bar n}(b_i)\to\mathbf{\Phi}_{-n}\mathbf{\Phi}^\dagger_{-\bar n}(b_i+n b_i^-+\bar n b_i^+)}$ [@Li:2016axz]. In order to prevent the formation of another $(-n)_\perp^\infty$ (with different vector $s^\mu$), all parameters $b_i^\pm$ should be different. In this case the end-points of Wilson lines in $\mathbf{\Omega}$ do not meet at $r$ and $\bar r$ but distributed along light-cone axes with coordinates $r+nb_i^-/\lambda^2$ and $\bar r+\bar n b_i^+/\bar \lambda^2$.
In all cases the Wilson lines do not join[^4] together at points $r$ and $\bar r$. Therefore, we conclude that rapidity divergences of $\mathbf{\Sigma}$ turn into the UV multi-cusp divergences of $\mathbf{\Omega}$. Moreover, the rapidity divergence associated with $(-n)_\perp^\infty$ ($(-\bar n)_\perp^\infty$) turns into the separate UV divergences at $r$ ($\bar r$).
The UV divergences of soft factor $\mathbf{\Omega}$ at points $r$ and $\bar r$ are removed by renormalization factors $\mathbf{Z}(\{v\})$ and $\mathbf{Z}^\dagger(\{\bar v\})$ independently [@Dotsenko:1979wb; @Brandt:1981kf]. In other words, $$\begin{aligned}
\label{RTRD:1}
\mathbf{\Omega}^{\text{UV-finite at $r,\bar r$}}(\{v,\bar v\},\mu,\bar \mu)=\mathbf{Z}(\{v\},\mu)\mathbf{\Omega}(\{v,\bar v\})\mathbf{Z}^\dagger(\{\bar v\},\bar\mu),\end{aligned}$$ where $\mu$ and $\bar \mu$ are renormalization scales. Applying $C^{-1}_{n\bar n}$ to the right-hand-side we transform each factor independently and obtain the correspondence $$\begin{aligned}
\label{RTRD:2}
C^{-1}_{n\bar n}{\left(}\mathbf{Z}(\{v\},\mu){\right)}=\mathbf{R}_n(\{b\},\nu^+),\qquad
C^{-1}_{n\bar n}{\left(}\mathbf{Z}^\dagger(\{\bar v\},\bar \mu){\right)}=\mathbf{R}^\dagger_{\bar n}(\{b\},\nu^-).\end{aligned}$$ The scale $\nu^+$($\nu^-$) is a function of $\mu$($\bar \mu$). Applying inverse transformation $C^{-1}_{n\bar n}$ to the function on the left-hand-side of (\[RTRD:1\]) we obtain the function $\mathbf{\Sigma}^R$ which is free from rapidity divergences. Therefore, the product $\mathbf{R}_{n}\mathbf{\Sigma}\mathbf{R}^\dagger_{\bar n}$ is free from rapidity divergences.
According to the renormalization theorem, we can define a (rapidity divergence) finite rapidity anomalous dimension (RAD) $$\begin{aligned}
\label{RTRD:D_def}
\mathbf{D}(\{b\})=\frac{1}{2}\mathbf{R}_n^{-1}(\{b\},\nu^+)\nu^+\frac{d}{d\nu^+}\mathbf{R}_{n}(\{b\},\nu^+),\end{aligned}$$ where the factor $1/2$ is set to meet the common definition of $\mathbf{D}$. The solution of this equation is $$\begin{aligned}
\label{RTRD:Rn}
\mathbf{R}_n(\{b\},\nu^+)&=&\mathbf{A}e^{-2\mathbf{D}(\{b\}) \ln(\delta^+/\nu^+)},\end{aligned}$$ where $\mathbf{A}$ is a $\nu$-independent matrix, which represents the scheme dependant part and is set to unity in the following. The explicit form of the rapidity renormalization factor (\[RTRD:Rn\]) together with RTRD give the explicit form of the soft factor $\mathbf{\Sigma}$. It can be written as $$\begin{aligned}
\label{RTRD:SIGMA_FAC}
\mathbf{\Sigma}(\{b\},\delta^+,\delta^-)&=&\mathbf{R}_n^{-1}(\{b\},\nu^+)\mathbf{\Sigma}_{0}(\{b\},\nu^2)(\mathbf{R}_{\bar n}^{\dagger})^{-1}(\{b\},\nu^-)
\\{\nonumber}&=&e^{2\mathbf{D}(\{b\}) \ln(\delta^+/\nu^+)}\mathbf{\Sigma}_{0}(\{b\},\nu^2)e^{2\mathbf{D}^\dagger(\{b\}) \ln(\delta^-/\nu^-)},\end{aligned}$$ where $\nu^2=\nu^+\nu^-$, $\delta^+$($\delta^-$) represents the regulator of rapidity divergences coupled to the scale $\nu^+$ ($\nu^-$), and the matrix $\mathbf{\Sigma}_0$ is a rapidity divergent free matrix. The equation (\[RTRD:SIGMA\_FAC\]) is an alternative form of RTRD (\[RTRD\]). Although it is written in the $\delta$-regularization, it can be written in any rapidity regulator by replacing $\ln\delta$ by the corresponding rapidity divergent function.
The subscripts $n$ and $\bar n$ on the normalization factors $\mathbf{R}$ label the type of rapidity divergences (and hence the regulator), which are collected in the factors. The renormalization scales $\nu^\pm$ are not boost invariant, but transforms as corresponding components of a vector. It can be seen by considering an effect of the rescaling of geometrical regulators onto parameters $\nu$. The function $\mathbf{\Sigma}_0$ depends only on the product of $\nu^2=\nu^+\nu^-$ in the consequence of Lorentz invariance.
Next, we promote the theorem to QCD. We start with the consideration of QCD in the critical regime, where its conformal invariance is restored, and hence the equation (\[RTRD:SIGMA\_FAC\]) holds. There are several possibilities to turn QCD to the critical regime in the perturbation theory, see e.g. [@Braun:2014vba; @Banks:1981nn]. We found it convenient to use the critical number of space-time dimension, $d^*=4-2\epsilon^*$. The value of $\epsilon^*$ is determined by the relation $\beta(\epsilon^*)=0$ order-by-order in the perturbation theory. Using the expression for the $\beta$-function in the dimensional regularization we find $$\begin{aligned}
\label{RTRD:e^*}
\epsilon^*=-a_s \beta_0-a_s^2 \beta_1-a_s^3\beta_2-...~.\end{aligned}$$ Note, that the UV divergences of $\mathbf{\Omega}$ at $r$ and $\bar r$ should be regularized by a non-dimensional regulator (e.g. by the cut of $\omega$). At the critical number of space-time dimension, the theorem holds up to an arbitrary order of the perturbation theory. The physical QCD is defined at $\epsilon=0$. To obtain the theorem in the physical QCD we push the $\epsilon^*$ to the $0$ order-by-order in the perturbation theory. So, at the first step the $\epsilon^*$ is shifted by $\epsilon^*\to \epsilon^{**}+\beta_0 a_s$. Since QCD is conformal invariant at one-loop level, and the counting of rapidity divergences is not affected by dimensional regularization, the form of the soft factor (\[RTRD:SIGMA\_FAC\]) is preserved, with slightly changed values of $\mathbf{D}$ and $\mathbf{\Sigma}_0$. Such shift can be repeated $K$ times, with increasing perturbative order. This defines constants $\mathbf{R}$ at $(0+\mathcal{O}(a_s^{K+1}))$-number of dimension. Alternatively, this statements can be checked by solving the renormalization group equation order-by-order in a shift parameter. Thus, we have proved the theorem in conformal theory and at arbitrary order of QCD perturbation theory.
So far we do not specify the renormalization factors for cusps. The cusp renormalization can be done before or after the rapidity renormalization. The order of renormalization affects the value of $\mathbf{R}$, due to the presence of double poles. These double poles have a geometrical origin, see e.g. [@Korchemskaya:1992je; @Drummond:2007aua], and do not influence the combinatorics of the subtractions. However, the order influences the relative compositions of renormalization scales. So the completely renormalized soft factor takes form $$\begin{aligned}
\mathbf{\Sigma}^{R,R}(\{b\},\nu^+,\nu^-,\mu)=\prod_{i=1}^N Z_{i,\text{cusp}}(\mu)
\mathbf{R}^\dagger_{n}(\{b\},\nu^+)\mathbf{\Sigma}(\{b\})\mathbf{R}_{\bar n}(\{b\},\nu^-).\end{aligned}$$ Since the renormalization factors $Z$ are scalars, it is more convenient to present RTRD in the symmetric form combining the singular factors together $$\begin{aligned}
\label{RDRT:sym_rep}
\mathbf{\Sigma}^{R,R}(\{b\},\zeta,\bar \zeta,\mu)={\left(}\prod_{i=1}^N Z^{1/2}_{i,\text{cusp}}(\mu)
\mathbf{R}^\dagger_{n}(\{b\},\nu^+){\right)}\mathbf{\Sigma}(\{b\}){\left(}\mathbf{R}_{\bar n}(\{b\},\nu^-)\prod_{i=1}^N Z^{1/2}_{i,\text{cusp}}(\mu){\right)}.\end{aligned}$$ The equation (\[RTRD:SIGMA\_FAC\]) transforms into $$\begin{aligned}
\label{RTRD:SIGMA_FAC_Z}
&&\mathbf{\Sigma}(\{b\})=
\\{\nonumber}&&\prod_{i=1}^N Z^{1/2}_{i,\text{cusp}}(\mu) e^{2\mathbf{D}(\{b\},\mu) \ln(\delta^+/\nu^+)}\mathbf{\Sigma}_{0}(\{b\},\nu^2,\mu)e^{2\mathbf{D}^\dagger(\{b\},\mu) \ln(\delta^-/\nu^-)}Z^{1/2}_{i,\text{cusp}}(\mu).\end{aligned}$$ The $\mu$-dependence of RAD can be found by combining equations (\[RTRD:D\_def\]), (\[RTRD:SIGMA\_FAC\]), and (\[RDRT:sym\_rep\]) into $$\begin{aligned}
\label{RTRD:3}
\mu^2\frac{d}{d\mu^2} e^{2\mathbf{D}(\{b\},\mu) \ln(\delta^+/\nu^+)}=\frac{1}{4}\sum_{i=1}^N\Gamma^i_{\text{cusp}}e^{2\mathbf{D}(\{b\},\mu) \ln(\delta^+/\nu^+)},\end{aligned}$$ where $\Gamma^i_{\text{cusp}}$ is $$\begin{aligned}
\Gamma^i_{\text{cusp}}=(Z^{i}_{\text{cusp}})^{-1}\mu\frac{d}{d\mu}Z^i_{\text{cusp}}.\end{aligned}$$ The equation (\[RTRD:3\]) can be written in the convenient form $$\begin{aligned}
\label{RTRD:RGE}
\mu^2 \frac{d}{d\mu^2}\mathbf{D}(\{b\},\mu)=\frac{1}{4}\sum_{i=1}^N\Gamma^i_{\text{cusp}}\mathbf{I}.\end{aligned}$$ This is the generalization of the well-known Collins-Soper (CS) equation [@Collins:1984kg] to the MPS case. Note, that the values of $\Gamma^i_{\text{cusp}}$ differ only by color factors, since the angles of all cusps are the same. In the scalar case $N=2$, which corresponds to the TMD RAD, the color representation of both cusps are the same. In this case the equation (\[RTRD:RGE\]) is reduced to the original CS equation [@Collins:1984kg] $$\begin{aligned}
\label{TMD:D_RGE}
\mu^2 \frac{d}{d\mu^2}\mathcal{D}^i(b,\mu)=\frac{\Gamma^i_{\text{cusp}}}{2}.\end{aligned}$$ For $N=4$ it has been checked in [@Vladimirov:2016qkd] at NNLO (see also discussion in sec.\[sec:non-dipole\]).
Let us mention that there is also a possibility to leave the UV divergences unrenormalized since practically the soft factor is always combined with parton distributions. The obtained combination can be renormalized as a whole. This approach requires less algebra and thus is more convenient practically. For example, it has been used in [@Echevarria:2015usa; @Echevarria:2016scs] for NNLO calculations.
Some consequences and extensions {#sec:consiquences}
================================
MPS factorization[^5] {#sec:MPS_fac}
---------------------
The RTRD states that the rapidity divergences related to different directions are factorizable. Thus, we can finalize the factorization theorem for the multi-DY process and define a divergence-free multi-parton distribution (multiPD). Note, that all expressions presented in this section are easily reduced to the case of TMD factorization. To obtain the TMD expressions, one should only remove the $\{\}$-brackets from variables and release the color structure (see also sec.\[sec:scheme\]).
The multi-DY scattering is characterized by momenta of produced hard particles $q_i$, with $q_i^2=Q_i^2+q_{iT}^2$. In the regime $Q_i^2\gg q_{Ti}^2$ the hadron tensor of the MPS has can be written in the factorized form [@Diehl:2011yj; @Diehl:2015bca] $$\begin{aligned}
\label{MDY:cross}
W(\{q\})&=&\prod_{i=1}^{N/2} \sum_{f,\bar f} H_{i,f_i\bar f_i} {\left(}\frac{Q^2_i}{\mu^2}{\right)}\\{\nonumber}&& \int d^2 b_i d^2 b_{N-i}e^{-i (q_{i}\cdot (b_i-b_{N-i}))_\perp}
\tilde F^T_{\{\bar f\}{\leftarrow}h_2}(\{\bar x\},\{b\},\mu)
\times
\mathbf{\Sigma}(\{b\},\mu)
\times
\tilde F_{\{f\}{\leftarrow}h_1}(\{x\},\{b\},\mu),\end{aligned}$$ where $H$ are hard scattering coefficient functions, $x$ and $\bar x$ are Bjorken variables, $\mu$ is a common hard-factorization scale $\mu\sim Q_{1,..,N/2}$. The multiPD is given by the following matrix element $$\begin{aligned}
\label{MDY:MPD}
&&\tilde F_{\{f\}{\leftarrow}h}(\{x\},\{b\})=\int \Big(\prod_{i=1}^{N/2} \frac{dy_i^-dy_{N-i}^-}{(2\pi)^2} e^{i x_i (y^-_i-y^-_{N-i}) p^+}\Big)
\\&&{\nonumber}\langle h|\bar T\{\bar \xi_{f_1}(y_1^-,b_1)
...\bar \xi_{f_{N/2}}(y_{N/2}^-,b_{N/2})\}T\{
\xi_{f_{N/2+1}}(y_{N/2+1}^-,b_{N/2+1})
...\xi_{f_{N}}(y_{N}^-,b_{N})
\}|h \rangle.\end{aligned}$$ The Lorentz structure of multiPDs is omitted for simplicity. In both formulas, a single variable $b_i$ and a single variable $y_i^-$ can be set to zero by the translation invariance, and corresponding integrals eliminated. The fields $\xi$ can be quark, anti-quark and gluon fields with adjusted half-infinite line Wilson lines, e.g. $\xi_{q}(x)=\mathbf{\Phi}_{-n}(x)q(x)$. Therefore, the multi-parton distribution $\tilde F$ is the vector in the color space. Consequently, the multiPD $\tilde F^T$ is a row in the color space. The multiPDs $F$ are non-zero only for a color singlet combinations of indices. It automatically eliminates the non-gauge-invariant parts of the soft factor $\mathbf{\Sigma}$.
The fields participated in the definition (\[MDY:MPD\]) are collinear fields. It implies that the soft modes of these fields should be subtracted (so-called zero-bin subtractions, see e.g[@Becher:2014oda]). The procedure of subtraction is dependent on the rapidity-divergences regularization. In the convenient regulator, it can be presented by an inverse soft factor see e.g.[@Echevarria:2016scs] (or product of soft factors, see e.g. [@Collins:2011ca]). Till the end of this section we use the $\delta$-regularization for explicitness. However, the derivation can be performed in any other regularization scheme in the same manner and with the same final result. In the $\delta$-regularization, the zero-bin subtraction take the form of the inverse soft factor [@GarciaEchevarria:2011rb; @Echevarria:2016scs] $$\begin{aligned}
\label{MDY:F-Fus}
\tilde F_{\{f\}{\leftarrow}h_1}(\{x\},\{b\},\mu,\delta^-)=\mathbf{\Sigma}^{-1}(\mu;\delta^+,\delta^-)\times \tilde F^{\text{us}}_{\{f\}{\leftarrow}h_1}(\{x\},\{b\},\mu,\delta^+),\end{aligned}$$ where $\tilde F^{\text{us}}$ is the unsubtracted multiPD, i.e. evaluated directly as it stands in (\[MDY:MPD\]).
The factorization theorem (\[MDY:cross\]) is not complete in the sense that it does not express the cross-section via finite quantities, which depend only on a single hadron. The problem here is rapidity divergences which are presented in every constituent of the theorem. The multiPD $F$ ($F^T$) has rapidity divergences due to the interaction of far end points of Wilson lines, i.e. divergences are localized at $(-n)_\perp^\infty$ ($(-\bar n)_\perp^\infty$), and regularized by $\delta^+$ ($\delta^-$). The rapidity divergences cancel in the product $F^T(\delta^+)\times \mathbf{\Sigma}(\delta^+,\delta^-)\times F(\delta^-)$ by the statement of the factorization theorem. To complete the factorization theorem, we apply RTRD, and insert the soft factor in the form (\[RTRD:SIGMA\_FAC\]). Since the multiPD $F_{\{f\}{\leftarrow}h_1}$ contains only rapidity divergences regularized by $\delta^-$ the following combination is free from rapidity divergences $$\begin{aligned}
\label{MDY:F_finite}
F_{\{f\}{\leftarrow}h_1}(\{x\},\{b\},\nu^+)&=&\mathbf{\Sigma}_0(\nu^2) \mathbf{R}_{\bar n}^{\dagger\,-1}(\{b\},\nu^-)\times \tilde F_{\{f\}{\leftarrow}h}(\{x\},\{b\},\delta^-)
\\{\nonumber}&=&\mathbf{\Sigma}_0(\nu^2)e^{2\mathbf{D}^\dagger(\{b\}) \ln(\delta^-/\nu^-)}\times \tilde F_{\{f\}{\leftarrow}h}(\{x\},\{b\},\delta^-),\end{aligned}$$ where the finite prefactor $\mathbf{\Sigma}_0$ is put for the future convenience. Note, that the left-hand-side of this equation is independent on $\nu^-$. It became explicit in the terms of unsubtracted multiPDs (\[MDY:F-Fus\]), where $$\begin{aligned}
\label{MDY:F_finite_us}
F_{\{f\}{\leftarrow}h_1}(\{x\},\{b\},\nu^+)&=&\mathbf{R}_n(\{b\},\nu^+)\times \tilde F^{\text{us}}_{\{f\}{\leftarrow}h}(\{x\},\{b\},\delta^+)
\\{\nonumber}&=&e^{-2\mathbf{D}(\{b\}) \ln(\delta^+/\nu^+)}\times \tilde F^{\text{us}}_{\{f\}{\leftarrow}h}(\{x\},\{b\},\delta^+).\end{aligned}$$ Making the similar redefinition of $\tilde F^T$ we obtain the factorization theorem in the form $$\begin{aligned}
{\nonumber}&&\tilde F^T(\{\bar x\},\{b\},\mu)\times \mathbf{\Sigma}(\{b\},\mu)\times \tilde F^T(\{x\},\{b\},\mu)=
\\&&\label{MDY:rap_FAC}
\qquad\qquad\qquad\qquad\qquad
F^T(\{\bar x\},\{b\},\mu,\nu^-)\times \mathbf{\Sigma}^{-1}_0(\{b\},\mu,\nu^2)\times F(\{x\},\{b\},\mu,\nu^+).\end{aligned}$$ Here all components are finite. And thus, the factorization theorem is completed.
The dependence of a multiPD on the rapidity scales follows from the equations (\[MDY:F\_finite\_us\]) and (\[RTRD:D\_def\]), $$\begin{aligned}
\label{MDY:nu-evol}
\nu^+ \frac{d}{d\nu^+}F_{\{f\}{\leftarrow}h}(\{x\},\{b\},\mu,\nu^+)=\frac{1}{2}\mathbf{D}^{\{f\}}(\{b\},\mu)\times F_{\{f\}{\leftarrow}h}(\{x\},\{b\},\mu,\nu^+).\end{aligned}$$ The factorized expression (\[MDY:rap\_FAC\]) contains the multiPDs that depend on the variables $\nu^+$ and $\nu^-$, which seems to contradict the Lorentz invariance. Nonetheless, there is no contradiction, because the multiPDs are defined on the states with momenta oriented along $n$ or $\bar n$. It allows to pass to a more convenient (and standard) boost invariant variables $\zeta$ and $\bar \zeta$, which is done in the next section.
We also note that the rapidity divergences are independent on the kind of states. They are the part of the operator, similarly to UV divergences. Therefore, the factor $\mathbf{R}$ applies directly to the multiPD operator. Such composition greatly simplifies the study of properties of multiPD operators without reference to the parton model consistently. For example, to perform the operator product expansion in the background field technique.
### Boost invariant variables and scheme dependence {#sec:scheme}
Let us introduce the boost invariant variables $$\begin{aligned}
\label{zeta-def}
\zeta=2(p^+)^2\frac{\nu^-}{\nu^+},\qquad \bar \zeta=2(p^-)^2\frac{\nu^+}{\nu^-},\qquad \zeta\bar \zeta=(2p^+p^-)^2\end{aligned}$$ where $p^+$ and $p^-$ are components of a vector $p^\mu$. Vector $p^\mu$ can be selected arbitrary, but it is convenient to associate it with the momentum of the produced particle (e.g. with the momentum of the produced photon for the DY processes). In this case, we have $\zeta\bar \zeta=Q^4$ where $Q$ is the typical virtuality of the process. Assuming this, the multiPD becomes a function of $\zeta$ and $\nu^2$, i.e. $F(\{x\},\{b\},\mu,\zeta,\nu^2)$. The $\zeta$ dependence follows from equation (\[MDY:nu-evol\]), $$\begin{aligned}
\zeta \frac{d}{d\zeta}F_{\{f\}{\leftarrow}h}(\{x\},\{b\},\mu,\zeta,\nu^2)=-\mathbf{D}^{\{f\}}(\{b\},\mu)\times F_{\{f\}{\leftarrow}h}(\{x\},\{b\},\mu,\zeta,\nu^2).\end{aligned}$$ This equation coincides with the standard definition of the rapidity evolution (see e.g.[@Vladimirov:2016qkd; @Diehl:2015bca; @Aybat:2011zv]).
In the presented above construction defers from usual constructions, e.g. in refs.[@Echevarria:2016scs; @Vladimirov:2016qkd; @Diehl:2011yj; @Diehl:2015bca], by the presence of an extra parameter $\nu^2$. This parameter decouples from the equations and, therefore, is unrestricted. We stress that it also appears in the remnant of the soft factor $\mathbf{\Sigma}_0(\nu^2)$, which is scheme dependent. In this way, the parameter $\nu^2$ is a part of the scheme definition.
We recall that the rapidity renormalization factors $\mathbf{R}$ are defined up to an arbitrary matrix, see (\[RTRD:Rn\]). Therefore, the definition of the multiPD is not unique. We can introduce an alternative multiPD with the multiplication by an arbitrary finite matrix $\mathbf{S}$, i.e. $$\begin{aligned}
\label{MDY:eqeq1}
F_{\{f\}{\leftarrow}h_1}(\{x\},\{b\},\zeta,\nu^2)\to \mathbf{S}\times F_{\{f\}{\leftarrow}h_1}(\{x\},\{b\},\zeta,\nu^2).\end{aligned}$$ Such procedure does not damage the factorization theorem (\[MDY:rap\_FAC\]), and leads only to the replacement $$\begin{aligned}
\label{MDY:eqeq2}
\mathbf{\Sigma}^{-1}_0(\{b\},\mu,\nu^2)\to (\mathbf{S}^{-1})^{T}\mathbf{\Sigma}^{-1}_0(\{b\},\mu,\nu^2) \mathbf{S}^{-1}.\end{aligned}$$ Compare equations (\[MDY:eqeq1\],\[MDY:eqeq2\],\[MDY:F\_finite\_us\]) and (\[RTRD:Rn\]) we conclude that the the matrix $\mathbf{S}$ can be recasted to the matrix $\mathbf{A}$, and thus, is a part of scheme definition. Since the matrix $\mathbf{S}$ is a part of the rapidity renormalization factor is can depend on any variables except $\zeta$.
The expression for matrix $\mathbf{S}$ should be fixed conveniently by some regularization-independent condition. Let us discuss the fixation of the scheme in the TMD case. The conventional form of the TMD factorization theorem (see e.g. [@Collins:2011zzd; @Collins:2011ca; @GarciaEchevarria:2011rb; @Echevarria:2012js; @Chiu:2012ir; @Echevarria:2015byo; @Echevarria:2016scs; @Luebbert:2016itl; @Li:2016axz]) defines the hadron tensor as a product of two TMD distributions without any remnant of the soft factor matrix $\Sigma_0$. The TMD hadron tensor reads $$\begin{aligned}
\label{TMD:fffff}
W_{\text{TMD}}&=&\sum_{\bar f,f}H_{\bar f f}{\left(}\frac{Q^2}{\mu^2}{\right)}\int \frac{d^2b}{(2\pi)^2} e^{i (q\cdot b)_T}
F_{\bar f{\leftarrow}h_2}(\bar x,b,\mu,\bar \zeta)
F_{f{\leftarrow}h_1}(x,b,\mu,\zeta).\end{aligned}$$ This form of the factorization theorem agrees with the parton model picture, since the hard coefficient can be interpreted as the cross-section of parton scattering, and at small-$b$ $F(x,b\to 0)\to f(x)$, where $f(x)$ is the usual parton distribution function. The expression (\[TMD:fffff\]) implies the following relation $$\begin{aligned}
\label{TMD:scheme}
S^{-1}(b,\mu, \nu^2)\Sigma_0^{-1\,\text{TMD}}(b,\mu, \nu^2)S^{-1}(b,\mu,\nu^2)=1.\end{aligned}$$ Using this scheme we obtain the following expression for TMD distribution $$\begin{aligned}
F_{f{\leftarrow}h}(x,b,\mu,\zeta,\nu^2)&=&\sqrt{\Sigma_0^{\text{TMD}}(b,\mu,\nu^2)}e^{-2\mathcal{D}^f(b,\mu)\ln(\delta^+/\nu^+)}\tilde F^{\text{us}}_{f{\leftarrow}h}(x,b,\delta^+).\end{aligned}$$ Recalling the simple structure of the TMD soft factor (\[TMD:structure\]) we arrive to the standard expression for the TMD distribution $$\begin{aligned}
F_{f{\leftarrow}h}(x,b,\mu,\zeta)&=&\sqrt{\Sigma_{\text{TMD}}{\left(}b,\frac{\delta^+}{\sqrt{2}p^+}\sqrt{\zeta},\frac{\delta^+}{\sqrt{2}p^+}\sqrt{\zeta}{\right)}}\tilde F_{f{\leftarrow}h}(x,b,\delta^+).\end{aligned}$$ The $\nu^2$ parameter is not presented in this definition.
The natural generalization of the TMD scheme fixation condition (\[TMD:scheme\]) for the MPS case is $$\begin{aligned}
\mathbf{S}(b,\mu,\nu^2)\mathbf{\Sigma}_0(\{b\},\mu,\nu^2) \mathbf{S}^T(b,\mu,\nu^2)=\mathbf{I}.\end{aligned}$$ In the recent paper [@Buffing:2017mqm], it has been shown that in the $N=4$ case the solution of this equation exists and naturally expresses in the terms of matrices $s$ (\[DPD:small\_s\]). In this scheme the MPS factorization theorem is $$\begin{aligned}
\label{MDY:cross2}
W(\{q\})&=&\prod_{i=1}^{N/2} \sum_{f,\bar f} H_{i,f_i\bar f_i} {\left(}\frac{Q^2_i}{\mu^2}{\right)}\\{\nonumber}&& \int d^2 b_i d^2 b_{N-i}e^{-i (q_{i}\cdot (b_i-b_{N-i}))_\perp}
F^T_{\{\bar f\}{\leftarrow}h_2}(\{\bar x\},\{b\},\mu,\zeta)
\times
F_{\{f\}{\leftarrow}h_1}(\{x\},\{b\},\mu,\zeta).\end{aligned}$$ Such scheme is equivalent to the decomposition of the soft factor (\[DPD:FAC\]). In the case of DPDs this decomposition has been explicitly demonstrated at NNLO in [@Vladimirov:2016qkd]. Note, that generally speaking the matrix $\mathbf{S}$ does not commute with $\mathbf{D}$ and therefore, the rapidity anomalous dimension is scheme dependent $$\begin{aligned}
\mathbf{D}_{S}=\mathbf{S}\mathbf{D}\mathbf{S}^{-1}\sim \mathbf{D}+a_s[\mathbf{s},\mathbf{D}]+\mathcal{O}(a_s^2),\end{aligned}$$ where for the last equality we substitute $\mathbf{S}=\exp(a_s\mathbf{s})$. The explicit evaluation of color structure presented in sec.\[sec:color\] shows that $[\mathbf{s},\mathbf{D}]\sim \mathcal{O}(a_s^3)$ at least.
Correspondence between soft and rapidity anomalous dimensions {#sec:corespondance}
-------------------------------------------------------------
The relation between the rapidity and UV singularities give rise to the correspondence between RAD and SAD [@Vladimirov:2016dll]. The correspondences between anomalous dimensions are highly interesting, since they connect different regimes of physics. To our best knowledge, nowadays there are only two examples of such correspondences in QCD: the discussed here SAD-to-RAD correspondence, and the BK/JIMWLK-to-BMS correspondence [@Hatta:2008st]. The check of SAD-to-RAD correspondence gives a non-trivial confirmation of RTRD.
The soft anomalous dimension (SAD) is defined as $$\begin{aligned}
\pmb{\gamma}_s(\{v\})=\mathbf{Z}^{-1}(\{v\},\mu)\mu\frac{d}{d\mu}\mathbf{Z}(\{v\},\mu),\end{aligned}$$ where $\mathbf{Z}$ is the UV renormalization factor for multi-cups non-analyticity of Wilson lines, that appear in $\mathbf{\Omega}$ (\[RTRD:1\]). Comparing to (\[RTRD:D\_def\]) we obtain the exact relation in the conformal field theory $$\begin{aligned}
\label{RAD-SAD:conformal}
\pmb{\gamma}_s(\{v\})=2\mathbf{D}(\{b\}),\end{aligned}$$ where vectors $v$ and $b$ are related by $C_{n\bar n}$ transformation. This relation has been observed for the TMD case in the conformal invariant $\mathcal{N}=4$ super-Yang-Mills theory at three-loop order [@Li:2016ctv].
In QCD the equality (\[RAD-SAD:conformal\]) holds at the critical point (\[RTRD:e\^\*\]). The UV anomalous dimension is $\epsilon$-independent, in the contrast to the RAD. Therefore, we have $$\begin{aligned}
\label{RAD-SAD}
\pmb{\gamma}_s(\{v\})=2\mathbf{D}(\{b\},\epsilon^*).\end{aligned}$$ Using this expression the physical value of RAD (SAD) can be obtained at a given perturbative order using the finite part of the previous perturbative order and the know expression for SAD (RAD). Indeed, substituting $\epsilon^*$ in the form (\[RTRD:e\^\*\]) and comparing the coefficients for different powers of $a_s$ we obtain $$\begin{aligned}
\label{RAD-SAD:1}
\mathbf{D}_1(\{b\})&=&\frac{1}{2}\pmb{\gamma}_1(\{v\}),
\\\label{RAD-SAD:2}
\mathbf{D}_2(\{b\})&=&\frac{1}{2}\pmb{\gamma}_2(\{v\})+\beta_0 \mathbf{D}'_1(\{b\}),
\\\label{RAD-SAD:3}
\mathbf{D}_3(\{b\})&=&\frac{1}{2}\pmb{\gamma}_3(\{v\})+\beta_0 \mathbf{D}'_2(\{b\})+\beta_1 \mathbf{D}'_1(\{b\})-\frac{\beta_0^2}{2}\mathbf{D}''_1(\{b\}),\end{aligned}$$ and so on. Here, we use the notation $\pmb{\gamma}=\sum a_s^n \pmb{\gamma}_{n}$ and $\mathbf{D}=\sum a_s^n \mathbf{D}_n$, and primes denote the derivatives with respect to $\epsilon$ at $\epsilon=0$.
### TMD rapidity anomalous dimension at three-loop order {#sec:SADRAD}
The practically most interesting case is the TMD RAD. It is corresponded to the dipole part of the SAD, or to the lightlike cusp anomalous dimension. The expression for the cusp anomalous dimension is known up to three-loop order [@Moch:2004pa], which allows us to learn the three-loop RAD, using the two-loop calculation.
The dipole contribution to the SAD has the form $$\begin{aligned}
C_i\gamma_{\text{dipole}}(v_i\cdot v_j)
=\ln{\left(}\frac{(v_i\cdot v_j)\mu^2}{\nu_{ij}^2}{\right)}\Gamma^i_{cusp}-\tilde \gamma^i_s,\end{aligned}$$ where $\nu_{ij}^2$ is a IR scale which regularizes the lightlike cusp angle, and $C_i$ is the quadratic Casimir eigenvalue. The coefficients of the perturbative expansion for $\Gamma$ and $\gamma_s$ can be found in [@Moch:2004pa], and are given in the appendix \[app:ADs\]. The NLO TMD anomalous dimension at arbitrary $\epsilon$ is [@Echevarria:2015byo] (see also (\[SF:1loop\])) $$\begin{aligned}
\label{RAD-SAD:D1}
\mathcal{D}_1^i(b,\epsilon)&=&-2a_sC_i {\left(}B^\epsilon\Gamma(-\epsilon)+\frac{1}{\epsilon}{\right)},\end{aligned}$$ where $B=b^2\mu^2/4 e^{-2\gamma_E}$. Using the equation (\[RAD-SAD:1\]) we obtain the equality $$\begin{aligned}
\frac{\gamma_{1,\text{dipole}}}{2}=2\ln{\left(}\frac{(v_1\cdot v_2)\mu^2}{\nu_{12}^2}{\right)}=\frac{\mathcal{D}^i_1}{C_i}=2\ln{\left(}\frac{b_{12} \mu^2}{4 e^{-2\gamma_E}}{\right)}.\end{aligned}$$ The vectors $v$ and $b$ are related by (\[Cnn:vectors\]) which gives $$\begin{aligned}
\label{RAD-SAD:transform}
C_{nn}(v_i\cdot v_j)=\frac{b_{12}}{(\lambda \bar \lambda+ a \bar a b_1^2)(\lambda \bar \lambda+ a \bar a b_2^2)}.\end{aligned}$$ It fixes the relative scheme dependence between rapidity renormalization and UV renormalization $$\begin{aligned}
\nu_{ij}=4 e^{-2\gamma_E}(\lambda \bar \lambda+ a \bar a b_i^2)(\lambda \bar \lambda+ a \bar a b_j^2).\end{aligned}$$
At the order $a_s^2$, RAD has an extra logarithm structure which is produced by the expansion of $B^\epsilon$ in (\[RAD-SAD:D1\]). Therefore, comparing left and right hand sides of (\[RAD-SAD:2\]) we find $$\begin{aligned}
\mathcal{D}_2=d^{(2,2)}L^2_b+d^{(2,1)}L_b+d^{(2,0)}=\beta_0 L_b^2+2\Gamma_1 L_b-\frac{\tilde \gamma_1}{2}+\beta_0 \zeta_2,\end{aligned}$$ where $$\begin{aligned}
d^{(2,2)}=\beta_0,\qquad d^{(2,1)}=2\Gamma_1,\qquad d^{(2,0)}=-\frac{\tilde \gamma_1}{2}+\beta_0\zeta_2.\end{aligned}$$ These numbers coincide with RAD coefficients calculated directly, see e.g. [@Echevarria:2015byo; @Becher:2010tm; @Luebbert:2016itl].
To obtain the RAD at NNLO the $\epsilon$-dependent NLO expression is required. It has been evaluated in [@Echevarria:2015byo], and reads $$\begin{aligned}
\label{RAD-SAD:D2}
\mathcal{D}_2^i(b,\epsilon)&=&2 C_i\Bigg\{ B^{2\epsilon}\Gamma^2(-\epsilon)\Big[C_A(2\psi(-2\epsilon)-2\psi(-\epsilon)+\psi(\epsilon)+\gamma_E)
\\&&{\nonumber}+\frac{1-\epsilon}{(1-2\epsilon)(3-2\epsilon)}{\left(}\frac{3(4-3\epsilon)}{2\epsilon}C_A-N_f{\right)}\Big] +B^\epsilon \frac{\Gamma(-\epsilon)}{\epsilon}\beta_0+\frac{\beta_0}{2\epsilon^2}-\frac{\Gamma_1}{2\epsilon}\Bigg\}.\end{aligned}$$ Substituting it into equation (\[RAD-SAD:3\]) we obtain $$\begin{aligned}
\mathcal{D}_3&=&d^{(3,3)}L_b^3+d^{(3,2)}L_b^3+d^{(3,1)}L_b+d^{(3,0)}=2 \Gamma_2 L_b-\frac{\tilde \gamma_2}{2}
\\{\nonumber}&&\quad-\frac{\beta_0^2}{3}L_b^3+\beta_1 L_b^2-\beta_0^2 \zeta_2L_b-\frac{2}{3}\beta_0^2 \zeta_3+\beta_1\zeta_2
\\{\nonumber}&&\quad+\beta_0^2 L_b^3+2\beta_0\Gamma_1L_b^2+\beta_0(2d^{(2,0)}+\beta_0\zeta_2)L_b+\beta_0\Gamma_1\zeta_2+
\beta_0{\left[}C_A{\left(}\frac{2428}{81}-26\zeta_4{\right)}-N_f\frac{328}{81}{\right]},\end{aligned}$$ where the second line comes from the expansion of $\mathcal{D}_1$ (\[RAD-SAD:D1\]), and the third line comes from the expansion of $\mathcal{D}_2$ (\[RAD-SAD:D2\]). The coefficients $d^{(n,k)}$ are $$\begin{aligned}
d^{(3,3)}&=&\frac{2}{3}\beta_0^2,\qquad d^{(3,2)}=2\Gamma_1\beta_0+\beta_1,\qquad d^{(3,1)}=2\beta_0d^{(2,0)}+2\Gamma_2,
\\{\nonumber}d^{(3,0)}&=&-\frac{\tilde \gamma_2}{2}+(\beta_1+\beta_0\Gamma_1) \zeta_2-\frac{2}{3}\beta_0^2 \zeta_3+\beta_0{\left[}C_A{\left(}\frac{2428}{81}-26\zeta_4{\right)}-N_f\frac{328}{81}{\right]}.\end{aligned}$$ Substituting the explicit expressions anomalous dimensions we obtain $$\begin{aligned}
d^{(3,0)}&=&C_A^2{\left(}\frac{297029}{1458}-\frac{3196}{81}\zeta_2-\frac{6164}{27}\zeta_3-\frac{77}{3}\zeta_4+\frac{88}{3}\zeta_2\zeta_3+96\zeta_5{\right)}\\{\nonumber}&&+C_AN_f{\left(}-\frac{31313}{729}+\frac{412}{81}\zeta_2+\frac{452}{27}\zeta_3-\frac{10}{3}\zeta_4{\right)}\\{\nonumber}&&+C_FN_f{\left(}-\frac{1711}{54}+\frac{152}{9}\zeta_3+8\zeta_4{\right)}+N_f^2{\left(}\frac{928}{729}+\frac{16}{9}\zeta_3{\right)}.\end{aligned}$$ This expression coincides with the expression obtained in [@Vladimirov:2016dll; @Li:2016ctv].
The obtained expressions satisfy the renormalization group equation for TMD RAD (\[TMD:D\_RGE\]). On one hand side, it gives an extra check for the calculation. On another hand side, it is not accidental. The UV anomalous dimensions are $\epsilon$-independent by definition, and therefore the equation (\[RTRD:RGE\]) holds at arbitrary $\epsilon$.
### Leading non-dipole contribution to rapidity anomalous dimension {#sec:non-dipole}
The leading contributions to the non-dipole SAD has been calculated in [@Almelid:2015jia]. In accordance to (\[RAD-SAD:1\]), the leading non-dipole contribution to RAD can be obtained by the direct transformation.
The leading non-dipole contribution to SAD appears at the three-loop level. The SAD at this order has the form [@Almelid:2015jia] $$\begin{aligned}
\label{SAD:color}
\pmb{\gamma}_s(\{v\})&=&-\frac{1}{2}\sum_{[i,j]}
\mathbf{T}^A_i\mathbf{T}^A_j \gamma_{\text{dipole}}(v_i\cdot v_j)-
\sum_{[i,j,k,l]}if^{ACE}if^{EBD}\mathbf{T}_i^A\mathbf{T}_j^B\mathbf{T}_k^C\mathbf{T}_l^D\mathcal{F}_{ijkl}
\\&&{\nonumber}-
\sum_{[i,j,k]}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^C\mathbf{T}_k^D if^{ACE}if^{EBD}C+\mathcal{O}(a_s^4),\end{aligned}$$ where we use the same notation as in sec.\[sec:color\].
It is important that the SAD depends only on the conformal rations $\rho$ of vectors $v$ [@Aybat:2006mz; @Gardi:2009qi]. In contrast to the transformation of the scalar product (\[RAD-SAD:transform\]), the conformal ratios $\rho$ do not obtain any scheme factors under the transformation $C_{n\bar n}$. E.g. at N$^3$LO only the following ratios arise $$\begin{aligned}
\label{SAD:rho->rho}
\rho_{ijkl}=\frac{(v_i\cdot v_j)(v_k\cdot v_l)}{(v_i\cdot v_k)(v_j\cdot v_l)},\qquad C_{n\bar n}(\rho_{ijkl})=\tilde \rho_{ijkl}=\frac{(b_i-b_j)^2(b_k-b_l)^2}{(b_i-b_k)^2(b_j-b_l)^2}.\end{aligned}$$
The color structure of the MPS soft factor is elaborated in the appendix \[app:color\] and presented in sec.\[sec:color\]. Taking into account that the dipole part is the TMD soft factor with the structure (\[TMD:structure\]) and the definition (\[RTRD:SIGMA\_FAC\]) we find that up to three-loop order the RAD has the following expression $$\begin{aligned}
\label{RAD:allcolor}
\mathbf{D}(\{b\})&=&-\frac{1}{2}\sum_{[i,j]}
\mathbf{T}^A_i\mathbf{T}^A_j \mathcal{D}(b_{ij})-
\sum_{[i,j,k,l]}if^{ACE}if^{EBD}\mathbf{T}_i^A\mathbf{T}_j^B\mathbf{T}_k^C\mathbf{T}_l^D\tilde{\mathcal{F}}_{ijkl}(\{b\})
\\&&{\nonumber}-
\sum_{[i,j,k]}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^C\mathbf{T}_k^D if^{ACE}if^{EBD}\tilde C+\mathcal{O}(a_s^4).\end{aligned}$$ The color structure literally coincides with (\[SAD:color\]). Therefore, the functions $C$ and $\mathcal{F}$ could be obtained by replacing $\rho\to \tilde \rho$ (\[SAD:rho->rho\]). Comparing with the parametrization of [@Almelid:2015jia] we obtain $$\begin{aligned}
\label{RAD:non-dipole1}
\tilde C&=&a_s^3{\left(}\zeta_2\zeta_3+\frac{\zeta_5}{2}{\right)}+\mathcal{O}(a_s^4),
\\\label{RAD:non-dipole2}
\tilde{\mathcal{F}}_{ijkl}(\{b\})&=&8a_s^3\mathcal{F}(\tilde \rho_{ikjl},\tilde \rho_{iljk})+\mathcal{O}(a_s^4),\end{aligned}$$ where function $\mathcal{F}$ is given in [@Almelid:2015jia] in the terms of single-valued harmonic polylogarithms.
Using the color decomposition (\[RAD:allcolor\]) we can test the renormalization group equation (\[RTRD:RGE\]). Differentiating (\[RAD:allcolor\]) with respect to $\mu$ and using the renormalization group equation for the dipole part (\[TMD:D\_RGE\]) we find $$\begin{aligned}
\mu^2 \frac{d}{d\mu^2}\mathbf{D}(\{b\})&=&-\frac{1}{2}\sum_{[i,j]}\mathbf{T}^A_i\mathbf{T}^A_j \frac{\Gamma^i_{\text{cusp}}}{2 C_i}+\mu^2\frac{d}{d\mu^2}(\text{\textbf{non-dipole terms}})
\\ {\nonumber}&=&\frac{1}{4}\sum_{i=1}^N\Gamma^i_{\text{cusp}}\mathbf{I}_i+\mu^2\frac{d}{d\mu^2}(\text{\textbf{non-dipole terms}}),\end{aligned}$$ where the non-dipole terms include all possible non-dipole color structures starting from the leading terms presented in (\[RAD:allcolor\]). To obtain the last line we have used the color neutrality condition (\[def:colorless\]). Thus we conclude that at all orders of the perturbation theory $$\begin{aligned}
\label{RADSAD:nondipoleRGE}
\mu^2\frac{d}{d\mu^2}(\text{\textbf{non-dipole terms}})=0,\end{aligned}$$ which agrees with results (\[RAD:non-dipole1\]), (\[RAD:non-dipole2\]).
### All-order constraint on the color-structure of soft anomalous dimension
The absence of the color-tripole in the SAD is well-known. It has been shown in [@Aybat:2006mz; @Gardi:2009qi; @Dixon:2009ur], that tripole contribution is absent at all-orders in the consequence of permutation and rescaling symmetries. Using the correspondence between SAD and RAD we can make a more restrictive statement.
The MPS soft factor has a peculiar color-structure which follows from the generating function decomposition, see sec.\[sec:color\] and the derivation in appendix \[app:color\]. Namely, it has only even-color contributions at all orders (\[MPS:all\_order\_color\]). The decomposition of the soft factor (\[RTRD:SIGMA\_FAC\]) does not violate such structure. It is the consequence of commutativity of generators with different indices. Indeed, commuting odd-number of generators we necessary obtain an anti-symmetric structure in some sub-set of indices, which is eliminated by the symmetric sum over all Wilson lines (the examples of such structures up to fourth order are demonstrated in appendix \[app:color\]). Therefore, the rapidity anomalous dimension has the same color-pattern as $\mathbf{\Sigma}$, $$\begin{aligned}
\mathbf{D}(\{b\})&=&
\sum_{\substack{n=2\\n\in\text{even}}}^{\infty}\sum_{i_1,...,i_n=1}^N\{\mathbf{T}^{A_1}_{i_1}...\mathbf{T}^{A_n}_{i_n}\}D^{n;i_1...i_n}_{A_1...A_n}(\{v\}).\end{aligned}$$ The explicit example for $n=2$ and $n=4$ is given in (\[RAD:allcolor\]).
The correspondence between SAD and RAD preserves the color structure. Thus, the SAD also contains only the even-number of color-generators $$\begin{aligned}
\pmb{\gamma}_s(\{v\})&=&
\sum_{\substack{n=2\\n\in\text{even}}}^{\infty}\sum_{i_1,...,i_n=1}^N\{\mathbf{T}^{A_1}_{i_1}...\mathbf{T}^{A_n}_{i_n}\}\gamma^{n;i_1...i_n}_{A_1...A_n}(\{v\}).\end{aligned}$$ The explicit structure involving four generators is given, e.g. in (\[app:MPS\_colored\]). The next-order color structures requires six generators, and thus appear only at fifth loop order.
Universality of TMD soft factor {#sec:TMD_UNIVERSAL}
-------------------------------
The RTRD is formulated for the DY-like geometry of the soft factor. Such kinematics is essential, since in this case the soft factor can be written as a matrix element of a single T-ordered operator, and thus the conformal transformation could be applied. The same is true for the soft factor in the kinematics of $e^+e^-$-annihilation. In contrast, the soft factor for the SIDIS-like processes could not be analyzed in this way. However, the TMD-soft factor has a peculiarly simple structure, which leads to the equality of DY and SIDIS soft factors. Let us present it in details.
The TMD soft factor for the SIDIS kinematics reads $$\begin{aligned}
\label{SF:TMD_SIDIS}
\Sigma_{\text{TMD}}^{\text{SIDIS}}(b) = \frac{1}{N_c}
\langle 0|\bar T\{\Phi^{dc_2}_{n}(b)\Phi^{\dagger c_2a}_{-\bar n}(b)\}T\{\Phi^{ac_1}_{-\bar n}(0)\Phi^{\dagger c_1d}_{n}(0)\} |0\rangle.\end{aligned}$$ The fields of $\Phi_{-\bar n}$ are separated by the timelike distances from the fields of $\Phi_{n}$. Thus, one cannot present the SIDIS soft factor as a matrix element of a single T-ordered operator.
However, the SIDIS soft factor can be factorized, as a consequence of the factorization theorem for the DY soft factor. Let us compare these soft factors within the $\delta$-regularization. On the level of Feynman diagrams the only difference between DY and SIDIS soft factors is the sign of $\delta^-$ contribution. I.e. a diagram with $n$-gluons coupled to Wilson lines $\Phi_{-\bar n}$ in the DY case has the form (in the momentum representation) $$\begin{aligned}
\label{FF:1}
I^{\text{DY}}=\int d^dk_1...d^dk_n F(\{k\},\delta^+)\frac{1}{(k^-_1+i\alpha_1\delta^-)...(k^-_n+i\alpha_n\delta^-)},\end{aligned}$$ where $\alpha_i$ are some integers. The same diagram in the SIDIS kinematics reads $$\begin{aligned}
\label{FF:2}
I^{\text{SIDIS}}=\int d^dk_1...d^dk_n F(\{k\},\delta^+)\frac{1}{(k^-_1-i\alpha_1\delta^-)...(k^-_n-i\alpha_n\delta^-)}.\end{aligned}$$ The function $F$ is the same in both cases. We split the integration measure as $d^dk=dk_+dk_-d^{d-2}k_\perp$, and integrate over $k^+$ components. The integration over the $k^+$ components can be done closing the integration contours on the poles of (anti-)Feynman propagators or by $\delta$-functions of cut propagators. Both cases restrict the integration over minus-components to finite or semi-infinite region of integration, $R$. Note, that contributions of eikonal poles do not restrict minus-components. Such contributions vanish in the sum of diagrams, because they result into the power-divergences in $\delta$, which necessarily cancel, see sec.\[sec:delta-structure\]. Therefore, the integral (\[FF:1\]) and (\[FF:2\]) became $$\begin{aligned}
\label{FF:3}
I^{\text{DY}(\text{SIDIS})}=\int d^{d-2}k_{1\perp}...d^{d-2}k_{n\perp}
\int dk^-_1...dk^-_n F'(\{k\},\delta^+)\frac{\theta(k_1^-,...,k_n^-\in R)}{(k^-_1\pm i\alpha_1\delta^-)...(k^-_n\pm i\alpha_n\delta^-)}.\end{aligned}$$ In this integral, the change $\delta^-\to-\delta^-$ can be done without the crossing of the integration contour. Therefore, the SIDIS integrals are related to the DY integral by the analytical continuation $\delta^-\to-\delta^-$. The rapidity divergences arises as $\ln(\delta^+\delta^-)$. The analytical continuation $\delta^-\to-\delta^-$ does not change the coefficient of the highest power of $\ln(\delta^+\delta^-)$, while the coefficients of lower powers can obtain extra terms proportional to $(i\pi)^k$.
Let us note that due to the absence of color-matrix structure in the TMD case the equation (\[RTRD:SIGMA\_FAC\]) reduces to $$\begin{aligned}
\Sigma^{\text{DY}}_{\text{TMD}}(b)=\exp{\left(}2\mathcal{D}(b,\mu)\ln{\left(}\frac{\delta^+\delta^-}{\mu^2}{\right)}+B(b,\mu){\right)},\end{aligned}$$ where $B$ is some rapidity divergences-free function. The logarithm contribution is not affected by analytical continuation. So the SIDIS soft factor is rapidity factorizable. The statement can be enforced. The only possible addition to the finite part $B$ should be proportional to $i\pi$. However, $\Sigma=\Sigma^\dagger$ and thus $$\begin{aligned}
\Sigma_{\text{TMD}}^{\text{DY}}=\Sigma_{\text{TMD}}^{\text{SIDIS}}=\exp{\left(}2\mathcal{D}(b,\mu)\ln{\left(}\frac{|\delta^+\delta^-|}{\mu^2}{\right)}+B(b,\mu){\right)}.\end{aligned}$$ This relation has been checked explicitly at NNLO in [@Echevarria:2015byo]. The method used here cannot be generalized to a $N>2$ case because the matrix MPS soft factor contains the higher powers of $\ln(\delta^+\delta^-)$.
Conclusion
==========
In this work, we have considered the structure of rapidity divergences of the multi-parton scattering (MPS) soft factor. We have proven the renormalization theorem for rapidity divergences (RTRD) for MPS soft factors and discussed some of its consequences. The RTRD states that the rapidity divergences of the MPS soft factor related to different lightlike directions do not mix and can be independently renormalized. It leads to a number of consequences. The main one is the generalization of the TMD factorization theorem for a larger class of processes, e.g. double-parton scattering.
The proof of RTRD relies on the observation that the MPS soft factor can be converted to another soft factor by a conformal transformation. The obtained soft factor has a compact spatial structure and completely defined set of UV divergences. Tracing the transformation of rapidity divergences we connect the UV renormalization factor with the rapidity divergences renormalization factor. This consideration, which is valid in the conformal field theory, can be promoted to QCD using the conformal invariance of QCD at one-loop, and that the rapidity divergences are insensitive to the dimensional regularization. In this way, the RTRD can be seen as a consequence of the renormalization theorem for ultraviolet (UV) divergences and the counting rules for rapidity divergences.
We have studied the rapidity renormalization for the soft factors typical for the Drell-Yan processes. The same procedure can be done for more general soft factors. The only requirement is the possibility to rewrite the soft factor as a matrix element of single T-ordered operator. In the article, we demonstrate an example where the absence of this requirement does not destroy RTRD. This is the TMD soft factor for SIDIS. In this case, the analytical continuation between the DY and SIDIS soft factors can be performed. As a result, these soft factors are equal to each other, what has been discussed in the literature for a long time, see e.g. [@Collins:2011zzd; @Echevarria:2015byo; @Echevarria:2016scs]. A similar study is not obviously possible for many other kinematic configurations. E.g. the soft factors for processes with jets that have restrictions on the integration phase-space [@Stewart:2010tn; @Jouttenus:2011wh]. Nonetheless, even in these cases the application of the conformal transformation $C_{n\bar n}$ (or its analogue) can give a hint on the structure of divergences.
In general, the graph-topological structure of rapidity divergences is the same as for UV divergences (sec.\[sec:rap\_div\]). In this light, RTRD can be seen as the rule for the subdiagram subtractions, which splits the divergences from each other. It suggests a stronger form the RTRD with the independent renormalization of each pack of lightlike Wilson lines that share the same transverse plane at light-cone infinity (see detailed description in sec.\[sec:geom\_rap\_div\]). The rigorous proof of this stronger form of RTRD requires the demonstration of iterative subtraction for rapidity divergences. We expect that it can be performed with the help of Ward identities for the rapidity divergent contributions. Nonetheless, we were not able to pass through this procedure, because in order to disentangle different rapidity divergences a special (singular) gauge fixation condition should be used, which greatly complicates the task.
The formulation of RTRD is made at a finite (although arbitrary) perturbative order. On one hand, it is a consequence of necessity to use the perturbation theory to pass from QCD at the critical coupling to the physical coupling. On another hand sending the order to infinity and studying the asymptote of the perturbation expansions one recovers a part of the non-perturbation corrections associated with renomalon contributions. Therefore, we expect that RTRD can be used non-perturbatively at least for the renormalon contributions. The explicit leading order evaluation confirms it [@Scimemi:2016ffw].
We have derived the all-order color structure of the MPS soft factor and presented its decomposition (up to three-loop order inclusively) in the terms of the generating function. In this way, we have checked the equivalence of the color structure of the soft anomalous dimension (SAD) and the rapidity anomalous dimension (RAD), which is predicted by RTRD. In turn the simple structure of MPS soft factor results to all-order constraints on the SAD. Namely, it predicts the absence of odd-color contributions at all orders, which is not known to our best knowledge. We have also presented in details the SAD-to-RAD correspondence discovered in [@Vladimirov:2016dll], which predicts the three-loop expression for RAD using the finite-$\epsilon$ two-loop calculation [@Echevarria:2015byo], and the three-loop expression for SAD [@Moch:2004pa; @Almelid:2015jia]. The obtained three-loop RAD coincides with the calculation made in [@Li:2016ctv] by bootstrapping the decompositions of TMD and fully differential soft factors. This agreement shows a non-trivial confirmation of RTRD.
The author gratefully acknowledges V.Braun, A.Manashov, and I.Scimemi for numerous stimulating discussions, and M.Diehl for important comments and help with the definition of multi-parton distributions.
$\delta$-regularization {#app:delta-reg}
=======================
The general part of the discussion presented in the article is not restricted to any regularization procedure. For the examples we use the $\delta$-regularization. The connection between $\delta$-regularization and the regularization by the tilted Wilson lines can be found in [@Buffing:2017mqm] (see Appendix B).
The $\delta$-regularization has been consistently formulated in [@Echevarria:2015byo], and used in NNLO calculation [@Echevarria:2015usa; @Echevarria:2016scs; @Vladimirov:2016qkd]. The $\delta$-regularization consists in the following modification of the Wilson line $$\begin{aligned}
\mathbf{\Phi}_v(x)\Big|_{\delta-\text{reg.}}=P\exp{\left(}ig \int_0^\infty d\sigma v^\mu A_\mu^A(v\sigma+x)\mathbf{T}^A e^{-|(v\cdot \delta)|\sigma}{\right)}.\end{aligned}$$ The $\delta$-regularization completely regularizes the rapidity divergences and IR-divergences associated with Wilson lines. To regularize the UV divergences the dimensional regularization is used with $d=4-2\epsilon$ (with $\epsilon>0$).
The $\delta$-regularization is convenient for practical evaluation. The first, it gives a clear separation of rapidity and IR divergences. The rapidity divergences arise as a logarithms of $\delta$. The IR-divergences arise as $\epsilon-$power of $\delta$, e.g. $(\delta^+\delta^-)^{-\epsilon}$. Since $\epsilon>0$, such contribution is explicitly singular. Note, that this separation is clear only at non-zero $\epsilon$. Therefore, we demand that the limit $\delta\to 0$ is taken prior to $\epsilon\to 0$. However, this demand is not necessary for IR-safe matrix-elements. The second, the $\delta$-regularization is defined as a modification of Wilson line operator (in contrast to regularizations which modify e.g. the loop-integral measure). Therefore, the $\delta$-regularization can be applied to any configuration of Wilson lines. At last, the loop calculus with the $\delta$-regularization is simple, due to the fact that it preserves the lightlike vectors.
The IR and rapidity divergences are clearly distinguished within the $\delta$-regularization. Let us demonstrate it for the generic one-loop integral $I_{ij}$ given in (\[rapdiv:1-loop\]). In the $\delta$-regularization the integral reads $$\begin{aligned}
I_{ij}&=&2^{2-2\epsilon}\Gamma(1-\epsilon) \int_0^\infty d L
\frac{2L(v_i\cdot v_j)}{(2(v_i\cdot v_j)L^2+b_{ij}^2+i0)^{1-\epsilon}}\int_0^\infty \frac{d\alpha}{\alpha}e^{-L \delta_i\alpha }e^{-L\delta_j/\alpha },\end{aligned}$$ where $\delta_i=(v_i\cdot \delta)$, and $b^2_{ij}=-(b_i-b_j)^2>0$. The rapidity-divergent regimes $\alpha\to 0$ or $\alpha \to \infty$ result into $\ln\delta_j$ and $\ln\delta_j$ correspondingly. In the IR-regime then $L\to \infty$ the integral has only single dimensional parameter $\delta^2=2\delta_i\delta_j$, and therefore is proportional to $(\delta^2)^{-\epsilon}$. This contribution is singular at $\epsilon>0$ and $\delta \to 0$, and represents the IR-singularity. Indeed, evaluating the integral $I_{ij}$ we obtain $$\begin{aligned}
I_{ij}&=&2\Gamma^2(\epsilon)\Gamma(1-\epsilon){\left(}\frac{2\delta_i \delta_j}{(v_i\cdot v_j)}{\right)}^{-\epsilon}-2 \Gamma(-\epsilon){\left(}\frac{b_{ij}^2}{4}{\right)}^\epsilon
{\left(}\ln{\left(}\frac{b_{ij}^2}{4}\frac{2\delta_i \delta_j}{(v_i\cdot v_j)}{\right)}-\psi(-\epsilon)+\gamma_E{\right)}.\end{aligned}$$ Such structure holds for arbitrary difficult loop integral, due to the fact that rapidity divergences insensitive to the dimensional regularization, while IR-divergences should be regularized at $\epsilon<0$.
The negative point of the $\delta$-regularization is the violation of the gauge-transformation properties of the Wilson line. However, these contributions are easy to trace, since gauge violating contributions are given by the positive powers of $\delta$. Therefore, in the calculation one should keep the parameter $\delta$ infinitesimal[^6], which makes loop-calculus even simpler.
Cancellation of mass-divergences in $\delta$-regularization {#sec:delta-structure}
-----------------------------------------------------------
Any $n$-loop diagram contributing to the MPS soft factor in the $\delta$-regularization has a generic form $$\begin{aligned}
\label{rad:gen_integral}
\mathbf{M}^{[n]}=(\delta^2)^{-n\epsilon}\mathbf{A}^{[n]}_n(\epsilon)+(\delta^2)^{-(n-1)\epsilon}(b^2)^{\epsilon}\mathbf{A}^{[n]}_{n-1}(\ln(\delta^2),\{b\},\epsilon)+...+
(b^2)^{n\epsilon}\mathbf{A}^{[n]}_{0}(\ln(\delta^2),\{b\},\epsilon),\end{aligned}$$ where $b^2$ is a transverse distance, say $b^2=(b_1-b_2)^2$, and $\mathbf{A}$ are dimensionless functions of transverse distances, rapidity divergent logarithms and parameter $\epsilon$. Note, that due to the Lorentz invariance the regularization parameters $\delta_{i,j}$ can appear only the combination $\delta^2$.
If color indices of MPS form a singlet, the IR-divergences cancel at each order of perturbation theory. It can be proven as following. Let us rescale $b_i\to l b_i$. If the color indices form singlets, the MPS soft factor should reduce to unity in the limit $\lambda\to 0$, $$\begin{aligned}
\label{rap:to_unity}
\lim_{\lambda\to 0}\mathbf{\Sigma}(\{\lambda b\},\delta)=\mathbf{I}.\end{aligned}$$ It is the consequence of operator identity $\mathbf{\Phi}^\dagger_v(z)\mathbf{\Phi}_v(z)=\mathbf{I}$, which holds at arbitrary $\delta$ (even not infinitesimal). Therefore, the sum over diagrams at any given order vanishes in this limit $$\begin{aligned}
\label{rap:11}
\lim_{\lambda\to 0}\sum_{\text{diag.}}\mathbf{M}^{[n]}(\{\lambda b\},\delta)=0.\end{aligned}$$ The functions $\mathbf{A}$ being dimensionless dependent on $\lambda$ only logarithmically. Therefore, all entires $\mathbf{A}_{i\neq n}$ in the expression (\[rad:gen\_integral\]) vanish in the limit $\lambda\to0$. Consequently, we have $$\begin{aligned}
\label{rap:22}
\sum_{\text{diag.}}\mathbf{A}^{[n]}_n(\epsilon)=0.\end{aligned}$$ Next, we rescale $\delta^2\to \delta^2 \lambda^{2/(n-1)}$. The relation (\[rap:to\_unity\]) holds. Considering (\[rap:11\]) we obtain $\sum_{\text{diag.}}\mathbf{A}^{[n]}_{n-1}(\epsilon)=0.$ On the next step we rescale $\delta^2\to \delta^2 l^{2/(n-2)}$, and demonstrate the absence of $\mathbf{A}^{[n]}_{n-2}$ constitutions. And so on. In this way, we obtain that $$\begin{aligned}
\sum_{\text{diag.}}\mathbf{A}^{[n]}_k(\epsilon)=0,\qquad k>0.\end{aligned}$$
The cancellation of IR divergences takes a place *only* for color-singlet components of the MPS soft factor. The colored contributions are IR divergent, which can be seen already at NLO (see e.g.(\[app:MPS\_colored\])). In the spirit of presented discussion, the colored contributions do not obey the relation (\[rap:to\_unity\]), and thus, should not cancel in the sum of diagrams. Practically, it is convenient to keep contributions $\mathbf{A}_{k>0}^{[n]}$ in the diagrams, since they cancellation presents a nice check of the calculation.
Generating function decomposition of MPS soft factor {#app:color}
====================================================
The generating function approach for the exponentiation of matrix elements of Wilson lines has been elaborated in [@Vladimirov:2015fea; @Vladimirov:2014wga]. It naturally generalizes the well-known non-Abelian exponentiation technique for Wilson loops [@Gatheral:1983cz; @Frenkel:1984pz], onto the arbitrary configuration of Wilson lines. It is a powerful method which decouples the external color structure (i.e. the color part related to the Wilson lines, but not to the intrinsic loops) from the momentum integration. In this approach the final expression is given in the term of color generators and generating functions: the connected matrix elements of operator $V$, which are discussed later.
The operators $V$ are by-products of Wilson lines, and inherit their geometrical structure. It is convenient to present the final result via generating functions $W$ defined on the most elementary geometrical structures. In the case of MPS soft factor these are straight lightlike ray or paths of individual $\Phi$’s. However, color indices are contracted between pairs of $\mathbf{\Phi}$’s and thus, from the point of color-decomposition, $\mathbf{\Phi}$ is not an elementary object.
There are two principal approaches in this situation. The first approach is to decouple the color indices at the transverse plane. The resulting object $\widetilde{\mathbf{\Sigma}}$ has $2N$-pairs of color indices. It can be straightforwardly written in the terms of elementary generating functions $W$ as $$\begin{aligned}
\widetilde{\mathbf{\Sigma}}=e^{\mathbf{T}^{A_1}_1\frac{\partial}{\partial \theta^{A_{1}}_1}}...e^{\mathbf{T}^{A_{2N}}_{2N}\frac{\partial}{\partial \theta^{A_{2N}}_{2N}}}e^{W[\theta]}\Big|_{\theta=0}.\end{aligned}$$ The color indices are coupled within the differential operator, which produces a more complicated operator, that act on the generating exponent, $$\begin{aligned}
\mathbf{\Sigma}=e^{\pmb{\mathcal{D}}[\frac{\partial}{\partial \theta_1},\frac{\partial}{\partial \theta_{N+1}}]}...
e^{\pmb{\mathcal{D}}[\frac{\partial}{\partial \theta_N},\frac{\partial}{\partial \theta_{2N}}]}e^{W[\theta]}\Big|_{\theta=0},\end{aligned}$$ with $$\begin{aligned}
{\nonumber}\pmb{\mathcal{D}}[\frac{\partial}{\partial \theta_1},\frac{\partial}{\partial \theta_{N+1}}]=\ln{\left(}e^{\mathbf{T}^A\frac{\partial}{\partial \theta^A_1}}e^{\mathbf{T}^B\frac{\partial}{\partial \theta^B_{N+1}}}{\right)},\end{aligned}$$ where matrices $\mathbf{T}$ are contracted. This approach has been used in [@Vladimirov:2016qkd] for the calculation of DPS soft factor.
The second approach applies the same procedures in the opposite order. The expression for the MPS soft factor reads $$\begin{aligned}
\label{app:formula1}
\mathbf{\Sigma}=e^{\mathbf{T}^{A_1}_1\frac{\partial}{\partial \theta^{A_{1}}_1}}...e^{\mathbf{T}^{A_{N}}_{N}\frac{\partial}{\partial \theta^{A_{N}}_{N}}}e^{\mathcal{W}[\theta]}\Big|_{\theta=0}=e^{\pmb{\mathcal{W}}+\pmb \delta \pmb{\mathcal{W}}[\mathcal{W}]},\end{aligned}$$ where $\mathcal{W}$ is the generating functions $\mathcal{W}$ for operators $\mathcal{V}$, which are defined on the cusped paths, $\pmb{\mathcal{W}}=\mathcal{W}[\mathbf{T}]$, and $\pmb \delta \pmb{\mathcal{W}}$ is the algebraic function of $\mathcal{W}$. The function $\delta\mathcal{W}$ is derived and discussed in details in ref.[@Vladimirov:2014wga], and is called the defect of exponential procedure. In the turn, the operators $\mathcal{V}$ defined on an arbitrary paths can be rewritten in the terms of operators $V$ defined on elementary segments. Consequently, the generating function $\mathcal{W}$ can be presented in the terms of elementary generating functions $W$, and substituted to (\[app:formula1\]).
In the following, we present in the details the calculation performed within the second approach.
Evaluation of color structure
-----------------------------
The MPS soft factor given in eq.(\[SF:MPS\]) can be conveniently presented in the form $$\begin{aligned}
\label{app:MPS_SF}
\mathbf{\Sigma}(b_1,b_2,...,b_N)=\langle 0|T\{\mathbf{\Lambda}(b_N)...\mathbf{\Lambda}(b_2)\mathbf{\Lambda}(b_1)\}|0\rangle,\end{aligned}$$ where $\mathbf{\Lambda}(z)$ is a single Wilson lines build from two segment that meet at the point $z$, $$\begin{aligned}
\mathbf{\Lambda}(z)=\mathbf{\Phi}_{-n}(z)\mathbf{\Phi}^\dagger_{-\bar n}(z).\end{aligned}$$ The color indices are contracted at the cusp, but remain open on the ends of Wilson lines.
The operator $\mathbf{\Lambda}$ can be written as $$\begin{aligned}
\mathbf{\Lambda}(z)=P\exp{\left(}ig\int_{\gamma} dy^\mu A_\mu^A(y+z)\mathbf{T}^A{\right)}=e^{\mathbf{T}^A \mathcal{V}_A},\end{aligned}$$ where $\gamma$ is the path of Wilson line. The expression for the operators $\mathcal{V}$ can be found in [@Vladimirov:2015fea; @Vladimirov:2014wga]. In the terms of generating functions for these operator the MPS soft factors takes the form $$\begin{aligned}
\mathbf{\Sigma}(\{b\})={\left(}\prod_{i=1}^N e^{\mathbf{T}^{A_i}\frac{\partial}{\partial \theta^{A_i}_i}}{\right)}e^{\mathcal{W}[\{\theta\},\{b\}]}\Big|_{\theta=0},\end{aligned}$$ where $$\begin{aligned}
e^{\mathcal{W}[\{\theta\},\{b\}]}=\langle 0|e^{\sum_{i=1}^N \theta_i^{A}\mathcal{V}_i}|0\rangle,\end{aligned}$$ where $\mathcal{V}_i=\mathcal{V}(b_i)$.
The generating function $\mathcal{W}$ contains only fully connected matrix element of various compositions of operators $\mathcal{V}$. It has the general form $$\begin{aligned}
\label{app:W[theta]}
\mathcal{W}[\theta]&=&\sum_i \theta_i^A \mathcal{W}_i^A+\frac{1}{2}\sum_{i,j} \theta_i^A \theta_j^B \mathcal{W}^{AB}_{ij}
+...+\frac{1}{n!}\sum_{i,j,..,k} \theta_i^A \theta_j^B..\theta_k^C \mathcal{W}^{AB..C}_{ij..k}+...~,\end{aligned}$$ where the summation runs from $1$ to $N$ for each summation label, and $$\begin{aligned}
\label{app:calW_def}
\mathcal{W}^{AB..C}_{ij..k}=\langle\langle \mathcal{V}_i^A\mathcal{V}_j^B...\mathcal{V}_k^C \rangle\rangle,\end{aligned}$$ where double brackets $\langle\langle..\rangle\rangle$ denote the connected part of the matrix element. Accordingly, $\mathcal{W}_{ij..}$ depends only on $\{b_i,b_j,...\}$. The functions $\mathcal{W}^{A..B}_{i..j}$ are necessarily symmetric over the permutations over the pairs of indices $(A,i)$. The matrix element $\langle\langle \mathcal{V}_1...\mathcal{V}_n\rangle\rangle$ has is proportional to $a_s^{n-1}$ at least.
The color indices of a generating function $\mathcal{W}^{A...B}_{i...j}$ are restricted to color-singlets, due to the global color-conservation. For the consideration of $a_s^3$-order the following function are required, $$\begin{aligned}
\mathcal{W}^A_i&=&0,{\nonumber}\\
\mathcal{W}^{AB}_{ij}&=&a_s\delta^{AB} \mathcal{W}_{ij},\label{app:W->Wsinglet}
\\
\mathcal{W}^{ABC}_{ijk}&=&a_s^2if^{ABC} \mathcal{W}_{ijk}+a_s^3 d^{ABC}\mathcal{W}^{(s)}_{ijk},{\nonumber}\\
\mathcal{W}^{ABCD}_{ijkl}&=&a_s^3 {\left(}if^{AB;CD}W_{ijkl}+if^{AC;BD}W_{ikjl}+if^{AD;BC}W_{iljk}{\right)}+...,{\nonumber}\end{aligned}$$ where $if^{AB;CD}=if^{AB\alpha}if^{\alpha CD}$, $d^{ABC}=2{\mathrm{Tr}}(\mathbf{T}_{adj}^A\{\mathbf{T}_{adj}^B\mathbf{T}_{adj}^C\})$. We have extracted the minimal perturbative order from functions $\mathcal{W}_{i...j}$, however, note that functions $\mathcal{W}_{i...j}$ have all perturbative orders. The dots in the last line of (\[app:W->Wsinglet\]) denote the contributions of order $a_s^4$ which are accompanied by different color structures, e.g. by $f^{AB\alpha}d^{\alpha CD}$.
The action of the derivative exponent can be presented as the sum of terms (\[app:formula1\]). The first term $\pmb{\mathcal{W}}$ is obtained from (\[app:W\[theta\]\]) by substitution of sources $\theta^A_i$ by $\mathbf{T}^A_i$. The result of substitution can be written in the form $$\begin{aligned}
\label{app:W_eq2}
\pmb{\mathcal{W}}&=&\frac{a_s}{2}\sum_i C_{i}\mathcal{W}_{ii}+\frac{a_s}{2}\sum_{[i,j]}\mathbf{T}^A_i\mathbf{T}^A_j\mathcal{W}_{ij}
+\frac{a_s^2}{3!}\sum_{[i,j,k]}\mathbf{T}_i^A\mathbf{T}_j^B\mathbf{T}_k^C if^{ABC}\mathcal{W}_{ijk}
\\
&&{\nonumber}+\frac{a_s^3}{3!}\sum_{[i,j,k]}\mathbf{T}_i^A\mathbf{T}_j^B\mathbf{T}_k^C d^{ABC}\mathcal{W}_{ijk}^{(s)}
+a_s^3\sum_{[i,j]}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^C d^{ABC}\frac{\mathcal{W}_{iij}^{(s)}}{2}
+a_s^3\sum_{i}\mathbf{T}_i^{\{ABC\}}d^{ABC}\mathcal{W}_{iii}^{(s)}
\\
&&{\nonumber}+a_s^3\sum_{[i,j]}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^{\{CD\}}if^{AC;BD}\frac{\mathcal{W}_{ijij}}{4}
+a_s^3\sum_{[i,j,k]}\mathbf{T}_i^{\{AB\}}\mathbf{T}^C_j\mathbf{T}_k^D if^{AC;BD}\frac{\mathcal{W}_{ijik}}{2}
\\&&{\nonumber}+a_s^3\sum_{[i,j,k,l]}\mathbf{T}_i^A\mathbf{T}_j^B\mathbf{T}_k^C\mathbf{T}_l^Dif^{AC;BD}\frac{\mathcal{W}_{ijkl}}{4}+\mathcal{O}(a_s^4),\end{aligned}$$ where $C_i$ is the quadratic Casimir eigenvalue of $i$’th representation $C_i=\mathbf{T}_i^A\mathbf{T}_i^A$, and the summations run from $1$ to $N$ for each summation label, and none of labels are equal (which we denote by square brackets). The symmetric combinations of generators are labeled by curly brackets, $\mathbf{T}^{\{AB\}}_i=(\mathbf{T}^A_i\mathbf{T}^B_i+\mathbf{T}^B_i\mathbf{T}^A_i)/2$, $\mathbf{T}^{\{ABC\}}_i=(\mathbf{T}^A_i\mathbf{T}^B_i\mathbf{T}^C_i+...+\mathbf{T}^C_i\mathbf{T}^B_i\mathbf{T}^A_i)/6$, etc. To present the expression (\[app:W\_eq2\]) in compact form, Jacobi identities have been used.
The derivation of the general form for the defect contribution in given in ref.[@Vladimirov:2015fea]. It can be presented as $\pmb{\delta \mathcal{W}}=\pmb{\delta}_2 \pmb{\mathcal{W}}+\pmb{\delta}_3 \pmb{\mathcal{W}}+...$, where $\pmb{\delta}_n \pmb{\mathcal{W}}$ contains algebraic combinations of $n$ entries of $\mathcal{W}$. At $a_s^3$-order only two leading terms contribute. They are $$\begin{aligned}
\pmb{\delta}_2 \pmb{\mathcal{W}}&=&\frac{\{\pmb{\mathcal{W}}^2\}}{2}-\frac{\pmb{\mathcal{W}}^2}{2},
\\
\pmb{\delta}_3 \pmb{\mathcal{W}}&=&\frac{\{\pmb{\mathcal{W}}^3\}}{6}-\frac{\pmb{\delta}_2 \pmb{\mathcal{W}}\,\pmb{\mathcal{W}}
+\pmb{\mathcal{W}}\,\pmb{\delta}_2 \pmb{\mathcal{W}}}{2}-\frac{\pmb{\mathcal{W}}^3}{6},\end{aligned}$$ where curly brackets denote the complete symmetrization of generators. Evaluation of these expressions is straightforward. The results are $$\begin{aligned}
\pmb{\delta}_2\pmb{\mathcal{W}}&=&-a_s^2\frac{C_A}{48}\sum_{i}C_i\mathcal{W}_{ii}^2+a_s^2\frac{C_A}{48}\sum_{[i,j]}\mathbf{T}_i^A\mathbf{T}_j^A{\left(}3\mathcal{W}_{ij}^2-4\mathcal{W}_{ij}\mathcal{W}_{ii}{\right)}\\{\nonumber}&&+
a_s^3\frac{C_A}{48}\sum_{[i,j,k]}\mathbf{T}_i^A\mathbf{T}_j^B\mathbf{T}_k^C if^{ABC}\mathcal{W}_{ijk}{\left(}3\mathcal{W}_{ij}
-2\mathcal{W}_{ii}{\right)}+\mathcal{O}(a_s^4),\end{aligned}$$ $$\begin{aligned}
\pmb{\delta}_3\pmb{\mathcal{W}}&=&a_s^3 \frac{C_A^2}{576}\sum_i C_i \mathcal{W}_{ii}^3
\\{\nonumber}&&+a_s^3\frac{C_A^2}{576}\sum_{[i,j]}\mathbf{T}_i^A\mathbf{T}_j^A
{\left(}6\mathcal{W}_{ii}^2\mathcal{W}_{ij}-12\mathcal{W}_{ii}\mathcal{W}^2_{ij}+2\mathcal{W}_{ii}\mathcal{W}_{jj}\mathcal{W}_{ij}+5\mathcal{W}_{ij}^3
{\right)}\\{\nonumber}&&
+\frac{a_s^3}{24}\sum_{[i,j]}\mathbf{T}^{\{AB\}}_i\mathbf{T}^{\{CD\}}_jif^{AC;BD}{\left(}\mathcal{W}_{ij}^3-\mathcal{W}_{ij}^2\mathcal{W}_{ii}{\right)}\\{\nonumber}&&
+\frac{a_s^3}{24}\sum_{[i,j,k]}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^C\mathbf{T}_k^D
if^{AC;BD}{\left(}-\mathcal{W}_{ij}\mathcal{W}_{ik}\mathcal{W}_{jk}+2\mathcal{W}_{ij}\mathcal{W}_{ik}^2-\mathcal{W}_{ii}\mathcal{W}_{ij}\mathcal{W}_{ik}{\right)}\\{\nonumber}&&
+\frac{a_s^3}{24}\sum_{[i,j,k,l]}\mathbf{T}_i^{A}\mathbf{T}_j^B\mathbf{T}_k^C\mathbf{T}_l^D
if^{AC;BD}{\left(}-\mathcal{W}_{ij}\mathcal{W}_{ik}\mathcal{W}_{jl}{\right)}+\mathcal{O}(a_s^4),\end{aligned}$$ where $C_A=N_c$.
Segment reduction of the generating function {#app:segment_decomposition}
--------------------------------------------
To reduce the generating function $\mathcal{W}$ (that are connected matrix elements of $\mathcal{V}$) to elementary generating functions $W$ (that are connected matrix elements of $V$) we recall that $$\begin{aligned}
\label{app:eV=eVeV}
\mathbf{\Lambda}(z)=e^{\mathbf{T}^A \mathcal{V}_A(z)}=\mathbf{\Phi}_{-n}(z)\mathbf{\Phi}^\dagger_{-\bar n}=e^{-\mathbf{T}^A V^{n}_A(z)}e^{\mathbf{T}^B V^{\bar n}_B(z)}.\end{aligned}$$ The different signs infront of $V^n$ and $V^{\bar n}$ are consequence of the definition of $V$ on the path from $0$ to infinity. Using Baker-Campbell-Hausdorff formula we obtain $$\begin{aligned}
\mathcal{V}_A&=&-V^n_A+V^{\bar n}_A-\frac{if^{ABC}}{2}V^n_BV^{\bar n}_C+\frac{if^{AB;CD}}{12}{\left(}V^n_BV^n_CV^{\bar n}_D-V^{\bar n}_BV^{\bar n}_CV^n_D{\right)}\\{\nonumber}&&
-\frac{if^{AB;C;DE}}{24}V^{\bar n}_BV^n_CV^n_DV^{\bar n}_E+
\frac{if^{AB;C;D;EF}}{720}\Big(V^{\bar n}_BV^{\bar n}_CV^{\bar n}_DV^{\bar n}_EV^n_F-V^n_BV^n_CV^n_DV^n_EV^{\bar n}_F
\\&&{\nonumber}+2 V^n_BV^{\bar n}_CV^{\bar n}_DV^{\bar n}_EV^n_F-2V^{\bar n}_BV^n_CV^n_DV^n_EV^{\bar n}_F
+6 V^{\bar n}_BV^n_CV^{\bar n}_DV^n_EV^{\bar n}_F-6V^n_BV^{\bar n}_CV^n_DV^{\bar n}_EV^n_F
\Big)
\\{\nonumber}&&+\mathcal{O}(g^6),\end{aligned}$$ where we omit the arguments $z$ and $$if^{AB;C;...;EF}=if^{AB\alpha}if^{\alpha C\beta}if^{\beta...}...if^{...\gamma}if^{\gamma EF}.$$
There is an important consequence of (\[app:eV=eVeV\]), which gives exact restrictions on generating functions. We observe that $$\begin{aligned}
\mathcal{V}_A=-\mathcal{V}_A(n\leftrightarrow \bar n).\end{aligned}$$ Alternatively, the exchange of $n$ and $\bar n$ can done by a rotation. It implies $$\begin{aligned}
\mathcal{W}=\mathcal{W}(n \leftrightarrow \bar n).\end{aligned}$$ Therefore, the generating functions of odd power of $\mathcal{V}$ are exactly zero: $$\begin{aligned}
\label{app:odd=0}
\mathcal{W}_{i_1...i_{2n+1}}^{A_1...A_{2n+1}}=0.\end{aligned}$$ For the case of generating functions $\mathcal{W}_{ijk}$ this statement was demonstrated in [@Vladimirov:2016qkd]. Important to note that absence of generating functions with odd-number of indices does not imply the absence of diagrams which connect odd number of Wilson lines. Such contributions are possible, but the number of connections will be even. The generators belonging to the same Wilson lines always appear in the symmetric composition. The general all-order structure can be written as $$\begin{aligned}
\label{app:color-hyirarchy}
\mathbf{\Sigma}(\{b\})=\exp{\left(}\sum_{\substack{n=2\\n\in\text{even}}}^{\infty}a_s^{n/2}\sum_{i_1,...,i_n=1}^N\{\mathbf{T}^{A_1}_{i_1}...\mathbf{T}^{A_n}_{i_n}\}\sigma^{n;i_1...i_n}_{A_1...A_n}(\{b\}){\right)}.\end{aligned}$$ In the next paragraph we present the first two entries of this expression.
The $\mathcal{W}$ is given by the connected matrix element of $\mathcal{V}$’s, see (\[app:calW\_def\]). However, the operators $V$ inside $\mathcal{V}$ are not necessary connected. Therefore, we have to decompose matrix elements over their connected parts, e.g. $$\langle\langle \{V_1\}\{V_2V_3V_4\}\rangle\rangle=\langle\langle V_1V_2V_3V_4\rangle\rangle+
\langle\langle V_1V_2\rangle\rangle\langle\langle V_3V_4\rangle\rangle
+\langle\langle V_1V_3\rangle\rangle\langle\langle V_2V_4\rangle\rangle
+\langle\langle V_1V_4\rangle\rangle\langle\langle V_2V_3\rangle\rangle.$$ In this example, $\{V_1\}$ and $\{V_2V_3V_4\}$ are resulted from separate $\mathcal{V}$’s and thus the connectivity between these operators should be preserved. By simple algebraic manipulations we arrive to the expressions for $\mathcal{W}$ in the terms of elementary generating functions. To present the result in the compact form let us introduce the notation which respect the symmetries of matrix elements $$\begin{aligned}
{\nonumber}&&\langle\langle V_A^{v_i}(b_i)V_B^{v_j}(b_j) \rangle\rangle=a_s \delta_{AB}(v_i\cdot v_j)W(b_{ij}),
\\\label{app:W_def}
&&\langle\langle V_A^{v_i}(b_i)V_B^{v_j}(b_j)V_C^{v_k}(b_k) \rangle\rangle =a_s^2 i f_{ABC}\Big[
(v_i\cdot v_j)(v_i\cdot v_k)W(b_{ij},b_{ik},b_{jk})
\\{\nonumber}&&\qquad\qquad\qquad\qquad\qquad\qquad
+(v_i\cdot v_j)(v_j\cdot v_k)W(b_{jk},b_{ij},b_{ik})
+(v_j\cdot v_k)(v_i\cdot v_k)W(b_{ik},b_{jk},b_{ij})\Big],\end{aligned}$$ where $b_{ij}=b_i-b_j$ and $$W(x,y,z)=-W(y,x,z).$$ In the parameterization (\[app:W\_def\]), we have taken into account that Wilson lines are lightlike. The parametrization of the generating function of the fourth order is cumbersome. Therefore, we simply denote $$\begin{aligned}
&&\langle\langle V_A^{v_i}(b_i)V_B^{v_j}(b_j)V_C^{v_k}(b_k)V_D^{v_l}(b_l) \rangle\rangle
\\{\nonumber}&&\qquad\qquad=a_s^3{\left(}i f^{AB;CD}W^{v_iv_jv_kv_l}_{ijkl}+if^{AC;BD}W^{v_iv_kv_jv_l}_{ikjl}+if^{AD;BC}W^{v_iv_lv_jv_k}_{iljk}{\right)}{\nonumber}+\mathcal{O}(a_s^4).\end{aligned}$$ The generating functions $\mathcal{W}$ reads $$\begin{aligned}
\mathcal{W}_{ij}&=&
-2W(b_{ij})+a_s C_A{\left(}-2 W(b_{ij},0,b_{ij})+\frac{W^2(b_{ij})}{4}-\frac{W(b_{ij})W(0)}{3}{\right)}\\&&{\nonumber}a_s^2 C_A^2 \Bigg(-\frac{W^3(b_{ij})}{48}+\frac{5 W^2(b_{ij})W(0)}{24}-\frac{7W(b_{ij})W^2(0)}{72}
\\&&{\nonumber}\frac{W_{iijj}^{n\bar n n\bar n}}{4}+\frac{W_{ijij}^{n n \bar n \bar n}+W_{ijij}^{n \bar n n \bar n}}{8}
+\frac{W_{ijjj}^{n n n \bar n}+W_{ijjj}^{n\bar n n\bar n}
+W_{jiii}^{n n n \bar n}+W_{jiii}^{n\bar n n\bar n}}{8}\Bigg)+\mathcal{O}(a_s),\end{aligned}$$ $$\begin{aligned}
{\nonumber}\mathcal{W}_{ijkl}=2\Big(W_{ijkl}^{nn\bar n\bar n}+W_{ijkl}^{n\bar nn\bar n}+W_{ijkl}^{\bar nnn\bar n}
-W_{ijkl}^{nnn\bar n}- W_{ijkl}^{nn\bar n n}-W_{ijkl}^{n\bar nn n}-W_{ijkl}^{\bar nnnn}\Big)+\mathcal{O}(a_s).\end{aligned}$$
Result {#app:results}
------
Finally, we combine the expression for the MPS soft factor in the form $$\begin{aligned}
\label{app:MPS_colored}
\mathbf{\Sigma}&=&\exp\Bigg[a_s\sum_i C_i X_{0}+a_s \sum_{[i,j]}\mathbf{T}_i^A\mathbf{T}_j^A X_{2}^{ij}+
a_s^3 \Big(\sum_{[i,j]}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^{\{CD\}}if^{AC;BD}X_{4}^{ij}
\\{\nonumber}&&+\sum_{[i,j,k]}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^C\mathbf{T}_k^D if^{AC;BD}X_{4}^{ijk}
+\sum_{[i,j,k,l]}\mathbf{T}_i^{A}\mathbf{T}_j^B\mathbf{T}_k^C\mathbf{T}_l^D if^{AC;BD}X_{4}^{ijkl}\Big)+\mathcal{O}(a_s^4)\Bigg].\end{aligned}$$ The functions $X$ are $$\begin{aligned}
X_{0}&=&-W(0)-a_s\frac{C_A}{8}W^2(0)+\frac{a_s^3 C_A^2}{16} {\left(}\frac{7}{18}W^3(0)+5W_{iiii}^{n\bar nn\bar n}{\right)}+\mathcal{O}(a_s^3),
\\
X_{2}^{ij}&=&-W(b_{ij})+\frac{a_s C_A}{2}{\left(}\frac{3}{4}W^2(b_{ij})-W(b_{ij})W(0)-2 W(b_{ij},0,b_{ij}){\right)}\\{\nonumber}&&+\frac{a_s^2C_A^2}{16}\Big(-\frac{41}{18}W^3(b_{ij})+\frac{19}{3}W^2(b_{ij})W(0)-\frac{11}{3}W(b_{ij})W^2(0)
\\{\nonumber}&&+8W(b_{ij})W(b_{ij},0,b_{ij})
-\frac{16}{3}W(b_{ij})W(b_{ij},0,b_{ij})+
\\{\nonumber}&&
2W_{iijj}^{n\bar n n\bar n}+W_{ijij}^{n n \bar n \bar n}+W_{ijij}^{n \bar n n \bar n}
+2W_{ijjj}^{n n n \bar n}+2W_{ijjj}^{n\bar n n\bar n}\Bigg)+\mathcal{O}(a_s^3)
\\
X_{4}^{ij}&=&-\frac{W^3(b_{ij})}{3}+\frac{W^2(b_{ij})W(0)}{3}
\\{\nonumber}&&+\frac{W_{ijij}^{nn\bar n\bar n}+W_{ijij}^{n\bar n n\bar n}
+W_{ijij}^{\bar nnn\bar n}}{2}-W_{ijij}^{nnn\bar n}-W_{ijij}^{nn\bar n n}+\mathcal{O}(a_s)
\\
X_{4}^{ijk}&=&\frac{W(b_{ij})W(b_{ik})}{3}{\left(}W(b_{jk})+W(0)-2W(b_{ik}){\right)}+\mathcal{W}_{ijik}+\mathcal{O}(a_s),
\\
X_{4}^{ijkl}&=&\frac{W(b_{ij})W(b_{ik})W(b_{jl})}{3}+\frac{\mathcal{W}_{ijkl}}{4}+\mathcal{O}(a_s).\end{aligned}$$ To present expression in this form, we have used the permutation symmetries, the fact that indices $i,j,k,l$ are summed, and the Jacobi identities.
If we impose the color conservation condition $$\begin{aligned}
\sum_{i=1}^N\mathbf{T}_i^A=0,\end{aligned}$$ we can eliminate some terms in favor of another terms. E.g. the following decomposition looks reasonable $$\begin{aligned}
\mathbf{\Sigma}&=&\exp\Bigg[-a_s \sum_{[i,j]}\mathbf{T}_i^A\mathbf{T}_j^A \sigma(b_{ij})
\\{\nonumber}&&
+a_s^3 \Big(\sum_{[i,j,k]}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^C\mathbf{T}_k^D if^{AC;BD}Y_{4}^{ijk}
+\sum_{[i,j,k,l]}\mathbf{T}_i^{A}\mathbf{T}_j^B\mathbf{T}_k^C\mathbf{T}_l^D if^{AC;BD}X_{4}^{ijkl}\Big)+\mathcal{O}(a_s^4)\Bigg].\end{aligned}$$ where $$\begin{aligned}
\sigma(b_{ij})&=&X_0-X_2^{ij}-\frac{C_A^2}{2}X_4^{ij},
\\
Y_4^{ijk}&=&X_4^{ijk}-\frac{X_4^{ij}+X_4^{ik}}{2}.\end{aligned}$$ This decomposition can be written in the form used in [@Almelid:2015jia] as $$\begin{aligned}
\mathbf{\Sigma}&=&\exp\Bigg[-2a_s \sum_{i<j}\mathbf{T}_i^A\mathbf{T}_j^A Y_{2}^{ij}
+a_s^3 \Big(2\sum_{\substack{j<k\\i\neq j,k}}\mathbf{T}_i^{\{AB\}}\mathbf{T}_j^C\mathbf{T}_k^D if^{AC;BD}Y_{4}^{ijk}
\\{\nonumber}&&
+4\sum_{i<j<k<l}\mathbf{T}_i^{A}\mathbf{T}_j^B\mathbf{T}_k^C\mathbf{T}_l^D
{\left(}if^{AB;CD}X_{4}^{ijkl}+if^{AC;BD}X_{4}^{ikjl}+if^{AD;BC}X_{4}^{iljk}{\right)}\Big)+\mathcal{O}(a_s^4)\Bigg].\end{aligned}$$
Explicit expressions for anomalous dimensions {#app:ADs}
=============================================
In this appendix, we collect the expressions for anomalous dimensions which are used in the text.
The QCD $\beta$ function and its leading coefficients are $$\begin{aligned}
\beta(a_s)&=&\sum_{n=0}^\infty \beta_n a_s^{n+1},
\\{\nonumber}\beta_0&=& \frac{11}{3}C_A-\frac{2}{3}N_f,\qquad \beta_1= \frac{34}{3}C_A^2-\frac{10}{3}C_AN_f-2C_FN_f,\end{aligned}$$ where $C_A=N_c$ and $C_F=(N_c^2-1)/2N_c$ are eigenvalues of quadratic Casimir operator for adjoint and fundamental representations, and $N_f$ is the number of flavors. The cusp-anomalous dimension and its leading coefficients [@Moch:2004pa] are $$\begin{aligned}
\Gamma_{cusp}^i &=& 4 C_i\sum_{n=0}^\infty a_s^{n+1}\Gamma_n,
\\{\nonumber}\Gamma_0&=&1,\qquad \Gamma_1={\left(}\frac{67}{9}-2\zeta_2{\right)}C_A-\frac{10}{9}N_f,
\\{\nonumber}\Gamma_2&=&C_A^2{\left(}\frac{245}{6}-\frac{268}{9}\zeta_2+22\zeta_4+\frac{22}{3}\zeta_3{\right)}+C_AN_f
{\left(}-\frac{209}{27}+\frac{40}{9}\zeta_2-\frac{56}{3}\zeta_3{\right)}\\{\nonumber}&&\qquad\qquad\qquad\qquad\qquad\qquad+C_FN_f{\left(}-\frac{55}{6}+8\zeta_3{\right)}-\frac{4 N_f^2}{27},\end{aligned}$$ where $i$ is a representation of Wilson lines. The non-cusp part of the SAD and its leading coefficients [@Moch:2004pa] are $$\begin{aligned}
\tilde \gamma^i_s&=&C_i\sum_{n=0}^\infty a_s^{n+1}\gamma_n,
\\{\nonumber}\gamma_0&=&0,\qquad
\gamma_1=C_A{\left(}-\frac{808}{27}+\frac{22}{3}\zeta_2+28 \zeta_3{\right)}+{\left(}\frac{112}{27}-\frac{4}{3}\zeta_2{\right)}N_f
\\{\nonumber}\gamma_2&=&C^2_A{\left(}-\frac{136781}{729}+\frac{12650}{81}\zeta_2+\frac{1316}{3}\zeta_3-176\zeta_4-\frac{176}{3}\zeta_2\zeta_3-192\zeta_5{\right)}\\{\nonumber}&&+
{\left(}\frac{11842}{729}-\frac{2828}{81}\zeta_2-\frac{728}{27}\zeta_3+48\zeta_4{\right)}C_AN_f+{\left(}\frac{1711}{27}-4\zeta_2-\frac{304}{9}\zeta_3-16\zeta_4{\right)}C_FN_f
\\{\nonumber}&&+{\left(}\frac{2080}{729}+\frac{40}{27}\zeta_2-\frac{112}{27}\zeta_3{\right)}N_f^2.\end{aligned}$$
The rapidity anomalous dimension has the form $$\begin{aligned}
\label{app:d}
\mathcal{D}^i&=&C_i \sum_{n=1}^\infty a_s^n \sum_{k=0}^n L_b^k d^{(n,k)},\qquad L_b=\ln{\left(}\frac{\mu^2 b^2}{4 e^{-2\gamma_E}}{\right)},
\\{\nonumber}d^{(1,0)}&=&0,\qquad d^{(2,0)}=C_A{\left(}\frac{404}{27}-14 \zeta_3{\right)}-\frac{56}{27}N_f,
\\{\nonumber}d^{(3,0)}&=&C_A^2{\left(}\frac{297029}{1458}-\frac{3196}{81}\zeta_2-\frac{6164}{27}\zeta_3-\frac{77}{3}\zeta_4+\frac{88}{3}\zeta_2\zeta_3+96\zeta_5{\right)}\\{\nonumber}&&+C_AN_f{\left(}-\frac{31313}{729}+\frac{412}{81}\zeta_2+\frac{452}{27}\zeta_3-\frac{10}{3}\zeta_4{\right)}\\{\nonumber}&&+C_FN_f{\left(}-\frac{1711}{54}+\frac{152}{9}\zeta_3+8\zeta_4{\right)}+N_f^2{\left(}\frac{928}{729}+\frac{16}{9}\zeta_3{\right)}.\end{aligned}$$ The coefficients $d^{(n,i>0)}$ can be expressed in the terms of other anomalous dimensions and are given in sec.\[sec:SADRAD\].
[^1]: In the case of Wilson lines with timelike directions the path ordering and time ordering contradict each other. It can result to the extra non-physical singularities, in the self-interacting diagrams, see e.g. discussion in [@Collins:2011zzd]. For lightlike Wilson lines, which are discussed here, there is no such problem.
[^2]: The regularizations of rapidity divergences of non-geometrical type cannot be consider directly, because typically, such regularizations explicitly violate conformal symmetry.
[^3]: It is important to perform the limit $\alpha\to 0$ prior to the limit $\sigma\to \infty$. I.e. to keep the deviation from the light-cone infinitesimal even at the light-cone infinity. If this requirement is not satisfied, then both points $r$ and $\bar r$ turn to $\{0^+,0^-,0_\perp\}$. In this case the rapidity divergences are not factorizable.
[^4]: To ensure the gauge-invariance one should add extra transverse links which connect end points. In the soft factor $\mathbf{\Omega}$ these links would turn into the curved links. There are not extra cusp UV divergences in this case since the directions of links at meeting points are perpendicular.
[^5]: I thank M.Diehl for the help in the elaboration of consistent definitions presented in this section.
[^6]: One should pay special attention to the power-like IR divergences, e.g. $\delta^{-1-\epsilon}$. These divergences can interfere with the higher-order terms in the $\delta$-expansion and compensate each other. This case leads to the gauge violating contributions. However, these divergences are simple to track. See detailed discussion is given in the appendix of [@Echevarria:2015byo].
|
---
abstract: 'The purpose of this paper is to analyze in detail the Hamiltonian formulation for the compact Gowdy models coupled to massless scalar fields as a necessary first step towards their quantization. We will pay special attention to the coupling of matter and those features that arise for the $\mathbb{S}^1\times\mathbb{S}^2$ and $\mathbb{S}^3$ topologies that are not present in the well studied $\mathbb{T}^3$ case –in particular the polar constraints that come from the regularity conditions on the metric. As a byproduct of our analysis we will get an alternative understanding, within the Hamiltonian framework, of the appearance of initial and final singularities for these models.'
author:
- 'J. Fernando'
- Daniel
- 'Eduardo J.'
date: 'July 23, 2007'
title: Hamiltonian Dynamics of Linearly Polarized Gowdy Models Coupled to Massless Scalar Fields
---
Introduction
============
[\[Intro\]]{}
Symmetry reductions are a way to gain useful insights for difficult problems in classical and quantum general relativity. In this respect the two Killing vector reductions provided by the so called Gowdy models are specially attractive because they have a cosmological interpretation and share some interesting features with similar reductions such as the Einstein-Rosen waves –in particular their solvability– both at the classical and quantum regimes. Most of the work on these models, after the initial papers by Gowdy [@Gowdy:1971jh; @Gowdy:1973mu], has profusely analyzed those corresponding to the 3-torus spatial topology and, in fact, this is by far the preferred choice to discuss quantization issues [@Misner; @Berger:1975kn; @Berger:1973; @Cortez; @Corichi:2006xi; @Corichi:2006zv; @Mena:1997; @Torre:2002xt; @Torre:2007zj; @Romano:1996ep; @Corichi:2002vy; @Pierri:2000ri; @BarberoG.:2006zw]. The other possible closed (compact and without boundary) topologies, the *three-handle* $\mathbb{S}^1\times\mathbb{S}^2$, the three-sphere $\mathbb{S}^3$, and the lens spaces $L(p,q)$, are interesting in their own right. From the physical point of view their most salient feature is the fact that they describe cosmological models with both initial and final singularities. For this reason they will become useful test beds for issues related to quantization in cyclic universes.
The first step towards quantization is the Hamiltonian formulation of the model at hand. This is specially so for constrained systems where the identification of the relevant constraints is a necessary first step either to attempt a phase space reduction, a gauge fixing, or a quantization *à la Dirac* where the physical Hilbert space is identified as the kernel of suitable self-adjoint operators representing the constraints.
Our goal in this paper is to perform a detailed Hamiltonian analysis of the compact Gowdy models coupled to massless scalar fields, extending in several ways previous results on this subject [@BarberoG.:2005ge; @BarberoG.:2006gd]. Adding matter fields to the system is a way to enrich these models and get closer to physically realistic situations. Here we will work in the spirit of previous treatments for other two Killing vector reductions [@Ashtekar:1996bb; @Beetle:1998iu], paying close attention to the constraint analysis, gauge fixing, and deparameterization. To our knowledge the Hamiltonian analysis, for the vacuum $\mathbb{S}^1\times\mathbb{S}^2$ and $\mathbb{S}^3$ Gowdy models, has only been addressed in a partial way in [@Hanquin] where the authors give Hamiltonians for these systems. However they do not provide the detailed phase space description (constraints, gauge fixing, and so on) necessary to understand relevant geometrical issues. Also their reduced phase space treatment does not allow to follow other roads to quantization such as Dirac’s approach or the viewpoint pioneered by Varadarajan in [@Varadarajan:2006am]. Among several issues we want to find out how the topology of the spatial slices affects the definition of the constraints, and how the coupling of massless scalar fields is realized in the different topologies. Along the way we also want to understand in a detailed way, and within the Hamiltonian setting, the mechanisms leading to the appearance of final singularities. Our results can be immediately particularized to the vacuum situation.
The paper is structured as follows. After this introduction we will review in section \[generalframework\] the main points concerning the Geroch reduction for polarized Gowdy models coupled to massless scalar matter fields. In particular we will show that this reduction, and a subsequent conformal transformation, allows us to interpret these models as 2+1 gravity coupled to a set of massless scalar fields with axial symmetry. Some details related to this reduction vary depending on the topologies (in particular those related to the quotient spatial manifold) and will be commented separately for each case.
Section \[Torus\] will be devoted to discuss the Gowdy $\mathbb{T}^3$ models coupled to massless scalar fields extending the previous treatments for the vacuum case [@Pierri:2000ri; @Mena:1997] (and similar models such as Einstein-Rosen waves [@BarberoG.:2005ge; @BarberoG.:2006gd]). Along the way we will clarify some issues related to deparameterization and the appearance of singularities in the $3+1$ dimensional metrics. This section will be the basis of the treatment that we will follow to study the other possible topologies.
Section \[handle\] will be devoted to the Hamiltonian formulation of Gowdy models in $\mathbb{S}^1\times\mathbb{S}^2$ coupled to massless scalars. Here we will have to pay special attention to the identification of the regularity conditions that the basic fields describing the model must satisfy as a consequence of the regularity conditions on the metric. As we will see the constraints that are relevant here are different from the ones present for the 3-torus due to the presence of a symmetry axis in the spatial manifold. In particular we will get what we will refer to as “polar constraints” involving the values of the basic fields at the poles of the two dimensional sphere that appears as the quotient space after performing a Geroch reduction. As we will show they are first class and play a relevant role to guarantee the differentiability of the other constraints. Another issue that will be discussed is how the deparameterization achieved by a partial gauge fixing works for this model and how one can arrive at a reduced phase space description. We will see that, as it also happens in the $\mathbb{T}^3$ case, the dynamics of the system is described by a time dependent Hamiltonian though the time dependence now is different and reflects the appearance of initial and final singularities. In fact, as a result of our analysis, we will get a geometric understanding of this fact in terms of the geometry of the constraint hypersurface in phase space.
After this we will perform a similar analysis in section \[S3\] for the $\mathbb{S}^3$ topology. Here the main difference stems from the fact that we will be forced to perform the Geroch reduction needed to describe the model in $2+1$ dimensions by using a Killing field whose norm vanishes on a circle $\mathbb{S}^1$. This will introduce some modifications in our description and will change the analysis of the relevant regularity conditions for the metric. Nevertheless we will find out that the final description is quite similar to the one corresponding to the three handle discussed above.
The detailed quantization of the $\mathbb{S}^1\times\mathbb{S}^2$ and $\mathbb{S}^3$ Gowdy models will be carried out elsewhere. A fact that will play a relevant role there is the possibility of describing the compact Gowdy models in the different topologies as field theories in certain conformally stationary curved backgrounds. As this point of view is also useful to understand some of the issues discussed in the paper from a different perspective we will show in section \[back\] how this can be done.
We end the paper in section \[conclusions\] with a discussion of the main results and suggestions for future work on this subject.
General features of compact Gowdy models: Geroch reduction and $2+1$ dimensional formulation
============================================================================================
[\[generalframework\]]{}
Let us consider a smooth, effective, and proper action of the biparametric Lie Group $G^{{\scriptstyle{(2)}}}:=U(1)\times
U(1)=\{(g_1,g_2)=(e^{ix_1},e^{ix_2})\,|
\,x_{1},x_{2}\in\mathbb{R}(\mathrm{mod}\,2\pi)\}$ on a compact, connected, and oriented 3-manifold ${^{\scriptstyle{{{\scriptstyle{(3)}}}}}}\Sigma$. It can be shown [@MOSTERT; @Chrusciel:1990zx] that this action is unique up to automorphisms of $G^{{\scriptstyle{(2)}}}$ and diffeomorphisms of ${^{\scriptstyle{{{\scriptstyle{(3)}}}}}}\Sigma$. The spatial manifold ${^{\scriptstyle{{{\scriptstyle{(3)}}}}}}\Sigma$ is then restricted to have the topology of a three-torus $\mathbb{T}^3$, a three-handle $\mathbb{S}^1\times \mathbb{S}^2$, the three-sphere $\mathbb{S}^3$, or the lens spaces $L(p,q)$ (that can be studied by imposing discrete symmetries on the $\mathbb{S}^3$ case).
Let us take a four manifold ${^{{{\scriptstyle{(4)}}}}}\mathcal{M}$ diffeomorphic to $\mathbb{R}\times{^{{{\scriptstyle{(3)}}}}}\Sigma$ and such that $({^{{{\scriptstyle{(4)}}}}}\mathcal{M}, {^{{{\scriptstyle{(4)}}}}}g_{ab})$ is a globally hyperbolic spacetime endowed with a Lorentzian metric[^1] ${^{{{\scriptstyle{(4)}}}}}g_{ab}$. Let us further require that $G^{{\scriptstyle{(2)}}}$ acts by isometries on the spatial slices of ${^{{{\scriptstyle{(4)}}}}}\mathcal{M}$. In this paper we will focus on the so called *linearly polarized case*, hence, the isometry group will be generated by a pair of mutually orthogonal, commuting, spacelike, and globally defined hypersurface-orthogonal Killing vector fields $(\xi^{a},\sigma^{a})$.
Let us consider now the Einstein-Klein-Gordon equations $$\label{EKG}
{^{{{\scriptstyle{(4)}}}}}R_{ab}=8\pi
G_{N}(\mathrm{d}\phi)_a(\mathrm{d}\phi)_b\,,\quad
{^{{{\scriptstyle{(4)}}}}}g^{ab}\,{^{{{\scriptstyle{(4)}}}}}\nabla_{a}{^{{{\scriptstyle{(4)}}}}}\nabla_{b}\phi=0\,$$ corresponding to (3+1)-dimensional gravity minimally coupled to a zero rest mass scalar field $\phi$ symmetric under the diffeomorphisms generated by the Killing fields ($\mathcal{L}_{\xi}\phi=\mathcal{L}_{\sigma}\phi=0$, $\mathcal{L}_{\xi}{^{{{\scriptstyle{(4)}}}}}g_{ab}=\mathcal{L}_{\sigma}{^{{{\scriptstyle{(4)}}}}}g_{ab}=0$). Here ${^{{{\scriptstyle{(4)}}}}}R_{ab}$ and ${^{{{\scriptstyle{(4)}}}}}\nabla_{a}$ denote the Ricci tensor and the metric connection associated to ${^{{{\scriptstyle{(4)}}}}}g_{ab}$, respectively. The exterior derivative of the scalar field $\phi$ is denoted by $(\mathrm{d}\phi)_a$ and $G_{N}$ is the Newton constant.
In order to get a simplified, lower dimensional description we will perform a Geroch reduction [@Geroch:1970nt] by taking advantage of the existence of Killing vector fields. The possibility of finding the necessary non-vanishing Killing field $\xi^a$ will depend, as we will see later, on the spatial topology that we consider. In some cases the appropriate Killing vectors vanish on 2-dimensional submanifolds but, nevertheless, we will be able to use Geroch’s procedure even in this situation. The idea is to find a suitable reduction on the manifold ${^{{{\scriptstyle{(3)}}}}}\mathcal{M}={^{{{\scriptstyle{(4)}}}}}\tilde{\mathcal{M}}/U(1)$, diffeomorphic to $\mathbb{R}\times{^{{{\scriptstyle{(2)}}}}}\Sigma$, where ${^{{{\scriptstyle{(4)}}}}}\tilde{\mathcal{M}}$ denotes the set of points in ${^{{{\scriptstyle{(4)}}}}}\mathcal{M}$ in which $\xi^{a}$ is nonvanishing, and reintroduce the removed points (the symmetry axis) as a boundary where the fields must satisfy certain regularity conditions. In the present situation hypersurface orthogonality will allow us to view ${^{{{\scriptstyle{(3)}}}}}\mathcal{M}$ as an embedded submanifold, everywhere orthogonal to the closed orbits of $\xi^a$, and endowed with the induced metric ${^{{{\scriptstyle{(3)}}}}}g_{ab}:={^{{{\scriptstyle{(4)}}}}}g_{ab}-\lambda_{\xi}^{-1}\xi_{a}\xi_{b}$, where $\lambda_{\xi}:={^{{{\scriptstyle{(4)}}}}}g_{ab}\xi^{a}\xi^{b}:=\xi_{a}\xi^{a}>0$.
In the linearly polarized case the twist of the Killing fields vanishes and the field equations can be written as those corresponding to a set of massless scalar fields coupled to (2+1)-gravity by performing the conformal transformation $g_{ab}:=\lambda_{\xi}{^{{{\scriptstyle{(3)}}}}}g_{ab}$. The system (\[EKG\]) is then equivalent to $$R_{ab}=\frac{1}{2}\sum_i
(\mathrm{d}\phi_i)_a(\mathrm{d}\phi_i)_b\,,\quad
g^{ab}\nabla_a\nabla_b\phi_i=0\,,\quad \mathcal{L}_\sigma
g_{ab}=0\,,\quad \mathcal{L}_\sigma\phi_i=0\,,\label{ecs}$$ where $R_{ab}$ and $\nabla_{a}$ denote, respectively, the Ricci tensor and the Levi-Civita connection associated to $g_{ab}$ (all of them three dimensional objects), we have defined[^2] $\phi_1:=\log\lambda_{\xi}$, $\phi_2:=\sqrt{16\pi G_{N}}\phi$, and we must remember that we have the additional symmetry generated by the remaining Killing vector field $\sigma^a$. Notice that (\[ecs\]) are formally symmetric under the exchange of the gravitational and matter scalars. However, it is important to realize that for some of the topologies that we will discuss, these fields may be subject to different regularity conditions in the gravitational and matter sectors that effectively break the symmetry among them. The relevant details for each spatial topology will be given in the corresponding section.
In order to obtain the Hamiltonian formulations for the models that we are considering here we want to derive the previous equations from an action principle. To this end we introduce the (2+1)-dimensional Einstein-Hilbert action corresponding to gravity coupled to massless scalars $$\begin{aligned}
\label{act}
\displaystyle {^{{{\scriptstyle{(3)}}}}}S(g_{ab},\phi_i)&=&\frac{1}{16\pi
G_{3}}\int_{(t_0,t_1)\times^{{{\scriptstyle{(2)}}}}\Sigma}\!\!\!^{{{\scriptstyle{(3)}}}}\mathrm{e}\,|g|^{1/2}\left(R-
\frac{1}{2}\sum_i g^{ab}(\mathrm{d}\phi_i)_a(\mathrm{d}\phi_i)_b\right)\nonumber\\
& & \displaystyle +\frac{1}{8\pi
G_{3}}\int_{\{t_{0}\}\times{^{{{\scriptstyle{(2)}}}}\Sigma\cup\{t_{1}\}\times{^{{{\scriptstyle{(2)}}}}}\Sigma}}
\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\,|h|^{1/2}K\,.\end{aligned}$$ Here $R$ denotes the Ricci scalar associated to $g_{ab}$. $K$ and $h_{ab}$ are, respectively, the trace of the second fundamental form $K_{ab}$ (defined by the exterior normal unit vector $n^a$), and the induced 2-metric on the boundary $\{t_{0}\}\times^
{{{\scriptstyle{(3)}}}}\Sigma\cup\{t_{1}\}\times ^{{{\scriptstyle{(3)}}}}\Sigma$. Finally $G_{3}$ denotes the Newton constant per unit length in the direction of the $\xi$-symmetric orbits. We have restricted the integration region to an interval $[t_0,t_1]$, where $t$ defines a global coordinate on $\mathbb{R}$. The action is written with the help of a fiducial (i.e. non dynamical) volume form ${^{{{\scriptstyle{(3)}}}}}\mathrm{e}$ compatible with the canonical volume form ${^{{{\scriptstyle{(3)}}}}}\epsilon$ defined by the metric $g_{ab}$. This is given by[^3] ${^{{{\scriptstyle{(3)}}}}}\epsilon=\sqrt{|g|}\,{^{{{\scriptstyle{(3)}}}}}\mathrm{e}$. The volume form ${^{{{\scriptstyle{(3)}}}}}\epsilon$ induces a 2-form ${^{{{\scriptstyle{(2)}}}}}\epsilon_{ab}={^{{{\scriptstyle{(3)}}}}}\epsilon_{abc}n^{c}$ on each slice $\{t\}\times{^{{{\scriptstyle{(2)}}}}}\Sigma$ that agrees with the volume associated to the 2-metric $h_{ab}$. We have also introduced a fixed volume 2-form ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}$ on $\{t\}\times{^{{{\scriptstyle{(2)}}}}}\Sigma$ such that ${^{{{\scriptstyle{(2)}}}}}\epsilon=\sqrt{|h|}\,{^{{{\scriptstyle{(2)}}}}}\mathrm{e}$, and verifies $\sqrt{|g|}\,{^{{{\scriptstyle{(3)}}}}}\mathrm{e}_{abc}n^{c}=\sqrt{|h|}\,{^{{{\scriptstyle{(2)}}}}}\mathrm{e}_{ab}$. We require that ${^{{{\scriptstyle{(3)}}}}}\mathrm{e}$ and ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}$ be time-independent, i.e. $\mathcal{L}_{t}{^{{{\scriptstyle{(3)}}}}}\mathrm{e}=0$, $\mathcal{L}_{t}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}=0$ where $\mathcal{L}_{t}$ denotes Lie derivative along $t^{a}:=(\partial/\partial t)^{a}$. We also demand them to be invariant under the action of the remaining Killing vector field. In particular, given the (2+1)-dimensional splitting of ${^{{{\scriptstyle{(3)}}}}}\Sigma$ it is natural to choose ${^{{{\scriptstyle{(3)}}}}}\mathrm{e}=\mathrm{d}t\wedge{^{{{\scriptstyle{(2)}}}}}\mathrm{e}$, with $\mathcal{L}_{t}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}=\mathcal{L}_{\sigma}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}=0$ [@Wald].
For all the different topologies, using the Stokes theorem, we get
$$\begin{aligned}
\label{actr}
\!\!\!\!\displaystyle
{^{{{\scriptstyle{(3)}}}}}S(g_{ab},\phi_i)\!=\!\frac{1}{16\pi
G_{3}}\int_{t_0}^{t_1}\!\!\!\mathrm{d}t\!\int_{{^{{{\scriptstyle{(2)}}}}}\Sigma}\!\!
\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\,|g|^{1/2}\Bigg(\,^{{{\scriptstyle{(2)}}}}R+K_{ab}K^{ab}-K^2
- \frac{1}{2}\sum_i g^{ab}
(\mathrm{d}\phi_{i})_a(\mathrm{d}\phi_{i})_b\Bigg)\end{aligned}$$
where we have used the relation $R={^{{{\scriptstyle{(2)}}}}}R+K_{ab}K^{ab}-K^{2}+2\nabla_a(n^{a}K-n^{b}\nabla_{b}n^{a})$ and ${^{{{\scriptstyle{(2)}}}}}R$ denotes the Ricci scalar associated to $h_{ab}$. Our strategy in the different topologies that we will study in the paper will be to write down an action of this type, adapted to the peculiarities of the different spatial topologies (in particular those originating in the different sets of regularity conditions that we will have to consider) and use it to derive a Hamiltonian formulation for the system.
$\mathbb{T}^3$ Gowdy models coupled to massless scalars
=======================================================
[\[Torus\]]{}
The $\mathbb{T}^3$ Gowdy model is, by far, the most studied to date both at the classical and quantum levels [@Gowdy:1971jh; @Gowdy:1973mu; @Misner; @Berger:1975kn; @Berger:1973; @Cortez; @Corichi:2006xi; @Corichi:2006zv; @Mena:1997; @Torre:2002xt; @Torre:2007zj; @Romano:1996ep; @Corichi:2002vy; @Pierri:2000ri; @BarberoG.:2006zw; @Chrusciel:1990zx]. We will consider in this section the coupling of some types of matter fields and the most important aspects of its Hamiltonian treatment paying special attention to the deparameterization and reduced Hamiltonian description.
Let us start by considering the orientable 3-manifold ${^{\scriptstyle{{{\scriptstyle{(3)}}}}}}\Sigma=\mathbb{T}^3
=\mathbb{S}^1\times\mathbb{S}^1\times\mathbb{S}^1$, whose points we parameterize in the form $(z_1,z_2,z_3)=(e^{i\theta},e^{i\sigma},e^{i\xi})$ with $\theta,\sigma,\xi\in\mathbb{R}(\mathrm{mod}\,2\pi)$. In particular, we endow ${^{\scriptstyle{{{\scriptstyle{(3)}}}}}}\Sigma$ with the standard volume form $\mathrm{d}\theta\wedge\mathrm{d}\sigma\wedge\mathrm{d}\xi$. We define the following (left) $G^{{\scriptstyle{(2)}}}$-group action $$(g_1,g_2)\cdot(z_1,z_2,z_3)=(e^{ix_1},e^{ix_2})\cdot(e^{i\theta},e^{i\sigma},e^{i\xi})
:=(e^{i\theta},e^{i(x_1+\sigma)},e^{i(x_2+\xi)})\,.$$ We can consider now the group orbits defined by the commuting subgroups $(g_1,g_2)=(e^{ix},1)$, $(g_1,g_2)=(1,e^{ix})$, $x\in\mathbb{R}(\mathrm{mod}\,2\pi)$ $$\begin{aligned}
&&(e^{ix},1)\cdot(e^{i\theta},e^{i\sigma},e^{i\xi})=(e^{i\theta},e^{i(x+\sigma)},e^{i\xi})\,,\nonumber\\
&&(1,e^{ix})\cdot(e^{i\theta},e^{i\sigma},e^{i\xi})=(e^{i\theta},e^{i\sigma},e^{i(x+\xi)})\,,\nonumber\end{aligned}$$ and their corresponding tangent vectors at each point of $\mathbb{T}^3$ obtained by differentiating the previous expressions with respect to $x$ at $x=0$ $$\begin{aligned}
&&(0,ie^{i\sigma},0)\,,\nonumber\\
&&(0,0,ie^{i\xi})\,.\nonumber\end{aligned}$$ Let us consider the four manifold ${^{\scriptstyle{{{\scriptstyle{(4)}}}}}}\mathcal{M}\simeq\mathbb{R}\times\mathbb{T}^3$. We introduce now three smooth vector fields $\theta^{a}$, $\sigma^a$ and $\xi^a$, tangent to the embedded submanifolds $\{t\}\times\mathbb{T}^3$ (here $t$ is a global coordinate on $\mathbb{R}$). In order to do this, let us fix $t_0\in\mathbb{R}$ and define on $\{t_0\}\times\mathbb{T}^3$ the 3-dimensional vector fields[^4] $(\partial/\partial\theta)^{a}$, $(\partial/\partial\sigma)^{a}$, and $(\partial/\partial\xi)^{a}$ given in the description of ${^{\scriptstyle{{{\scriptstyle{(3)}}}}}}\Sigma$ at the beginning of this section. We extend them to ${^{\scriptstyle{{{\scriptstyle{(4)}}}}}}\mathcal{M}$ by Lie dragging along a smooth vector field $t^a$ defined[^5] as the tangent vector to a smooth congruence of curves transverse to the slices $\{t\}\times\mathbb{T}^3$. Notice that given a one parameter family of diffeomorphisms $f_t$ we have $f_{t*}[\xi,\sigma]^a=[f_{t*}\xi,f_{t*}\sigma]^a$ so we guarantee that the extended fields commute everywhere. The 4-tuple $(t^{a},\theta^{a},\sigma^{a},\xi^{a})$ defines then a paralelization of ${^{\scriptstyle{{{\scriptstyle{(4)}}}}}}\mathcal{M}$. Here $\theta^a$ is the vector field obtained by extending $(\partial/\partial\theta)^a$ to the four-dimensional manifold ${^{\scriptstyle{{{\scriptstyle{(4)}}}}}}\mathcal{M}$; $\sigma^a$ and $\xi^a$ are obtained by the same procedure. Once we have introduced these vector fields on ${^{\scriptstyle{{{\scriptstyle{(4)}}}}}}\mathcal{M}$ as *background* objects we restrict ourselves to working with metrics $^{{{\scriptstyle{(4)}}}}g_{ab}$ satisfying the following conditions:
1. The action of the group $G^{{\scriptstyle{(2)}}}$ on ${^{\scriptstyle{{{\scriptstyle{(4)}}}}}}\mathcal{M}$ defined by $(g_1,g_2)\cdot(t,p)=(t,(g_1,g_2)\cdot p)$, $t\in\mathbb{R}$, $p\in\mathbb{T}^{3}$, with $(g_1,g_2)\cdot p$ defined above, is an action by isometries, i.e. $\xi^a$ and $\sigma^a$ are Killing vector fields ($\mathcal{L}_{\xi}^{{{\scriptstyle{(4)}}}}g_{ab}=0$, $\mathcal{L}_{\sigma}^{{{\scriptstyle{(4)}}}}g_{ab}=0$).
2. $t$ is a global time function, i.e. ${^{\scriptstyle{{{\scriptstyle{(4)}}}}}}g^{ab}(\mathrm{d}t)_b$ is a timelike vector field. From now on we will consider the manifold ${^{\scriptstyle{{{\scriptstyle{(4)}}}}}}\mathcal{M}$ to be endowed with a time orientation such that this vector field is past-directed.
3. $\{t\}\times\mathbb{T}^{3}$ are spacelike hypersurfaces for all $t\in\mathbb{R}$. In particular $\lambda_{\xi}:={^{\scriptstyle{{{\scriptstyle{(4)}}}}}}g_{ab}\xi^a\xi^b>0$, $\lambda_{\sigma}:={^{\scriptstyle{{{\scriptstyle{(4)}}}}}}g_{ab}\sigma^a\sigma^b>0$.
4. $\xi^a$ and $\sigma^a$ are hypersurface orthogonal (this defines the so called linearly polarized case). This condition means that the twist of the two fields vanishes. This will ultimately allow us to simplify the field equations and describe the system as a simple theory of scalar fields.
Two simple but important results that can be proved at this point as a consequence of the first are the following:
i\) If $\xi^a$ and $\sigma^a$ are Killing vectors and $[\xi,\sigma]^a=0$ then $\mathcal{L}_\sigma(^{{{\scriptstyle{(4)}}}}g_{ab}-\xi_a\xi_b/\lambda_{\xi})=0$;
ii\) Furthermore, if we define the vector $X^a$ orthogonal to $\xi^a$ as $X^a:=\sigma^a-\xi^a(\xi^b\sigma_b)/\lambda_{\xi}$ it satisfies $[\xi,X]^a=0$ and also $\mathcal{L}_X(^{{{\scriptstyle{(4)}}}}g_{ab}-\xi_a\xi_b/\lambda_{\xi})=0$. This means that, without loss of generality, we can work with everywhere orthogonal and commuting Killing vector fields $\xi^a$ and $\sigma^a$. In fact, we impose
5\. $(\theta^{a},\sigma^{a},\xi^{a})$ are mutually ${^{{{\scriptstyle{(4)}}}}}g$-orthogonal vector fields.
After we perform the Geroch reduction with respect to the field $\xi^{a}$ as described above we end up with a set of equations that can be obtained from a 2+1 dimensional action of the type (\[actr\]) with ${^{{{\scriptstyle{(2)}}}}}\Sigma=\mathbb{T}^2=\mathbb{S}^1\times\mathbb{S}^1$. Since the remaining Killing vector field $\sigma^{a}$ is still hypersurface orthogonal, and non-vanishing, the corresponding space of orbits ${^{{{\scriptstyle{(2)}}}}}\mathcal{M}:={^{{{\scriptstyle{(3)}}}}}\mathcal{M}/U(1)\simeq\mathbb{R}\times\mathbb{S}^{1}$ can be identified as an embedded hypersurface in $^{{{\scriptstyle{(3)}}}}\mathcal{M}$ everywhere orthogonal to the (closed) orbits of $\sigma^{a}$. The induced 2-metric of signature $(-+)$ on $^{{{\scriptstyle{(2)}}}}\mathcal{M}$ can be written $$s_{ab}=g_{ab}-\tau^{-2}\sigma_a\sigma_b\,,$$ where $\tau^2:=g_{ab}\sigma^{a}\sigma^{b}=\lambda_{\xi}\lambda_{\sigma}>0$ is the area density of the symmetry $G^{{\scriptstyle{(2)}}}$-group orbits. In the following we will use the notation $\tau=+\sqrt{\tau^2}$. We have now an induced foliation over $^{{{\scriptstyle{(2)}}}}\mathcal{M}$ defined by the global time function $t$ introduced before. Let $n^{a}$ be the $g$-unit and future-directed ($g^{ab}n_{a}(\mathrm{d}t)_b>0$) vector field normal to this foliation, and let $\hat{\theta}^{a}$ be the $g$-unit spacelike vector field of closed orbits tangent to the slices of constant $t$, such that $$\theta^{a}=e^{\gamma/2}\hat{\theta}^{a}$$ for some extra field $\gamma$. If we choose the congruence of curves with $t^{a}$ tangent to ${^{{{\scriptstyle{(2)}}}}}\mathcal{M}$. Then, the congruence is transverse to the foliation, and we can express $$\label{ta}
t^{a}=e^{\gamma/2}(Nn^{a}+N^{\theta}\hat{\theta}^{a})\,,$$ where $N>0$ and $N^{\theta}$ are proportional to the lapse and shift functions. The factor $e^{\gamma/2}$ will allow us to obtain a proper gauge algebra and simplify later calculations. We require that $N$, $N^{\theta}$, and $\gamma$ are smooth real-valued fields on ${^{{{\scriptstyle{(3)}}}}}\mathcal{M}$. As we will see in the following the symmetry generated by $\sigma^{a}$ will further constraint them, in particular they will be constant along the orbits defined by the remaining Killing vector field. The orthonormal basis $(n^{a},\hat{\theta}^{a},\sigma^{a}/\tau)$ is positively oriented with respect to the volume 3-form associated to the 3-metric $g_{ab}$, compatible with $\mathrm{d}t\wedge\mathrm{d}\theta\wedge\mathrm{d}\sigma$, satisfying ${^{{{\scriptstyle{(3)}}}}}\epsilon_{abc}n^{a}\hat{\theta}^{a}\sigma^{a}/\tau=1$.
The expression of the metric is $$\begin{aligned}
\label{metric}
&&g_{ab}=e^\gamma\bigg((N^{\theta
2}-N^{2})(\mathrm{d}t)_a(\mathrm{d}t)_b+
2N^{\theta}(\mathrm{d}t)_{(a}(\mathrm{d}\theta)_{b)} +
(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b\bigg)+
\tau^{2}(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b\,.\quad\quad\end{aligned}$$ The fact that the vectors $(t^a,\theta^a,\sigma^a)$ commute everywhere will translate into necessary conditions that the vectors $n^a$ and $\theta^a$ and the scalars $N$, $N^\theta$, and $\gamma$ must satisfy. These are $$\begin{aligned}
\mathcal{L}_\sigma N=0\,,\,&\mathcal{L}_\sigma
N^\theta=0\,,\,&\mathcal{L}_\sigma\gamma=0\,,\label{dersigma}\end{aligned}$$ $$\begin{aligned}
&(\mathcal{L}_\sigma n)^a=0\,,\,&
(\mathcal{L}_\sigma \hat{\theta})^a=0\,,\label{comm}\end{aligned}$$ and $$\begin{aligned}
\frac{1}{2}(\mathcal{L}_{\theta}\gamma)(Nn^a+N^\theta\hat{\theta}^a)
-e^{-\gamma/2}(\mathcal{L}_te^{\gamma/2})\hat{\theta}^a+N
e^{\gamma/2}[\hat{\theta},n]^a+(\mathcal{L}_{\theta}N)n^a
+(\mathcal{L}_{\theta}N^\theta)\hat{\theta}^a=0\,.\nonumber\end{aligned}$$ This last equation can be projected in the directions defined by the basis vectors to give $$\begin{aligned}
&&\frac{1}{2}N\mathcal{L}_\theta\gamma+\mathcal{L}_\theta
N+Ne^{\gamma/2}n^an^b\nabla_a\hat{\theta}_b=0\,,\label{proy_1}\\
&&\frac{1}{2}N^\theta\mathcal{L}_\theta\gamma+\mathcal{L}_\theta
N^\theta-\frac{1}{2}\mathcal{L}_t\gamma+
Ne^{\gamma/2}\hat{\theta}^a\hat{\theta}^b\nabla_a n_b=0\,,\label{proy_2}\\
&&\hat{\theta}^a\sigma^b\nabla_an_b=0\,.\label{proy_3}\end{aligned}$$ These equations are important because they relate the components of the extrinsic curvature of some surfaces with derivatives of $N$, $N^{\theta}$, and $\gamma$. Notice that the scalars $\phi_i$ are also constant on the orbits of $\sigma^a$ (the matter scalar $\phi_2$ because we have imposed this from the start and the gravitational scalar $\phi_1$ due to the fact that the two Killings $\xi^a$ and $\sigma^a$ commute: $\mathcal{L}_\sigma\lambda_{\xi}=0$). Therefore, as we will end up with an essentially two dimensional model with fields depending only on coordinates $t$ and $\theta$, we will eventually denote $\mathcal{L}_t$ with a dot and $\mathcal{L}_\theta$ with a prime. With this convention, we obtain: $$\begin{aligned}
{^{{{\scriptstyle{(2)}}}}}R&=&\tau^{-1}e^{-\gamma}(\tau'\gamma'-2\tau'')\,,\\
K_{ab}K^{ab}-K^2&=&
-\frac{e^{-\gamma}}{N^2\tau}(\dot{\gamma}
-N^{\theta}\gamma'-2N^{\theta\prime})(\dot{\tau}-N^{\theta}\tau')\,,\\
g^{ab}
(\mathrm{d}\phi_i)_a(\mathrm{d}\phi_i)_b&=&-\frac{e^{-\gamma}}{N^{2}}
(\dot{\phi}_{i}^{2}-2N^{\theta}\dot{\phi}_{i}\phi_{i}^{\prime}
+({N^{\theta}}^{2}-N^2)\phi_{i}^{\prime 2})\,,\end{aligned}$$ and then the action can be written as $$\begin{aligned}
&&\frac{1}{16\pi G_3}\int_{t_0}^{t_1}
\mathrm{d}t\int_{\mathbb{T}^2}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}|g|^{1/2}e^{-\gamma}\bigg(
\frac{1}{\tau}(\gamma^\prime\tau^\prime-2\tau^{\prime\prime})-\frac{1}{\tau
N^2} (\dot{\gamma}-2N^{\theta\prime}-\gamma^\prime
N^\theta)(\dot{\tau}-N^\theta\tau^\prime)\nonumber\\
&&\hspace{4cm}+\frac{1}{2N^2}\sum_{i}\big[\dot{\phi}^2_i-2N^\theta\dot{\phi}_i\phi_i^\prime+
(N^{\theta 2}-N^2)\phi^{\prime2}_i\big]\bigg)\,. \nonumber\end{aligned}$$ This will be the starting point for the Hamiltonian formalism. Notice that the previous expression is coordinate independent. The Lagrangian is written as an integral over the torus $\mathbb{T}^2$ of the 2-form obtained by multiplying the fiducial volume form and a scalar function. All the terms in this scalar are defined through the use of geometrical objects, in particular the derivatives are Lie derivatives along the fields introduced above. This will prove particularly important when dealing with other spatial topologies. In this case, it is natural to choose as fiducial 2-form ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}$ the one verifying ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}_{ab}\theta^{a}\sigma^{b}=Ne^{\gamma}\tau/|g|^{1/2}=1$, i.e. ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}=\mathrm{d}\theta\wedge\mathrm{d}\sigma$.
The Hamiltonian can be easily obtained by performing a Legendre transform. It has the form $$\begin{aligned}
H=C[N]+C_\theta[N^\theta]\,,\end{aligned}$$ where[^6] $$\begin{aligned}
&&C[N]=\int_{\mathbb{S}^{1}} NC:=\int_{\mathbb{S}^1}
N\bigg(\frac{1}{8G_3}(2\tau^{\prime\prime}-\gamma^\prime\tau^\prime)-8G_3p_\gamma
p_\tau+\frac{1}{2}\sum_{i}\Big(8G_3\frac{p_{\phi_{i}}^2}{\tau}+
\frac{\tau}{8G_3}\phi_{i}^{\prime2}\Big)\bigg)\,,\label{const_N}\\
&&C_\theta[N^\theta]=\int_{\mathbb{S}^{1}}
N^{\theta}C_{\theta}:=\int_{\mathbb{S}^1} N^\theta
\bigg(-2p_\gamma^\prime+p_\gamma \gamma^\prime+p_\tau
\tau^\prime+\sum_{i} p_{\phi_{i}}\phi_{i}^\prime
\bigg)\,.\label{const_N_theta}\end{aligned}$$ The lapse and the shift act as Lagrange multipliers and enforce the constraints $C=0$, $C_{\theta}=0$. The canonical phase space $(\Gamma,\omega)$ is coordinatized by the canonically conjugate pairs $(\gamma,p_{\gamma};\tau,p_{\tau};\phi_{i},p_{\phi_i})$ and $\omega$ denotes the standard (weakly) symplectic form $$\label{ws2f}
\omega=\int_{\mathbb{S}^{1}}\Big(\delta\gamma\wedge\delta
p_{\gamma}+\delta\tau\wedge\delta
p_{\tau}+\sum_{i}\delta\phi_{i}\wedge\delta p_{\phi_{i}}\Big)\,.$$ The dynamical variables are restricted to belong to a constraint submanifold $\Gamma_{c}\subset\Gamma$ globally defined by $C=0$, $C_{\theta}=0$. The constraints can be written in an equivalent way by taking “linear combinations” obtained by integrating them against suitable weight functions $N_{g}$ and $N^{\theta}_{g}$ in such a way that the vanishing of the weighted form of the constraints for all of them is equivalent to the vanishing of $C$ and $C_{\theta}$ at every point of $\mathbb{S}^1$. The gauge transformations generated by the (weighted) constraints are[^7] $$\begin{aligned}
&&\{\gamma,C[N_g]\}=-N_gp_\tau\,,\nonumber\\
&&\{\tau,C[N_g]\}=-N_gp_\gamma\,,\nonumber\\
&&\{\phi_{i},C[N_g]\}=N_g\frac{p_{\phi_i}}{\tau}\,,\nonumber\\
&&\{p_\gamma,C[N_g]\}=-(N_g\tau^\prime)^\prime\,,\nonumber\\
&&\{p_\tau,C[N_g]\}=
-(N_g\gamma^\prime)^\prime+\frac{1}{2}N_g\sum_{i}\Big(\frac{p_{\phi_{i}}^2}{\tau^2}-
\phi_{i}^{\prime2}\Big)\,,\nonumber\\
&&\{p_{\phi_i},C[N_g]\}=
(N_g\tau\phi^\prime_i)^\prime\,,\nonumber\end{aligned}$$ and $$\begin{aligned}
&&\{\gamma,C_\theta[N^\theta_g]\}=
2N_{g}^{\theta\prime}+N^\theta_g\gamma^\prime\,,
\nonumber\\
&&\{\tau,C_\theta[N^\theta_g]\}=N^\theta_g \tau^\prime\,,\nonumber\\
&&\{\phi_i,C_\theta[N^\theta_g]\}=N^\theta_g\phi_i^\prime\,,\nonumber\\
&&\{p_\gamma,C_\theta[N^\theta_g]\}=(N^\theta_g
p_\gamma)^\prime\,,\nonumber\\
&&\{p_\tau,C_\theta[N^\theta_g]\}=(N^\theta_g p_\tau)^\prime\,,\nonumber\\
&&\{p_{\phi_i},C_\theta[N^\theta_g]\}=(N^\theta_g p_{\phi_i})^\prime\,.\nonumber\\\end{aligned}$$ A straightforward calculation shows that the constraints are first class in Dirac terminology, or equivalently that $\Gamma_{c}$ is a coisotropic submanifold of $\Gamma$. Indeed, the Poisson algebra of the constraints is a proper Lie algebra $$\begin{aligned}
&&\{C[N_g],C[M_g]\}=C_\theta[N_g
M^\prime_g-M_g N^\prime_g]\,,\nonumber\\
&&\{C[N_g],C_\theta[N^\theta_g]\}=C[N_g
M^{\theta\prime}_g-M^\theta_g N_g^\prime]\,,\nonumber\\
&&\{C_\theta[N^\theta_g],C_\theta[M^\theta_g]\}=C_\theta[N^\theta_g
M^{\theta\prime}_g-M^\theta_g N^{\theta\prime}_g]\,.\nonumber\end{aligned}$$ Notice also that, as a consequence of the introduction of the suitable exponential factor $e^{\gamma/2}$ in (\[ta\]) we have a *closed* gauge algebra [@Romano:1996ep] (i.e. with structure *constants*).
In order to proceed we would like to isolate the true physical degrees of freedom of the model. As is well known there are several possible ways to do this. The usual ones are gauge fixing, i.e. the isolation of a single point per gauge orbit by imposing appropriate extra conditions on the phase space variables, and phase space reduction –that requires us to find a way to effectively quotient the phase space by the equivalence relation loosely defined as “belonging to the same orbit”. The successful implementation of the reduction allows us not only to label gauge orbits but also provides us with important mathematical structures (topological, symplectic,...) from the ones present in the initial phase space. Here we will see that a *partial* gauge fixing (*deparameterization*) can provide us with another interesting way to deal with the system because it can be described by a time-dependent, quadratic, Hamiltonian [@Misner; @Berger:1973; @Pierri:2000ri]. As we will show below this is also true for the other spatial topologies. If one is interested in quantizing the model one can alternatively use the Hamiltonian formulation described above to attempt a Dirac quantization.
The Hamiltonian vector fields associated to the weighted constraints $C[N_g]$, $C_{\theta}[N_{g}^{\theta}]$ are tangential to $\Gamma_c$ and define the degenerate directions of the presymplectic form $\omega|_{\Gamma_c}$. The deparameterization procedure is based on the choice of one of these Hamiltonian vector fields to define an evolution vector field $E_{H_R}$, generated by a reduced Hamiltonian $H_R$ of a generically non-autonomous system. With this aim in mind, we will impose gauge fixing conditions in such a way that at least one of the first class constraints $\mathcal{C}$ is not fixed. This will be used to define dynamics. Any remaining constraints left over by the (partial) gauge fixing will generate residual gauge symmetries.
Let $\iota:\Gamma_G\rightarrow\Gamma_c$ denote the embedding of the gauge fixed surface given by the first class constraints and the gauge fixing conditions; the pull-back of the presymplectic form to this surface, $\iota^{*}\omega$, has a single degenerate direction defined by the Hamiltonian vector field $E_{H_R}$. Select then a suitable phase space variable $T$ such that $E_{H_R}(T)=1$. The level surfaces of $T$ are all diffeomorphic to a manifold $\Gamma_{R}$ and transverse to $E_{H_R}$, defining a foliation of $\Gamma_{G}$ with $T$ as global time function. In that case, $\iota^{*}\omega=-\mathrm{d}T\wedge\mathrm{d}H_R+\omega_{R}$, $E_{H_R}=\partial_{T}+X_{H_R}$, where $\omega_{R}$ is a weakly non-degenerate form, and $(\Gamma_R,\omega_R,H_R(T))$ define a non-autonomous Hamiltonian system. Any remaining first class constraints will define a constrain submanifold in $\Gamma_{R}$.
The conditions that are usually considered for this problem [@Pierri:2000ri; @Romano:1996ep; @Corichi:2002vy; @Torre:2002xt] are $$\begin{aligned}
&&\tau^\prime=0\,,\label{gauge1}\\
&&p_\gamma^\prime=0\,.\label{gauge2}\end{aligned}$$ They mean that both $\tau$ and $p_\gamma$ take the same value irrespective of the point of $\mathbb{S}^1$ but they do not specify which one. Notice that conditions of the type $\tau=t_0$ or $p_\gamma=-p$ with $t_0,p\in\mathbb{R}$ not only would tell us that $\tau$ and $p_\gamma$ are independent of $\theta$ but also assign a fixed value to them, thus removing additional degrees of freedom.[^8]
This means that when using (\[gauge1\],\[gauge2\]) there is still a dynamical mode in $\tau$ that may vary in the evolution –at the end of the day it will be identified with the time parameter– but is constant on every spatial slice in the 3+1 decomposition. The fact that this class of models have an initial spacetime singularity suggests that there are interesting interpretive issues as far as the equivalence of the different choices of gauge fixing is concerned (how does this singularity manifest itself after a *full* gauge fixing? how does it show up if other gauge fixing conditions are used?). It should also be pointed out that although it is possible to think of the condition $\tau^\prime=0$ as a one parameter family of gauges $\tau=t$, with $t\in(0,\infty)$, it is dangerous to use it in this last form when computing Poisson brackets (it would be something like “mixing parametric and implicit equations”) as can be checked by explicit computations. In this case the correct attitude would be to work in the extended (“odd-dimensional”) phase space, mathematically described as a cosymplectic or contact manifold, incorporating a time variable and employ the usual techniques for non-autonomous Hamiltonian systems [@Leon1; @Leon2].
A convenient way to discuss gauge fixings is to describe our family of gauge conditions by introducing an orthonormal basis of weight functions on $\mathbb{S}^1$ $$Y_n(\theta):=\frac{1}{\sqrt{2\pi}}e^{in\theta}\,,\,\,\,n\in\mathbb{Z}\,,$$ and consider the family of constraints $C[Y_{n}]$, $C_\theta[Y_{n}]$. By expanding now $$\begin{aligned}
&&\tau=\sum_{n\in\mathbb{Z}}\tau_{n}Y_{n}\,, \quad
p_{\gamma}=\sum_{n\in\mathbb{Z}}p_{\gamma_n}Y_{n}\,,\end{aligned}$$ with $$\begin{aligned}
&&\tau_n=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{S}^1}e^{-in\theta}\tau\,,\quad
p_{\gamma_n}=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{S}^1}e^{-in\theta}p_{\gamma}\,,\end{aligned}$$ the previous gauge fixing conditions become $$\tau_n=0\,,\quad p_{\gamma_n}=0\,,\quad \forall\,n\in
\mathbb{Z}-\{0\}\,.$$ In order to see if this is a good gauge fixing (and, alternatively, find out if some gauge freedom is left) we compute $$\begin{aligned}
&&\{\tau_n,C[Y_m]\}
\approx-\frac{1}{\sqrt{2\pi}}\delta_{nm}p_{\gamma_0}\,,\quad
\{\tau_n,C_\theta[Y_m]\}\approx0\,,\nonumber\\
&&\{p_{\gamma_n},C[Y_m]\}\approx0\,,\quad \{p_{\gamma
n},C_\theta[Y_m]\}\approx\frac{in}{\sqrt{2\pi}}\delta_{nm}p_{\gamma_0}\,,\nonumber\end{aligned}$$ where $n\in\mathbb{Z}-\{0\}$, $m\in\mathbb{Z}$, and the symbol $\approx$ denotes equality on the hypersurface defined by the gauge fixing conditions and the constraints, the so-called gauge fixing surface $\Gamma_{G}\subset\Gamma_{c}$. Notice that with this way of writing the constraints (without the extra terms that would be present if we had not introduced the exponential prefactor in (\[ta\])) the gauge transformations of $\tau$ and $p_\gamma$ only involve these objects themselves. It is convenient to write the previous expressions in table form
$\tau_1=0$ $p_{\gamma_1}=0$ $\tau_{-1}=0$ $p_{\gamma_{-1}}=0$ $\dots$
-------------------- ------------------------------------- ------------------------------------ ------------------------------------- ------------------------------------- ----------
$C[Y_{0}]$ $0$ $0$ $0$ $0$ $\dots$
$C_\theta[Y_0]$ $0$ $0$ $0$ $0$ $\dots$
$C[Y_{1}]$ $-\frac{1}{\sqrt{2\pi}}p_{\gamma0}$ $0$ $0$ $0$ $\dots$
$C_\theta[Y_1]$ $0$ $\frac{i}{\sqrt{2\pi}}p_{\gamma0}$ $0$ $0$ $\dots$
$C[Y_{-1}]$ $0$ $0$ $-\frac{1}{\sqrt{2\pi}}p_{\gamma0}$ $0$ $\dots$
$C_\theta[Y_{-1}]$ $0$ $0$ $0$ $-\frac{i}{\sqrt{2\pi}}p_{\gamma0}$ $\dots$
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\ddots$
As we can see the only constraints that are not gauge-fixed by the conditions introduced above, as long as $p_{\gamma_0}\neq0$, are $C[1]=0$ and $C_\theta[1]=0$. From now on we will consider the sector $p_{\gamma_0}<0$. As we can see we have two first class constraints left over by our partial gauge fixing $$\begin{aligned}
&&\int_{\mathbb{S}^1}
\bigg((2\tau^{\prime\prime}-\gamma^\prime\tau^\prime)-p_\gamma
p_\tau+\frac{1}{2}\sum_{i}\Big(\frac{p_{\phi_{i}}^2}{\tau}
+\tau\phi_{i}^{\prime2}\Big)\bigg)\approx0\,,\label{const_res_N}\\
&&\int_{\mathbb{S}^1} \Big(-2p_\gamma^\prime+p_\gamma
\gamma^\prime+p_\tau \tau^\prime+\sum_{i}p_{\phi_{i}}\phi_{i}^\prime
\Big)\approx0\,.\label{const_res_N_theta}\end{aligned}$$ We can pullback the relevant geometric objects to the submanifold $\Gamma_{G}$ defined by the gauge fixing conditions to eliminate some of the variables in our model. Denoting by $\iota:\Gamma_{G}\rightarrow\Gamma$ the immersion map, the pullback of the (weakly) symplectic form (\[ws2f\]) becomes $$\begin{aligned}
\label{GG2-f}
\iota^{*}\omega=\mathrm{d}\gamma_0\wedge \mathrm{d}
p_{\gamma_0}+\mathrm{d}\tau_0\wedge \mathrm{d}
p_{\tau_0}+\sum_{i}\int_{\mathbb{S}^1} \delta\phi_i\wedge \delta
p_{\phi_i}\,.\label{symplectic1}\end{aligned}$$ The pullback of the constraints (\[const\_res\_N\],\[const\_res\_N\_theta\]) is $$\begin{aligned}
&&\mathcal{C}:=-p_{\gamma_0} p_{\tau_0}+\frac{1}{2}\sum_{i}
\int_{\mathbb{S}^1}\left(\sqrt{2\pi}\frac{p^2_{\phi_i}}{\tau_0}+
\frac{\tau_0}{\sqrt{2\pi}}\phi^{\prime2}_i\right)\approx0\,,\label{Cgf1}\\
&&\mathcal{C}_\theta:=\sum_{i}\int_{\mathbb{S}^1}p_{\phi_i}\phi^\prime_i\approx0\,.\label{Cgf2}\end{aligned}$$ Let us look now at the gauge transformations of $\tau_0$ generated by (\[Cgf1\]) $$\begin{aligned}
&&\{\tau_0,\mathcal{C}\}\approx-p_{\gamma_0}\,,\nonumber\\
&&\{p_{\gamma_0},\mathcal{C}\}\approx0\,.\nonumber\end{aligned}$$ If $s\in (0,\infty)$ parameterizes the gauge orbits we see that on them we have $\tau_0=ps$ and $p_{\gamma_0}=-p$, with $p>0$. This suggests that a simplification of our model will occur if we introduce a canonical transformation where $\tau_0$ and $p_{\gamma_0}$ are substituted for new canonical variables. Indeed, the canonical transformation [@Cortez] $$\begin{aligned}
&\displaystyle\tau_0=TP\,,\hspace{2.3cm}&p_{\tau_0}=\frac{p_T}{P}\,,\nonumber\\
&\displaystyle\gamma_0=-\frac{1}{\sqrt{2\pi}}\big(Q+\frac{p_T}{P}T\big)\,,
&p_{\gamma_0}=-\sqrt{2\pi}P\,,\nonumber\end{aligned}$$ with $(Q,P>0)$, and $(T,p_T)$ canonically conjugate pairs, allows us to write $$\mathcal{C}=p_T+\frac{1}{2}\sum_{i}\int_{\mathbb{S}^1}\left(\frac{p^2_{\phi_i}}{PT}+
PT\phi^{\prime2}_i\right)\approx0\,.\label{Cgf3}$$ Finally the canonical transformation (here $(\tilde{Q},\tilde{P})$ and $(\varphi_i,p_{\varphi_i})$ are new canonical pairs [@Cortez]) $$\begin{aligned}
&&\tilde{Q}:=PQ+\frac{1}{2}\sum_{i}\int_{\mathbb{S}^1}p_{\phi_i}\phi_i\,,
\quad \tilde{P}:=\log P\,,\nonumber\\
&&\varphi_i:=\sqrt{P}\phi_i\,,\quad
p_{\varphi_i}:=\frac{1}{\sqrt{P}}p_{\phi_i}\,,\end{aligned}$$ turns the constraints (\[Cgf3\],\[Cgf2\]) into $$\begin{aligned}
&&\mathcal{C}=p_T+\frac{1}{2}\sum_{i}\int_{\mathbb{S}^1}\bigg(\frac{p^2_{\varphi_i}}{T}+
T\varphi^{\prime2}_i\bigg)\approx0\,,\label{C1}\\
&&\mathcal{C}_\theta=\sum_{i}\int_{\mathbb{S}^1}p_{\varphi_i}\varphi^\prime_i\approx0\label{C2}\,,\end{aligned}$$ and the 2-form (\[GG2-f\]) becomes $$\label{omegaHR}
\iota^{*}\omega=\mathrm{d}\tilde{Q}\wedge\mathrm{d}\tilde{P}+\sum_{i}\int_{\mathbb{S}^{1}}
\delta\varphi_{i}\wedge\delta
p_{\varphi_i}+\mathrm{d}T\wedge\mathrm{d}p_T\,.$$ The fact that (\[C1\]) is linear in $p_T$ allows us to interpret the 4-tuple $((0,\infty)\times\Gamma_{R},\mathrm{d}t,\omega_R,H_{R})$ as a non-autonomous Hamiltonian system with $T=t$ as the time parameter, restricted to verify the global constraint (\[C2\]). The reduced phase space $\Gamma_{R}$ is coordinatized now by the canonical pairs $(\tilde{Q},\tilde{P};\varphi_{i},p_{\varphi_i})$ and is endowed with the (weakly) symplectic form $$\label{omegaR}
\omega_{R}:=\mathrm{d}\tilde{Q}\wedge\mathrm{d}\tilde{P}+\sum_{i}\int_{\mathbb{S}^{1}}
\delta\varphi_{i}\wedge\delta p_{\varphi_i}\,.$$ The reduced time-dependent Hamiltonian $H_{R}(t):\Gamma_{R}\rightarrow\mathbb{R}$ is given by $$\label{HR}
H_{R}(t)=\frac{1}{2}\sum_{i}\int_{\mathbb{S}^1}\bigg(\frac{p^2_{\varphi_i}}{t}+
t\varphi^{\prime2}_i\bigg)\,,$$ and the evolution vector field is given by $$E_{H_R}=\frac{\partial}{\partial
t}+\sum_{i}\int_{\mathbb{S}^{1}}\left(\frac{p_{\varphi_i}}{t}\frac{\delta}{\delta\varphi_i}+t\varphi_{i}^{\prime\prime}\frac{\delta}{\delta
p_{\varphi_i}}\right).$$ This defines the only degenerate direction of (\[omegaHR\]). Although the form of the Hamiltonian that we have just obtained seems to suggest that the gravitational and matter scalars are not coupled, in fact the constraint (\[C2\]) shows that this is not the case[^9]. Notice also that the canonical pair $(\tilde{Q},\tilde{P})$ describes a global degree of freedom even though they are constants of motion under the dynamics generated by (\[HR\]). The singularities that must be present in this case as a consequence of the Hawking-Penrose theorems [@Wald] can be understood as coming from the singular behavior at $t=0$ of the Hamiltonian (\[HR\]).
Finally, it is possible to recover the original 4-dimensional spacetime from this 3-dimensional formulation. First notice that the gauge fixing conditions defining the deparameterization are preserved under the dynamics if and only if the lapse and shift functions $N$ and $N^{\theta}$ are constant. By redefining the coordinate $\theta$ as in [@Mena:1997] we can eliminate the shift function from the metric. We can proceed in an analogous way for the lapse function to make it equal to 1. Once we integrate the Hamiltonian equations corresponding to (\[HR\]), undo the canonical transformation defined above, and solve the constraint $C_{\theta}=0$ in order to obtain the $\gamma$ function, we uniquely determine the 3-metric (\[metric\]), and hence the original 4-metric.
$\mathbb{S}^1\times\mathbb{S}^2$ Gowdy models coupled to massless scalars
=========================================================================
[\[handle\]]{}
Let us consider now the three-handle ${^{{{\scriptstyle{(3)}}}}}\Sigma=\mathbb{S}^1\times\mathbb{S}^2$, parameterized as $(e^{i\xi},e^{i\sigma}\sin\theta,\cos\theta)$ with $\theta\in[0,\pi]$, $\xi,\sigma\in\mathbb{R}(\mathrm{mod}\,2\pi)$. Using the group parametrization introduced above we can write the $G^{{\scriptstyle{(2)}}}$-group action in the form $$(g_1,g_2)\cdot(e^{i\xi},e^{i\sigma}\sin\theta,\cos\theta)=
(e^{x_1},e^{x_2})\cdot(e^{i\xi},e^{i\sigma}\sin\theta,\cos\theta)=
(e^{i(x_1+\xi)},e^{i(x_2+\sigma)}\sin\theta,\cos\theta)\,.$$ The action of the two $U(1)$ subgroup factors of $G^{{\scriptstyle{(2)}}}$ is $$\begin{aligned}
&&(1,e^{ix})\cdot(e^{i\xi},e^{i\sigma}\sin\theta,\cos\theta)=
(e^{i\xi}, e^{i(x+\sigma)}\sin\theta,\cos\theta)\,,\nonumber\\
&&(e^{ix},1)\cdot(e^{i\xi},e^{i\sigma}\sin\theta,\cos\theta)=
(e^{i(x+\xi)},e^{i\sigma}\sin\theta,\cos\theta)\,.\nonumber\end{aligned}$$ The corresponding tangent vectors at each point of ${^{{{\scriptstyle{(3)}}}}}\Sigma$, obtained by differentiating the previous expressions with respect to $x$ at $x=0$, are $$\begin{aligned}
&&(0,ie^{i\sigma}\sin\theta,0)\,,\nonumber\\
&&(ie^{i\xi},0,0)\,.\nonumber\end{aligned}$$ As we can see the second one is never zero but the first one vanishes at the poles of the sphere $\mathbb{S}^2$ where $\theta=0,\pi$. This corresponds to the circumferences given by $(e^{i\xi},0,1)$ and $(e^{i\xi},0,-1)$. It is straightforward to verify that both fields commute. In view of all this we perform a Geroch reduction by using the non-vanishing Killing. After a suitable conformal transformation the field equations can be derived from an action of the form (\[actr\]) with $^{\scriptscriptstyle{{{\scriptstyle{(2)}}}}}\Sigma=\mathbb{S}^2$. All the fields in this action are defined on $\mathbb{S}^2$ and are symmetric under the symmetries generated by the remaining Killing $\sigma^a$. Since this Killing vector vanishes at the poles of the sphere $\mathbb{S}^2$ we cannot build an everywhere orthonormal basis that involves this vector. In fact, we know that as $\mathbb{S}^2$ is not paralelizable this is impossible on general grounds. We nevertheless will consider the triplet of vectors $(n^a,\hat{\theta}^a,\sigma^a/\tau)$ whenever it is different from zero (for all $\theta\neq0,\pi$). Taking again the definition (\[ta\]), the form of the metric is the same as in the $\mathbb{T}^3$ case (\[metric\]). The symmetry of the problem implies also that $(N,N^\theta,\gamma,\tau,\phi_{i})$ are constant on the orbits of the Killing field $\sigma^a$.
A very important issue now is the regularity of the metric. From a classical point of view the final outcome of the Hamiltonian analysis of the system is a set of equations whose solutions allow us to reconstruct a four dimensional spacetime metric and a set of scalar fields satisfying the coupled Einstein-Klein Gordon equations. This means that once we decide the functional space to which this metric belongs this will imply that the objects that appear during the dimensional reduction, gauge fixing and so on may be subject to some regularity conditions. In the $\mathbb{T}^3$ case these are simple smoothness requirements but in the present case, due to the existence of a symmetry axis, these are more complicated. The regularity conditions that the metric components for an axially symmetric metric must verify can be deduced as in [@Chrusciel:1990zx; @Rinne:2005sk]. By using the coordinates $(t,\theta,\sigma,\xi)$, we can write the original 4-metric ${^{{{\scriptstyle{(4)}}}}}g_{ab}$ as $$\begin{aligned}
{^{{{\scriptstyle{(4)}}}}}g_{ab}&=&e^{(\gamma-\phi_1)}[(N^{\theta
2}-N^{2})(\mathrm{d}t)_a(\mathrm{d}t)_b+
2N^{\theta}(\mathrm{d}t)_{(a}(\mathrm{d}\theta)_{b)}+(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b]
\label{4metric}\\
&+& \tau^{2}e^{-\phi_1}(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b+
e^{\phi_1}(\mathrm{d}\xi)_a(\mathrm{d}\xi)_b\,.\nonumber\end{aligned}$$ This means that we have the following regularity conditions for $\theta\rightarrow0,\pi$ (if we impose analyticity, otherwise we need only to know the asymptotic behavior for small values of $\sin{\theta}$) $$\begin{aligned}
&&e^{(\gamma-\phi_1)}(N^{\theta 2}-N^2)=A(t,\cos\theta)\,,\label{reg1}\\
&&e^{(\gamma-\phi_1)}N^\theta=B(t,\cos\theta)\sin\theta\,, \label{reg2}\\
&&e^{\phi_1}=C(t,\cos\theta)\,,\label{reg3}\\
&&e^{\gamma-\phi_1}=D(t,\cos\theta)+E(t,\cos\theta)\sin^2\theta\,, \label{reg4}\\
&&\tau^2e^{-\phi_1}=\sin^2\theta[D(t,\cos\theta)-E(t,\cos\theta)\sin^2\theta
]\,,\label{reg5}\end{aligned}$$ where $A,\,B,\,C,\,D,\,E$ are analytic in their arguments (despite of the fact that they also depend on $t$, as we will use them in the Hamiltonian formulation of the model we will not write the $t$ dependence explicitly in the following). Notice, in particular, that the functions $D$ and $E$ both appear in the last two expressions. The conditions for the fields themselves (dropping the $t$ dependence) are $$\begin{aligned}
&&\phi_i=\hat{\phi}_{i}(\cos\theta)\,,\label{reg_1}\\
&&\gamma=\hat{\gamma}(\cos\theta)\,,\label{reg_2}\\
&&N^\theta=\hat{N}^{\theta}(\cos\theta)\sin\theta\,,\label{reg_3}\\
&&N=\hat{N}(\cos\theta)\,,\label{reg_4}\\
&&\tau=\hat{T}(\cos\theta)\sin\theta\,,\label{reg_5}\\
&&\tau^2e^{-\gamma}=\frac{D(\cos\theta)-E(\cos\theta)\sin^2\theta}{D(\cos\theta)
+E(\cos\theta)\sin^2\theta}\sin^2\theta\,,\label{reg_6}\end{aligned}$$ where $\hat{\phi}_{i},\hat{\gamma},\hat{N}^{\theta},\hat{N},\hat{T}:[-1,1]\rightarrow\mathbb{R}$ ($\hat{N}>0$) and can be written as functions of $A,\,B,\,C,\,D,\,E$. They must be differentiable in $(-1,1)$ and their right and left derivatives at $\pm1$ must be defined (equivalently they must be $\mathcal{C}^\infty$ in $(-1,1)$ with bounded derivative). Several comments are in order now. First we have been able to write all the relevant fields in such a way that their singular dependence has been factored out ($\sin\theta$ *is not* a smooth function on the sphere). The functions defined on $\mathbb{S}^2$ as $\hat{\phi}_{i}\circ\cos\theta,\hat{\gamma}\circ\cos\theta,
\hat{N}^{\theta}\circ\cos\theta,\hat{N}\circ\cos\theta,\hat{T}\circ\cos\theta$ can be alternatively viewed as analytic functions on the sphere invariant under rotations around its symmetry axis and can be considered as the basic fields to describe our system. In fact we will do so in the following. We will refer to these functions on the sphere as $\hat{\phi}_{i},\hat{\gamma},\hat{N}^{\theta},\hat{N},\hat{T}$ (without the $\circ\cos\theta$ that will only be used if the possibility of confusion arises) and collectively as the *hat*-fields. In the following we will write everything in terms of them. Second we can see that condition (\[reg\_6\]) implies that $$\hat{T}(\pm1)=e^{\hat{\gamma}(\pm1)/2}\,.\label{polarS2}$$ This means that the values of the fields $\hat{T}$ and $\hat{\gamma}$ at the poles of the sphere are not independent of each other. This is a new feature, not present for the $\mathbb{T}^3$ topology, that must be taken into account. As we will see these are necessary ingredients to guarantee the consistency of the model.
Given a smooth (and axially symmetric) function on $\mathbb{S}^2$ its Lie derivative along $\theta^a$, $\mathcal{L}_\theta$, cannot necessarily be extended as a smooth function on the sphere. The function $\cos\theta$ itself is an example of this because $\mathcal{L}_\theta\cos\theta=-\sin\theta$. We can, however, define a smooth derivative $f^\prime$ for a smooth axially symmetric function as the extension of $f^\prime:=-\frac{1}{\sin\theta}\partial_\theta f$ to $\mathbb{S}^2$ (this is formally done by considering $f$ as a function of $\cos\theta$ and differentiating). In the following the *prime* symbol will always refer to this derivative.
It is natural to consider in this case ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}$ as the fiducial 2-form associated to a round metric, such that ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}_{ab}\theta^{a}\sigma^{b}=\hat{N}e^{\hat{\gamma}}\hat{T}\sin\theta/|g|^{1/2}=\sin\theta$, i.e. ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}=\sin\theta\mathrm{d}\theta\wedge\mathrm{d}\sigma$. Taking this into account we get an action $$\begin{aligned}
&&\frac{1}{16\pi G_3}\int_{t_0}^{t_1}\!\!\!\!\!
\mathrm{d}t\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\bigg[
\hat{N}[(\hat{\gamma}^\prime\hat{T}^\prime-2\hat{T}^{\prime\prime})\sin^2\theta+
(6\hat{T}^{\prime}-\hat{\gamma}^\prime\hat{T})\cos\theta+2\hat{T}]\nonumber\\
&&\hspace{30mm}+\frac{1}{\hat{N}}[\hat{N}^\theta
\hat{T}\cos\theta-\dot{\hat{T}}-\hat{N}^\theta
\hat{T}^\prime\sin^2\theta]\,[\dot{\hat{\gamma}}
+(2\hat{N}^{\theta\prime}+\hat{N}^\theta\hat{\gamma}^\prime)\sin^2\theta-2\hat{N}^\theta\cos\theta]
\nonumber\\
&&\hspace{30mm}+\frac{\hat{T}}{2\hat{N}}\sum_{i}\bigg({\dot{\hat{\phi}}}\,^2_i
+2\hat{N}^\theta\dot{\hat{\phi}}_i\,\hat{\phi}_i^\prime\sin^2\theta+(\hat{N}^{\theta2}\sin^2\theta
-\hat{N}^2)\hat{\phi}_i^{\prime2}\sin^2\theta\bigg)\bigg]\,.\nonumber\end{aligned}$$ As we can see it is expressed as the integral of a smooth function on the sphere. This is so because all the fields that appear in the integrand are either the $hat$-fields, their prime derivatives or smooth functions of $\cos\theta$. The Hamiltonian can be readily derived from the previous action and, as in the $\mathbb{T}^{3}$ case, is of the form $H=C[\hat{N}]+C_\theta[\hat{N}^\theta]$ with $$\begin{aligned}
C[\hat{N}]&=&\int_{\mathbb{S}^2}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}\hat{N}C\label{C1S2}\\
&:=&\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\hat{N}\bigg[ -16\pi
G_3 p_{\hat{\gamma}}p_{\hat{T}}+\frac{1}{16\pi
G_3}\big[(2\hat{T}^{\prime\prime}-\hat{\gamma}^\prime\hat{T}^\prime)\sin^2\theta
+(\hat{\gamma}^\prime\hat{T}-6\hat{T}^\prime)\cos\theta-2\hat{T}\big]\nonumber\\
&&\hspace{1cm}+\frac{1}{2}\sum_{i}\bigg(\frac{16\pi
G_3}{\hat{T}}p_{\hat{\phi}_i}^2+\frac{\hat{T}}{16\pi
G_3}\hat{\phi}_i^{\prime2}\sin^2\theta\bigg)\bigg],\nonumber\\
C_\theta[\hat{N}^\theta]&=&\int_{\mathbb{S}^2}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}\hat{N}^{\theta}C_{\theta}\label{C2S2}
\\&:=&\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\hat{N}^\theta
\bigg([2p^\prime_{\hat{\gamma}}-\hat{\gamma}^\prime
p_{\hat{\gamma}}- \hat{T}^\prime
p_{\hat{T}}-\sum_{i}\hat{\phi}_i^\prime
p_{\hat{\phi}_i}]\sin^2\theta +[\hat{T}
p_{\hat{T}}-2p_{\hat{\gamma}}]\cos\theta\bigg).\nonumber\end{aligned}$$ Again, the dynamical variables are restricted to belong to a constraint surface $\Gamma_{c}\subset\Gamma$ in the canonical phase space of the system $(\Gamma,\omega)$, globally defined by the constraints $C=0$, $C_{\theta}=0$. $\Gamma$ is coordinatized by the conjugated pairs $(\hat{\gamma},p_{\hat{\gamma}};\hat{T},p_{\hat{T}};\hat{\phi}_{i},p_{\hat{\phi}_i})$ and endowed with the standard (weakly) symplectic form $$\label{omegaS2XS1}
\omega:=\int_{\mathbb{S}^2}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}\bigg(\delta\hat{\gamma}\wedge\delta
p_{\hat{\gamma}}+\delta\hat{T}\wedge\delta
p_{\hat{T}}+\sum_{i}\delta\hat{\phi}_i\wedge\delta
p_{\hat{\phi}_i}\bigg)\,.$$ The gauge transformations generated by these constraints are[^10] $$\begin{aligned}
&&\{\hat{\gamma},C[\hat{N}_g]\}=-\hat{N}_gp_{\hat{T}}\,,
\nonumber\\
&&\{\hat{T},C[\hat{N}_g]\}=-\hat{N}_g p_{\hat{\gamma}}\,,\nonumber\\
&&\{\hat{\phi_i},C[\hat{N}_{g}]\}=\frac{\hat{N}^g}{\hat{T}}p_{\hat{\phi}_i}\,,\nonumber\\
&&\{p_{\hat{\gamma}},C[\hat{N}_g]\}= \hat{N}_{g}^{\prime}(\hat{T}\cos\theta-\hat{T}^\prime\sin^2\theta)+
\hat{N}_g(\hat{T}+3\hat{T}^\prime\cos\theta-\hat{T}^{\prime\prime}\sin^2\theta),\nonumber\\
&&\{p_{\hat{T}},C[\hat{N}_{g}]\}=\hat{N}_{g}^{\prime}(2\cos\theta-\hat{\gamma}^\prime\sin^2\theta)+
\hat{N}_{g}(\hat{\gamma}^\prime\cos\theta-\hat{\gamma}^{\prime\prime}\sin^2\theta)-2\hat{N}_{g}^{\prime\prime}
\sin^2\theta\nonumber\\
&&\hspace{2.6cm}+\frac{\hat{N}_g}{2}\sum_{i}\bigg(\frac{p_{\hat{\phi}_i}^2}{\hat{T}^2}-
\sin^2\theta\hat{\phi}_i^{\prime2}\bigg)\,,\nonumber\\
&&\{p_{\hat{\phi}_i},C[\hat{N}_g]\}=
\hat{N}_{g}^{\prime}\hat{T}\hat{\phi}_i^{\prime}\sin^2\theta+\hat{N}_{g}[(\hat{T}^\prime\hat{\phi}_i^\prime+
\hat{T}\hat{\phi}_i^{\prime\prime})\sin^2\theta-2\hat{T}\hat{\phi_i^\prime}\cos\theta]\,,\nonumber\\\end{aligned}$$ and $$\begin{aligned}
&&\{\hat{\gamma},C_\theta[\hat{N}_{g}^\theta]\}=-2\hat{N}_{g}^{\theta\prime}\sin^2\theta+
\hat{N}_{g}^\theta(2\cos\theta-\hat{\gamma}^\prime\sin^2\theta)\,,\nonumber\\
&&\{\hat{T},C_\theta[\hat{N}_{g}^\theta]\}=\hat{N}_{g}^\theta(\hat{T}\cos\theta-\hat{T}^\prime\sin^2\theta)\,,\nonumber\\
&&\{\hat{\phi}_i,C_\theta[\hat{N}_{g}^\theta]\}=-\hat{N}_{g}^\theta\hat{\phi}_i^\prime\sin^2\theta\,,\nonumber\\
&&\{p_{\hat{\gamma}},C_\theta[\hat{N}_{g}^\theta]\}=
\hat{N}_{g}^\theta(2p_{\hat{\gamma}}\cos\theta-p^\prime_{\hat{\gamma}}\sin^2\theta)-
\hat{N}_{g}^{\theta\prime}p_{\hat{\gamma}}\sin^2\theta\,,\nonumber\\
&&\{p_{\hat{T}},C_\theta[\hat{N}_{g}^\theta]\}=\hat{N}_{g}^\theta(p_{\hat{T}}\cos\theta-p^\prime_{\hat{T}}\sin^2\theta)-
\hat{N}_{g}^{\theta\prime}p_{\hat{T}}\sin^2\theta\,,\nonumber\\
&&\{p_{\hat{\phi}_i},C_\theta[\hat{N}_{g}^\theta]\}=\hat{N}_{g}^\theta(2p_{\hat{\phi}_i}\cos\theta-
p^\prime_{\hat{\phi}_i}\sin^2\theta)-\hat{N}_{g}^{\theta\prime}p_{\hat{\phi}_i}\sin^2\theta\,.\nonumber\\\end{aligned}$$ As is the $\mathbb{T}^{3}$ case, $\Gamma_c\subset\Gamma$ is a first class submanifold as can be seen by computing the Poisson algebra of the (weighted) constraints $$\begin{aligned}
&&\hspace{-2mm}\{C[\hat{N}_g],C[\hat{M}_g]\}=C_\theta[\hat{M}_g
\hat{N}_g^{\prime}-\hat{N}_g
\hat{M}_g^{\prime}]\,,\nonumber\\
&&\hspace{-2mm}\{C[\hat{N}_g],C_\theta[\hat{M}_g^\theta]\}=C[(\hat{M}_{g}^\theta
\hat{N}_{g}^{\prime}-\hat{N}_g
\hat{M}_{g}^{\theta\prime})\sin^2\theta+\hat{N}_g\hat{M}_{g}^\theta\cos\theta]\,,\nonumber\\
&&\hspace{-2mm}\{C_\theta[\hat{N}_{g}^\theta],C_\theta[\hat{M}_{g}^\theta]\}=C_\theta[(\hat{M}_{g}^\theta
\hat{N}_{g}^{\theta\prime}-\hat{N}_{g}^\theta
\hat{M}_{g}^{\theta\prime})\sin^2\theta]\,.\nonumber\end{aligned}$$ We want to check now the stability of the “polar constraints” $(\hat{T}e^{-\hat{\gamma}/2})(\pm1)=1$. To this end we compute $$\begin{aligned}
&&\hspace{-4mm}\{\hat{T}e^{-\hat{\gamma}/2},C[\hat{N}_{g}]\}=8\pi
G_3\hat{N}_{g}e^{-\hat{\gamma}/2}
\left(\hat{T}p_{\hat{T}}-2p_{\hat{\gamma}}\right),\nonumber\\
&&\hspace{-4mm}\{\hat{T}e^{-\hat{\gamma}/2},C_\theta[\hat{N}_{g}^\theta]\}=
e^{-\hat{\gamma}/2}\big[\hat{T}\hat{N}_g^{\theta\prime}+
\hat{N}_{g}^{\theta}(\frac{1}{2}\hat{T}\hat{\gamma}^\prime-\hat{T}^\prime)\big]\sin^2\theta\,.
\nonumber\end{aligned}$$ The first expression vanishes at the poles as a consequence of the constraint (\[C2S2\]) for $\theta=0,\,\pi$ ($\sin\theta=0$ and $|\cos\theta|=1$) whereas the second vanishes because of the $\sin^2\theta$ factor. We then conclude that there are no secondary constraints coming from the stability of the polar constraints. An interesting point to highlight here is the fact that these polar constraints are necessary conditions for the differentiability of the constraints (\[C1S2\],\[C2S2\]).
Deparameterization in this case is carried out by basically following the same steps as in the $\mathbb{T}^3$ case. Again, in view of the gauge transformations, we begin by choosing gauge fixing conditions similar to (\[gauge1\],\[gauge2\]) $$\begin{aligned}
&&\hat{T}^\prime=0\,,\label{gauge1S2}\\
&&p_{\hat{\gamma}}^\prime=0\,.\label{gauge2S2}\end{aligned}$$ We introduce now an orthonormal basis of functions on the subspace of axially symmetric functions on $\mathbb{S}^2$ $$Y_{n}(\theta)=
\left({\frac{2n+1}{4\pi}}\right)^{1/2}P_n(\cos\theta)\,,\,\,\,n\in\mathbb{N}\cup\{0\}\,,$$ where $P_n$ are the Legendre polynomials. By expanding now $$\begin{aligned}
\quad\hat{T}=\sum_{n=0}^\infty \hat{T}_nY_{n}\,,\quad
p_{\hat{\gamma}}=\sum_{n=0}^\infty p_{\hat{\gamma}_n}Y_n\,,\end{aligned}$$ with $$\begin{aligned}
&&\hat{T}_n=\left({\frac{2n+1}{4\pi}}\right)^{1/2}\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}
P_{n}(\cos\theta)\hat{T}\,,\quad
p_{\hat{\gamma}_n}=\left({\frac{2n+1}{4\pi}}\right)^{1/2}\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}
P_n(\cos\theta)p_{\hat{\gamma}}\,,\end{aligned}$$ the conditions (\[gauge1S2\],\[gauge2S2\]) become $$\hat{T}_n=0=p_{\hat{\gamma}_n}\,,\,\,\, \forall\,n\in
\mathbb{N}\,.\label{gauge3S2}$$ In order to see if this is a good gauge fixing (and, alternatively, find out if some gauge freedom is left) we compute $$\begin{aligned}
&&\{\hat{T}_n,C[Y_m]\}\approx-\sqrt{\frac{(2n+1)(2m+1)}{(4\pi)^3}}p_{\hat{\gamma}0}\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}P_n(\cos\theta)P_m(\cos\theta)=\frac{1}{\sqrt{4\pi}}p_{\hat{\gamma}0}\delta_{nm}
\nonumber\\
&&\{\hat{T}_n,C_\theta[Y_m]\}\approx\frac{\hat{T}_0}{4\pi}\sqrt{\frac{(2n+1)(2m+1)}{4\pi}}
\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\cos\theta P_n(\cos\theta)P_m(\cos\theta)\nonumber\\
&&\hspace{2cm}=\left\{\begin{array}{l}\displaystyle-\frac{(m+1)\hat{T}_0}{\sqrt{4\pi(2m+1)(2m+3)}}
\quad\mathrm{if}\,n=m+1\\
\displaystyle-\frac{m\hat{T}_0}{\sqrt{4\pi(2m+1)(2m-1)}}\quad\mathrm{if}\,n=m-1\\
0\quad\quad\mathrm{otherwise}\end{array}\right.\nonumber\\
&&\{p_{\gamma
n},C[Y_m]\}\approx\hat{T}_0\sqrt{\frac{(2n+1)(2m+1)}{(4\pi)^3}}
\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}P_n(\cos\theta)[P_m(\cos\theta)+
\cos\theta P_m^\prime(\cos\theta)]\nonumber\\
&&\hspace{2cm}=\left\{\begin{array}{l}\displaystyle0\quad\quad\mathrm{if}\,\,m=0\,\, \mathrm{or}\,\,m<n\nonumber\\
\displaystyle-\hat{T}_0\frac{n+1}{\sqrt{4\pi}}\quad\mathrm{if}\,\,m=n\\
\displaystyle\star\quad \quad\mathrm{otherwise}\end{array}\right.\nonumber\\
&&\{p_{\gamma
n},C_\theta[Y_m]\}\approx p_{\gamma0}\sqrt{\frac{(2n+1)(2m+1)}{(4\pi)^3}}
\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}P_n(\cos\theta)[P_m(\cos\theta)-\sin^2\theta
P_m^\prime(\cos\theta)]\nonumber\\
&&\hspace{2cm}=\left\{\begin{array}{l}\displaystyle0\quad\quad\mathrm{if}\,\,m=0\,\,
\mathrm{or}\,\,m<n-1\nonumber\\
\displaystyle\ast\quad\quad\mathrm{otherwise}\end{array}\right.\nonumber\end{aligned}$$ where the symbol $\approx$ denotes that we are restricting ourselves to points in the hypersurface $\Gamma_{G}\subset\Gamma_c$ defined by the gauge fixing conditions and the constraints. The $\star$ and $\ast$ symbols denote terms (computable in closed form but with somewhat complicated expressions) that are not needed in the following discussion. As before it helps to display the previous result in table form
$\hat{T}_1=0$ $p_{\gamma,1}=0$ $\hat{T}_{2}=0$ $p_{\gamma,2}=0$ $\hdots$
----------------- ------------------------------------ -------------------------------- ------------------------------------ ------------------------------------- ----------
$C[Y_0]$ $0$ $0$ $0$ $0$ $\hdots$
$C_\theta[Y_0]$ $\frac{\hat{T}_0}{2\sqrt{3\pi}}$ $0$ $0$ $0$ $\hdots$
$C[Y_1]$ $-\frac{p_{\gamma0}}{2\sqrt{\pi}}$ $\frac{\hat{T}_0}{\sqrt{\pi}}$ $0$ $0$ $\hdots$
$C_\theta[Y_1]$ $0$ $\ast$ $\frac{\hat{T}_0}{\sqrt{15\pi}}$ $0$ $\hdots$
$C[Y_2]$ $0$ $\star$ $-\frac{p_{\gamma0}}{2\sqrt{\pi}}$ $\frac{3\hat{T}_0}{\sqrt{4\pi}} $ $\hdots$
$C_\theta[Y_2]$ $\frac{\hat{T}_0}{\sqrt{15\pi}}$ $\ast$ $0$ $\ast$ $\hdots$
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\ddots$
One must also check if the polar constraints are gauge fixed by our conditions (\[gauge3S2\]). To this end we compute $$\begin{aligned}
&&\{\hat{T}_n,\hat{T}e^{-\hat{\gamma}/2}\}\approx0\,,\nonumber\\
&&\{p_{\hat{\gamma}n},\hat{T}e^{-\hat{\gamma}/2}\}\approx
\frac{1}{2}\hat{T}e^{-\hat{\gamma}/2}\sqrt{\frac{2n+1}{4\pi}}P_n(\cos\theta)\,.\nonumber\end{aligned}$$ The last Poisson bracket is different from zero at the poles $(\theta=0,\pi)$ for all values of $n\in\mathbb{N}$. As we can see the only constraint that is not gauge-fixed by the conditions introduced above, as long as $p_{\gamma_0}\neq0$ and $\hat{T}_0\neq0$, is $C[1]$. This is different from the situation in the $\mathbb{T}^2$ case where we were left with two constraints instead of just one.
As we did before we can pullback everything to the phase space hypersurface defined by the gauge fixing conditions. The induced 2-form becomes $$\begin{aligned}
\iota^{*}\omega=\mathrm{d}\hat{\gamma}_0\wedge \mathrm{d}
p_{\hat{\gamma}_0}+\mathrm{d}\hat{T}_0\wedge \mathrm{d}
p_{\hat{T}_0}+\sum_{i}\int_{\mathbb{S}^2}{^{{{\scriptstyle{(2)}}}}}e\,
\delta\phi_i\wedge \delta p_{\phi_i}\,\label{symplecticS2}\end{aligned}$$ and the remaining constraint is $$\begin{aligned}
\label{remCS2}\mathcal{C}&:=&-p_{\hat{\gamma}_0}p_{\hat{T}_0}+\hat{T}_0
\big[4\sqrt{\pi}\big(\log\frac{\hat{T}_0}{\sqrt{4\pi}}-1\big)-\hat{\gamma_0}\big]
\\&+&\frac{1}{2}\sum_{i}\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\left(\frac{\sqrt{4\pi}}{\hat{T}_0}p_{\hat{\phi}_i}^2+
\frac{\hat{T}_0}{\sqrt{4\pi}}\hat{\phi}_i^{\prime2}\sin^2\theta\right)\approx0\,.\nonumber\end{aligned}$$ The gauge transformations generated by this constraint in the variables $\hat{T}_0$ and $p_{\hat{\gamma}_0}$ are $$\begin{aligned}
&&\{\hat{T}_0,\mathcal{C}\}=-p_{\hat{\gamma}_0}\,,\nonumber\\
&&\{p_{\hat{\gamma}_0},\mathcal{C}\}=\hat{T}_0\,,\nonumber\end{aligned}$$ so if we parameterize the gauge orbits as before with $s\in(0,\pi)$ we find now $\hat{T}_0=p\sin s$ and $p_{\hat{\gamma}_0}=-p\cos s$, $p\neq0$. In the spirit of the previous section we introduce now the following canonical transformation ($(Q,P)$ and $(T,p_{T})$ denote canonically conjugate pairs) $$\begin{aligned}
&\displaystyle\hat{T}_0=P\sin T\,,\hspace{1.7cm}&p_{\hat{T}_0}=\frac{p_T}{P}\cos T-Q\sin T\,,\nonumber\\
&\displaystyle\hat{\gamma}_0=-Q\cos T-\frac{p_T}{P}\sin
T\,,&p_{\hat{\gamma}_0}=-P\cos T\,.\nonumber\end{aligned}$$ In addition, as we did in the $\mathbb{T}^3$ case, it is possible to write the remaining constraint in a more pleasant form by performing a further canonical transformation (here, again, $(\tilde{Q},\tilde{P})$ and $(\varphi_i,p_{\varphi_i})$ are canonical pairs) $$\begin{aligned}
&\displaystyle\tilde{Q}:=PQ+\frac{1}{2}\sum_{i}\int_{\mathbb{S}^2}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}\,p_{\hat{\phi}_i}\hat{\phi}_i\,,
&\displaystyle\tilde{P}:=\log P\,,\nonumber\\
&\displaystyle\varphi_i=(4\pi)^{-1/4}\sqrt{P}\hat{\phi}_i\,,\hspace{1.7cm}&p_{\varphi_i}=(4\pi)^{1/4}\frac{p_{\hat{\phi}_i}}{\sqrt{P}}\,,\end{aligned}$$ giving $$\mathcal{C}=p_{T}+4\sqrt{\pi}e^{\tilde{P}}(\log \frac{\sin
T}{\sqrt{4\pi}}+\tilde{P}-1)\sin
T+\frac{1}{2}\sum_{i}\int_{\mathbb{S}^{2}}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}\left(\frac{p_{\varphi_{i}}^{2}}{\sin
T}+\varphi_{i}^{\prime2}\sin T\sin^{2}\theta\right)\approx 0\,.$$ It is now obvious the interpretation of the system as a non-autonomous Hamiltonian system $((0,\pi)\times\Gamma_R,\mathrm{d}t,\omega_R,H_R)$, where $\Gamma_{R}$ denotes the reduced phase space coordinatized by the canonical pairs $(\tilde{Q},\tilde{P};\varphi_{i},p_{\varphi_i})$, endowed with the standard (weakly) simplectic form (\[omegaR\]). The dynamics is given by the time dependent Hamiltonian $H_{R}(t):\Gamma_{R}\rightarrow\mathbb{R}$ $$H_{R}(t)=4\sqrt{\pi}e^{\tilde{P}}(\log \frac{\sin
t}{\sqrt{4\pi}}+\tilde{P}-1)\sin
t+\frac{1}{2}\sum_{i}\int_{\mathbb{S}^{2}}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}\left(\frac{p_{\varphi_{i}}^{2}}{\sin
t}+\varphi_{i}^{\prime2}\sin t\sin^{2}\theta\right),\label{4.24}$$ with the evolution vector field
$$\begin{aligned}
E_{H_R}&=&\frac{\partial}{\partial
t}+4\sqrt{\pi}e^{\tilde{P}}(\log\frac{\sin
t}{\sqrt{4\pi}}+\tilde{P})\sin
t\frac{\partial}{\partial\tilde{Q}}\nonumber\\
&&+\sum_{i}\int_{\mathbb{S}^{2}}{^{(2)}}\mathrm{e}\left(\frac{p_{\varphi_{i}}}{\sin
t}\frac{\delta}{\delta \varphi_{i}}+(\sin^{2}\theta\varphi_i^{\prime})^{\prime}\sin
t\frac{\delta}{\delta
p_{\varphi_{i}}}\right).\end{aligned}$$
Several comments are in order at this point. First we can see now that the final description of our system is somewhat simpler that in the $\mathbb{T}^3$ case because we do not have any remaining constraints and the fields $\varphi_1$ and $\varphi_2$ are decoupled (at variance with the previous case). On the other hand we see now that the dynamics of the global modes, though easy to get in explicit form, is not as simple as the one found for the torus. Notice also that the Hamiltonian (\[4.24\]) is singular whenever $\sin t=0$. This means that if we pick the initial time $t_0\in(0,\pi)$ in order to write the Cauchy data we meet a past singularity at $t=0$ and a future singularity at $t=\pi$.
$\mathbb{S}^3$ Gowdy models coupled to massless scalars
=======================================================
[\[S3\]]{}
Let us finally consider the case where the spatial slices ${^{{{\scriptstyle{(3)}}}}}\Sigma$ have the topology of a 3-sphere $\mathbb{S}^3$, described as $\mathbb{S}^3=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^2+|z_2|^2=1\}$. Let us define the following action of $G^{{{\scriptstyle{(2)}}}}$ on ${^{{{\scriptstyle{(3)}}}}}\Sigma$ $$(g_1,g_2)\cdot(z_1,z_2)=(e^{ix_1},e^{ix_2})\cdot(z_1,z_2)=(e^{ix_1}z_1,e^{ix_2}z_2)\,.
\label{S3action_1}$$ The action of the two $U(1)$ subgroup factors is $$\begin{aligned}
&&(e^{ix},1)\cdot(z_1,z_2)=(e^{ix}z_1,z_2)\nonumber\\
&&(1,e^{ix})\cdot(z_1,z_2)=(z_1,e^{ix}z_2).\nonumber\end{aligned}$$ The corresponding tangent vectors at each point of $\mathbb{S}^3$, obtained by differentiating the previous expressions with respect to $x$ at $x=0$, are now $$\begin{aligned}
&&(iz_1,0)\nonumber\\
&&(0,iz_2).\nonumber\end{aligned}$$ As we can see they vanish at $z_1=0$ and $z_2=0$ (i.e. at the circumferences $(0,e^{i\xi})$ and $(e^{i\sigma},0)$, $\xi,
\sigma\in\mathbb{R}(\mathrm{mod}\,2\pi)$). This fact poses now the question of how one can possibly use them to perform a Geroch reduction that requires us to have at least a non-vanishing Killing vector field. On some other respects they present no problems, in particular they are commuting fields. A useful parametrization of $\mathbb{S}^3$ is $z_1=e^{i\sigma}\sin(\theta/2)$, $z_2=e^{i\xi}\cos(\theta/2)$ with $\theta\in[0,\pi]$, $\xi,\sigma\in\mathbb{R}(\mathrm{mod}\,2\pi)$, with the commuting Killing fields $\sigma^a$ and $\xi^a$ given by $\sigma^a=(\partial/\partial\sigma)^{a}$ and $\xi^a=(\partial/\partial\xi)^{a}$. This allows us to view the three-sphere as a filled torus in which the points on the same parallel of the surface are identified (so that the surface itself can be viewed as a circle $\mathbb{S}^1$). This is helpful to perform the Geroch reduction.
The fact that the Killing vectors that we have chosen vanish alternatively in two different circles poses a problem as far as the Geroch reduction is concerned because to perform it one should use a non-vanishing vector. We will show now that the the fact that $\xi^{a}$ only vanishes in a one dimensional submanifold will effectively allow us to use them to carry out this reduction. To this end we start form an action in four dimensions defined on ${^{{{\scriptstyle{(4)}}}}}\mathcal{M}$, topologically $\mathbb{R}\times
\mathbb{S}^3$, and remove the circle where the Killing vanishes from the integration region. As this is a zero-measure set the integral will not change. Of course one must take now into account the fact that the fields in the new integration region cannot be completely arbitrary but should be subject to some restrictions (regularity conditions) reflecting the fact that they should extend to the full ${^{{{\scriptstyle{(4)}}}}}\mathcal{M}$ in a smooth way.
Topologically the two dimensional manifold that appears in the action (\[act\]) is ${^{{{\scriptstyle{(2)}}}}}\Sigma=D$, where $D$ denotes an open disk. The regularity conditions on the disk boundary are such that the fields (of any tensor type) behave in a “radial coordinate” $\theta$ exactly as an axially symmetric field would do in the axis. Eventually this will allow us to change the disk by a two sphere.
As in the previous cases we are going to use $(t^{a},\theta^{a},\sigma^{a})$ as coordinate vector fields. We will write now $\theta^a=f\hat{\theta}^a$, $t^a=f(Nn^a+N^\theta\hat{\theta}^a)$ with $N>0$ and $f>0$. The scalars $f$, $N$, and $N^\theta$ are supposed to be smooth fields on $\mathbb{R}\times D$ subject to some regularity conditions that will be specified later. Notice that we write now $f$ instead of $e^\gamma/2$ as in previous cases because we want to allow $f$ to go to zero at the disk boundary. The same argument that we used for the two previous cases tells us now that $$\begin{aligned}
&&N\mathcal{L}_\theta f+f\mathcal{L}_\theta
N+Nf^2n^an^b\nabla_a\hat{\theta}_b=0\,,\label{proy_1_3sph}\\
&&N^\theta\mathcal{L}_\theta f+f\mathcal{L}_\theta
N^\theta-\mathcal{L}_t f+
Nf^2\hat{\theta}^a\hat{\theta}^b\nabla_a n_b=0\,,\label{proy_2_3sph}\\
&&\hat{\theta}^a\sigma^b\nabla_an_b=0\,.\label{proy_3_3sph}\end{aligned}$$ The form of the 3-metric $g_{ab}$ is basically the same as in the other cases $$\begin{aligned}
\label{metric_S3}
g_{ab}=f^2[(N^{\theta 2}-N^{2})(\mathrm{d}t)_a(\mathrm{d}t)_b+
2N^{\theta}(\mathrm{d}t)_{(a}(\mathrm{d}\theta)_{b)}+
(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b]+
\tau^{2}(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b\,\end{aligned}$$ and the determinant is now given by $|g|=\tau^2N^2f^4$. Again $(N,N^\theta,\gamma,\phi_{i})$, are constant on the orbits of the remaining Killing field $\sigma^a$ and hence they only depend on the coordinates $(t,\theta)$. Using the coordinates system $(t,\theta,\sigma,\xi)$ we write the original 4-metric ${^{{{\scriptstyle{(4)}}}}}g_{ab}$ as $$\begin{aligned}
\label{4metricS3}{^{{{\scriptstyle{(4)}}}}}g_{ab}&=&\frac{f^2}{\lambda_\xi}[(N^{\theta
2}-N^{2})(\mathrm{d}t)_a(\mathrm{d}t)_b+
2N^{\theta}(\mathrm{d}t)_{(a}(\mathrm{d}\theta)_{b)}+
(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b]\\
&+&
\frac{\tau^{2}}{\lambda_\xi}(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b+
\lambda_{\xi}(\mathrm{d}\xi)_a(\mathrm{d}\xi)_b.\nonumber\end{aligned}$$ We have to find out the regularity conditions satisfied by this metric. At $\theta=0$ the regularity conditions should be of the same type as the ones that we have already used in the $\mathbb{S}^1\times\mathbb{S}^2$ case. Here, however, we also have to impose regularity conditions when we approach the boundary of the filled torus that we obtained by removing the circle where the Killing used to perform the Geroch reduction vanishes. This can be formally achieved by changing $\theta$ for $\pi-\theta$. By doing this we find $$\begin{aligned}
&&\frac{f^2}{\lambda_{\xi}}(N^{\theta 2}-N^2)=A(t,\cos\theta)\,,\label{reg1_S3}\\
&&\frac{f^2}{\lambda_{\xi}}N^\theta=B(t,\cos\theta)\sin\theta\,, \label{reg2_S3}\\
&&\lambda_{\xi}=4\cos^2(\theta/2)[F(t,\cos\theta)-G(t,\cos\theta)\cos^2(\theta/2)]\,,\label{reg3_S3}\\
&&\frac{f^2}{\lambda_{\xi}}=D(t,\cos\theta)+E(t,\cos\theta)\sin^2(\theta/2)=\nonumber\\
&&\hspace{1cm}F(t,\cos\theta)+G(t,\cos\theta)\cos^2(\theta/2)\,,\label{reg4_S3}\\
&&\frac{\tau^2}{\lambda_{\xi}}=4\sin^2(\theta/2)[D(t,\cos\theta)-E(t,\cos\theta)\sin^2(\theta/2)
]\,.\label{reg5_S3}\end{aligned}$$ Here the functions $A$, $B$, $D$, $E$, $F$, and $G$ are analytic in their arguments. Notice that they are not independent because they are constrained to satisfy $$D(t,\cos\theta)+E(t,\cos\theta)\sin^2(\theta/2)=
F(t,\cos\theta)+G(t,\cos\theta)\cos^2(\theta/2)\,.$$ We have used the functions $\sin(\theta/2)$ and $\cos(\theta/2)$ because they alternatively vanish on the circles where the Killings themselves become zero and have the dependence of a regular scalar function in terms of the “radial” coordinates $\theta$ or $\pi-\theta$ on the circles where they do not vanish. The cosine dependence of the other functions is dictated by regularity at the two circles. This is very important because we will be able to write down our model in terms of them and, having $\cos\theta$ as their argument they can be interpreted as functions on $\mathbb{S}^2$ as in the $\mathbb{S}^1\times\mathbb{S}^2$ case. The conditions that the fields must satisfy (dropping the $t$-dependence) are $$\begin{aligned}
&&\lambda_{\xi}=e^{\hat{\phi}_1(\cos\theta)}\cos^2(\theta/2)\,,\label{condfields2_S3}\\
&&\phi_2=\hat{\phi}_2(\cos\theta)\,,\label{condfields1_S3}\\
&&f=\cos(\theta/2)e^{\hat{\gamma}(\cos\theta)/2}\,,\label{condfields3_S3}\\
&&N^\theta=\hat{N}^\theta(\cos\theta)\sin\theta\,,\label{condfields4_S3}\\
&&N=\hat{N}(\cos\theta)\,,\label{condfields5_S3}\\
&&\tau=\hat{T}(\cos\theta)\sin\theta\,,\label{condfields6_S3}\\
&&\hat{T}^2e^{-\hat{\gamma}}=\frac{D(\cos\theta)-E(\cos\theta)\sin^2(\theta/2)}
{D(\cos\theta)+E(\cos\theta)\sin^2(\theta/2)}\,,\label{condfields7_S3}\\
&&e^{2\hat{\phi}_1-\hat{\gamma}}=4\frac{F(\cos\theta)-G(\cos\theta)\cos^2(\theta/2)}
{F(\cos\theta)+G(\cos\theta)\cos^2(\theta/2)}\,,\label{condfields8_S3}\end{aligned}$$ where we have used $ \sin\theta=2\sin(\theta/2)\cos(\theta/2)$. Here, as in the $\mathbb{S}^1\times\mathbb{S}^2$ case, we have that $\hat{\phi}_{i},\hat{\gamma},\hat{N}^{\theta},\hat{N},\hat{T}:[-1,1]\rightarrow\mathbb{R}$ ($\hat{N}>0$). They can be written as functions of $A,\,B,\,D,\,E,\,F,$ and $G$. They must be differentiable in $(-1,1)$, and their right and left derivatives at $\pm1$ must be defined (equivalently they must be $\mathcal{C}^\infty$ in $(-1,1)$ with bounded derivative). Conditions (\[condfields7\_S3\],\[condfields8\_S3\]) imply that $$\hat{T}(+1)e^{-\hat{\gamma}(+1)/2}=1\,\quad \mathrm{and}\quad
e^{2\hat{\phi}_1(-1)-\hat{\gamma}(-1)}=4\,.$$ These are the polar constraints for the $\mathbb{S}^3$ topology. This is slightly different from previous examples because now these conditions involve different pairs of objets at the two poles of $\mathbb{S}^2$.
Our starting point is now the action $$\begin{aligned}
\label{act3esf}
\displaystyle {^{{{\scriptstyle{(3)}}}}}S(g_{ab},\phi_i)&=&\frac{1}{16\pi
G_{3}}\int_{t_0}^{t_1}\mathrm{d}t\int_{\mathbb{S}^2}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}|g|^{1/2}\left(\,^{{{\scriptstyle{(2)}}}}\!R+K_{ab}K^{ab}-K^2-
\frac{1}{2}g^{ab}\sum_{i}(\mathrm{d}\phi_{i})_a(\mathrm{d}\phi_{i})_b\right),\nonumber\end{aligned}$$ with $\phi_1=\log\lambda_{\xi}=\hat{\phi}_1+\log\cos^2(\theta/2)$. Notice that we have changed the integration region to $\mathbb{S}^2$ because, as we will see, it can be written in terms of the *hat*-fields that are smoothly extended to $\mathbb{S}^2$.
As in the case of the three-handle we choose the fiducial volume element $^{{{\scriptstyle{(2)}}}}\mathrm{e}$ to be compatible with the auxiliary round metric on the 2-sphere $\mathbb{S}^2$, i.e. ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}=\sin\theta\mathrm{d}\theta\wedge\mathrm{d}\sigma$, with ${^{{{\scriptstyle{(2)}}}}}\mathrm{e}_{ab}\theta^{a}\sigma^{b}=Nf^2\tau/|g|^{1/2}=\sin\theta$. In terms of the fields $(\hat{N},\hat{N}^{\theta},\hat{\gamma},\hat{T},\hat{\phi}_{i})$ the action becomes $$\begin{aligned}
&&\hspace{-.5cm}{^{{{\scriptstyle{(3)}}}}}S(\hat{N},\hat{N}^\theta,\hat{\gamma},\hat{T},\hat{\phi}_i)=\nonumber\\
&&\hspace{-.5cm}=\frac{1}{16\pi G_3}\int_{t_0}^{t_1}\!\!\!\!\!
\mathrm{d}t\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\bigg(
\hat{N}[(\hat{\gamma}^\prime\hat{T}^\prime-2\hat{T}^{\prime\prime})\sin^2\theta+
(5\hat{T}^{\prime}-\hat{\gamma}^\prime\hat{T})\cos\theta+\hat{T}^\prime+\frac{3}{2}\hat{T}]\nonumber\\
&&+\frac{1}{\hat{N}}[\hat{N}^\theta
\hat{T}\cos\theta-\dot{\hat{T}}-\hat{N}^\theta
\hat{T}^\prime\sin^2\theta][\dot{\hat{\gamma}}
+(2\hat{N}^{\theta\prime}+\hat{N}^\theta\hat{\gamma}^\prime)\sin^2\theta+(1-3\cos\theta)\hat{N}^\theta]
\label{act_esf}\\
&&+\frac{\hat{T}}{2\hat{N}}\sum_{i}\bigg[\dot{\hat{\phi}}^2_i
+2\hat{N}^\theta\dot{\hat{\phi}}_i\,\hat{\phi}_i^\prime\sin^2\theta+
(\hat{N}^{\theta2}\sin^2\theta-\hat{N}^2)\hat{\phi}_i^{\prime2}\sin^2\theta\bigg]\nonumber\\
&&+\frac{\hat{T}}{2\hat{N}}[2(1-\cos\theta)(\hat{N}^\theta
\dot{\hat{\phi}}_1+(\hat{N}^{\theta2}\sin^2\theta-\hat{N}^2)\hat{\phi}_1^\prime)
+(1-\cos\theta)^2\hat{N}^{\theta2}] \bigg)\,.\nonumber\end{aligned}$$ It is important to remark at this point that the action is the integral of a smooth function on the sphere. We arrive at this result after several non-trivial cancelations of terms that would diverge at the poles. This reflects the fact that indeed, by removing the circle where the Killing vector field used in the Geroch reduction vanishes, we arrive at a consistent description of the model. It is also worthwhile pointing out that the structure of the action is very similar to the one found in the $\mathbb{S}^1\times\mathbb{S}^2$ case but not exactly the same, in fact we will see later that the differences are important to guarantee, for example, the stability of the polar constraints in this case.
The Hamiltonian of the system can be readily obtained. As in previous cases it can be written as a sum of constraints $H=C[\hat{N}]+C_\theta[\hat{N}^\theta]$ with $$\begin{aligned}
C[\hat{N}]&=& \int_{\mathbb{S}^2}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}\hat{N}C
\\
&
&\hspace{-1.2cm}=\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\hat{N}\left(
-16\pi G_3 p_{\hat{\gamma}}p_{\hat{T}}+\frac{1}{16\pi
G_3}\big[(2\hat{T}^{\prime\prime}-\hat{\gamma}^\prime\hat{T}^\prime)\sin^2\theta
+(\hat{\gamma}^\prime\hat{T}-5\hat{T}^\prime)\cos\theta-\frac{3}{2}\hat{T}-\hat{T}^\prime\big]\right.\nonumber\\
&&+\frac{1}{2}\sum_{i}\bigg(\frac{16\pi
G_3p_{\hat{\phi}_i}^2}{\hat{T}} +\frac{\hat{T}}{16\pi
G_3}\hat{\phi}_i^{\prime2}\sin^2\theta\bigg)
+(1-\cos\theta)\frac{\hat{T}}{16\pi G_3}\hat{\phi}_1^\prime\bigg)\nonumber\\
\label{esc_const_S3} C_\theta[\hat{N}^\theta]&=&
\int_{\mathbb{S}^2}{^{{{\scriptstyle{(2)}}}}}\mathrm{e}\hat{N}^{\theta}C_{\theta}
\\
&&\hspace{-1.2cm}
=\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}\hat{N}^\theta
\bigg([2p^\prime_{\hat{\gamma}}-\hat{\gamma}^\prime
p_{\hat{\gamma}}- \hat{T}^\prime
p_{\hat{T}}-\sum_{i}\hat{\phi}_i^\prime
p_{\hat{\phi}_i}]\sin^2\theta +[\hat{T}
p_{\hat{T}}-p_{\hat{\gamma}}+p_{\hat{\phi_1}}]\cos\theta-p_{\hat{\gamma}}-p_{\hat{\phi}_1}\bigg).
\nonumber\end{aligned}$$ The two previous expressions, together with the conditions at the poles $\hat{T}(+1)e^{-\hat{\gamma}(+1)/2}=1$ and $e^{2\hat{\phi}_1(-1)-\hat{\gamma}(-1)}=4$, define the constraints of the system. As before, the polar constraints are necessary conditions to guarantee the differentiability of the (weighted) constraints $C[N_g]$ and $C_\theta[N_g^\theta]$. The gauge transformations defined by $C[\hat{N}_g]$ and $C_\theta[\hat{N}_g^\theta]$ are[^11] $$\begin{aligned}
&&\{\hat{\gamma},C[\hat{N}_g]\}=-\hat{N}_g p_{\hat{T}}\,,
\nonumber\\
&&\{\hat{T},C[\hat{N}_g]\}=-\hat{N}_g p_{\hat{\gamma}}\,,\nonumber\\
&&\{\hat{\phi_i},C[\hat{N}_g]\}=\hat{N}_g\frac{p_{\hat{\phi}_i}}{\hat{T}}\,,\nonumber\\
&&\{p_{\hat{\gamma}},C[\hat{N}_g]\}=\hat{N}_{g}^{\prime}(\hat{T}\cos\theta-\hat{T}^\prime\sin^2\theta)
+\hat{N}_g(3\hat{T}^\prime\cos\theta+\hat{T}-\hat{T}^{\prime\prime}\sin^2\theta),\nonumber\\
&&\{p_{\hat{T}},C[\hat{N}_g]\}=
\hat{N}_g\big[\frac{1}{2}-\hat{\phi}_1^\prime+(\hat{\gamma}^\prime+\hat{\phi}_1^\prime)\cos\theta
-\hat{\gamma}^{\prime\prime}\sin^2\theta\big]+
\hat{N}_g^\prime(3\cos\theta-1-\hat{\gamma}^\prime\sin^2\theta)\nonumber\\
&&\hspace{2.5cm}-
2\hat{N}_g^{\prime\prime}\sin^2\theta+\frac{\hat{N}_g}{2}\sum_{i}\bigg(\frac{p_{\hat{\phi}_i}^2}{\hat{T}^2}
-\sin^2\theta\hat{\phi}_i^{\prime2}\bigg)\,,\nonumber\\
&&\{p_{\hat{\phi}_1},C[\hat{N}_g]\}=[\hat{N}_g\hat{T}(\hat{\phi}^\prime_2\sin^2\theta+1-\cos\theta)]^\prime\,,\nonumber\\
&&\{p_{\hat{\phi}_2},C[\hat{N}_g]\}=
(\hat{N}_g\hat{T}\hat{\phi}^\prime_2\sin^2\theta)^\prime\,,\nonumber\end{aligned}$$ and $$\begin{aligned}
&&\{\hat{\gamma},C_\theta[\hat{N}^\theta_g]\}=-2\hat{N}_g^{\theta\prime}\sin^2\theta+\hat{N}_g^\theta(3\cos\theta-\hat{\gamma}^\prime\sin^2\theta-1)\,,\nonumber\\
&&\{\hat{T},C_\theta[\hat{N}^\theta_g]\}=\hat{N}_g^\theta(\hat{T}\cos\theta-\hat{T}^\prime\sin^2\theta)\,,\nonumber\\
&&\{\hat{\phi}_1,C_\theta[\hat{N}^\theta_g]\}=\hat{N}_g^\theta(\cos\theta-1-\hat{\phi}_2^\prime\sin^2\theta)\,,\nonumber\\
&&\{\hat{\phi}_2,C_\theta[\hat{N}^\theta_g]\}=-\hat{N}_g^\theta\hat{\phi}_2^\prime\sin^2\theta\,,\nonumber\\
&&\{p_{\hat{\gamma}},C_\theta[\hat{N}^\theta_g]\}=-(\hat{N}_g^\theta p_{\hat{\gamma}}\sin^2\theta)^\prime\,,\nonumber\\
&&\{p_{\hat{T}},C_\theta[\hat{N}^\theta_g]\}=\hat{N}_g^\theta(p_{\hat{T}}\cos\theta-p_{\hat{T}}^\prime\sin^2\theta)
-\hat{N}_g^{\theta\prime}p_{\hat{T}}\sin^2\theta\,,
\nonumber\\
&&\{p_{\hat{\phi}_i},C_\theta[\hat{N}^\theta_g]\}=-(\hat{N}_g^\theta p_{\hat{\phi_i}}\sin^2\theta)^\prime\,.\nonumber\\\end{aligned}$$ The Poisson brackets of these constraints give exactly the same result that we obtained for the $\mathbb{S}^1\times\mathbb{S}^2$ topology and, hence, define a fist class constrained surface $\Gamma_c\subset\Gamma$. Here $(\Gamma,\omega)$ denotes the canonical phase space of the system, coordinatized by the canonical pairs $(\hat{\gamma},p_{\hat{\gamma}};\hat{T},p_{\hat{T}};\hat{\phi}_{i},p_{\hat{\phi}_i})$, and endowed with the standard (weakly) symplectic form (\[omegaS2XS1\]). We must check now the stability of the polar constraints. We do this by computing $$\begin{aligned}
&&\{\hat{T}e^{-\hat{\gamma}/2},C[\hat{N}_g]\}=\frac{1}{2}\hat{N}_ge^{-\hat{\gamma}/2}(\hat{T}p_{\hat{T}}-2p_{\hat{\gamma}})\,,\label{pb1}\\
&&\{\hat{T}e^{-\hat{\gamma}/2},C_\theta[\hat{N}_g^\theta]\}=e^{-\hat{\gamma}/2}
\bigg(\frac{1}{2}\hat{N}_g^\theta\hat{T}(1-\cos\theta)+(\hat{N}_g^{\theta\prime}\hat{T}
-\hat{N}_g^\theta\hat{T}^\prime+\frac{1}{2}\hat{N}_g^\theta\hat{T}\hat{\gamma}^\prime)\sin^2\theta\bigg),\quad\quad\label{pb2}\\
&&\{e^{2\hat{\phi}_1-\hat{\gamma}},C[\hat{N}_g]\}=\frac{\hat{N}^g}{\hat{T}}
e^{2\hat{\phi}_1-\hat{\gamma}}(2p_{\hat{\phi}_1}+\hat{T}p_{\hat{T}})\,,\label{pb3}\\
&&\{e^{2\hat{\phi}_1-\hat{\gamma}},C_\theta[\hat{N}_g^\theta]\}=e^{2\hat{\phi}_1-\hat{\gamma}}
\Big(-\hat{N}_g^\theta(1+\cos\theta)+
(2\hat{N}_g^{\theta\prime}-2\hat{N}_g^\theta\hat{\phi}_2^\prime
+\hat{N}_g^\theta\hat{\gamma}^\prime)\sin^2\theta\Big)\,.\label{pb4}\end{aligned}$$ The constraint (\[esc\_const\_S3\]) at the poles $\theta=0,\pi$ gives respectively, $\hat{T}(+1)p_{\hat{T}}(+1)-2p_{\hat{\gamma}}(+1)=0$, and $\hat{T}(-1)p_{\hat{T}}(-1)+2p_{\hat{\phi}_1}(-1)=0$. These guarantee that the Poisson bracket (\[pb1\]), vanishes at $\theta=0$ and (\[pb3\]) vanishes at $\theta=\pi$. The vanishing of (\[pb2\]) at $\theta=0$ is due to the presence of the factors $1-\cos\theta$ and $\sin^2\theta$ and, finally, (\[pb4\]) is zero at $\theta=\pi$ due to the factors $1+\cos\theta$ and $\sin^2\theta$. As in the $\mathbb{S}^1\times\mathbb{S}^{2}$ we conclude that there are no secondary constraints coming from the
stability of these polar constraints.
The deparameterization in this case follows closely the one for $\mathbb{S}^1\times\mathbb{S}^2$. The same gauge fixing conditions work in our case now. The only new element now is checking if the polar constraints are gauge fixed or not and this only requires the computation of $$\begin{aligned}
&&\{p_{\hat{\gamma}n},e^{2\hat{\phi}_1-\hat{\gamma}}\}=
e^{2\hat{\phi}_1-\hat{\gamma}}\sqrt{\frac{2n+1}{4\pi}}P_n(\cos\theta)\end{aligned}$$ which is different from zero at the poles. As we see the situation now is completely analogous to the previous case. The pull-back of the symplectic form to the phase space hypersurface defined by the gauge fixing conditions is given again by (\[symplecticS2\]). We are left only with the constraint $$\begin{aligned}
\label{constS3}\mathcal{C} &:=& -p_{\hat{\gamma}_{0}}p_{\hat{T}_{0}}+\hat{T}_{0}\left(\sqrt{4\pi}
\log\frac{\hat{T}_{0}}{\sqrt{4\pi}}-\hat{\gamma}_{0}-\sqrt{\pi}(2\log2+3)
+\hat{\phi}_{1_{0}}\right)\\
&+&\frac{1}{2}\sum_{i}\int_{\mathbb{S}^{2}}
{^{\scriptstyle{{{\scriptstyle{(2)}}}}}}\mathrm{e}\left(\frac{\sqrt{4\pi}p_{\hat{\phi}_{i}}^{2}}{\hat{T}_{0}}
+\frac{\hat{T}_{0}}{\sqrt{4\pi}}\hat{\phi}_i^{\prime2}\right)\approx0\,.\nonumber\end{aligned}$$ The gauge transformations generated by this constraint on the variables $\hat{T}_0$ and $p_{\hat{\gamma}0}$ are the same as for the three-handle and, hence, we can use the canonical transformations introduced at the end of the previous section to rewrite (\[constS3\]) as $$\begin{aligned}
\label{finconstS3}p_T&+&(4\pi)^{1/4}e^{\tilde{P}/2}\hat{\varphi}_{1_0}\sin
T+2\sqrt{\pi}e^{\tilde{P}}(\log \frac{\sin
T}{\sqrt{4\pi}}+\tilde{P}-\log2-\frac{3}{2})\sin
T\\
&+&\frac{1}{2}\sum_{i}\int_{\mathbb{S}^2}\!\!\!^{{{\scriptstyle{(2)}}}}\mathrm{e}
\bigg(\frac{p_{\hat{\varphi}_i}^2}{\sin
T}+\hat{\varphi}_i^{\prime2}\sin
T\sin^2\theta\bigg)\approx0.\nonumber\end{aligned}$$ The description of the system by a time-dependent Hamiltonian is now straightforward. It is interesting at this point to compare the dynamics of this model and the $\mathbb{S}^1\times\mathbb{S}^2$ one. First of all we see that the global mode have a different behavior now, in particular couples to $\varphi_{1_0}$ through the term $e^{\tilde{P}/2}\hat{\varphi}_{1_0}\sin T$ in (\[finconstS3\]). As we see the gravitational and matter modes cease to play a symmetric role in this particular description, at variance with the other topologies. However, by writing the regularity condition (\[condfields2\_S3\]) with an extra $\hat{T}$ (as will be justified in the next section) it is possible to restore the symmetry between the gravitational and matter scalars in a straightforward way.
As in the previous cases, it is possible to interpret the system as a non-autonomous Hamiltonian system $((0,\pi)\times\Gamma_R,\mathrm{d}t,\omega_R,H_R)$, where $\Gamma_{R}$ denotes the reduced phase space coordinatized by the canonical pairs $(\tilde{Q},\tilde{P};\varphi_{i},p_{\varphi_i})$, endowed with the standard (weakly) simplectic form (\[omegaR\]). The dynamics is given by the time dependent Hamiltonian $H_{R}(t):\Gamma_{R}\rightarrow\mathbb{R}$ $$\begin{aligned}
H_{R}(t)&=&(4\pi)^{1/4}e^{\tilde{P}/2}\varphi_{1_0}\sin
t+2\sqrt{\pi}e^{\tilde{P}}(\log \frac{\sin
t}{\sqrt{4\pi}}+\tilde{P}-\log2-\frac{3}{2})\sin
t\\
&+&\frac{1}{2}\sum_{i}\int_{\mathbb{S}^2} {^{{{\scriptstyle{(2)}}}}}\mathrm{e}\,
\bigg(\frac{p_{\varphi_i}^2}{\sin t}+\varphi_i^{\prime2}\sin
t\sin^2\theta\bigg),\end{aligned}$$ with the evolution vector field given by $$\begin{aligned}
E_{H_R}&=&\frac{\partial}{\partial
t}+\left[(4\pi)^{1/4}e^{\tilde{P}/2}\varphi_{1_0}\sin
t+2\sqrt{\pi}e^{\tilde{P}}\sin t\left(\log\frac{\sin
t}{\sqrt{4\pi}}+\tilde{P}-\log2-\frac{1}{2}\right)\right]
\frac{\partial}{\partial\tilde{Q}}\\
&-&(4\pi)^{1/4}e^{\tilde{P}/2} \sin t\frac{\partial}{\partial
p_{\varphi_{1_0}}}+\sum_{i}\int_{\mathbb{S}^{2}}{^{(2)}}\mathrm{e}\left(\frac{p_{\varphi_{i}}}{\sin
t}\frac{\delta}{\delta
\varphi_{i}}+(\sin^{2}\theta\varphi_i^{\prime})^{\prime}\sin
t\frac{\delta}{\delta p_{\varphi_{i}}}\right).\end{aligned}$$ The singularities in this case show up in the same way as for the $\mathbb{S}^1\times\mathbb{S}^2$ topology.
Gowdy models as scalar field theories in 2+1 curved background
==============================================================
[\[back\]]{}
The purpose of this section is to reinterpret the reduced models presented in the previous sections as certain simple massless scalar field theories in conformally stationary backgrounds. We will show how the metrics obtained after the specific gauge fixing and deparameterization used in the previous sections can be employed to reinterpret the meaning (and solution) of the field equations for each topology. This will allow us to use well-known techniques of quantum field theory in curved backgrounds to quantize these systems [@BarberoG.:2007].
Let us start by giving a simple way to solve equations (\[ecs\]) $$\begin{aligned}
&&R_{ab}=\frac{1}{2}\sum_i
(\mathrm{d}\phi_i)_a(\mathrm{d}\phi_i)_b\,,\label{e1}\\
&&g^{ab}\nabla_a\nabla_b\phi_i=0\,,\label{e2}\\
&&\mathcal{L}_\sigma\phi_i=0.\label{e3}\end{aligned}$$ If a specific solution $(\mathring{g}_{ab},\mathring{\phi}_1,\mathring{\phi}_2)$ is known it is possible to decouple (\[e1\]) and (\[e2\],\[e3\]) because, when (\[e3\]) is satisfied, we have the equivalence $$\begin{aligned}
g^{ab}\nabla_a\nabla_b\phi_i=0 \Leftrightarrow \mathring{g}^{ab}
\mathring{\nabla}_a\mathring{\nabla}_b\phi_i=0.\end{aligned}$$ The idea is then to solve the last equation in the background $\mathring{g}_{ab}$ and then equation (\[e1\]) just gives integrability conditions allowing us to recover $g_{ab}$. We will discuss next the specific form of $\mathring{g}_{ab}$ for each of the spatial topologies considered in the paper.
- **Background metric for $\mathbb{T}^3$**
In this case the form of the metric $g_{ab}$ found after the deparameterization is $$g_{ab}=e^\gamma\Big(-(\mathrm{d}t)_a(\mathrm{d}t)_b+(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b\Big)
+\frac{P^2t^2}{2\pi}(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b$$ defined on $(0,\infty)\times \mathbb{T}^2$. A possible (non unique) choice for $(\mathring{g}_{ab},\mathring{\phi}_1,\mathring{\phi}_2)$ is $$\begin{aligned}
\mathring{g}_{ab}&=&t^2\Big(-(\mathrm{d}t)_a(\mathrm{d}t)_a+(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b
+(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b\Big)\\
\mathring{\phi}_1&=&\log t\\
\mathring{\phi}_2&=&0\end{aligned}$$ where it is important to notice that even though $\mathring{g}_{ab}$ is not stationary it is conformal to a (flat) stationary metric on $(0,\infty)\times\mathbb{T}^2$.
- **Background metric for $\mathbb{S}^1\times\mathbb{S}^2$**
After deparameterization we get now $$g_{ab}=e^\gamma\Big(-(\mathrm{d}t)_a(\mathrm{d}t)_b+(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b\Big)+
\frac{P^2}{4\pi}\sin^2 t
\sin^2\theta(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b$$ defined on $(0,\pi)\times \mathbb{S}^2$. In this case a convenient choice for $(\mathring{g}_{ab},\mathring{\phi}_1,\mathring{\phi}_2)$ is $$\begin{aligned}
\mathring{g}_{ab}&=&\sin^2t\Big(-(\mathrm{d}t)_a(\mathrm{d}t)_a+(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b+\sin^2\theta(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b\Big)\\
\mathring{\phi}_1&=&\log\sin(t/2)-\log\cos(t/2)\\
\mathring{\phi}_2&=&0.\end{aligned}$$ Again this metric is not stationary but it is equal to a time dependent conformal factor times the Einstein static metric on $(0,\pi)\times\mathbb{S}^2$.
- **Background metric for $\mathbb{S}^3$**
Finally we have now $$g_{ab}=\cos^2(\theta/2)e^\gamma\bigg(-(\mathrm{d}t)_a(\mathrm{d}t)_b
+(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b\bigg)+\frac{P^2}{4\pi}\sin^2t\sin^2\theta
(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b$$ defined on $(0,\pi)\times D$ where $D$ denotes the open disk introduced in the previous section. In this case, a possible choice of $(\mathring{g}_{ab},\mathring{\phi}_1,\mathring{\phi}_2)$ is $$\begin{aligned}
\mathring
g_{ab}&=&\cos^2(\theta/2)e^{\mathring{\gamma}}\bigg(-(\mathrm{d}t)_a(\mathrm{d}t)_b
+(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b\bigg)+\sin^2t\sin^2\theta (\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b\\
\mathring{\phi}_1&=&\cos\theta \cos t\log(\tan(t/2))+\cos\theta+\log(\cos^2(\theta/2))+\log(2\sin t)\\
\mathring{\phi}_2&=&0\end{aligned}$$ where $$\begin{aligned}
\mathring{\gamma}&=& \frac{\sin^2\theta}{4}\Big(\sin^2 t\log^2(\tan
t/2)-2\cos t\log(\tan t/2)-1\Big)+\log\left(\sin^2 t\right)\\
&-&\cos t\log(\tan(t/2))+\cos\theta \cos
t\log(\tan(t/2))+\cos\theta -1.\end{aligned}$$ It is important to realize that the concrete functional form of $\mathring{\gamma}$ is irrelevant because, whenever $\mathcal{L}_\sigma \phi_i=0$, we have the following equivalence in $(0,\pi)\times(\mathbb{S}^2-\{\theta=\pi\})$ $$\mathring g^{ab}\mathring \nabla_a\mathring \nabla_b
\phi_i=0\Leftrightarrow \breve{g}^{ab} \breve{\nabla}_a
\breve{\nabla}_b \phi_i=0$$ with $$\breve{g}_{ab}=\sin^2t\Big(-(\mathrm{d}t)_a(\mathrm{d}t)_b
+(\mathrm{d}\theta)_a(\mathrm{d}\theta)_b+\sin^2\theta(\mathrm{d}\sigma)_a(\mathrm{d}\sigma)_b\Big)\,.$$ Notice that the metric $ \breve{g}_{ab}$ is the one that we found for $(0,\pi)\times \mathbb{S}^2$ restricted to the manifold $(0,\pi)\times D$ obtained by removing a point from the sphere.
It is important to point out that $\phi_1$ cannot be extended to the boundary of the disk, parameterized as $\theta=\pi$, because (\[condfields2\_S3\]) forces $\phi_1$ to behave as $\log(\cos^2(\theta/2))$ for $\theta\rightarrow\pi$. However if we split $\phi_1$ as $\phi_1=\phi_1^{\mathrm{sing}}+\phi_1^{\mathrm{reg}}$ with $
\phi_1^{\mathrm{sing}}=\log(\cos^2(\theta/2))+\log(2\sin t)$, satisfying $$\breve{g}^{ab} \breve{\nabla}_a \breve{\nabla}_b
\phi^{\mathrm{sing}}_1=0\,,$$ we guarantee that the degrees of freedom contained in $\phi^{\mathrm{reg}}_1$ still satisfy $\breve{g}^{ab}
\breve{\nabla}_a \breve{\nabla}_b \phi_1^{\mathrm{reg}}=0$ (just the same equation as the matter field $\phi_2$) and can be extended to $(0,\pi)\times \mathbb{S}^2$. Notice that the role of the scalar fields $\phi_1^{\mathrm{reg}}$ and $\phi_2$, both well behaved on $(0,\pi)\times\mathbb{S}^2$, is symmetric just as in the description of the previous topologies.
It is important to notice that the scalar field dynamics generated by the time dependent Hamiltonians that we have obtained in the previous sections corresponds exactly to the one defined by the Klein-Gordon equations on the backgrounds given by $\mathring{g}_{ab}$.
To end this section we want to point out that there are certain obstructions to the unitary implementation of quantum dynamics for these systems. Specifically, it can be shown that it is impossible to find a Fock space representation in which time evolution is unitarily implementable [@Corichi:2002vy; @Torre:2002xt]. The solution to this problem for the torus case relies on certain field redefinitions involving functions of time [@Corichi:2006xi; @Corichi:2006zv]. These can be understood in the present scheme as coming from the time dependent conformal factors appearing in $\mathring{g}_{ab}$ (or $\breve{g}_{ab}$). As we will show in a forthcoming paper, the solution to the unitarity problem for the topologies considered here relies on field redefinitions involving precisely the conformal factors shown above. Indeed, by performing a redefinition of the scalar fields at the Lagrangian level, such that the conformal factor relating both metrics is traded by a time-dependent potential term, we expect to find unitary dynamics if this potential is well behaved. In these cases the background metric corresponds to a simple, fixed stationary background.
Conclusions and comments
========================
[\[conclusions\]]{}
We have studied in this paper the Hamiltonian formalism for the compact, linearly polarized Gowdy models coupled to massless scalar fields. The purpose of the analysis is to have a Hamiltonian formulation of the models that can be a starting point for their quantization either *à la Dirac* or by gauge fixing and deparameterization. The results for the $\mathbb{T}^3$ topology reproduce the known ones for the gravitational sector and show that the interaction of the matter fields occur though the constraints left over by the deparameterization of the system. In the other two cases the coupling of matter and gravity degrees of freedom can only be seen when the four metric is reconstructed.
The description of the $\mathbb{S}^1\times\mathbb{S}^2$ and $\mathbb{S}^3$ models requires a careful discussion of the regularity conditions that the metric must satisfy in the symmetry axis left over after the Geroch reduction performed to describe the systems in 2+1 dimensions. These regularity conditions are responsible for the appearance of the so called *polar constraints*. These can be shown to be first class and are necessary conditions to guarantee the differentiability of the other constraints present in the models. Of course they must be taken into account in a Dirac quantization of the Gowdy models corresponding to these topologies.
An interesting feature of both the $\mathbb{S}^1\times\mathbb{S}^2$ and $\mathbb{S}^3$ cases is the fact that after the deparameterization introduced in the paper (which is a straightforward generalization of the ones used in the literature for the familiar $\mathbb{T}^3$ case) there are no constraints left so that the system can be completely described by the time dependent Hamiltonians that we have found. This is in contrast with the situation for the 3-torus where in addition to the dynamics generated by the time dependent Hamiltonian there is an additional constraint in the system that must be appropriately taken into account.
A somewhat surprising fact is the possibility to describe both the $\mathbb{S}^1\times\mathbb{S}^2$ and $\mathbb{S}^3$ models by using smooth fields on the sphere $\mathbb{S}^2$. An interesting possibility that may teach us something in the case of $\mathbb{S}^3$ is to use a Hopf fibration to perform the Geroch reduction to get a 2+1 dimensional description. This may be the subject of future work.
The dynamics of the global modes for the different spatial topologies is easy to obtain but there are significant differences depending on the topologies. Whereas in the $\mathbb{T}^3$ case the value of $\tilde{Q}$ and $\tilde{P}$ are just constants in the other cases $\tilde{P}$ is constant but $\tilde{Q}$ is a function of time.
We have been able to understand in very simple terms the appearance of both initial and final singularities in the spacetime metrics that solve the Einstein-Klein Gordon equations for these models from the point of view of the phase space description of the dynamics, in particular after the deparameterization process that we have followed. As we have seen there are natural variables with very simple gauge transformations (“time dynamics”) that suggest canonical transformations that lead to the time dependent Hamiltonians describing the dynamics and explicitly show how the singularities appear. In the $\mathbb{S}^1\times\mathbb{S}^2$ and $\mathbb{S}^3$ topologies the function $\sin t$ in a denominator of the time-dependent Hamiltonian shows that both final and initial singularities are present whereas the $t$ denominator in the Hamiltonian for the 3-torus shows that only an initial (or final) singularity appears in this case.
The authors want to thank I. Garay and J. M. Martín García for discussions. Daniel Gómez Vergel acknowledges the support of the Spanish Research Council (CSIC) through a I3P research assistantship. This work is also supported by the Spanish MEC under the research grant FIS2005-05736-C03-02.
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[^1]: Throughout the paper we will use the Penrose abstract index convention with tangent space indices belonging to the beginning of the Latin alphabet [@Penrose]. Lorentzian spacetime metrics will have signature $(-+++)$ and the conventions for the curvature tensors are those of Wald [@Wald].
[^2]: 2+1 massless scalar fields will be denoted be the subindex $i=1,2$. The subindex $i=1$ will label the gravitational scalar that encodes the local gravitational degrees of freedom in Gowdy models and the subindex $i=2$ will label the original 3+1 matter scalar. It is completely straightforward to couple any number $N$ of massless scalar fields, in practice this can be done by supposing that the index $i$ runs from 1 to $N$.
[^3]: In any basis where the nonvanishing components of ${^{{{\scriptstyle{(3)}}}}}\epsilon$ have the values $\pm1$, $|g|^{1/2}$ is equivalent to the square of the absolute value of the determinant of the matrix of the metric in that basis.
[^4]: Notice that even though we use a coordinate notation these are globally defined vector fields on the spatial manifolds $\{t_0\}\times\mathbb{T}^3$.
[^5]: In particular, take $t^{a}:=(\partial/\partial t)^{a}$.
[^6]: Here and in the following $\displaystyle\int_{\mathbb{S}^1}
F:=\int_{\mathbb{S}^1} F \,\mathrm{d}\theta$.
[^7]: In the rest of this section we will choose units such that $8G_3=1$.
[^8]: A useful example to appreciate the difference between taking some derivatives to be zero and fixing the values of the functions is to consider the straight line $x_1=x_2=x_3$ in $\mathbb{R}^3$ where all the points have equal coordinates in contrast with the point $x_1=x_2=x_3=1$.
[^9]: The matter fields act as sources for the gravitational field, hence, the solutions to the Einstein equations should depend on the matter content.
[^10]: In the following $16\pi G_3=1$.
[^11]: Again we take $16\pi
G_3=1$.
|
---
abstract: 'In this paper we provide a preliminary analysis of Google+ privacy. We identified that Google+ shares photo metadata with users who can access the photograph and discuss its potential impact on privacy. We also identified that Google+ encourages the provision of other names including maiden name, which may help criminals performing identity theft. We show that Facebook lists are a superset of Google+ circles, both functionally and logically, even though Google+ provides a better user interface. Finally we compare the use of encryption and depth of privacy control in Google+ versus in Facebook.'
author:
- |
Shah Mahmood\
\
\
\
Yvo Desmedt\
\
\
\
bibliography:
- 'References\_GooglePlus.bib'
title: 'Preliminary Analysis of Google+’s Privacy'
---
=10000 = 10000
=8.5in
Introduction {#sec:Introduction}
============
Google launched its latest social networking site Google+ on June 28$^{\rm th}$, 2011. According to comScore, an Internet traffic watcher, Google+ registered 25 million users in its first 5 weeks [@ComScore11], which motivates a close scrutiny. Current leader of social networking market and the key rival of Google+, Facebook, has over 750 million registered users [@FacebookStatistics11]. Facebook users share more than 30 billion pieces of content (photos, videos, web links, notes, blog posts etc.) every month.
Google+ like other social networks is used for sharing private information including status updates, occupation, employment history, home and work addresses, contact numbers, relationship status, photos, videos, etc. As Google+’s market penetration grows, so will the amount of data shared by its users. With the enormous amount of data produced on social networks, privacy is one of the issues widely discussed both in media and academia [@Anderson08]. Considering the importance of protection of the private information of its users Google+ has introduced circles as a new concept to address the issue.
Use of social networks has resulted in disclosure of embarrassing information, loss of employment, suspension from school, and blackmail [@Weiner11]. Social networks are also used for social phishing attacks. Phishers harvest email addresses to find the real names and social network profiles of their victims [@Polakis10]. This harvest is possible because both Google+ and Facebook require its users to use their real names and allow search based on email addresses. Once the real names and social network profiles are found, phishers extract more information including people in the circles (or friend list) of the victim, any comments, events attended etc. This information is then used to craft personalized phishing attacks, called social phishing [@Jagatic07]. Identity theft is costing US economy \$15.6 billion a year [@IdentityTheft11]. Moreover, social network status updates facilitated robberies on several occasions, where the owner announced absence from their property for a certain duration [@FacebookRobbery10]. Furthermore, the large amount of data is also of interest to advertisers and marketers. According to a survey by Social Media Examiner over 92% marketers use social networks as a tool [@FacebookMarketer11].
In view of the above discussion, it is very important and timely to analyze Google+ and identify any privacy related issues. This is the main goal of this paper.
**Our contributions:**
We provide a preliminary analysis of privacy in Google+. We identify that Google+ shares the metadata of photos uploaded which could lead to privacy violations, discussed in Section \[PhotoMetadata\]. Moreover, Google+ encourages its users to provide their past addresses and other names e.g. maiden name which could be used for identity theft. For further details see Section \[OtherNames\].
We compare Google+ circles (it’s main privacy selling point) to Facebook lists. We show that, although Google+ circles have a better graphical user interface, they are logically and functionally a subset of Facebook lists. Details are provided in Section \[CirclesVSLists\].
We also make other comparisons between Facebook and Google+ including the use of encryption and the ability to disable comments and message sharing. Further details are provided in Section \[OtherComparisons\]
Google+ Privacy
===============
In this section we present some privacy related problems and features of Google+. We also make a comparison with Facebook, when applicable.
Google+’s photo metadata {#PhotoMetadata}
------------------------
When a user uploads a photo on Google+, some metadata including the name of the photo owner, the date and time the photo was taken, the make and model of the camera etc. are made available to those with whom the photo is shared. This set of information, in particular the date and time, may at first look relatively innocent and trivial, but could in reality lead to some serious privacy concerns. On August 10, 2007, in Pennsylvania (USA), a divorce lawyer proved the spouse of the client being unfaithful to his partner, when the electronic toll records showed him in New Jersey (USA) on that night and not in a business meeting in Pennsylvania [@Divorce11]. With the metadata revealed by Google+ a user might leak enough information to be legally held liable on similar accounts.
Similarly, the make of the camera could be another concern for privacy. Higher end cameras cost thousands of dollars. There have been past incidents where the victims were killed for their cameras. In May 2011, a Greek citizen, 44, was killed for his camera when taking his wife to the hospital for child birth [@Camcorder11].
Just to give an example of the level of information a picture exposes about the camera, look at the metadata of the publicly shared pictures (from his Google+ profile) of Google co-founder Larry Page, shown in Figure \[fig:Larry\]. It reveals that they he used a Canon EOS 5D Mark II camera to shoot his vacation photographs. This camera is worth approximately USD 2000. This gives the robber incentives.
Cities lived in and other names on profile {#OtherNames}
------------------------------------------
In the “About” section of personal information, Google+ encourages its user to provide the names of cities the user lived in and other names. In the text box for other names, they write *“For example: maiden name, alternative spelling”*. Messages, photos and comments on social networks and other online sources can be used to infer family relationships. So, if someone can link a profile to the profile of the mother and if the mother provides the maiden name, then this could be used for identity theft, as mother’s maiden name is one of the most widely used secret question [@Berghel00]. Moreover, the past addresses can only help the attacker with such attacks.
Google+ circles vs Facebook lists {#CirclesVSLists}
---------------------------------
Paul Adams, then a Google employee, introduced the concept of social circles [@Adams10]. These social circles act as the foundation of circles in Google+. In Google+, by default there are four circles: “friends”, “family”, “acquaintances” and “following”. We can remove/ rename any of the default circles or add new circles. A user can add any of her contacts to one or more circles just by a simple drag and drop. Figure \[fig:GoogleCircle\] shows the graphical interface of Google+ circles. The intersection of two or more circles can be a non-empty set.
A user can share the content of her choice with a specific set of her circles, all her circles, her extended circles(people in all her circles and all people in the circles of the people in her circles) and with the public (everyone). Google+ does not allow any exceptions, i.e. , if some content is shared with a larger circle, there is no way to exclude any subset of that circle. Anything shared with the public is shared with all circles including the family and friends circle, which might not be what the user may require.
Facebook on the other hand calls all the user’s connections as “friends”. Friends could be divided into groups called “lists”. There is no default list, so any structure has to be created from scratch. Content on Facebook can be shared with one or more lists, exactly like Google+ circles. But, there is one difference that makes Facebook lists more robust than Google+ circles i.e. the possibility of making exceptions. In Facebook, we can limit access of our content to a list which is a subsets of a set of lists with whom the content is shared. This means, we can share a message with a list called “All” (containing all our contacts) and still make the content invisibile to our “CoWorkers”, as shown in Figure \[fig:FBExceptions\].
As Facebook’s list creation was relatively cumbersome, recently a Facebook application called “Circle Hack” [@CircleHack11] has been launched which provides the Google+ circles graphical interface for Facebook lists. The possibility and use of this application further proves our claim that Facebook lists are logically and functionally a superset of Google+ circles.
Google+ vs Facebook: other comparisons {#OtherComparisons}
--------------------------------------
Facebook uses an encrypted channel only for user authentication (login) while Google+ uses it throughout the connection. This makes it harder to launch a man in the middle attack against Google+. Moreover, Google+ allows finer control of the content shared by a user. A user can disable comments on a post at any time and enable it again later. This could be a useful option to calm down any heated discussions, on the users wall, between two contacts over the shared content or anything else. Facebook, on the other hand, provides its users only with coarser control i.e. they can only block a user from the entire wall but not on an individual content basis (if it was initially shared with them). Furthermore, Google+ allows disabling the resharing of a content at any instant on a content by content basis, again its not possible in Facebook. Finally, Google+ allows its users to edit their comments whenever they want. The time stamp of the last editing remains visible on a comment, so users may modify or backtrack their comments at any time. This too is not possible in Facebook.
Related work
============
Bradshaw identified the first privacy flaw in Google+ [@Bradshaw11]. The flaw was that any content shared with a particular circle could be reshared with anyone by someone from those circles. Although resharing of information is always possible in the electronic world, if someone downloads a copy and upload it again. But, the simplicity and provision of a share button without proper authorization is a privacy problem. This problem is now fixed by Google+.
Social networks privacy and its potential threats have been widely studied in recent years. One of the earliest works on potential threats to individual’s privacy including stalking, embarrassment and identity theft was done by Gross *et al.* [@Gross05].
Felt [@Felt07] presented a vulnerability in Facebook Markup Language which lead to session hijacking. Bonneau and Dhingra independently presented conditional and limited unauthorized access to Facebook photos [@Bonneau09b; @Dhingra08].
Conclusion {#Conclusion}
==========
To conclude, we provided a preliminary analysis of Google+ privacy. We expressed concern that Google+ shares the metadata of the photos uploaded by its users. We also showed that Google+ encourages its users to provide their other names, e.g. , maiden names which may help in identity theft. Moreover, we provided a comparison of Google+ circles with Facebook lists and showed that the latter is a superset of the former, both logically and functionally even though Google+ provides a better graphical interface. Finally, we provided other comparisons, including the use of encryption and the possibility of modifying comments at a later stage, between Facebook and Google+.
|
---
abstract: |
In 2000, an attractive new quantum cryptography was discovered by H.P.Yuen based on quantum communication theory. It is applicable to direct encryption, for example quantum stream cipher based on Yuen protocol(Y-00), with high speeds and for long distance by sophisticated optical devices which can work under the average photon number per signal light pulse:$<n> = 1000 \sim 10000$. In addition, it may provide information-theoretic security against known/chosen plaintext attack, which has no classical analogue. That is, one can provide secure communication, even the system has $H(K) << H(X)$.
In this paper, first, we give a brief review on the general logic of Yuen’s theory. Then, we show concrete security analysis of quantum stream cipher to quantum individual measurement attacks. Especially by showing the analysis of Lo-Ko known plaintext attack, the feature of Y-00 is clarified. In addition, we give a simple experimental result on the advantage distillation by scheme consisting of intensity modulation/direct detection optical communication.
author:
- |
Osamu Hirota, Kentaro Kato, Masaki Sohma, Tsuyoshi S. Usuda,\
Katsuyoshi Harasawa Research Center for Quantum Information Science, Tamagawa University, Tokyo, Japan\
21st century COE program, Chuo University, Tokyo, Japan\
Aichi Prefectural University, Aichi, Japan\
Hitachi Hybrid Network Co. Ltd. Yokohama, Japan.
title: Quantum stream cipher based on optical communications
---
INTRODUCTION
============
There is no encryption scheme with provable security in the conventional cryptography. One of methods to provide “provable security” is quantum cryptography. A quantum key generation scheme for two legitimate users(Alice and Bob) as the quantum cryptography is one of the most interesting subjects in quantum information science, which was pioneered by C.Bennett and G.Brassard in 1984\[1\]. We emphasize that such results are great achievement and open a new science. Many researchers believe that the key distribution by single photon is on the verge of commercial application. However we should take into account the fact that the societies of electronics and communication, and of cryptography are basically not interested in the practical use of quantum cryptography based on single photon schemes, because of extremely low performance in the sense of communication science. Although there is no means of solving such a serious argument, we would like to make the following comment. The key generation is a very important, but it is very narrow sense that one defines quantum cryptography by only BB-84 and similar principle\[2\]. In addition, still it requires “one time pad” to provide secure communication in principle.
Yuen, and his group have pointed out that the quantum cryptography should involve other aspects, and called quantum information scientist’s attention to quantum cryptography based on another principle\[3\]\[4\]. The basic protocol is called Yuen protocol 2000(Y-00). The fundamental structure of Y-00 is organized by a shared initial key like symmetric key scheme in the conventional cryptography, but it is constructed as a physical cryptography. In addition, this gives a generalization of conventional unconditionally secure key generation based on Maurer\[5\] and similar theory in which a shared seed key is not used to establish an advantage distillation. The research like Gisin’s work\[6\] and coherent state based BB-84\[7\] to cope with the low performance should be encouraged, and also research like Northwestern University’s group to investigate another scheme\[8,9\] for achieving the same function should be welcome. The authors believe that collaboration of both types of the quantum cryptography brings real applications in communication networks, because the combination of BB-84 and AES has no meaning in the sense of secure communication.
In this paper, to reveal an excellent potential of Y-00, we shall discuss on concrete security analysis of quantum stream cipher as an example of Y-00. The paper is organized as follows. We introduce the new quantum cryptography in the section 2 and give concrete performance of quantum stream cipher in the section 3. In addition, an experimental result of quantum stream cipher based on intensity modulation and direct detection scheme as conventional optical communication is reported.
Yuen protocol:Y-00
==================
In this section, we will survey a theory of Y-00\[4\]. First we assume that Alice and Bob share a seed key $K$. The key is stretched by a pseudo random number generator to $K'$. The data bit is modulated by $M$-ary keying driven by random decimal number generated from the block :$K'/log M=\bar{K}'=(k_1,k_2,\dots)$ of pseudo random number with the seed key $K$. The $M$-ary keying has $M$ different basis based on 2$M$ coherent states. So the data bit is mapped into one of 2$M$ coherent states randomly, but of course its modulation map has a definite relationship based on key, which is opened.
Let us mention first what is basic principle to guarantee the security. There are many fundamental theorems in quantum information theory. The most important theorem for information processing of classical information by quantum states is the following:\
[**Theorem 1**]{}:\
[*Signals with non-orthogonal states cannot be distinguished without error and optimum lower bounds for error rate exist.*]{}\
This means that if we assign non-orthogonal states for bit values 1 and 0, then one cannot distinguish 1 and 0 without error. When the error probability is 1/2 based on quantum noise, there is no way to distinguish them, from quantum signal detection theory pioneered by Helstrom, Holevo, and Yuen\[10\]. On the other hand, there is no-cloning theorem developed by Wootters-Zurek, and Yuen\[11\] as follows.\
[**Theorem 2**]{}:\
[*Non-orthogonal states cannot be cloned without error.*]{}\
This is an essential basis for BB-84 and others. However Y-00 does not require this theorem explicitly .
Direct data encryption
----------------------
An application of Y-00 is, first, direct data encryption like a stream cipher in conventional cryptography, and then it is extended to key generation, but not one-time pad which is very inefficient. Here it is natural that we should employ different security criteria for direct encryption and key generation. For direct encryption, the criteria are given as follows.
- Ciphertext-only attack on data and on key: To get plaintext or key, Eve knows only the ciphertext from her measurement.
- Known/chosen plaintext attack: To get key, Eve inserts her known or chosen plaintext data into modulation system( for example, inserts all 0 sequence as text). Then Eve tries to determine key from input-output. Using the key, Eve can determine the data from the ciphertext.
In the conventional theory, we have $H(X|Y_E, R_M) \le H(K)$ known as Shannon bound for ciphertext only attack on data, and $H(K|Y_E,R_M)\ge 0$ for ciphertext only attack on key which is relevant with “unicity distance”, where $X$ is data sequence, $Y_E$ is Eve’s data, and $R_M$ is public mathematical randomization, respectively. In addition, for known plaintext attack, we have $H(K|X, Y_E, R_M) =0$ which means a computational complexity based security. In the following, we will see that one may overcome such limitations in the conventional direct encryption by Y-00.
A fundamental requirement of secure communication by “one way scheme” is, first, to establish that the channel between Alice and Eve is very noisy, but the channel of Alice and Bob is kept as a normal communication channel by physical structure. To realize it, a combination of a shared short key for the legitimate users and a kind of stream cipher with specific modulation scheme is employed, following the theorem 1. We note that a main idea of this protocol is the explicit use of a shared short key and physical nature of scheme for cryptographic objective of secure communication and key generation. This is called initial seed key advantage.\
[**Principle of security**]{}: [*The origin of security comes from difference between optimum quantum measurement performances with key and without key.*]{}\
In general, a quantum information system is described by a density operator. The density operator of the output of the coding/modulation system of Y-00 for Eve depends on attacks. For ciphertext-only quantum individual attack, the density operator is $$\rho_{T}=p_0\rho_0 + p_1\rho_1$$ where $$\rho_0 = \sum q_j|\alpha_j\rangle\langle \alpha_j|, \quad
\rho_1 = \sum q_k|\alpha_k\rangle\langle \alpha_k|$$ The probability $p_i$ depends on the statistics of the data, and $q_j$, $q_k$ depend on the pseudo random number with $j$, and $k$ being even and odd number, for example. Eve has to extract the data from the quantum system with such a density operator. However, according to one of the most fundamental theorem(theorem 1) in quantum information theory, the accuracy of Eve’s measurement is limited. Thus, to induce error in Eve’s measurement is essential in Y-00. Error probability of Eve and its requirement for secure system depend on attack methods. A typical measurement in the quantum individual attack is direct measurement of transmitting signal, which corresponds to discrimination of information bit or discrimination of $M$-ary states. In this case, for ciphertext only attack on the information data, the best way of Eve is of course given by the quantum optimum detection for two mixed states :$\rho_0$ and $\rho_1$. That is, the limitation for accuracy of measurement of Eve is given by Helstrom bound as follows\[10\]: $$\bar{P}_e= \min_{\Pi}(p_1Tr\rho_1\Pi_0 + p_0Tr\rho_0\Pi_1)$$ where $\Pi$ is POVM(positive operator valued measure) or quantum detection operator which corresponds to general optical receiver in optical communications. The error probability of Eve becomes $\sim 1/2$ from the appropriate choice of the number $M$ and signal energy. It means that Eve’s data $Y_E$ is completely inaccurate. On the ciphertext only attack on key, the best way for Eve is to detect $M$ basis based on 2$M$ coherent states. In this case, the limitation for accuracy of Eve’s data is also given by the minimax quantum detection\[12\] for 2$M$ pure coherent states. That is, the measured data on the running key involve unavoidable error given by $$\bar{P}_e = \max_{p_i}\min_{\Pi} (1 - \sum p_iTr \rho_i\Pi_i)$$ For appropriate $M$ and signal energy, we have $\bar{P}_e \sim 1$. As a result, Eve’s data have almost complete error by irreducible noise.
When Eve can know or insert some input data $X$, one can apply also the above equation, but the number for detection is reduced to $M$ coherent states. On the other hand, Eve can devise certain known/chosen plaintext attack in this situation. That is, Eve can use the known plaintext after all trial of her measurement.
If Eve takes another type of attack like combination with measurement of [*indirect observable* ]{} and structure information of $M$-ary modulator, then density operators of Eve may be redefined and the error probability is also calculated by quantum detection theory as shown in later sections.
Let us turn to issue of randomization which is essential in Y-00 for getting ultimate performance in practical sense. The structure of Y-00 is formed by physical processes with specific modulator performance and so on. This means that the protocol is different with conventional cryptography formulated by mathematical relation only, even the scheme is constructed by devices based on classical physics. It is called “physical cryptography”. Since the protocol is constructed by combination of physical processes and mathematical encryption, one can devise a new randomization that is possible with physics to increase the security. Such a randomization $R_P$ is called “physical randomization”. A general theory of such randomizations was already given, which involves a theory of [**DSR**]{}: deliberate signal randomization, [**DER**]{}:deliberate error randomization and so on \[4\]. Thus, this is a new type of quantum cryptography based on quantum detection theory. That is, the security is guaranteed by quantum noise, but the system can be implemented by sophisticated optical devices or quantum optimum receiver, depending on the required performance. As an advantage, Y-00 may provide information-theoretic security against known/chosen plaintext attack based on several quantum measurements.
We can summarize the properties as follows:\
In the cipher-text only attack on data, Y-00 can exceed the classical Shannon limit, even we use a system with $H(K) << H(X)$. That is, $$H(X|Y_E, R_M, R_P) > H(K)$$ where $X$ is information data, $Y_E$ is ciphertext as “measured value” for Eve, $R_P$ is physical randomization and $K$ is initial seed key.
For known/chosen plaintext attack, it will be expected that $$H(K|Y_E,R_M, R_P, X) > 0$$ which corresponds to [**information-theoretic security**]{}. If one has $H(K|Y_E,R_M, R_P, X) =H(K)$, then it is full security. These are not realized by the conventional theory. It means that Y-00 breaks the limitation of the conventional cryptography theory. Furthermore, by introducing several new randomizations, this will provide information-theoretically secure scheme with highly efficient performance even if the advantage of the legitimate user is small in principle.
When the signal power is very strong and the system has a bad design, there may be some algorithm for computing to find the key based on the measured data including “small error”. It may have $H(K|Y_E, R_M, R_P, X) \sim 0$. Even so, it requires additional exponential search.
Quantum key generation
----------------------
In the case of key generation protocol, data is a true random number sequence. So there is no criterion like known plaintext attack. In the conventional theory of key generation, if one has $$I(X_A; Y_E) < I(X_A; Y_B),$$ then key generation is possible. That is, an information-theoretic existence proof is given under “the condition of no public discussion”. However, so far there was no theoretical consideration on scheme with shared seed key between Alice and Bob. In the above subsection, we explained the basic scheme. By this scheme or more generalized scheme, one can also make a key generation scheme. The use of shared seed key between Alice and Bob that determine the quantum states generated for the data bit sequences in a detection/coding scheme gives them a better error performance over Eve who does not know $K$. Based on the above scheme, the conditions for key generation were discussed to include the use of [*a shared seed key*]{}. Let us introduce the basic result here.\
[**Remark**]{}\
[*For the scheme with the seed key, one has to make sure that the generated key:$K_g$ between Alice and Bob is fresh. That is, the generated key is independent of the initial seed key, and*]{} $I(K_g; Y_E |K) \sim 0$.
In order to characterize this situation, one can allow that Eve can know the key only after she has made her measurement. So her information is described by $I(X_A, Y_E|K)$. As a result, the condition for secure key generation is $$I(X_A; Y_E|K) < I(X_A; Y_B)$$ or $$H(X_A|Y_E, K) > H(X_A|Y_B)$$ where $Y_B$ is Bob’s observation with knowledge of the seed key. The above result means that Eve can get the key $K$ after Eve’s measurement as a kind of side information. In quantum signal detection theory, the difference between “before measurement” and “after measurement” on the knowledge of key which control the quantum measurement is essential. In the classical channel, there may be no difference for the order of the knowledge of key. That is, $$H(X_A|Y_E, K) =0$$ From the above result, one may denote $$R^*_g = \max_{p(X_A)}[I(X_A; Y_B) - I(X_A; Y_E|K)]$$ By choosing a rate below the key rate, one is able to force the Eve’s information to be zero as a consequence of the well known coding theorem.
Basis for concrete security analysis
====================================
Quantum measurements
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Let us denote here several kinds of quantum processing which can be used by Eve. We can divide broadly into two categories: Individual and Collective measurements, respectively. Both of them are well defined by quantum detection theory based on Helstrom-Holevo-Yuen formalism. The former is that Eve prepares the probe/interaction to each qubit of the quantum signal sequence individually and identically, and processes the resulting information independently from one qubit to the other. Then she employs classical joint processing. On the other hand, the qubits may be correlated through the running key $K'$. So one can take a joint attack which requires correlated qubit measurement so called collective measurement. In the each category, one has two kinds of measurement method. That is, if Eve tries to measure directly the information bits $\{0,1\}$, or $M$ values to $M$-ary modulation scheme, then it is called [*direct observable attack*]{}. On the other hand, if Eve tries to use some information on the structure of modulation and so on, and measure the indirect observable which can be derived from the structure, then it is called [*indirect observable attack*]{}.
In addition, there is another quantum measurement scheme so called unambiguous measurement. This has no advantage in communication theory, but it is applicable to cryptography \[13,14\]. However, here we have the following property\[15\].\
[**Theorem 3**]{}\
[*The lower bound of inconclusive probability in unambiguous measurement is given by the quantum optimum solution in quantum detection theory for the same state ensemble.*]{}\
As a result, we can evaluate the limitation of unambiguous measurement attack by quantum detection theory.
Randomization in physical layer
-------------------------------
In general, one uses additional several randomizaions in the conventional stream cipher. The first demonstration of direct encryption as Y-00 consists of LFSR as PRNG and $M$-ary modulator\[8,9\]. In addition, new randomizaions for Y-00 are employed, which involves very different concept. We introduce more concrete one here. The conventional randomization is of data, and it is designed by a mathematical relation. However, Y-00 is designed by both mathematical relation and physical layer as modulation scheme. So one can take new parameters for randomization. For example, modulation, synchronization and so on which are parameters of physical layer in the communication protocol. A new function of this randomization in physical layer is to stimulate making random error for Eve’s measurement. This type of randomization provides very good security performance even the system is classical and noiseless. The system consisting of only such a new randomization is called classical Y-00. Although the classical Y-00 is stronger than conventional stream cipher, it cannot have information-theoretic security or unconditional security, because it will be insecure under specific side information. For our purpose, one needs irreversible noise effect based on quantum mechanics.\
[**Proposition 1**]{}:\
[*The sufficient condition for information-theoretic security on known/chosen plaintext attack is that irreversible error for $Y_E$ is induced by physical randomization with quantum effect.*]{}\
Proof:\
In the known plaintext attack, when $Y_E$ does not involve error(no error by measurement), we have $Y_E=Y_B$. So $H(K|Y_E, R_P, X)=0$. The condition for nonzero of key equivocation is that the key is not determined uniquely by $Y_E, R_P, X$. To realize such a situation, the key equivocation should be nonzero even when Eve is allowed to get the randomization schemes used by users after measurement. If error is recovered by side information, the key equivocation is zero. If error is irreversible, then the key is not uniquely assigned by any side information. Since the classical noise is, in principle, removable, we need really irreversible quantum effect which corresponds to the projection postulate.
In conventional system, it is assumed that error in measurement process is zero, and error by lack of information for mathematical relation(mathematical randomization and so on) can be recovered by side information. In quantum system, the randomization and quantum noise effect help each other to make a secure communication scheme by inducing irreversible error in the measurement. We can show a concrete example here so called [*Overlap Selection Keying*]{}:[**OSK**]{} \[16\] which is defined as making Eve’s density operators $\rho_1=\rho_0$ in the case of direct observable, and $\rho_{up}=\rho_{down}$ in the case of indirect observable as shown in later sections. So Eve cannot get any information by her measurement. But the realization methods of OSK are adapted to Eve’s attack. OSK which produces the identical density operators for Eve’s any kind of measurement is indeed physical randomization. Thus, the effect of quantum noise is diffused by the randomization. On the other hand, we can introduce an error inducement randomization([**EIR**]{}) by physical process also. The EIR means to induce error in the discrimination for the signals based on [*physical force*]{} in communication scheme. The typical one is to break the synchronization rule between Alice and Eve by changing the synchronization rule between Alice and Bob. Even if the synchronization errors are few slots, then the error is induced as measurement error. Such errors are not recovered by the knowledge of the EIR. We will show some examples of the effect of OSK and EIR in the section 4.
Design of number of basis in M-ary modulation
---------------------------------------------
One of features in Y-00 is that one can use conventional laser light as the transmitter which has mesoscopic energy. Since Eve can attack at the transmitter side, we have to design the number of basis to keep the non-orthogonality among quantum states at the transmitter. In the phase modulation scheme, the coherent states are described by positions on circle in the phase space representation. The radius corresponds to the amplitude or average photon number per pulse at the transmitter.The positions on the circle correspond to phase information of the light wave. If the number of basis is $M$, then the signal distance between neighbor states is about $\frac{2\pi |\alpha|}{2M}$. In the practical sense, we can design the number of basis which satisfies $$P_e(i+1, i) = \frac{1}{2} - \frac{1}{\sqrt{2\pi}}\int_{0}^{t_0}\exp(-t^2/2) dt
=0.45 \sim 0.5$$ where $t_0= \frac{\pi |\alpha|}{2M}$. This corresponds to the error probability between neighbor states. As a result, we have to use at least $M=10^3$ for $<n> =|\alpha|^2=10^4$, and $M=10^4$ for $<n> =10^5$.
Quantum individual measurement attack
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In the case of direct encryption, two attacks are important. One is ciphertext only attack on data and key. Other is known/chosen plaintext attack. We will discuss on both attacks to the system implemented by Northwestern university group so called $\alpha\eta$ scheme \[8,9\] based on $M$-ary phase shift keying by coherent states. In general, the phase spaces for Alice-Bob, and Alice-Eve are different, because the phase used in the communication means relative phase between transmitter light and reference light at receiver. Although the phase spaces are not same in general, we first assume that the phase spaces of three parties are same.
Ciphertext only attack
----------------------
Now the seed key in the system has the relation $H(X) >> H(K)$. In the conventional theory, we have always $H(X_A|Y_E, R_M) \le H(K)$ for ciphertext only attack on data. But this is not true in the case of Y-00. In the following we will show some examples.
### Direct observable attack
When Eve employs direct measurement on data or on key, the ultimate error probability is given by Eq(3) based on quantum detection theory for each bit slot. When we design the system by appropriate photon number and $M$ based on Eq(12), the error probability is almost $\frac{1}{2}$. In addition, by using OSK, it becomes exactly $\frac{1}{2}$ because of $\rho_1 =\rho_0$. These fact mean that Eve cannot get any information on data by her measurement.
For the property of [*reuse of the key*]{}, one can check the processing $Y(1)\oplus Y(2)=X(1)\oplus X(2)$, where $Y(1)=X(1)\oplus K', Y(2)=X(2)\oplus K'$, and where $K'$ is PRN. Thus, in general, the key disappears in the stream cipher, while the one time pad has $Y(1)\oplus Y(2)=X(1)\oplus X(2)\oplus K_1 \oplus K_2$. $K_1$ and $K_2$ are true random number. However, in Y-00, the measured values of Eve have $Y_E \ne Y(1 {\rm or} 2)$, but $Y_E=Y \oplus \bf{e}$, where $\bf{e}$ is error vector and it is completely random by quantum noise effect. So we have $Y_E(1)\oplus Y_E(2)=X(1) \oplus X(2)\oplus \bf{e}_1 \oplus \bf{e}_2$. Since $\bf{e}_1 \oplus \bf{e}_2$ is completely random when $\bar{P}_e = 1/2$, the performance is equivalent to the one time pad.
### Indirect observable attack
Since the modulator has definite structure, Eve can use an information on such a scheme. Indeed, the randomly selected phase shift keying used in $M$-ary cipher scheme as Y-00 is taken to be $$l_i=x_i\oplus \tilde{k}_i$$ on the phase space, where $l_i$ is one of two regions separated by appropriate axis on the phase space. If the fundamental axis is horizontal, $l_0$ is upper plain, $l_1$ is down plain, $x_i$ is data bit. $\tilde{k}_i$ is 0 for even number and 1 for odd number in the running key of $M$-ary assignment \[15,16\]. However, we should denote that $\tilde{k}_i$ is the result of the mapping from the running key of decimal number:$k_i = (1 \sim M)$. $$K'_j =(k_1, k_2, k_3, \dots) \mapsto \tilde{K}_j =(\tilde{k}_{1}, \tilde{k}_{2}, \tilde{k}_{3}, \dots), \quad \{j=1 \sim 2^{|K|}\}$$ where $ \tilde{k}_{i}= even$ or $odd$. For example, $(l_i=up,\quad \tilde{k}_i=even) \longrightarrow x=1$, $(up,\quad odd) \longrightarrow x=0$, $(down, \quad even) \longrightarrow x=0$, $(down,\quad odd )\longrightarrow x=1$. Let us define the sequences of numbers $l$, $x$, $\tilde{k}$ as follows: $L=(l_1, l_2, l_3, \dots), X=(x_1, x_2, x_3, \dots),
\tilde{K}=(\tilde{k}_{1}$. Here $K$, and $N$ are an initial key with length $|K|$, and length $|N|$ of pseudo random number, respectively. The essential point of the attack is to measure indirect observable $L$. However, since the observable does not contain the information of the data bit, Eve is asked to use the brute force attack for key to find a correct sequence of the data. Here we can define that $\cal{R_A}$ is a set of data sequence with the length of $|N|$. Alice sends a sequence $R_T$ in $\cal{R_A}$, and it is coded based on Eq(13)\[8,9\]. $\tilde{K}_j$ corresponds to pseudo random number sequence which has at most a number of possibilities of $2^{|K|}$ and the length $|N|$. Let $R_T$, $L_T$, $\tilde{K}_T$ be true sequences used and defined on the phase space for Alice and Bob. She tries to assign all kind of $\tilde{K}_j$ to her measured sequence $L_m$ of $L$. So she gets a set $\cal{R}_{\rm{E}}$ based on $l_i=x_i \oplus \tilde{k}_i$. If $L_m$ is error free, then it is guaranteed that one of $\cal{R}_{\rm{E}}$ is the data bits sequence. At this stage, we have $H(X|Y_E)=0$ based on the brute force attack. Here, if there is one bit error in $L_m$ by some reasons, then Eve has $L_T\oplus \bf{e}$, where ${\bf{e}}=(0,0,1,0,0, \dots)$ is error sequence. The position of the error is unknown and uniformly distributed. When Eve applies $\tilde{K}_j$ to $L_T\oplus \bf{e}$, then it is not guaranteed that the true $R_T$ exists in $\cal{R}_{\rm{E}}$. She has to try $2^{2|K|}$ greater than the initial one. If there are many error, then it becomes $\sim 2^{2^{|K|}}$. So one may obtain $H(X|Y_E) > H(K)$.
Indeed, let us show that the error of the measurement for $l_i$ is unavoidable. The density operators of signal sets for up and down measurement are $$\begin{aligned}
\rho_{up}&=&\sum_{up} \frac{1}{M}|\alpha_{i}\rangle \langle \alpha_{j}|, \\
\rho_{down}&=&\sum_{down} \frac{1}{M}|\alpha_{j}\rangle \langle \alpha_{j}|\end{aligned}$$ It is easy to show the quantum limit, which is the most rigorous lower bound of error probability for this signal\[17,18\]. When the coherent state is mesoscopic$<n>\sim 10000$ and one thousand of $M$ based on Eq(12), the error is several percents: $P_e \sim 0.01$. This means that the number of error bits is $P_e \times 2^{|K|} \gg 1$ which is enough to prevent this type of attack with the brute force search. That is, we have $H(X|Y_E)>H(K)$. For the performance of reuse of key, we have $L_1\oplus L_2=X_1\oplus X_2 \oplus \bf{e}_1 \oplus \bf{e}_2$. If the number of error is small, the security may depend on $H(X)$. However, the error probability can be increased by arranging the signal state assignment.
Furthermore, again we can employ OSK. Since the best way for indirect observable attack is to divide the phase space into two regions like up plain and down plain, Alice can employs [**OSK**]{} which provides $\rho_{up}=\rho_{down}$. As a result, $\bf{e}_1 \oplus \bf{e}_2$ of $L$ is completely random, and it has no information on transmitting states. Such an OSK is made by changing the fundamental axis of the phase space of Alice-Bob based on a part of running key, for example $0-\pi$, and $\pi-0$. Thus, if one uses randomizations as OSK, then the scheme is secure to the ciphertext only attack on data even in the noiseless. If users require a security against only the ciphertext only attack, then “quantum effect” in Y-00 may be not essential, though it is helpful. That is, [*the classical Y-00 with several randomizations is sufficient enough for ciphertext only attack*]{}.
Known/chosen plaintext attack
-----------------------------
Since the ciphertext only attack on key is absorbed into the known/chosen plaintext attack, we skip it. In the conventional cryptography, the known/chosen plaintext attack means that Eve can get many pairs between known data bit sequence and corresponding ciphertext. In Y-00, a ciphertext corresponds to a sequence of quantum states. Thus, in this case, Eve knows several set of plaintext and corresponding quantum state sequences. So Eve has an additional problem such as quantum state identification which is done by quantum measurement. Eve will have two methods which utilize such an additional knowledge for improving her attack. One is to reduce the number of selection of quantum measurement, and other is to use it after measurement which corresponds to Lo-Ko attack\[19\].
### Conventional method
Let us assume that Eve can insert known sequences as data. So the output quantum state sequence of the $M$-ary modulator correspond to running key information. In the case of individual attack, Eve can try to discriminate $M$ coherent states by quantum optimum receiver. It is well known that one can find a seed key or solve next bit prediction for LFSR if one knows exactly bit sequence of 2$|K|$ as the running key. However, in Y-00, the error in the measurement is unavoidable. Although there is no general theory of lower bound of algorithm for finding key based on running key information with error, in the limit, we may have the fact that the error probability is given by Eq(4), which converges 1 with respect to large number of $M$. As a result, by appropriate number of $M$ and photon number, we have $H(K|Y_E, X) > 0$ which means information-theoretic security.
As an example, we here give more concrete scheme. Y-00 has the following structure. The output of LFSR is divided by $log M$ block, and the bit sequence of the length $log M$ is changed into the number $k_i \in {\cal M}$ which corresponds to the numbering of basis. So the output state is a coherent state with the same numbering. Here the number is assigned as regular order on the circle of the phase space. Let us assume that Eve knows plaintext with length of $2|K|$, say $X_a$. By heterodyne receiver, Eve measures the quadrature amplitude $x_c$ and $x_s$ putting known plaintext:$X_a$ to decide which basis is used. The errors of measured data of Eve are induced mainly for the neighbor quantum states. That is, when the measured number is 5, then it has possibility of 4 or 6 as true number. Since the number of slot is $2|K|/log M$, the measured sequence will be, at least, one of the number of sequences $2^{2|K|/log M}$. This may correspond to 1 or 2 bit random error per $log M$ bit block in the bit sequence of the output of LFSR. If the error occurs among several neighbors, then Eve suffers more error in corresponding bit sequence. Here, since the error bits of measured data in bit sequence are $\sim (2|K|/log M)=T$ bits, Eve has to launch $W=2^T$ times the next bit prediction algorithm. By design of $M$ and $<n>$ based on Eq(12), we have $W>>2^{|K|}$.\
### Lo-Ko attack
Here let us assume that Eve will divide the light wave conveying known plaintext $X_a$ into $W$ beams by beam splitter, then she will prepare $W$ receivers with different key and measures each beam by each receiver. Since Eve knows the plaintext, the receiver which outputs the same plaintext is correct one. Such a method is called Lo-Ko attack\[19\]. Let us assume that the amplitude attenuation parameter of channel between Alice and Bob is $\kappa=1/(t+1)$. The amplitude of Bob is given by $\alpha/(t+1)$. Eve makes $(t+1)$ copies by means of division of the output light from Alice by $(t+1)$ beam splitters. $$\begin{aligned}
|\Psi \rangle &=&|\frac{\alpha_i}{t+1} \rangle |\frac{\alpha_j}{t+1} \rangle |\frac{\alpha_k}{t+1} \rangle \dots \nonumber \\
|\Psi \rangle &=&|\frac{\alpha_i}{t+1} \rangle |\frac{\alpha_j}{t+1} \rangle |\frac{\alpha_k}{t+1} \rangle \dots \nonumber \\
|\Psi \rangle &=&|\frac{\alpha_i}{t+1} \rangle |\frac{\alpha_j}{t+1} \rangle |\frac{\alpha_k}{t+1} \rangle \dots \\
\vdots \nonumber\end{aligned}$$ Then the first sequence is sent to Bob by a lossless channel. Bob cannot notice the existence of Eve, because his coherent state sequence is exactly same as that of regular communication. Bob will employ appropriate receiver with assigned key in which the performance will be error free(actually it is not error free, and they use error correcting code). Eve has to try all kinds of receiver. So she needs exactly $W=2^{|K|}$ copies. Since she knows plaintext as input data, she can compare her measurement result with the plaintext. The receiver which showed the same result with plaintext is Bob’s receiver, and she can know the key. However, the amplitude of each coherent state is scale down by factor $1/(t+1)$. In this case, in order to get $W$ copies, the requirement for loss between Alice and Bob is $1/t =1/W$. It is easy to show that Lo-Ko attack does not work for practical situations, for example, if the communication length by conventional optical fiber is 100 km, then $t=100 << W=2^{|K|}$. This means that Eve cannot get enough copies which is able to decide the key.\
### Modified Lo-Ko attack
Let us assume that the loss is realistic such as $\kappa >> 2^{-|K|}$. By first splitter, Eve makes $$|{\Psi}' \rangle =|\kappa \alpha_i \rangle |\kappa \alpha_j \rangle |\kappa \alpha_k \rangle \dots$$ This is sent to Bob by a lossless channel. Then she makes $2^{|K|}$ copies as follows: $$\begin{aligned}
|\Psi \rangle &=&|(1-\kappa)\frac{\alpha_i}{W}\rangle |(1-\kappa)\frac{\alpha_j}{W}\rangle |(1-\kappa)\frac{\alpha_k}{W} \rangle \dots \nonumber \\
|\Psi \rangle &=&|(1-\kappa)\frac{\alpha_i}{W}\rangle |(1-\kappa)\frac{\alpha_j}{W}\rangle |(1-\kappa)\frac{\alpha_k}{W} \rangle \dots \nonumber \\
|\Psi \rangle &=&|(1-\kappa)\frac{\alpha_i}{W}\rangle |(1-\kappa)\frac{\alpha_j}{W}\rangle |(1-\kappa)\frac{\alpha_k}{W} \rangle
\dots \\
\vdots \nonumber\end{aligned}$$ However, in this case, the amplitude of Eve’s quantum state sequences is smaller than that of Bob. So even if Eve has $W$ copies, since the performance of her measurement is less than that of Bob, her results involve unavoidable error. As a result, the knowledge of plaintext cannot help to decide Bob’s receiver.\
### Indirect observable attack and others
Let us discuss known/chosen plaintext attack based on indirect observable. Eve knows certain relation of the modulator like $l_i=x_i\oplus \tilde{k}_i$. Assume that Eve has several known plaintexts. For measured data of $l_i$, she applies the known plaintext. So she can get the sequence of $\tilde{k}_i$ based on the above relation. However, the running key sequence cannot be determined by the sequence of $\tilde{k}_i$, because Eq(14) is not one to one correspondence. In addition, as we mentioned in the case of ciphertext only attack, the number of error slots on the measurement of $l_i$ are not so small even the error probability for $l_i$ is small, when the system is well designed based on Eq(12).
Then, let us apply unambiguous measurement attack. On the prediction of the key sequence based on known plaintext, in the conventional stream cipher by LFSR, the requirement of length of known plaintext is about the key length. In order to realize the unambiguous measurement attack, we need collective measurement. The collective measurement means that a discrimination problem of many quantum state sequences is treated such that a quantum state sequence can be regarded as one pure state in the tensor product Hilbert space of time slot modes, and the design of the detection operator:$\Pi^{N}$ as POVM is formulated on the extended space. Since Eve knows structure of PRNG and modulation scheme, she can apply the unambiguous measurement:$\Pi_{un}^{N}$ for $2^{|K|}$ quantum state sequences to one copy of the transmitting sequence on the extended space. When the length of key is enough long, the success probability is given based on the theorem 3 as follows: $$P_d \sim 2^{-2|K|}$$ where we use $|K| > 100$. So with this probability, Eve will get an exact running key sequence. However, in principle, Eve cannot get a situation which the success probability is 1.\
Randomization for long distance communication
---------------------------------------------
In order to realize secure long distance communication, one can use more strong light power at the transmitter. In such a case, the error may be very small in Eve’s any measurements. However, we can cope with such a situation by introducing several kinds of physical randomization pioneered by Yuen.
To prevent the several attacks, we can introduce the randomization for numbering to assign the basis state which is one example of DSR. As a result, the error position in the bit sequence of running key can be uniformly diffused into the total length of the running key. On the other hand, in the above sections, we assumed that Eve can synchronize the phase spaces between Alice-Bob, and Alice-Eve. In general situation, it is difficult. Even it can do at the first step, legitimate users can break easily the synchronization by randomization or based on seed key advantage. Let us denote more detail nature. Since Y-00 is a physical cryptography, we should clarify the physical property. In general, phase spaces of Alice-Bob and Alice-Eve are not same. The phase space is formed by the relative phase based on local phases of Bob and Eve. For example, quadrature amplitudes are $\{x_c=A\cos(\phi_S-\phi_{L(Bob)}), x_s=A\sin(\phi_S-\phi_{L(Bob)})\}$, $\{x_c=A\cos(\phi_S-\phi_{L(Eve)}), x_s=A\sin(\phi_S-\phi_{L(Eve)})\}$. The Eq(13) is defined for the phase space of Alice-Bob. In general Eve does not know the correct phase space. Thus it is easy to control the fundamental axis of the phase space to prevent Eve’s locking. This fact is one of characters of physical cryptography. So Eve never understands what is the axis decided by herself. This is one of EIR. As a result, Eve’s data involves many errors, even when the measurement is noiseless. This is a concrete example why Y-00 is stronger than that of the conventional scheme even the system is classical one. When such randomizations and quantum noise effect are used, the quantum noise effect is diffused. As a result, the quantum noise effect is enhanced.
Key generation
==============
In the key generation, since data is true random number, Eve’s strategy is mainly to apply direct observable attack. In addition, the attack to get the running key (or seed key) based on direct observable of ciphertext is hopeless for Eve as mentioned in the previous section. So Eve has to try to get information of random number directly.
Key rate based on entropy evaluation
------------------------------------
In this section, we would like to obtain certain example of the key rate in the framework of Yuen’s general discussions. From lemma 1 in the reference \[4\] and Holevo bound theorem, we have $$I(X_A(n), Y_E(n)|K)/n < I(X_A(n), Y_E(n))/n + H(K)/n < S(\rho_T) - \sum p(x) S(\rho_{x}) + H(K)/n$$ where $I(X_A(n), Y_E(n))$ refers to the mutual information between $n$-bit sequences, $S(\rho)$ is von Neumann entropy, and where $$\rho_T = p(0)\rho_0 +p(1)\rho_1$$ $\rho_0$ and $\rho_1$ are given by Eq(2). Let $\epsilon$ be cp-map between Alice and Bob, and let us assume that the channel between Alice and Eve is ideal channel(cp-map is identity map), because Eve can make her measurement at the close Alice. As a result, we can read as follows: $$r_{g(n)} \ge \max_{{p}(i)} [\{S(\epsilon ({\rho}^{B}_T)) - \sum {p}(i)S(\epsilon ({\rho}^{B}_i)\} -
\{S({\rho}^{E}_T) - \sum {p}(i)S({\rho}^{E}_i) + H(K)/n \}]$$ When the cp-map corresponds to ideal linear attenuation with attenuation parameter $\kappa$, Bob’s density operator is ${\rho}^{B}_0 = |\kappa\alpha\rangle\langle \kappa\alpha|$, ${\rho}^{B}_1 = |-\kappa\alpha\rangle\langle -\kappa\alpha|$, and that of Eve is Eq(2) for the direct observable. When $n$ is enough large, we can calculate it for the concrete scheme described above. For example, we assume that $<n>=1000$, $M=100$, $\kappa=1$, and the data rate is 1 Gbps. Then the secure key bits are $10^7$. However, with respect to $\kappa$, the secure key bits decrease. It has only 6 dB energy loss advantage to keep the secure non-zero key bits.
When we apply OSK for direct observable, we have $$I(X_A;Y_E)=I(X_A;Y_E|K) = 0$$ As a result, key rate is equal to the Holevo capacity of the channel between Alice and Bob. Thus we can see the great advantage of the diffusion of quantum noise by randomizatons.
Combined attack to quantum and classical channels
-------------------------------------------------
Let us assume that the generated key is used as symmetric key in classical channel communication like one time pad, and Eve can get perfect ciphertext on the classical channel.
If Eve takes direct observable on the quantum channel,then Alice and Bob can share the key bits as mentioned above. Even Eve gets all ciphertext on the classical channel, the key in the ciphertext has no correlation between Eve’s prediction and real key for one time pad, because of privacy amplification.
On the other hand, we can consider the indirect observable in this case. Nishioka et al discussed on the following attack\[20,21\]. Eve measures $l_i$, and she does not assign $\tilde{K}_j$, $j \in 2^{|K|}$ at the first stage for quantum communication used Y-00. Then she get the data $c_i$ on the classical channel. In this model, one has the relations as follows: $$\begin{aligned}
c_i&=&x_i \oplus r_i \\
c_i \oplus l_i &=& x_i \oplus \tilde{k}_i\end{aligned}$$ where $c_i$ is ciphertext, $x_i$ is plaintext. The random number $r_i \in R$ as the key disappears. Since Eve does not know $\tilde{k}_i$, she has to assign to all the different running keys in which the number of possibilities is $2^{|K|}$ . If Eve can get the data $l_i$ and $c_i$ without error, then it corresponds to conventional noiseless stream cipher with exponential search. However, by the effect of quantum noise and randomizations like OSK and DSR, Eve’s data of $l_i$ are completely random. So such an attack has no meaning. We emphasize that the original experiment\[8\] of Y-00 did not claim this kind of protocol. So this kind of attack is a fiction.
Intensity modulation scheme and its experiment
==============================================
We would like to devise an attractive technology based on quantum communication theory which is applicable to the real world. It is well known that intensity modulation based optical communication is widely used in the conventional optical fiber network system. Of course Y-00 is applicable to it. However, in general, it is difficult to increase the number of $M$ in the intensity modulation scheme. So we need appropriate randomizations. In the reference \[16\], we proposed Y-00 based on intensity modulation and direct detection(IMDD) scheme with appropriate randomization so called OSK which was reviewed in the section 3. Let us give again a brief explanation of the scheme. The maximum amplitude of the transmitter is fixed as $\alpha_{max}=\alpha_{2M}$. We divide it into 2$M$. So we have $M$ sets of basis state $\{(A_1,A_2),(B_1,B_2), \dots \}$. The total set of basis states becomes as shown in Fig.1.
----------------------------------------------------------------------------------------
![M-ary scheme based on amplitude(intensity)[]{data-label="figure1"}](Fig1.eps "fig:")
----------------------------------------------------------------------------------------
The error probabilities of Bob and Eve in this case can be derived from our papers \[17,18\]. Here let us employ OSK for data bit sequence and also for the numbering of the basis $M$ related up and down on the axis for amplitude. Each set of basis state is used for $\{1, 0\}$, and $\{0, 1\}$, depending on sub-running key. $$\begin{aligned}
Set\quad A_1 : 0 &\rightarrow& |\alpha_{(1)}\rangle,
\quad 1 \rightarrow |\alpha_{(M+1)}\rangle\\
Set\quad A_2 : 0 &\rightarrow& |\alpha_{(M+1)}\rangle,
\quad 1 \rightarrow|\alpha_{(1)}\rangle\end{aligned}$$ So the density operators of 1 and 0 for Eve are $\rho_1=\rho_0$. Furthermore, by changing the numbering of the basis from up to down by sub-running key, the density operators for Eve become $\rho_{up}=\rho_{down}$.
In order to verify the advantage distillation of Y-00 based on the intensity modulation scheme, we show experimental result on error performances of receivers of Bob and Eve. The system consists of the conventional laser diode and photo diode which work under the 5 Mbps and room temperature. The number of coherent states are 4, but the amplitude difference is very small. We assume that the technology level of Bob and Eve are the same one. Figure 2 shows the error probabilities of Bob who knows key and of Eve who does not know key, respectively, when the legitimate users do not use the randomizations. Y-00 can work under the distance corresponding to the difference of the error probability. On the other hand, when we employ the OSK, the error probability of Eve is 1/2. So the communication distance is limited only by the error performance of Bob. The effect of the OSK is very clear in this case.
Conclusions
===========
In this paper, it has been discussed that Y-00 based on IMDD realizes a scheme with provable security in the sense of information theoretic security against at least several proposed attacks, which is not realized by conventional cryptography. If one requires the security for ciphertext only attack, then one does not need “quantum”, but classical Y-00 is enough. Thus, it is clear that recent criticisms \[19, 20\] on Y-00 have no meaning in the sense of cryptography.
Acknowledgment {#acknowledgment .unnumbered}
==============
OH is grateful to H.P.Yuen, P.Kumar and many colleagues of Northwestern University for discussions.
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![Error probabilities of Eve and Bob[]{data-label="figure2"}](Fig2.eps)
|
---
abstract: 'The effects of the first nonlinear corrections to the DGLAP equations are studied in light of the HERA data. Saturation limits are determined in the DGLAP+GLRMQ approach for the free proton and for the Pb nucleus.'
author:
- 'K.J. Eskola $^{\rm a,b,}$, H. Honkanen $^{\rm a,b}$, V.J. Kolhinen $^{\rm a,b}$, Jianwei Qiu $^{\rm c}$, C.A. Salgado $^{\rm d}$'
title: ' Nonlinear corrections to the DGLAP equations; looking for the saturation limits[^1]'
---
HIP-2003-08/TH
Introduction
============
Parton distribution functions (PDF) of the free proton, $f_i(x,Q^2)$, are needed for the calculation of the cross sections of hard processes in hadronic collisions. Once they are determined at certain initial scale $Q_0^2$, the DGLAP equations [@Gribov:ri] describe well their scale evolution at large scales. Based on the global fits to the available data several different parametrizations of PDF have been obtained [@Martin:2001es; @Lai:1999wy; @Pumplin:2002vw]. The older PDF sets do not describe adequately the recent HERA data [@Adloff:2000qk] on the structure function $F_2$ at the perturbative scales $Q^2$ at small $x$. In the analysis of newer PDF sets, such as CTEQ6 [@Pumplin:2002vw] and MRST2001 [@Martin:2001es], these data have been taken into account. However, difficulties arise when fitting both small and large scale data simultaneously. In the MRST set, the entire H1 data set [@Adloff:2000qk] has been used in the analysis, leading to a good average fit at all scales, but at the expense of allowing for a negative NLO gluon distribution at small $x$ and $Q^2{\,{\buildrel < \over {_\sim}}\,}1$ GeV$^2$. In the CTEQ6 set only the large scale ($Q^2>4$ GeV$^2$) data have been included, giving a good fit at large $Q^2$, but leaving the fit at small-$x$ and small $Q^2$ ($Q^2<4$ GeV$^2$) region worse. Moreover, the gluon distribution at the values of small $x$ and $Q^2 {\,{\buildrel < \over {_\sim}}\,}1.69$ GeV$^2$ has been set to zero.
These problems are interesting as they can be signs of a new QCD phenomenon: at small values of momentum fraction $x$ and scales $Q^2$, gluon recombination terms, which lead to nonlinear corrections to the evolution equations, can become significant. First of these nonlinear terms have been calculated by Gribov, Levin and Ryskin [@Gribov:tu], and, Mueller and Qiu [@Mueller:wy]. In the following these correction terms shall be referred to as GLRMQ terms for short. With the modifications, the evolution equations become [@Mueller:wy] $$\begin{aligned}
\frac{\partial xg(x,Q^2) }{\partial \log Q^2}
&=& \frac{\partial xg(x,Q^2) }{\partial \log Q^2}\bigg|_{\rm DGLAP}
- \quad \frac{9\pi}{2} \frac{\alpha_s^2}{Q^2}
\int_x^1 \frac{dy}{y} y^2 G^{(2)}(y,Q^2), \label{gl-evol} \\
\frac{\partial x\bar{q}(x,Q^2)}{\partial \log Q^2} & = &
\frac{\partial x\bar{q}(x,Q^2)}{\partial \log Q^2}\bigg|_{\rm DGLAP}
- \quad \frac{3\pi}{20}\frac{\alpha_s^2}{Q^2}
x^2 G^{(2)}(x,Q^2)
+ \ldots G_{\rm HT}, \label{sea-evol}\end{aligned}$$ where the two-gluon density can be modelled as $ x^2G^{(2)}(x,Q^2)=
\frac{1}{\pi R^2}[xg(x,Q^2)]^2, $ with the radius of the proton $R=1$ fm. The higher dimensional gluon term $ G_{\rm HT}$ [@Mueller:wy] is here assumed to be zero. The effects of the nonlinear corrections to the DGLAP evolution of the PDF of the free proton were studied in [@Eskola:2002yc] in view of the recent H1 data; the results are discussed below.
The analysis
============
![[ $F_2(x,Q^2)$ calculated using CTEQ6L [@Pumplin:2002vw] (dotted-dashed) and the DGLAP+GLRMQ results with set 1 (solid) and set 2a (double dashed) [@Eskola:2002yc], compared with the H1 data [@Adloff:2000qk]. [**Right:**]{} The $Q^2$ dependence of the gluon distribution function at fixed $x$, from set 1 evolved with DGLAP+GLRMQ (solid), and directly from CTEQ6L (dotted-dashed).]{} []{data-label="F2_vs_cteq6"}](F2vsQ2-p6d.eps "fig:"){width="8cm"} ![[ $F_2(x,Q^2)$ calculated using CTEQ6L [@Pumplin:2002vw] (dotted-dashed) and the DGLAP+GLRMQ results with set 1 (solid) and set 2a (double dashed) [@Eskola:2002yc], compared with the H1 data [@Adloff:2000qk]. [**Right:**]{} The $Q^2$ dependence of the gluon distribution function at fixed $x$, from set 1 evolved with DGLAP+GLRMQ (solid), and directly from CTEQ6L (dotted-dashed).]{} []{data-label="F2_vs_cteq6"}](gluons.eps "fig:"){width="6.9cm"}
The goal of the analysis in [@Eskola:2002yc] was (1) to possibly improve the (LO) fit of the calculated $F_2(x,Q^2)$ to the H1 data [@Adloff:2000qk] at small $Q^2$, while (2) at the same time maintain the good fit at large $Q^2$, and finally (3) to study the interdependence between the initial distributions and the evolution.
In CTEQ6L a good fit to the H1 data is obtained (see Fig. \[F2\_vs\_cteq6\]) with a flat small-$x$ gluon distribution at $Q^2\sim 1.4$ GeV$^2$. As can be seen from Eqs. (\[gl-evol\]-\[sea-evol\]), the GLRMQ corrections slow down the scale evolution. Now one may ask whether the H1 data can be reproduced equally well with different initial conditions (i.e. assuming larger initial gluon distributions) and the GLRMQ corrections included in the evolution. This question has been studied in [@Eskola:2002yc] by generating three new sets of initial distributions using DGLAP + GLRMQ evolved CTEQ5 [@Lai:1999wy] and CTEQ6 distributions as guidelines. The initial scale was chosen to be $Q_0^2=1.4$ GeV$^2$, slightly below the smallest scale of the data points. The modified distributions at $Q_0^2$ were constructed piecewise from CTEQ5L and CTEQ6L distributions evolved down from $Q^2$ = 3 and 10 GeV$^2$ (CTEQ5L) and $Q^2$ = 5 GeV$^2$ (CTEQ6L). A power law form was used in the small-$x$ region to tune the initial distributions until a good agreement with the H1 data was found.
The difference between the three sets in [@Eskola:2002yc] is that in set 1 there is still a nonzero charm distribution at $Q_0^2=1.4$ GeV$^2$, which is slightly below the charm mass treshold, taken to be $m_c=1.3$ GeV in CTEQ6. In sets 2 the charm distribution has been removed at the initial scale and the resulting deficit in $F_2$ has been compensated by slightly increasing the other sea quarks at small $x$. Moreover, the effect of the charm was studied by using different mass tresholds: $m_{\rm c}=1.3$ GeV in set 2a whereas in set 2b it is $m_{\rm c}=\sqrt{1.4}$ GeV, i.e. charm begins to evolve immediately from the initial scale.
The results from the DGLAP+GLRMQ evolution with the new initial distributions are shown in Figs. \[F2\_vs\_cteq6\]. The left panel shows the comparison between the H1 data and the (LO) structure function $F_2(x,Q^2)=\sum_i e_i^2 x[q_i(x,Q^2)+\bar q_i(x,Q^2)]$ calculated from set 1 (solid lines), set 2a (double dashed) and the CTEQ6L parametrization (dotted-dashed lines). As can be seen, the results are very similar, which shows that with modified initial conditions and DGLAP+GLRMQ evolution, one obtains as good or even a better fit to the HERA data ($\chi/N = 1.13$, 1.17, 0.88 for the sets 1, 2a, 2b, correspondingly) as with the CTEQ6L distributions ($\chi/N
= 1.32$).
The evolution of the gluon distribution functions in the DGLAP+GLRMQ and DGLAP cases is illustrated more explicitly in the right panel of Fig. \[F2\_vs\_cteq6\], in which the absolute distributions for fixed $x$ are plotted as a function of $Q^2$ for set 1 and for CTEQ6L. The figure shows how the differences which are large at initial scale vanish during the evolution due to the GLRMQ effects. At scales $Q^2
{\,{\buildrel > \over {_\sim}}\,}4$ GeV$^2$ the GLRMQ corrections fade out rapidly and the DGLAP terms dominate the evolution.
Saturation
==========
The DGLAP+GLRMQ approach also offers a way to study the gluon saturation limits. For each $x$ in the small-$x$ region, the saturation scale $Q_{\rm sat}^2$ can be defined as the value of the scale $Q^2$ where the DGLAP and GLRMQ terms in the nonlinear evolution equation become equal, $\frac{\partial xg(x,Q^2)}{\partial \log
Q^2}|_{Q^2=Q_{\rm sat}^2(x)}=0$. The region of applicability of the DGLAP+GLRMQ is at $Q^2>Q^2_{\rm sat}(x)$ where the linear DGLAP part dominates the evolution. In the saturation region, at $Q^2<Q^2_{\rm
sat}(x)$, the GLRMQ terms dominate, and all nonlinear terms become important.
In order to find the saturation scales $Q^2_{\rm sat}(x)$ for the free proton, the obtained initial distributions (set 1) at $Q_0^2=1.4$ GeV$^2$ have to be evolved downwards in scale using the DGLAP+GLRMQ equations. As discussed in [@Eskola:2002yc], since only the first correction term has been taken into account, the gluon distribution near the saturation region should be considered as an upper limit. Consequently, the obtained saturation scale is an upper limit as well. The result is shown in Fig. \[satur\] (asterisks). The saturation line for the free proton from the geometric saturation model by Golec-Biernat and Wüsthoff (G-BW) [@Golec-Biernat:1998js] is also plotted (dashed line) for comparison. It is interesting to note that although the DGLAP+GLRMQ and G-BW approaches are very different, the slopes of the curves are very similar at the smallest values of $x$.
Saturation scales for nuclei can also be determined in a similar manner. For a nucleus $A$, the two-gluon density can be modelled as $x^2G^{(2)}(x,Q^2)= \frac{A}{\pi R_A^2}[xg(x,Q^2)]^2$, i.e. the effect of the correction is enhanced by a factor of $A^{1/3}$. Now a first estimate for the saturation limit can be obtained by starting the downwards evolution at high enough scales, $Q^2=100 \ldots 200$ GeV$^2$, where the GLRMQ terms are negligible. The result, which similarly to the proton case is an upper limit, is shown for Pb in Fig. \[satur\] (dots). The effect of the nuclear modifications was also studied by applying the EKS98 [@Eskola:1998df] parametrization at the high starting scale. As a result, the saturation scales $Q_{\rm sat}^2(x)$ are somewhat reduced, as shown in Fig. \[satur\] (crosses). The saturation limit obtained for a Pb nucleus by Armesto in a Glauberized geometric saturation model [@Armesto:2002ny] is shown (dotted-dashed) for comparison. Again, despite of the differences between the approaches, the slopes of the curves are strikingly similar.
For further studies and for more accurate estimates of $Q_{\rm
sat}^2(x)$ in the DGLAP+GLRMQ approach, a full global fit analysis for the nuclear parton distribution functions should be performed, along the same lines as in EKRS [@Eskola:1998iy; @Eskola:1998df] and in HKM [@Hirai:2001np].
![[The gluon saturation limits in the DGLAP+GLRMQ approach for proton (asterisks) and Pb ($A=208$), with (crosses) and without (dots) nuclear modifications [@Eskola:2002yc]. The saturation line for the proton from the geometric saturation model [@Golec-Biernat:1998js] (dashed line), and for Pb from [@Armesto:2002ny] (dotted-dashed) are also plotted. ]{}[]{data-label="satur"}](pbsat2bw.eps){width="7.5cm"}
[**Acknowledgements.**]{} We thank N. Armesto, P.V. Ruuskanen, I. Vitev and other participants of the CERN Hard Probes workshop for discussions. We are grateful for the Academy of Finland, Project 50338, for financial support. J.W.Q. is supported in part by the United States Department of Energy under Grant No. DE-FG02-87ER40371. C.A.S. is supported by a Marie Curie Fellowship of the European Community programme TMR (Training and Mobility of Researchers), under the contract number HPMF-CT-2000-01025.
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[^1]: Contribution to CERN Yellow Report on Hard Probes in Heavy Ion Collisions at the LHC.
|
---
abstract: |
Drawdown (resp. drawup) of a stochastic process, also referred as the reflected process at its supremum (resp. infimum), has wide applications in many areas including financial risk management, actuarial mathematics and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (resp. drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a Lévy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to [study various drawdown quantities]{} for the general class of time-homogeneous Markov processes.
*Keywords*: Drawdown; Integral equation; Reflected process; Time-homogeneous Markov process
*MSC*(2000): Primary 60G07; Secondary 60G40
author:
- 'David Landriault[^1]'
- 'Bin Li[^2]'
- 'Hongzhong Zhang[^3]'
title: ' Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes'
---
15.5pt
Introduction
============
We consider a time-homogeneous, real-valued, non-explosive, [càdlàg]{} Markov process $X=(X_{t})_{t\geq0}$ with state space $\mathbb{R}$ [^4] defined on a filtered probability space $(\Omega,\mathcal{F},\boldsymbol{F}=(\mathcal{F}_{t})_{t\geq
0},\mathbb{P})$ [Throughout, we silently assume that $X$ satisfies the strong Markov property (see Section III.8,9 of Rogers and Williams [@RW00]), and exclude Markov processes with monotone paths.]{} The first passage time of $X$ above (below) a level $x\in\mathbb{R}
$ is denoted by $$\textcolor[rgb]{0.0,0.0,0.0}{T_{x}^{+(-)}=\inf\left\{ t\geq0:X_{t}>(<)x\right\},}$$ with the common convention that $\inf\emptyset=\infty$. The drawdown process of $X$ (also known as the reflected process of $X$ at its supremum) is denoted by $Y=(Y_{t})_{t\geq0}$ with $Y_{t}=M_{t}-X_{t},$ where $M_{t}=\sup_{0\leq s\leq t}X_{t}$. Let $\tau_{a}=\inf\{t>0:Y_{t}>a\}$ be the first time the magnitude of drawdowns exceeds a given threshold $a>0$. Note that $\left( \textcolor[rgb]{0.0,0.0,0.0}{\sup_{0\leq s\leq t}}Y_{s}>a\right) =\left( \tau_{a}\leq t\right) $ $\mathbb{P}$-a.s. Hence, the distributional study of the maximum drawdown of $X$ is equivalent to the study of the stopping time $\tau_{a}$. Similarly, the drawup process of $X$ is defined as $\hat{Y}_{t}=X_{t}-m_{t}$ for $t\geq0,$ where $m_{t}=\inf_{0\leq
s\leq t}X_{t}$. However, given that the drawup of $X$ can be investigated via the drawdown of $-X$, we exclusively focus on the drawdown process $Y$ in this paper.
Applications of drawdowns can be found in many areas. For instance, drawdowns are widely used by mutual funds and commodity trading advisers to quantify downside risks. Interested readers are referred to Schuhmacher and Eling [@SE11] for a review of drawdown-based performance measures. An extensive body of literature exists on the assessment and mitigation of drawdown risks (e.g., Grossman and Zhou [@GZ93], Carr et al. [@CZH11], Cherny and Obloj [@CO13], and Zhang et al. [@ZLH13]). Drawdowns are also closely related to many problems in mathematical finance, actuarial science and statistics such as the pricing of Russian options (e.g., Shepp and Shiryaev [@SS93], Asmussen et al. [@AAP04] and Avram et al. [@AKP04]), [De Finetti’s]{} dividend problem (e.g., Avram et al. [@APP07] and Loeffen [@L08]), loss-carry-forward taxation models (e.g., Kyprianou and Zhou [@KZ09] and Li et al. [@LTZ13]), and change-point detection methods (e.g., Poor and Hadjiliadis [@PH09]). [More specifically, in De Finetti’s dividend problem under a fixed dividend barrier $a>0$, the underlying surplus process with dividend payments is a process obtained from reflecting $X$ at a fixed barrier $a$ (the reflected process’ dynamics may be different than the drawdown process $Y$ when the underlying process $X$ is not spatial homogeneous). However, the distributional study of ruin quantities in De Finetti’s dividend problem can be transformed to the study of drawdown quantities for the underlying surplus process; see Kyprianou and Palmowski [@KP07] for a more detailed discussion. Similarly, ruin problems in loss-carry-forward taxation models can also be transformed to a generalized drawdown problem for classical models without taxation, where the generalized drawdown process is defined in the form of $Y_t=\gamma(M_t)-X_t$ for some measurable function $\gamma(\cdot)$.]{}
The distributional study of drawdown quantities is not only of theoretical interest, but also plays a fundamental role in the aforementioned applications. Early distributional studies on drawdowns date back to Taylor [@T75] on the joint Laplace transform of $\tau_{a}$ and $M_{\tau_{a}}$ for Brownian motions. This result was later generalized by Lehoczky [@L77] to time-homogeneous diffusion processes. Douady et al. [@DSY00] and Magdon et al. [@MAPA04] derived infinite series expansions for the distribution of $\tau_{a}$ for a standard Brownian motion and a drifted Brownian motion, respectively. For spectrally negative Lévy processes, Mijatovic and Pistorius [@MP12] obtained a sextuple formula for the joint Laplace transform of $\tau_{a}$ and the last reset time of the maximum prior to $\tau_{a}$, together with the joint distribution of the running maximum, the running minimum, and the overshoot of $Y$ at $\tau_{a}$. For some studies on the joint law of drawdown and drawup of spectrally negative Lévy processes or diffusion processes, please refer to Pistorius [@P04], Pospisil et al. [@PVH09], Zhang and Hadjiliadis [@ZH10], and Zhang [@Z15].
As mentioned above, Lévy processes[^5] and time-homogeneous diffusion processes are two main classes of Markov processes for which various drawdown problems have been extensively studied. The treatment of these two classes of Markov processes has typically been considered distinctly in the literature. For Lévy processes, Itô’s excursion theory is a powerful approach to handle drawdown problems (e.g., Avram et al. [@AKP04], Pistorius [@P04], and Mijatovic and Pistorius [@MP12]). However, the excursion-theoretic approach is somewhat specific to the underlying model, and additional care is required when a more general class of Markov processes is considered. On the other hand, for time-homogeneous diffusion processes, Lehoczky [@L77] introduced an ingenious approach which has recently been generalized by many researchers (e.g., Zhou [@Z07], Li et al. [@LTZ13], and Zhang [@Z15]). Here again, Lehoczky’s approach relies on the continuity of the sample path of the underlying model, and hence is not applicable for processes with upward jumps. Also, other general methodologies (such as the martingale approach in, e.g., Asmussen [@AAP04] and the occupation density approach in, e.g., Ivanovs and Palmowski [@IP12]) are well documented in the literature but they strongly depend on the specific structure of the underlying process. To the best of our knowledge, no unified treatment of drawdowns (drawups) for general Markov processes has been proposed in the literature.
In this paper, we propose a general and unified approach to study the joint law of $(\tau_{a},M_{\tau_{a}},Y_{\tau_{a}})$ for time-homogeneous Markov processes with possibly two-sided jumps. Under mild regularity conditions, the joint law is expressed as the solution to an integral equation which involves two-sided exit quantities of the underlying process $X$. The uniqueness of the integral equation for the joint law is also investigated. In particular, the joint law possesses explicit forms when $X$ has only one-sided jumps or is a Lévy process (possibly with two-sided jumps). In general, our main result reduces the drawdown problem to fundamental two-sided exit quantities.
The main idea of our proposed approach is briefly summarized below. By analyzing the evolution of sample paths over a short time period following time $0$ and using renewal arguments, we first establish tight upper and lower bounds for the joint law of $(\tau_{a},M_{\tau_{a}},Y_{\tau_{a}})$ in terms of the two-sided exit quantities. Then, under mild regularity conditions, we use a Fatou’s lemma with varying measures to show that the upper and lower bounds converge when the length of the time interval approaches $0$. This leads to an integro-differential equation satisfied by the desired joint law. Finally, we reduce the integro-differential equation to an integral equation. When $X$ is a spectrally negative Markov process or a general Lévy process, the integral equation can be solved and the joint law of $(\tau_{a},M_{\tau_{a}},Y_{\tau_{a}})$ is hence explicitly expressed in terms of two-sided exit quantities.
The rest of the paper is organized as follows. In Section 2, we introduce some fundamental two-sided exit quantities and present several preliminary results. In Section 3, we derive the joint law of $(\tau_{a},Y_{\tau_{a}},M_{\tau_{a}})$ for general time-homogeneous Markov processes. Several Markov processes for which the proposed regularity conditions are met are further discussed. [Some numerical examples are investigated in more detail in Section 4]{}. Some technical proofs are postponed to Appendix.
Preliminary
===========
For ease of notation, we adopt the following conventions throughout the paper. We denote by $\mathbb{P}_{x}$ the law of $X$ given $X_{0}=x\in\mathbb{R}
$ and write $\mathbb{P}\equiv\mathbb{P}_{0}$ for brevity. We write $u\wedge
v=\min\{u,v\}$, $\mathbb{R}
_{+}=[0,\infty)$, and $\int_{x}^{y}\cdot\mathrm{d}z$ for an integral on the open interval $z\in(x,y)$.
For $q,s\geq0$, $u\leq x\leq v$ and $z>0$, we introduce the following two-sided exit quantities of $X$: $$\begin{aligned}
B_{1}^{(q)}(x;u,v) & :=\mathbb{E}_{x}\left[ e^{-qT_{v}^{+}}1_{\left\{T_{v}^{+}<\infty,
T_{v}^{+}<T_{u}^{-},X_{T_{v}^{+}}=v\right\} }\right] ,\\
B_{2}^{(q)}(x,\mathrm{d}z;u,v) & :=\mathbb{E}_{x}\left[ e^{-qT_{v}^{+}}1_{\left\{T_{v}^{+}<\infty, T_{v}^{+}<T_{u}^{-},X_{T_{v}^{+}}-v\in\mathrm{d}z\right\}
}\right] ,\\
C^{(q,s)}(x;u,v) & :=\mathbb{E}_{x}\left[ e^{-qT_{u}^{-}-s(u-X_{T_{u}^{-}})}1_{\left\{ T_{u}^{-}<\infty, T_{u}^{-}<T_{v}^{+}\right\} }\right] .\end{aligned}$$ We also define the joint Laplace transform $$B^{(q,s)}(x;u,v):=\mathbb{E}_{x}\left[ e^{-qT_{v}^{+}-s(X_{T_{v}^{+}}-v)}1_{\left\{ T_{v}^{+}<\infty, T_{v}^{+}<T_{u}^{-}\right\} }\right] =B_{1}^{(q)}(x;u,v)+B_{2}^{(q,s)}(x;u,v), \label{BBB}$$ where $B_{2}^{(q,s)}(x;u,v):=\int_{0}^{\infty}e^{-sz}B_{2}^{(q)}(x,\mathrm{d}z;u,v)$.
The following pathwise inequalities are central to the construction of tight bounds for the joint law of the triplet $(\tau_{a},M_{\tau_{a}},Y_{\tau_{a}})$.
\[prop path\]For $q,s\geq0$, $x\in\mathbb{R}$ and $\varepsilon\in(0,a)$, we have $\mathbb{P}_{x}$-a.s. $$1_{\{T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon}^{+}<T_{x+\varepsilon-a}^{-}\}}\leq1_{\{T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon
}^{+}<\tau_{a}\}}\leq1_{\{T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon}^{+}<T_{x-a}^{-}\}}, \label{eq.up}$$ and $$\begin{aligned}
e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\left\{\tau_{a}<\infty, \tau_{a}<T_{x+\varepsilon}^{+}\right\} } & \geq e^{-qT_{x-a}^{-}-s(x-a-X_{T_{x-a}^{-}})-s\varepsilon
}1_{\{T_{x-a}^{-}<\infty, T_{x-a}^{-}<T_{x+\varepsilon}^{+}\}},\label{eq.down1}\\
e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\left\{ \tau_{a}<\infty, \tau_{a}<T_{x+\varepsilon}^{+}\right\} } & \leq e^{-qT_{x+\varepsilon-a}^{-}-s(x-a-X_{T_{x+\varepsilon-a}^{-}})}1_{\{T_{x+\varepsilon-a}^{-}<\infty, T_{x+\varepsilon-a}^{-}<T_{x+\varepsilon}^{+}\}}. \label{eq.down2}$$
By analyzing the sample paths of $X$, it is easy to see that, for any path $\omega\in(T_{x+\varepsilon}^{+}<\infty)$, we have ${\mathbb{P}}_x\{\tau_a\le T_{x-a}^-\}=1$, so $$(T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon}^{+}<\tau_{a})=(T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon
}^{+}<\tau_a\le T_{x-a}^{-})\subset(T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon}^{+}<T_{x-a}^-)\quad\mathbb{P}_{x}\text{-a.s.}\nonumber
\label{w1}$$ and similarly, ${\mathbb{P}}_x$-a.s.$$(T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon}^{+}<T_{x+\varepsilon-a}^-)=(T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon
}^{+}< T_{x+\varepsilon-a}^{-}, T_{x+\varepsilon
}^{+}< \tau_a)\subset(T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon}^{+}<\tau_a),\nonumber$$ which immediately implies (\[eq.up\]). On the other hand, by using the same argument, we have $$(T_{x-a}^{-}<\infty, T_{x-a}^{-}<T_{x+\varepsilon}^{+})=(T_{x-a}^{-}<\infty, \tau_{a}\leq T_{x-a}^{-}<T_{x+\varepsilon
}^{+})\subset(\tau_{a}<\infty, \tau_{a}<T_{x+\varepsilon}^{+})\quad\mathbb{P}_{x}\text{-a.s.}
\label{w1}$$ and$$(\tau_{a}<\infty, \tau_{a}<T_{x+\varepsilon}^{+})=(\tau_{a}<\infty, T_{x+\varepsilon-a}^{-}\leq\tau
_{a}<T_{x+\varepsilon}^{+})\subset(T_{x+\varepsilon-a}^{-}<\infty, T_{x+\varepsilon-a}^{-}<T_{x+\varepsilon
}^{+})\quad\mathbb{P}_{x}\text{-a.s.} \label{w2}$$ For any path $\omega\in(T_{x-a}^{-}<\infty, T_{x-a}^{-}<T_{x+\varepsilon}^{+})$, we know from (\[w1\]) that $\omega\in(T_{x-a}^{-}<\infty, \tau_{a}\leq T_{x-a}^{-}<T_{x+\varepsilon}^{+})$. This implies $M_{\tau_{a}}(\omega)\leq x+\varepsilon$ and $X_{\tau
_{a}}(\omega)\geq X_{T_{x-a}^{-}}(\omega)$, which further entails that $Y_{\tau_{a}}(\omega)=M_{\tau_{a}}(\omega)-X_{\tau_{a}}(\omega)\leq
x+\varepsilon-X_{T_{x-a}^{-}}(\omega)$. Therefore, by the above analysis and the second inequality of (\[eq.up\]), $$e^{-qT_{x-a}^{-}-s(x+\varepsilon-X_{T_{x-a}^{-}})}1_{\left\{ T_{x-a}^{-}<\infty, T_{x-a}^{-}<T_{x+\varepsilon}^{+}\right\} }\leq e^{-q\tau_{a}-sY_{\tau_{a}}}1_{\left\{\tau_{a}<\infty, \tau_{a}<T_{x+\varepsilon}^{+}\right\} }\quad\mathbb{P}_{x}\text{-a.s.}$$ which naturally leads to (\[eq.down1\]).
Similarly, for any sample path $\omega\in(\tau_{a}<\infty, \tau_{a}<T_{x+\varepsilon}^{+})$, we know from (\[w2\]) that $\omega\in(\tau_{a}<\infty, T_{x+\varepsilon-a}^{-}\leq\tau
_{a}<T_{x+\varepsilon}^{+})$, which implies that $x-X_{T_{x+\varepsilon
-a}^{-}}(\omega)\leq Y_{T_{x+\varepsilon-a}^{-}}(\omega)\leq Y_{\tau_{a}}(\omega).$ Therefore, by the first inequality of (\[eq.up\]), we obtain $$e^{-q\tau_{a}-sY_{\tau_{a}}}1_{\left\{ \tau_{a}<\infty, \tau_{a}<T_{x+\varepsilon}^{+}\right\} }\leq e^{-qT_{x+\varepsilon-a}^{-}-s(x-X_{T_{x+\varepsilon
-a}^{-}})}1_{\{T_{x+\varepsilon-a}^{-}<\infty, T_{x+\varepsilon-a}^{-}<T_{x+\varepsilon}^{+}\}}\quad
\mathbb{P}_{x}\text{-a.s.}$$ This implies the second inequality of (\[eq.down2\]).
By Proposition \[prop path\], we easily obtain the following useful estimates.
\[cor bd\]For $q,s\geq0$, $x\in\mathbb{R}
,z>0$ and $\varepsilon\in(0,a)$,$$\begin{aligned}
B_{1}^{(q)}(x;x+\varepsilon-a,x+\varepsilon) & \leq\mathbb{E}_{x}\left[
e^{-qT_{x+\varepsilon}^{+}}1_{\{T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon}^{+}<\tau_{a},X_{T_{x+\varepsilon}^{+}}=x+\varepsilon\}}\right] \leq B_{1}^{(q)}(x;x-a,x+\varepsilon),\\
B_{2}^{(q)}(x,\mathrm{d}z;x+\varepsilon-a,x+\varepsilon) & \leq
\mathbb{E}_{x}\left[ e^{-qT_{x+\varepsilon}^{+}}1_{\{T_{x+\varepsilon}^{+}<\infty, T_{x+\varepsilon}^{+}<\tau_{a},X_{T_{x+\varepsilon}^{+}}-x-\varepsilon\in\mathrm{d}z\}}\right]
\leq B_{2}^{(q)}(x,\mathrm{d}z;x-a,x+\varepsilon),\end{aligned}$$ and$$e^{-s\varepsilon}C^{(q,s)}(x;x-a,x+\varepsilon)\leq\mathbb{E}_{x}\left[
e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\left\{\tau_{a}<\infty, \tau_{a}<T_{x+\varepsilon}^{+}\right\} }\right] \leq e^{s\varepsilon}C^{(q,s)}(x;x+\varepsilon
-a,x+\varepsilon).$$
It is not difficult to check that the results of Proposition \[prop path\] and Corollary \[cor bd\] still hold if the first passage times and the drawdown times are only observed discretely or randomly (such as the Poisson observation framework in Albrecher et al. [@AIZ16] for the latter). Further, explicit relationship between Poisson observed first passage times and Poisson observed drawdown times (similar as for Theorem \[thm markov\] below) can be found by exploiting the same approach as laid out in this paper.
The later analysis involves the weak convergence of measures which is recalled here. Consider a metric space $S$ with the Borel $\sigma$-algebra on it. We say a sequence of finite measures $\{\mu_{n}\}_{n\in\mathbb{N}
}$ is weakly convergent to a finite measure $\mu$ as $n\rightarrow\infty$ if $$\lim_{n\rightarrow\infty}\int_{S}\phi(z)\mathrm{d}\mu_{n}(z)=\int_{S}\phi(z)\mathrm{d}\mu(z),$$ for any bounded and continuous function $\phi(\cdot)$ on $S$.
In the next lemma, we show some forms of Fatou’s lemma for varying measures under weak convergence. Similar results are proved in Feinberg et al. [@FKZ14] for probability measures. For completeness, a proof for general finite measures is provided in Appendix.
\[lem fatou\]Suppose that $\{\mu_{n}\}_{n\in\mathbb{N}
}$ is a sequence of finite measures on $S$ which is weakly convergent to a finite measure $\mu$, and $\{\phi_{n}\}_{n\in\mathbb{N}
}$ is a sequence of uniformly bounded and nonnegative functions on $S$. Then,$$\int_{S}\liminf_{n\rightarrow\infty,w\rightarrow z}\phi_{n}(w)\mathrm{d}\mu(z)\leq\liminf_{n\rightarrow\infty}\int_{S}\phi_{n}(z)\mathrm{d}\mu
_{n}(z)\text{,} \label{inf}$$ and $$\int_{S}\limsup_{n\rightarrow\infty,w\rightarrow z}\phi_{n}(w)\mathrm{d}\mu(z)\geq\limsup_{n\rightarrow\infty}\int_{S}\phi_{n}(z)\mathrm{d}\mu_{n}(z).
\label{sup}$$
Main results
============
In this section, we study the joint law of $(\tau_{a},M_{\tau_{a}},Y_{\tau
_{a}})$ for a general Markov process with possibly two-sided jumps. The following assumptions on the two-sided exit quantities of $X$ are assumed to hold, which are sufficient (but not necessary) conditions for the applicability of our proposed methodology. Weaker assumptions might be assumed for special Markov processes; see, for instance, Remark \[rk levy\] and Corollary \[cor snm\] below.
For all $q,s\geq0$, $z>0$ and $x>X_{0}$, we assume the following limits exist and identities hold: $$\begin{aligned}
\text{\textbf{(A1)} }b_{a,1}^{(q)}(x) & :=\lim_{\varepsilon\downarrow0}\frac{1-B_{1}^{(q)}(x;x-a,x+\varepsilon)}{\varepsilon}=\lim_{\varepsilon
\downarrow0}\frac{1-B_{1}^{(q)}(x;x+\varepsilon-a,x+\varepsilon)}{\varepsilon
}\\
& =\lim_{\varepsilon\downarrow0}\frac{1-B_{1}^{(q)}(x-\varepsilon
;x-a,x)}{\varepsilon}=\lim_{\varepsilon\downarrow0}\frac{1-B_{1}^{(q)}(x-\varepsilon;x-\varepsilon-a,x)}{\varepsilon},\end{aligned}$$ and $\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w<\infty$ for any $x,y\in\mathbb{R}
$;$$\begin{aligned}
\text{\textbf{(A2)} }b_{a,2}^{(q,s)}(x) & :=\lim_{\varepsilon\downarrow
0}\frac{1}{\varepsilon}B_{2}^{(q,s)}(x;x-a,x+\varepsilon)=\lim_{\varepsilon
\downarrow0}\frac{1}{\varepsilon}B_{2}^{(q,s)}(x;x+\varepsilon-a,x+\varepsilon
)\\
& =\lim_{\varepsilon\downarrow0}\frac{1}{\varepsilon}B_{2}^{(q,s)}(x-\varepsilon;x-a,x)=\lim_{\varepsilon\downarrow0}\frac{1}{\varepsilon}B_{2}^{(q,s)}(x-\varepsilon;x-\varepsilon-a,x),\end{aligned}$$ and $s\longmapsto b_{a,2}^{(q,s)}(x)$ is right continuous at $s=0$;$$\begin{aligned}
\text{\textbf{(A3)} }c_{a}^{(q,s)}(x) & :=\lim_{\varepsilon\downarrow0}\frac{C^{(q,s)}(x;x-a,x+\varepsilon)}{\varepsilon}=\lim_{\varepsilon
\downarrow0}\frac{C^{(q,s)}(x;x+\varepsilon-a,x+\varepsilon)}{\varepsilon}\\
& =\lim_{\varepsilon\downarrow0}\frac{C^{(q,s)}(x-\varepsilon;x-a,x)}{\varepsilon}=\lim_{\varepsilon\downarrow0}\frac{C^{(q,s)}(x-\varepsilon
;x-\varepsilon-a,x)}{\varepsilon}.\end{aligned}$$
Under Assumptions (**A1**) and (**A2**), it follows from (\[BBB\]) that $$b_{a}^{(q,s)}(x):=\lim_{\varepsilon\downarrow0}\frac{1-B^{(q,s)}(x;x-a,x+\varepsilon)}{\varepsilon}=b_{a,1}^{(q)}(x)-b_{a,2}^{(q,s)}(x).
\label{bbb}$$
\[rmk31\] Due to the general structure of $X$, it is difficult to refine Assumptions **(A1)**-**(A3)** unless a specific structure for $X$ is given. [A necessary condition for Assumptions **(A1)**-**(A3**) to hold is that, $$T_{x}^{+}=0\text{ and }X_{T_{x}^{+}}=x,\text{ }\mathbb{P}_{x}\text{-a.s. for
all }x\in\mathbb{R}\text{.}$$ In other words, $X$ must be upward regular and creeping upward at every $x$.]{}[^6] In the later part of this section, we provide some examples of Markov processes which satisfy Assumptions **(A1)**-**(A3)**, including spectrally negative Lévy processes, linear diffusions, piecewise exponential Markov processes, and jump diffusions.
\[rk weak\]By Theorem 5.22 of Kallenberg [@K02] or Proposition 7.1 of Landriault et al. [@LLZ16], we know that Assumption (**A2**) implies that the measures $\frac{1}{\varepsilon}B_{2}^{(q)}(x,\mathrm{d}z;x-a,x+\varepsilon)$, $\frac{1}{\varepsilon}B_{2}^{(q)}(x,\mathrm{d}z;x+\varepsilon-a,x+\varepsilon)$, $\frac{1}{\varepsilon
}B_{2}^{(q)}(x-\varepsilon,\mathrm{d}z;x-a,x)$ and $\frac{1}{\varepsilon}B_{2}^{(q)}(x-\varepsilon,\mathrm{d}z;x-\varepsilon-a,x)$ weakly converge to the same measure on $\mathbb{R}
_{+}$, denoted as $b_{a,2}^{(q)}(x,\mathrm{d}z)$, such that $\int_{\mathbb{R}
_{+}}e^{-sz}b_{a,2}^{(q)}(x,\mathrm{d}z)=b_{a,2}^{(q,s)}(x)$. We point out that it is possible that $b_{a,2}^{(q)}(x,\{0\})>0$, though the measure $B_{2}^{(q)}(x,\mathrm{d}z;u,v)$ is only defined on $z\in(0,\infty)$.
We are now ready to present the main result of this paper related to the joint law of $(\tau_{a},Y_{\tau_{a}},M_{\tau_{a}})$.
\[thm markov\]Consider a general time-homogeneous Markov process $X$ satisfying Assumptions (**A1**)-(**A3**). For $q,s\geq0$ and $K\in\mathbb{R}$, let $$h(x)=\mathbb{E}_{x}\left[ e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\{\tau_a<\infty, M_{\tau_{a}}\leq K\}}\right] ,\quad x\leq K.$$ Then $h(\cdot)$ is differentiable in $x<K$ and solves the following integral equation $$h(x)=\int_{x}^{K}e^{-\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w}\left(
c_{a}^{(q,s)}(y)+\int_{[0,K-y)}h(y+z)b_{a,2}^{(q)}(y,\mathrm{d}z)\right)
\mathrm{d}y\text{,}\quad x\leq K. \label{triple LT}$$
By the strong Markov property of $X$, for any $X_{0}=x\leq y<K$ and $0<\varepsilon<(K-y)\wedge a$, we have $$\begin{aligned}
h(y) & =\mathbb{E}_{y}\left[ e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\left\{\tau_{a}<\infty,
\tau_{a}<T_{y+\varepsilon}^{+}\right\} }\right] +\mathbb{E}_{y}\left[
e^{-qT_{y+\varepsilon}^{+}}1_{\{T_{y+\varepsilon}^{+}<\infty, T_{y+\varepsilon}^{+}<\tau_{a},X_{T_{y+\varepsilon}^{+}}=y+\varepsilon\}}\right] h(y+\varepsilon)\\
& +\int_{0}^{K-y-\varepsilon}\mathbb{E}_{y}\left[ e^{-qT_{y+\varepsilon}^{+}}1_{\{T_{y+\varepsilon}^{+}<\infty, T_{y+\varepsilon}^{+}<\tau_{a},X_{T_{y+\varepsilon}^{+}}-y-\varepsilon\in\mathrm{d}z\}}\right] h(y+\varepsilon+z).\end{aligned}$$ By Corollary \[cor bd\], it follows that$$\begin{aligned}
h(y+\varepsilon)-h(y) & \geq-e^{s\varepsilon}C^{(q,s)}(y;y+\varepsilon
-a,y+\varepsilon)+\left( 1-B_{1}^{(q)}(y;y-a,y+\varepsilon)\right)
h(y+\varepsilon)\nonumber\\
& -\int_{0}^{K-y-\varepsilon}h(y+\varepsilon+z)B_{2}^{(q)}(y,\mathrm{d}z;y-a,y+\varepsilon), \label{down}$$ and$$\begin{aligned}
h(y+\varepsilon)-h(y) & \leq-e^{-s\varepsilon}C^{(q,s)}(y;y-a,y+\varepsilon
)+\left( 1-B_{1}^{(q)}(y;y+\varepsilon-a,y+\varepsilon)\right)
h(y+\varepsilon)\nonumber\\
& -\int_{0}^{K-y-\varepsilon}h(y+\varepsilon+z)B_{2}^{(q)}(y,\mathrm{d}z;y+\varepsilon-a,y+\varepsilon). \label{up}$$ By Assumptions (**A1**)-(**A3**) and $h(\cdot)\in\lbrack0,1]$, it is clear that both the lower bound of $h(y+\varepsilon)-h(y)$ in (\[down\]) and the upper bound in (\[up\]) vanish as $\varepsilon\downarrow0$. Hence, $h(y)$ is right continuous for $y\in\lbrack x,K)$. Replacing $y$ by $y-\varepsilon$ in (\[down\]) and (\[up\]), and using Assumptions (**A1**)-(**A3**) again, it follows that $h(y)$ is also left continuous for $y\in(x,K]$ with $h(K)=0$. Therefore, $h(y)$ is continuous for $y\in\lbrack x,K]$ (left continuous at $x$ and right continuous at $K$).
To consecutively show the differentiability, we divide inequalities (\[down\]) and (\[up\]) by $\varepsilon$. It follows from Assumptions (**A1**)-(**A3**), Remark \[rk weak\], Lemma \[lem fatou\] and the continuity of $h$ that $$\begin{aligned}
& \liminf_{\varepsilon\downarrow0}\frac{h(y+\varepsilon)-h(y)}{\varepsilon}\\
& \geq-c_{a}^{(q,s)}(y)+b_{a,1}^{(q)}(y)h(y)-\limsup_{\varepsilon\downarrow
0}\int_{0}^{K-y-\varepsilon}h(y+\varepsilon+z)\frac{B_{2}^{(q)}(y,\mathrm{d}z;y-a,y+\varepsilon)}{\varepsilon}\\
& \geq-c_{a}^{(q,s)}(y)+b_{a,1}^{(q)}(y)h(y)-\int_{[0,K-y)}h(y+z)b_{a,2}^{(q)}(y,\mathrm{d}z)\text{,}$$ and similarly, $$\limsup_{\varepsilon\downarrow0}\frac{h(y+\varepsilon)-h(y)}{\varepsilon}\leq-c_{a}^{(q,s)}(y)+b_{a,1}^{(q)}(y)h(y)-\int_{[0,K-y)}h(y+z)b_{a,2}^{(q)}(y,\mathrm{d}z).$$ Since the two limits coincide, one concludes that $h(y)$ is right differentiable for $y\in(x,K)$. Moreover, by replacing $y$ by $y-\varepsilon$ in (\[down\]) and (\[up\]), and using similar arguments, we can show that $h(y)$ is also left differentiable for $y\in(x,K)$. Since the left and right derivatives coincide, we conclude that $h(y)$ is differentiable for any $y\in(x,K)$ and solves the following ordinary integro-differential equation (OIDE),$$h^{\prime}(y)-b_{a,1}^{(q)}(y)h(y)=-c_{a}^{(q,s)}(y)-\int_{[0,K-y)}h(y+z)b_{a,2}^{(q)}(y,\mathrm{d}z). \label{h'}$$
Multiplying both sides of (\[h’\]) by $e^{-\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w}$, integrating the resulting equation (with respect to $y$) from $x$ to $K$, and using $h(K)=0$, this completes the proof of Theorem \[thm markov\].
When the Markov process $X$ is spectrally negative (i.e., with no upward jumps), the upward overshooting density $b_{a,2}^{(q)}(x,\mathrm{d}z)$ is trivially $0$. Theorem \[thm markov\] reduces to the following corollary.
\[cor snm\]Consider a spectrally negative time-homogeneous Markov process $X$ satisfying Assumptions (**A1**) and (**A3**). For $q,s\geq0$ and $K>0$, we have$$\mathbb{E}_{x}\left[ e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\{\tau_{a}<\infty, M_{\tau_{a}}\leq
K\}}\right] =\int_{x}^{K}e^{-\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w}c_{a}^{(q,s)}(y)\mathrm{d}y\text{,}\quad x\leq K.$$
When $X$ is a general Lévy process (possibly with two-sided jumps), we have the following result for the joint Laplace transform of the triplet $(\tau_{a},Y_{\tau_{a}},M_{\tau_{a}})$. Note that Corollary \[cor levy\] should be compared to Theorem 4.1 of Baurdoux [@B09], in which, under the Lévy framework, the resolvent density of $Y$ is expressed in terms of the resolvent density of $X$ using excursion theory.
\[cor levy\]Consider a Lévy process $X$ satisfying Assumptions (**A1**)-(**A3**). For $q,s,\delta\geq0$, we have[^7] $$\mathbb{E}\left[ e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)-\delta M_{\tau_{a}}}\right]
=\frac{c_{a}^{(q,s)}(0)}{\delta+b_{a}^{(q,\delta)}(0)}. \label{levy}$$
By the spatial homogeneity of the Lévy process $X$, Eq. (\[triple LT\]) at $x=0$ reduces to $$h(0)=\frac{c_{a}^{(q,s)}(0)}{b_{a,1}^{(q)}(0)}\left( 1-e^{-b_{a,1}^{(q)}(0)K}\right) +\int_{0}^{K}e^{-b_{a,1}^{(q)}(0)y}\int_{[0,K-y)}h(y+z)b_{a,2}^{(q)}(0,\mathrm{d}z)\mathrm{d}y. \label{h}$$ Let $$\hat{h}(0):=\mathbb{E}\left[ e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)-\delta
M_{\tau_{a}}}\right] =\mathbb{E}\left[ e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\{M_{\tau_{a}}\leq e_{\delta}\}}\right] \text{,}$$ where $e_{\delta}$ is an independent exponential random variable with finite mean $1/\delta>0$. Multiplying both sides of (\[h\]) by $\delta e^{-\delta
K}$, integrating the resulting equation (with respect to $K$) from $0$ to $\infty$, and using integration by parts, one obtains $$\begin{aligned}
\hat{h}(0) & =\frac{c_{a}^{(q,s)}(0)}{\delta+b_{a,1}^{(q)}(0)}+\int
_{0}^{\infty}\delta e^{-\delta K}\int_{0}^{K}e^{-b_{a,1}^{(q)}(0)y}\int_{[0,K-y)}h(y+z)b_{a,2}^{(q)}(0,\mathrm{d}z)\mathrm{d}y\mathrm{d}K\\
& =\frac{c_{a}^{(q,s)}(0)}{\delta+b_{a,1}^{(q)}(0)}+\int_{0}^{\infty
}e^{-b_{a,1}^{(q)}(0)y}\mathrm{d}y\int_{\mathbb{R}
_{+}}b_{a,2}^{(q)}(0,\mathrm{d}z)\int_{z+y}^{\infty}\delta e^{-\delta
K}\mathbb{E}\left[ e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\{M_{\tau_{a}}\leq
K-y-z\}}\right] \mathrm{d}K\\
& =\frac{c_{a}^{(q,s)}(0)}{\delta+b_{a,1}^{(q)}(0)}+\hat{h}(0)\frac{\int_{\mathbb{R}
_{+}}e^{-\delta z}b_{a,2}^{(q)}(0,\mathrm{d}z)}{\delta+b_{a,1}^{(q)}(0)}.\end{aligned}$$ Solving for $\hat{h}(0)$ and using (\[bbb\]), it follows that$$\hat{h}(0)=\frac{c_{a}^{(q,s)}(0)}{\delta+b_{a,1}^{(q)}(0)-\int_{\mathbb{R}
_{+}}e^{-\delta z}b_{a,2}^{(q)}(0,\mathrm{d}z)}=\frac{c_{a}^{(q,s)}(0)}{\delta+b_{a}^{(q,\delta)}(0)}.$$ It follows from the monotone convergence theorem that (\[levy\]) also holds for $\delta=0$.
\[rk levy\] We point out that Assumptions (**A1**)-(**A3**) are not necessary to yield (\[levy\]) in the Lévy framework. In fact, by the spatial homogeneity of $X$, similar to (\[down\]) and (\[up\]), we have $$\frac{e^{-(s+\delta)\varepsilon}C^{(q,s)}(0;-a,\varepsilon)}{1-e^{-\delta
\varepsilon}B^{(q,\delta)}(0;\varepsilon-a,\varepsilon)}\leq\mathbb{E}\left[
e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)-\delta M_{\tau_{a}}}\right] \leq
\frac{e^{s\varepsilon}C^{(q,s)}(0;\varepsilon-a,\varepsilon)}{1-e^{-\delta
\varepsilon}B^{(q,\delta)}(0;-a,\varepsilon)},$$ for any $\varepsilon\in(0,a)$. Suppose that the following condition holds: $$\lim_{\varepsilon\downarrow0}\frac{C^{(q,s)}(0;-a,\varepsilon)}{1-e^{-\delta
\varepsilon}B^{(q,\delta)}(0;\varepsilon-a,\varepsilon)}=\lim_{\varepsilon
\downarrow0}\frac{C^{(q,s)}(0;\varepsilon-a,\varepsilon)}{1-e^{-\delta
\varepsilon}B^{(q,\delta)}(0;-a,\varepsilon)}:=D_{a}^{(q,s,\delta)}$$ Then, $$\mathbb{E}\left[ e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)-\delta M_{\tau_{a}}}\right]
=D_{a}^{(q,s,\delta)}.$$
Theorem \[thm markov\] shows that the joint law $\mathbb{E}_{x}\left[
e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\{M_{\tau_{a}}\leq K\}}\right] $ is a solution to Eq. (\[triple LT\]). Furthermore, the following theorem shows that Eq. (\[triple LT\]) admits a unique solution.
Suppose that Assumptions (**A1**)-(**A3**) hold. For $q,s\geq0$ and $K>0$, Eq. (\[triple LT\]) admits a unique solution.
From Theorem \[thm markov\], we know that $h(x):=\mathbb{E}_{x}\left[
e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)}1_{\{\tau_{a}<\infty, M_{\tau_{a}}\leq K\}}\right] $ is a solution of (\[triple LT\]). We also notice that any continuous solution to (\[triple LT\]) must vanish when $x\uparrow$ $K$. For any fixed $L\in(-\infty,K)$, we define a metric space $(\mathbb{A}_{L},\boldsymbol{d}_{L})$, where $\mathbb{A}_{L}=\left\{ f\in C[L,K],f(K)=0\right\} $ and the metric $\boldsymbol{d}_{L}(f,g)=\sup_{x\in\lbrack L,K]}|f(x)-g(x)|$ for $f,g\in\mathbb{A}_{L}$. We then define a mapping $\mathcal{L}$ on $\mathbb{A}_{L}$ by $$\mathcal{L}f(x)=\int_{x}^{K}e^{-\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w}\left( c_{a}^{(q,s)}(y)+\int_{[0,K-y)}f(y+z)b_{a,2}^{(q)}(y,\mathrm{d}z)\right) \mathrm{d}y,\text{\quad}x\in\lbrack L,K],$$ where $f\in\mathbb{A}_{L}$. It is clear that $\mathcal{L}(\mathbb{A}_{L})\subset\mathbb{A}_{L}$.
Next we show that $\mathcal{L}:\mathbb{A}_{L}\rightarrow\mathbb{A}_{L}$ is a contraction mapping. By the definitions of the two-sided exit quantities, for any $y\in\mathbb{R}
$, it follows that$$C^{(q,s)}(y;y-a,y+\varepsilon)+\int_{\mathbb{R}
_{+}}B_{2}^{(q)}(y,\mathrm{d}z;y-a,y+\varepsilon)\leq1-B_{1}^{(q)}(y;y-a,y+\varepsilon). \label{C1B}$$ Dividing each term in (\[C1B\]) by $\varepsilon\in(0,a)$ and letting $\varepsilon\downarrow0$, it follows from Assumptions (**A1**)-(**A3**) that$$0\leq c_{a}^{(q,s)}(y)+\int_{\mathbb{R}
_{+}}b_{a,2}^{(q)}(y,\mathrm{d}z)\leq b_{a,1}^{(q)}(y),\quad y\in\mathbb{R}
. \label{ineq}$$ By (\[ineq\]), we have for any $f,g\in\mathbb{A}_{L}$, $$\begin{aligned}
\boldsymbol{d}_{L}\left( \mathcal{L}f,\mathcal{L}g\right) & \leq\sup
_{t\in\lbrack L,K]}\left\vert f(t)-g(t)\right\vert \sup_{x\in\lbrack L,K]}\int_{x}^{K}e^{-\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w}\int_{\mathbb{R}
_{+}}b_{a,2}^{(q)}(y,\mathrm{d}z)\mathrm{d}y\\
& \leq\boldsymbol{d}_{L}(f,g)\sup_{L\leq x\leq K}\int_{x}^{K}e^{-\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w}b_{a,1}^{(q)}(y)\mathrm{d}y\\
& \leq\boldsymbol{d}_{L}(f,g)\left( 1-e^{-\int_{L}^{K}b_{a,1}^{(q)}(w)\mathrm{d}w}\right) .\end{aligned}$$ Since $\int_{L}^{K}b_{a,1}^{(q)}(w)\mathrm{d}w<\infty$ by Assumption (**A1**), one concludes that $\mathcal{L}:\mathbb{A}_{L}\rightarrow
\mathbb{A}_{L}$ is a contraction mapping. By Banach fixed point theorem, there exists a unique fixed point in $\mathbb{A}_{L}$. By a restriction of domain, it is easy to see that $\mathbb{A}_{L_{1}}\subset\mathbb{A}_{L_{2}}$ for $-\infty<L_{1}<L_{2}<K$. By the arbitrariness of $L$, the uniqueness holds for the space $\cap_{L<K}\mathbb{A}_{L}$. This completes the proof.
For the reminder of this section, we state several examples of Markov processes satisfying Assumptions (**A1**)-(**A3**). Note that the joint law of drawdown estimates for Examples \[eg SNLP\] and \[eg diffusion\] were solved by Mijatovic and Pistorius [@MP12] and Lehoczky [@L77], respectively (using different approaches). Assumption verifications for Examples \[eg PEMP\] and \[eg JD\] are postponed to Appendix.
\[Spectrally negative Lévy processes\]\[eg SNLP\] Consider a spectrally negative Lévy process $X$. Let $\psi(s):=\frac{1}{t}\log\mathbb{E}[e^{sX_{t}}]$ $\left( s\geq0\right) $ be the Laplace exponent of $X$. Further, let $W^{(q)}:\mathbb{R}
\rightarrow\lbrack0,\infty)$ be the well-known $q$-scale function of $X$; see, for instance Chapter 8 of Kyprianou [@K14]. The second scale function is defined as $Z^{(q)}(x)=1+q\int_{0}^{x}W^{(q)}(y)\mathrm{d}y$. Under some mild conditions (e.g., Lemma 2.4 of Kuznetsov et al. [@KKR12]), the scale functions are continuously differentiable which further implies that Assumptions (**A1**) and (**A3**) hold with $$b_{a,1}^{(q)}(0)=\frac{W^{(q)\prime}(a)}{W^{(q)}(a)}\text{ and }c_{a}^{(q,s)}(0)=e^{sa}\frac{Z_{s}^{(p)}(a)W_{s}^{(p)\prime}(a)-Z_{s}^{(p)\prime
}(a)W_{s}^{(p)}(a)}{W_{s}^{(p)}(a)}, \label{bc}$$ where $p=q-\psi(s)$, and $W_{s}^{(p)}$ ($Z_{s}^{(p)}$) is the (second) scale function of $X$ under a new probability measure $\mathbb{P}^{s}$ defined by the Radon-Nikodym derivative process $\left. \frac{\mathrm{d}\mathbb{P}^{s}}{\mathrm{d}\mathbb{P}}\right\vert _{\mathcal{F}_{t}}=e^{sX_{t}-\psi(s)t}$ for $t\geq0$. Therefore, by Corollary \[cor levy\] and (\[bc\]), we have$$\mathbb{E}\left[ e^{-q\tau_{a}-s(Y_{\tau_{a}}-a)-\delta M_{\tau_{a}}}\right]
=\frac{e^{sa}W^{(q)}(a)}{\delta W^{(q)}(a)+W^{(q)\prime}(a)}\frac{Z_{s}^{(p)}(a)W_{s}^{(p)\prime}(a)-pW_{s}^{(p)}(a)^{2}}{W_{s}^{(p)}(a)},$$ which is consistent with Theorem 3.1 of Landriault et al. [@LLZ16], and .
\[Refracted Lévy processes\]\[eg refracted\] Consider a refracted spectrally negative Lévy process $X$ of the form $$X_{t}=U_{t}-\lambda\int_{0}^{t}1_{\{X_{s}>b\}}\mathrm{d}s, \label{RLEVY}$$ where $\lambda\geq0$, $b>0$, and $U$ is a spectrally negative Lévy process (see Kyprianou and Loeffen [@KL10]). Let $W^{(q)}$ ($Z^{(q)}$) be the (second) $q$-scale function of $U$, and $\mathbb{W}^{(q)}$ be the $q$-scale function of the process $\{U_{t}-\lambda t\}_{t\geq0}$. Similar to Example \[eg SNLP\], all the scale functions are continuously differentiable under mild conditions.
For simplicity, we only consider the quantity with $b>x-a$ (otherwise the problem reduces to Example \[eg SNLP\] for $X_{t}=U_{t}-\lambda t$). By Theorem 4 of Kyprianou and Loeffen [@KL10], one can verify that Assumptions (**A1**) and (**A3**) hold. For $b>x$, from (\[bc\]) with $s=0$, we have $$b_{a,1}^{(q)}(x)=\frac{W^{(q)\prime}(a)}{W^{(q)}(a)}\text{ and }c_{a}^{(q,0)}(x)=\frac{Z^{(q)}(a)W^{(q)\prime}(a)-Z^{(q)\prime}(a)W^{(q)}(a)}{W^{(q)}(a)}.$$ For $x>b>x-a$,$$b_{a,1}^{(q)}(x)=\frac{\left( 1+\lambda\mathbb{W}^{(q)}(0)\right)
W^{(q)\prime}(a)+\lambda\int_{b-x+a}^{a}\mathbb{W}^{(q)\prime}(a-y)W^{(q)\prime}(y)\mathrm{d}y}{W^{(q)}(a)+\lambda\int_{b-x+a}^{a}\mathbb{W}^{(q)}(a-y)W^{(q)\prime}(y)\mathrm{d}y}$$ and $$c_{a}^{(q,0)}(x)=\frac{k_{a}^{(q)}(x)}{W^{(q)}(a)+\lambda\int_{b-x+a}^{a}\mathbb{W}^{(q)}(a-y)W^{(q)\prime}(y)dy},$$ where$$\begin{aligned}
k_{a}^{(q)}(x) & =(1+\lambda\mathbb{W}^{(q)}(0))\left( Z^{(q)}(a)W^{(q)\prime}(a)-qW^{(q)}(a)^{2}\right) \\
& +\lambda q(1+\lambda\mathbb{W}^{(q)}(0))\int_{b-x+a}^{a}\mathbb{W}^{(q)}(a-y)\left( W^{(q)\prime}(a)W^{(q)}(y)-W^{(q)}(a)W^{(q)\prime
}(y)\right) \mathrm{d}y\\
& -\lambda q\left[ W^{(q)}(a)+\lambda\int_{b-x+a}^{a}\mathbb{W}^{(q)}(a-y)W^{(q)\prime}(y)\mathrm{d}y\right] \int_{b-x+a}^{a}\mathbb{W}^{(q)\prime}(a-y)W^{(q)}(y)\mathrm{d}y\\
& +\lambda\left[ Z^{(q)}(a)+\lambda q\int_{b-x+a}^{a}\mathbb{W}^{(q)}(a-y)W^{(q)}(y)\mathrm{d}y\right] \int_{b-x+a}^{a}\mathbb{W}^{(q)\prime}(a-y)W^{(q)\prime}(y)\mathrm{d}y.\end{aligned}$$ By Corollary \[cor snm\], we obtain $$\mathbb{E}_{x}\left[ e^{-q\tau_{a}}1_{\{M_{\tau_{a}}\leq K\}}\right]
=\int_{x}^{K}e^{-\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w}c_{a}^{(q,0)}(y)\mathrm{d}y\text{,}\quad x\leq K,$$ which is a new result for the refracted Lévy process (\[RLEVY\]).
\[Linear diffusion processes\]\[eg diffusion\] Consider a linear diffusion process $X$ of the form $$\mathrm{d}X_{t}=\mu(X_{t})\mathrm{d}t+\sigma(X_{t})\mathrm{d}W_{t},$$ where $(W_{t})_{t\geq0}$ is a standard Brownian motion, and the drift term $\mu(\cdot)$ and local volatility $\sigma(\cdot)>0$ satisfy the usual Lipschitz continuity and linear growth conditions. As a special case of the jump diffusion process of Example \[eg JD\], it will be shown later that Assumptions (**A1**) and (**A3**) hold for linear diffusion processes. By Corollary \[cor snm\], we obtain $$\mathbb{E}_{x}\left[ e^{-q\tau_{a}}1_{\{\tau_a<\infty, M_{\tau_{a}}\leq K\}}\right]
=\int_{x}^{K}e^{-\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w}c_{a}^{(q,0)}(y)\mathrm{d}y\text{,}\quad x\leq K,$$ which is consistent with Eq. (4) of Lehoczky [@L77].
\[Piecewise exponential Markov processes\]\[eg PEMP\] Consider a piecewise exponential Markov process (PEMP) $X$ of the form $$\mathrm{d}X_{t}=\mu X_{t}\mathrm{d}t+\mathrm{d}Z_{t}, \label{PEMP}$$ where $\mu>0$ is the drift coefficient and $Z=(Z_{t})_{t\geq0}$ is a compound Poisson process given by $Z_{t}=\sum_{i=1}^{N_{t}}J_{i}$. Here, $(N_{t})_{t\geq0}$ is a Poisson process with intensity $\lambda>0$ and $J_{i}$’s are iid copies of a real-valued random variable $J$ with cumulative distribution function $F$. [We also assume the initial value $X_0\geq a$ which ensures that $X_t\geq 0$ for all $t<\tau_a$. In this case, as discussed in Remark \[rmk31\], $X$ is upward regular and creeps upward before $\tau_a$.]{} The first passage times of $X$ have been extensively studied in applied probability; see, e.g., Tsurui and Osaki [@TO76] and Kella and Stadje [@KS01]. For the PEMP (\[PEMP\]), semi-explicit expressions for the two-sided exit quantities $B_{1}^{(q)}(\cdot)$, $B_{2}^{(q)}(\cdot,\cdot)$ and $C^{(q,s)}(\cdot)$ are given in Section 6 of Jacobsen and Jensen [@JJ07]. As will be shown in Section \[ver pemp\], Assumptions (**A1**)-(**A3**) and Theorem \[thm markov\] hold for the PEMP $X$ with a continuous jump size distribution $F$.
\[Jump diffusion\]\[eg JD\] Consider a jump diffusion process $X$ of the form $$\mathrm{d}X_{t}=\mu(X_{t})\mathrm{d}t+\sigma(X_{t})\mathrm{d}W_{t}+\int_{-\infty}^{\infty}\gamma(X_{t-},z)N(\mathrm{d}t,\mathrm{d}z), \label{JD}$$ where $\mu(\cdot)$ and $\sigma(\cdot)>0$ are functions on $\mathbb{R}$, $(W_{t})_{t\geq0}$ is a standard Brownian motion, $\gamma(\cdot,\cdot)$ is a real-valued function on $\mathbb{R}^{2}$ modeling the jump size, and $N(\mathrm{d}t,\mathrm{d}z)$ is an independent Poisson random measure on $\mathbb{R}_{+}\times\mathbb{R}$ with a finite intensity measure . For specific $\mu(\cdot)$ and $\sigma(\cdot)$, the jump diffusion (\[JD\]) can be used to model the surplus process of an insurer with investment in risky assets; see, e.g., Gjessing and Paulsen [@GP97] and Yuen et al. [@YWN04]. We assume the same conditions as Theorem 1.19 of Øksendal and Sulem-Bialobroda [@OS07] so that (\[JD\]) admits a unique càdlàg adapted solution. Under this setup, we show in Section \[ver JD\] that Assumptions (**A1**)–(**A3**) and thus Theorem \[thm markov\] hold for the jump diffusion (\[JD\]).
Numerical examples
==================
[The main results of Section 3 rely on the analytic tractability of the two-sided exit quantities. To further illustrate their applicability, we now consider the numerical evaluation of the joint law of $(Y_{\tau_{a}},M_{\tau_{a}})$ for two particular spatial-inhomogeneous Markov processes with (positive) jumps through Theorem \[thm markov\]. For simplicity, we assume that the discount rate $q=0$ throughout this section.]{}
PEMP
----
[In this section, we consider the PEMP $X$ in Example \[eg PEMP\] with $\mu=1$, $\lambda=3$, and the generic jump size $J$ with density $$p(x)=\left\{
\begin{array}
[c]{ll}\frac{1}{3}e^{-x}, & x>0,\\
\frac{1}{3}(e^{x}+2e^{2x}), & x<0.
\end{array}
\right. \label{p1}$$ We follow Section 6 of Jacobsen and Jensen [@JJ07] to first solve for the two-sided exit quantities. Define the integral kernel $$\psi_{0}(z):=\frac{1}{z(z+1)(z-1)(z-2)},\quad z\in\mathbb{C}
,$$ and the linearly independent functions$$\begin{array}
[c]{ll}g_{1}(x):=\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_{1}}\psi_{0}(z)e^{-xz}dz=\frac{1}{6}e^{-2x}, & g_{2}(x):=\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_{2}}\psi_{0}(z)e^{-xz}dz=-\frac{1}{2}e^{-x},\\
g_{3}(x):=\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_{3}}\psi_{0}(z)e^{-xz}dz=\frac{1}{2}, & g_{4}(x):=\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_{4}}\psi
_{0}(z)e^{-xz}dz=-\frac{1}{6}e^{x},
\end{array}$$ for $x>0,$ where $\Gamma_{i}$ ($i=1,2,3,4$) is a small counterclockwise circle centered at the pole $\mu_{i}=3-i$ of $\psi_{0}(z)$. Moreover, for $0<u<v$, we consider the matrix-valued function $$(M_{i,k}({u,v}))_{1\leq i,k\leq4}:=\begin{pmatrix}
-\frac{1}{3}e^{-2u}(u+\frac{11}{6}) & \frac{e^{-2u}}{6} & \frac{e^{-2v}}{18} &
g_{1}(v)\\
e^{-u} & \frac{e^{-u}}{2}(u+\frac{1}{2}) & -\frac{e^{-v}}{4} & g_{2}(v)\\
-\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & g_{3}(v)\\
\frac{e^{u}}{9} & \frac{e^{u}}{12} & \frac{e^{v}}{6}(v-\frac{11}{6}) &
g_{4}(v)
\end{pmatrix}
,$$ where the matrix $M$ entries are chosen according to $$\left\{
\begin{array}
[c]{l}M_{i,k}(u,v)=\frac{\mu_{k}}{2\pi\sqrt{-1}}\int_{\Gamma_{i}}\frac{\psi_{0}(z)}{z-\mu_{k}}e^{-uz}\mathrm{d}z,\quad1\leq i\leq4, k=1,2,\\
M_{i,3}(u,v)=\frac{|\mu_{4}|}{2\pi\sqrt{-1}}\int_{\Gamma_{i}}\frac{\psi
_{0}(z)}{z-\mu_{4}}e^{-vz}\mathrm{d}z,\quad1\leq i\leq4.
\end{array}
\right.$$ Let $(N_{k,j}(u,v))_{1\leq k,j\leq4}$ be the inverse of $(M_{i,k}(u,v))_{1\leq
i,k\leq4}$. Combining Eq. (46) and a generalized Eq. (48) of Jacobsen and Jensen [@JJ07] (with $\zeta=s\geq0$ and $\rho\geq0$), we obtain the linear system of equations $$(c_{1},c_{2},c_{3},c_{4})(M_{i,k})=\left( -\frac{2\underline{C}}{s+2},-\frac{\underline{C}}{s+1},\frac{\overline{C}}{\rho+1},f(v)\right) ,
\label{CCCC}$$ where $\underline{C}$ and $\overline{C}$ are constants specified later, and $f(x)$ could stand for any of $B_{1}^{(0)}(x;u,v)$, $B_{2}^{(0,\rho)}(x;u,v)$, or $C^{(0,s)}(x;u,v)$ and has the representation $$f(x)=\sum\limits_{i=1}^{4}c_{i}g_{i}(x),\quad x\in\lbrack u,v].$$ ]{}
[To solve for $B_{1}^{(0)}(x;u,v)$, $B_{2}^{(0,\rho)}(x;u,v)$, or $C^{(0,s)}(x;u,v)$, we only need to solve (\[CCCC\]) with different assigned values of $\underline{C}$, $\overline{C}$, and $f(v)$ according to Eq. (45) of Jacobsen and Jensen [@JJ07]. By letting $\underline{C}=\overline{C}=0$ and $f(v)=1$, we obtain $$B_{1}^{(0)}(x;u,v)=\sum_{i=1}^{4}N_{4,i}(u,v)g_{i}(x).$$ Similarly, by letting $\underline{C}=f(v)=0$ and $\overline{C}=1$, for $\rho\geq0$, we obtain $$B_{2}^{(0,\rho)}(x;u,v)=\frac{1}{1+\rho}\sum_{i=1}^{4}N_{3,i}(u,v)g_{i}(x).$$ A Laplace inversion with respect to $\rho$ yields, for $z>0$, $$B_{2}^{(0)}(x,\mathrm{d}z;u,v)=e^{-z}\sum_{i=1}^{4}N_{3,i}(u,v)g_{i}(x)\mathrm{d}z.$$ By letting $\underline{C}=1$ and $\overline{C}=f(v)=0$, for $s\geq0$, we obtain $$C^{(0,s)}(x;u,v)=\sum_{i=1}^{4}\left( \frac{-2}{s+2}N_{1,i}(u,v)+\frac
{-1}{s+1}N_{2,i}(u,v)\right) g_{i}(x).$$ By the definitions, we have $$\begin{aligned}
b_{a,1}^{(0)}(x) & =-\sum_{i=1}^{4}D_{4,i}(x-a,x)g_{i}(x),\\
b_{a,2}^{(0)}(x,\mathrm{d}z) & =e^{-z}\left( \sum_{i=1}^{4}D_{3,i}(x-a,x)g_{i}(x)\right) \mathrm{d}z,\\
c_{a}^{(0,s)}(x) & =\sum_{i=1}^{4}\left( \frac{-2}{s+2}D_{1,i}(x-a,x)+\frac{-1}{s+1}D_{2,i}(x-a,x)\right) g_{i}(x),\end{aligned}$$ where we denote $D_{k,j}(u,v):=\frac{\partial}{\partial v}N_{k,j}(u,v)$.]{}
[In Figure \[fig1\] below, we use to numerically solve the integral equation (\[triple LT\]).]{}
A jump diffusion model
----------------------
[In this section, we consider a generalized PEMP $(X_{t})_{t\geq0}$ with diffusion whose dynamics is governed by $$\mathrm{d}X_{t}=X_{t}\mathrm{d}t+\sqrt{2}\mathrm{d}W_{t}+\mathrm{d}Z_{t},\quad
t>0, \label{eq:jd}$$ where the initial value $X_{0}=x\in\mathbb{R}
$, $(W_{t})_{t\geq0}$ is a standard Brownian motion, and $(Z_{t})_{t\geq0}$ is an independent compound Poisson process with a unit jump intensity and a unit mean exponential jump distribution. The two-sided exit quantities of this generalized PEMP can also be solved using the approach described in Sections 6 and 7 of Jacobsen and Jensen [@JJ07].]{}
[We define an integral kernel $$\psi_{1}(z)=\frac{e^{\frac{z^{2}}{2}}}{z(z+1)},\quad z\in\mathbb{C}
\text{.}$$ Let $\Gamma_{i}$ $\left( i=1,2\right) $ be small counterclockwise circles around the simple poles $\mu_{1}=0$ and $\mu_{2}=-1$, respectively, and define the linearly independent functions$$\begin{aligned}
g_{1}(x) & :=\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_{1}}\psi_{1}(z)e^{-xz}dz=1,\\
g_{2}(x) & :=\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_{2}}\psi_{1}(z)e^{-xz}dz=-e^{x+\frac{1}{2}},\end{aligned}$$ for $x\in\mathbb{R}
$. To find another linearly independent partial eigenfunction, we consider the vertical line $\Gamma_{3}=\{1+t\sqrt{-1},t\in\mathbb{R}
\}$ and define $$g_{3}(x):=\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_{3}}\psi_{1}(z)e^{-xz}\mathrm{d}z. \label{eq:G3last}$$ Next we derive an explicit expression for $g_{3}(x)$. We know from (\[eq:G3last\]) that $\lim_{x\rightarrow\infty}g_{3}(x)=0$ and $g_{3}$ is continuously differentiable with $$g_{3}^{\prime}(x)=-\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_{3}}\frac
{e^{\frac{z^{2}}{2}}}{z+1}e^{-xz}\mathrm{d}z. \label{eq:last1}$$ Notice that the bilateral Laplace transform functions (e.g., Chapter VI of [@W46]) of a standard normal random variable $U_{1}$ and an independent unit mean exponential random variables $U_{2}$ are given respectively by $$\int_{-\infty}^{\infty}e^{-zy}\cdot\frac{1}{\sqrt{2\pi}}e^{-\frac{y^{2}}{2}}\mathrm{d}y=e^{\frac{z^{2}}{2}},\quad\int_{0}^{\infty}e^{-zy}\cdot
e^{-y}\mathrm{d}y=\frac{1}{z+1},$$ for all complex $z$ such that $\Re(z)\geq0$. Hence, the bilateral Laplace transform of the density function of $U_{1}+U_{2}$, i.e., $$\int_{0}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{(x-y)^{2}}{2}}e^{-y}\mathrm{d}y$$ is given by $e^{\frac{z^{2}}{2}}/(z+1)$ for all complex $z$ such that $\Re(z)\geq0$. Since the right hand side of (\[eq:last1\]) is just the Bromwich integral for the inversion of the bilateral Laplace transform $-e^{\frac{z^{2}}{2}}/(z+1)$, evaluated at $-x$, we deduce that $$g_{3}^{\prime}(x)=-\int_{0}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{(x+y)^{2}}{2}}e^{-y}\mathrm{d}y.$$ It follows that $$g_{3}(x)=-\int_{x}^{\infty}g_{3}^{\prime}(y)\mathrm{d}y=1-\int_{0}^{\infty
}N(x+y)e^{-y}\mathrm{d}y.$$ where $N(\cdot)$ is the cumulative distribution function of standard normal distribution.]{}
[For any fixed $-\infty<u<v<\infty$, we define a matrix-valued function $$(M_{i,k}(u,v))_{1\leq i,k\leq3}:=\begin{pmatrix}
1 & g_{1}(v) & g_{1}(u)\\
ve^{v+\frac{1}{2}} & g_{2}(v) & g_{2}(u)\\
1-\int_{0}^{\infty}N(v+y)ye^{-y}\mathrm{d}y & g_{3}(v) & g_{3}(u)
\end{pmatrix}
,$$ where the first row is computed according to $$M_{i,1}(u,v)=\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_{i}}\frac{\psi_{0}(z)}{z+1}e^{-vz}\mathrm{d}z.$$ Notice that $M_{3,1}(u,v)$ can be calculated in the same way as $g_{3}(x)$. We also denote by $(N_{k,j}(u,v))_{1\leq k,j\leq3}$ the inverse of $(M_{i,k}(u,v))_{1\leq i,k\leq3}$.]{}
[By Eq. (46) and a generalized Eq. (48) of Jacobsen and Jensen [@JJ07] (with $\zeta=s=0$ and $\rho\geq0$), we obtain the linear system of equations $$(c_{1},c_{2},c_{3})(M_{i,k})=\left( \frac{\overline{C}}{\rho+1},f(v),f(u)\right) , \label{CCC}$$ where $\overline{C}$ is a constant specified later, and $f(x)$ could stand for any of $B_{1}^{(0)}(x;u,v)$, $B_{2}^{(0,\rho)}(x;u,v)$, or $C^{(0,0)}(x;u,v)$ and has the representation $$f(x)=\sum\limits_{i=1}^{3}c_{i}g_{i}(x),\quad x\in\lbrack u,v].$$ By letting (1) $\overline{C}=f(u)=0$ and $f(v)=1$, (2) $\overline{C}=1$ and $f(v)=f(u)=0$, (3) $\overline{C}=f(v)=0$ and $f(u)=1$, for any $\rho\geq0$ and $z>0$, and solving the linear system (\[CCC\]), we respectively obtain $$\begin{aligned}
B_{1}^{(0)}(x;u,v) & =\sum_{i=1}^{3}N_{2,i}(u,v)g_{i}(x),\\
B_{2}^{(0,\rho)}(x;u,v) & =\frac{1}{1+\rho}\sum_{i=1}^{3}N_{1,i}(u,v)g_{i}(x),\quad B_{2}^{(0)}(x,\mathrm{d}z;u,v)=e^{-z}\sum_{i=1}^{3}N_{1,i}(u,v)g_{i}(x)\mathrm{d}z,\\
C^{(0,0)}(x;u,v) & =\sum_{i=1}^{3}N_{3,i}(u,v)g_{i}(x).\end{aligned}$$ Furthermore, this implies $$\begin{aligned}
b_{a,1}^{(0)}(x) & =-\sum_{i=1}^{3}D_{2,1}(x-a,x)g_{i}(x),\\
b_{a,2}^{(0)}(x,\mathrm{d}z) & =e^{-z}\left( \sum_{i=1}^{3}D_{1,i}(x-a,x)g_{i}(x)\right) ,\\
c_{a}^{(0,0)}(x) & =\sum_{i=1}^{3}D_{3,i}(x-a,x)g_{i}(x),\end{aligned}$$ where we denote $D_{k,j}(u,v)=\frac{\partial}{\partial v}N_{k,j}(u,v)$.]{}
[In Figure 2 below, we plot $h(x)=\mathbb{P}_{x}\{M_{\tau_{a}}\leq K\}$ by numerically solving the integral equation (\[triple LT\]) using .]{}
Acknowledgments
===============
The authors would like to thank two anonymous referees for their helpful comments and suggestions. Support from grants from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by David Landriault and Bin Li (grant numbers 341316 and 05828, respectively). Support from a start-up grant from the University of Waterloo is gratefully acknowledged by Bin Li, as is support from the Canada Research Chair Program by David Landriault.
Appendix
========
Proof of Lemma \[lem fatou\]
----------------------------
We define $\psi _{n}(z)=\inf_{m\geq n}\phi _{m}(z)$ for $z\in S$. Further, we define $\underline{\psi }_{n}(z)=\liminf_{w\rightarrow z}\psi _{n}(w)$ which is lower semi-continuous (see, e.g., Lemma 5.13.4 of Berberian [B99]{}). Note that $\underline{\psi }_{n}$ is increasing in $n$, and by the definition of $\underline{\psi}_n$, we have $$\begin{aligned}
\lim_{n\rightarrow \infty }\underline{\psi }_{n}(z)=&\lim_{n\rightarrow
\infty }\lim_{r\downarrow 0}\inf_{w\in (z-r,z+r)}\inf_{m\geq n}\phi
_{m}(w)\\
=&\lim_{n\rightarrow
\infty }\lim_{r\downarrow 0}\inf_{m\geq n,w\in (z-r,z+r)}\phi
_{m}(w)\equiv\liminf_{n\rightarrow \infty ,w\rightarrow z}\phi _{n}(w),\end{aligned}$$where the second equality is because there is no ambiguity in switching the order of two infimums. By the monotone convergence theorem, we have $$\int_{S}\liminf_{n\rightarrow \infty ,w\rightarrow z}\phi _{n}(w)\mathrm{d}\mu (z)=\lim_{n\rightarrow \infty }\int_{S}\underline{\psi }_{n}(z)\mathrm{d}\mu (z). \label{lim1}$$By Portmanteau theorem of weak convergence and the fact that $\underline{ \psi }_{n}(z)$ is nonnegative and lower semi-continuous, it follows that $$\int_{S}\underline{\psi }_{n}(z)\mathrm{d}\mu (z)\leq \liminf_{m\rightarrow
\infty }\int_{S}\underline{\psi }_{n}(z)\mathrm{d}\mu _{m}(z) \label{lim2}$$for any $n\in
\mathbb{N}
$. Moreover, since $\psi _{n}(z)$ is monotone increasing in $n$, we have$$\liminf_{m\rightarrow \infty }\int_{S}\underline{\psi }_{n}(z)\mathrm{d}\mu
_{m}(z)\leq \liminf_{m\rightarrow \infty }\int_{S}\underline{\psi }_{m}(z)\mathrm{d}\mu _{m}(z). \label{lim3}$$By (\[lim1\])-(\[lim3\]),$$\int_{S}\liminf_{n\rightarrow \infty ,w\rightarrow z}\phi _{n}(w)\mathrm{d}\mu (z)\leq \liminf_{m\rightarrow \infty }\int_{S}\underline{\psi }_{m}(z)\mathrm{d}\mu _{m}(z)\leq \liminf_{m\rightarrow \infty }\int_{S}\phi _{m}(z)\mathrm{d}\mu _{m}(z),$$where the last inequality is due to $\underline{\psi }_{m}(z)\leq \psi
_{m}(z)\leq \phi _{m}(z)$.
Suppose that $\{\phi _{n}\}_{n\in
\mathbb{N}
}$ is uniformly bounded by $K>0$, by applying (\[inf\]) to $\{K-\phi
_{n}\}_{n\in
\mathbb{N}
}$, we obtain $$\begin{aligned}
K\mu (S)-\int_{S}\limsup_{n\rightarrow \infty , w\rightarrow z}\phi _{n}(w)\mathrm{d}\mu
(z)& =\int_{S}\liminf_{n\rightarrow \infty, w\rightarrow z}(K-\phi _{n}(w))\mathrm{d}\mu (z)
\\
& \leq \liminf_{n\rightarrow \infty }\int_{S}(K-\phi _{n}(z))\mathrm{d}\mu
_{n}(z) \\
& =K\liminf_{n\rightarrow \infty }\mu _{n}(S)-\limsup_{n\rightarrow \infty
}\int_{S}\phi _{n}(z)\mathrm{d}\mu _{n}(z).\end{aligned}$$Therefore, inequality (\[sup\]) follows immediately by the weak convergence of $\mu _{n}$ and $\mu (S)<\infty $.
Assumption verification for Example \[eg PEMP\]\[ver pemp\]
-----------------------------------------------------------
\[lem pemp\]Consider the PMEP (\[PEMP\]) with a continuous jump size distribution $F(\cdot)$. For $q,s\geq0$ and $0<u_{0}<x_{0}<v_{0}$, we have$$\lim_{(u,v)\downarrow(u_{0},v_{0})}g(x_{0};u,v)=\lim_{(x,u)\uparrow
(x_{0},u_{0})}g(x;u,v_{0})=g(x_{0},u_{0},v_{0}),$$ where the function $g(x;u,v)$ is any of the following three functions: $B_{1}^{(q)}(x;u,v)$, $B_{2}^{(q,s)}(x;u,v)$ and $C^{(q,s)}(x;u,v)$.
[Note that the condition $0<u_{0}<x_{0}<v_{0}$ is to ensure the process $X$ remains positive before exiting these finite intervals, which further implies $X$ is upward regular and creeps upward.]{} We limit our proof to $$\lim_{(u,v)\downarrow(u_{0},v_{0})}B_{1}^{(q)}(x_{0};u,v)=B_{1}^{(q)}(x_{0};u_{0},v_{0}). \label{B1eq}$$ The other results can be proved in a similar manner. [By the relationship $v>v_0>u>u_0$, we have]{} $$\begin{aligned}
& \left\vert B_{1}^{(q)}(x_{0};u_{0},v_{0})-B_{1}^{(q)}(x_{0};u,v)\right\vert
\nonumber\\
& \leq\left\vert \mathbb{E}_{x_{0}}\left[ e^{-qT_{v_{0}}^{+}}1_{\{T_{v_{0}}^{+}<T_{u_{0}}^{-},X_{T_{v_{0}}^{+}}=v_{0}\}}\right] -\mathbb{E}_{x_{0}}\left[ e^{-qT_{v}^{+}}1_{\{T_{v}^{+}<T_{u}^{-},X_{T_{v}^{+}}=v,X_{T_{v_{0}}^{+}}=v_{0}\}}\right] \right\vert \nonumber\\
& +\mathbb{P}_{x_{0}}\left\{ v_{0}<X_{T_{v_{0}}^{+}}\leq v\right\} .
\label{B1 in}$$ It is clear that the last term of (\[B1 in\]) vanishes as $v\downarrow
v_{0}$ by the right-continuity of the distribution function of $X_{T_{v_{0}}^{+}}$. Also,$$\begin{aligned}
& \left\vert \mathbb{E}_{x_{0}}\left[ e^{-qT_{v_{0}}^{+}}1_{\{T_{v_{0}}^{+}<T_{u_{0}}^{-},X_{T_{v_{0}}^{+}}=v_{0}\}}\right] -\mathbb{E}_{x_{0}}\left[ e^{-qT_{v}^{+}}1_{\{T_{v}^{+}<T_{u}^{-},X_{T_{v}^{+}}=v,X_{T_{v_{0}}^{+}}=v_{0}\}}\right] \right\vert \nonumber\\
& =\mathbb{E}_{x_{0}}\left[ e^{-qT_{v_{0}}^{+}}1_{\{T_{v_{0}}^{+}<T_{u}^{-},X_{T_{v_{0}}^{+}}=v_{0}\}}\right] -\mathbb{E}_{x_{0}}\left[
e^{-qT_{v}^{+}}1_{\{T_{v}^{+}<T_{u}^{-},X_{T_{v}^{+}}=v,X_{T_{v_{0}}^{+}}=v_{0}\}}\right] \nonumber\\
& +\mathbb{E}_{x_{0}}\left[ e^{-qT_{v_{0}}^{+}}1_{\{T_{u}^{-}<T_{v_{0}}^{+}<T_{u_{0}}^{-},X_{T_{v_{0}}^{+}}=v_{0}\}}\right] \nonumber\\
& \leq1-\mathbb{E}_{v_{0}}\left[ e^{-qT_{v}^{+}}1_{\{T_{v}^{+}<T_{u}^{-},X_{T_{v}^{+}}=v\}}\right] +\mathbb{P}_{x_{0}}\left\{ T_{u}^{-}<T_{v_{0}}^{+}<T_{u_{0}}^{-}\right\} . \label{10}$$ Let $\zeta$ be the time of the first jump of the compound Poisson process $Z$ with jump rate $\lambda>0$. [Note that $X$ will increase continuously up to time $\zeta$ as long as the initial value is positive. Since $v>v_0>0$, we have]{}$$1-\mathbb{E}_{v_{0}}\left[ e^{-qT_{v}^{+}}1_{\{T_{v}^{+}<T_{u}^{-},X_{T_{v}^{+}}=v\}}\right] \leq1-\mathbb{E}_{v_{0}}\left[ e^{-qT_{v}^{+}}1_{\{\textcolor[rgb]{0.0,0.0,0.0}{\zeta}>T_{v}^{+}\}}\right] =1-\left(
\frac{v}{v_{0}}\right) ^{-(q+\lambda)/\mu}. \label{11}$$ [By conditioning on $X_{T_{u}^{-}-}$, one obtains]{} $$\begin{aligned}
\mathbb{P}_{x_{0}}\left\{ T_{u}^{-}<T_{v_{0}}^{+}<T_{u_{0}}^{-}\right\} &
\leq\int_{u}^{v_{0}}\mathbb{P}_{x_{0}}\left\{ X_{T_{u}^{-}-}\in
\mathrm{d}y\right\} \mathbb{P}\left\{ y-u<J\leq y-u_{0}\right\} \nonumber\\
& \leq\max_{u_{0}\leq y\leq v_{0}}\left( F(y-u_{0})-F(y-u)\right) .
\label{12}$$ Since $F(\cdot)$ is continuous, and hence uniformly continuous for $y\in\lbrack0,v_{0}-u_{0}]$, it follows that the right-hand side of (\[12\]) vanishes as $u\downarrow u_{0}$. From (\[B1 in\])–(\[12\]), we conclude that (\[B1eq\]) holds.
[Note that although (\[12\]) only uses the continuity of $F$ on $[0,\infty)$, the proof for $C^{(q,s)}(x;u,v)$ will use the continuity of $F$ on $(-\infty,0]$.]{}
\[propA1\] Assumptions (**A1**)-(**A3**) hold for the piecewise exponential Markov process (\[PEMP\]) with a continuous jump size distribution $F(\cdot)$ [and initial value $X_0\geq a$]{}.
For $0<u<x<v$, by the strong Markov property, we have $$\begin{aligned}
B_{1}^{(q)}(x;u,v) & =\mathbb{E}_{x}\left[ e^{-qT_{v}^{+}}1_{\{T_{v}^{+}<T_{u}^{-},X_{T_{v}^{+}}=v,\zeta>T_{v}^{+}\}}\right] +\mathbb{E}_{x}\left[ e^{-qT_{v}^{+}}1_{\{T_{v}^{+}<T_{u}^{-},X_{T_{v}^{+}}=v,\zeta<T_{v}^{+}\}}\right] \nonumber\\
& =\left( \frac{v}{x}\right) ^{-(q+\lambda)/\mu}+\lambda\int_{0}^{\frac
{1}{\mu}\ln\frac{v}{x}}e^{-(q+\lambda)t}\mathrm{d}t\int_{u-xe^{\mu t}}^{v-xe^{\mu t}}B_{1}^{(q)}(xe^{\mu t}+w;u,v)F(\mathrm{d}w). \label{B1}$$ By Lemma \[lem pemp\], [Eq. (\[B1\])]{}, and the dominated convergence theorem, it is straightforward to verify that Assumption (**A1**) holds and [for $x>a$,]{}$$b_{a,1}^{(q)}(x)=\frac{q+\lambda}{\mu x}-\frac{\lambda}{\mu x}\int_{-a}^{0}B_{1}^{(q)}(x+w;x-a,x)F(\mathrm{d}w).$$ [Note that we require $x>a$ as otherwise $x+w $ in the above equation could be negative for $w\in(-a,0)$, and then Lemma \[lem pemp\] does not apply.]{} Obviously, $\int_{x}^{y}b_{a,1}^{(q)}(w)\mathrm{d}w<\infty$ for all $0<x<y<\infty$. Similarly, by conditioning on the first jump of $Z$, [for $0<u<x<v$,]{}$$\begin{aligned}
B_{2}^{(q)}(x,\mathrm{d}z;u,v) & =\lambda\int_{0}^{\frac{1}{\mu}\ln\frac
{v}{x}}e^{-(q+\lambda)t}F(v-xe^{\mu t}+\mathrm{d}z)\mathrm{d}t\\
& +\lambda\int_{0}^{\frac{1}{\mu}\ln\frac{v}{x}}e^{-(q+\lambda)t}\mathrm{d}t\int_{u-xe^{\mu t}}^{v-xe^{\mu t}}B_{2}^{(q)}(xe^{\mu
t}+w,\mathrm{d}z;u,v)F(\mathrm{d}w),\end{aligned}$$ and $$C^{(q,s)}(x;u,v)=\lambda\int_{0}^{\frac{1}{\mu}\ln\frac{v}{x}}e^{-(q+\lambda
)t}\mathrm{d}t\int_{-\infty}^{v-xe^{\mu t}}C^{(q,s)}(xe^{\mu t}+w;u,v)F(\mathrm{d}w),$$ where it is understood that $C^{(q,s)}(xe^{\mu t}+w;u,v)=e^{s(xe^{\mu t}+w-u)}$ for $w<u-xe^{\mu t}$. One can verify from Lemma \[lem pemp\] and the dominated convergence theorem that Assumptions (**A2**) and (**A3**) hold, [and for $x>a,$]{} $$\textcolor[rgb]{0.0,0.0,0.0}{b_{a,2}^{(q)}(x,\mathrm{d}z)}=\frac{\lambda
}{\mu x}F(\mathrm{d}z)+\frac{\lambda}{\mu x}\int_{-a}^{0}B_{2}^{(q)}(x+w,\mathrm{d}z;x-a,x)F(\mathrm{d}w),$$ and $$c_{a}^{(q,s)}(x)=\frac{\lambda}{\mu x}\int_{-\infty}^{0}C^{(q,s)}(x+w;x-a,x)F(\mathrm{d}w).$$ This ends the proof.
Assumption verification for Example \[eg JD\]\[ver JD\]
-------------------------------------------------------
Let $U$ be the continuous component of $X$, which is a linear diffusion process with the infinitesimal generator $\mathcal{L}_{U}=\frac{1}{2}\sigma^{2}(y)\frac{\mathrm{d}^{2}}{\mathrm{d}y^{2}}+\mu(y)\frac{\mathrm{d}}{\mathrm{d}y}.$ It is well-known that, for any $q>0$, there exist two independent and positive solutions, denoted as $\phi_{q}^{\pm}(y)$, to the Sturm-Liouville equation $$\mathcal{L}_{U}\phi_{q}^{\pm}(y)=q\phi_{q}^{\pm}(y), \label{SL}$$ where $\phi_{q}^{+}(\cdot)$ is strictly increasing and $\phi_{q}^{-}(\cdot)$ is strictly decreasing. By the Lipschitz assumption on $\mu(\cdot)$ and $\sigma(\cdot)$, it follows from the Schauder estimates (e.g., Theorem 6.14 of Gilbarg and Trudinger [@GT01]) of Eq. (\[SL\]) that $\phi_{q}^{\pm
}(\cdot)\in C^{2,\alpha}(\bar{\Omega})$ for any $\alpha\in(0,1]$ and any compact set $\bar{\Omega}\subset\mathbb{R}
$. Interested readers can refer to Section 4.1 of Gilbarg and Trudinger [@GT01] for more detail on the Hölder space $C^{2,\alpha}(\bar{\Omega
})$.
We denote the first hitting time of $U$ to level $z\in\mathbb{R}
$ by $H_{z}=\inf\{t>0:U_{t}=z\}$. It is well-known that, for $u\leq x\leq v$,$$\mathbb{E}_{x}\left[ e^{-qH_{u}}1_{\left\{ H_{u}<H_{v}\right\} }\right]
=\frac{f_{q}(x,v)}{f_{q}(u,v)}\quad\text{and\quad}\mathbb{E}_{x}\left[
e^{-qH_{v}}1_{\left\{ H_{v}<H_{u}\right\} }\right] =\frac{f_{q}(u,x)}{f_{q}(u,v)}, \label{2sd}$$ where $f_{q}(x,y):=\phi_{q}^{+}(x)\phi_{q}^{-}(y)-\phi_{q}^{+}(y)\phi_{q}^{-}(x)$. Note that $f_{q}(x,y)$ is strictly decreasing in $x$ and strictly increasing in $y$ with $f_{q}(x,x)=0$. In particular, for $u\leq x\leq v$, we have $$\mathbb{E}_{x}\left[ {e}^{-qH_{u}}\right] =\frac{\phi_{q}^{-}(x)}{\phi
_{q}^{-}(u)}\quad\text{and\quad}\mathbb{E}_{x}\left[ {e}^{-qH_{v}}\right]
=\frac{\phi_{q}^{+}(x)}{\phi_{q}^{+}(v)}. \label{1sd}$$ For $\mathbf{e}_{q}$ an independent exponential random variable with mean $1/q<\infty$, the $q$-potential measure of $U$ is given by $$r_{q}(x,y):=\frac{1}{q}\mathbb{P}_{x}\left\{ U_{\mathbf{e}_{q}}\in
\mathrm{d}y\right\} /\mathrm{d}y=\left\{
\begin{array}
[c]{l}\frac{2}{q\sigma^{2}(y)}\frac{\phi_{q}^{+}(x)\phi_{q}^{-}(y)}{f_{q,1}(y,y)},\quad x\leq y,\\
\frac{2}{q\sigma^{2}(y)}\frac{\phi_{q}^{+}(y)\phi_{q}^{-}(x)}{f_{q,1}(y,y)},\quad x>y,
\end{array}
\right.$$ where $f_{q,1}(x,y):=\frac{\partial}{\partial x}f_{q}(x,y).$ Furthermore, the $q$-potential measure of $U$ killed on exiting the interval $[u,v]$, for $u\leq x,y\leq v$, is given by $$\begin{aligned}
\theta^{(q)}(x,y;u,v) & :=\frac{1}{q}\mathbb{P}_{x}\left( U_{\mathbf{e}_{q}}\in\mathrm{d}y,\mathbf{e}_{q}<H_{u}\wedge H_{v}\right) /\mathrm{d}y\nonumber\\
& =r_{q}(x,y)-\frac{f_{q}(x,v)}{f_{q}(u,v)}r_{q}(u,y)-\frac{f_{q}(u,x)}{f_{q}(u,v)}r_{q}(v,y). \label{theta}$$
The next lemma is an analogy of Lemma \[lem pemp\]. Thanks to the diffusion term in the jump diffusion model (\[JD\]), we now allow for the presence of atoms in the jump intensity measure $\nu(\cdot)$.
\[lem JD\]Consider the jump diffusion model (\[JD\]). For $q,s\geq0$ and $u_{0}<x_{0}<v_{0}$, we have$$\lim_{(u,v)\downarrow(u_{0},v_{0})}g(x_{0};u,v)=\lim_{(x,u)\uparrow
(x_{0},u_{0})}g(x;u,v_{0})=g(x_{0},u_{0},v_{0}),$$ where $g(x;u,v)$ is any of the following functions: $B_{1}^{(q)}(x;u,v)$, $B_{2}^{(q,s)}(x;u,v)$ and $C^{(q,s)}(x;u,v)$.
We can follow the same proof as Lemma \[lem pemp\] except for the term $\mathbb{P}_{x_{0}}\left\{ T_{u}^{-}<T_{v_{0}}^{+}<T_{u_{0}}^{-}\right\} $ in (\[12\]), which will be handled distinctly here. We have $X_{t}=U_{t}$ a.s. for $t<\zeta$, where $\zeta$ is the first time a jump occurs which follows an exponential distribution with mean $1/\lambda=1/\nu(\mathbb{R})>0$. For any $u_{0}<u<x_{0}<v_{0}$, by (\[2sd\]) and (\[1sd\]), we have $$\begin{aligned}
\mathbb{P}_{x_{0}}\left\{ T_{u}^{-}<T_{v_{0}}^{+}<T_{u_{0}}^{-}\right\} &
\leq\mathbb{P}_{u}\left\{ T_{v_{0}}^{+}<T_{u_{0}}^{-}\right\} \\
& =\mathbb{P}_{u}\left\{ T_{v_{0}}^{+}<T_{u_{0}}^{-},\xi>T_{v_{0}}^{+}\right\} +\mathbb{P}_{u}\left\{ \xi\leq T_{v_{0}}^{+}<T_{u_{0}}^{-}\right\} \\
& \leq\mathbb{E}_{u}\left[ e^{-\lambda H_{v_{0}}}1_{\left\{ H_{v_{0}}<H_{u_{0}}\right\} }\right] +1-\mathbb{E}_{u}\left[ e^{-\lambda H_{u_{0}}}\right] \\
& =\frac{f_{q}(u_{0},u)}{f_{q}(u_{0},v_{0})}+1-\frac{\phi_{q}^{-}(u)}{\phi_{q}^{-}(u_{0})}.\end{aligned}$$ Therefore, it follows that $\lim_{u\downarrow u_{0}}\mathbb{P}_{x_{0}}\left\{
T_{u}^{-}<T_{v_{0}}^{+}<T_{u_{0}}^{-}\right\} =0$ by $f_{q}(u_{0},u_{0})=0$.
Assumptions (**A1**)-(**A3**) hold for the jump diffusion model (\[JD\]).
By the strong Markov property, (\[2sd\]) and (\[theta\]), for $u<x<v$, it follows that$$\begin{aligned}
& B_{1}^{(q)}(x;u,v)\\
& =\mathbb{E}_{x}\left[ e^{-qT_{v}^{+}}1_{\{T_{v}^{+}<T_{u}^{-},T_{v}^{+}=v,\zeta>T_{v}^{+}\}}\right] +\mathbb{E}_{x}\left[ e^{-qT_{v}^{+}}1_{\{T_{v}^{+}<T_{u}^{-},T_{v}^{+}=v,\zeta<T_{v}^{+}\}}\right] \\
& =\mathbb{E}_{x}\left[ e^{-(q+\lambda)H_{v}}1_{\{H_{v}<H_{u}\}}\right]
+\int_{u}^{v}\mathbb{E}_{x}\left[ e^{-q\zeta}1_{\{\zeta<H_{u}\wedge
H_{v},U_{\zeta}\in\mathrm{d}y\}}\right] \int_{\mathbb{R}}B_{1}^{(q)}(y+\gamma(y,w);u,v)\frac{\nu(\mathrm{d}w)}{\lambda}\\
& =\frac{f_{q+\lambda}(u,x)}{f_{q+\lambda}(u,v)}+\int_{u}^{v}\theta
^{(q+\lambda)}(x,y;u,v)\mathrm{d}y\int_{\mathbb{R}}B_{1}^{(q)}(y+\gamma
(y,w);u,v)\nu(\mathrm{d}w),\end{aligned}$$ where it is understood that $B_{1}^{(q)}(y+\gamma(y,w);u,v)=0$ if $\gamma(y,w)>v-y$ or $\gamma(y,w)<u-y$. By Lemma \[lem JD\], the dominated convergence theorem, and the identity $f_{q+\lambda}(u,v)=-f_{q+\lambda}(v,u)$, we can verify that Assumption (**A1**) holds with $$b_{a,1}^{(q)}(x)=\frac{-f_{q+\lambda,1}(x-a,x)}{f_{q+\lambda}(x-a,x)}-\int_{x-a}^{x}\tilde{\theta}_{a}^{(q+\lambda)}(x,y)\mathrm{d}y\int
_{\mathbb{R}}B_{1}^{(q)}(y+\gamma(y,w);x-a,x)\nu(\mathrm{d}w),$$ where we write $\tilde{\theta}_{a}^{(q+\lambda)}(x,y):=-\frac{f_{q+\lambda
,1}(x-a,x)}{f_{q+\lambda}(x-a,x)}r_{q+\lambda}(x,y)-r_{q+\lambda,1}(x,y)+\frac{f_{q+\lambda,1}(x,x)}{f_{q+\lambda}(x-a,x)}r_{q+\lambda}(x-a,y)$ and $r_{q+\lambda,1}(x,y):=\frac{\partial}{\partial x}r_{q+\lambda}(x,y).$ The integrability of $b_{a,1}^{(q)}(\cdot)$ follows from the continuity of the $\phi_{q}^{+}(\cdot)$ and $\phi_{q}^{-}(\cdot)$.
Similarly, by the strong Markov property of $X$, (\[2sd\]) and (\[theta\]), we have$$B_{2}^{(q)}(x,\mathrm{d}z;u,v)=\int_{u}^{v}\theta^{(q+\lambda)}(x,y;u,v)\mathrm{d}y\int_{\mathbb{R}}B_{2}^{(q)}(y+\gamma(y,w),\mathrm{d}z;u,v)\nu(\mathrm{d}w),$$ and$$C^{(q,s)}(x;u,v)=\frac{f_{q+\lambda}(x,v)}{f_{q+\lambda}(u,v)}+\int_{u}^{v}\theta^{(q+\lambda)}(x,y;u,v)\mathrm{d}y\int_{\mathbb{R}}C^{(q,s)}(y+\gamma(y,w);u,v)\nu(\mathrm{d}w).$$ One can verify from Lemma \[lem JD\] that Assumptions (**A2**) and (**A3**) hold with $$b_{2,a}^{(q)}(x,\mathrm{d}z)=\int_{x-a}^{x}\tilde{\theta}_{a}^{(q+\lambda
)}(x,y)\mathrm{d}y\int_{\mathbb{R}}B_{2}^{(q)}(y+\gamma(y,w),\mathrm{d}z;x-a,x)\nu(\mathrm{d}w),$$ and $$c_{a}^{(q,s)}(x)=\frac{-f_{q+\lambda,1}(x,x)}{f_{q+\lambda}(x-a,x)}+\int
_{x-a}^{x}\tilde{\theta}^{(q+\lambda)}(x,y)\mathrm{d}y\int_{\mathbb{R}}C^{(q,s)}(y+\gamma(y,z);x-a,x)\nu(\mathrm{d}w).$$ This completes the proof.
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[^1]: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada ([email protected])
[^2]: Corresponding Author: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada ([email protected])
[^3]: Department of IEOR, Columbia University, New York, NY, 10027, USA ([email protected])
[^4]: The state space can [sometimes be relaxed]{} to an open interval of $\mathbb{R}
$ [(e.g., (0,+$\infty$) for geometric Brownian motions)]{}. It is also possible to treat some general state space with complex boundary behaviors. However, for simplicity, we choose $\mathbb{R}
$ as the state space of $X$ in this paper.
[^5]: Most often, one-sided Lévy processes (an exception to this is Baurdoux [@B09] for general Lévy processes)
[^6]: See page 142 and page 197 of [@K14] for definitions of regularity and creeping for Lévy processes.
[^7]: For Lévy processes ${\mathbb{P}}\{\tau_a<\infty\}=1$ as long as $X$ is not monotone.
|
---
abstract: 'A recent article uncovered a surprising dynamical mechanism at work within the (vacuum) Einstein ‘flow’ that strongly suggests that many closed 3-manifolds that do not admit a locally homogeneous and isotropic metric *at all* will nevertheless evolve, under Einsteinian evolution, in such a way as to be *asymptotically* compatible with the observed, approximate, spatial homogeneity and isotropy of the universe [@Moncrief:2015]. Since this previous article, however, ignored the potential influence of *dark-energy* and its correspondent accelerated expansion upon the conclusions drawn, we analyze herein the modifications to the foregoing argument necessitated by the inclusion of a *positive* cosmological constant — the simplest viable model for dark energy.'
author:
- Vincent Moncrief
- Puskar Mondal
title: 'Could the Universe have an Exotic Topology?'
---
Acknowledgement {#acknowledgement .unnumbered}
===============
Moncrief is grateful to Robert Bartnik for making possible his two, very enriching professional visits to Australia and for conveying to him numerous valuable insights into the properties of CMC slicings of Einsteinian spacetimes. This article is dedicated to Bartnik on the occasion of his 60th birthday.
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|
---
author:
- Timothy Eller
bibliography:
- 'thesisbib.bib'
nocite: '[@*]'
title: Chiral vector bundles
---
Given a smooth $G$-vector bundle $E\to M$ with a connection $\nabla$, we propose the construction of a sheaf of vertex algebras $\mathcal{E}^{ch(E,\nabla)}$, which we call a *chiral vector bundle*. $\mathcal{E}^{ch(E,\nabla)}$ contains as subsheaves the sheaf of superalgebras $\Omega \otimes \Gamma (SE \otimes \Lambda E)$ and the sheaf of Lie algebras generated by certain endomorphisms of these superalgebras: $\nabla$, the infinitesimal gauge transformations of $E$, and the contraction operators $\iota_X$ on differential forms $\Omega$. Another subsheaf of primary importance is the chiral vector bundle $\mathcal{E}^{ch(M\times {\textbf{C}},d)}$, which is closely related to the chiral de Rham sheaf of Malikov et alii.
Introduction
============
Overview
--------
The advent of the chiral de Rham complex [@MSV-1999] has introduced vertex algebras to the context differential and algebraic geometry. Namely, the vertex algebra analogs of the Heisenberg and Clifford algebras (known in other literature either as the $\beta\gamma$ and $bc$ systems, or fermionic and bosonic ghost systems) patch together to form a sheaf, over a smooth manifold $M$, that contains the classical de Rham complex as a subsheaf. In the past decade, several aspects of geometry have subsequently been lifted to their vertex algebra analogs, a process we might call *chiralization*. Recent such examples include the chiralization of *equivariant* cohomology by [@LL-2007] and its implications, such as [@tan-2009-03]; and the chiralization of differential operators on $M$ [@kapranov-2006], among others.
In this paper, we continue this trend, extending *chiralization* to vector bundles and their geometry.
### Relation to the chiral de Rham sheaf {#sec:CDRrelationship}
Just as the ordinary de Rham complex resides inside the chiral de Rham sheaf as a subsheaf, our final goal is to take a $G$-vector bundle $E$ with connection $\nabla$ and exhibit the sheaf of sections of $\Omega \otimes \Gamma (SE \otimes \Lambda E)$ ($\Omega$ being the exterior algebra of differential forms, and $SE$ and $\Lambda E$ being the full symmetric and antisymmetric tensor algebras of $E$) as a subsheaf of $\mathcal{E}^{ch(E,\nabla)}$, a sheaf of vertex algebras which we call a *chiral vector bundle*. In fact $\mathcal{E}^{ch(E,\nabla)}$ will contain the Lie algebra of endomorphisms of $\Omega \otimes \Gamma (SE \otimes \Lambda E)$ generated by $\nabla$, the infinitesimal gauge Lie algebra, and the odd Lie algebra of contraction operators $\iota_X$ ($X \in \mathfrak{X}$, the set of smooth vector fields), as well.
The construction of the vertex algebra $\mathcal{E}^{ch(E,\nabla)}(U)$, $U\subset M$, is outlined as follows. Restricting all objects to $U$, we enlarge the space $\Omega \otimes \Gamma E$ to the supersymmetric algebra $\mathfrak{a} = \Omega \otimes \Gamma (SE \otimes \Lambda E)$. We then capture the connection $\nabla$ in a robust Lie superalgebra $\mathfrak{s} \mathfrak{X}_{\nabla}$ of endomorphisms of the aforementioned superalgebra $\mathfrak{a}$.
This pair is combined into a single entity $\mathfrak{s} \equiv \mathfrak{s} (\mathfrak{a} \otimes \mathfrak{s} \mathfrak{X}_{\nabla}, \mathfrak{a})$ (the tensor products taken over the ring $\Omega^0$ of smooth functions) which we generically call a *souped-up Lie algebra*. It is the semidirect sum Lie algebra of $\mathfrak{a} \otimes \mathfrak{s} \mathfrak{X}_{\nabla}$ and $\mathfrak{a}$, also with the structure of a left module for $\mathfrak{a}$. The souped-up Lie algebra $\mathfrak{s}$ is a classical construction; we have not mentioned vertex algebras so far.
The impetus for constructing the souped-up Lie algebra is that a vertex algebra generated by the underlying Lie algebra *almost* has the structure of a souped-up Lie algebra already. That is, even if we were to chiralize an arbitrary Lie algebra $\mathfrak{g}$ and some arbitrary $\mathfrak{g}$-module $\mathfrak{v}$, assembled into the semidirect sum Lie algebra $\mathfrak{g} \oplus \mathfrak{v}$, the resulting vertex algebra would *almost* contain the symmetric or antisymmetric algebras generated by $\mathfrak{v}$, and the coefficient ring of $\mathfrak{g}$ would *almost* include $S\mathfrak{v} \otimes \Lambda \mathfrak{v}$.
Very little effort is needed to eliminate the recurring word “almost" in the preceding paragraph. There is an obvious set of relations that must be added to the vertex algebra so that it possesses the souped-up Lie algebra structure. Including these relations is a crucial choice, not without consequences, and marks the departure of our construction from the construction of the chiral de Rham complex. As a result, our construction becomes a replacement for the chiral de Rham complex, rather than an extension of it.
The constituent vertex algebras in the chiral de Rham sheaf do not have the structure of a souped-up Lie algebra, which this author deems unsatisfactory. While the chiral de Rham sheaf appropriately transfers the action of a vector field $X$ on a smooth function $f$ to the vertex algebra (specifically, $Xf = X \circ_0f$), it does not transfer the *product* of $f$ with a vector field $X$ to the vertex algebra (so $fX \neq f\circ_{-1}X$ in general). Our amendment is to impose the relation $fX = f\circ_{-1}X$. However, this does not come without a trade-off. To wit, although we are able to prove in some specific cases that a souped-up Lie algebra injects into the vertex algebra it generates, this injectivity is only *conjectured* to hold for arbitrary souped-up Lie algebras, including for $\mathfrak{s} (\mathfrak{a} \otimes \mathfrak{s} \mathfrak{X}_{\nabla}, \mathfrak{a})$.
### Relation to other vertex algebra bundles
Ours is not the first discussion of vertex algebra bundles. There are two others of which the author is aware, and both of those are substantially different from the one at hand. The one more closely related is that of [@DLMZ-2004], who have defined a notion of a $K$-theory associated to vertex algebras. Their idea is to begin with the concept of a $V$-bundle on $M$, which is akin to a $G$-bundle but with the group $G$ replaced by a vertex algebra $V$. The difference between their bundle and ours is that in ours, not only is the structure group $G$ (whose action on the fibers is *vertical*) replaced by a vertex algebra, but the fibers themselves are represented in this vertex algebra. And still furthermore, the Lie algebra of vector fields on $M$ (whose action on the fibers is *horizontal*) is part of our vertex algebra. This final aspect (chiralizing the Lie algebra of vector fields) is what likens our construction to the chiral de Rham complex, and makes the situation both more intricate and robust than the $V$-bundles of [@DLMZ-2004].
There is a second notion of a vertex algebra bundle that predates even the one already discussed. Frenkel and Ben-Zvi [@FBZ-2004] speak of a vertex algebra bundle over the base space ${\textbf{C}}$ (some Riemann surface) in the quantum field theoretic point of view of conformal field theory, rather than over the target space $M$ as is done in our situation. Thus our bundles are placed in a very different context from theirs.
Organization of this manuscript
-------------------------------
In Chapter \[sec:VA\], we give a definition of a vertex algebra, portrayed as the quotient of an algebra $F$ with an infinity of totally nonassociative noncommutative products $\circ_n$. The usual relations of a vertex algebra are then represented by an ideal $I \subset F$. This is accompanied by a change in philosophy regarding the nonnegative products $\circ_n$, $n\geq 0$. Whereas these products are typically chosen in accordance with the Ward identity (which relates these products to the transformation properties of the underlying classical geometry; see [@Polchinski-1998]), we now shed this obligation and permit these particular products to be specified arbitrarily, or even not at all.
Perhaps this new stance distances vertex algebras from their origins in string theory, but the result is a new object of interest in mathematics.
In Chapter \[sec:soupedupLA\], we define a new classical object, the *souped-up Lie algebra*, which combines a Lie algebra $\mathfrak{g}$ and a module $\mathfrak{a}$ (an associative commutative unital algebra) into the semidirect sum Lie algebra $(\mathfrak{a} \otimes \mathfrak{g}) \oplus \mathfrak{a}$. We have enlarged the ring of coefficients of $\mathfrak{g}$ to include the algebra $\mathfrak{a}$. The most important souped-up Lie algebra we construct is the one associated to a $G$vector bundle $E$ with connection $\nabla$, discussed above in \[sec:CDRrelationship\].
In Chapter \[sec:soupedupVA\], we construct a vertex algebra from a souped-up Lie algebra. For some basic souped-up Lie algebras $\mathfrak{s}$, we prove that the resulting vertex algebra contains $\mathfrak{s}$ as a subspace. However, the viability of this construction for a *generic* souped-up Lie algebra is ultimately a conjecture; the resulting vertex algebra might be zero (and thus not a vertex algebra at all). That said, we are confident that the resulting vertex algebra contains the entire generating souped-up Lie algebra. We prove some necessary conditions for these conjectures to be true.
The importance of Chapter \[sec:uVecttoVA\] is the definition of the functor $\mathcal{V}$ from souped-up Lie algebras to vertex algebras. This functor will be used to associate a sheaf of souped-up Lie algebras to a sheaf of vertex algebras.
Next, in Chapter \[sec:sheaf\], we construct a sheaf of vertex algebras from a sheaf of souped-up Lie algebras.
Finally, in Chapter \[sec:chiralDG\], we apply this construction to the sheaf of souped-up Lie algebras associated to $(E,\nabla)$, giving us a sheaf of vertex algebras containing all the geometric and algebraic information from $(E,\nabla)$.
### A note on superspaces and supercommutativity
Throughout this manuscript, we will assume that every vector space is in fact a *superspace*, meaning it has a ${{\textbf{Z}}}_2$-grading. Of course, it is certainly possible that the odd component is trivial. Furthermore, we will use *commutative* to mean *supercommutative*. Thus a commutative ring has a ${{\textbf{Z}}}_2$-grading in which odd elements $x,y$ satisfy $xy + yx = 0$. Similarly, in an abelian Lie algebra $\mathfrak{g}$, the associative multiplication on the universal enveloping algebra for $\mathfrak{s}$ is given by $[x,y] = xy + yx$ for odd elements $x,y$.
Despite this convention, for the sake of clarity, we will still occasionally apply the prefix *super-* when labeling certain vector spaces or products.
We will make explicit reference to the parity $p$ of an element only in Definition \[def:VA\] of a vertex algebra ideal. After that, all definitions and results hold for both even and odd elements, but to simplify the presentation we will only prove them for the even case.
Vertex algebras {#sec:VA}
===============
The notion of a vertex algebra was pinned down by Borcherds in 1986 [@borcherds-1986] to abstract the quantum fields prevalent in string theory. Although the axioms for a vertex algebra have been expressed in several different ways since their advent (see [@borcherds-1986], [@kac-1997]), we have chosen to recast the definition yet again with a mind to the work that appears in later chapters. Because ours is not the orthodox definition, we will prove that it is equivalent to the original definition in Proposition \[prop:B-equivalence\].
Definition of a vertex algebra as a quotient of an infinite free algebra
------------------------------------------------------------------------
We begin with an important preliminary definition of an *infinite free algebra*. In our rendition, a vertex algebra is some quotient of this algebra.
We will often use the following notion of a *unital vector space*.
A *unital vector space* is a vector space with a distinguished vector ${\textbf{1}}$.
\[def:IFA\] An *infinite free algebra* is a unital vector space that is closed under an infinite number of totally nonassociative noncommutative linear products $\circ_n$, $n\in {{\textbf{Z}}}$. We will generally abbreviate the product $x \circ_n y$ by $x_ny$.
An *ideal* in $F$ is closed with respect to *all* products $\circ_n$.
The element ${\textbf{1}}$ is *not* an identity for the products $\circ_n$ in general. However, upon taking a quotient of $F$ in the upcoming definition of a vertex algebra, ${\textbf{1}}$ will become the two-sided identity for the product $\circ_{-1}$.
Every element in an infinite free algebra $F$ can be represented as a sum of full binary rooted trees in which each leaf represents another element of $F$, and each of the remaining nodes is labeled by an integer. As we have defined $F$, it is possible that a tree may be infinitely deep. This awkwardness is removed by assuming $F$ is generated by some unital vector space $\mathfrak{v}$, which we put in a definition.
\[def:generatedIFA\] We write $F(\mathfrak{v})$ when an infinite free algebra $F$ is generated by a unital vector space $\mathfrak{v}$. That is, $F(\mathfrak{v})$ is the closure of $\mathfrak{v}$ under the products $\circ_n$ and addition.
Any additional structure on $\mathfrak{v}$, beyond its unital vector space structure, is forgotten in $F(\mathfrak{v})$.
In a *generated* infinite free algebra $F(\mathfrak{v})$, one can now identify when a leaf node of some element $x$ is terminal: the leaf node is an element of $\mathfrak{v}$. We thus have the notion of a *monomial* in $F(\mathfrak{v})$, which is an element that can be written as a product of elements of $\mathfrak{v}$, or can be depicted as a single tree in which every leaf node is in $\mathfrak{v}$.
While a vertex algebra may defined as a particular quotient of an infinite free algebra $F$ coming from Definition \[def:IFA\], virtually all examples in the literature, including those of importance in the current paper, are quotients of *generated* infinite free algebras $F(\mathfrak{v})$.
\[note:notation\] In $F$ and $F(\mathfrak{v})$, we can speak of *multiples* and *factors*. When we refer to a *multiple* $y$ of some element $x$, we mean that $y$ is obtained from $x$ by a sequence products on the left and right. In this case we also say that $x$ is a *factor* of $y$. Equivalently, $x$ is a subtree of $y$.
We will often use the notation $y{[\![x]\!]}$ to denote the dependence of an element $y$ on another element $x$. In particular, $y{[\![x]\!]}$ means $x$ is a factor of some term in $y$.
We will follow the convention that the particular product $\circ_{-2}{\textbf{1}}$ is denoted $D$, so $Dx \equiv x_{-2}{\textbf{1}}$. We will see shortly that $D$ is a derivation over all products in a vertex algebra.
We now define a vertex algebra as a particular quotient of an infinite free algebra $F(\mathfrak{v})$.
\[def:VA\] Let $F(\mathfrak{v})$ be an infinite free algebra, and fix some function $N(u,v) \geq 0$ on $\mathfrak{v} \times \mathfrak{v}$. The *vertex algebra ideal* $I(\mathfrak{v})$ is the ideal generated by the following sets:
identity:
: $\textbf{i}{[\![x;n]\!]} \triangleq {\textbf{1}}_nx - \delta_{n+1}x$
locality:
: \
$\textbf{c}{[\![u,v;n]\!]} \triangleq u_nv \quad \text{for all } n \geq N(u,v)$
derivation:
: \
$\textbf{d}{[\![x,y;n]\!]} \triangleq D(x_ny) - (Dx)_ny - x_n(Dy)$\
$\textbf{e}{[\![x,y;n]\!]} \triangleq (Dx)_ny + nx_{n-1}y \quad \text{for all } n \in {{\textbf{Z}}}$
quasi-commutativity:
: \
$\textbf{qc}{[\![x,y;n]\!]} \triangleq x_ny + (-1)^{p(x) p(y)}\sum_{k\geq 0} \frac{(-1)^{n+k}}{k!} D^k(y_{n+k}x)$
quasi-associativity:
: \
$\textbf{qa}{[\![x,y,z;m,n]\!]} \\ \triangleq (x_my)_nz - \sum_{k\geq 0}\binom{m}{k}(-1)^k \left(x_{m-k}(y_{n+k}z) - (-1)^{m+ p(x) p(y)} y_{m+n-k}(x_kz) \right)$
for $u,v\in \mathfrak{v}$ and $x,y,z \in F(\mathfrak{v})$, for $m,n=-1$, and for all $m,n \gg 0$ unless otherwise indicated. The binomial coefficient $\binom{m}{k}$ is defined as usual for all $m\in {{\textbf{Z}}}$ and $k \in {{\textbf{Z}}}_{\geq 0}$, and extends to all $k\in {{\textbf{Z}}}$ by setting $\binom{m}{k} = 0$ if $k<0$. $p$ is the parity induced by the ${{\textbf{Z}}}_2$-grading of $\mathfrak{v}$.
A vertex algebra $V(\mathfrak{v})$ is the quotient $F(\mathfrak{v})/J$, where $J$ any proper ideal containing the vertex algebra ideal $I(\mathfrak{v})$.
By linearity, we may assume that the variables $x,y,z$ represent *monomials*. This assumption simplifies future arguments.
Although we will not be needing it much, we give the definition of a vertex algebra in the more general case that $F$ is not assumed to be generated by $\mathfrak{v}$.
\[def:generalVA\] In the case that $F$ is not generated by some $\mathfrak{v}$, we define the vertex algebra ideal $I$ more generally by replacing the function $N(u,v)$ with a function $N(x,y) \geq 0$ for all $x,y \in F$. Then a vertex algebra is the quotient $V = F/J$, where $J$ is any proper ideal containing $I$.
Upon forming the quotient $V$ (or $V(\mathfrak{v})$), the elements $\textbf{i}{[\![x;n]\!]}$ imply that ${\textbf{1}}$ is a *left* identity for the product $\circ_{-1}$ on $V$. Making use of the quasi-commutativity relation $\textbf{qc}{[\![x,{\textbf{1}};-1]\!]}$ on $V$, we see that ${\textbf{1}}$ is a *right* identity for $\circ_{-1}$ as well. The elements $\textbf{c}{[\![x,y;n]\!]}$ ensure that the tail ends of the series appearing in $\textbf{qc}{[\![x,y;n]\!]}$ and $\textbf{qa}{[\![x,y,z;m,n]\!]}$ get killed in $V$, so that the series are in fact convergent. The elements $\textbf{d}{[\![x,y;n]\!]}$ express that $D \equiv \circ_{-2}{\textbf{1}}$ is a derivation of all products in $V$. The elements $\textbf{qc}{[\![x,y;n]\!]}$ and $\textbf{qa}{[\![x,y,z;m,n]\!]}$ express the extent to which each product is not commutative or associative on $V$, as seen by comparing the leading term with the first term of the summation.
We will write $F$ and $V$ when we are making no assumptions about the existence of an underlying vector space $\mathfrak{v}$.
We will use the letters $u,v$ to denote elements of $\mathfrak{v}$ and their inclusions in $F(\mathfrak{v})$; the letters $x,y$ denote either general elements of some infinite free algebra. Thus when we refer to $N(u,v)$, we mean a function $N$ defined on $\mathfrak{v} \times \mathfrak{v}$, whereas $N(x,y)$ denotes a function defined on all of $F\times F$ or $F(\mathfrak{v}) \times F(\mathfrak{v})$. Also, an element $x\in F, F(\mathfrak{v})$ will also be used to denote its equivalence class $x+J \in V,V(\mathfrak{v})$.
From now on, to ease the notation, we will assume all elements are even, so that the parity $p(x)=0$ for all $x\in F, F(\mathfrak{v})$. Even so, with an appropriate adjustment of signs, every subsequent definition and result still applies to both even and odd elements.
The impact of the $u,v$-dependence of the integer $N(u,v)$ in the second set of generators $\textbf{c}{[\![u,v;n]\!]}$ is subtle but important. We are *not* guaranteed that there is a single number $n$ such that $u_nv \in I(\mathfrak{v})$ for all $u,v \in \mathfrak{v}$, but rather that for each pair $u,v$ there is a number $N(u,v)$ such that $u_nv \in I(\mathfrak{v})$ for all $n \geq N(u,v)$. In contrast, in all of the other sets of generators, the choice of $n$ is independent of $x$ and $y$. It is the content of Dong’s Lemma (proved in Proposition \[prop:dong\]) that in fact for any pair of elements $x,y \in F(\mathfrak{v})$ there is some number $N(x,y)\geq 0$ such that $x_ny \in I(\mathfrak{v})$ for all $n\geq N(x,y)$.
The function $N(u,v)$ has not been specified, meaning that the notation ideal $I(\mathfrak{v})$ has some ambiguity. Thus when we speak of *the* vertex algebra ideal $I(\mathfrak{v})$, we really mean *some* vertex algebra ideal $I(\mathfrak{v})$ for which $N(u,v)$ has been specified.
In contrast, the lack of specification of the integers $m$ and $n$ is harmless. As we will see in Proposition \[prop:B-equivalence\], the inclusion of any generator for $n=-1$ and $n\gg 0$ implies the inclusion of that family of generators for all $n \in {{\textbf{Z}}}$.
Equivalence to Borcherds’ definition
------------------------------------
For this section, we increase our scope to $F$ and $V$ as in Definition \[def:generalVA\], not assumed to be generated by $\mathfrak{v}$. Thus $I$ includes the larger set of generators $\textbf{c}{[\![x,y;n]\!]} = x_ny$ for $n \geq N(x,y)$ for some function $N(x,y) \geq 0$ on $F\times F$.
The key difference between our definition and others is the range of the integers $m, n$ in the products $\circ_m, \circ_n$ appearing in the generators of $I$. In our definition, the number of generators is severely reduced in that generally we have only $m,n=-1$ and all $m,n \gg 0$, whereas in the standard definitions $m$ and $n$ usually range over all of ${{\textbf{Z}}}$. Our only set of generators in which the product $\circ_n$ is indexed by all $n \in {{\textbf{Z}}}$ is $\textbf{e}{[\![x,y;n]\!]}$. As we will see in the upcoming Proposition \[prop:B-equivalence\], with the aid of this particular set of generators, all standard generators can be recovered.
\[prop:B-equivalence\] Definition \[def:generalVA\] of a vertex algebra is equivalent to Borcherds’ original definition.
Borcherds’ relations in a vertex algebra are almost the same as our generators for $I$. On one hand, his versions of $\textbf{i}{[\![x;n]\!]}$, $\textbf{d}{[\![x,y;n]\!]}$, $\textbf{qc}{[\![x,y;n]\!]}$, and $\textbf{qa}{[\![x,y,z;m,n]\!]}$ hold for *all* $m,n \in {{\textbf{Z}}}$. But offsetting these extra relations, his set lacks our $\textbf{e}{[\![x,y;n]\!]}$.
To see that Borcherds’ relations imply ours, one can check explicitly that $\textbf{e}{[\![x,y;n]\!]}$ is the sum of elements from Borcherds’ relations: $$\begin{aligned}
\textbf{e}{[\![x,y;n]\!]} &= \textbf{qa}{[\![x,{\textbf{1}},y;-2,n]\!]} \\
&\quad + \sum_{k\geq 0}\binom{-2}{k}(-1)^k \left(x_{-2-k}\textbf{i}{[\![y;n+k]\!]} - \textbf{i}{[\![x_ky;-2+n-k]\!]} \right).\end{aligned}$$
For the other direction, we must show that $I$ contains the elements $\textbf{d}{[\![x,y;n]\!]}$, $\textbf{qc}{[\![x,y;n]\!]}$, and $\textbf{qa}{[\![x,y,z;m,n]\!]}$ for all $m,n \in {{\textbf{Z}}}$. The argument is a downward induction.
We begin by showing that $\textbf{d}{[\![x,y;n-1]\!]} \in I$ whenever $\textbf{d}{[\![x,y;n]\!]} \in I$. As a base case, we are given that $\textbf{d}{[\![x,y;n]\!]} \in I$ for all $n \gg 0$. We have the readily checked identity $$\begin{aligned}
n\textbf{d}{[\![x,y;n-1]\!]} = D\textbf{e}{[\![x,y;n]\!]} - \textbf{d}{[\![Dx,y;n]\!]} - \textbf{e}{[\![Dx,y;n]\!]} - \textbf{e}{[\![x,Dy;n]\!]}.\end{aligned}$$ We see that on the left the index of $\textbf{d}$ is $n-1$ whereas on the right the index of $\textbf{d}$ is $n$, so $I$ contains $\textbf{d}{[\![x,y;n-1]\!]}$ whenever it contains $\textbf{d}{[\![x,y;n]\!]}$. This is precisely the (downward) inductive argument. This breaks down when $n=0$ since the left side vanishes. The base case is re-founded with the inclusion of $\textbf{d}{[\![x,y;-1]\!]}$ in our set of generators. In conclusion, $\textbf{d}{[\![x,y;n]\!]} \in I$ for all $n \in {{\textbf{Z}}}$.
For the rest of the proof we abbreviate $\textbf{d}{[\![x,y;n]\!]} + \textbf{e}{[\![x,y;n]\!]}$ as $\textbf{f}{[\![x,y;n]\!]}$ for all $n \in {{\textbf{Z}}}$.
To show that $\textbf{i}{[\![x;n]\!]}, \textbf{qc}{[\![xy;n]\!]}, \textbf{qa}{[\![xy,z;m,n]\!]} \in I$ for all $n \in {\textbf{Z}}$, $I$ already includes the base cases for all $m,n \gg 0$ and $m,n=-1$. The separate inductive arguments are then given by the equalities $$\begin{aligned}
n\textbf{i}{[\![x;n-1]\!]} &= \textbf{i}{[\![Dx;n]\!]} - D\textbf{i}{[\![{\textbf{1}},x;n]\!]} + \textbf{f}{[\![{\textbf{1}},x;n]\!]}, \\
n\textbf{qc}{[\![x,y;n-1]\!]} &= -\textbf{qc}{[\![Dx,y;n]\!]} + \textbf{e}{[\![x,y;n]\!]} - \sum_{k\geq 0} \frac{(-1)^{n+k}}{k!} D^k\textbf{f}{[\![y,x;n+k]\!]},\end{aligned}$$
$$\begin{aligned}
m \textbf{qa}{[\![x,y,z;&m-1,n]\!]} = -\textbf{qa}{[\![Dx,y,z;m,n]\!]} + \textbf{e}{[\![x,y;m]\!]}_nz \\
& - \sum_{k\geq 0}\binom{m}{k}(-1)^k \left(\textbf{e}{[\![x,y_{n+k}z;m-k]\!]} - (-1)^m y_{m+n-k} \textbf{e}{[\![x,z;k]\!]} \right),\end{aligned}$$
and $$\begin{aligned}
n \textbf{qa}{[\![x,y,z;&m,n-1]\!]} = -D\textbf{qa}{[\![x,y,z;m,n]\!]} + \textbf{qa}{[\![x,y,Dz;m,n]\!]} + \textbf{f}{[\![x_my,z;n]\!]} \\
& - \sum_{k \geq 0}\binom{m}{k}(-1)^k \left( \textbf{f}{[\![x,y_{n+k}z;m-k]\!]} - (-1)^m \textbf{f}{[\![y,x_kz;m+n-k]\!]} \right) \\
& - \sum_{k \geq 0}\binom{m}{k}(-1)^k \left( x_{m-k} \textbf{f}{[\![y,z;n+k]\!]} - (-1)^m y_{m+n-k} \textbf{f}{[\![x,z;k]\!]}\right).\end{aligned}$$
\[rmk:fields\] Other axioms for a vertex algebra feature a collection of *quantum fields* $x(\zeta) \in ({\operatorname{End}}V)[[\zeta, \zeta^{-1}]]$ (formal power series in $\zeta$ and $\zeta^{-1}$ with coefficients in ${\operatorname{End}}V$), and an even endomorphism $D$ of $V$ satisfying $[D,x(\zeta)] = \frac{d}{d\zeta} x(\zeta)$. Our formulation captures this data via the definitions $$\begin{aligned}
x(\zeta) &\triangleq \sum_{n\in {\textbf{Z}}}\frac{x \circ_n}{\zeta^{n+1}} \\
D &\triangleq \circ_{-2}{\textbf{1}}.\end{aligned}$$
Decompositions of F(v) {#sec:decompositions}
----------------------
$F(\mathfrak{v})$ has some useful linear decompositions. One of the decompositions we discuss, the *degree grading*, descends to a decomposition of the quotient $V(\mathfrak{v}) = F(\mathfrak{v}) / I(\mathfrak{v})$, precisely because each generator of $I(\mathfrak{v})$ has homogenous degree.
First we will discuss a set of decompositions of $F(\mathfrak{v})$ that are ordered by coarseness. To begin, we define a *monomial* as any element that can be written as a single (full binary rooted) tree whose leaves are elements of $\mathfrak{v}$. Then a very fine decomposition is of $F(\mathfrak{v})$ into a direct sum of subspaces spanned by single monomials. As examples, $\{v,\, v_2w,\, (u_{-2}v)_1w\}$ are monomials provided $u,v,w \in \mathfrak{v}$. The element $u_1v + w_1v$ is also a monomial because it can be written as $(u+w)_1v$. Two monomials are in the same summand precisely when corresponding leaf nodes are scalar multiples of one another.
Slightly coarser, we can group monomials by *product shape*, which pays attention no not only to be the shape of the tree, but also to the integer at each node. The product shape captures the sequence of products $\circ_n$ used to construct this element. Examples of elements of homogeneous product shape are $\{v_2w,\, u_5(v_2w) + x_5(y_2u),\, (u+v)_{-1}w + x_{-1}y\}$, where the variables represent monomials. Note that any monomial has homogeneous product shape. An element of homogeneous product shape can be represented by replacing all leaves with asterisks, in which case our three examples are now written $\{\ast_2\ast,\, \ast_5(\ast_2\ast) + \ast_5(\ast_2\ast),\, \ast_{-1}\ast + \ast_{-1}\ast\}$.
Coarser still is the *shape* of the tree alone, without regard to the integers at each node. In this case, we can ignore the products $\circ_n$ altogether, and simply group leaves with parentheses. Then the previous example becomes $\{\ast\ast,\, \ast(\ast\ast) + \ast (\ast\ast),\, \ast\ast + \ast\ast\}$.
And coarsest of all, we have the *length* of a monomial, which is simply its number of leaves $\ast$. Altogether, we have the ordering, from coarse to fine, $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ \text{length} & \text{shape} & \text{product shape} & \text{monomial} \\};
\path[->]
(m-1-1) edge (m-1-2)
(m-1-2) edge (m-1-3)
(m-1-3) edge (m-1-4);
\end{tikzpicture}$$
Note that the subspace $\mathfrak{v} \subset F(\mathfrak{v})$ is precisely the subspace of length 1. Thus in the decomposition by length, the subspaces of lengths 2 or greater are orthogonal to $\mathfrak{v}$.
We now present a degree grading that leads to a decomposition descending nicely onto the quotient $V(\mathfrak{v})$. Equivalently, the vertex algebra ideal $I(\mathfrak{v})$ decomposes according to the degree grading.
Given a direct sum decomposition of $\mathfrak{v}$, and assigning each summand a weight (not necessarily in ${{\textbf{Z}}}$), there is a number of ways a degree grading $|\cdot|$ is induced on $F(\mathfrak{v})$. Insisting that the element ${\textbf{1}}\in \mathfrak{v}$ has degree 0, and insisting that the grading be additive over the products $\circ_n$, meaning $$|x_ny| = |x| + |\circ_n| + |y|,$$ then such a grading will descend nicely to the vertex algebra $F(\mathfrak{v})/I(\mathfrak{v})$ only when the degree of the product $\circ_n$ is given by $$|\circ_n| \triangleq - n - 1.$$ Indeed, in this case it is easily verified that every generator of $I$ has homogeneous degree.
The decomposition of $F(\mathfrak{v})$ by degree is neither finer nor coarser than the decompositions above, but a common refinement can be found, giving the partial ordering diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=1.3em,
text height=1.5ex, text depth=0.25ex]
{ & \text{length} & \text{shape} & \text{prod. shape} & \text{monomial} \\ {|\cdot|} & \text{length} \cap |\cdot| & \text{shape} \cap |\cdot| & \text{prod. shape} \cap |\cdot| & \text{monomial} \cap |\cdot| \\};
\path[->]
(m-1-2) edge (m-1-3)
(m-1-3) edge (m-1-4)
(m-1-4) edge (m-1-5)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3)
(m-2-3) edge (m-2-4)
(m-2-4) edge (m-2-5)
(m-1-2) edge (m-2-2)
(m-1-3) edge (m-2-3)
(m-1-4) edge (m-2-4)
(m-1-5) edge (m-2-5);
\end{tikzpicture}$$
The generators of the ideal $I(\mathfrak{v})$ are homogeneous in none of these decompositions except for the degree grading $|\cdot|$. As a result, only the degree grading descends onto the vertex algebra $V(\mathfrak{v})$. Despite this fact, the we are still able to speak of *monomials* and *lengths* in $V(\mathfrak{v})$.
\[def:VAmonomiallength\] Let $V(\mathfrak{v}) = F(\mathfrak{v}) / J$ for some ideal $J \supset I(\mathfrak{v})$. We say an element $x \in V(\mathfrak{v})$ is a *monomial* if the coset $x + J$ contains a monomial in $F(\mathfrak{v})$. The *length* of $x$ is the length, in $F(\mathfrak{v})$, of the shortest element in the coset $x + J$.
Injectivity of *v* into *V(v) = F(v)/I(v)*
------------------------------------------
A reassuring feature of a vertex algebra generated by a unital vector space $\mathfrak{v}$ is that $\mathfrak{v}$, considered as a subspace of $F(\mathfrak{v})$, survives intact upon taking the quotient by $I(\mathfrak{v})$. We will delay the proof of this statement until Chapter \[sec:soupedupVA\], where it will be a corollary to the slightly stronger Theorem \[thm:Ainjectivity\]. In that theorem, we will prove that $\mathfrak{v}$ survives for a particular enlargement of the ideal $I(\mathfrak{v})$.
For now, we state this fact as a theorem without proof.
\[thm:Vinjectivity\] The map taking $\mathfrak{v}$ to its image in $V(\mathfrak{v}) = F(\mathfrak{v})/I(\mathfrak{v})$ is a monomorphism. Equivalently, $I(\mathfrak{v}) \cap \mathfrak{v} = \{0\}$.
This injectivity is independent of choice of the function $N(u,v)$. In particular, we may choose the most constrictive function possible, setting $N(u,v) \equiv 0$, which has the effect of enlarging $I(\mathfrak{v})$ and shrinking $V(\mathfrak{v})$ simultaneously. In particular, this choice implies that in $V(\mathfrak{v})$, $u_nv = 0$ for all $n\geq 0$ and all $u,v \in \mathfrak{v}$.
To keep our perspective, we point out that this injectivity could be destroyed if we enlarge $I(\mathfrak{v})$ further in other ways. We could easily collapse a part of $\mathfrak{v}$ by replacing $I(\mathfrak{v})$ with a larger ideal $J \supset I(\mathfrak{v})$ that, say, includes some element $v \in \mathfrak{v}$.
Commutator formula and Dong’s Lemma
-----------------------------------
In this section, for completeness, we present the long-established commutator formula between the *mode operators* $x_m$ and $y_n$ on general $F$ and $V$ (as in Definition \[def:generalVA\]). This formula helps us prove Dong’s Lemma \[prop:dong\] (originally proven in [@Li:1994sp]; see [@kac-1997] for another version), which is essential for the convergence of the quasi-commutativity and quasi-associativity relations in a vertex algebra $V(\mathfrak{v})$. In particular, it guarantees that vanishing of the products $u_nv$ for $u,v \in \mathfrak{v}$ and sufficiently large $n$ implies the vanishing of *all* products $x_ny$ for all $x,y \in V(\mathfrak{v})$ for sufficiently large $n$.
\[prop:commutator\] In a vertex algebra $V$, we have the commutator formula $$[x_m,y_n]z = \sum_{k\geq 0} \binom{m}{k} (x_ky)_{m+n-k}z.$$
It is straightforward to verify the identity $$\begin{aligned}
[x_m,y_n]z - \sum_{k\geq 0} \binom{m}{k}& (x_ky)_{m+n-k}z = \textbf{qa}{[\![x,y,z;-1,-1]\!]} - \textbf{qa}{[\![y,x,z;-1,-1]\!]} \\
& - \textbf{qc}{[\![y,x;-1]\!]} + \sum_{0\leq i \leq j} \frac{(-1)^{j+1} i!}{(j+1)!} \textbf{e}{[\![D^{j-i}(x_jy),z; -1-i]\!]}.\end{aligned}$$ The proposition then follows since all terms on the right side are in $I$, and therefore so is the left side, which is our desired relation.
\[lem:Nsymmetric\] $N(x,y)$ is symmetric. That is, if $x_ny \in I$ for all $n\geq N(x,y)$, then also $y_nx \in I$ for $n\geq N(x,y)$.
Assuming that $x_ny \in I$ for $n \geq N(x,y)$, we have $$y_nx = \textbf{qc}{[\![y,x;n]\!]} - \sum_{k\geq 0} \frac{(-1)^{n+k}}{k!} D^k(x_{n+k}y).$$ Every term on the right is in $I$, thus so is $y_nx$.
We now prove Dong’s Lemma for a vertex algebra $V(\mathfrak{v})$ generated by a unital vector space $\mathfrak{v}$.
\[prop:dong\] For any pair of elements $x,y \in F(\mathfrak{v})$, there exists some $N(x,y)\geq 0$ such that $x_ny \in I(\mathfrak{v})$ for all $n\geq N(x,y)$.
By linearity, we may assume that $x$ and $y$ are monomials. We will perform an induction on the length of $x_ny$.
The base case, length 2, is given to us by the inclusion $\textbf{c}{[\![u,v;n]\!]} \in I(\mathfrak{v})$ for all $n \geq N(u,v)$ and $u,v \in \mathfrak{v}$.
Now assume that the lemma is true for all monomials of length $p$. We wish to show that for a monomial $w_nz$ having length $p+1$ there exists some $N$ such that $w_nz \in I(\mathfrak{v})$ for all $n\geq N$. At least one of $w$ and $z$ necessarily factors further. By Lemma \[lem:Nsymmetric\], we may assume that $w$ factors, so that $w_nz = (x_ry)_nz$. By hypothesis, since any pair among $x,y$, and $z$ has length not exceeding $p$, there exists a sufficiently large $M$ such that all products $x_my, y_mz, x_mz$, for $m \geq M$, are in $I(\mathfrak{v})$. ($M$ may be the maximum of $N(x,y), N(x,z)$, and $N(y,z)$.)
We will proceed in two steps: first we will prove the proposition for $r\geq 0$, and then for $r<0$.
Consider the commutator formula from Proposition \[prop:commutator\] in the case $m \geq 2M, n=m-M$. The two terms $x_m(y_{m-M}z)$ and $y_{m-M}(x_mz)$ on the left side both vanish in $V(\mathfrak{v})$ since (as elements of $F(\mathfrak{v})$) they are in $I(\mathfrak{v})$ by hypothesis. Similarly, on the right side, all terms with $k\geq M$ vanish, leaving altogether $$\sum_{k=0}^{M-1} \binom{m}{k} (x_ky)_{2m-M-k}z = 0.$$ Consider as well the cases $(m-1,m-M+1), (m-2, m-M+2), \ldots, (m-M, m)$, giving the equations $$\sum_{k=0}^{M-1} \binom{m-1}{k} (x_ky)_{2m-M-k}z = 0,\,\ldots,\, \sum_{k=0}^{M-1} \binom{m-M}{k} (x_ky)_{2m-M-k}z = 0.$$ Altogether we have a system of $M+1$ homogeneous equations in $M$ unknowns, and the binomial coefficients force the solution to be trivial: $$(x_0y)_{2m-M}z = (x_1y)_{2m-M-1}z = \cdots = (x_{M-1}y)_{2m-2M+1}z = 0.$$ Equivalently, each monomial above is in the ideal $I(\mathfrak{v})$. This holds for all $m \geq 2M$, so for a monomial of length $p+1$ of the form $(x_ry)_nz$ with $r\geq 0$, the proposition is satisfied by the choice $n \geq N = 3M - r$. By Lemma \[lem:Nsymmetric\], this also holds for elements $z_n(x_ry)$, $r\geq 0$.
To prove the lemma for length $p+1$ elements of the form $(x_ry)_nz$ for $r<0$, we note that when $n \geq N = 3M-r$, then every term on the right side of the equation $$(x_ry)_nz = \textbf{qa}{[\![x,y,z;r,n]\!]} + \sum_{k\geq 0}\binom{r}{k}(-1)^k \left(x_{r-k}(y_{n+k}z) - (-1)^r y_{r+n-k}(x_kz) \right)$$ is in the ideal $I(\mathfrak{v})$. Indeed, the terms $x_{r-k}(y_{n+k}z)$ in the summation are in the ideal by hypothesis, since $n+k \geq M$; for the same reason, the terms $y_{r+n-k}(x_kz)$ for $k \geq M$ are in the ideal; and the terms $y_{r+n-k}(x_kz)$ for $0 \leq k < M$ are in the ideal in light of the result above for $r \geq 0$. Thus $(x_ry)_nz \in I(\mathfrak{v})$ for all $n \geq N = 3M-r$ when $r < 0$.
This concludes the proof of Dong’s Lemma.
Souped-up Lie algebras {#sec:soupedupLA}
======================
In the well-known *current algebra construction* (see [@kac-1997], Chapter 2.5), a vertex algebra is generated from a Lie algebra $\mathfrak{g}$. This vertex algebra actually contains a one-dimensional central extension of $\mathfrak{g}$, with the Lie bracket played by the product $\circ_0$, and the central extension generated by the vacuum element ${\textbf{1}}$.
With some additional simple relations (addressed next in Chapter \[sec:soupedupVA\]), a vertex algebra may contain a much richer classical structure, combining a Lie algebra $\mathfrak{g}$ and a $\mathfrak{g}$-module $\mathfrak{a}$. In this chapter we define this structure, a *souped-up Lie algebra*, and give several popular examples. Our final example will be a combination of the exterior algebra of differential forms $\Omega(M)$, the Lie algebra of contraction operators $\iota_X$ by vector fields $X \in \mathfrak{X}(M)$, sections of a vector bundle $E\to M$, the Lie algebra of infinitesimal gauge transformations on the space of sections, and a connection $\nabla$. This robust souped-up Lie algebra will be the basis of our final construction: a chiral vector bundle, in Chapter \[sec:chiralDG\].
Definition of a souped-up Lie algebra
-------------------------------------
We begin by defining a *souped-up Lie algebra*.
\[def:soupedupLA\] Let $\mathfrak{a}$ be an associative commutative unital algebra and let $\mathfrak{g}$ be a Lie algebra such that
1. $\mathfrak{g}$ is a left $\mathfrak{a}$-module, this product written as juxtaposition.
2. $\mathfrak{a}$ is a two-sided $\mathfrak{g}$-module in which $\mathfrak{g}$ acts by derivations, written as the Lie bracket $[\cdot, \cdot]$.
3. The two products are compatible in that $[g,ah] = [g,a]h + a[g,h]$ for $a \in \mathfrak{a}$ and $g,h \in \mathfrak{g}$.
Interpreting $\mathfrak{a}$ as an abelian Lie algebra, the *souped-up Lie algebra* $\mathfrak{s}(\mathfrak{g}, \mathfrak{a})$ is defined as the semidirect sum Lie algebra $\mathfrak{g} \oplus \mathfrak{a}$ with the left multiplication by $\mathfrak{a}$.
It is permissible that a souped-up Lie algebra has additional relations, and it will still be considered a souped-up Lie algebra as long as ${\textbf{1}}\neq 0$.
\[rmk:soupedupLA\] It might be tempting to extend the left multiplication of $\mathfrak{a}$ on $\mathfrak{g}$ to a two-sided multiplication by symmetry, effectively adding the relation $s_{-1}a = as$, but as we will see below in Proposition \[prop:leftmult\], this relation is too strong to carry over into a vertex algebra, even though it poses no problems to the underlying souped-up Lie algebra.
Examples of souped-up Lie algebras {#sec:SLAexamples}
----------------------------------
Souped-up Lie algebras include the following examples:
An associative commutative unital algebra $\mathfrak{a}$ is isomorphic to the souped-up Lie algebra $\mathfrak{s} (0, \mathfrak{a})$.
Let $\mathfrak{g}$ be a Lie algebra, and let a new element $c$ generate a central extension of $\mathfrak{g}$. Letting $\mathfrak{a} = {\operatorname{span}}\{c\}$, we have a souped-up Lie algebra $\mathfrak{s}(\mathfrak{a} \otimes \mathfrak{g}, \mathfrak{a})$.
Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{v}$ a $\mathfrak{g}$-module. Let $\mathfrak{a}$ be the associative commutative unital algebra $S\mathfrak{v} \otimes \Lambda \mathfrak{v}$, which is also a module for $\mathfrak{g}$. We may convert $\mathfrak{g}$ to a left module for $\mathfrak{a}$ by extending its coefficients: $\mathfrak{a} \otimes \mathfrak{g}$. Altogether, we have the souped-up Lie algebra $\mathfrak{s}(\mathfrak{a} \otimes \mathfrak{g}, \mathfrak{a})$. The bracket on the Lie algebra underlying $\mathfrak{s}$ is given by $$[a\otimes s, b] \triangleq a \otimes [s,b].$$
Let $\mathfrak{X}$ be the Lie algebra of vector fields on a smooth manifold $M$, and let $\Omega^0$ be the algebra of smooth functions on $M$. $\mathfrak{X}$ acts on $\Omega^0$ by the Lie derivative and $\mathfrak{X}$ is a left module for ${\Omega}^0$, the latter multiplication compatible with the Lie derivative action. Then $\mathfrak{s}(\mathfrak{X}, {\Omega}^0)$ is a souped-up Lie algebra.
Let $\mathfrak{sX}$ be the Lie superalgebra of vector fields on $M$, and let $\Omega$ be the superalgebra of differential forms on $M$. $\mathfrak{sX}$ is the semidirect sum Lie algebra formed by $\mathfrak{X}$ acting on $\Omega$ as Lie derivatives ($(X,\omega) \mapsto \mathcal{L}_X \omega)$, and a second copy of $\mathfrak{X}$ acting as the *odd* Lie algebra of contraction operators ($(Y,\omega) \mapsto \iota_Y \omega$). The Lie bracket between any two elements of $\mathfrak{sX}$ is given by their commutator as operators on $\Omega$. Altogether, this leads to the bracket $$\begin{aligned}
[ \mathcal{L}_X, \mathcal{L}_Y] &= \mathcal{L}_{[X,Y]} \\
[\mathcal{L}_X, \iota_Y] &= \iota_{[X,Y]} \\
[\iota_X, \iota_Y] &= 0.\end{aligned}$$ We elevate this Lie algebra and its module to a souped-up Lie algebra by extending the coefficients of $\mathfrak{sX}$ to include $\Omega$, giving altogether $\mathfrak{s} ( \Omega \otimes_{\Omega^0} \mathfrak{s} \mathfrak{X}, \Omega)$.
\[ex:deRham\] We can add the exterior derivative $d$, regarded as an *odd* element, to the Lie algebra $\mathfrak{s}\mathfrak{X}$ from the previous example, forming the Lie algebra $\mathfrak{s}\mathfrak{X}_d$. In fact, this Lie algebra is entirely generated by the elements $\{\iota_X \mid X\in \mathfrak{X}\}$ and $d$, since we have the odd bracket $[d, \iota_X] = \mathcal{L}_X$ (the famous Cartan formula for $\mathcal{L}_X$). Altogether $\mathfrak{s}\mathfrak{X}_d$ has the brackets defined on $\mathfrak{s}\mathfrak{X}$, plus $$\begin{aligned}
[\mathcal{L}_X, d] &= 0 \\
[d, \iota_X] &= \mathcal{L}_X \\
[d,d] &= 0.\end{aligned}$$ The associated souped-up Lie algebra is then $\mathfrak{s} ( \Omega \otimes_{\Omega^0} \mathfrak{s} \mathfrak{X}_d, \Omega)$.
\[ex:gauge\] Consider a smooth vector bundle $E\to M$ with structure Lie group $G$. The group of gauge transformations $\mathcal{G}$ is a subbundle of $\Gamma ({\operatorname{Aut}}E)$. The Lie algebra $L\mathcal{G}$ to the gauge group is then a subbundle of $\Gamma ({\operatorname{End}}E)$. The action of $L\mathcal{G}$ on the space of sections $\Gamma E$ induces an action on the spaces of sections on the symmetric and antisymmetric tensor algebras $SE$ and $\Lambda E$. Then we can form the souped-up Lie algebra $\mathfrak{s}(\Gamma (SE \otimes \Lambda E) \otimes_{\Omega^0} L\mathcal{G}, \Gamma (SE \otimes \Lambda E))$.
\[ex:connectionLA\] This final example is of primary importance to this paper. We combine Examples \[ex:deRham\] and \[ex:gauge\] to form what we call the *connection Lie algebra* $\mathfrak{s} \mathfrak{X}_{\nabla}$.
Consider a $G$-vector bundle with connection $(E,\nabla)$. We designate as $\mathfrak{a}$ the algebra $\Omega \otimes \Gamma (SE \otimes \Lambda E)$, the tensor product of the exterior algebra of differential forms with the supersymmetric tensor algebra of $E$.
The Lie algebra $\mathfrak{g}$ is the subalgebra of ${\operatorname{End}}\mathfrak{a}$ generated by the contraction operators $\{\iota_X \mid X \in\mathfrak{X}\}$, the connection $\nabla$, and the gauge Lie algebra $L\mathcal{G}$ from the previous example. The first set of operators (all odd elements) acts on $\omega \otimes s \in \Omega \otimes \Gamma (SE \otimes \Lambda E)$ by performing as usual on the first factor and ignoring the second. The connection, also an odd element, acts as usual on the entire product, and an infinitesimal gauge transformation $A$, an even element, acts as usual on the second factor and ignores the first. Altogether, we have the actions (for $\omega$ with homogeneous degree $p$) $$\begin{aligned}
\iota_X (\omega \otimes s) &= \iota_X\omega \otimes s \\
\nabla (\omega \otimes s) &= d\omega \otimes s + (-1)^p \omega \otimes \nabla s\\
A (\omega \otimes s) &= \omega \otimes As.\end{aligned}$$ The Lie bracket between any pair is simply their commutator as operators on $\mathfrak{a}$. We denote this Lie algebra $\mathfrak{s}\mathfrak{X}_{\nabla}$.
It is straightforward to check that $[\iota_X,\iota_Y] = [\iota_X, A] = 0$. Of course, the bracket of $\nabla$ with itself is $[\nabla,\nabla] = 2\nabla^2$, twice the curvature 2-form.
Finally, we combine the algebra $\Omega \otimes \Gamma (SE \otimes \Lambda E)$ with the Lie algebra $\mathfrak{s} \mathfrak{X}_{\nabla}$ to form the souped-up Lie algebra $$\mathfrak{s} \equiv \mathfrak{s}( \Omega \otimes \Gamma (SE \otimes \Lambda E) \otimes \mathfrak{s} \mathfrak{X}_{\nabla}, \Omega \otimes \Gamma (SE \otimes \Lambda E)),$$ where all tensor products are over $\Omega^0$.
The souped-up connection Lie algebra contains all the standard algebraic and geometric information regarding the underlying manifold $M$. The subspace corresponding to $i=j=k=0$ in the first term in the direct sum decomposition $$\begin{aligned}
\label{eq:soupconnLAdecomp}
&\mathfrak{s} = \bigoplus_{0 \leq i,j,k} \Omega^i\otimes \Gamma (S^jE) \otimes \Gamma (\Lambda^k E) \otimes \mathfrak{s} \mathfrak{X}_{\nabla} \oplus \bigoplus_{0 \leq i,j,k} \Omega^i \otimes \Gamma (S^j E) \otimes \Gamma (\Lambda^k E)\end{aligned}$$ is the connection Lie algebra $\mathfrak{s} \mathfrak{X}_{\nabla}$. The subspace corresponding to $j=k=0$ in the second grouping is precisely the exterior algebra of differential forms $\Omega$. Noting that the actions of $\mathfrak{s}\mathfrak{X}_{\nabla}$ and $\mathfrak{s}\mathfrak{X}$ coincide when restricted to $\Omega$, then altogether we can see that the souped-up Lie algebra $\mathfrak{s}(\Omega \otimes \mathfrak{s}\mathfrak{X}_d, \Omega)$ is a summand of $\mathfrak{s}$. We will refer to this containment when we construct a sheaf of vertex algebras from this souped-up Lie algebra.
The particular element $\mathring{\nabla}_X \triangleq [\nabla, \iota_X] \equiv \nabla\iota_X + \iota_X\nabla$ acts on $\omega \otimes s$ by the beautiful formula $$\mathring{\nabla}_X (\omega\otimes s) = \mathcal{L}_X \omega \otimes s + \omega \otimes \nabla s.$$ This is a symmetrized version of the usual covariant derivative $\nabla_X \triangleq \iota_X\nabla$, and can be though of as the covariant analog of the Cartan formula $\mathcal{L}_X = d\iota_X + \iota_X d$. From this formula one can then derive the nice bracket $$[\mathring{\nabla}_X, \iota_Y] = \iota_{[X,Y]}.$$
Souped-up Lie algebra modules
-----------------------------
Upon generating a vertex algebra from a souped-up Lie algebra $\mathfrak{s}$, we will show that $\mathfrak{s}$ has a natural action on the vertex algebra. This will be an instance of a *souped-up Lie algebra module*.
\[def:smodule\] A vector space $\mathfrak{v}$ is a module for a souped-up Lie algebra $\mathfrak{s}(\mathfrak{g}, \mathfrak{a})$ if it is a left module for the underlying semidirect sum Lie algebra $\mathfrak{g} \oplus \mathfrak{a}$ and a left module for the algebra $\mathfrak{a}$. That is, we require the two representation $\rho$ and $\sigma$: $$\begin{aligned}
\rho([s,t], v) &\mapsto \rho(s, \rho(t,v)) - \rho(t, \rho(s,v)) \\
\sigma(ab,v) &\mapsto \sigma (a,\sigma(b,v))\end{aligned}$$ for $a,b\in \mathfrak{a}$, $s,t \in \mathfrak{s}$, and $v \in \mathfrak{v}$.
In this case, we say $\mathfrak{v}$ is an $\mathfrak{s}$-module.
Although in $\mathfrak{s}$ we have the relation $[ag,h] = a[g,h] + [a,h]g$, an $\mathfrak{s}$-module does not include the relation $\rho(ag, v) = \sigma(a, \rho(g,v)) + \sigma(\rho(a,v), g)$, because in the final term, $\rho(a,v)$ is not an element of $\mathfrak{a}$, so the notation makes no sense.
Vertex algebras generated by souped-up Lie algebras {#sec:soupedupVA}
===================================================
We now generate a vertex algebra $V(\mathfrak{s})$ from a souped-up Lie algebra $\mathfrak{s} \equiv \mathfrak{s}(\mathfrak{g}, \mathfrak{a})$. The procedure is first to generate the infinite free algebra $F(\mathfrak{s})$, and then to take the quotient by a particular ideal containing not only the vertex algebra ideal from Definition \[def:VA\], but also a set of generators that capture the souped-up structure of $\mathfrak{s}$.
Definition of a vertex algebra generated by a souped-up Lie algebra
-------------------------------------------------------------------
\[def:SVA\] Let $\mathfrak{s} \equiv \mathfrak{s}(\mathfrak{g},\mathfrak{a})$ be a souped-up Lie algebra and let $F(\mathfrak{s})$ be the infinite free algebra generated by the vector space underlying $\mathfrak{s}$, with grading induced by $|\mathfrak{g}| = 1$ and $|\mathfrak{a}| = 0$. Choosing some function $N(s,t) \geq 0$ on $\mathfrak{s} \times \mathfrak{s}$, the $I(\mathfrak{s})$ is generated by the generators from Definition \[def:VA\] (using $N(s,t)$) and the sets
Lie algebra $\mathfrak{s}$:
: $\textbf{s}{[\![s,t]\!]} \triangleq s_0t - [s,t]$
algebra $\mathfrak{a}$:
: $\textbf{a}{[\![a,s]\!]} \triangleq a_{-1}s - as$
$\mathfrak{a}$-module:
: $\textbf{am}{[\![a,b,x]\!]} \triangleq (ab)_{-1}x - a_{-1}(b_{-1}x)$
for $a,b\in \mathfrak{a}$, $s,t \in \mathfrak{s}$, and monomials $x \in F(\mathfrak{s})$.
The vertex algebra $V(\mathfrak{s})$ is defined as the quotient $F(\mathfrak{s})/J$ where $J$ is any proper ideal that contains $I(\mathfrak{s})$ and (as is true for $I(\mathfrak{s})$) is decomposable into $|\cdot|$-homogeneous subspaces.
The requirement that ideal $J \supset I(\mathfrak{s})$ be decomposable into $|\cdot|$-homogeneous subspaces implies that $F(\mathfrak{s}) / J$ retains a $|\cdot|$-grading.
As we mentioned earlier, any structure beyond the underlying unital vector space of $\mathfrak{s}$ is forgotten in $F(\mathfrak{s})$. In contrast, the ideals $I(\mathfrak{v})$ and $I(\mathfrak{s})$ are distinct, the former being properly contained in the latter.
Regarding the function $N$, we will use $s,t$ to denote elements of $\mathfrak{s}$. Thus $N(s,t)$ is defined only on $\mathfrak{s} \times \mathfrak{s}$.
The elements $\textbf{s}{[\![s,t]\!]}$ in $I(\mathfrak{s})$ will ensure that the bracket between elements of $\mathfrak{s}$ carries over to $V(\mathfrak{s})$ as the product $\circ_0$. Similarly, the elements $\textbf{a}{[\![a,x]\!]}$ will transfer the associative commutative product on $\mathfrak{s}$ to the product $\circ_{-1}$. The elements $\textbf{am}{[\![a,b,x]\!]}$ will ensure that $V(\mathfrak{s})$ is a left module for $\mathfrak{a}$, as we will prove in Theorem \[thm:smodule\].
For the record, there is some innocuous redundancy between $\textbf{a}{[\![a,s]\!]}$ and $\textbf{am}{[\![a,b,x]\!]}$. For example, one can check that $$\textbf{am}{[\![a,b,s]\!]} = \textbf{a}{[\![ab,s]\!]} - \textbf{a}{[\![a,bs]\!]} - a_{-1}\textbf{a}{[\![b,s]\!]}$$ for $a,b \in \mathfrak{a}$ and $s \in \mathfrak{s}$. There is also marginal redundancy between $\textbf{i}{[\![x;n]\!]}$, $\textbf{s}{[\![s,t]\!]}$, and $\textbf{a}{[\![a,s]\!]}$, since $\textbf{i}{[\![s;0]\!]} = \textbf{s}{[\![{\textbf{1}},s]\!]}$ and $\textbf{i}{[\![s;-1]\!]} = \textbf{a}{[\![{\textbf{1}},s]\!]}$. Lastly, it interesting to note that $\textbf{i}{[\![x;n]\!]}$ may be replaced by the requirement that $N(s,{\textbf{1}}) = 1$ for $s \in \mathfrak{s}$ in $\textbf{c}{[\![s,t;n \geq N(s,t)]\!]}$. We will not prove or use this fact. The proof uses induction on the length of an element.
\[rmk:CDRcomparison\] The inclusion of the generators $\textbf{s}{[\![s,t]\!]}$ in $I(\mathfrak{s})$ likens this construction to the current algebra construction (see [@kac-1997]), in which a Lie algebra is used to generate a vertex algebra with the equivalence $[s,t] = s_0t$. This construction is also used to define the chiral de Rham sheaf, in which each vertex algebra captures the Lie superalgebra $\mathfrak{s} \mathfrak{X}$ of vector fields in a semidirect sum with its module $\Omega$. This is akin to our souped-up Lie algebra $\mathfrak{s}(\Omega \otimes \mathfrak{s} \mathfrak{X}, \Omega)$, but without the enhanced ring of coefficients for $\mathfrak{s} \mathfrak{X}$.
Our inclusion of the generators $\textbf{a}{[\![a,s]\!]}$ and $\textbf{am}{[\![a,b,x]\!]}$ in $I(\mathfrak{s})$ is where a our construction departs from the construction of the chiral de Rham sheaf. In fact, this set of generators shrinks the vertex algebra in comparison, and in fact *kills* it outright unless, as we discuss in Proposition \[prop:Nconstraint\], we compensate by removing generators from $I(\mathfrak{s})$ elsewhere. Specifically, we permit $N(s,t)$ to be greater than 1, whereas the chiral de Rham sheaf uses $N \equiv 1$.
To summarize, in comparison to the construction of the chiral de Rham sheaf, we include the extra generators $\textbf{a}{[\![a,s]\!]}$ and $\textbf{am}{[\![a,b,x]\!]}$, but simultaneously toss out some generators $s_nt$ for as many $n$ as we need or want.
\[thm:smodule\] The vertex algebra $V(\mathfrak{s})$ is an $\mathfrak{s}$-module (Definition \[def:smodule\]), with the action given by $$\begin{aligned}
(s,x) &\mapsto s_0x \\
(a,x) &\mapsto a_{-1}x\end{aligned}$$ for $a \in \mathfrak{a}$, $s \in \mathfrak{s}$, and $x\in V(\mathfrak{s})$.
We must verify the two equalities $$\begin{aligned}
[s,t]_0x &= s_0(t_0x) - t_0(s_0x) \\
(ab)_{-1}x &= a_{-1}(b_{-1}x)\end{aligned}$$ in $V(\mathfrak{s})$.
The second follows directly from $\textbf{am}{[\![a,b,x]\!]} \in I(\mathfrak{s})$, and the first follows from $$\textbf{qa}{[\![s,t,x;0,0]\!]} - \textbf{s}{[\![s,t]\!]}_0x \equiv [(s_0t)_0x - s_0(t_0x) + t_0(s_0x)] - (s_0t - [s,t])_0x$$ in $I(\mathfrak{s})$.
Injectivity of *s* into *V(s) = F(s)/I(s)*
------------------------------------------
The feature we most desire in a vertex algebra $V(\mathfrak{s})$ generated from a souped-up Lie algebra $\mathfrak{s}$ is that it contains $\mathfrak{s}$ as a subalgebra, with the bracket $[\cdot,\cdot]$ played by $\circ_0$ and the scalar multiplication by $\mathfrak{a}$ played by $\circ_{-1}$. We know this to be true in two basic cases: when $\mathfrak{g} = 0$, in which case $\mathfrak{s}(0, \mathfrak{a}) \cong \mathfrak{a}$ (Theorem \[thm:Ainjectivity\]); and $\mathfrak{s}(\mathfrak{g}, {\textbf{C}})$, where ${\textbf{C}}$ is the trivial representation for $\mathfrak{g}$. In this case, $\mathfrak{s}$ is really just a central extension of $\mathfrak{g}$ by the ring of coefficients ${\textbf{C}}$ (Theorem \[thm:Ginjectivity\]). However, for a general souped-up Lie algebra $\mathfrak{s}$, it is only *conjectured* that $\mathfrak{s}$ survives intact in $V(\mathfrak{s})$ (Conjecture \[conj:injectivity\]). We will discuss compelling reasons for this conjecture below.
First we state and prove the two known cases.
\[thm:Ainjectivity\] The map taking the souped-up Lie algebra $\mathfrak{s} (0, \mathfrak{a}) \cong \mathfrak{a}$ to its image in $V(\mathfrak{s}) = F(\mathfrak{s}) / I(\mathfrak{s})$, with $N(s,t) \equiv 0$, is injective. Equivalently, $I(\mathfrak{s}) \cap \mathfrak{s} = \{0\}$.
This proof would require very little effort if every generator of $I(\mathfrak{s})$ had homogeneous length at least 2. In that case, the *length* decomposition of $F(\mathfrak{s})$ would descend onto $V(\mathfrak{s})$, and then it would suffice to note that $\mathfrak{s}$ and the ideal $I(\mathfrak{s})$ intersect trivially since the former space in the length 1 summand, while $I(\mathfrak{s})$ is orthogonal to that. Thus nothing in $\mathfrak{s}$ is killed in the quotient $F(\mathfrak{s}) / I(\mathfrak{s})$.
Alas, this is not the case, since the generators $\textbf{a}{[\![a,s]\!]} \equiv a_{-1}s - as$ do not have homogeneous length. (All other generators do have homogeneous length, including $\textbf{s}{[\![s,t]\!]} \equiv s_0t - [s,t]$, since $[s,t] = 0$ in this setup.)
We will construct a homomorphism $R$ (of vector spaces, not of infinite free algebras) from $F(\mathfrak{s})$ to itself that fixes every element of $\mathfrak{s}$ and projects $I(\mathfrak{s})$ onto the orthogonal complement of the length 1 subspace of $F(\mathfrak{s})$. Since $\mathfrak{s}$ is the length 1 subspace of $F(\mathfrak{s})$, it follows that the intersection $I(\mathfrak{s}) \cap \mathfrak{s}$ is trivial, since apparently any element in the intersection has length 1 (being in $\mathfrak{s}$) and length other than 1 (being in the image of $R$) simultaneously.
To begin, we define the homomorphism of vector spaces $r$ by the rules $$\begin{aligned}
r(a) &\triangleq a \\
r(a_0b) &\triangleq [a,b] \equiv 0 \\
r(a_{-1}b) &\triangleq ab \\
r({\textbf{1}}_{-1}x) &\triangleq x \\
r(x_{-1}{\textbf{1}}) &\triangleq x \\
r(x_ny) &\triangleq r(x)_n r(y) \text{ otherwise},\end{aligned}$$ for $a,b \in \mathfrak{s} = \mathfrak{a}$ and $x,y \in F(\mathfrak{s})$, and extending linearly.
We then define the projection $R \triangleq r^{\infty}$, the recursive application of $r$. Since $r$ systematically shortens the syntactic strings, deleting any monomial with the substring “$a_0b$," removing “$_{-1}$" from “$a_{-1}b$," eliminating the substrings “$_{-1}{\textbf{1}}$" and “${\textbf{1}}_{-1}$" altogether, and leaving the string alone otherwise, we see that any element stabilizes after a finite number of applications of $r$. This shows that $R$ is indeed a projection.
The fact that $R$ fixes $\mathfrak{a}$ follows precisely from the first rule.
It remains to show that $R$ projects $I(\mathfrak{s})$ onto the orthogonal complement of the subspace of elements with length 1. It suffices to show that this is true for the generators of $I(\mathfrak{s})$. Indeed, if a generator is killed by $R$, then so is any multiple of that generator. On the other hand, if the image of a generator has length at least 2, then so does the image of any multiple. Altogether, it follows that the entire ideal $I(\mathfrak{s})$ is also mapped under $R$ to an element orthogonal to the subspace of length 1, and hence orthogonal to $\mathfrak{s}$.
We begin with the sets of generators $\textbf{s}{[\![a,b]\!]} \equiv s_0t - [a,b]$, $\textbf{a}{[\![a,b]\!]} \equiv a_{-1}b - ab$, $\textbf{am}{[\![a,b,x]\!]} \equiv (ab)_{-1}x - a_{-1}(b_{-1}x)$, and $\textbf{c}{[\![a,b;n]\!]}$. Applying $R$, we have $$\begin{aligned}
R(\textbf{s}{[\![a,b]\!]}) &= 0 \\
R(\textbf{a}{[\![a,b]\!]}) &= 0 \\
R(\textbf{am}{[\![a,b,x]\!]}) &= \textbf{am}{[\![a,b,R(x)]\!]} \\
R(\textbf{c}{[\![a,b;n]\!]}) &= \textbf{c}{[\![a,b;n]\!]}.\end{aligned}$$ In the third case, by inspection we can see that the image is actually 0 in those cases where $x \in \mathfrak{s}$. Regardless, in all cases above, we see that the image is orthogonal to the subspace $\mathfrak{s}$, since all elements have length $\neq 1$.
Next, we address the generators $\textbf{d}{[\![x,y;n]\!]} \equiv (x_ny)_{-2}{\textbf{1}}- (x_{-2}{\textbf{1}})_ny - x_n(y_{-2}{\textbf{1}})$. There are the following cases: $$\begin{aligned}
R(\textbf{d}{[\![a,b;0]\!]}) &= 0 \\
R(\textbf{d}{[\![a,b;-1]\!]}) &= (ab)_{-2}{\textbf{1}}- (a_{-2}{\textbf{1}})_{-1}b - a_{-1}(b_{-2}{\textbf{1}}) \\
R(\textbf{d}{[\![x,{\textbf{1}};-1]\!]}) &= R(x)_{-2}{\textbf{1}}- R(x)_{-2}{\textbf{1}}- R(x)_{-1}({\textbf{1}}_{-2}{\textbf{1}}) \\
R(\textbf{d}{[\![{\textbf{1}},x;-1]\!]}) &= R(y)_{-2}{\textbf{1}}- ({\textbf{1}}_{-2}{\textbf{1}})_{-1}R(y) - R(y)_{-2}{\textbf{1}}\\
R(\textbf{d}{[\![x,y;n]\!]}) &= (R(x)_nR(y))_{-2}{\textbf{1}}- (R(x)_{-2}{\textbf{1}})_nR(y) - R(x)_n(R(y)_{-2}{\textbf{1}}).\end{aligned}$$ In each case, the image is again orthogonal to $\mathfrak{s}$.
The proofs for the remaining generators are of a similar spirit, and left to the reader.
This argument was independent of specification of $N(s,t)$. In this case, we may choose $N(s,t) \equiv 0$, which is as strong as possible on $\mathfrak{s} \times \mathfrak{s}$.
If we let $\mathfrak{v}$ be the vector space underlying $\mathfrak{s}$, then this also proves Theorem \[thm:Vinjectivity\] since $I(\mathfrak{v}) \subset I(\mathfrak{s})$ and hence $I(\mathfrak{v}) \cap \mathfrak{v} \subset I(\mathfrak{s}) \cap \mathfrak {s} = \{0\}$.
The other basic case for which we can prove that $\mathfrak{s}$ injects into $V(\mathfrak{s})$ is a complement to the first case. In this second case, the Lie algebra may be robust, while the associative commutative unital algebra is played by ${\textbf{C}}$.
\[thm:Ginjectivity\] Let ${\textbf{C}}$ be regarded as a trivial module for a Lie algebra $\mathfrak{g}$ with coefficients in ${\textbf{C}}$. The map taking the souped-up Lie algebra $\mathfrak{s}(\mathfrak{g}, {\textbf{C}})$ to its image in $V(\mathfrak{s}) = F(\mathfrak{s}) / I(\mathfrak{s})$, with $N(s,t) \equiv 1$ on $\mathfrak{s} \times \mathfrak{s}$, is injective. Equivalently, $I(\mathfrak{s}) \cap \mathfrak{s} = \{0\}$.
For this souped-up Lie algebra, since the algebra $\mathfrak{a}$ is played by ${\textbf{C}}$, the generators $\textbf{a}{[\![a,s]\!]}$ and $\textbf{am}{[\![a,b,x]\!]}$ are easily seen to be combinations of ${\textbf{1}}_{-1}s - s$. In this case, we may toss out those sets of generators. If we then choose $N(s,t) \equiv 1$ so that $I(\mathfrak{s})$ contains all elements $s_{\geq 1}t$, it follows that $I(\mathfrak{s})$ is precisely the ideal of relations in the well-known current algebra construction ([@kac-1997], [@LL-2007]) with vanishing bilinear form $B$ on $\mathfrak{g}$. In a current algebra, it is known that the entire centrally extended generating Lie algebra survives intact in the quotient $V(\mathfrak{s})$, and the theorem is proved.
These two theorems leads us to conjecture that this injectivity holds for an arbitrary souped-up Lie algebra.
\[conj:injectivity\] There exists some constant $N$ such that, for any souped-up Lie algebra $\mathfrak{s}$, we have $I(\mathfrak{s}) \cap \mathfrak{s} = \{0\}$ with $N(s,t) \equiv N$.
For the remainder of this paper, we will assume Conjecture \[conj:injectivity\]. The following two propositions are necessary conditions that are implied by this conjecture.
\[prop:leftmult\] In general, even if $\mathfrak{s}$ is regarded as a two-sided module for $\mathfrak{a}$ with the relation $as = sa$, the set of elements $\textbf{a}{[\![a,s]\!]} \equiv a_{-1}s - as \in I(\mathfrak{s})$ may not be enlarged to include the elements $s_{-1}a - as$.
There are cases where such an addition would not affect the vertex algebra $V(\mathfrak{s})$ adversely. These include the two examples in Theorems \[thm:Ainjectivity\] and \[thm:Ginjectivity\]. But there are important cases in which adding the generators corresponding to a right multiplication by $\mathfrak{a}$ actually collapses $V(\mathfrak{s})$ to 0
It suffices to exhibit a souped-up Lie algebra in which the inclusion of a particular element $s_{-1}a - as$ forces $I(\mathfrak{s})$ to include ${\textbf{1}}$. The crux is to find three elements $a \in \mathfrak{a}$, $g,h \in \mathfrak{g}$ such that $[h,[g,a]] = {\textbf{1}}$.
We choose $\mathfrak{g}$ to be the Lie algebra of vector fields on the real line ${{\textbf{R}}}$ and $\mathfrak{a}$ to be the algebra of differentiable functions on ${{\textbf{R}}}$. Then letting $b$ be a coordinate function on ${{\textbf{R}}}$, we let $a = b^2/2$ and $g$ be the coordinate vector field $\partial/\partial b$. With these choices, we have defined a souped-up Lie algebra in which $[g,[g,a]] = {\textbf{1}}$.
Suppose we include the generators $\{ g_{-1}a - ag\}$ in $I(\mathfrak{s})$. Then we can express ${\textbf{1}}$ as a combination of elements of $I(\mathfrak{s})$ by the equation $$\begin{aligned}
{\textbf{1}}&= [g,[g,a]] \\
&= \textbf{qc}{[\![a,g;-1]\!]}_1 h - \textbf{a}{[\![a,g]\!]} + (g_{-1}a - ag) - \textbf{e}{[\![g_0a,g;1]\!]} + \textbf{s}{[\![g,a]\!]}_0g + \textbf{s}{[\![[g,a],g]\!]} \\
& \quad - \sum_{k \geq 2} \frac{(-1)^{-1+k}}{k!} \left( \textbf{e}{[\![D^{k-1}(g_{-1+k}a),g;1]\!]} + \textbf{e}{[\![D^{k-2}(g_{-1+k}a),g;0]\!]} \right).\end{aligned}$$
\[prop:Nconstraint\] For any pair of elements $a \in \mathfrak{a}$ and $g \in \mathfrak{g}$ satisfying $[g,a] = {\textbf{1}}$, the generators $\textbf{c}{[\![s,t;n \geq N(s,t)]\!]}$ must omit $a_ng$ for at least two positive values of $n$. In particular $N \geq 3$ in Conjecture \[conj:injectivity\].
Permitting elements $a_ng$ to survive for some $n \geq 1$ illustrates a difference between the construction at hand and the current algebra construction.
As in Proposition \[prop:leftmult\], we work with the souped-up Lie algebra in which $\mathfrak{g}$ is the Lie algebra of vector fields on ${{\textbf{R}}}$ and $\mathfrak{a}$ is the algebra of differentiable functions. We let $a$ be a coordinate function on ${{\textbf{R}}}$, and let $g$ be the coordinate vector field $\partial/\partial a$. Note that $[g,a] = {\textbf{1}}$.
Suppose to the contrary that $\textbf{c}{[\![a,g;n]\!]} \in I(\mathfrak{s})$ for all $n\geq 1$ except for some single positive value $N$. To begin with, this hypothesis implies that $Da - a_{-N-2}(a_Ng) \in I(\mathfrak{s})$, since $$\label{eq:Da}
Da - a_{-N-2}(a_Ng) = \frac{1}{2}\textbf{qa}{[\![a,a,g;-1,-1]\!]} + a_{-2}\textbf{s}{[\![a,g]\!]} + \sum_{\substack{k\geq 1 \\ k\neq N}} a_{-k-2}\textbf{c}{[\![a,g;k]\!]},$$ every expression on the right being in $I(\mathfrak{s})$.
Additionally, our hypothesis implies that $I(\mathfrak{s})$ contains $D^N(a_Ng)$ since it can be written as a combination of elements of $I(\mathfrak{s})$: $$D^N(a_N g) = \left(a_0 g + {\textbf{1}}\right) + \textbf{s}{[\![g,a]\!]} - \textbf{qc}{[\![g,a;0]\!]} + \sum_{\substack{k\geq 1 \\ k \neq N}} \frac{(-1)^k}{k!} D^k \textbf{c}{[\![a,g;k]\!]} .$$
This implies furthermore that $(a_Ng)_m x \in I(\mathfrak{s})$ for all $m \leq -N-1$ and $m \geq 0$, and all $x$. Indeed, one can check the identity $$\begin{aligned}
\begin{split}
\label{eq:aNg}
&\binom{m+N}{N} (a_Ng)_m x \\
= &\left( D^N(a_Ng)\right)_{m+N} x - \sum_{k=0}^{N-1} \binom{m+N}{k} \textbf{e}{[\![D^{N-1-k} (a_Ng),x;m+N-k]\!]},
\end{split}\end{aligned}$$ which expresses $(a_ng)_mx$ as a combination of elements in $I(\mathfrak{s})$. The restrictions on $m$ follow from demanding that the binomial coefficient on the left does not vanish.
Finally, we are able to express the element ${\textbf{1}}$ as a combination of elements of $I(\mathfrak{s})$, as seen in the following equation: $$\begin{aligned}
{\textbf{1}}&= \textbf{s}{[\![a,g]\!]} - \textbf{e}{[\![a,g;1]\!]} + \left(Da - a_{-N-2}(a_Ng)\right)_1g + \textbf{qa}{[\![a,a_Ng,g;-N-2,1]\!]} \\
& \quad + \sum_{k \geq 0} \binom{-N-2}{k}(-1)^k \left( a_{-N-2-k} ((a_Ng)_{1+k}g) - (-1)^N (a_Ng)_{-N-1-k}(a_kg)\right)\end{aligned}$$ On the right side, the third term is contained in $I(\mathfrak{s})$ due to equation (\[eq:Da\]), and the terms in the final summation are (multiples of) elements of the form $(a_Ng)_mx$, which are in $I(\mathfrak{s})$ due to equation (\[eq:aNg\]).
A functor from unital vector spaces and souped-up Lie algebras to vertex algebras {#sec:uVecttoVA}
=================================================================================
Thus far, we have focused on generating a vertex algebra first from a unital vector space and next from a souped-up Lie algebra. But as we have seen, there are many vertex algebras that can arise from either classical structure, depending both on the choice of ideal $J$ used to form the vertex algebra quotient and on the choice of function $N$.
In order to turn this procedure into a functor, we must single out an appropriate choice of $J$ and $N$ so that the resulting vertex algebra behaves well under morphisms. The first choice is easy: just use the vertex algebra ideal $I(\mathfrak{v})$ (or $I(\mathfrak{s})$). For simplicity, we may choose the function $N$ to be the constant posited in Conjecture \[conj:injectivity\]. In Proposition \[prop:functorV\], we will show that these choices lead to a veritable functor from either classical category to the category of vertex algebras.
We begin with the definitions of the unital vector space category $\textbf{uVect}$, the souped-up Lie algebra category $\textbf{SLA}$, and the vertex algebra category $\textbf{VA}$.
An object in $\textbf{uVect}$ is a unital vector space. A morphism $\phi \colon \mathfrak{v} \to \mathfrak{v}'$ is a homomorphism such that $\phi({\textbf{1}}_{\mathfrak{v}}) = {\textbf{1}}_{\mathfrak{v}'}$.
\[def:SLAcategory\] An object in the category $\textbf{SLA}$ is a souped-up Lie algebra. A morphism between two souped-up Lie algebras $\mathfrak{s}(\mathfrak{g},\mathfrak{a})$ and $\mathfrak{s}(\mathfrak{g}', \mathfrak{a}')$ is a linear map $\phi$ sending $\mathfrak{g}$ to $\mathfrak{g}'$ and $\mathfrak{a}$ to $\mathfrak{a}'$ such that $\phi({\textbf{1}}_{\mathfrak{s}}) = {\textbf{1}}_{\mathfrak{s}'}$, $\phi[s,t] = [\phi(s),\phi(t)]$, and $\phi(as) = \phi(a)\phi(s)$ for $a \in \mathfrak{a}$ and $s,t \in \mathfrak{s}$.
An object in $\textbf{VA}$ is a vertex algebra, and a morphism between two vertex algebras $V,V'$ is a linear map $\Phi$ that takes ${\textbf{1}}_V$ to ${\textbf{1}}_{V'}$ and satisfies $\Phi(x_ny) = \Phi(x)_n\Phi(y)$.
We now define the map $\mathcal{V}$ from either $\textbf{uVect}$ or $\textbf{SLA}$ to $\textbf{VA}$, and prove that it is a functor.
\[prop:functorV\]
The map $\mathcal{V}$ taking $\mathfrak{v} \in {\operatorname{ob}}(\textbf{uVect})$ to the vertex algebra $V(\mathfrak{v}) = F(\mathfrak{v}) / I(\mathfrak{v})$ with $N(u,v) \equiv 0$, and taking the morphism $\phi \colon \mathfrak{v} \to \mathfrak{v}'$ to the map $\Phi \colon V(\mathfrak{v}) \to V(\mathfrak{v}')$ defined by $$\begin{aligned}
\Phi(x) &\triangleq \phi(x) + I(\phi(\mathfrak{v})) \text{ for } x\in \mathfrak{v} \subset F(\mathfrak{v})\\
\Phi(x_ny) &\triangleq \Phi(x)_n\Phi(y) \text{ for } x,y \in F(\mathfrak{v}),\end{aligned}$$ is a functor.
$\mathcal{V}$, defined analogously on $\textbf{SLA}$, but with the constant $N(s,t) \equiv N$ from Conjecture \[conj:injectivity\], is also a functor.
In the case of $\mathcal{V} \colon \textbf{SLA} \to \textbf{VA}$, there might be other viable choices for $N(s,t)$ other than a constant. At the very least, since $\Phi$ necessarily takes $I(\mathfrak{s})$ to $I(\phi(\mathfrak{s}))$, we must have $$\Phi(\textbf{c}{[\![s,t;n]\!]}) = \Phi(s_nt) = \Phi(s)_n\Phi(t) = \textbf{c}{[\![\Phi(s),\Phi(t);n]\!]},$$ in which case $N(s,t)$ must satisfy $N(\Phi(s),\Phi(t)) \leq N(s,t)$. This is certainly satisfied when $N$ is the constant from Conjecture \[conj:injectivity\], but it is possible that $N(s,t)$ is merely constant on each space $\mathfrak{a} \times \mathfrak{a}$, $\mathfrak{a} \times \mathfrak{g}$, and $\mathfrak{g} \times \mathfrak{g}$. In the best case, by Theorems \[thm:Ainjectivity\] and \[thm:Ginjectivity\] and Proposition \[prop:Nconstraint\], we might be able to choose $N(s,t) \equiv 0$ on the first space, 3 on the second, and 1 on the third. This is an open question.
We will sketch the proof on $\textbf{uVect}$. The proof on $\textbf{SLA}$ is the same.
Let us verify first of all that $\mathcal{V}$ is indeed a map from $\hom(\textbf{uVect})$ to $\hom(\textbf{VA})$. Let $\phi$ be a morphism between unital vector spaces $\mathfrak{v}$ and $\mathfrak{v}'$. We must check that $\mathcal{V}(\phi) \equiv \Phi$ is really a morphism of vertex algebras, meaning that $\Phi$ is a homomorphism of infinite free algebras, and that $\Phi$ takes the ideal $I \subset F(\mathfrak{v})$ gets mapped to the ideal $I\subset F(\mathfrak{v}')$. The first statement is clear from the definition of $\Phi$. For the second statement, it suffices to show that generators get taken to generators. Checking this is straightforward (using the comments above when checking this on the generators $\textbf{c}{[\![x,y;n]\!]}$) and is left to the reader.
Having established that $\mathcal{V}(\phi) \in \hom(\textbf{VA})$, we may proceed by verifying that $\mathcal{V}$ takes the identity morphism ${\operatorname{id}}_\textbf{uVect}$ to the identity morphism ${\operatorname{id}}_\textbf{VA}$, and that $\mathcal{V}(\phi \circ \psi) = \mathcal{V}(\phi) \circ \mathcal{V}(\psi)$. These two statements are straightforward to verify, and left to the reader.
Sheaves of vertex algebras generated by sheaves of modules over smooth functions {#sec:sheaf}
================================================================================
From Chapter \[sec:uVecttoVA\], we have a functor $\mathcal{V}$ from both $\textbf{uVect}$ and $\textbf{SLA}$ to $\textbf{VA}$, taking $\mathfrak{v}$ to $F(\mathfrak{v})/I(\mathfrak{v})$, where $N(u,v)$ is constant on $\mathfrak{v} \times \mathfrak{v}$ (respectively, $\mathfrak{s}$ to $F(\mathfrak{s})/I(\mathfrak{s})$, and $N(s,t)$ constant on $\mathfrak{s} \times \mathfrak{s}$). In the current chapter, we will extend these associations to sheaves of $\Omega^0$-modules, where $\Omega^0$ is the structure sheaf of smooth functions on a manifold $M$. Given any sheaf $\mathcal{E}$ of $\Omega^0$-modules, we will construct a sheaf $\mathcal{E}^{va}$ of vertex algebras. If $\mathcal{E}$ is additionally a sheaf of souped-up Lie algebras, then $\mathcal{E}^{va}$ is additionally a sheaf of $\mathcal{E}$-modules, following Theorem \[thm:smodule\].
This construction will apply to all of the souped-up Lie algebras from Section \[sec:SLAexamples\] that pertain to structures on a smooth manifold $M$, since they are sheaves of $\Omega^0$-modules (as well as sheaves of souped-up Lie algebras).
Throughout this chapter, $\mathfrak{s}$ will denote a souped-up Lie algebra that is additionally a $\Omega^0(U)$-module, where $U$ is some open set of a smooth manifold $M$. Every result applies also to the more general case in which $\mathfrak{s}$ is only a unital vector space. Unless we say explicitly otherwise, we will tacitly assume that a definition or result applies to both a unital vector space and a souped-up Lie algebra.
The supported vertex algebra ideal *K(s)* {#sec:supportedidealK}
-----------------------------------------
On $\mathfrak{s}$ we have the notion of *support* of the element of $\mathfrak{s}$. We shall extend the notion of support to the infinite free algebra $F(\mathfrak{s})$, and then define an operator that kills elements that have no support.
\[def:commonsupport\] Consider the infinite free algebra $F(\mathfrak{s})$ generated by $\mathfrak{s}$. Extending the definition of support of a section to products in the algebra $F(\mathfrak{s})$ by the recursive formula $${\operatorname{supp}}(x_ny) \triangleq \overline{{\operatorname{Int}}({\operatorname{supp}}x) \cap {\operatorname{Int}}({\operatorname{supp}}y) }$$ and then extending by linearity, we define the *common support projection operator* $\pi$ on $F(\mathfrak{s})$ by $$\begin{aligned}
\pi (s) &\triangleq x \text{ for }x \in \mathfrak{s} \\
\pi (x_ny) &\triangleq \begin{cases} 0 & \text{ if } {\operatorname{supp}}(x_ny) = \emptyset \\ x_ny & \text{ otherwise} \end{cases}\end{aligned}$$
The interpretation of the operator $\pi$ is that if the length 1 factors of a monomial $x$ have no common support, then $\pi$ kills $x$. Otherwise, $x$ is untouched by $\pi$. This makes $\pi$ indeed a projection operator.
In the next definition, we use the operator $\pi$ to kill monomials in $V(\mathfrak{s})$ (monomials, in the sense of Definition \[def:VAmonomiallength\]) in which the constituent factors from $\mathfrak{s}$ have no common support. In an example, suppose $f$ and $g$ are two functions on ${{\textbf{R}}}$ with no common support. If we were to multiply them pointwise, their product would vanish. We also want the product $f_ng$ to vanish in a vertex algebra. This will eventually be necessary to turn a sheaf of $\Omega^0$-modules into a sheaf of vertex algebras.
\[def:supportedSVAideal\] The *supported vertex algebra ideal* $K(\mathfrak{s}) \subset F(\mathfrak{s})$ is the ideal generated by $I(\mathfrak{s})$ (with constant $N(s,t) \equiv N$) and the additional generators
common support:
: $\textbf{k}{[\![x]\!]} \triangleq x - \pi(x)$.
The ideal $K(\mathfrak{s})$ thus contains all monomials the constituent factors of which have no common support, since $\pi$ vanishes on such products. The impact is that when one forms the quotient $F(\mathfrak{s}) / K(\mathfrak{s})$, two monomials whose corresponding factors agree on their common support are now equal.
Upon adding the generators $\textbf{k}{[\![x]\!]}$ to $I(\mathfrak{s})$, there is the possibility that the resulting ideal $K(\mathfrak{s})$ no longer has trivial intersection with $\mathfrak{s}$, as is true for $I(\mathfrak{s})$ for particular souped-up Lie algebras or when $\mathfrak{s}$ is only a unital vector space by Theorems \[thm:Ainjectivity\], \[thm:Ginjectivity\], and \[thm:Vinjectivity\]; and which is conjectured to be true for a general souped-up Lie algebra $\mathfrak{s}$. We are assured by the next theorem that this situation does not occur.
\[thm:Kinjectivity\] The map taking $\mathfrak{s}$ to its image in $V(\mathfrak{s}) = F(\mathfrak{s}) / I(\mathfrak{s})$ is injective. Equivalently, $K(\mathfrak{s}) \cap \mathfrak{s} = \{0\}$.
This requires Conjecture \[conj:injectivity\] in the case that $\mathfrak{s}$ is an arbitrary souped-up Lie algebra.
To begin, it follows from the definition of the supported vertex algebra ideal $K(\mathfrak{s})$ that we may write $$K(\mathfrak{s}) = I(\mathfrak{s}) + (\textbf{k}),$$ where $(\textbf{k})$ is the ideal generated by the elements $\textbf{k}{[\![x]\!]}$.
There are three important facts to note: First, by Definition \[def:commonsupport\], $\pi$ acts as the identity on the subspace $\mathfrak{s}$. Second, $I(\mathfrak{s})$ is invariant under $\pi$, which follows from the fact that each generator of $I(\mathfrak{s})$ is a sum of monomials with the same factors, and hence $\pi$ either kills or preserves all terms alike. Third, $(\textbf{k})$ is killed by $\pi$, which follows since $\pi$ kills each generator $\textbf{k}{[\![x]\!]}$ of $(\textbf{k})$.
Then we have $$\begin{aligned}
K(\mathfrak{s}) \cap \mathfrak{s} &= \pi(K(\mathfrak{s}) \cap \mathfrak{s}) \\
&\subset \pi (K(\mathfrak{s})) \cap \pi (\mathfrak{s}) \\
&= \pi (I(\mathfrak{s}) + (\textbf{k})) \cap \mathfrak{s} \\
&\subset I(\mathfrak{s}) \cap \mathfrak{s} \\
&= \{0\}.\end{aligned}$$ The final step follows either from Conjecture \[conj:injectivity\] or Theorems \[thm:Ainjectivity\], \[thm:Ginjectivity\], or \[thm:Vinjectivity\].
From presheaves to sheaves of vertex algebras {#sec:VAsheaf}
---------------------------------------------
As before, we will continue tacitly to acknowledge that the forthcoming definitions and results apply equally to souped-up Lie algebras and unital vector spaces, unless we state otherwise.
Having established the viability of the supported vertex algebra ideal $K(\mathfrak{s})$, we use it to define the presheaf $\mathcal{E}^{va}$ of vertex algebras associated to a sheaf of $\Omega^0$-modules $\mathcal{E}$. Using $K(\mathfrak{s})$ instead of $I(\mathfrak{s})$ will be of utmost importance when it comes time to converting this presheaf into a sheaf.
\[def:VAsheaf\] For a sheaf $\mathcal{E}$ of $\Omega^0$-modules, we define the presheaf $\mathcal{E}^{va}$ of vertex algebras by the following data. On an open set $U$ we define $$\mathcal{E}^{va}(U) \triangleq \mathcal{V}(\mathcal{E}(U)) \equiv F(\mathcal{E}(U)) / K(\mathcal{E}(U))$$ where $\mathcal{V}$ is the functor featured in Proposition \[prop:functorV\].
A restriction morphism ${\operatorname{res}}_{U,U'} \colon \mathcal{E}(U) \to \mathcal{E}(U')$ induces a morphism $${\operatorname{Res}}_{U,U'} \colon F(\mathcal{E}(U)) \to F(\mathcal{E}(U'))$$ by the formula $$\begin{aligned}
{\operatorname{Res}}_{U,U'} x &\triangleq {\operatorname{res}}_{U,U'}x \text{ for } x\in \mathcal{E}(U) \subset F(\mathcal{E}(U))\\
{\operatorname{Res}}_{U,U'}(x_ny) &\triangleq ({\operatorname{Res}}_{U,U'}x)_n({\operatorname{Res}}_{U,U'}y) \text{ for } x,y \in F(\mathcal{E}(U))\end{aligned}$$
Since ${\operatorname{Res}}_{U,U'}$ also takes $K(\mathcal{E}(U))$ to $K(\mathcal{E}(U'))$, it follows that ${\operatorname{Res}}_{U,U'}$ descends to the quotient $F(\mathcal{E}(U)) / K(\mathcal{E}(U))$. Then a restriction morphism $${\operatorname{Res}}_{U,U'} \colon \mathcal{E}^{va}(U) \to \mathcal{E}^{va}(U')$$ is induced by ${\operatorname{res}}_{U,U'}$ according to the commutative diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ \mathcal{E}(U) & (F(\mathcal{E}(U)), K(\mathcal{E}(U))) & \mathcal{E}^{va}(U) \\ \mathcal{E}(U') & (F(\mathcal{E}(U')), K(\mathcal{E}(U'))) & \mathcal{E}^{va}(U') \\};
\path[->]
(m-1-1) edge node[left]{${\operatorname{res}}_{U,U'}$} (m-2-1)
(m-1-2) edge node[left]{${\operatorname{Res}}_{U,U'}$} (m-2-2)
(m-1-3) edge node[left]{${\operatorname{Res}}_{U,U'}$} (m-2-3)
(m-1-1) edge (m-1-2)
(m-1-2) edge (m-1-3)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3);
\end{tikzpicture}$$
We could very well have defined a presheaf of more general vertex algebras (not necessarily $\Omega^0$-modules), ignoring the support generators $\textbf{k}{[\![x]\!]}$ altogether. However, such a presheaf cannot in general be made into a sheaf, for otherwise the uniqueness axiom is violated, as we will see in the proof of Theorem \[thm:VAsheaf\].
We now prove that the presheaf $\mathcal{E}^{va}$ is a sheaf, beginning with a technical lemma.
\[lem:bumpsupport\] Let $U \subset M$ be open and let $T \subset U$ be an open subset such that $\overline{T} \subset U$. Let $\sigma$ be a smooth bump function on $U$ that has height 1 on $T$ and vanishes smoothly at the boundary of $U$. Lastly, consider an element $x = x{[\![s, t, \ldots, u]\!]} \in F(\mathcal{E}(U))$ with $s,t,\ldots, u \in \mathcal{E}(U) \subset F(\mathcal{E}(U))$. Then using the notation $$\sigma \ast x \triangleq x{[\![\sigma s, \sigma t, \ldots, \sigma u]\!]},$$ we have $${\operatorname{supp}}(x - (\sigma \ast x)) \subset U\backslash T .$$
The proof is an induction on the length of $x$. As the base case, we assume $x$ itself has length 1, and therefore is in $\mathcal{E}(U)$. Then $$x - (\sigma \ast x) \equiv x - \sigma x = ({\textbf{1}}-\sigma) x.$$ The factor ${\textbf{1}}- \sigma$, and therefore the product $({\textbf{1}}-\sigma)x$, clearly has support only in $U\backslash T$.
For the inductive case, suppose that ${\operatorname{supp}}(x - (\sigma \ast x)) \subset U\backslash T$ for all elements $x$ with length not exceeding $p$. Without loss of generality, consider a monomial element $w$ of length $p+1$. $w$ necessarily factors as $w = x_ny$ for some elements $x$ and $y$ with lengths not exceeding $p$. Then we have $$\begin{aligned}
x_ny - (\sigma \ast (x_ny)) &= x_ny - (\sigma \ast x)_n (\sigma \ast y) \\
&= \left[ x_ny - x_n (\sigma \ast y) \right] + \left[ x_n(\sigma \ast y) - (\sigma \ast x)_n (\sigma \ast y) \right] \\
&= x_n(y - (\sigma \ast y)) + (x - (\sigma \ast x))_n (\sigma \ast y).\end{aligned}$$
Then taking the support of both sides, we have $$\begin{aligned}
{\operatorname{supp}}\left( x_ny - (\sigma \ast (x_ny))\right) &= {\operatorname{supp}}\left( x_n(y - \sigma \ast y) + (x - (\sigma \ast x))_n (\sigma \ast y) \right) \\
&= {\operatorname{supp}}\left( x_n(y - \sigma \ast y)\right) \cup {\operatorname{supp}}\left((x - (\sigma \ast x))_n (\sigma \ast y) \right) \\
&\subset \left( {\operatorname{supp}}x \cap (U\backslash T) \right) \cup \left( (U\backslash T) \cap {\operatorname{supp}}(\sigma \ast y) \right) \\
&\subset U\backslash T.\end{aligned}$$
\[cor:bumpsupport\] Let $U,T$ and $\sigma$ be as in Lemma \[lem:bumpsupport\]. Let $\rho$ be a function on $U$ with support in $T$. Then in the quotient $F(\mathcal{E}(U)) / K(\mathcal{E}(U))$, we have, for all $x$, $$\rho_n x = \rho_n (\sigma \ast x).$$
We see that the support of the difference between the two sides is $$\begin{aligned}
{\operatorname{supp}}\left(\rho_nx - \rho_n(\sigma \ast x)\right) &= {\operatorname{supp}}\left( \rho_n (x - (\sigma \ast x))\right) \\
&\subset \overline{ {\operatorname{Int}}({\operatorname{supp}}\rho) \cap {\operatorname{Int}}({\operatorname{supp}}(x - (\sigma \ast x)))} \\
&= \overline{T \cap (U\backslash T)} \\
&= \emptyset.\end{aligned}$$ Thus by Definition \[def:commonsupport\], $\pi(\rho_nx - \rho_n(\sigma \ast x)) = 0$, and finally $$\rho_nx - \rho_n(\sigma \ast x) = \rho_nx - \rho_n(\sigma \ast x) + \pi(\rho_nx - \rho_n(\sigma \ast x)) \in K.$$
\[thm:VAsheaf\] The presheaf $\mathcal{E}^{va}$ is a sheaf.
We must verify the existence and uniqueness axioms.
$\textbf{Existence.}$ To demonstrate the existence axiom, we must show that given a covering $\{U^i\}$ of any open set $U \subset M$, and given a section $x^i \in \mathcal{E}^{va}(U^i)$ for each $i$ such that the common restrictions agree, meaning $$\label{eq:commonrestriction}
{\operatorname{Res}}_{U^i,U^i \cap U^j} x^i = {\operatorname{Res}}_{U^j,U^i \cap U^j} x^j,$$ then there exists a section $x \in \mathcal{E}^{va}(U)$ with ${\operatorname{Res}}_{U,U^i} x = x^i$ for all $i$.
By the Shrinking Lemma, each open set $U^i$ contains an open set $T^i$ such that $\overline{T^i} \subset U^i$ and $\{T^i\}$ is also an open cover for $U$. Let $\{\rho^i\}$ be a partition of unity subordinate to the open covering $\{T^i\}$, and let $\{\sigma^i\}$ be a set of smooth bump functions (*not* a partition of unity) such that $\sigma^i$ takes the value 1 on $T^i$ and vanishes smoothly at the boundary of $U^i$. That is, starting from the interior of some $T^i$, each function $\rho^i$ dies smoothly as we approach the boundary of $T^i$, while $\sigma^i$ stays constant at 1. Then $\sigma^i$ dies smoothly as we move from the boundary of $T^i$ to the boundary of $U^i$.
The salient feature of the sets $U^i, T^i$ and the functions $\rho^i, \sigma^i$ is that they satisfy the conditions for Corollary \[cor:bumpsupport\], which we will use below.
The global section $x \in \mathcal{E}^{va}(U)$ we seek is $$x \triangleq \sum_j \rho^j_{-1} \left(\sigma^j \ast x^j\right).$$ Ignoring the “$\sigma^j \ast$" for a moment, this definition resembles the usual breakdown of a smooth function into a sum of smooth functions supported on the various open sets $U^j$. The reason we must incorporate the smoothing factors $\sigma^j$ is because the elements $\{\rho^j_{-1} x^j\}$ are not contained in $\mathcal{E}^{va}(U)$, whereas each $\rho^j_{-1} \left(\sigma^j\ast x^j\right)$ is. The length 1 factors of the former elements are $\rho^j$ and various elements of $\mathcal{E}(U^j)$ which are not in general directly multiplied by $\rho^j$ (in the multiplication in $\mathcal{E}(U^j)$), and therefore do not vanish smoothly themselves at the boundary of $U^j$.
We check that $x$ restricts to $x^i$ on each open set $U^i$ as it ought: $$\begin{aligned}
{\operatorname{Res}}_{U,U^i} x &= {\operatorname{Res}}_{U,U^i} \sum_j \rho^j_{-1} \left(\sigma^j \ast x^j\right) \\
&= {\operatorname{Res}}_{U,U^i} \sum_j \rho^j_{-1} x^j \\
&= {\operatorname{Res}}_{U,U^i} \sum_j \rho^j_{-1} x^i \\
&= {\operatorname{Res}}_{U,U^i} {\textbf{1}}_{-1} x^i \\
&= x^i.\end{aligned}$$
To go from the first line to the second, we have noted that the $j$th term is supported on $U^i \cap T^j$, enabling us to apply Corollary \[cor:bumpsupport\]. Going to the third line, we have used the hypothesis that $x^i$ and $x^j$ agree on $U^i \cap T^j$ (equation (\[eq:commonrestriction\])).
$\textbf{Uniqueness.}$ For uniqueness, we must show that if two elements $x,y \in \mathcal{E}^{va}(U)$ agree when restricted to each $U^i$ in some open covering $\{U^i\}$ of $U$, then $x = y$. By linearity, we may specialize to $y = 0$. The task is then to show that whenever ${\operatorname{Res}}_{U,U^i} x = 0$ for all $i$, then $x = 0$ as well.
For a fixed index $i$, viewing ${\operatorname{Res}}_{U,U^i}$ as a map from $F(\mathcal{E}(U))$ to $F(\mathcal{E}(U^i))$, the kernel of ${\operatorname{Res}}_{U,U^i}$ is spanned by all monomials $z$ such that ${\operatorname{supp}}(z) \subset (U^i)^c$, the complement of $U^i$ within $U$. Descending to the quotient $\mathcal{E}^{va}(U) = V(\mathcal{E}(U))$, the kernel of ${\operatorname{Res}}_{U,U^i}$ is any element of the form $z + K \subset F(\mathcal{E}(U))$ with $z$ as before.
Since the inclusion ${\operatorname{supp}}(z) \subset (U^i)^c$ holds for all $i$, we have $${\operatorname{supp}}(z) \subset \cap_i (U^i)^c = (\cup_i U^i)^c = U^c = \emptyset.$$ By Definition \[def:commonsupport\], this implies that $\pi(z) = 0$ so that $z = z - \pi(z) \in K$, and thus that $z + K = K$, which is 0 in $\mathcal{E}^{va}(U)$. Thus we have shown that if an element is killed by all restriction morphisms ${\operatorname{Res}}_{U,U^i}$, then that element must be 0.
In the proof of Uniqueness above, the inclusion $z - \pi(z) \in K$ is precisely where we have relied on the presence of the additional generators $\textbf{k}{[\![x]\!]} = x - \pi (x)$ in $K$.
To conclude this chapter, we note that the sheaf of vertex algebras $\mathcal{E}^{va}$ contains the underlying classical sheaf $\mathcal{E}$ as a subsheaf. This is our analog of the containment of the de Rham complex in the chiral de Rham sheaf.
\[thm:sheafinjectivity\] The sheaf of vertex algebras $\mathcal{E}^{va}$ contains $\mathcal{E}$ as a subsheaf. Moreover, if $\mathcal{E}$ is a sheaf of souped-up Lie algebras, then $\mathcal{E}^{va}$ is a sheaf of $\mathcal{E}$-modules.
The first statement follows from the injectivity of each $\Omega^0$-module $\mathcal{E}(U)$ into $\mathcal{E}^{va}(U)$, in accordance with Theorem \[thm:Kinjectivity\]. The second statement follows from Theorem \[thm:smodule\].
Chiral vector bundles and chiral differential geometry {#sec:chiralDG}
======================================================
Finally, we are able to assemble the results of the previous chapters to define a *chiral vector bundle*. We will continue to assume Conjecture \[conj:injectivity\].
The later examples of souped-up Lie algebras in Section \[sec:SLAexamples\] are actually sheaves of modules for the structure sheaf $\Omega^0$ of smooth functions. By the work done in Chapter \[sec:sheaf\], each such sheaf $\mathcal{E}$ generates a sheaf of vertex algebras which is moreover a sheaf of $\mathcal{E}$-modules, by Theorem \[thm:sheafinjectivity\]. We are particularly interested in Example \[ex:connectionLA\].
Given a $G$-vector bundle with connection $(E,\nabla)$, the *chiral vector bundle* $\mathcal{E}^{ch(E,\nabla)}$ is the sheaf of vertex algebras generated by the sheaf of souped-up Lie algebras $$\mathcal{E}^{E,\nabla} \triangleq \mathfrak{s}( \Omega \otimes \Gamma (SE \otimes \Lambda E) \otimes \mathfrak{s} \mathfrak{X}_{\nabla}, \Omega \otimes \Gamma (SE \otimes \Lambda E)).$$
In the notation of the previous chapter, $\mathcal{E}^{ch(E,\nabla)} \equiv (\mathcal{E}^{E,\nabla})^{va}$.
We use the term *chiral* to highlight its close relation to the chiral de Rham sheaf, as we will see below.
By Theorem \[thm:sheafinjectivity\], $\mathcal{E}^{ch(E,\nabla)}$ contains $\mathcal{E}^{E,\nabla}$ as a subsheaf and is a sheaf of $\mathcal{E}^{E,\nabla}$-modules. By the decomposition in equation (\[eq:soupconnLAdecomp\]), $\mathcal{E}^{E,\nabla}$ itself notably contains the subsheaf $$\mathcal{E}^{M\times {\textbf{C}},d} \triangleq \mathfrak{s}( \Omega \otimes \mathfrak{s} \mathfrak{X}_d, \Omega) .$$
In a diagram, we have the sheaf containments $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4em, column sep=4em,
text height=1.5ex, text depth=0.25ex]
{ \mathcal{E}^{ch(M\times {\textbf{C}},d)} & \mathcal{E}^{ch(E,\nabla)} \\ \mathcal{E}^{M\times {\textbf{C}},d} & \mathcal{E}^{E, \nabla} \\};
\path[right hook->]
(m-2-1) edge (m-1-1)
(m-1-1) edge (m-1-2)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-1-2);
\end{tikzpicture}$$ The sheaf $\mathcal{E}^{ch(M\times {\textbf{C}},d)}$ is our analog of the chiral de Rham sheaf. As we have pointed out in Remark \[rmk:CDRcomparison\], the vertex algebra relations between the constituent vertex algebras of $\mathcal{E}^{ch(M\times {\textbf{C}},d)}$ and those in the chiral de Rham sheaf do not quite align.
Every feature of the vector bundle $E$ makes an appearance in $\mathcal{E}^{ch(E,\nabla)}$. For example, the connection $\nabla$ on $E$ induces a connection on $\mathcal{E}^{ch(E,\nabla)}(U)$ by $$\nabla \colon x \mapsto \nabla_0x.$$ In accordance with Theorem \[thm:smodule\], this is a derivation of all products $\circ_n$ in that $$\nabla_0(x_ny) = (\nabla_0x)_ny + x_0(\nabla_ny).$$ This map respects the grading decomposition of $\mathcal{E}^{ch(E,\nabla)}$. One can also define a covariant derivative $\nabla_X \equiv \iota_X \nabla$ by $$\nabla_X \colon x \mapsto (\iota_X)_0 (\nabla_0 x).$$
The curvature operator $\nabla^2 \equiv \frac{1}{2}[\nabla,\nabla]$ then becomes an operator on $\mathcal{E}^{ch(E\nabla)}(U)$ by any of the following equivalent expressions: $$\nabla^2 s = \frac{1}{2} [\nabla,\nabla] s = \frac{1}{2} (\nabla_0\nabla)_0 s = \nabla_0 (\nabla_0 s).$$ When restricted to $\Omega \otimes \Gamma E \subset \mathcal{E}^{E,\nabla}$, we may compute the trace of the curvature and arrive at the usual Chern-Weil map to the cohomology of $M$. An open question is how to extend this to all of $\mathcal{E}^{ch(E,\nabla)}$.
An advantage we enjoy over the chiral de Rham sheaf is that in our analog $\mathcal{E}^{ch(M \times {\textbf{C}}, d)}$, the element $d$ gives rise to a global field $$d(\zeta) \triangleq \sum_{n\in {\textbf{Z}}}\frac{d \circ_n}{\zeta^{n+1}}$$ under the state-field correspondence (see Remark \[rmk:fields\]). In the chiral de Rham sheaf on the other hand, the corresponding element exists globally only when the underlying manifold is Calabi-Yau [@MSV-1999].
The construction of the chiral vector bundle $\mathcal{E}^{ch(E,\nabla)}$ is a starting point for extending *classical* differential geometry to its string theoretic analog.
|
---
abstract: 'In this work, we investigate the stability region, the throughput performance, and the queueing delay of an asymmetric relay-assisted cooperative random access wireless network with multipacket reception (MPR) capabilities. We consider a network of $N$ saturated source users that transmit packets to a common destination node with the cooperation of two relay nodes. The relays are equipped with infinite capacity buffers, and assist the users by forwarding the packets that failed to reach the destination. Moreover, the relays have also packets of their own to transmit to the destination. We assume random access of the medium and slotted time. With the relay-assisted cooperation, the packets originated by each user can be transmitted to the destination through multiple relaying paths formed by the relays. We assume that the relays employ an adaptive retransmission control mechanism. In particular, a relay node is aware of the status of the other relay, and accordingly adapts its transmission probability. Such a protocol is towards self-aware networks and leads to substantial performance gains in terms of delay. We investigate the stability region and the throughput performance for the full MPR model. Moreover, for the two-user, two-relay case we derive the generating function of the stationary joint queue-length distribution at the relays by solving a Riemann-Hilbert boundary value problem. Finally, for the symmetric case we obtain explicit expressions for the average queueing delay in a relay node without the need of solving a boundary value problem. Numerical examples are presented providing insights on the system performance.'
author:
-
bibliography:
- 'bibliography.bib'
title: 'Performance Analysis of a Cooperative Wireless Network with Adaptive Relays: A Network-Level Study'
---
Adaptive transmission, Stability region, Queueing analysis, Boundary value problem
Introduction
============
Over the past few decades, wireless communications and networking have witnessed an unprecedented growth. The growing demands require high data rates, considerably large coverage areas and high reliability. Relay-assisted wireless networks have been proposed as a candidate solution ot fulfill these requirements [@rab], since relays can decrease the delay and can also provide increased reliability and higher energy efficiency [@cov; @nos]. A relay-based cooperative wireless system operates as follows: There is a number of source users that transmit packets to a common destination node, and a number of relay nodes that assist the source users by storing and retransmitting the packets that failed to reach the destination; e.g., [@PappasTWC2015; @pap; @pap2; @sad]. A cooperation strategy among sources and relays specifies which of the relays will cooperate with the sources. This problem gives rise to the usage of a cooperative space diversity protocol [@send], where each user has a number of “partners" (i.e., relays) that are responsible for retransmitting its failed packets.
Related work
------------
Cooperative relaying is mostly considered at the physical layer, and is based on information-theoretic considerations. The classical relay channel was first examined in [@mul] and later in [@cov]. Recently cooperative communications have received renewed attention, as a powerful technique to combat fading and attenuation in wireless networks; e.g., [@send; @lan]. Most of the research has concentrated on information-theoretic studies. Recent works [@PappasTWC2015; @pap; @sad; @rong] shown that similar gains can be achieved by network-layer cooperation. By network-layer cooperation, relaying is assumed to take place at the protocol level avoiding physical layer considerations.
In addition, random access recently re-gained interest due to the increased number of communicating devices in 5G networks, and the need for massive uncoordinated access [@Laya2014]. Random access and alternatives schemes and their effect on the operation of LTE and LTE-A are presented in [@Laya2014], [@KoseogluTCOM2016], [@LeungTWC2012]. In [@PopovskiSPL2017], the effect of random access in Cloud-Radio Access Network is considered.
The characterization of the stable throughput region, i.e. the stability region, which gives the set of arrival rates such that there exist transmission probabilities under which the system is stable, is a meaningful metric to measure the impact of bursty traffic and the concept of interacting nodes in a network; e.g., [@LuoAE1999; @Rao_TIT1988; @szpa].
Except throughput, delay is another important metric, which recently received considerable attention due to the rapid growth on supporting real-time applications, which in turn require delay-based guarantees. However, due to the interdependence among queues, the characterization of the delay even in small networks with random access is a rather difficult task, even for the non-cooperative collision channel model [@nain]. In [@nain], queueing delay was studied with the aid of the theory of boundary value problems. The traditional collision model is accurate for modeling wire-line communication, however, it is not an appropriate model for probabilistic reception in wireless multiple access networks. For the **non**-cooperative multipacket reception (MPR) model, delay analysis was performed in [@NawareTong2005], based on the assumption of a symmetric network. Recently, the authors in [@dimpap] generalized the model in [@NawareTong2005; @nain] by considering time-varying links between nodes where the channel state information was modeled according to a Gilbert-Elliot model. The study of queueing systems using the theory of boundary value problems was initiated in [@fay1], and a concrete methodological approach was given in [@coh; @fay]. The vast majority of queueing models are analyzed with the aid of the theory of boundary value problems referring to continuous time models, e.g., [@avr; @box; @dim1; @dim2; @fr; @guillemin2004; @Guillemin2013; @van]. On the contrary, there are very few works on the analysis of discrete time models [@nain; @dimpap; @szpa; @szpa1; @szpa2]. This is mainly due to the complex boundary behavior of the underlying random walk, which reflects the interdependence and coupling among the queues.
Contribution
------------
Our contribution is summarized as follows. We consider a cooperative wireless network with $N$ saturated source users, two relay nodes with adaptive transmission control, and a common destination. Our primary interest is to investigate the stability conditions, throughput performance, and provide expressions for the queueing delay experienced at the buffers of relay nodes. The time is slotted, corresponding to the duration of a transmission of a packet, and the sources/relay nodes access the medium in a random access manner. The sources transmit packets to the destination with the cooperation of the two relays. If a transmission of a user’s packet to the destination fails, the relays store it in their queues and try to forward it to the destination at a subsequent time slot. Moreover, the relays have also external bursty arrivals that are stored in their infinite capacity queues. We consider MPR capabilities at the destination node.
We assume that there is no coordination among relays and sources, but the destination node can sense both of them. Here, we assume that the destination node gives “priority" to the sources when it senses that they will transmit. If it senses that all of the sources will remain silent then it switches to the relays. The relays are accessing the wireless channel randomly and employ a state-dependent transmission protocol. More precisely, a relay adapts its transmission characteristics based on the status of the other relay in order to exploit its idle slots, and to increase its transmission efficiency, which in turn leads towards self-aware networks [@mah]. More specifically, we assume that each relay node is aware of the state of the other one. Note that this feature is common in cognitive radios [@sad; @mah]. To the best of our knowledge this variation of random access has not been reported in the literature. The contribution of this work has two main parts focused on the stable throughput region, and the detailed analysis of the queueing delay at relay nodes.
### Stability analysis and throughput performance
We provide the throughput analysis of the general two-user network with MPR capabilities and the symmetric $N$-user network under random access. The performance characterization for $N$ symmetric users can provide insights on scalability of the network. In addition, we provide the stability conditions for the queues at the relays.
### Delay Analysis
The second part of the contribution of this work is the delay analysis. Except its practical implications, our work is also theoretically oriented. To the best of our knowledge there is no other work in the related literature that deals with the detailed delay analysis of an asymmetric random access cooperative wireless system with adaptive transmissions and MPR capabilities.
To enhance the readability of our work we consider the case of $N=2$ source users, and focus on a subclass of MPR models, called the “capture" channel, under which at most one packet can be successfully decoded by the receiver of the node $D$, even if more than one nodes transmit. We need to mention, that the assumption of two users is not restrictive, and our analysis can be extended to the general case of $N$ users. Moreover, our analysis remains valid even for the case of general MPR model. However, in both cases some important technical requirements must be further taken into account, which in turn will worse the readability of the paper. Besides, our aim here is to focus on the fundamental problem of characterizing the delay in a cooperative wireless network with two relay nodes, and our model and analysis can serve as a building block for the more general case.
Our system is modeled as a two-dimensional discrete time Markov chain, and we show that the generating function of the stationary joint relay queue length distribution by solving a fundamental functional equation with the aid of a Riemann-Hilbert boundary value problem. Furthermore, each relay node employs an adaptive transmission policy, under which it adapts its transmission probabilities based on the status of the queue of the other relay. Moreover, the kernel of this functional equation has never been treated in the related literature. More precisely,
- Based on a relation among the values of the transmission probabilities we distinguish the analysis in two cases, which are different both in the modeling, and in the technical point of view. In particular, the analysis leads to the formulation of two boundary value problems [@ga] (i.e., a Dirichlet, and a Riemann-Hilbert problem), the solution of which will provide the generating function of the stationary joint distribution of the queue size for the relays. This is the key element for obtaining expressions for the average delay at each user node. To our best knowledge, it is the first time in the related literature on cooperative networks with MPR capabilities, where such an analysis is performed.
- Furthermore, for the two-user, two-relay symmetric system, we provide explicit expression for the average queueing delay, without the need of solving a boundary value problem.
Concluding, the analytical results in this work, to the best of our knowledge, have not been reported in the literature.
The rest of the paper is organized as follows. In Section \[mod\] we describe the system model in detail. Section \[stab-anal-2\] is devoted to the investigation of the throughput and the stability conditions for the asymmetric MPR model of $N=2$, while in Section \[stab-anal-n\] we generalize our previous results for the general case of $N$ users with MPR capabilities. In Section \[anal\] we focus on the delay analysis for the general asymmetric two-user with two-relays network. The fundamental functional equation is derived, and some preparatory results in view of the resolution of the functional equation are obtained. We formulate and solve two boundary value problems, the solution of which provide the generating function of the stationary joint queue length distribution of relay nodes. The basic performance metrics are obtained, and important hints regarding their numerical evaluation are also given. In Section \[sym\] we obtain explicit expressions for the average delay at each relay node for the symmetrical system without solving a boundary value problem. Finally, numerical examples that shows insights in the system performance are given in Section \[num\].
Model description and notation {#mod}
==============================
In this work, we consider a network consisting of $N$ saturated users-sources, two relays, and one destination node. In this section, we will describe the case of $N=2$ saturated users assisted by two relays as depicted in Fig. \[mood\]. We focus on the two-user scenario in order to facilitate the presentation and the description of the cooperation protocol.
Network Model
-------------
We consider a network of $N=2$ saturated source users, i.e. sources $1$ and $2$, two relay nodes, denoted by $R_{1}$ and $R_{2}$, and a common destination node $D$ depicted in Fig. \[mood\].[^1] The sources transmit packets to the node $D$ with the cooperation of the relays. The packets have equal length and the time is divided into slots corresponding to the transmission time of a packet.
We assume that the relays and the destination have multipacket reception (MPR) capabilities and the success probabilities for the transmissions will be provided in Section \[sec:PHY\]. MPR is a more suitable model than the collision channel since it can capture better the wireless transmissions. The source-users have random access to the medium with no coordination among them. At the beginning of a slot, the source user $P_k$ attempts to transmit a packet with a probability $t_{k}$, $k=1,2$, i.e., with probability $\bar{t}_{k}=1-t_{k}$ remains silent. The nodes are assume to have priority over the relays. More specifically, the sources and the relays transmit in different channels, however, the destination node can overhear both of the channels. However, the destination node gives “priority" to the sources if it senses that they will transmit. If the destination senses no activity from the sources, it switches to the relays. We assume that this sensing time is negligible. If a packet transmission from a source to the destination fails and at the same time if at least one of the relays will be able to decode this packet, then will store it in its queue with a probability, and it will forward it to the destination at a subsequent time slot. The queues at the relays are assumed to have infinite size.
![An instance of the two-relay cooperative wireless network with two users. In addition, the relays $R_1$ and $R_2$ have their own traffic $\widehat{\lambda}_{1}$ and $\widehat{\lambda}_{2}$ respectively, and they are assisting the users $P_1$ and $P_2$ by forwarding part of users’ packets to the destination $D$. The relays are assumed to have infinite capacity buffers. The case of pure relays can be obtained by replacing $\widehat{\lambda}_{1}=\widehat{\lambda}_{2}=0$.[]{data-label="mood"}](SystemModel-2users.pdf)
Description of Relay Cooperation
--------------------------------
If a transmission of a user’s packet to the destination fails, the relays overhear the wireless transmission, they can store it in their queues with a probability, and try to forward it to the destination at a subsequent time slot. In case that both relays receive the same packet from a user, they choose randomly which will store the packet. In particular, we define the probability $p_{a_{i,j}}$ that a transmitted packet from the $i$-th source will be stored at the queue of $j$-th relay if the relay is able to decode it. This probability captures two scenarios, (i) the partial cooperation of a relay, which was introduced in [@PappasISIT2012] and (ii) when both relays receive the failed packet from node $i$, then the first one will keep in its queue with probability $p_{a_{i,1}}$ and with probability $p_{a_{i,2}}=1-p_{a_{i,1}}$ will be stored in the queue of the second relay. We would like to emphasize that in case that only one relay, i.e the first relay, will receive correctly a failed packet, then it will store it in its queue with probability $p_{a_{i,1}}$. This probability, controls the amount of the cooperation that this relay provides. However, in this work we assume that if only one relay receives successfully a packet that fails to reach the destination, then this packet will be stored in its queue. When both relays decode correctly a failed packet, if we assume that $p_{a_{i,1}}+p_{a_{i,2}}=1$, then the packet will enter one queue only, either the first or the second one. If we will assume that $p_{a_{i,1}}+p_{a_{i,2}}<1$, then there is a probability that the failed packet will not be accepted in the queues of the relays and it has to be retransmitted in a future timeslot by its source.
Let $N_{i,n}$ be the number of packets in the buffer of relay node $R_{i}$, $i=1,2$, at the beginning of the $n$th slot. Moreover, during the time interval $(n,n+1]$ (i.e., during a time slot) the relay $R_{i}$, $i=1,2$ generates also packets of its own (i.e., exogenous traffic). Let $\{A_{i,n}\}_{n\in N}$ be a sequence of i.i.d. random variables where $A_{i,n}$ represents the number of packets which arrive at $R_i$ in the interval $(n,n+1]$, with $E(A_{i,n})=\widehat{\lambda}_{i}<\infty$. The network with pure relays can be obtained by replacing $\widehat{\lambda}_{1}=\widehat{\lambda}_{2}=0$.
In case node $D$ senses no activity from the source users at the beginning of a slot, it switches to the channel of relay nodes. If there are stored packets in the buffers of the relays, they will also attempt to transmit a packet to the node $D$ with a probability.
Due to the interference among the relays, we consider the following opportunistic access policy: If both relays are non empty, $R_{i}$, $i=1,2,$ transmits a packet with probability $\alpha_{i}$. If $R_{1}$ (resp. $R_{2}$) is the only non-empty, it adapts its transmission probability. More specifically, it transmits a packet with a probability $\alpha_{i}^{*}>\alpha_{i}$, in order to utilize the idle slot of the neighbor relay node.[^2] Note that in such a case, a relay node is aware about the state of its neighbor.[^3]
Physical Layer Model {#sec:PHY}
--------------------
The MPR channel model used in this work is a generalized form of the packet erasure model. In wireless networks, a transmission over a link is successful with a probability. We denote $P_s(i,k,A)$ the success probability of the link between nodes $i$ and $k$ when the set of active transmitters are in $A$. For example, ${P_{s}(1,R_1,\{1,2\})}$ denotes the success probability for the link between the first source and the first relay when both sources are transmitting. The probability that the transmission fails is denoted by ${\overline{P}_{s}(1,R_1,\{1,2\})}$. In order to take also into account the interference among the relays, we have to distinguish the success probabilities when a relay transmits and the other is active or inactive (i.e., it is empty). Thus, when $i\in\{R_{1},R_{2}\}$, the success probability of the link between relay node $i$ and node $D$ when relay node $i$ is the only non empty is denoted by $P_s^{*}(i,D,\{i\})$. In this work we distinguish this case, in order to have more general results that can capture scenarios that one relay can increase its transmission power when the other relay is empty, thus silent, in order to achieve a higher success probability. Thus, ${P_{s}^{*}(R_1,D,\{R_1\})}\geq{P_{s}(R_1,D,\{R_1\})}$. The probabilities of successful packet reception can be obtained using the common assumption in wireless networks that a packet can be decoded correctly by the receiver if the received SINR (Signal-to-Interference-plus-Noise-Ratio) exceeds a certain threshold. The SINR depends on the modulation scheme, the target bit error rate and the number of bits in the packet [@tse] and the expressions for the success probabilities can be found in several papers, i.e for the case of Raleigh fading refer in [@PappasTWC2015]. On the other hand, if source user $k=1,2$, is the only that transmits, ${P_{s}(k,D,\{k\})}$ denotes the probability that its packet is successfully decoded by the destination, while with probability ${\overline{P}_{s}(k,D,\{k\})}=1-{P_{s}(k,D,\{k\})}$ this transmission fails.
We now provide the service rates $\mu_{1}$, $\mu_{2}$ seen at relay nodes. For the first relay we have $$\label{eq:mu1}
\mu_1 = \overline{t}_1 \overline{t}_2 \left[ \mathrm{Pr} (N_2 = 0) \alpha_{1}^{*} {P_{s}^{*}(R_1,D,\{R_1\})}+ \mathrm{Pr} (N_2 > 0) \alpha_{1} \left( \alpha_{2} {P_{s}(R_1,D,\{R_1,R_2\})}+ \overline{\alpha}_{2} {P_{s}(R_1,D,\{R_1\})}\right) \right].$$
Similarly we have the service rate at the second relay
$$\label{eq:mu2}
\mu_2 = \overline{t}_1 \overline{t}_2 \left[ \mathrm{Pr} (N_1 = 0) \alpha_{2}^{*} {P_{s}^{*}(R_2,D,\{R_2\})}+ \mathrm{Pr} (N_1 > 0) \alpha_{2} \left( \alpha_{1} {P_{s}(R_2,D,\{R_1,R_2\})}+ \overline{\alpha}_{1} {P_{s}(R_2,D,\{R_2\})}\right) \right].$$
Note that the success probability ${P_{s}(R_1,D,\{R_1,R_2\})}$ (resp. ${P_{s}(R_2,D,\{R_1,R_2\})}$) refers to the case where a submitted packet from relay $R_{1}$ (resp. $R_{2}$) is successfully decoded by node $D$, and includes both the case where only a packet from $R_{1}$ (resp. $R_{2}$) is decoded, both the case where both relays have successful transmissions (i.e. MPR case).
We define the following two variables in order to simplify the presentation in the analysis
$$\label{eq:Delta1}
\Delta_1={P_{s}(R_1,D,\{R_1,R_2\})}-{P_{s}(R_1,D,\{R_1\})},$$
and $$\label{eq:Delta2}
\Delta_2={P_{s}(R_2,D,\{R_1,R_2\})}-{P_{s}(R_2,D,\{R_2\})}.$$ These variables can be seen as an indication regarding the MPR capability for each user. If $\Delta_i \to 0$, then the interference caused by the other user is negligible.
--------------------------------------------------------------------------------------------------------------------------------------- --
**Symbol & **Explanation\
$N_{i,n}$ & The number of packets in relay node $R_{i}$ at the beginning of slot $n$\
$A_{i,n}$ &The number of packets arriving during $(n,n+1]$ in relay node $R_{i}$, $i=1,2$\
$\widehat{\lambda}_{i}$ &The expected number of external arrivals in relay node $R_{i}$, $i=1,2$, during a slot\
$t_{k}$ &Transmission probability of source $k$, $k=1,2$\
$a_{i}$ & Transmission probability of relay node $R_{i}$, $i=1,2$, when both users are active (i.e., non-empty)\
$a^{*}_{i}$ & Transmission probability of relay node $R_{i}$, $i=1,2$, when it is the only active (i.e., non-empty) node\
$P_{s}(k,m,A)$& Success probability of the link between node $k$ and $m$ when the set of the transmitted nodes are in $A$\
$P_{s}^{*}(R_{i},D,\{R_{i}\})$&Success probability of relay node $R_{i}$, $i=1,2$, when it is the only active (i.e., non-empty) node\
$P_{s,k}(k,R_{i},\{1,2\})$& Success probability of the link between source $k$ and $R_{i}$ when both sources transmit,\
& but source $k$ fails to directly reach node $D$, $k=1,2$, $i=1,2.$\
****
--------------------------------------------------------------------------------------------------------------------------------------- --
: Basic Notation
\[table:notation\]
Throughput and Stability Analysis for the two-user case – General MPR case {#stab-anal-2}
==========================================================================
In this section, we provide the analysis for the two-user case under the MPR channel model. More specifically, we provide the throughput analysis for the two users and in addition we derive the stability conditions for the queues at the relays.
Based on the definition in [@szpa], a queue is said to be *stable* if $\lim_{n \rightarrow \infty} {Pr}[N_{i,n}< {x}] = F(x)$ and $\lim_{ {x} \rightarrow \infty} F(x) = 1$. Loynes’ theorem [@Loynes] states that if the arrival and service processes of a queue are strictly jointly stationary and the average arrival rate is less than the average service rate, then the queue is stable. If the average arrival rate is greater than the average service rate, then the queue is unstable and the value of $N_{i,n}$ approaches infinity almost surely. The stability region of the system is defined as the set of arrival rate vectors $\boldsymbol{\lambda}=(\lambda_1, \lambda_2)$, for which the queues in the system are stable.
We start the analysis by deriving the throughput per user which allow us to calculate the endogenous arrivals at the relays.
Throughput per user
-------------------
Here we will consider the throughput per (source) user when both queues of the relays are stable. Conditions for stability are given in a subsequent subsection. *When the queues at the relays are not stable the throughput per user can be obtained using the approach in [@PappasTWC2015]*.
The throughput per user $k$, $T_k$, is the direct throughput when the transmission to the destination is successful plus the throughput contributed by the relays (if they can decode the transmission) in case of a failed transmission to the destination. Thus, the throughput seen by the first user is given by
$$\label{eq:T1-2users}
T_1=T_{1,D}+T_{1,R},$$
where $$\label{eq:T1D-2users}
T_{1,D} = t_1 \bar{t}_2 {P_{s}(1,D,\{1\})}+ t_1 t_2 {P_{s}(1,D,\{1,2\})},$$ and $$\begin{array}{rl}
T_{1,R}= t_1 (1-t_2) {\overline{P}_{s}(1,D,\{1\})}{P_{s}(1,R_1,\{1\})}{\overline{P}_{s}(1,R_2,\{1\})}+t_1 t_2 {\overline{P}_{s}(1,D,\{1,2\})}{P_{s}(1,R_1,\{1,2\})}{\overline{P}_{s}(1,R_2,\{1,2\})}+ \nonumber \\
+ t_1 (1-t_2) {\overline{P}_{s}(1,D,\{1\})}{\overline{P}_{s}(1,R_1,\{1\})}{P_{s}(1,R_2,\{1\})}+t_1 t_2 {\overline{P}_{s}(1,D,\{1,2\})}{\overline{P}_{s}(1,R_1,\{1,2\})}{P_{s}(1,R_2,\{1,2\})}+ \nonumber \\
+ t_1 (1-t_2) {\overline{P}_{s}(1,D,\{1\})}{P_{s}(1,R_1,\{1\})}{P_{s}(1,R_2,\{1\})}+t_1 t_2 {\overline{P}_{s}(1,D,\{1,2\})}{P_{s}(1,R_1,\{1,2\})}{P_{s}(1,R_2,\{1,2\})}.
\end{array}
\label{eq:T2D}$$ Similarly we can obtain the throughout for the second user.
The aggregate or network-wide throughput of the network when the queues at the relays are both stable is $$T_{aggr}=T_1+T_2+\widehat{\lambda}_{1}+\widehat{\lambda}_{2}.$$
Endogenous arrivals at the relays
---------------------------------
Here, we will derive the internal (or endogenous) arrival rate from the users to each relay. We would like to mention that the relays have also their own traffic (exogenous) denoted by $\widehat{\lambda}_{i}$ for the $i$-th relay.
A packet from a user can enter a queue at one relay if the transmission to the destination fails and at the same time at least one relay decodes correctly the packet. In the two-user case with MPR capabilities, in a relay up to two packets can enter the queue.
Here we will derive the endogenous arrival from user $1$ to the first relay denoted by $\lambda_{1,1}$. The $\lambda_{1,1}$ is also the probability that a transmitted packet by the first user will enter the queue at the first relay. So, the term $\lambda_{i,j}$ denotes the endogenous arrival probability from $i$-th user, $i=1,2$, to the queue at the $j$-th relay, $j=1,2$.
The endogenous arrival rate $\lambda_{1,1}$ is given by
$$\begin{array}{rl}
\lambda_{1,1} = t_1 (1-t_2) {\overline{P}_{s}(1,D,\{1\})}{P_{s}(1,R_1,\{1\})}{\overline{P}_{s}(1,R_2,\{1\})}+t_1 t_2 {\overline{P}_{s}(1,D,\{1,2\})}{P_{s}(1,R_1,\{1,2\})}{\overline{P}_{s}(1,R_2,\{1,2\})}+ \nonumber \\
+ t_1 (1-t_2) {\overline{P}_{s}(1,D,\{1\})}{P_{s}(1,R_1,\{1\})}{P_{s}(1,R_2,\{1\})}p_{a_{1,1}}
+t_1 t_2 {\overline{P}_{s}(1,D,\{1,2\})}{P_{s}(1,R_1,\{1,2\})}{P_{s}(1,R_2,\{1,2\})}p_{a_{1,1}}.
\end{array}
\label{lambda11}$$
The endogenous arrival rate $\lambda_{1,2}$ is given by
$$\begin{array}{rl}
\lambda_{1,2} = t_1 (1-t_2) {\overline{P}_{s}(1,D,\{1\})}{\overline{P}_{s}(1,R_1,\{1\})}{P_{s}(1,R_2,\{1\})}+t_1 t_2 {\overline{P}_{s}(1,D,\{1,2\})}{\overline{P}_{s}(1,R_1,\{1,2\})}{P_{s}(1,R_2,\{1,2\})}+ \nonumber \\
+ t_1 (1-t_2) {\overline{P}_{s}(1,D,\{1\})}{P_{s}(1,R_1,\{1\})}{P_{s}(1,R_2,\{1\})}p_{a_{1,2}}
+t_1 t_2 {\overline{P}_{s}(1,D,\{1,2\})}{P_{s}(1,R_1,\{1,2\})}{P_{s}(1,R_2,\{1,2\})}p_{a_{1,2}}.
\end{array}
\label{lambda12}$$
Similarly we can define $\lambda_{2,1}$, and $\lambda_{2,2}$. Note that $T_{1,R} = \lambda_{1,1} + \lambda_{1,2}$, which is the relayed throughput for the first user defined in the previous subsection.
The average arrival rate at the relay $i$ is given by
$$\lambda_{i} = \widehat{\lambda}_{i} + \lambda_{1,i} + \lambda_{2,i}.$$
Recall that $p_{a_{1,1}}$ denotes the probability that the transmitted by the first source packet which is correctly received by both relays and failed to reach the destination will enter the queue at the first relay. The term $p_{a_{1,2}}=1-p_{a_{1,1}}$ denotes the probability that the packet will enter the queue at the second relay. Thus, a packet can enter only one queue so we avoid wasting resources by transmitting the same packet twice.
Stability conditions for the queues at the relays
-------------------------------------------------
We now proceed with the investigation of the stability conditions, based on the concept of stochastic dominant systems developed in [@Rao_TIT1988; @szpa]. The stability region of the system is defined as the set of arrival rate vectors $\boldsymbol{\lambda}=(\lambda_1, \lambda_2)$, for which the queues of the relay nodes are stable. Here, we will derive the stability analysis for the total average arrival rate at each relay, $\lambda_{i}$.
The next theorem provides the stability criteria for the two-user general MPR case.
\[Thm2users\] The stability region $\mathcal{R}$ is given by $\mathcal{R}=\mathcal{R}_1 \bigcup \mathcal{R}_2$ where $$\begin{aligned}
\label{eq:R1_2}
\mathcal{R}_1= \left\lbrace (\lambda_{1},\lambda_{2}):\lambda_{1} < \overline{t}_1 \overline{t}_2 \alpha_{1}^{*} {P_{s}^{*}(R_1,D,\{R_1\})}- \frac{\lambda_{2} \left[\alpha_{1}^*{P_{s}^{*}(R_1,D,\{R_1\})}- \alpha_{1} \left[{\alpha_{2}\Delta_1+{P_{s}(R_1,D,\{R_1\})}}\right]\right]}{{\alpha_{2} \left[\alpha_{1}\Delta_2 + {P_{s}(R_2,D,\{R_2\})}\right]}}, \right. \notag \\
\left. \lambda_2 < \overline{t}_1 \overline{t}_2 \alpha_{2} \left[ \alpha_{1} \Delta_2+ {P_{s}(R_2,D,\{R_2\})}\right]
\right\rbrace.\end{aligned}$$ and $$\begin{aligned}
\label{eq:R2_2}
\mathcal{R}_2= \left\lbrace (\lambda_{1},\lambda_{2}):\lambda_{2} < \overline{t}_1 \overline{t}_2 \alpha_{2}^{*} {P_{s}^{*}(R_2,D,\{R_2\})}- \frac{\lambda_{1} \left[\alpha_{2}^*{P_{s}^{*}(R_2,D,\{R_2\})}- \alpha_{2} \left[{\alpha_{1}\Delta_2+{P_{s}(R_2,D,\{R_2\})}}\right]\right]}{{\alpha_{1} \left[\alpha_{2}\Delta_1 + {P_{s}(R_1,D,\{R_1\})}\right]}}, \right. \notag \\
\left. \lambda_1 < \overline{t}_1 \overline{t}_2 \alpha_{1} \left[ \alpha_{2} \Delta_1+ {P_{s}(R_1,D,\{R_1\})}\right]
\right\rbrace.\end{aligned}$$
The average service rates of the first and second relay are given by (\[eq:mu1\]) and (\[eq:mu2\]), respectively.
Since the average service rate of each relay depends on the queue size of the other relay, the stability region cannot be computed directly. Thus, we apply the stochastic dominance technique introduced in [@Rao_TIT1988], i.e. we construct hypothetical dominant systems, in which the relay with the empty queue transmits dummy packets, while the non-empty relay transmits according to its traffic.
In the first dominant system, the first relay transmit dummy packets and the second relay behaves as in the original system. All the rest operational aspects remain unaltered in the dominant system. Thus, in this dominant system, the first queue never empties, hence the service rate for the second relay is
$$\mu_2 = \overline{t}_1 \overline{t}_2 \alpha_{2} \left[ \alpha_{1} {P_{s}(R_2,D,\{R_1,R_2\})}+ \overline{\alpha}_{1} {P_{s}(R_2,D,\{R_2\})}\right].$$
Which can be rewritten as $$\label{eq:mu2-d1}
\mu_2 = \overline{t}_1 \overline{t}_2 \alpha_{2} \left[ \alpha_{1} \Delta_2+ {P_{s}(R_2,D,\{R_2\})}\right].$$
Then, we can obtain stability conditions for the second relay by applying Loynes’ criterion [@Loynes]. The queue at the second source is stable if and only if $\lambda_2 < \mu_2$, that is $\lambda_2 < \overline{t}_1 \overline{t}_2 \alpha_{2} \left[ \alpha_{1} \Delta_2+ {P_{s}(R_2,D,\{R_2\})}\right]$. Then we can obtain the probability that the second relay is empty by applying Little’s theorem, i.e.
$$\begin{aligned}
\label{eq:Pr2empty_D1}
\mathrm{Pr}\left(N_2 = 0 \right) = 1-\frac{\lambda_2}{\overline{t}_1 \overline{t}_2 \alpha_{2} \left[ \alpha_{1} \Delta_2+ {P_{s}(R_2,D,\{R_2\})}\right]}.\end{aligned}$$
After replacing into we obtain $$\mu_1 = \overline{t}_1 \overline{t}_2 \alpha_{1}^{*} {P_{s}^{*}(R_1,D,\{R_1\})}- \frac{\lambda_{2}\alpha_{1}^*{P_{s}^{*}(R_1,D,\{R_1\})}}{{\alpha_{2} \left[\alpha_{1}\Delta_2 + {P_{s}(R_2,D,\{R_2\})}\right]}} + \frac{\lambda_{2}\alpha_{1} \left[{\alpha_{2}\Delta_1+{P_{s}(R_1,D,\{R_1\})}}\right]}{{\alpha_{2} \left[\alpha_{1}\Delta_2 + {P_{s}(R_2,D,\{R_2\})}\right]}}.$$
Thus, after applying Loynes’ criterion, the stability condition for the first relay in the first dominant system is
$$\lambda_{1} < \overline{t}_1 \overline{t}_2 \alpha_{1}^{*} {P_{s}^{*}(R_1,D,\{R_1\})}- \frac{\lambda_{2} \left[\alpha_{1}^*{P_{s}^{*}(R_1,D,\{R_1\})}- \lambda_{2}\alpha_{1} \left[{\alpha_{2}\Delta_1+{P_{s}(R_1,D,\{R_1\})}}\right]\right]}{{\alpha_{2} \left[\alpha_{1}\Delta_2 + {P_{s}(R_2,D,\{R_2\})}\right]}}.$$
The stability region $\mathcal{R}_1$ obtained from the first dominant system is given by
$$\begin{aligned}
\mathcal{R}_1= \left\lbrace (\lambda_{1},\lambda_{2}):\lambda_{1} < \overline{t}_1 \overline{t}_2 \alpha_{1}^{*} {P_{s}^{*}(R_1,D,\{R_1\})}- \frac{\lambda_{2} \left[\alpha_{1}^*{P_{s}^{*}(R_1,D,\{R_1\})}- \alpha_{1} \left[{\alpha_{2}\Delta_1+{P_{s}(R_1,D,\{R_1\})}}\right]\right]}{{\alpha_{2} \left[\alpha_{1}\Delta_2 + {P_{s}(R_2,D,\{R_2\})}\right]}}, \right. \notag \\
\left. \lambda_2 < \overline{t}_1 \overline{t}_2 \alpha_{2} \left[ \alpha_{1} \Delta_2+ {P_{s}(R_2,D,\{R_2\})}\right]
\right\rbrace.\end{aligned}$$
Similarly, we construct a second dominant system where the second relay transmits a dummy packet when it is empty and the first relay behaves as in the original system. All other operational aspects remain unaltered in the dominant system. Following the same steps as in the first dominant system, we obtain the stability region, $\mathcal{R}_2$, of the second dominant system.
$$\begin{aligned}
\mathcal{R}_2= \left\lbrace (\lambda_{1},\lambda_{2}):\lambda_{2} < \overline{t}_1 \overline{t}_2 \alpha_{2}^{*} {P_{s}^{*}(R_2,D,\{R_2\})}- \frac{\lambda_{1} \left[\alpha_{2}^*{P_{s}^{*}(R_2,D,\{R_2\})}- \alpha_{2} \left[{\alpha_{1}\Delta_2+{P_{s}(R_2,D,\{R_2\})}}\right]\right]}{{\alpha_{1} \left[\alpha_{2}\Delta_1 + {P_{s}(R_1,D,\{R_1\})}\right]}}, \right. \notag \\
\left. \lambda_1 < \overline{t}_1 \overline{t}_2 \alpha_{1} \left[ \alpha_{2} \Delta_1+ {P_{s}(R_1,D,\{R_1\})}\right]
\right\rbrace.\end{aligned}$$
An important observation made in [@Rao_TIT1988] is that the stability conditions obtained by the stochastic dominance technique are not only sufficient but also necessary for the stability of the original system.
The *indistinguishability* argument [@Rao_TIT1988] applies to our problem as well. Based on the construction of the dominant system, *it is easy to see that the queue sizes in the dominant system are always greater than those in the original system, provided they are both initialized to the same value and the arrivals are identical in both systems*. Therefore, given $\lambda_{2}<\mu_{2}$, if for some $\lambda_{1}$, the queue at the first relay is stable in the dominant system, then the corresponding queue in the original system must be stable. Conversely, if for some $\lambda_{1}$ in the dominant system, the queue at the first relay saturates, then it will not transmit dummy packets, and as long as the first relay has a packet to transmit, the behavior of the dominant system is identical to that of the original system since dummy packet transmissions are eliminated as we approach the stability boundary. Therefore, the original and the dominant systems are indistinguishable at the boundary points.
The stability region obtained in Theorem \[Thm2users\] is depicted in Fig. \[region2\]. To simplify presentation, we denote the points $A_1, A_2, B_1, B_2$ with the following expressions
$$\begin{aligned}
A_1=\overline{t}_1 \overline{t}_2 \alpha_{1}^{*} {P_{s}^{*}(R_1,D,\{R_1\})}, \text{ }A_2=\overline{t}_1 \overline{t}_2 \alpha_{1} \left[{\alpha_{2}\Delta_1+{P_{s}(R_1,D,\{R_1\})}}\right]
\\
B_1=\overline{t}_1 \overline{t}_2 \alpha_{2}^{*} {P_{s}^{*}(R_2,D,\{R_2\})}, \text{ }
B_2=\overline{t}_1 \overline{t}_2 \alpha_{2} \left[{\alpha_{1}\Delta_2+{P_{s}(R_2,D,\{R_2\})}}\right].\end{aligned}$$
![The stability region described in Theorem \[Thm2users\].[]{data-label="region2"}](Stability_Region2)
The stability region is a convex polyhedron when the following condition holds $\frac{\alpha_1\left(P_{s}(R_1,D,\{R_1\}+\alpha_2\Delta_1) \right)}{\alpha_{1}^{*}P_{s}^{*}(R_1,D,\{R_1\})}+\frac{\alpha_2\left(P_{s}(R_2,D,\{R_2\}+\alpha_1\Delta_2) \right)}{\alpha_{2}^{*}P_{s}^{*}(R_2,D,\{R_2\})}\geq1$. In the previous condition, when equality holds, the region becomes a triangle and coincides with the case of time-sharing of the channel between the relays. Convexity is an important property since it corresponds to the case when parallel concurrent transmissions are preferable to a time-sharing scheme. Additionally, convexity of the stability region implies that if two rate pairs are stable, then any rate pair lying on the line segment joining those two rate pairs is also stable.
The case of pure relays can be obtained easily by replacing $\widehat{\lambda}_{1}=\widehat{\lambda}_{2}=0$.
The network without relay’s assistance can be obtained by $p_{a_{1,2}}=p_{a_{1,1}}=0$. In this case, we have a network with saturated users and also two users with bursty traffic that transmit packets only when the saturated users are silent.
One can connect the endogenous arrivals from the users to the relays with the stability conditions, obtained in Theorem \[Thm2users\], by replacing the relevant expressions of $\widehat{\lambda}_{1}$ and $\widehat{\lambda}_{2}$ into $\lambda_{1}$ and $\lambda_{2}$.
A slightly different scenario is captured by the case where the relays can transmit in a different channel than the users and the destination can hear both channels at the same time. The receivers at the relays are operating at the same channels where the users are transmitting. In this case, we can have a full duplex operation at the relays on different bands. Thus, we have the following average service rate for the first relay
$$\mu_1 = \mathrm{Pr} (N_2 = 0) \alpha_{1}^{*} {P_{s}(R_1,D,\{R_1\})}+ \mathrm{Pr} (N_2 > 0) \alpha_{1} \left( \alpha_{2} {P_{s}(R_1,D,\{R_1,R_2\})}+ \overline{\alpha}_{2} {P_{s}(R_1,D,\{R_1\})}\right).$$
Similarly we have the service rate at the second relay
$$\mu_2 =\mathrm{Pr} (N_1 = 0) \alpha_{2}^{*} {P_{s}(R_2,D,\{R_2\})}+ \mathrm{Pr} (N_1 > 0) \alpha_{2} \left( \alpha_{1} {P_{s}(R_2,D,\{R_1,R_2\})}+ \overline{\alpha}_{1} {P_{s}(R_2,D,\{R_2\})}\right).$$
The stability analysis for this case can be trivially obtained by the presented analysis thus, it is omitted. However, this scenario has applicability in nowadays relay-assisted networks.
Throughput and Stability Analysis – The symmetric $N$-user case for the General MPR case {#stab-anal-n}
========================================================================================
Here we will generalize the analysis provided in the previous section for the $N$-user case. However, due to presentation clarity we will focus on the symmetric user case. The users attempt to transmit with probability $t$. The success probability from a user to the destination is the same for all the users, thus, in order to characterize it we just need the number of active users, i.e. the interference. This probability is denoted by $P_s(D,i)$ to capture the case that $i$ users are attempting transmission (including the user we intend to study its performance), similarly we define $P_s(R_j,i), j=1,2$.
Endogenous arrivals at the relays and throughput performance
------------------------------------------------------------
The direct throughput of a user to the destination in the case of $N$ symmetric users is given by $$T_D=\sum_{i=1}^{N} t^i (1-t)^{N-i} P_s(D,i)$$
We will derive the endogenous arrivals at the first and the second relay respectively in order to calculate the relayed throughput in the network. For the symmetric $N$-user case we denote the endogenous arrivals from the users at the first (second) relay as $\lambda_{1,u}$ ($\lambda_{2,u}$).
We need to characterize the average number of packet arrivals from the users at each relay. Thus, we define as $r_{k,1}$ the probability that $k$ packets will arrive in a timeslot at the first relay. Similarly, we define $r_{k,2}$. Then, the average endogenous arrival rate at the $j$-th relay is given by $$\label{eq:lambdaju}
\lambda_{j,u} = \sum_{k=1}^{N} k r_{k,j}, \text{ }j=1,2.$$
The probability $r_{k,1}$ where $1 \leq k \leq N$ is given by $$\begin{aligned}
r_{k,1}=\sum_{i=k}^{N}\sum_{l=0}^{k}{N \choose i}{i \choose k}{k \choose l} {t^{i}\overline{t}^{N-i}} \left(P_s(R_1,i)\right)^{k} \left(\overline{P}_s(D,i)\right)^{k} \left(\overline{P}_s(R_2,i)\right)^{k-l}\left(P_s(R_2,i)\right)^{l} p_{a_1}^{l}\left[1-P_s(R_1,i)\overline{P}_s(D,i)\right]^{i-k}.
\end{aligned}$$
Similarly, we obtain $r_{k,2}$ for the second relay. Note that the *network-wide relayed throughput* when both relays are stable is given by $\lambda_{1,u}+\lambda_{2,u}$. Thus, the aggregate or network-wide throughput of the network when both relays are stable is given by
$$T_{aggr}=N T_D + \lambda_{1,u}+\lambda_{2,u} + \widehat{\lambda}_{1}+\widehat{\lambda}_{2}$$
Recall that the total arrival rate at relay $i$ is $\lambda_{i}=\lambda_{i,u}+\widehat{\lambda}_{i}$, consisting of the endogenous arrivals from the users and the external traffic.
Below provide the stability conditions at the relays.
Stability conditions for the queues at the relays
-------------------------------------------------
The service rates at the at the first relay is given by $$\label{eq:mu1n}
\mu_1 = \overline{t}^{N} \left[ \mathrm{Pr} (N_2 = 0) \alpha_{1}^{*} {P_{s}^{*}(R_1,D,\{R_1\})}+ \mathrm{Pr} (N_2 > 0) \alpha_{1} \left( \alpha_{2} {P_{s}(R_1,D,\{R_1,R_2\})}+ \overline{\alpha}_{2} {P_{s}(R_1,D,\{R_1\})}\right) \right].$$
Similarly, we have the service rate at the second relay is given by $$\label{eq:mu2n}
\mu_2 = \overline{t}^{N} \left[ \mathrm{Pr} (N_1 = 0) \alpha_{2}^{*} {P_{s}^{*}(R_2,D,\{R_2\})}+ \mathrm{Pr} (N_1 > 0) \alpha_{2} \left( \alpha_{1} {P_{s}(R_2,D,\{R_1,R_2\})}+ \overline{\alpha}_{1} {P_{s}(R_2,D,\{R_2\})}\right) \right].$$
Following the same methodology as in the proof of Theorem \[Thm2users\], we obtain the stability conditions for the symmetric $N$-user case. The stability conditions are given by $\mathcal{R}=\mathcal{R}_1 \cup \mathcal{R}_2$ where
$$\begin{aligned}
\label{eq:R1_n}
\mathcal{R}_1= \left\lbrace (\lambda_{1},\lambda_{2}):\lambda_{1} < \overline{t}^N \alpha_{1}^{*} {P_{s}^{*}(R_1,D,\{R_1\})}- \frac{\lambda_{2} \left[\alpha_{1}^*{P_{s}^{*}(R_1,D,\{R_1\})}- \alpha_{1} \left[{\alpha_{2}\Delta_1+{P_{s}(R_1,D,\{R_1\})}}\right]\right]}{{\alpha_{2} \left[\alpha_{1}\Delta_2 + {P_{s}(R_2,D,\{R_2\})}\right]}}, \right. \notag \\
\left. \lambda_2 < \overline{t}^N \alpha_{2} \left[ \alpha_{1} \Delta_2+ {P_{s}(R_2,D,\{R_2\})}\right]
\right\rbrace.\end{aligned}$$
and $$\begin{aligned}
\label{eq:R2_n}
\mathcal{R}_2= \left\lbrace (\lambda_{1},\lambda_{2}):\lambda_{2} < \overline{t}^N \alpha_{2}^{*} {P_{s}^{*}(R_2,D,\{R_2\})}- \frac{\lambda_{1} \left[\alpha_{2}^*{P_{s}^{*}(R_2,D,\{R_2\})}- \alpha_{2} \left[{\alpha_{1}\Delta_2+{P_{s}(R_2,D,\{R_2\})}}\right]\right]}{{\alpha_{1} \left[\alpha_{2}\Delta_1 + {P_{s}(R_1,D,\{R_1\})}\right]}}, \right. \notag \\
\left. \lambda_1 < \overline{t}^N \alpha_{1} \left[ \alpha_{2} \Delta_1+ {P_{s}(R_1,D,\{R_1\})}\right]
\right\rbrace.\end{aligned}$$
Delay analysis: The two-user case {#anal}
=================================
This section is devoted to the analysis of the queueing delay experienced at the relays. Our aim is to obtain the generating function of the joint stationary distribution of the number of packets at relay nodes. In the following we consider the case of $N=2$ users, and focus on a subclass of MPR models, called the “capture" channel, under which at most one packet can be successfully decoded by the receiver of the node $D$, even if more than one nodes transmit.
In order to proceed, we have to provide some more information regarding the success probabilities of a transmission between nodes that were defined in subsection \[sec:PHY\]. More precisely, we have to take into account the number as well as the type of nodes that transmit (i.e., source or relay node). This is due to several reasons, such as the fact that generally the channel quality between relay nodes and destination node is usually better than between sources and destination, as well as due to the wireless interference, since the channel quality is severely affected by the the number of nodes that attempt a transmission. Moreover, it is crucial to take into account the possibility that a failed packet can be successfully decoded by both relays, as well as the ability of the “smart" relay nodes to be aware of the status of the others, which in turn leads to self-aware networks. With that in mind we consider the following cases:
1. Both sources transmit
1. When both sources *failed* to transmit directly to the node $D$, the failed packet of source $k$ is successfully decoded by relay $R_{i}$ with probability ${P_{s}(k,R_i,\{1,2\})}$, $k=1,2$, $i=1,2$, where with probability ${\overline{P}_{s}(0,R_i,\{1,2\})}$, the relay $R_{i}$ failed to decode both packets. Note also that $\overline{P}_{s}(1,R_{i},\{1,2\})=\overline{P}_{s}(0,R_{i},\{1,2\})+P_{s}(2,R_{i},\{1,2\})$, $\overline{P}_{s}(2,R_{i},\{1,2\})=\overline{P}_{s}(0,R_{i},\{1,2\})+P_{s}(1,R_{i},\{1,2\})$, $i=1,2$. Due to the total probability law we have $$({\overline{P}_{s}(0,R_1,\{1,2\})}+{P_{s}(1,R_1,\{1,2\})}+{P_{s}(2,R_1,\{1,2\})})({\overline{P}_{s}(0,R_2,\{1,2\})}+{P_{s}(1,R_2,\{1,2\})}+{P_{s}(2,R_2,\{1,2\})})=1.$$
2. When source 1 (resp. source 2) is the only that succeeds to transmit a packet at node $D$, i.e., its transmission was successfully decoded by node $D$, then with probability $P_{s,2}(2,R_{i},\{1,2\})$ (resp. $P_{s,1}(1,R_{i},\{1,2\})$), $i=1,2$, the failed packet of source $2$ (resp. source $1$) is successfully decoded by the relay $R_{i}$. On the contrary, with probability $\overline{P}_{s,2}(2,R_{i},\{1,2\})$ (resp. $\overline{P}_{s,1}(1,R_{i},\{1,2\})$), the relay $R_{i}$ failed to decode the packet from source $2$ (resp. source $1$), and thus, it is considered lost. Due to the total probability law we have, $$\begin{array}{rl}
(P_{s,2}(2,R_{1},\{1,2\})+\overline{P}_{s,2}(2,R_{1},\{1,2\}))(P_{s,2}(2,R_{2},\{1,2\})+\overline{P}_{s,2}(2,R_{2},\{1,2\}))&=1,\\
(P_{s,1}(1,R_{1},\{1,2\})+\overline{P}_{s,1}(1,R_{1},\{1,2\}))(P_{s,1}(1,R_{2},\{1,2\})+\overline{P}_{s,1}(1,R_{2},\{1,2\}))&=1.
\end{array}$$
2. Only one source transmit, say source $k$, and the other remains silent. When source $k$ fails to transmit directly to node $D$, its failed packet is successfully decoded by relay $R_{i}$ with probability ${P_{s}(k,R_i,\{k\})}$, $k=1,2$, $i=1,2$, where with probability ${\overline{P}_{s}(0,R_i,\{k\})}$, the relay $R_{i}$ fails to decode the packet. Due to the total probability law we have $$({\overline{P}_{s}(0,R_1,\{k\})}+{P_{s}(k,R_1,\{k\})})({\overline{P}_{s}(0,R_2,\{k\})}+{P_{s}(k,R_2,\{k\})})=1.$$
Note that the cases $(1,b)$ and $(2)$ refer to the case where only one source cooperate with a relay. However, we have to distinguish it in two cases because in the former one, there is an interaction among sources since both of them transmit, while in the latter one, only one source transmit and the other remains silent (i.e., there is no interaction). Such an interaction, plays a crucial role on the values of the success probabilities. In wireless systems, the feature of interference and interaction among transmitting nodes is of great importance and have to be taken into account.
If both relays transmit simultaneously, with probability ${P_{s}(R_i,D,\{R_1,R_2\})}$, the packet transmitted from $R_{i}$ is successfully received by node $D$, while with probability ${\overline{P}_{s}(R_i,D,\{R_1,R_2\})}=1-\sum_{i=1,2}{P_{s}(R_i,D,\{R_1,R_2\})}$, both of them failed to be received by the node $D$, and have to be retransmitted in a later time slot. Recall also the success probabilities ${P_{s}^{*}(R_i,D,\{R_i\})}$, ${P_{s}(R_i,D,\{R_i\})}$ of $R_i$ when the other relay node is active (i.e., non-empty), and inactive (i.e., empty) respectively. We assume that ${P_{s}^{*}(R_i,D,\{R_i\})}>{P_{s}(R_i,D,\{R_i\})}>{P_{s}(R_i,D,\{R_1,R_2\})}$. Denote the counter probabilities ${\overline{P}_{s}^{*}(R_i,D,\{R_i\})}=1-{P_{s}^{*}(R_i,D,\{R_i\})}$, ${\overline{P}_{s}(R_i,D,\{R_i\})}=1-{P_{s}(R_i,D,\{R_i\})}$, $i=1,2$.
In the following we proceed with the derivation of a fundamental functional equation, the solution of which, will provide the generating function of the stationary joint queue length distribution at relay nodes. The solution of this functional equation is the key element for obtaining expressions for the queueing delay at relay nodes.
Functional equation and preparatory results
-------------------------------------------
Clearly, $Y_{n}=(N_{1,n},N_{2,n})$ is a discrete time Markov chain with state space $\mathcal{S}=\{(k_{1},k_{2}):k_{1},k_{2}=0,1,2,...\}$. The queues of both relay nodes evolve as: $$\begin{array}{c}
N_{i,n+1}=[N_{i,n}+F_{i,n}]^{+}+A_{i,n},\,i=1,2,
\end{array}
\label{x}$$ where $F_{i,n}$ is either the number of arrivals (in this case $F_{i,n}$ equals 0 or 1) at relay $R_{i}$ at time slot $n$ (in case both source users transmit simultaneously, and the unsuccessful packet is stored in $R_{i}$, or only a single source user transmits, but its transmission was unsuccessful), or the number of departures (in this case $F_{i,n}$ equals $0$ or $-1$) from $R_i$ (this is because when the sources do not transmit, and $R_i$ attempts to transmit a packet at node $D$) at time slot $n$. Recall that the relays have their own traffic, and $A_{i,n}$ represents the number of arrivals (of such generated traffic) in the the time interval $(n,n+1]$. Let $H(x,y)$ be the generating function of the joint stationary queue process and $Z(x,y)$ the generating function of the joint distribution of the number of arriving packets in any slot (i.e., self-generated traffic of the relays), viz. $$\begin{array}{c}
H(x,y)=\lim_{n\to\infty}E(x^{N_{1,n}}y^{N_{2,n}}),\,|x|\leq1,|y|\leq1,\\
Z(x,y)=\lim_{n\to\infty}E(x^{A_{1,n}}y^{A_{2,n}}),\,|x|\leq1,|y|\leq1.
\end{array}$$ In the following we assume for sake of convenience only a particular distribution for the self-generated arrival processes at both relays, namely the geometric distribution[^4] [@nain]. We also assume that both arrival processes are independent. More precisely we assume hereon that $$\begin{array}{c}
Z(x,y)=[(1+\widehat{\lambda}_{1}(1-x))(1+\widehat{\lambda}_{2}(1-y))]^{-1}.
\end{array}$$ Then, by exploiting (\[x\]), and using (\[fc\]) (see Appendix \[po\]), we obtain after lengthy calculations $$\begin{array}{c}
R(x,y)H(x,y)=A(x,y)H(x,0)+B(x,y)H(0,y)+C(x,y)H(0,0),
\end{array}
\label{we}$$ where, $$\begin{array}{rl}
R(x,y)=&Z^{-1}(x,y)-1+\bar{t}_{1}\bar{t}_{2}[\alpha_{1}\widehat{\alpha}_{2}(1-\frac{1}{x})+\alpha_{2}\widehat{\alpha}_{1}(1-\frac{1}{y})]+(1-x)L_{1}+(1-y)L_{2}+(1-xy)L_{3},
\end{array}$$ and $$\begin{array}{rl}
L_{1}=& t_{1}\bar{t}_{2}{\overline{P}_{s}(1,D,\{1\})}{\overline{P}_{s}(1,R_2,\{1\})}{P_{s}(1,R_1,\{1\})}+t_{2}\bar{t}_{1}{\overline{P}_{s}(2,D,\{2\})}{\overline{P}_{s}(2,R_2,\{2\})}{P_{s}(2,R_1,\{2\})}\\&+t_{1}t_{2}[{\overline{P}_{s}(0,D,\{1,2\})}{\overline{P}_{s}(0,R_2,\{1,2\})}({P_{s}(1,R_1,\{1,2\})}+{P_{s}(2,R_1,\{1,2\})})\\
&+{P_{s}(1,D,\{1,2\})}\overline{P}_{s,2}(2,R_{2},\{1,2\})P_{s,2}(2,R_{1},\{1,2\})+{P_{s}(2,D,\{1,2\})}\overline{P}_{s,1}(1,R_{2},\{1,2\})P_{s,1}(1,R_{1},\{1,2\})],
\end{array}$$
$$\begin{array}{rl}
L_{2}=& t_{1}\bar{t}_{2}{\overline{P}_{s}(1,D,\{1\})}{P_{s}(1,R_2,\{1\})}{\overline{P}_{s}(1,R_1,\{1\})}+t_{2}\bar{t}_{1}{\overline{P}_{s}(2,D,\{2\})}{P_{s}(2,R_2,\{2\})}{\overline{P}_{s}(2,R_1,\{2\})}\\&+t_{1}t_{2}[{\overline{P}_{s}(0,D,\{1,2\})}{\overline{P}_{s}(0,R_1,\{1,2\})}({P_{s}(1,R_2,\{1,2\})}+{P_{s}(2,R_2,\{1,2\})})\\
&+{P_{s}(1,D,\{1,2\})}\overline{P}_{s,2}(2,R_{1},\{1,2\})P_{s,2}(2,R_{2},\{1,2\})+{P_{s}(2,D,\{1,2\})}\overline{P}_{s,1}(1,R_{1},\{1,2\})P_{s,1}(1,R_{2},\{1,2\})],
\end{array}$$
$$\begin{array}{rl}
L_{3}=& t_{1}\bar{t}_{2}{\overline{P}_{s}(1,D,\{1\})}{P_{s}(1,R_2,\{1\})}{P_{s}(1,R_1,\{1\})}+t_{2}\bar{t}_{1}{\overline{P}_{s}(2,D,\{2\})}{P_{s}(2,R_2,\{2\})}{P_{s}(2,R_1,\{2\})}\\&+t_{1}t_{2}[{\overline{P}_{s}(0,D,\{1,2\})}({P_{s}(1,R_2,\{1,2\})}+{P_{s}(2,R_2,\{1,2\})})({P_{s}(1,R_1,\{1,2\})}+{P_{s}(2,R_1,\{1,2\})})\\
&+{P_{s}(1,D,\{1,2\})}P_{s,2}(2,R_{1},\{1,2\})P_{s,2}(2,R_{2},\{1,2\})+{P_{s}(2,D,\{1,2\})}P_{s,1}(1,R_{1},\{1,2\})P_{s,1}(1,R_{2},\{1,2\})],
\end{array}$$
$$\begin{array}{rl}
A(x,y)=&\bar{t}_{1}\bar{t}_{2}[d_{1}(1-\frac{1}{x})+\alpha_{2}\widehat{\alpha}_{1}(1-\frac{1}{y})],\\
B(x,y)=&\bar{t}_{1}\bar{t}_{2}[d_{2}(1-\frac{1}{y})+\alpha_{1}\widehat{\alpha}_{2}(1-\frac{1}{x})],\\
C(x,y)=&\bar{t}_{1}\bar{t}_{2}[d_{1}(\frac{1}{x}-1)+d_{2}(\frac{1}{y}-1)],
\end{array}$$
$$\begin{array}{rl}
\widehat{\alpha}_{i}=&\bar{\alpha}_{i}{P_{s}(R_i,D,\{R_i\})}+\alpha_{i}{P_{s}(R_i,D,\{R_1,R_2\})},\,i=1,2,\\
d_{1}=&\alpha_{1}\widehat{\alpha}_{2}-\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})},\\
d_{2}=&\alpha_{2}\widehat{\alpha}_{1}-\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}.
\end{array}$$
Note that $L_{i}$, $i=1,2,3$, has a clear probabilistic interpretation. Indeed, $L_{1}$ (resp. $L_{2}$) is the probability that a (failed) transmitted source packet will be decoded and stored at relay $R_{i}$. Moreover, $L_{3}$ is the probability that a failed transmitted source packet will be decoded and stored at both relays.
Some interesting relations can be obtained directly from (\[we\]). Taking $y = 1$, dividing by $x-1$ and taking $x\to 1$ in (\[we\]) and vice versa yield the following “conservation of flow" relations: $$\lambda_{1}=\bar{t}_{1}\bar{t}_{2}\{\alpha_{1}\widehat{\alpha}_{2}(1-H(0,1))-d_{1}(H(1,0)-H(0,0))\},
\label{r1}$$ $$\lambda_{2}=\bar{t}_{1}\bar{t}_{2}\{\alpha_{2}\widehat{\alpha}_{1}(1-H(1,0))-d_{2}(H(0,1)-H(0,0))\},
\label{r2}$$ where for $i=1,2,$ $$\begin{array}{l}
\lambda_{i}=\widehat{\lambda}_{i}+\lambda_{1,i}+\lambda_{2,i},
\end{array}$$ where now, $$\begin{array}{rl}
\lambda_{1,1}=&t_{1}\bar{t}_{2}{\overline{P}_{s}(1,D,\{1\})}{P_{s}(1,R_1,\{1\})}({P_{s}(1,R_2,\{1\})}+{\overline{P}_{s}(1,R_2,\{1\})})\\
&+t_{1}t_{2}[{\overline{P}_{s}(0,D,\{1,2\})}({\overline{P}_{s}(1,R_2,\{1,2\})}+{P_{s}(1,R_2,\{1,2\})}){P_{s}(1,R_1,\{1,2\})}\\
&+{P_{s}(2,D,\{1,2\})}(\overline{P}_{s,1}(1,R_{2},\{1,2\})+P_{s,1}(1,R_{2},\{1,2\}))P_{s,1}(1,R_{1},\{1,2\})],\vspace{2mm}\\
\lambda_{2,1}=&t_{2}\bar{t}_{1}{\overline{P}_{s}(2,D,\{2\})}{P_{s}(2,R_1,\{2\})}({P_{s}(2,R_2,\{2\})}+{\overline{P}_{s}(2,R_2,\{2\})})\\
&+t_{1}t_{2}[{\overline{P}_{s}(0,D,\{1,2\})}({\overline{P}_{s}(2,R_2,\{1,2\})}+{P_{s}(2,R_2,\{1,2\})}){P_{s}(2,R_1,\{1,2\})}\\
&+{P_{s}(1,D,\{1,2\})}(\overline{P}_{s,2}(2,R_{2},\{1,2\})+P_{s,2}(2,R_{2},\{1,2\}))P_{s,2}(2,R_{1},\{1,2\})],
\end{array}$$ $$\begin{array}{rl}
\lambda_{1,2}=&t_{1}\bar{t}_{2}{\overline{P}_{s}(1,D,\{1\})}{P_{s}(1,R_2,\{1\})}({P_{s}(1,R_1,\{1\})}+{\overline{P}_{s}(1,R_1,\{1\})})\\
&+t_{1}t_{2}[{\overline{P}_{s}(0,D,\{1,2\})}({\overline{P}_{s}(1,R_1,\{1,2\})}+{P_{s}(1,R_1,\{1,2\})}){P_{s}(1,R_2,\{1,2\})}\\
&+{P_{s}(2,D,\{1,2\})}(\overline{P}_{s,1}(1,R_{1},\{1,2\})+P_{s,1}(1,R_{1},\{1,2\}))P_{s,1}(1,R_{2},\{1,2\})],\vspace{2mm}\\
\lambda_{2,2}=&t_{2}\bar{t}_{1}{\overline{P}_{s}(2,D,\{2\})}{P_{s}(2,R_2,\{2\})}({P_{s}(2,R_1,\{2\})}+{\overline{P}_{s}(2,R_1,\{2\})})\\
&+t_{1}t_{2}[{\overline{P}_{s}(0,D,\{1,2\})}({\overline{P}_{s}(1,R_1,\{1,2\})}+{P_{s}(1,R_1,\{1,2\})}){P_{s}(2,R_2,\{1,2\})}\\
&+{P_{s}(1,D,\{1,2\})}(\overline{P}_{s,2}(2,R_{1},\{1,2\})+P_{s,2}(2,R_{1},\{1,2\}))P_{s,2}(2,R_{2},\{1,2\})].
\end{array}$$ From (\[r1\]), (\[r2\]) we realize that the analysis is distinguished in two cases:
1. For $\frac{\alpha_{1}\widehat{\alpha}_{2}}{\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}}+\frac{\alpha_{2}\widehat{\alpha}_{1}}{\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}}=1$, (\[r1\]), (\[r2\]) yield $$\begin{array}{c}
H(0,0)=1-\frac{\lambda_{1}}{\bar{t}_{1}\bar{t}_{2}\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}}-\frac{\lambda_{2}}{\bar{t}_{1}\bar{t}_{2}\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}}=1-\rho.
\end{array}$$
2. For $\frac{\alpha_{1}\widehat{\alpha}_{2}}{\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}}+\frac{\alpha_{2}\widehat{\alpha}_{1}}{\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}}\neq1$, (\[r1\]), (\[r2\]) yield $$\begin{array}{l}
H(1,0)=\frac{d_{2}\lambda_{1}+\alpha_{1}\widehat{\alpha}_{2}(\bar{t}_{1}\bar{t}_{2}\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}-\lambda_{2})+d_{2}\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}H(0,0)}{\bar{t}_{1}\bar{t}_{2}(\alpha_{1}\widehat{\alpha}_{2}\alpha_{2}\widehat{\alpha}_{1}-d_{1}d_{2})},\vspace{2mm}\\
H(0,1)=\frac{d_{1}\lambda_{2}+\alpha_{2}\widehat{\alpha}_{1}(\bar{t}_{1}\bar{t}_{2}\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}-\lambda_{1})+d_{1}\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}H(0,0)}{\bar{t}_{1}\bar{t}_{2}(\alpha_{1}\widehat{\alpha}_{2}\alpha_{2}\widehat{\alpha}_{1}-d_{1}d_{2})}.
\end{array}
\label{rd}$$
Our primary interest in the following is to obtain expressions for the queueing delay at relay nodes. The key element for doing this is to solve the functional equation (\[we\]) and obtain $H(x,y)$. In order to this, we have to obtain the boundary functions $H(x,0)$, $H(0,y)$ and the term $H(0,0)$. The basic tool for obtaining these functions is the theory of boundary value problems [@coh; @fay]. Since we are dealing with a quite technical approach we summarized in the following the basic steps.
#### Step 1. {#step-1. .unnumbered}
From the functional equation (\[we\]), we prove that $H(x,0)$ and $H(0,y)$ satisfy certain boundary value problems of Riemann-Hilbert-Carleman type [@fay], i.e., with boundary conditions on closed curves. These curves are studied in Lemma \[SQ\]. The proof of this lemma (Appendix \[a1\]) requires the investigation of the kernel $R(x,y)$ (see subsection \[ker\]). All the required results are given in Lemmas \[LEM\], \[lem1\] (the proof of Lemma \[LEM\] is given in the Appendix \[a0\]). Note that based on the values of the parameters the unit disc may lie inside the region bounded by these contours. Clearly, the functions $H(x,0)$, $H(0,y)$ are analytic inside the unit disc, but they might have poles in the region bounded by the unit disc and these closed curves. The position of these poles (if exist) are investigated in the Appendix \[ap4\]. With that in mind, the boundary functions admit analytic continuations in the whole interiors of the curves above; see also Chapter 3 in [@fay]. Then, we have to obtain the precise boundary conditions on these curves. This is done in subsubsections \[dir\], \[rh\]; see (\[p1\]), (\[df3\]) respectively.
#### Step 2. {#step-2. .unnumbered}
Next, we conformally transform these problems into boundary value problems of Riemann-Hilbert type on the unit disc; see [@coh]. This conversion is motivated by the fact that the latter problems are more usual and by far more treated in the literature. It is done using conformal mappings in subsubsections \[dir\], \[rh\]; see (\[zx\]).
#### Step 3. {#step-3. .unnumbered}
Finally we solve these new problems and we deduce an explicit integral representation of the unknown boundary functions. This will conclude subsubsections \[dir\], \[rh\]; see (\[soll\]), (\[fin\]) respectively.
Analysis of the kernel {#ker}
----------------------
In the following we focus on the kernel $R(x,y)$, and provide some important properties for the following analysis. To the best of our knowledge, this type of kernel has never been treated in the related literature. Clearly, $$R(x,y)=a(x)y^{2}+b(x)y+c(x)=\widehat{a}(y)x^{2}+\widehat{b}(y)x+\widehat{c}(y),$$ where $$\begin{array}{rl}
a(x)=&-x[\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1}(1-x))+L_{2}+L_{3}x],\\
b(x)=&x[\widehat{\lambda}+\widehat{\lambda}_{1}\widehat{\lambda}_{2}+\bar{t}_{1}\bar{t}_{2}(\alpha_{1}\widehat{\alpha}_{2}+\alpha_{2}\widehat{\alpha}_{1})+L_{1}+L_{2}+L_{3}]-\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}-[\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1}]x^{2},\\
c(x)=&-\bar{t}_{1}\bar{t}_{2}\alpha_{2}\widehat{\alpha}_{1}x,
\end{array}$$ $$\begin{array}{rl}
\widehat{a}(y)=&-y[\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2}(1-y))+L_{1}+L_{3}y],\\
\widehat{b}(y)=&y[\widehat{\lambda}+\widehat{\lambda}_{1}\widehat{\lambda}_{2}+\bar{t}_{1}\bar{t}_{2}(\alpha_{1}\widehat{\alpha}_{2}+\alpha_{2}\widehat{\alpha}_{1})+L_{1}+L_{2}+L_{3}]-\bar{t}_{1}\bar{t}_{2}\alpha_{2}\widehat{\alpha}_{1}-[\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+L_{2}]y^{2},\\
\widehat{c}(y)=&-\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}y.
\end{array}$$ The roots of $R(x,y)=0$ are $X_{\pm}(y)=\frac{-\widehat{b}(y)\pm\sqrt{D_{y}(y)}}{2\widehat{a}(y)}$, $Y_{\pm}(x)=\frac{-b(x)\pm\sqrt{D_{x}(x)}}{2a(x)}$, where $D_{y}(y)=\widehat{b}(y)^{2}-4\widehat{a}(y)\widehat{c}(y)$, $D_{x}(x)=b(x)^{2}-4a(x)c(x)$.
\[LEM\] For $|y|=1$, $y\neq1$, the kernel equation $R(x,y)=0$ has exactly one root $x=X_{0}(y)$ such that $|X_{0}(y)|<1$. For $\lambda_{1}<\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}$, $X_{0}(1)=1$. Similarly, we can prove that $R(x,y)=0$ has exactly one root $y=Y_{0}(x)$, such that $|Y_{0}(x)|\leq1$, for $|x|=1$.
See Appendix \[a0\].
Using simple algebraic arguments we can prove the following lemma, which provides information about the location of the branch points of the two-valued functions $Y(x)$, $X(y)$.
\[lem1\] The algebraic function $Y(x)$, defined by $R(x,Y(x)) = 0$, has four real branch points $0< x_{1}<x_{2}\leq1<x_{3}<x_{4}<\frac{1+\tilde{\lambda}_{1}}{\tilde{\lambda}_{1}}$. Moreover, $D_{x}(x)<0$, $x\in(x_{1},x_{2})\cup(x_{3},x_{4})$ and $D_{x}(x)>0$, $x\in(-\infty,x_{1})\cup(x_{2},x_{3})\cup(x_{4},\infty)$. Similarly, $X(y)$, defined by $R(X(y),y) = 0$, has four real branch points $0\leq y_{1}<y_{2}\leq1<y_{3}<y_{4}<\frac{1+\tilde{\lambda}_{2}}{\tilde{\lambda}_{2}}$, and $D_{x}(y)<0$, $y\in(y_{1},y_{2})\cup(y_{3},y_{4})<$ and $D_{x}(y)>0$, $y\in(-\infty,y_{1})\cup(y_{2},y_{3})\cup(y_{4},\infty)$.
To ensure the continuity of the two valued function $Y(x)$ (resp. $X(y)$) we consider the following cut planes: $\tilde{C}_{x}=\mathbb{C}_{x}-([x_{1},x_{2}]\cup[x_{3},x_{4}]$, $\tilde{C}_{y}=\mathbb{C}_{y}-([y_{1},y_{2}]\cup[y_{3},y_{4}]$, where $\mathbb{C}_{x}$, $\mathbb{C}_{y}$ the complex planes of $x$, $y$, respectively. In $\tilde{C}_{x}$ (resp. $\tilde{C}_{y}$), denote by $Y_{0}(x)$ (resp. $X_{0}(y)$) the zero of $R(x,Y(x))=0$ (resp. $R(X(y),y)=0$) with the smallest modulus, and $Y_{1}(x)$ (resp. $X_{1}(y)$) the other one.
Define the image contours, $\mathcal{L}=Y_{0}[\overrightarrow{\underleftarrow{x_{1},x_{2}}}]$, $\mathcal{L}_{ext}=Y_{0}[\overrightarrow{\underleftarrow{x_{3},x_{4}}}]$, $\mathcal{M}=X_{0}[\overrightarrow{\underleftarrow{y_{1},y_{2}}}]$, $\mathcal{M}_{ext}=X_{0}[\overrightarrow{\underleftarrow{y_{3},y_{4}}}]$, where $[\overrightarrow{\underleftarrow{u,v}}]$ stands for the contour traversed from $u$ to $v$ along the upper edge of the slit $[u,v]$ and then back to $u$ along the lower edge of the slit. The following lemma shows that the mappings $Y(x)$, $X(y)$, for $x\in[x_{1},x_{2}]$, $y\in[y_{1},y_{2}]$ respectively, give rise to the smooth and closed contours $\mathcal{L}$, $\mathcal{M}$ respectively:
\[SQ\]
1. For $y\in[y_{1},y_{2}]$, the algebraic function $X(y)$ lies on a closed contour $\mathcal{M}$, which is symmetric with respect to the real line and defined by $$\begin{array}{l}
|x|^{2}=m(Re(x)),\,m(\delta)=\frac{\widehat{c}(\zeta(\delta))}{\widehat{a}(\zeta(\delta))},\,|x|^{2}\leq\frac{\widehat{c}(y_{2})}{\widehat{a}(y_{2})},
\end{array}$$ where, $\zeta(\delta)=\frac{k_{2}(\delta)-\sqrt{k_{2}^{2}(\delta)-4\bar{t}_{1}\bar{t}_{2}\alpha_{2}\widehat{\alpha}_{1}k_{1}(\delta)}}{2k_{1}(\delta)}$, $$\begin{array}{rl}
k_{1}(\delta):=&\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+L_{2}+2\delta(L_{3}-\widehat{\lambda}_{1}\widehat{\lambda}_{2}),\\
k_{2}(\delta):=&(1+2\delta)(L_{1}+\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2}))+\widehat{\lambda}_{2}+L_{2}+L_{3}+\bar{t}_{1}\bar{t}_{2}(\alpha_{1}\widehat{\alpha}_{2}+\alpha_{2}\widehat{\alpha}_{1}).
\end{array}$$ Set $\beta_{0}:=\sqrt{\frac{\widehat{c}(y_{2})}{\widehat{a}(y_{2})}}$, $\beta_{1}:=-\sqrt{\frac{\widehat{c}(y_{1})}{\widehat{a}(y_{1})}}$ the extreme right and left point of $\mathcal{M}$, respectively.
2. For $x\in[x_{1},x_{2}]$, the algebraic function $Y(x)$ lies on a closed contour $\mathcal{L}$, which is symmetric with respect to the real line and defined by $$\begin{array}{l}
|y|^{2}=v(Re(y)),\,v(\delta)=\frac{c(\theta(\delta))}{a(\theta(\delta))},\,|y|^{2}\leq\frac{c(x_{2})}{a(x_{2})},
\end{array}$$ where $\theta(\delta)=\frac{l_{2}(\delta)-\sqrt{l_{2}^{2}(\delta)-4\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}l_{1}(\delta)}}{2l_{1}(\delta)}$, $$\begin{array}{rl}
l_{1}(\delta):=&\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1}+2\delta(L_{3}-\widehat{\lambda}_{1}\widehat{\lambda}_{2}),\\
l_{2}(\delta):=&(1+2\delta)(L_{2}+\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1}))+\widehat{\lambda}_{1}+L_{2}+L_{3}+\bar{t}_{1}\bar{t}_{2}(\alpha_{1}\widehat{\alpha}_{2}+\alpha_{2}\widehat{\alpha}_{1}).
\end{array}$$ Set $\eta_{0}:=\sqrt{\frac{c(x_{2})}{a(x_{2})}}$, $\eta_{1}=-\sqrt{\frac{c(x_{1})}{a(x_{1})}}$, the extreme right and left point of $\mathcal{L}$, respectively.
See Appendix \[a1\].
The boundary value problems {#bound}
---------------------------
As indicated in the previous section the analysis is distinguished in two cases, which differ both from the modeling and the technical point of view.
### A Dirichlet boundary value problem {#dir}
Consider the case $\frac{\alpha_{1}\widehat{\alpha}_{2}}{\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}}+\frac{\alpha_{2}\widehat{\alpha}_{1}}{\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}}=1$. Then, $$\begin{array}{c}
A(x,y)=\frac{d_{1}}{\alpha_{1}\widehat{\alpha}_{2}}B(x,y)\Leftrightarrow A(x,y)=\frac{\alpha_{2}\widehat{\alpha}_{1}}{d_{2}}B(x,y).
\end{array}$$ Therefore, for $y\in \mathcal{D}_{y}=\{x\in\mathcal{C}:|y|\leq1,|X_{0}(y)|\leq1\}$, $$\begin{array}{l}
\alpha_{2}\widehat{\alpha}_{1}H(X_{0}(y),0)+d_{2}H(0,y)+\frac{\alpha_{2}\widehat{\alpha}_{1}(1-\rho)C(X_{0}(y),y)}{A(X_{0}(y),y)}=0.
\end{array}
\label{con}$$ For $y\in \mathcal{D}_{y}-[y_{1},y_{2}]$ both $H(X_{0}(y),0)$, and $H(0,y)$ are analytic and thus, by means of analytic continuation, we can also consider (\[con\]) for $y\in[y_{1},y_{2}]$, or equivalently, for $x\in\mathcal{M}$ $$\begin{array}{c}
\alpha_{2}\widehat{\alpha}_{1}H(x,0)+d_{2}H(0,Y_{0}(x))+\frac{\alpha_{2}\widehat{\alpha}_{1}(1-\rho)C(x,Y_{0}(x))}{A(x,Y_{0}(x))}=0.
\end{array}
\label{con2}$$ Then, multiplying both sides of (\[con2\]) by the imaginary complex number $i$, and noticing that $H(0,Y_{0}(x))$ is real for $x\in\mathcal{M}$, since $Y_{0}(x)\in[y_{1},y_{2}]$, we have $$\begin{array}{c}
Re(iH(x,0))=Re(-i\frac{C(x,Y_{0}(x))}{A(x,Y_{0}(x))})(1-\rho),\,x\in\mathcal{M}.
\end{array}
\label{p1}$$ Clearly, some analytic continuation considerations must be made in order to be everything well defined. To do this, we have to check for poles of $H(x,0)$ in $S:=G_{\mathcal{M}}\cap\bar{D}_{x}^{c}$, where $G_{\mathcal{U}}$ be the interior domain bounded by $\mathcal{U}$, and $D_{x}=\{x:|x|<1\}$, $\bar{D}_{x}=\{x:|x|\leq1\}$, $\bar{D}_{x}^{c}=\{x:|x|>1\}$. These poles, if exist, they coincide with the zeros of $A(x,Y_{0}(x))$ in $S_{x}$; see Appendix \[ap4\]. Note that equation (\[p1\]) is defined on $\mathcal{M}$. In order to solve (\[p1\]) we must firstly conformally transform the problem from $\mathcal{M}$ to the unit circle. Let the conformal mapping, $z=\gamma(x):G_{\mathcal{M}}\to G_{\mathcal{C}}$, and its inverse $x=\gamma_{0}(z):G_{\mathcal{C}}\to G_{\mathcal{M}}$.
Then, we have the following problem: Find a function $\tilde{T}(z)=H(\gamma_{0}(z),0)$ regular for $z\in G_\mathcal{C}$, and continuous for $z\in\mathcal{C}\cup G_\mathcal{C}$ such that, $Re(i\tilde{T}(z))=w(\gamma_{0}(z))$, $z\in\mathcal{C}$. In case $H(x,0)$ has no poles in $S$, the solution of the Dirichlet problem with boundary condition (\[p1\]) is: $$\begin{array}{c}
H(x,0)=-\frac{1-\rho}{2\pi}\int_{|t|=1}f(t)\frac{t+\gamma(x)}{t-\gamma(x)}\frac{dt}{t}+C,\,x\in\mathcal{M},
\end{array}
\label{sol1}$$ where $f(t)=Re(-i\frac{C(\gamma_{0}(t),Y_{0}(\gamma_{0}(t)))}{A(\gamma_{0}(t),Y_{0}(\gamma_{0}(t)))})$, $C$ a constant to be defined by setting $x=0\in G_{\mathcal{M}}$ in (\[sol1\]) and using the fact that $H(0,0)=1-\rho$, $\gamma(0)=0$ (In case $H(x,0)$ has a pole, say $\bar{x}$, we still have a Dirichlet problem for the function $(x-\bar{x})H(x,0)$).
Following the discussion above, $$\begin{array}{c}
C=(1-\rho)(1+\frac{1}{2\pi}\int_{|t|=1}f(t)\frac{dt}{t}),
\end{array}$$ Setting $t=e^{i\phi}$, $\gamma_{0}(e^{i\phi})=\rho(\psi(\phi))e^{i\psi(\phi)}$, we obtain after some algebra, $$\begin{array}{c}
f(e^{i\phi})=\frac{d_{1}\alpha_{2}^{*}\sin(\psi(\phi))(1-Y_{0}(\gamma_{0}(e^{i\phi}))^{-1})}{\rho(\psi(\phi))k(\phi)},\end{array}$$ which is an odd function of $\phi$, and $$\begin{array}{c}
k(\phi)=[\alpha_{2}\widehat{\alpha}_{1}(1-Y_{0}^{-1}(\gamma_{0}(e^{i\phi})))+d_{1}(1-\frac{\cos(\psi(\phi))}{\rho(\psi(\phi))})]^{2}+(d_{1}\frac{\sin(\psi(\phi))}{\rho(\psi(\phi))})^{2}.
\end{array}$$ Thus, $C=1-\rho$. Substituting in (\[sol1\]) we obtain after simple calculations an integral representation of $H(x,0)$ on a real interval, i.e., $$\begin{array}{c}
H(x,0)=(1-\rho)\{1+\frac{2\gamma(x)i}{\pi}\int_{0}^{\pi}\frac{f(e^{i\phi})\sin(\phi)}{1-2\gamma(x)\cos(\phi)-\gamma(x)^{2}}d\phi\},\,x\in G_{\mathcal{M}}.
\end{array}
\label{soll}$$
Similarly, we can determine $H(0,y)$ by solving another Dirichlet boundary value problem on the closed contour $\mathcal{L}$. Then, using the fundamental functional equation (\[we\]) we uniquely obtain $H(x,y)$.
### A homogeneous Riemann-Hilbert boundary value problem {#rh}
In case $\frac{\alpha_{1}\widehat{\alpha}_{2}}{\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}}+\frac{\alpha_{2}\widehat{\alpha}_{1}}{\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}}\neq1$, consider the following transformation: $$\begin{array}{rl}
G(x):=H(x,0)+\frac{\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}d_{2}H(0,0)}{d_{1}d_{2}-\alpha_{1}\widehat{\alpha}_{2}\alpha_{2}\widehat{\alpha}_{1}},&
L(y):=H(0,y)+\frac{\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}d_{1}H(0,0)}{d_{1}d_{2}-\alpha_{1}\widehat{\alpha}_{2}\alpha_{2}\widehat{\alpha}_{1}}.
\end{array}$$ Then, for $y\in \mathcal{D}_{y}$, $$\begin{array}{c}
A(X_{0}(y),y)G(X_{0}(y))=-B(X_{0}(y),y)L(y).
\end{array}
\label{po31}$$ For $y\in \mathcal{D}_{y}-[y_{1},y_{2}]$ both $G(X_{0}(y))$, $L(y)$ are analytic and the right-hand side can be analytically continued up to the slit $[y_1, y_2]$ or equivalently, for $x\in\mathcal{M}$ $$\begin{array}{c}
A(x,Y_{0}(x))G(x)=-B(x,Y_{0}(x))L(Y_{0}(x)).
\end{array}
\label{za1}$$ Clearly, $G(x)$ is holomorphic for $D_{x}$, continuous for $\bar{D}_{x}$. However, $G(x)$ might has poles in $S_{x}$, based on the values of the system parameters. These poles (if exist) coincide with the zeros of $A(x,Y_{0}(x))$ in $S_{x}$; see Appendix \[ap4\]. For $y\in[y_{1},y_{2}]$, let $X_{0}(y)=x\in\mathcal{M}$ and realize that $Y_{0}(X_{0}(y))=y$ so that $y=Y_{0}(x)$. Taking into account the possible poles of $G(x)$, and noticing that $L(Y_{0}(x))$ is real for $x\in\mathcal{M}$, since $Y_{0}(x)\in[y_{1},y_{2}]$, we have $$\begin{array}{c}
Re[iU(x)\tilde{G}(x)]=0,\,x\in\mathcal{M},\vspace{2mm}\\
U(x)=\frac{A(x,Y_{0}(x))}{(x-\bar{x})^{r}B(x,Y_{0}(x))},\,\tilde{G}(x)=(x-\bar{x})^{r}G(x),
\end{array}
\label{df3}$$ where $r=0,1$, whether $\bar{x}$ is zero or not of $A(x,Y_{0}(x))$ in $S_{x}$. Thus, $\tilde{G}(x)$ is regular for $x\in G_{\mathcal{M}}$, continuous for $x\in\mathcal{M}\cup G_{\mathcal{M}}$, and $U(x)$ is a non-vanishing function on $\mathcal{M}$. As usual, we must firstly conformally transform the problem (\[df3\]) from $\mathcal{M}$ to the unit circle, using the mapping $z=\gamma(x):G_{\mathcal{M}}\to G_{\mathcal{C}}$, and its inverse given by $x=\gamma_{0}(z):G_{\mathcal{C}}\to G_{\mathcal{M}}$.
Then, the problem in (\[df3\]) is reduced to the following: Find a function $F(z):=\tilde{G}(\gamma_{0}(z))$, regular in $G_{\mathcal{C}}$, continuous in $G_{\mathcal{C}}\cup\mathcal{C}$ such that, $Re[iU(\gamma_{0}(z))F(z)]=0,\,z\in\mathcal{C}$.
A crucial step in the solution of the boundary value problem is the determination of the index $\chi=\frac{-1}{\pi}[arg\{U(x)\}]_{x\in \mathcal{M}}$, where $[arg\{U(x)\}]_{x\in \mathcal{M}}$, denotes the variation of the argument of the function $U(x)$ as $x$ moves along the closed contour $\mathcal{M}$ in the positive direction, provided that $U(x)\neq0$, $x\in\mathcal{M}$. Following the lines in [@fay] we have,
1. If $\lambda_{2}<\lambda_{2}^{*}$, then $\chi=0$ is equivalent to $$\begin{array}{l}
\frac{d A(x,Y_{0}(x))}{dx}|_{x=1}<0\Leftrightarrow\lambda_{1}<\bar{t}_{1}\bar{t}_{2}[\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}+\frac{d_{1}\lambda_{2}}{\bar{t}_{1}\bar{t}_{2}\alpha_{2}\widehat{\alpha}_{1}}],\vspace{2mm}\\ \frac{d B(X_{0}(y),y)}{dy}|_{y=1}<0\Leftrightarrow\lambda_{2}<\bar{t}_{1}\bar{t}_{2}[\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}+\frac{d_{2}\lambda_{1}}{\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}}].
\end{array}$$
2. If $\lambda_{2}\geq \lambda_{2}^{*}$, $\chi=0$ is equivalent to $\frac{d B(X_{0}(y),y)}{dy}|_{y=1}<0\Leftrightarrow \lambda_{2}<\bar{t}_{1}\bar{t}_{2}[\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}+\frac{d_{2}\lambda_{1}}{\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}}]$.
Thus, under stability conditions (see Lemma \[Thm2users\]) the problem defined in (\[df3\]) has a unique solution given by, $$\begin{array}{rl}
H(x,0)=&K(x-\bar{x})^{-r}\exp[\frac{1}{2i\pi}\int_{|t|=1}\frac{\log\{J(t)\}}{t-\gamma(x)}dt]-\frac{\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}d_{2}H(0,0)}{d_{1}d_{2}-\alpha_{1}\widehat{\alpha}_{2}\alpha_{2}\widehat{\alpha}_{1}},\,x\in G_{\mathcal{M}},
\end{array}
\label{sool1}$$ where $K$ is a constant and $J(t)=\frac{\overline{U_{1}(t)}}{U_{1}(t)}$, $U_{1}(t)=U(\gamma_{0}(t))$, $|t|=1$. Setting $x=0$ in (\[sool1\]) we derive a relation between $K$ and $H(0,0)$. Then, for $x=1\in G_{\mathcal{M}}$, and using the first in (\[rd\]) we can obtain $K$ and $H(0,0)$. Substituting back in (\[sool1\]) we obtain for $x\in G_{\mathcal{M}}$, $$\begin{array}{rl}
H(x,0)=&\frac{\lambda_{1}d_{2}+\alpha_{1}\widehat{\alpha}_{2}(\bar{t}_{1}\bar{t}_{2}\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}-\lambda_{2})}{(\bar{x}-1)^{r}\bar{t}_{1}\bar{t}_{2}(\alpha_{1}\widehat{\alpha}_{2}\alpha_{2}\widehat{\alpha}_{1}-d_{1}d_{2})}\left((\bar{x}-x)^{r}\exp[\frac{\gamma(x)-\gamma(1)}{2i\pi}\int_{|t|=1}\frac{\log\{J(t)\}}{(t-\gamma(x))(t-\gamma(1))}dt]\right.\vspace{2mm}\\&\left.+\frac{\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}d_{2}\bar{x}^{r}}{\alpha_{1}\widehat{\alpha}_{2}\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}}\exp[\frac{-\gamma(1)}{2i\pi}\int_{|t|=1}\frac{\log\{J(t)\}}{t(t-\gamma(1))}dt]\right).
\end{array}
\label{fin}$$ Similarly, we can determine $H(0,y)$ by solving another Riemann-Hilbert boundary value problem on the closed contour $\mathcal{L}$. Then, using the fundamental functional equation (\[we\]) we uniquely obtain $H(x,y)$.
Construction of the conformal mappings and numerical issues
-----------------------------------------------------------
The construction of the conformal mapping $\gamma(x)$ is not a trivial task. However, we can construct the inverse of it in order to obtain expressions for the expected value of the queue lengths in each relay node. To proceed, we need a representation of $\mathcal{M}$ in polar coordinates, i.e., $\mathcal{M}=\{x:x=\rho(\phi)\exp(i\phi),\phi\in[0,2\pi]\}.$ This procedure is described in detail in [@coh].
In the following we summarize the basic steps: Since $0\in G_{\mathcal{M}}$, for each $x\in\mathcal{M}$, a relation between its absolute value and its real part is given by $|x|^{2}=m(Re(x))$ (see Lemma \[SQ\]). Given the angle $\phi$ of some point on $\mathcal{M}$, the real part of this point, say $\delta(\phi)$, is the zero of $\delta-\cos(\phi)\sqrt{m(\delta)}$, $\phi\in[0,2\pi].$ Since $\mathcal{M}$ is a smooth, egg-shaped contour, the solution is unique. Clearly, $\rho(\phi)=\frac{\delta(\phi)}{\cos(\phi)}$, and the parametrization of $\mathcal{M}$ in polar coordinates is fully specified.
Then, the mapping from $z\in G_{\mathcal{C}}$ to $x\in G_{\mathcal{M}}$, where $z = e^{i\phi}$ and $x= \rho(\psi(\phi))e^{i\psi(\phi)}$, satisfying $\gamma_{0}(0)=0$, $\gamma_{0}(z)=\overline{\gamma_{0}(\bar{z})}$ is uniquely determined by (see [@coh], Section I.4.4), $$\begin{array}{rl}
\gamma_{0}(z)=&z\exp[\frac{1}{2\pi}\int_{0}^{2\pi}\log\{\rho(\psi(\omega))\}\frac{e^{i\omega}+z}{e^{i\omega}-z}d\omega],\,|z|<1,\\
\psi(\phi)=&\phi-\int_{0}^{2\pi}\log\{\rho(\psi(\omega))\}\cot(\frac{\omega-\phi}{2})d\omega,\,0\leq\phi\leq 2\pi,
\end{array}
\label{zx}$$ i.e., $\psi(.)$ is uniquely determined as the solution of a Theodorsen integral equation with $\psi(\phi)=2\pi-\psi(2\pi-\phi)$. This integral equation has to be solved numerically by an iterative procedure. For the numerical evaluation of the integrals we split the interval $[0,2\pi]$ into $M$ parts of length $2\pi/M$, by taking $M$ points $\phi_{k}=\frac{2k\pi}{M}$, $k=0,1,...,M-1$. For the $M$ points given by their angles $\left\{\phi_{0},...,\phi_{M-1}\right\}$ we should solve the second in (\[zx\]) to obtain the corresponding points $\left\{\psi(\phi_{0}),...,\psi(\phi_{M-1})\right\}$, iteratively from, $$\begin{array}{rl}
\psi_{0}(\phi_{k})=&\phi_{k},\\
\psi_{n+1}(\phi_{k})=&\phi_{k}-\frac{1}{2\pi}\int_{0}^{2\pi}\log\left\{\frac{\delta(\psi_{n}(\omega))}{\cos(\psi_{n}(\omega))}\right\}\cot[\frac{1}{2}(\omega-\phi_{k})]d\omega,
\end{array}
\label{uip}$$ where $\lim_{n\to\infty}\psi_{n+1}(\phi)=\psi(\phi)$, and $\delta(\psi_{n}(\omega))$ is determined by, $$\delta(\psi_{n}(\omega))=cos(\psi_{n}(\omega))\sqrt{m(\delta(\psi_{n}(\omega)))},$$ using the Newton-Raphson root finding method. For each step, the integral in (\[uip\]) is numerically determined by again using the trapezium rule with $M$ parts of equal length $2\pi/M$. For the iteration, we have used the following stopping criterion $\max_{k\in\left\{0,1,...,M-1\right\}}\left|\psi_{n+1}(\phi_{k})-\psi_{n}(\phi_{k})\right|<10^{-6}$
Having obtained $\psi(\phi)$ numerically, the values of the conformal mapping $\gamma_{0}(z)$, $\left|z\right|\leq 1$, can be calculated by applying the Plemelj-Sokhotski formula to the first in (\[zx\]) $$\gamma_{0}(e^{i\phi})=e^{i\psi(\phi)}\frac{\delta(\psi(\phi))}{\cos(\psi(\phi))}=\delta(\psi(\phi))[1+i \tan(\psi(\phi))],\,0\leq\phi\leq 2\pi.$$ We further need to find $\gamma(1)$, $\gamma^{\prime}(1)$. To do this, one needs to use the Newton’s method and solve $\gamma_{0}(z_{0})=1$, in $[0,1]$, i.e., $z_{0}$ is the zero in $[0,1]$ of $\gamma_{0}(z)=1$. Then, $\gamma(1)=z_{0}$. Moreover, using the first in (\[zx\]) $$\gamma^{\prime}(1)=(\gamma_{0}^{\prime}(z_{0}))^{-1}=\{\frac{1}{\gamma(1)}+\frac{1}{2\pi i}\int_{0}^{2\pi}\log\{\rho(\psi(\omega))\}\frac{2e^{i\omega}}{(e^{i\omega}-\gamma(1))^{2}}d\omega\}^{-1},
\label{cv}$$ which can be obtained numerically by using the Trapezoidal rule for the integral on the right-hand side of (\[cv\]).
Clearly, the numerical computation of the exact conformal mappings is generally time consuming. Since $\mathcal{M}$, $\mathcal{L}$ are close to ellipses, alternatively, we can approximate them by conformal mappings that map the interior of ellipses to $G_{\mathcal{C}}$ [@neh]. In particular, we can approximate the contour $\mathcal{M}$ by ellipse $\mathcal{E}$ with semi-axes $\rho(0)$, $\rho(\pi/2)$. Then, $\epsilon(x)$ maps $G_{\mathcal{E}}$ to $G_{\mathcal{C}}$ [@neh], where $$\begin{array}{rl}
\epsilon(x)=\sqrt{k}sn\left(\frac{2Q}{\pi}\sin^{-1}(\frac{x}{\sqrt{\rho^{2}(0)-\rho^{2}(\pi/2)}});k^{2}\right),&k=16q\prod_{n=1}^{\infty}\left(\frac{1+q^{2n}}{1+q^{2n-1}}\right)^{8},\\
q=\left(\frac{\rho(0)-\rho(\pi/2)}{\rho(0)+\rho(\pi/2)}\right)^{2},&Q=\int_{0}^{1}\frac{dt}{\sqrt{(1+t^{2})(1-k^{2}t^{2})}},
\end{array}$$ where $sn(w;l)$ is the Jacobian elliptic function. Our approximation for $\gamma(x)$ is $\epsilon(x)$, $x\in \mathcal{M}\cup G_{\mathcal{M}}$.
Performance metrics {#per}
-------------------
In the following we derive formulas for the expected number of packets and the average delay at each user node in steady state, say $E_{i}$ and $D_{i}$, $i=1,2,$ respectively. Denote by $H_{1}(x,y)$, $H_{2}(x,y)$ the derivatives of $H(x,y)$ with respect to $x$ and $y$ respectively. Then, $E_{i}=H_{i}(1,1)$, and using Little’s law $D_{i}=H_{i}(1,1)/\lambda_{i}$, $i=1,2$. Using the functional equation (\[we\]), (\[r1\]) and (\[r2\]) we arrive after simple calculations in $$\begin{array}{lccr}
E_{1}=\frac{\lambda_{1}+d_{1}H_{1}(1,0)}{\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}},&&&E_{2}=\frac{\lambda_{2}+d_{2}H_{2}(0,1)}{\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}}.
\end{array}
\label{perf}$$ We only focus on $E_{1}$, $D_{1}$ (similarly we can obtain $E_{2}$, $D_{2}$). Note that $H_{1}(1,0)$ can be obtained using either (\[fin\]) or (\[soll\]). For the general case $\frac{\alpha_{1}\widehat{\alpha}_{2}}{\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}}+\frac{\alpha_{2}\widehat{\alpha}_{1}}{\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}}\neq1$, and using (\[fin\]), $$\begin{array}{rl}
H_{1}(1,0)=&\frac{\lambda_{1}d_{2}+\alpha_{1}\widehat{\alpha}_{2}(\bar{t}_{1}\bar{t}_{2}\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}-\lambda_{2})}{\bar{t}_{1}\bar{t}_{2}(\alpha_{1}\widehat{\alpha}_{2}\alpha_{2}\widehat{\alpha}_{1}-d_{1}d_{2})}\frac{\gamma^{\prime}(1)}{2\pi i}\int_{|t|=1}\frac{\log\{J(t)\}}{(t-\gamma(1))^{2}}dt.
\end{array}
\label{xz}$$ Substituting (\[xz\]) in (\[perf\]) we obtain $E_{1}$, and dividing with $\lambda_{1}$, the average delay $D_{1}$. Note that the calculation of (\[perf\]) requires the evaluation of integrals (\[zx\]), (\[cv\]), (\[xz\]) using the trapezoid rule, and $\gamma(1)$, $\gamma^{\prime}(1)$, as described above.
Explicit expressions for the symmetrical model {#sym}
==============================================
In the following we consider the symmetrical model and obtain exact expressions for the average delay without solving a boundary value problem.
As symmetrical, we mean the case where $\widehat{\lambda}_{k}=\widehat{\lambda}$, ${\overline{P}_{s}(k,D,\{k\})}=1-{P_{s}(k,D,\{k\})}=1-q=\bar{q}$, ${P_{s}(k,D,\{1,2\})}=\tilde{q}$, $t_{k}=t$, $k=1,2,$ $\alpha_{i}^{*}=\alpha^{*}$, $\alpha_{i}=\alpha$, $P_{s}(R_{i},D,\{R_{i}\})=\bar{s}$, $P_{s}(R_{i},D,\{R_{1},R_{2}\})=s_{1,2}$, $P_{s}^{*}(R_{i},D,\{R_{i}\})=\tilde{s}$, $P_{s}(k,R_{i},\{k\})=P_{s}(1,R_{i},\{1,2\})+P_{s}(2,R_{i},\{1,2\})=P_{s,k}(k,R_{i},\{1,2\})=r$, $k=1,2$, $i=1,2$. As a result, $d_{1}=d_{2}=d$.
Then, by exploiting the symmetry of the model we clearly have $H_{1}(1,1)=H_{2}(1,1)$, $H_{1}(1,0)=H_{2}(0,1)$. Recall that $E_{i}=H_{i}(1,1)$ the expected number of packets in relay node $R_{i}$. Therefore, after simple calculations using (\[we\]) we obtain, $$\begin{array}{c}
E_{1}=\frac{\widehat{\lambda}+t(t+2\bar{t}\bar{q})r+\bar{t}^{2}dH_{1}(1,0)}{\bar{t}^{2}\alpha
\widehat{\alpha}-(\widehat{\lambda}+t(t+2\bar{t}\bar{q})r)}.
\end{array}
\label{t1}$$ Setting $x=y$ in (\[we\]), differentiating it with respect to $x$ at $x=1$, and using (\[r1\]) we obtain $$\begin{array}{c}
E_{1}+E_{2}=2E_{1}=\frac{2(\widehat{\lambda}+t(t+2\bar{t}\bar{q})-\widehat{\lambda}^{2}+2H_{1}(1,0)\bar{t}^{2}(\alpha\widehat{\alpha}+d)}{2[\bar{t}^{2}\alpha
\widehat{\alpha}-(\widehat{\lambda}+t(t+2\bar{t}\bar{q})r]}.
\end{array}
\label{t2}$$ Using (\[t1\]), (\[t2\]) we finally obtain $$\begin{array}{c}
E_{1}=E_{2}=\frac{\widehat{\lambda}^{2}d+2\widehat{\lambda}\alpha\widehat{\alpha}+\lambda(\lambda+2\bar{\lambda}\bar{q})r(2\alpha\widehat{\alpha}-rd)}{2\alpha^{*}\tilde{s}[\bar{\lambda}^{2}\alpha
\widehat{\alpha}-(\widehat{\lambda}+\lambda(\lambda+2\bar{\lambda}\bar{q})r)]}.
\end{array}
\label{rt}$$ Therefore, using Little’s law the average delay in a node is given by $$\begin{array}{c}
D_{1}=D_{2}=\frac{\widehat{\lambda}^{2}d+2\widehat{\lambda}\alpha\widehat{\alpha}+t(t+2\bar{t}\bar{q})r(2\alpha\widehat{\alpha}-rd)}{2\tilde{\lambda}\alpha^{*}\tilde{s}[\bar{t}^{2}\alpha
\widehat{\alpha}-\lambda]},
\end{array}
\label{rt1}$$ where $\lambda=\widehat{\lambda}+t(t+2\bar{t}\bar{q})r$, and $\bar{t}^{2}\alpha
\widehat{\alpha}-\lambda>0$ due to the stability condition.
Numerical results {#num}
=================
In this section we evaluate numerically the theoretical results obtained in the previous sections. We will focus on a symmetric users setup in order to simplify the presentation. In particular, we consider the case where $\alpha^*=1$, $\alpha=0.7$, and $t=0.1$. The distance between the users and the destination is $110m$, the distance between the users and the relays is $80m$ and between the destination and the relays the distance is $80m$. The path-loss exponent is assumed to be four, we also consider Raleigh fading for the channel gain. The transmit power for both relays is $10mW$ and for the users is $1mW$. We consider two cases for the SINR threshold, $SINR_t=0.2, 1$, when $SINR_t=0.2$ the MPR capability is stronger thus, we can have more than two concurrent successful transmissions. We can compute the success probabilities using Equation (1) in [@PappasTWC2015], for $SINR_t=0.2$ we obtain $P_s(D,1)=0.74$, $P_s(R_1,1)=P_s(R_2,1)=0.92$, ${P_{s}(R_1,D,\{R_1\})}={P_{s}(R_2,D,\{R_2\})}=0.99$, ${P_{s}(R_1,D,\{R_1,R_2\})}={P_{s}(R_2,D,\{R_1,R_2\})}=0.83$. For $SINR_t=1$ we obtain $P_s(D,1)=0.23$, $P_s(R_1,1)=P_s(R_2,1)=0.66$, ${P_{s}(R_1,D,\{R_1\})}={P_{s}(R_2,D,\{R_2\})}=0.96$, ${P_{s}(R_1,D,\{R_1,R_2\})}={P_{s}(R_2,D,\{R_1,R_2\})}=0.5$.
Stability
---------
Here we present the effect of the number of users on the stability region. We consider the cases where the number of users is varying from $1$ to $11$, i.e. $N=1,...,11$. In Fig. \[stab02\], we consider the case where $SINR_t=0.2$, the outer curve in the plot represents the case where $N=1$, the inner the case corresponds to $N=11$. Since we have stronger MPR capabilities at the receivers we observe that the region for up to four users is convex thus, it the performance is better than a time division scheme as also described in Sections \[stab-anal-2\] and \[stab-anal-n\].
![The effect of number of users on the stability region for $SINR_t=0.2$.[]{data-label="stab02"}](stability_2.pdf)
In Fig. \[stab1\], we consider the case where $SINR_t=1$, the outer curve in the plot represents the case where $N=1$ and the inner the case for $N=11$. In this plot, we observe that above two users the region is not a convex set. Thus, a time division scheme would be preferable as also described in Sections \[stab-anal-2\] and \[stab-anal-n\].
![The effect of number of users on the stability region for $SINR_t=1$.[]{data-label="stab1"}](stability_1.pdf)
In both cases, we observe that as the number of users is increasing, then the number of slots that the relays can transmit packets from their queues is decreasing. Thus, when $N=11$, the stability region is approaching the empty set, which is an indication that the relays cannot sustain the traffic in their queues.
Throughput performance
----------------------
We provide numerical evaluation regarding throughput per user and aggregate throughput for the case of pure relays, i.e. $\widehat{\lambda}_{1}=\widehat{\lambda}_{2}=0$.
The throughput per user as the number of users increasing in the network is depicted in Fig. \[thrU\]. The throughput per user is decreasing as the number of users is increasing, in addition we can observe for $SINR_t=0.2$, the system becomes unstable after $12$ users, for $SINR_t=1$ the system remains stable when the number of users is up to $6$. The aggregate throughput is depicted in Fig. \[athr\], the maximum aggregate throughput for $SINR_t=0.2$ and $SINR_t=1$ is achieved for twelve and six users respectively.
![Throughput per user versus the number of users.[]{data-label="thrU"}](thrU.pdf)
![Aggregate throughput versus the number of users.[]{data-label="athr"}](athr.pdf)
Average Delay and Stability Region for the capture model
--------------------------------------------------------
In this part we will evaluate the average delay performance. The setup will be different from the previous two subsection due to the capture channel model assumed in the analysis.
#### Example 1. The symmetrical system
In the following we consider the symmetric system and we investigate the effect of the system parameters on the average delay. We assume that $q=0.5$, $\bar{s}=0.8$, $\tilde{s}=0.9$, $s_{12}=0.4$. In Fig. \[f153\] we can see the effect of $r$ (i.e., the reception probability of blocked packet by a relay node) on the average delay for increasing values of $\widehat{\lambda}$ (i.e., the average number of of external packet arrivals at a relay node during a time slot) and $\alpha$ (i.e., the transmission probability of a relay). As expected, the increase in $\widehat{\lambda}$ increases the expected delay, and that decrease becomes more apparent as $\alpha$ takes small values and $r$ increases.
![The average delay vs $\alpha$ and $\widehat{\lambda}$ for different values of $r$.[]{data-label="f153"}](itc1c.pdf)
Figure \[f152\] illustrate how sensitive is the average delay as we increase the probability of a direct transmission (at the beginning of a slot) of a source. In particular, as $t$ remains small, the increase in $\widehat{\lambda}$ from $0.1$ to $0.15$ will not affect the average delay. As $t$ increases, the simultaneous increase in $\widehat{\lambda}$ will cause a rapid increase in the average delay, even when we set the transmission probability $\alpha=\alpha^{*}$. This is expected, since at the beginning of a slot, source users have precedence over the relays.
![Effect of $\widehat{\lambda}$ on average delay.[]{data-label="f152"}](itc2c.pdf)
Similar observations can be deduced by Fig. \[f151\], where we can observe the average delay as a function of $\alpha^{*}$ and $\widehat{\lambda}$. Clearly, as $t$ increases from $0.3$ to $0.4$, the average delay increases rapidly, especially when, $\widehat{\lambda}$ tends to $0.1$.
![Effect of $t$ on average delay.[]{data-label="f151"}](itc3c.pdf)
#### Example 2. Stability region
We now focus on the general model, and specifically on the case $\frac{\alpha_{1}\widehat{\alpha}_{2}}{\alpha_{1}^{*}\tilde{s}_{1/\{1\}}}+\frac{\alpha_{2}\widehat{\alpha}_{1}}{\alpha_{2}^{*}\tilde{s}_{2/\{2\}}}\neq1$. We investigate the effect of parameters on the stability region, i.e., the set of arrival rate vectors $(\lambda_{1},\lambda_{2})$, for which the queues of the system are stable. In what follows, let $\alpha_{1}=0.7$, $\alpha_{2}=0.6$, $\alpha_{2}^{*}=0.9$, ${P_{s}^{*}(R_1,D,\{R_1\})}={P_{s}^{*}(R_2,D,\{R_2\})}=0.9$, ${P_{s}(R_1,D,\{R_1\})}={P_{s}(R_2,D,\{R_2\})}=0.8$, $P_{s}(R_{i},D,\{R_{1},R_{2}\})=0.4$, $i=1,2$, $t_{2}=0.3$.
In Fig. \[f15s\] we observe the impact of $t_{1}$, i.e., the probability of transmission of a packet of source user $1$ at the beginning of a slot, on the stability region. Note that this factor plays a crucial role in the performance of the system, since although the destination node hears both source users and the relays, but gives priority to the source users at this time slot. We can easily observe that the increase of $t_{1}$ from $0.2$ to $0.4$, will cause a deterioration of the stability region. Moreover, that increase will affect both relays, i.e., both adequate the values of $\widehat{\lambda}_{1}$, and $\widehat{\lambda}_{2}$ will be decreased in order to sustain stability.
![Effect of $t_{1}$ on the stability region for $\alpha_{1}^{*}=0.9$.[]{data-label="f15s"}](itc4c.pdf)
![Effect of transmission control on the stability region.[]{data-label="f15z"}](itc5c.pdf)
In Fig. \[f15z\] we set $t_{1}=0.2$, and investigate the impact of the adaptive transmission control in relay node $1$ on the stability region. In particular, first we assume that $\alpha_{1}^{*}=\alpha_{1}=0.7$, i.e., the relay $1$ does not adapt its transmission probability when it senses relay $2$ inactive. In such case, the stability region is the part in Fig. \[f15z\] colored in blue and yellow. When relay $1$ adapts its transmission probability to $\alpha_{1}^{*}=0.9$ (when it senses relay $2$ inactive) the stability region changes and is given by the part of Fig. \[f15z\] colored in blue and red. Note that the increase of $\alpha_{1}^{*}$ will affect the relay $2$, since a packet in relay $1$ is more likely to be transmitted. Thus, we expect that adequate values of $\widehat{\lambda}_{2}$ to be lower in order to ensure stability.
Conclusions and future work
===========================
In this work we obtained the stable throughput region, and investigated the delay analysis of a relay-assisted cooperative wireless network with MPR capabilities and adaptive transmission policy. By applying the stochastic dominance technique we obtained the stability region under general MPR both for the asymmetric network of two source-users, two-relay nodes, and for the symmetric model of $N$ users. In addition, we provided the aggregate throughput and the throughput per user in terms of closed form expressions.
We investigated the fundamental problem of delay analysis, and for the asymmetric network of two-users, two-relays we performed a detailed mathematical analysis, which led to the determination of the generating function of the stationary joint queue length distribution of relay nodes in terms of a solution of a Riemann-Hilbert boundary value problem. For the symmetrical model, closed form expressions for the expected delay at each relay node were also derived without the need of solving a boundary value problem. Extensive numerical results were obtained providing insights in the system performance.
In a future work we plan to investigate the stable throughput and delay for the case of a completely random access network, where there is no coordination between source users and relays. A challenging task will be the extension to the case of more than two relay nodes. The investigation of such a network is an open problem, and using our current work as a building block we plan to investigate the possibility of obtaining at least some bounds for the expected delay at relay nodes.
Derivation of the Functional equation {#po}
=====================================
The queue evolution equation (\[x\]) implies $$\begin{array}{l}
E(x^{N_{1,n+1}}y^{N_{2,n+1}})=D(x,y)\left\{\bar{t}_{1}\bar{t}_{2}[P(N_{1,n}=N_{2,n}=0)+E(x^{N_{1,n}}1_{\{N_{1,n}>0,N_{2,n}=0\}})(1+\alpha_{1}^{*}{P_{s}^{*}(R_1,D,\{R_1\})}(\frac{1}{x}-1))\right.\vspace{2mm}\\
\left.+E(y^{N_{2,n}}1_{\{N_{1,n}=0,N_{2,n}>0\}})(1+\alpha_{2}^{*}{P_{s}^{*}(R_2,D,\{R_2\})}(\frac{1}{y}-1))+E(x^{N_{1,n}}y^{N_{2,n}}1_{\{N_{1,n}>0,N_{2,n}>0\}})(1+\alpha_{1}\widehat{\alpha}_{2}(\frac{1}{x}-1)\right.\vspace{2mm}\\
\left.+\alpha_{2}\widehat{\alpha}_{1}(\frac{1}{y}-1))]+E(x^{N_{1,n}}y^{N_{2,n}})[t_{1}\bar{t}_{2}(1+S_{1}(x,y))+t_{2}\bar{t}_{1}(1+S_{2}(x,y))+t_{2}t_{1}(1+S_{3}(x,y))]\right\},
\end{array}
\label{fc}$$ where $1_{\{A\}}$ denotes the indicator function of the event $A$, and $$\begin{array}{rl}
S_{k}(x,y)=&{\overline{P}_{s}(k,D,\{k\})}[{P_{s}(k,R_1,\{k\})}\overline{P}_{s}(k,R_{2},\{k\})(x-1)+\overline{P}_{s}(k,R_{1},\{k\}) P_{s}(k,R_{2},\{k\})(y-1)\\
&+P_{s}(k,R_{1},\{k\}) P_{s}(k,R_{2},\{k\})(xy-1)],\,k=1,2,\vspace{2mm}\\
S_{3}(x,y)=&(x-1)\{{\overline{P}_{s}(0,D,\{1,2\})}{\overline{P}_{s}(0,R_2,\{1,2\})}({P_{s}(1,R_1,\{1,2\})}+{P_{s}(2,R_1,\{1,2\})})\\
&+{P_{s}(1,D,\{1,2\})}\overline{P}_{s,2}(2,R_{2},\{1,2\})P_{s,2}(2,R_{1},\{1,2\})+{P_{s}(2,D,\{1,2\})}\overline{P}_{s,1}(1,R_{2},\{1,2\})P_{s,1}(2,R_{1},\{1,2\})\}\\
&+(y-1)\{{\overline{P}_{s}(0,D,\{1,2\})}{\overline{P}_{s}(0,R_1,\{1,2\})}({P_{s}(1,R_2,\{1,2\})}+{P_{s}(2,R_2,\{1,2\})})\\
&+{P_{s}(1,D,\{1,2\})}\overline{P}_{s,2}(2,R_{1},\{1,2\})P_{s,2}(2,R_{2},\{1,2\})+{P_{s}(2,D,\{1,2\})}\overline{P}_{s,1}(1,R_{1},\{1,2\})P_{s,1}(2,R_{2},\{1,2\})\}\\
&+(xy-1)\{{\overline{P}_{s}(0,D,\{1,2\})}({P_{s}(1,R_1,\{1,2\})}+{P_{s}(2,R_1,\{1,2\})})({P_{s}(1,R_2,\{1,2\})}+{P_{s}(2,R_2,\{1,2\})})\\
&+{P_{s}(1,D,\{1,2\})}P_{s,2}(2,R_{2},\{1,2\})P_{s,2}(2,R_{1},\{1,2\})+{P_{s}(2,D,\{1,2\})}P_{s,1}(1,R_{2},\{1,2\})P_{s,1}(2,R_{1},\{1,2\})\}.
\end{array}$$ Note that $$\begin{array}{rl}
H(0,0)=&\lim_{n\to\infty}P(N_{1,n}=N_{2,n}=0),\\
H(x,0)-H(0,0)=&\lim_{n\to\infty}E(x^{N_{1,n}}1_{\{N_{1,n}>0,N_{2,n}=0\}}),\\
H(0,y)-H(0,0)=&\lim_{n\to\infty}E(y^{N_{2,n}}1_{\{N_{1,n}=0,N_{2,n}>0\}}),\\
H(x,y)-H(x,0)-H(0,y)+H(0,0)=&\lim_{n\to\infty}E(x^{N_{1,n}}y^{N_{2,n}}1_{\{N_{1,n}>0,N_{2,n}>0\}}).
\end{array}$$
Then, using (\[fc\]) we obtain the functional equation (\[we\]).
Proof of Lemma \[LEM\] {#a0}
======================
It is easily seen that $R(x,y)=\frac{xy-\Psi(x,y)}{xyD(x,y)}$, where $\Psi(x,y)=D(x,y)[xy(1+L_{3}(xy-1))+y(1-x)(\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}-L_{1}x)+x(1-y)(\bar{t}_{1}\bar{t}_{2}\alpha_{2}\widehat{\alpha}_{1}-L_{2}y)]$, where for $|x|\leq1$, $|y|\leq1$, $\Psi(x,y)$ is a generating function of a proper probability distribution. Now, for $|y|=1$, $y\neq1$ and $|x|=1$ it is clear that $|\Psi(x,y)|<1=|xy|$. Thus, from Rouché’s theorem, $xy-\Psi(x,y)$ has exactly one zero inside the unit circle. Therefore, $R(x,y)=0$ has exactly one root $x=X_{0}(y)$, such that $|x|<1$. For $y=1$, $R(x,1)=0$ implies $$\begin{array}{c}
(x-1)\left(\lambda_{1}-\frac{\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}}{x} \right)=0.
\end{array}$$ Therefore, for $y=1$, and since $\lambda_{1}<\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}$, the only root of $R(x,1)=0$ for $|x|\leq1$, is $x=1$.$\square$
Proof of Lemma \[SQ\] {#a1}
=====================
We will prove the part related to $\mathcal{M}$. Similarly, we can also prove the other. For $y\in[y_{1},y_{2}]$, $D_{y}(y)$ is negative, so $X_{0}(y)$, $X_{1}(y)$ are complex conjugates. It also follows that $$\begin{array}{l}
Re(X(y))=\frac{-\widehat{b}(y)}{2\widehat{a}(y)}.
\end{array}
\label{rd1}$$ Therefore, $|X(y)|^{2}=\frac{\widehat{c}(y)}{\widehat{a}(y)}=g(y)$. Clearly, $g(y)$ is an increasing function for $y\in[0,1]$ and thus, $|X(y)|^{2}\leq g(y_{2})=\beta_{0}$. Using simple algebraic considerations we can prove that, $X_{0}(y_{1}):=\beta_{1}=-g(y_{1})$ is the extreme left point of $\mathcal{M}$. Finally, $\zeta(\delta)$ is derived by solving (\[rd1\]) for $y$ with $\delta = Re(X(y))$, and taking the solution such that $y\in[0,1]$.$\square$
Intersection points of the curves {#ap4}
=================================
In the following, we focus on the location of the intersection points of $R(x,y)=0$, $A(x,y)=0$ (resp. $B(x,y)$). These points (if they exist) are potential singularities for the functions $H(x,0)$, $H(0,y)$, and thus, their investigation is crucial regarding the analytic continuation of $H(x,0)$, $H(0,y)$ outside the unit disk; see also Lemma 2.2.1 and Theorem 3.2.3 in [@fay1] for alternative approaches.
Intersection points between $R(x,y)=0$, $A(x,y)=0$.
---------------------------------------------------
Let $R(x,y) = 0$, $x = X_{\pm}(y)$, $y\in \tilde{C}_{y}$. We can easily show that the resultant in $x$ of the two polynomials $R(x,y)$ and $A(x,y)$ is $Res_{x}(R,A;y)=y(y-1)Q(y)$, where $$\begin{array}{rl}
Q(y)=&-d_{1}[\lambda_{2}d_{1}+a_{2}\widehat{a}_{1}(\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+L_{2})]y^{2}+ya_{2}\widehat{a}_{1}[d_{1}(\lambda_{1}+\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+L_{2})\\&-\bar{t}_{1}\bar{t}_{2}a_{1}^{*}P_{s}^{*}(R_{1},D,\{R_{1}\})(a_{2}\widehat{a}_{1}+d_{1})]+\bar{t}_{1}\bar{t}_{2}a_{1}^{*}P_{s}^{*}(R_{1},D,\{R_{1}\})(a_{2}\widehat{a}_{1})^{2}.
\end{array}$$ Note that $Q(0)=\bar{t}_{1}\bar{t}_{2}a_{1}^{*}P_{s}^{*}(R_{1},D,\{R_{1}\})(a_{2}\widehat{a}_{1})^{2}>0$ and $Q(1)=d_{1}[\lambda_{1}\alpha_{2}\widehat{\alpha}_{1}-\lambda_{2}d_{1}-\bar{t}_{1}\bar{t}_{2}\alpha_{2}\widehat{\alpha}_{1}a_{1}^{*}P_{s}^{*}(R_{1},D,\{R_{1}\})>0$, since $d_{1}<0$ and due to the stability condition.
Similarly, for $R(x,y) = 0$, $y = Y_{\pm}(x)$, $x\in \tilde{C}_{x}$, the resultant in $y$ of the two polynomials $R(x,y)$, $A(x,y)$ is $Res_{y}(R,A;x)=x(x-1)\bar{t}_{1}\bar{t}_{2}\alpha_{2}\widehat{\alpha}_{1}\tilde{Q}(x)$, where, $$\begin{array}{rl}
\tilde{Q}(x)=&-[\alpha_{2}\widehat{\alpha}_{1}\lambda_{1}+(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1})d_{1}]x^{2}+x[(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+\lambda_{2}+L_{1})d_{1}\\&+(\alpha_{2}\widehat{\alpha}_{1}+d_{1})\alpha_{1}^{*}P_{s}^{*}(R_{1},D,\{R_{1}\})\bar{t}_{1}\bar{t}_{2}]-\alpha_{1}^{*}P_{s}^{*}(R_{1},D,\{R_{1}\})d_{1}\bar{t}_{1}\bar{t}_{2}.
\end{array}$$
Note also that $\tilde{Q}(0)=-\alpha_{1}^{*}P_{s}^{*}(R_{1},D,\{R_{1}\})d_{1}\bar{t}_{1}\bar{t}_{2}>0$ since $d_{1}<0$ and $\tilde{Q}(1)=\bar{t}_{1}\bar{t}_{2}\alpha_{2}\widehat{\alpha}_{1}\alpha_{1}^{*}P_{s}^{*}(R_{1},D,\{R_{1}\})-\lambda_{1}\alpha_{2}\widehat{\alpha}_{1}+\lambda_{2}d_{1}>0$ due to the stability conditions (see Lemma \[Thm2users\]). If $\alpha_{1}^{*}\leq min\{1,\frac{\alpha_{2}\widehat{\alpha}_{1}\lambda_{1}+(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1})\alpha_{1}\widehat{\alpha}_{2}}{(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1})P_{s}^{*}(R_{1},D,\{R_{1}\})}\}$, then $\lim_{x\to\infty}\tilde{Q}(x)=-\infty$, and $\tilde{Q}(x)=0$ has two roots of opposite sign, say $x_{*}<0<1<x^{*}$. If $\frac{\alpha_{2}\widehat{\alpha}_{1}\lambda_{1}+(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1})\alpha_{1}\widehat{\alpha}_{2}}{(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1})P_{s}^{*}(R_{1},D,\{R_{1}\})}<\alpha_{1}^{*}\leq 1$, then $\lim_{x\to\infty}\tilde{Q}(x)=+\infty$, and $\tilde{Q}(x)=0$ has two positive roots, say $1<\tilde{x}_{*}<x_{3}<x_{4}<\tilde{x}^{*}$, due to the stability conditions. In the former case we have to check if $x^{*}$ is in $S_{x}$, while in the latter case if $\tilde{x}_{*}$ is in $S_{x}$. These zeros, if they lie in $S_{x}$ such that $|Y_{0}(x)|\leq1$, are poles of $H(x,y)$. Denote by $$\bar{x}=\left\{\begin{array}{rl}
x^{*},&\alpha_{1}^{*}\leq min\{1,\frac{\alpha_{2}\widehat{\alpha}_{1}\lambda_{1}+(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1})\alpha_{1}\widehat{\alpha}_{2}}{(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1})P_{s}^{*}(R_{1},D,\{R_{1}\})}\},\\
\tilde{x}_{*},&\frac{\alpha_{2}\widehat{\alpha}_{1}\lambda_{1}+(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1})\alpha_{1}\widehat{\alpha}_{2}}{(\widehat{\lambda}_{1}(1+\widehat{\lambda}_{2})+L_{1})P_{s}^{*}(R_{1},D,\{R_{1}\})}<\alpha_{1}^{*}\leq 1.
\end{array}\right.$$
Intersection points between $R(x,y)=0$, $B(x,y)=0$.
---------------------------------------------------
Let $y\in \tilde{C}_{y}$ and $R(x,y) = 0$, $x = X_{\pm}(y)$. It is easily shown that the resultant in $x$ of $R(x,y)$, $B(x,y)$ is $Res_{x}(R,B,y)=y(y-1)T(y)$, where $$\begin{array}{rl}
T(y)=&-[\alpha_{1}\widehat{\alpha}_{2}\lambda_{2}+(\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+L_{2})d_{2}]y^{2}+y[(\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+\lambda_{1}+L_{2})d_{2}\\&+(\alpha_{1}\widehat{\alpha}_{2}+d_{2})\alpha_{2}^{*}P_{s}^{*}(R_{2},D,\{R_{2}\})\bar{t}_{1}\bar{t}_{2}]-\alpha_{2}^{*}P_{s}^{*}(R_{2},D,\{R_{2}\})d_{2}\bar{t}_{1}\bar{t}_{2}.
\end{array}$$
Note that $T(0)=-\alpha_{2}^{*}P_{s}^{*}(R_{2},D,\{R_{2}\})d_{2}\bar{t}_{1}\bar{t}_{2}>0$, $T(1)=\bar{t}_{1}\bar{t}_{2}\alpha_{1}\widehat{\alpha}_{2}\alpha_{2}^{*}P_{s}^{*}(R_{2},D,\{R_{2}\})-\lambda_{2}\alpha_{1}\widehat{\alpha}_{2}+\lambda_{1}d_{2}>0$, since $d_{2}<0$ and due to the stability conditions. If $\alpha_{2}^{*}<min\{1,\frac{\alpha_{1}\widehat{\alpha}_{2}\lambda_{2}+(\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+L_{2})\alpha_{2}\widehat{\alpha}_{1}}{(\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+L_{2})P_{s}^{*}(R_{2},D,\{R_{2}\})}\}$, $\lim_{y\to \infty}T(x)=-\infty$, and $T(x)$ has two roots of opposite sign, say $y_{*}$, $y^{*}$ such that $y_{*}<0<1<y^{*}$, which in turn implies that $B(X_{0}(y),y)\neq0$, $y\in[y_{1},y_{2}]\subset(0,1)$, or equivalently $B(x,Y_{0}(x))\neq0$, $x\in\mathcal{M}$. When $\frac{\alpha_{1}\widehat{\alpha}_{2}\lambda_{2}+(\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+L_{2})\alpha_{2}\widehat{\alpha}_{1}}{(\widehat{\lambda}_{2}(1+\widehat{\lambda}_{1})+L_{2})P_{s}^{*}(R_{2},D,\{R_{2}\})}<\alpha_{2}^{*}\leq1$, $\lim_{y\to \infty}T(y)=+\infty$, and $T(y)$ has two positive roots, say $\widehat{y}_{*}$, $\widehat{y}^{*}$ such that $1<\widehat{y}_{*}<y_{3}<y_{4}<\widehat{y}^{*}$, which in turn implies that $B(X_{0}(y),y)\neq0$, $y\in[y_{1},y_{2}]$, i.e., $B(x,Y_{0}(x))\neq0$, $x\in\mathcal{M}$.$\square$
[^1]: In this section we will present the system model for the case of two users. However, in Section \[stab-anal-n\], we consider the case where there are $n$-symmetric users
[^2]: We consider the general case for $\alpha_{i}^{*}$ instead of assuming directly $\alpha_{i}^{*}=1$. This can handle cases where the node cannot transmit with probability one even if the other node is silent, e.g., when the nodes are subject to energy limitations. It is outside of the scope of this work to consider specific cases and we intent to keep the proposed analysis general.
[^3]: In such a shared access network, it is practical to assume a minimum exchanging information of one bit between the nodes.
[^4]: Note that such a distribution is natural in radio-packet networks.
|
---
abstract: 'We present a microscopic theory of coherent quantum transport through a superconducting film between two ferromagnetic electrodes. The scattering problem is solved for the general case of ferromagnet/superconductor/ferromagnet (FSF) double-barrier junction, including the interface transparency from metallic to tunnel limit, and the Fermi velocity mismatch. Charge and spin conductance spectra of FSF junctions are calculated for parallel (P) and antiparallel (AP) alignment of the electrode magnetization. Limiting cases of nonmagnetic normal-metal electrodes (NSN) and of incoherent transport are also presented. We focus on two characteristic features of finite size and coherency: subgap tunneling of electrons, and oscillations of the differential conductance. Periodic vanishing of the Andreev reflection at the energies of geometrical resonances above the superconducting gap is a striking consequence of the quasiparticle interference. Also, the non-trivial spin-polarization of the current is found for FSF junctions in AP alignment. This is in contrast with the incoherent transport, where the unpolarized current is accompanied by excess spin accumulation and destruction of superconductivity. Application to spectroscopic measurements of the superconducting gap and the Fermi velocity is also discussed.'
address: 'Department of Physics, University of Belgrade, P.O. Box 368, 11001 Belgrade, Yugoslavia'
author:
- 'M. Božović[^1] and Z. Radović[^2]'
title: 'Spin-polarized currents in superconducting films'
---
Introduction
============
During the past decade, there has been a growing interest in various electronic systems driven out of equilibrium by the injection of spin-polarized carriers. Such systems can be realized by current-biasing structures consisting of ferromagnetic and non-ferromagnetic (e.g. superconducting) layers, due to the difference in population of majority and minority spin subbands.[@Prinz] The concept of spin-polarized current nowadays has attracted considerable interest in ferromagnetic heterostructures, in particular for applications in spintronics.[@Osofsky]
Charge transport through a normal metal/superconductor (NS) junction, with an insulating barrier of arbitrary strength at the interface, has been studied by Blonder, Tinkham, and Klapwijk (BTK),[@BTK] and the Andreev reflection is recognized as the mechanism of normal-to-supercurrent conversion.[@Andreev; @Furusaki; @Tsukada] The BTK theory has been extended by Tanaka and Kashiwaya to include the anisotropy of the pair potential in $d$-wave superconductors.[@Tanaka; @95; @Tanaka; @00] The modification of the Andreev reflection by the spin injection from a ferromagnetic metal into a superconductor in ferromagnet/superconductor (FS) junctions was first analyzed by de Jong and Beenakker.[@dJB] More recently, the effects of unconventional $d$-wave and $p$-wave pairing and of the exchange interaction in FS systems, such as the zero-bias conductance peak and the virtual Andreev reflection, have been clarified by Kashiwaya [*et al*]{}.[@Beasley] and Yoshida [*et al*]{}.[@Yoshida] The Fermi velocity mismatch between two metals can also significantly affect the Andreev reflection by altering the subgap conductance,[@Zutic] which is similar to the presence of an insulating barrier.[@Zhu]
In experiments, a superconductor is used to determine the spin polarization of the current injected from (or into) a ferromagnet by measuring the differential conductance. These measurements have been performed on tunnel junctions in an external magnetic field,[@Tedrow; @Platt] metallic point contacts,[@Soulen; @Novo] nano-contacts formed by microlithography,[@Upad] and FS junctions with $d$-wave superconductors, grown by molecular beam epitaxy.[@Vasko] In diffusive FS junctions, the excess resistance may be induced by spin accumulation near the insulating interface,[@Jedema] and by the proximity effect.[@Gueron; @Petrashov; @Sillanpaa]
When electrons pass incoherently through the interfaces, the BTK model can be successfully applied to normal metal/superconductor/normal metal (NSN) or ferromagnet/superconductor/ferromagnet (FSF) double junctions.[@Takahashi; @Kinezi] However, the properties of coherent quantum transport in clean superconducting heterostructures are strongly influenced by size effects, which are not included in the BTK model. Well-known examples are the current-carrying Andreev bound states[@Tanaka; @00; @Nazarov] and multiple Andreev reflections[@KBT; @Basel; @Ingerman] in superconductor/normal metal/superconductor (SNS) junctions. Since early experiments by Tomasch,[@Tomasch] the geometric resonance nature of the differential conductance oscillations in SNS and NSN tunnel junctions has been ascribed to the electron interference in the central film.[@Anderson; @Rowell; @McMillan; @Kanadjani] Recently, the McMillan-Rowell oscillations were observed in SNS edge junctions of $d$-wave superconductors, and used for measurements of the superconducting gap and the Fermi velocity.[@Nesher]
Here we present a comperhensive microscopic theory of coherent transport in FSF double junctions (with NSN as a special case).[@Milos; @ZZM] We limit ourselves to clean conventional ($s$-wave) superconductors, and neglect, for simplicity, the self-consistency of the pair potential,[@Geers; @Buzdin] and nonequlibrium effects of charge and spin accumulation at the interfaces.[@FalkoB; @McCann] When two interfaces are recognized by electrons simultaneously, characteristic features of finite size and coherency are the subgap tunneling of electrons and oscillations of both charge and spin differential conductances above the gap. One consequence of the quasiparticle interference is the periodic vanishing of the Andreev reflection at the energies of geometrical resonances. The other is the existence of a non-trivial spin-polarization of the current not only for the parallel (P), but also for the antiparallel (AP) alignment of the electrode magnetizations. Previous analysis of incoherent transport in FSF double junctions predict the absence of the spin current and suppression of superconductivity with increasing voltage for AP alignment, as a result of spin imbalance in the superconducting film.[@Takahashi; @Kinezi]
The scattering problem
======================
255=0.7
We consider an FSF double junction consisting of a clean superconducting layer of thickness $l$, connected to ferromagnetic electrodes by thin, insulating interfaces, Fig. \[sch\]. For the ferromagnetic metal we adopt the Stoner model describing the spin-polarization effect by the usual one-electron Hamiltonian with an exchange potential. The quasiparticle propagation is described by the Bogoliubov–de Gennes equation $$\begin{aligned}
\left(
\begin{array}{ccc}
H_0({\bf r})-\rho_{\sigma}h({\bf r}) && \Delta({\bf r}) \\
\Delta^{*}({\bf r}) && -H_0({\bf r})+\rho_{\bar{\sigma}}h({\bf
r})
\end{array}
\right) \Psi_\sigma({\bf r})~=~E\Psi_\sigma({\bf r}), \label{BdG}\end{aligned}$$ with $H_{0}({\bf r})=-\hbar^{2}\nabla^{2}/2m+W({\bf r})+U({\bf
r})-\mu$, where $U({\bf r})$ and $\mu$ are the Hartree and the chemical potential, respectively. The interface potential is modeled by $W({\bf r})=\hat{W}\{\delta(z)+\delta(z-l)\}$, where $z$-axis is perpendicular to the layers and $\delta(z)$ is the Dirac $\delta$-function. Neglecting the self-consistency of the superconducting pair potential, $\Delta({\bf r})$ is taken in the form $\Delta \Theta(z) \Theta(l-z)$, where $\Delta$ is the bulk superconducting gap and $\Theta(z)$ is the Heaviside step function. In Eq. (\[BdG\]), $\sigma$ is the quasiparticle spin ($\sigma =\uparrow ,\downarrow$ and $\bar{\sigma}=\downarrow,\uparrow$), $E$ is the energy with respect to $\mu$, $h({\bf r})$ is the exchange potential given by $h_{0}\{\Theta(-z)+[-]\Theta(z-l)\}$ for the P \[AP\] alignment, and $\rho_{\sigma}$ is 1 (-1) for spins up (down). The electron effective mass $m$ is assumed to be the same for the whole junction. Here, $\mu-U({\bf r})$ is the Fermi energy of the superconductor, $E^{(S)}_F$, or the mean Fermi energy of a ferromagnet, $E^{(F)}_F=(E^\uparrow_F+E^\downarrow_F)/2$. Moduli of the Fermi wave vectors, $k^{(F)}_F= \sqrt{2mE^{(F)}_F/\hbar^2}$ and $k^{(S)}_F= \sqrt{2mE^{(S)}_F/\hbar^2}$, can be different in general, and in the following, the Fermi wave vector mismatch (FWVM) will be taken into account through the parameter $\kappa=k^{(F)}_F/k^{(S)}_F$. The parallel component of the wave vector ${\bf k}_{||,\sigma}$ is conserved, and the wave function $$\Psi_\sigma({\bf r})=\exp(i{\bf k}_{||,\sigma} \cdot {\bf r}
)~\psi_\sigma(z),$$ satisfies the boundary conditions $$\begin{aligned}
\label{bc1}
\psi_\sigma (z)|_{z=0_-}&=&\psi_\sigma (z)|_{z=0_+},\\
\frac{d\psi_\sigma (z)}{dz}\Big|_{z=0_-}&=&\frac{d\psi_\sigma
(z)}{dz}\Big|_{z=0_+} -\frac{2m\hat{W}}{\hbar ^2}\psi_\sigma
(0),\\
\psi_\sigma (z)|_{z=l_-}&=&\psi_\sigma (z)|_{z=l_+},\\
\frac{d\psi_\sigma (z)}{dz}\Big|_{z=l_-}&=&\frac{d\psi_\sigma
(z)}{dz}\Big|_{z=l_+}-\frac{2m\hat{W}}{\hbar ^2}\psi_\sigma (l)
\label{bc4}.\end{aligned}$$ Four independent solutions of Eq. (\[BdG\]) correspond to the four types of injection: an electron or a hole from either the left or from the right electrode.[@Furusaki; @Tsukada]
For the injection of an electron from the left, with energy $E>0$, spin $\sigma$, and angle of incidence $\theta$ (measured from the $z$-axis), solution for $\psi_\sigma (z)$ in various regions has the following form:
in the left ferromagnet ($z<0$) $$\psi_\sigma
(z)=\{\exp(ik^+_{\sigma}
z)+b_{\sigma}(E,\theta)\exp(-ik^+_{\sigma}
z)\}\left(\begin{array}{c}
1 \\
0 \\
\end{array}\right)+
a_{\sigma}(E,\theta)\exp(ik^-_{\bar{\sigma}} z)\left(\begin{array}{c}
0 \\
1 \\
\end{array}\right),
\label{psiL}$$ in the superconductor ($0<z<l$) $$\begin{aligned}
\psi_\sigma (z)&=&\{
c_{1}(E,\theta)\exp(iq^+_{\sigma}z)+c_{2}(E,\theta)\exp(-iq^+_{\sigma}z)\}
\left(\begin{array}{c}
\bar{u} \\
\bar{v}
\end{array}\right) \nonumber \\
&~&+\{
c_{3}(E,\theta)\exp(iq^-_{\sigma}z)+c_{4}(E,\theta)\exp(-iq^-_{\sigma}z)\}
\left(\begin{array}{c}
\bar{v}^{*} \\
\bar{u}^{*}
\end{array}\right),
\label{psiS}\end{aligned}$$ and in the right ferromagnet ($z>l$), for the P \[AP\] alignment of the magnetizations $$\psi_\sigma(z)=c_{\sigma}(E,\theta)\exp(ik^+_{\sigma[{\bar\sigma}]}z)\left(\begin{array}{c}
1 \\
0 \\
\end{array}\right)+
d_{\sigma}(E,\theta)\exp(-ik^-_{{\bar\sigma}[\sigma]}z)\left(\begin{array}{c}
0 \\
1 \\
\end{array}\right).
\label{psiR}$$
Here, $\bar{u}=\sqrt{(1+\Omega/E)/2}$ and $\bar{v}=\sqrt{(1-\Omega/E)/2}$ are the BCS coherence factors, and $\Omega=\sqrt{E^2-\Delta^2}$. The $z$-components of the wave vectors are $k^\pm_{\sigma}=\sqrt{(2m/\hbar
^2)(E^{(F)}_F+\rho_{\sigma}h_0 \pm E)-{\bf k}^2_{||,\sigma}}$, and $q^\pm_{\sigma}=$\
$\sqrt{(2m/\hbar ^2)(E^{(S)}_F\pm\Omega)-{\bf
k}^2_{||,\sigma}}$, where $|{\bf k}_{||,\sigma}|=\sqrt{(2m/\hbar
^2)(E^{(F)}_F+\rho_{\sigma}h_0+E)}~\sin\theta$. The coefficients $a_{\sigma}$, $b_{\sigma}$, $c_{\sigma}$, and $d_{\sigma}$ are, respectively, the probability amplitudes of: (1) Andreev reflection as a hole of the opposite spin (AR); (2) normal reflection as an electron (NR); (3) transmission to the right electrode as an electron (TE); (4) transmission to the right electrode as a hole of the opposite spin (TH). Processes (1) and (4) are equivalent to the formation of a Cooper pair in the superconductor by taking one more electron from either the left or the right electrode, respectively. Amplitudes of the Bogoliubov electron-like and hole-like quasiparticles, propagating in the superconducting layer, are given by the coefficients $c_1$ through $c_4$.
From the probability current conservation, the probabilities of outgoing particles satisfy the normalization condition $$\label{ABCD}
A_\sigma(E,\theta)+B_\sigma(E,\theta)+C_\sigma(E,\theta)+D_\sigma(E,\theta)=1,$$ where, $$\begin{aligned}
\label{maliA}
A_\sigma(E,\theta)&=&\Re\left(\frac{\tilde{k}_{\bar{\sigma}}}
{\tilde{k}_{\sigma}}\right)|a_\sigma (E,\theta)|^2, \\
\label{maliB} B_\sigma(E,\theta)&=&|b_\sigma (E,\theta)|^2, \\
\label{maliC}
C_\sigma(E,\theta)&=&\Re\left(\frac{\tilde{k}_{\sigma[{\bar{\sigma}}]}}{\tilde{k}_{\sigma}}\right)
|c_\sigma(E,\theta)|^2, \\ \label{maliD} D_\sigma(E,\theta)&=&
\Re\left(\frac{\tilde{k}_{{\bar{\sigma}}[\sigma]}}{\tilde{k}_{\sigma}}\right)
|d_\sigma(E,\theta)|^2.\end{aligned}$$
Neglecting small terms $E/E^{(F)}_F\ll 1$ and $\Delta/E^{(S)}_F\ll
1$ in the wave vectors, except in the exponents $$\label{zeta} \zeta_\pm =l\left(q^+_{\sigma}\pm
q^-_{\sigma}\right),$$ solutions of Eqs. (\[bc1\])-(\[bc4\]) for the probability amplitudes can be written in the following form $$\begin{aligned}
\label{a general} a_\sigma(E,\theta)&=&\frac{4
(\tilde{k}_{\sigma}/\tilde{q}_\sigma) \Delta
\sin(\zeta_-/2)}{\Gamma}\left[{\cal A}^R_+ E \sin(\zeta_-/2)+i
{\cal B}^R_+ \Omega \cos(\zeta_-/2)\right], \\ \label{b general}
b_\sigma(E,\theta)&=&\frac{1}{\Gamma}[{\cal A}^R_+{\cal
C}_+\Delta^2 - \left({\cal A}^R_+{\cal C}_+E^2 + {\cal B}^R_+{\cal
D}_+\Omega^2 \right)\cos(\zeta_-) + \left({\cal A}^R_-{\cal C}_- +
{\cal B}^R_-{\cal D}_-\right)\Omega^2\cos(\zeta_+) \nonumber\\ &~&
+ i\left({\cal B}^R_+{\cal C}_+ + {\cal A}^R_+{\cal
D}_+\right)E\Omega\sin(\zeta_-) - i\left({\cal B}^R_-{\cal C}_- +
{\cal A}^R_-{\cal D}_-\right)\Omega^2\sin(\zeta_+)], \\ \label{c
general} c_\sigma(E,\theta)&=&\frac{4
(\tilde{k}_{\sigma}/\tilde{q}_\sigma) \Omega
e^{-ik^+_{\bar{\sigma}}l } }{\Gamma}\times \nonumber\\
&~&\times\{ i\left[{\cal F}_+ \cos(\zeta_+/2)+i {\cal E}_+
\sin(\zeta_+/2)\right]E\sin(\zeta_-/2) - \left[{\cal
E}_+\cos(\zeta_+/2)+i {\cal F}_+\sin(\zeta_+/2)
\right]\Omega\cos(\zeta_-/2)\},
\\ \label{d general} d_\sigma(E,\theta)&=&\frac{4
(\tilde{k}_{\sigma}/\tilde{q}_\sigma) \Delta\Omega
e^{ik^-_{\sigma}l}}{\Gamma}\times \nonumber\\ &~&\times
i\left[{\cal F}_- \cos(\zeta_+/2)+i {\cal
E}_-\sin(\zeta_+/2)\right]\sin(\zeta_-/2),\end{aligned}$$ where $$\begin{aligned}
\label{Gamma} {\Gamma}={\cal A}^L_+{\cal A}^R_+\Delta^2 -
\left({\cal A}^L_+{\cal A}^R_+E^2 + {\cal B}^L_+{\cal
B}^R_+\Omega^2 \right)\cos(\zeta_-) + \left({\cal A}^L_-{\cal
A}^R_- + {\cal B}^L_-{\cal B}^R_-\right)\Omega^2\cos(\zeta_+)
\nonumber\\ + i\left({\cal A}^L_+{\cal B}^R_+ + {\cal B}^L_+{\cal
A}^R_+\right)E\Omega\sin(\zeta_-) - i\left({\cal A}^L_-{\cal
B}^R_- + {\cal B}^L_-{\cal A}^R_-\right)\Omega^2\sin(\zeta_+).\end{aligned}$$ In Eqs. (\[a general\])-(\[Gamma\]) $$\begin{aligned}
{\cal A}^{L(R)}_{\pm}=K^{L(R)}_1 \pm K^{L(R)}_2,&~~~ {\cal
B}^{L(R)}_{\pm}=1 \pm K^{L(R)}_1 K^{L(R)}_2,&~~~ {\cal C}_{\pm}=
{K^{L}_1}^* \mp K^{L}_2, \\ {\cal D}_{\pm}=-(1 \mp {K^{L}_1}^*
K^{L}_2),&~~~ {\cal E}_{\pm}=K^{L}_2 \pm K^{R}_2,&~~~ {\cal
F}_{\pm}= 1 \pm K^{L}_2 K^{R}_2,\end{aligned}$$ with $K^L_1=\left(\tilde{k}_{\sigma}+iZ\right)/\tilde{q}_\sigma$, $K^L_2=\left(\tilde{k}_{\bar{\sigma}}-iZ\right)/\tilde{q}_\sigma$, $K^R_1=\left(\tilde{k}_{\sigma[{\bar{\sigma}}]}+iZ\right)/\tilde{q}_\sigma$, $K^R_2=\left(\tilde{k}_{{\bar{\sigma}}[\sigma]}-iZ\right)/\tilde{q}_\sigma$, for the P \[AP\] alignment. Here, ${K^{L}_1}^*=(\tilde{k}_{\sigma}-iZ)/\tilde{q}_\sigma$ is the complex conjugate of $K^L_1$, and $Z={2m\hat{W}}/\hbar ^2
k^{(S)}_F$ is the parameter measuring the strength of each interface barrier. Approximated wave-vector components, in units of $k^{(S)}_F$, are $\tilde{q}_\sigma=\sqrt{1-\tilde{\bf
k}^2_{||,\sigma} }$, $\tilde{k}_{\sigma}=\lambda_\sigma
\cos\theta$, and $|\tilde{\bf k}_{||,\sigma}|=\lambda_\sigma
\sin\theta$, where $\lambda_\sigma=\kappa\sqrt{1+\rho_\sigma X}$, $X=h_0 /E^{(F)}_F\geq 0$, and $\kappa\neq 1$ is measuring FWVM.
In the corresponding FNF double junction, AR and TH processes are absent, and the expression for NR amplitude, Eq. (\[b general\]), reduces to $$\label{bN} b^N_{\sigma}(E,\theta)=\frac{ ({K^{L}_1}^* -
K^{R}_1)\cos(lq^N_{\sigma})+i (1-{K^{L}_1}^*
K^{R}_1)\sin(lq^N_{\sigma}) }{ (K^{L}_1 +
K^{R}_1)\cos(lq^N_{\sigma})-i(1+K^{L}_1
K^{R}_1)\sin(lq^N_{\sigma}) },$$ where $q^N_{\sigma}=\sqrt{(2m/\hbar ^2)(E^{(S)}_F+E)-{\bf
k}^2_{||,\sigma}}$.
To complete our considerations, we also present the probability amplitudes for an FS single junction in the same notation, $$\begin{aligned}
\label{a FS} a_\sigma(E,\theta)&=&\frac{2
(\tilde{k}_{\sigma}/\tilde{q}_\sigma) \Delta }{{\cal A}^L_+E +
{\cal B}^L_+\Omega},\\ \label{b FS}
b_\sigma(E,\theta)&=&\frac{{\cal C}_+E + {\cal
D}_+\Omega}{{\cal A}^L_+E + {\cal B}^L_+\Omega},\\
\label{c FS} c_\sigma(E,\theta)&=&\frac{2
(\tilde{k}_{\sigma}/\tilde{q}_\sigma) E\bar{u}(1+K^L_2) }{{\cal
A}^L_+E + {\cal B}^L_+\Omega},\\ \label{d FS}
d_\sigma(E,\theta)&=&\frac{2 (\tilde{k}_{\sigma}/\tilde{q}_\sigma)
E\bar{v}(1-K^L_2)}{{\cal A}^L_+E + {\cal B}^L_+\Omega}.\end{aligned}$$ Note that $c_\sigma$ and $d_\sigma$ now describe the transmission of the Bogoliubov electron-like and hole-like quasiparticle, respectively. The well-known BTK results can be reproduced by taking $X=0$, $\kappa=1$, and $\theta=0$ in Eqs. (\[a FS\])-(\[d FS\]).[@foot]
In case of the simplest NSN metallic junction, taking $X=0$, $Z=0$, and $\kappa=1$ in Eqs. (\[a general\])-(\[d general\]), the scattering probabilities can be written in an explicit form[@ZZM] $$\begin{aligned}
\label{aNSN 0}
A_\sigma(E,\theta)&=&\left|\frac{\Delta\sin(\zeta_-/2)}{E\sin(\zeta_-/2)+i\Omega\cos(\zeta_-/2)}\right|^2,\\
\label{bNSN 0} B_\sigma(E,\theta)&=&D_\sigma(E,\theta)=0,\\
\label{cNSN 0}
C_\sigma(E,\theta)&=&\left|\frac{\Omega}{E\sin(\zeta_-/2)+i\Omega\cos(\zeta_-/2)}\right|^2.\end{aligned}$$
Solutions for the other three types of injection can be obtained by the same procedure. In particular, if a hole with energy $-E$, spin $\sigma$, and angle of incidence $\theta$ is injected from the left, the substitution $q^+_{\sigma} \rightleftharpoons
q^-_{\sigma}$ holds, and the scattering probabilities are the same as for the injection of an electron with $E$, $\sigma$, and $\theta$. Therefore, in order to include the description of both electron and hole injection, the calculated probabilities should be regarded as even functions of $E$. Also, for an electron or a hole, injected from the right, the probabilities are the same as for the injection from the left, except $\sigma\to\bar{\sigma}$ for the AP alignment.
255=0.55
255=0.55
255=0.55
Following the conservation of ${\bf k}_{||,\sigma}$, transmission of an electron (hole) with $\sigma=\uparrow$, injected from the left electrode into the superconductor, is possible only for angles of incidence $\theta$ satisfying $\theta <\theta _{c1}$, where $\theta _{c1}=\arcsin(1/\lambda_\uparrow)$ is the angle of total reflection. Then, $A_\uparrow(E,\theta)=0$ and $B_\uparrow(E,\theta)=1$ for $\theta >\theta _{c1}$. On the other hand, $\tilde{k}_{\downarrow}$, which corresponds to the hole (electron) created by the Andreev reflection, is real only for $\theta<\theta _{c2}=
\arcsin(\lambda_\downarrow/\lambda_\uparrow)$. The virtual Andreev reflection occurs for $\theta _{c2}<\theta<\theta _{c1}$, since $\tilde{k}_{\downarrow}$ becomes imaginary in that case. For injection of an electron (hole) with $\sigma=\downarrow$, transmission into the superconductor is possible for any $\theta<\pi/2$, and $\tilde{k}_{\uparrow}$ is always real.[@Beasley]
From Eqs. (\[a general\]) and (\[d general\]) it follows that $A_\sigma(E,\theta)=D_\sigma(E,\theta)=0$ when $$\label{resonance} \zeta_-=2n\pi$$ for $n=0,\pm 1,\pm 2,\ldots$. Therefore, the Andreev reflection at both interfaces vanishes at the energies of geometrical resonances in quasiparticle spectrum. The effect is similar to the over-the-barrier resonances in the simple problem of one-particle scattering against a step-function potential,[@CohenT] the superconducting gap playing the role of a finite-width barrier (as in the semiconductor model[@Tinkham]). The absence of AR and TH processes means that all quasiparticles with energies satisfying Eq. (\[resonance\]) will pass unaffected from one electrode to another, without creation or annihilation of Cooper pairs.
Characteristic features of coherent quantum transport through clean superconducting layers are the subgap tunneling and oscillations of the scattering probabilities. For $E<\Delta$, the subgap tunneling suppresses the Andreev reflection, thereby enhancing the transmission. For $E>\Delta$, all probabilities oscillate with $E$ and $l$ due to the interference effect.
The interface resistance reduces AR and TE, and enhances NR and TH probabilities. In contrast to the positions of zeros of $A_\sigma(E,0)$, given by Eq. (\[resonance\]), the positions of maxima of $A_\sigma(E,0)$, as well as that of zeros and maxima of $B_\sigma(E,0)$, $C_\sigma(E,0)$, and $D_\sigma(E,0)$, are $Z$-dependent. Approaching the tunnel limit ($Z\to\infty$), peaks in the scattering probabilities gradually split into two spikes belonging to consecutive pairs with positions defined by the quantization conditions $$\label{n} lq^+_{\sigma}=n_1\pi,~~~lq^-_{\sigma}=n_2\pi.$$ Here, $n_1-n_2=2n$, with $n$ coresponding to that of Eq. (\[resonance\]). The exception is the spike at the gap edge, originating from the singularity in the BCS density of states. Note that Eq. (\[n\]) gives the bound state energies of an isolated superconducting film. This is illustrated in Fig. \[tunnel10\] for an NSN junction, showing a simple connection between the resonances in metallic junctions ($Z=0$) and the bound states in the corresponding tunnel junctions ($Z\to\infty$).
The influence of exchange interaction is illustrated in Figs. \[thin\] and \[thick\] for an FSF double junction in P alignment with $Z=0$ and $\kappa=1$. Taking $\Delta/E^{(S)}_F=10^{-3}$, in a thin superconducting film, $lk^{(S)}_F\sim 10^3$, the Andreev reflection is strongly suppressed, since the subgap transmission is considerable, Fig. [\[thin\]]{}. In this case, the oscillations are less pronounced, with the period much larger than $\Delta$. For a thick film, $lk^{(S)}_F\sim 10^4$, the subgap tunneling is irrelevant (except for small ’tails’ in $A_\sigma(E,0)$ and $C_\sigma(E,0)$ at $E\lesssim\Delta$) and above the gap the oscillations are pronounced, with the period on the order of $\Delta$, Fig. [\[thick\]]{}. The scattering probabilities for AP alignment differ very slightly in the case of normal incidence, $\theta=0$. Although spin-independent due to the singlet-state pairing, $A_\sigma(E,0)$ is suppressed in comparison with the corresponding NSN junction, and $D_\sigma(E,0)$ becomes non-trivial. The spin-dependent normal reflection also occurs, $B_\sigma(E,0)$ having zeros at the same energies as $A_\sigma(E,0)$ and $D_\sigma(E,0)$, so that maxima in $C_\sigma(E,0)$ are still equal to unity due to the interface transparency.
Differential conductances
=========================
When voltage $V$ is applied to the junction, the charge current density is given by $$\label{j} j_q(V)=\sum_{\sigma}\int\frac{d^3{\bf k}}{(2\pi)^3}
e{\bf v}\cdot {\bf \hat{z}}~\delta f({\bf k},V),$$ where ${\bf v}=(\hbar/m)\Im[u_\sigma^*({\bf r})\nabla
u_\sigma({\bf r})+v_{\bar{\sigma}}^*({\bf r})\nabla
v_{\bar{\sigma}}({\bf r})]$ is the velocity, and $\delta f({\bf
k},V)$ is the asymmetric part of the nonequilibrium distribution function of current carriers. Using the solution of the scattering problem for the injection of an electron from the left, described in the previous section, and the dispersion relation ${\bf
k}(E)=k^{+}_{\sigma}{\bf \hat{z}}+{\bf k}_{||,\sigma}$, Eq. (\[j\]) can be rewritten in the form $$j_q(V)=\frac{e{k^{(S)}_{F}}^2}{\pi
h}\int\limits_{-\infty}^{\infty}dE \sum_{\sigma}\lambda^2_\sigma
\int\limits_{0}^{\pi/2}d\theta \sin\theta \cos\theta
\left[1+A_\sigma(E,\theta)-B_\sigma(E,\theta)\right]\delta f({\bf
k},V).$$ In accordance with BTK, without solving the suitable transport equation, we take $\delta f({\bf k},V)=f_0(E-eV/2)-f_0(E+eV/2)$, where $f_0(E)$ is the Fermi-Dirac equilibrium distribution function.
255=0.7
255=0.7
In this approach, the charge current per electron is given by $$\label{Iq} I_q(V)=\frac{1}{e}\int\limits_{-\infty}^{\infty}dE
\left[f_0(E-eV/2)-f_0(E+eV/2)\right]G_q(E),$$ where the spin-averaged differential charge conductance at zero temperature is $$\label{3D q} G_q(E)=\frac{e^2}{2h}\sum_\sigma \lambda^2_\sigma
\int\limits_{0}^{\pi/2}d\theta~\sin\theta\cos\theta
\left[1+A_\sigma(E,\theta)-B_\sigma(E,\theta)\right].$$ By analogy, the corresponding spin current (proportional to the probability current) is given by $$\label{Is}
I_s(V)=\frac{1}{e}\int\limits_{-\infty}^{\infty}dE\left[f_0(E-eV/2)-f_0(E+eV/2)\right]G_s
(E),$$ where the differential spin conductance at zero temperature is $$\label{3D s} G_s(E)=\frac{e^2}{2h}\sum_\sigma\rho_\sigma
\lambda^2_\sigma
\int\limits_{0}^{\pi/2}d\theta~\sin\theta\cos\theta
\left[1-A_\sigma(E,\theta)-B_\sigma(E,\theta)\right].$$ Note that in Eqs. (\[3D q\]) and (\[3D s\]) the upper limit of integration over $\theta$ for $\sigma=\uparrow$ is the angle of total reflection $\theta_{c1}\leq\pi/2$. Avoiding integration over $\theta$, the differential charge and spin conductances of a point-contact FSF double junction are simply expressed by $$\label{PC q} G_q(E)=\frac{e^2}{2h}\sum_\sigma \lambda^2_\sigma
\left[1+A_\sigma(E,0)-B_\sigma(E,0)\right]$$ and $$\label{PC s} G_s(E)=\frac{e^2}{2h}\sum_\sigma\rho_\sigma
\lambda^2_\sigma \left[1-A_\sigma(E,0)-B_\sigma(E,0)\right].$$
The influence of the exchange interaction on the conductance spectra is shown for $X=0.5$ on the example of thin, $lk^{(S)}_F=10^3$, and thick, $lk^{(S)}_F=10^4$, superconducting films (Figs. \[l3\]-\[GZ1\]). Besides the case of transparent interfaces, $Z=0$ (Figs. \[l3\] and \[GZ0\]), the effect of interface resistance is illustrated in the tunnel limit, $Z=10$, and for weak non-transparency, $Z=1$, in Figs. \[Z10 thin\] and \[GZ1\]. The influence of FWVM on the conductance spectra, $\kappa\neq 1$, is similar to that of the interface resistance.[@Milos] The values of normal conductances, $G^N_q$ and $G^N_s$ of the corresponding FNF double planar junction, indicated by arrows, are obtained by setting $A_\sigma(E,\theta)=0$ and $B_{\sigma}(E,\theta)=\left|b^N_{\sigma}(E,\theta)\right|^2$ in Eqs. (\[3D q\]) and (\[3D s\]), where $b^N_{\sigma}(E,\theta)$ is given by Eq. (\[bN\]).
The spin-polarized subgap tunneling of quasiparticles, and strong suppression of the Andreev reflection as a consequence, is significant in thin superconducting films, whereas the conductance oscillations above the gap are pronounced in the thick films. The magnetoresistance is apparent, as charge and spin conductances are larger for the P than for the AP alignment. An important consequence of the coherent transport is that the spin conductance is non-trivial for the AP alignment, approaching its normal value $G^N_s=0$ either for $E/\Delta\gg 1$, or in the tunnel limit ($Z\to\infty$) for all energies. We emphasize that the amplitudes of the oscillations are considerably larger for the point-contact FSF than for the planar FSF double junction, Fig. \[1D\].
Incoherent transport through an FSF double junction is described as a transport through the corresponding FS and SF junctions in series. In that case, the conductance spectra are calculated using the generalized BTK probabilities, obtained from Eqs. (\[a FS\]) and (\[b FS\]). Numerical results for the incoherent transport are also presented in Figs. \[GZ0\] and \[GZ1\] for comparison. It is evident that in thick films the only difference comes from the interference-effect oscillations for the energies above the gap. In contrast with the coherent transport, for the AP alignment $G_s(E)\equiv 0$, and nonequilibrium spin density accumulation changes the chemical potential of two spin subbands in the superconductor. This reduces the superconducting gap with increasing voltage, and destroys the superconductivity at a critical voltage on the order of $\Delta/e$.[@Takahashi] The effect of incoherency is less pronounced in metallic than in the tunnel junctions due to the Andreev reflection.[@Kinezi]
255=0.55
The results can be applied to reliable spectroscopic measurements of $\Delta$ and $v^{(S)}_F$ in superconducting films. From Eq. (\[resonance\]), for $\theta=0$, the energy $E_n$ of the $n$-th geometrical resonance (conductance minimum) satisfies a simple relation $$\begin{aligned}
\label{En} E_n^2=\Delta^2+\Big(\frac{hv^{(S)}_F}{2l}\Big)^2n^2.\end{aligned}$$ Therefore, the linear plot of $E_n^2$ vs $n^2$ has the intercept equal to $\Delta^2$ and the slope equal to $(hv^{(S)}_F/2l)^2$. An example is shown in Fig. \[E2n2\]. Note that even the points obtained for planar (3D) double junctions lie almost on the same straight line given by Eq. (\[En\]) for the particular case of a point-contact (1D) double junction. The numerical results show that the method is almost independent on dimensionality of the junction and on parameters of the electrodes.
The net spin polarization of the current is defined as $\Pi
(V)={I_s (V)}/{I_q (V)}$. In thin superconducting films, $\Pi (V)$ is almost constant, considerably smaller than $X$, which is the polarization in the corresponding FNF junction. Below the gap, $eV/2\Delta<1$, the contribution of subgap tunneling to spin polarization becomes negligibly small for large $l$. Above the gap, $eV/2\Delta>1$, the polarization increases with $V$, the increase becoming steeper with the interface non-transparency, Fig. \[Pi\]. On the other hand, in a tunnel FS junction the polarization abruptly changes from $\Pi=0$ to $\Pi=X$ at $eV/2\Delta=1$. The same result holds for incoherent transport through an FSF double tunnel junction.
255=0.7
Summary
=======
We have analyzed transport properties of FSF double-barrier junctions, taking into account the influence of the exchange interaction, the resistance of the interfaces, and the Fermi velocity mismatch on the scattering probabilities and the conductance spectra. We have shown that subgap tunneling and oscillations of differential conductances are the main features of the coherent quantum transport through a superconducting layer in both FSF and NSN double-barrier junctions. The subgap tunneling suppresses the Andreev reflection, thereby enhancing the transmission, especially in thin films. The scattering probabilities and conductances oscillate as a function of the layer thickness and of the quasiparticle energy above the gap.
Periodic vanishing of the Andreev reflection at the energies of geometrical resonances is found as an important consequence of the quasiparticle interference. Insulating barriers at the interfaces reduce the Andreev reflection and transmission, mainly for energies below the gap. The Fermi velocity mismatch has a similar effect. Results are directly accessible to experiments. In principle, oscillations of differential conductances with the period of geometrical resonances could be used for reliable spectroscopy of quasiparticle excitations in superconductors.
In conclusion, finite-size effects, along with the difference between coherent and incoherent transport, are essential for spin-currents in FSF junctions. For the coherent transport, besides the spin-polarized subgap tunneling in thin superconducting films, pronounced oscillations of spin conductance occur in thick films. As a consequence of the quasiparticle interference, a non-trivial spin current without the excess spin accumulation and destruction of superconductivity is possible even for AP alignment of the electrode magnetizations.
Acknowledgment
==============
We are grateful to Ivan Božović for pointing out the significance of the problem treated in this paper, and for help at the initial stage of this work. Furthermore, we thank Irena Knežević for useful discussions. This work has been supported by the Serbian Ministry of Science, Technology, and Development, grant ${\rm N}^{\circ} 1899$.
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[^1]: [*E-mail*]{}: [[email protected]]{}
[^2]: [*E-mail*]{}: [[email protected]]{}
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---
abstract: 'This deep, extended solar minimum and the slow start to Cycle 24 strongly suggest that Cycle 24 will be a small cycle. A wide array of solar cycle prediction techniques have been applied to predicting the amplitude of Cycle 24 with widely different results. Current conditions and new observations indicate that some highly regarded techniques now appear to have doubtful utility. Geomagnetic precursors have been reliable in the past and can be tested with 12 cycles of data. Of the three primary geomagnetic precursors only one (the minimum level of geomagnetic activity) suggests a small cycle. The Sun’s polar field strength has also been used to successfully predict the last three cycles. The current weak polar fields are indicative of a small cycle. For the first time, dynamo models have been used to predict the size of a solar cycle but with opposite predictions depending on the model and the data assimilation. However, new measurements of the surface meridional flow indicate that the flow was substantially faster on the approach to Cycle 24 minimum than at Cycle 23 minimum. In both dynamo predictions a faster meridional flow should have given a shorter cycle 23 with stronger polar fields. This suggests that these dynamo models are not yet ready for solar cycle prediction.'
author:
- 'David H. Hathaway'
title: 'Does the current minimum validate (or invalidate) cycle prediction methods?'
---
Introduction
============
As each sunspot cycle wanes solar astronomers with widely different interests take their turn at predicting the size and timing of the next cycle. The average length of the previous 22 sunspot cycles is 131.7 months - almost exactly 11 years. However, with one exception, the last 8 cycles have been short cycles with periods closer to 10 years. The minimum preceeding Cycle 23 was in August or September of 1996 so many were expecting the minimum preceeding Cycle 24 to come in 2007 or even 2006. Instead, minimum came in November of 2008 (Fig. 1). This delayed start of Cycle 24, and the depth of the minimum (smoothed sunspot number at its lowest in nearly 100 years) stirred up additional interest and even more predictions [@Pesnell08] including talk of an impending grand minimum like the Maunder Minimum [@Schatten03 e.g.].
Prediction Methods
==================
Predicting the size and timing of a sunspot cycle is very reliable once a cycle is well underway. Auto-regression techniques [@McNish49] and parametric curve fitting techniques [@Hathaway94] even give smoothed month-to-month behavior. However, they only become reliable 2-3 years after minimum - at about the inflection point in the rise of the sunspot number toward maximum (Fig. 2).
Predictions made prior to the start of a cycle or shortly after minimum require methods other than auto-regression or curve-fitting. The simplest method, and the one used as a benchmark for predictive capablilty, is to use an average cycle (maximum smoothed sunspot number $114\pm40$ for cycles 1-23). Many predictions are based on trends or periodicities percieved in the history of cycle amplitudes (eg. the @Gleissberg39 8-cycle periodicity) . Others are based on the characteristics of the previous cycle or of the cycle minimum itself. In the latter category two characteristics stand out - the Amplitude-Period relation [@Wilson98] and the Maximum-Minimum relation [@Brown76].
With the Amplitude-Period relation the amplitude of a cycle is related to the period (length) of the previous cycle - small cycles start late and leave behind a long period cycle. With the Maximum-Minimum relation the amplitude (maximum) of a cycle is related to the level of the minimum preceeding it - small cycles start late and leave behind a low minimum. These two relations are shown in Fig. 3 along with the associated predictions for Cycle 24.
@Hathaway99 examined many of these prediction methods and tested them by backing-up to the minimum predeeding Cycle 19 and using each method to predict Cycles 19-22 but only using data obtained prior to the minimim of each of those cycles. The predictions were examined for both accuracy and stability - stability in the sense of how stable the predicting relation was from cycle-to-cycle. For example, using the average cycle as a predictor gave an RMS error of about 60 and the predicting relation (the size of the average cycle) varied by 12% from 104 prior to Cycle 19 to 112 prior to Cycle 22. The conclusion from this study was that the most accurate and stable prediction methods were based on geomagnetic precursors - geomagnetic activity near or before the time of sunspot cycle minimum. It was also noted that predictions based on the strength of the Sun’s polar fields [see @Schatten78] were promising but could not be adequately tested due to the lack of direct data prior to Cycle 21. Since 2006 predictions based on Flux Transport Dynamo Models with assimilated data have been offered - @Dikpati06 and @Choudhuri07. These three promising methods - Geomagnetic Precursors, Polar Field Precursors, and Flux Transport Dynamos - are examined in the following sections.
Geomagnetic Precursor Predictions
=================================
@Ohl66 was among the first to note that geomagnetic activity around the time of sunspot cycle minimum was a good predictor for the size of the following cycle. In particular, he noted that the minimum in the smoothed monthly geomagnetic index *aa* was well correlated with the amplitude of the following cycle. The *aa* index is a measure of the geomagnetic field variations obtained at 3-hour intervals since 1868 from two nearly antipodal observatories - one in England and one in Austrailia [@Mayaud72]. Each observatory was relocated at least once since 1868 and the change made in 1957 seems to have had a significant effect on the data [@Svalgaard04]. Fig. 4 illustrates the method described by @Ohl66. The minima in the smoothed monthly *aa* index are very well correlated with the maximum sunspot number of the following cycle. As of December 2009 the smoothed *aa* index (for June 2009) was still falling and at a record low of 8.8.
It has been known for many decades[@Bartels32] that there are two solar sources of geomagnetic activity. One source (now known to be CMEs) has a frequency of occurance that is in phase with the sunspot cycle while the second source (now known to be high-speed solar wind streams) is out of phase with the sunspot cycle and tends to peak late in each cycle. @Feynman82 suggested a method for separating these two components. She noted that background level of geomagnetic activity rose and fell with the sunspot numbers. Removing this sunspot cycle background level of activity leaves behind a component of geomagnetic activity that is out of phase with the sunspot cycle (Fig. 5). @Hathaway99 noted that the peaks in this second component that occur just prior to sunspot cycle minimum were well correlated with the amplitude of the following cycle (Fig. 6). This led @Hathaway06 to a prediction of $160\pm25$ for Cycle 24 based on the assumption that sunspot cycle minimum was eminent in 2006. It is now clear the minimum was still over two years off. The maximum in $aa_I$ used for that prediction was from the fall of 2003 and obviously associated with the 2003 Haloween events. Since this activity was not reflected in significantly higher sunspot numbers, it shows up as a huge (and probably misleading) peak in the $aa_I$ component.
@Thompson93 also recognized that some geomagnetic activity in a sunspot cycle was indicative of the amplitude of the following cycle. However, instead of trying to separate the geomagnetic activity into components he found that the total number of geomagnetically disturbed days (defined as days with geomagnetic index $Ap \geq 25$) during a cycle was proportional to the sum of the amplitudes of the current cycle and the future cycle. This is shown in Fig. 7. Here again, as with the Feynman Method prediction, the Halloween events of 2003 significantly impact the process and results. Removing these events lowers the predicted amplitude of Cycle 24 from $130\pm28$ to $95\pm28$.
When we extend the prediction method testing of @Hathaway99 to include Cycle 23 we still find that these Geomagnetic Precursor methods are substantially better than other methods. Although this testing indicates that the Thompson and Feynman Methods faired slightly better than the Ohl Method for Cycles 19-23, the impact of the Halloween 2003 events on those methods suggest that greater weight should be given to the Ohl Method prediction for Cycle 24 - an amplitude of $70\pm18$.
Polar Field Precursor Predictions
=================================
The strength of the Sun’s polar magnetic fields near sunspot cycle minimum has been used to predict the last three cycles - Cycle 21 [@Schatten78], Cycle 22 [@Schatten87], and cycle 23 [@Schatten96], with considerable success. This method is based on the dynamo model described by @Babcock61 and @Leighton69 in which the Sun’s poloidal field at minimum is amplified and converted into the toroidal field (that erupts in active regions) by differential rotation. While several questions remain about the implementation of this method (What precise phase of the solar cycle should the measurement be taken? Is the relationship between polar fields and sunspot cycle amplitude linear?) the success with predicting the last three cycles places this method on par with the geomagnetic precursor methods. The polar fields as measured at the Wilcox Solar Observatory (Fig. 8) have remained substantially weaker since 2004 leading to a prediction of $78\pm8$ for the amplitude of Cycle 24 [@Svalgaard04].
Flux Transport Dynamo Predictions
=================================
In a ground-breaking paper @Dikpati06 used a dynamo model with assimilated data to predict a solar cycle. Their dynamo model is a kinematic flux transport dynamo in which the axisymmetric flows in the convection zone (differential rotation and meridional circulation) are prescribed along with a diffusivity (representing the effects of the non-axisymmetric convective flows) and a field regenerating term (representing the stretching and twisting of magnetic field lines by instabilities and the effect of rotation on rising magnetic flux tubes). Historical sunspot cycle data was assimilated into the model by adding magnetic sources at the surface representative of observed sunspot areas and positions. The strength of the toroidal fields in the model were found to accurately reflect the strength of the last 10 sunspot cycles. They concluded that Cycle 24 would be 30-50% larger than cycle 23 i.e. an amplitude of 160-180 for Cycle 24. They went on to note that the speed of the meridional flow had apparently slowed as Cycle 23 approached maximum [@Basu03]. In their model a slow meridional flow produces long cycles and weak polar fields. From this they concluded that Cycle 24 would start late.
Shortly after the publication of this paper @Choudhuri07 presented their own prediction based on a similar Flux Transport Dynamo model. Their model had one substantial difference from the @Dikpati06 model - a significantly larger diffusivity. In addition, instead of assimilating sunspot area data they reset the poloidal field at minimum for the last three cycles and found a good fit to the observed cycle amplitudes. Putting in the weak polar fields at the current minimum predicted a Cycle 24 about 35% weaker than Cycle 23 - an amplitude of about 80 for Cycle 24 - right in line with the Polar Field Precursor prediction of @Svalgaard04.
Meridional Flow Variations
==========================
In a recent paper @Hathaway10 measured the changes in the speed of the surface meridional flow over the completed Cycle 23. Over 60,000 fulldisk magnetograms from the MDI instrument on SOHO were used to determine the meridional motion of weak magnetic features that are carried by the flow. They found that while the speed of the meridional flow did indeed slow on the approach to Cycle 23 maximum in 2000/2001, it then sped up to substantially faster speeds for the remainder of the cycle (Fig. 9). This type of variation was also seen in Cycles 21 and 22 by @Komm93. However the faster speed on the approach to cycle 24 minimum should have produced *stronger* polar fields and a *shorter* cycle 23 with the Flux Transport Dynamo models.
Conclusions
===========
Ohl’s Geomagnetic Precursor, the Polar Field Precursors, the Amplitude-Period relation, and the Maximum-Minimum relation all indicate that Cycle 24 will be small with an amplitude of about 75. The other two geomagnetic precursor methods appear to be unduly impacted by the activity associated with the Halloween events of 2003 and give larger cycles. We conclude with @Wang09 that the more appropriate geomagnetic precursor is that of @Ohl66 - the minimum level of geomagnetic activity.
The predictions based on Flux Transport Dynamos gave very different predictions but they both predict behavior in conflict with the observed meridional flow variations. The faster meridional flow after Cycle 23 maximum sould give a short cycle with strong polar fields acording to these models. Instead we find a long cycle with weak polar fields. We must conclude with @Tobias06 that these dynamo modes are not yet ready for cycle predictions.
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---
abstract: 'It is known that there exists a network, called as the M-network, which is not scalar linearly solvable but has a vector linear solution for message dimension two. Recently, a generalization of this result has been presented where it has been shown that for any integer $m\geq 2$, there exists a network which has a $(m,m)$ vector linear solution, but does not have a $(w,w)$ vector linear solution for $w<m$. This paper presents a further generalization. Specifically, we show that for any positive integers $k,n,$ and $m\geq 2$, there exists a network which has a $(mk,mn)$ fractional linear solution, but does not have a $(wk,wn)$ fractional linear solution for $w<m$.'
author:
-
title: 'On Achievability of an $(r,l)$ Fractional Linear Network Code'
---
Introduction
============
The concept of network coding emerged in year 2000 where it was shown that by performing operations on incoming packets and then forwarding by nodes in a network may increase the throughput significantly[@alshwede]. Since then there have been extensive studies on various theoretical aspects and applications of network coding. The benefits of network coding can be best seen in a multicast network where the min-cut bound can be achieved using network coding which otherwise may be impossible through only routing [@alshwede].
Linear operations by the nodes is the most natural setting to consider due to the mathematical traceability of linear operations. The network coding involving only linear operations over a finite field is referred as the linear network coding. An $(r,l)$ fractional linear network code considers the information generated by the sources as $r$ length vectors of symbols and the information passing through all the edges as $l$ length vectors of symbols. A network is said to have an $(r,l)$ fractional linear network code solution if the sources can supply $r$ symbols to their respective terminals in $l$ uses of the network. If $r = l$, then an $(r,l)$ fractional linear network code is referred to as an $(r,r)$ vector linear network code (also as an $r$-dimensional vector linear code). An $1$-dimensional vector linear network code is referred to as a scalar linear network code. An $(r,l)$ factional linear network code for a network is achievable if the network has an $(r,l)$ fractional linear network code solution. A rate $r/l$ linear network code is said to be achievable if there exists an $(mr,ml)$ factional linear network code solution over some finite field where $m$ is a positive rational number such that $mr$ and $ml$ are integers.
In this paper, we show that achievability of an arbitrary rate $r/l$ linear network code does not necessarily imply the existence of an $(r,l)$ fractional linear network code solution for any positive integers $r$ and $l$. Specifically, we show that for any integer $m\geq 2$, there exists a network where a rate $r/l$ linear network code is achievable but it does not have a $(wr,wl)$ fractional linear network code solution for any $w$ less than $m$. For $r=l$, the result of this paper specializes to the result of [@das].
Related Work
------------
In this subsection, we provide a brief survey of the studies related to the work of this paper. Scalar linear network codes over sufficiently large finite fields have been shown to be sufficient to achieve the capacity for multicast networks [@li]. It has also been shown that such scalar linear codes can be efficiently designed [@jaggi]. Moreover, capacity achieving vector linear network codes can also be designed where the requirement of field size can be reduced as compared to capacity achieving scalar linear network codes[@ebra]. It has been shown in [@riis] and [@koetter] that all multicast networks are vector linearly solvable over the binary field $\mathbb{F}_2$ if the vector length is sufficiently large. For the multicast networks, it is also known that solution over an alphabet might not guarantee a solution over all larger alphabets [@doug2]. Although multicast networks have been extensively studied since the advent of network coding and there are many nice results such as characterising the capacity, dependency on the size of the alphabet etc., yet not everything is known and new results such as [@sun] continue to enhance the knowledge about multicast networks. Sun *et al.* [@sun] presented an instance of a multicast network which has an $L$-dimensional vector linear network code solution over $\mathbb{F}_q$ but has no scalar linear network code solution over $\mathbb{F}_{q^\prime}$ where $q^\prime \leq q^L$. They also presented an explicit instance to show that the existence of a vector linear network code solution for a certain message dimension does not necessarily guarantee a vector linear network code solution for all higher message dimensions.
For non-multicast networks, the results are fundamentally different from multicast networks. Linear network codes are shown to be insufficient [@doug]. Among linearly solvable networks, there are networks which are vector linearly solvable only for certain vector dimensions. In [@medard], it has been shown that there exists a network which has a 2-dimensional vector linear network code solution but has no scalar linear network code solution. This network is known as the M-network. In fact, the M-network has no vector linear solution for any odd message dimensions [@dougherty]. The results of [@medard] and [@dougherty] have been generalized in [@das]. In [@das], it has been shown that for any integer $m\geq 2$, there exists a network which has an $m$-dimensional vector linear solution but has no $w$-dimensional vector linear solution for $w<m$.
Matroid theory, a branch of mathematics, has been shown to be useful to study the linear solvability of networks. Indeed, some of the important results, such as [@dougherty; @doug; @doug2; @doug3; @doug4; @doug5; @sun], have been obtained by exploiting the connection between matroids and networks. It has been shown that a network has a scalar linear network code solution over $\mathbb{F}_q$ if and only if it is a matroidal network with respect to a matroid representable over $\mathbb{F}_q$ [@dougherty; @kim]. Using this result the authors of [@doug] constructed the Fano network from the Fano matroid to show that there exists a network which has a vector linear solution for any message dimension over any finite field of even characteristics but has no vector linear solution for any message dimension when the characteristics of the finite field is odd. Similarly, from the the non-Fano matroid the non-Fano network was constructed to show that there exists a network which has a vector linear solution for any message dimension over a finite field of odd characteristics but has no vector linear solution for any message dimension when the characteristics of the finite field is even. In [@dougherty] the V[á]{}mos network was constructed from the V[á]{}mos matroid and it was shown that Shannon-type inequalities are insufficient to compute the network coding capacity of the V[á]{}mos network. [@dougherty] was generalized in [@sundar] to show that similar to the correspondence between matroids and scalar linear network code solution of a network, discrete polymatroids corresponds to both vector linear network code solution and fractional linear network code solution of a network. It has been shown that a network has an $(r,l)$ fractional linear network code solution over $\mathbb{F}_q$ if and only if it is a $(r,l)$-discrete polymatroidal network with respect to a discrete polymatroid representable over $\mathbb{F}_q$ [@sundar].
Organization of the paper
-------------------------
The organization of the paper is as follows. Section \[prelims\] presents the formal definitions related to network coding. In Section \[sec2\], we present a key result (Theorem \[thm1\]): for any integer $m\geq 2$, there exists a network which has an $(m,mn)$ fractional linear network code solution but has no $(w,wn)$ fractional linear network code solution for $w<m$. Using this result, we present the main result of the paper (Theorem \[thm2\]): there exists a network which has an $(mk,mn)$ fractional linear network code solution but does not have a $(wk,wn)$ fractional linear network code solution for $w<m$. The proof of Theorem \[thm1\] is presented in Section \[sec4\]. Section \[conclusion\] concludes the paper.
Preliminaries {#prelims}
=============
We represent a network by a directed acyclic graph. The sources of a network are represented by the nodes in the graph with no incoming edges; and the terminals of a network are represented by the nodes in the graph with no outgoing edges. The sources are assumed to generate an i.i.d. random process uniformly distributed over over $\mathbb{F}_q$. The random process at one source is independent of any collection of the random processes generated at all other sources. Each edge in the graph is assumed to be of unit capacity. For any node $v$, the set of edges incoming to a node $v$ and the set of edges outgoing from node $v$ are denoted by $In(v)$ and $Out(v)$ respectively. $(u,v)$ represents an edge $e$ directed from $u$ to $v$. Each terminal demands a set of source processes. Let the set of sources and the set of terminals be denoted by $S$ and $T$ respectively.
An $(r,l)$ fractional linear network code is defined as follows. In such a code, $r$ symbols are considered at every source. Let the source process generated by the source $s_i\in S$ be denoted by $X_i\in \mathbb{F}_q^r$. Each edge is used for $l$ units of time. Let the message symbol carried by an edge $e$ be denoted by $Y_e$ where $Y_e \in \mathbb{F}_q^l$. For an edge $e$, if $tail(e)=s_i$, then $Y_e = A_{\{i,e\}}X_i$ where $A_{\{i,e\}}\in \mathbb{F}_q^{l\times r}$. Else, if $tail(e)$ is an intermediate node, then for $Y_e = \sum_{e^\prime \in In(tail(e))} A_{\{e^\prime,e\}}Y_e^\prime$ where $A_{\{e^\prime,e\}} \in \mathbb{F}_q^{l\times l}$. Matrices $A_{\{i,e\}}$ and $A_{\{e^\prime,e\}}$ are called the local coding matrices. The terminals compute a set of $r$ length vectors from its incoming edges. If $X_{t_i}\in\mathbb{F}_q^r$ denotes a vector that is computed by a terminal $t_i\in T$ then, $X_{t_i} = \sum_{e\in In(t_i)} A_{\{e,t_i\}}Y_{e}$ where $A_{\{e,t_i\}}\in \mathbb{F}_q^{r\times l}$. A network is said to have an $(r,l)$ fractional linear network code solution if the sources are able to send its symbols to the respective terminals in $l$ usages of the network using an $(r,l)$ fractional linear network code. An $(r,l)$ fractional linear network code is said to be achievable if there exists an $(r,l)$ fractional linear network code solution.
A network is said to have an $(r,l)$ routing solution if the sources are able to send its symbols to the respective terminals in $l$ usages of the network only through routing.
Fractional Linear Network Code Solutions {#sec2}
========================================
![A communication network $\mathcal{N}$. For any integer $m\geq 2$, $\mathcal{N}$ has an $(m,mn)$ fractional linear network code solution but has no $(w,wn)$ fractional linear network code solution when $w$ is less than $m$.[]{data-label="general"}](general12){width="48.00000%"}
Consider the network $\mathcal{N}$ shown in Fig. \[general\]. We note that the network $\mathcal{N}$ is a further generalization of the “generalized M-network” presented in [@das]. $\mathcal{N}$ has $m^2n$ sources and $\binom {mn}{n}^m$ terminals. The sources are partitioned into $m$ sets $S_1,S_2,\ldots,S_m$. Each set has $mn$ many sources. The $j^{th}$ source in the set $S_i$ is denoted by $s_{ij}$. Each terminal demands information from $n$ sources of each set. If $T_{i}$ is the set of all sources demanded by $t_i$, then $T_{i}\neq T_{j}$ if $i\neq j$. Without loss of generality assume that the terminal $t_1$ demands the messages from the sources $s_{ij}$ for $1\leq i\leq m$, $1\leq j\leq n$. There are following sets of edges in the network $\mathcal{N}$: $\{(s_{ij},u_i)\} | 1\leq i\leq m, 1\leq j \leq mn$; $\{e_{ii} = (u_i,v_i)| 1\leq i\leq m\}$; $\{e_{ij} = (u_i,v_j) | 1\leq i\leq m, m+1\leq j\leq 2m-1\}$; and $\{(v_i$,$t_j) | 1\leq i\leq 2m-1, 1\leq j\leq \binom {mn}{n}^m\}$. The message generated at the source $s_{ij}$ for $1\leq i\leq m$ and $1\leq j\leq mn$ is denoted by $X_{ij}$. All the source symbols are from the finite field $\mathbb{F}_q$. The information transmitted over the edge $e_{ii}$ for $1\leq i\leq m$ is denoted by $Y_{ii}$, the information transmitted over the edge $e_{ij}$ for $1\leq i\leq m$ and $m+1\leq j\leq 2m-1$ is denoted by $Y_{ij}$, and the information transmitted over the edge $(v_{i},t_{j})$ for $1\leq i\leq 2m-1$ and $1\leq j\leq \binom {mn}{n}^m$ is denoted by $Z_{ij}$.
Note that for $n=1$, the network $\mathcal{N}$ reduces to the “generalized M-network” presented in [@das].
\[le1\] The capacity of the network $\mathcal{N}$ is upper bounded by $1/n$.
*Proof.* Consider an $(r,l)$ fractional network code solution for $\mathcal{N}$. Since there is no source which is not demanded by any terminal, all the source messages in $S_1$ must get computed from the set $\{Y_{11},Y_{1(m+1)},Y_{1(m+2)},\ldots,Y_{1(2m-1)} \}$. Hence,
[l]{} H(Y\_[11]{},Y\_[1(m+1)]{},Y\_[1(m+2)]{},…,Y\_[1(2m-1)]{})\
H(X\_[11]{},X\_[12]{},…,X\_[1(mn)]{})\
H(Y\_[11]{}) + H(Y\_[1(m+1)]{}) + + H(Y\_[1(2m-1)]{})\
H(X\_[11]{}) + H(X\_[12]{}) + + H(X\_[1(mn)]{})\
ml\_2 q r(mn)\_2 q\
Let us assume that the network $\mathcal{N}$ has a $(d,dn)$ fractional linear network code solution. The following theorem puts a constraint on $d$.
\[thm1\] For an arbitrary finite field $\mathbb{F}_q$, there exists a network which has a $(d,dn)$ fractional linear network code solution over $\mathbb{F}_q$ if and only if $d$ is an integer multiple of $m$.
We prove the theorem by using the network $\mathcal{N}$. We defer the proof of this theorem to Section \[sec4\].
The network $\mathcal{N}$ has an $(m,mn)$ fraction linear network code solution but has no $(w,wn)$ fractional linear network code solution for ${w<m}$.
For a special case of $n=1$, Theorem \[thm1\] specializes to the following Corollary:
For an arbitrary finite field $\mathbb{F}_q$, there exists a network which has a $(d,d)$ fractional linear network code solution over $\mathbb{F}_q$ if and only if $d$ is an integer multiple of $m$.
We note that this corollary is the main result of the paper [@das].
Consider $k$ copies of $\mathcal{N}$, named as $\mathcal{N}_1,\mathcal{N}_2,\ldots,\mathcal{N}_k$. Denote the source $s_{ij}$ in $\mathcal{N}$ as the source $s_{ijp}$ in $\mathcal{N}_p$ for $1 \leq p \leq k$. Denote all the edges and terminals in $\mathcal{N}_p$ in a similar manner. We now construct a network $\mathcal{N}_{||}$ by connecting the $k$ copies of $\mathcal{N}$ in parallel such that for every $i$ and $j$ where $1 \leq i \leq m, 1 \leq j \leq mn$, $k$ sources $s_{ij1}$, $s_{ij2}$, $\ldots, s_{ijk}$ are combined and the combined source is denoted as $s_{ij}$. Similar combining is done with the terminals as well. Apart from sources and terminals, the rest of the nodes and edges of the $k$ copies of $\mathcal{N}$ remain disjoint in $\mathcal{N}_{||}$.
The capacity of $\mathcal{N}_{||}$ is upper bounded by $\frac{k}{n}$.
Consider an $(r,l)$ fractional linear network code solution for $\mathcal{N}_{||}$. Since from information carried by the edges in the set $\{\{e_{11p},e_{1(m+1)p},e_{1(m+2)p},\ldots,e_{1(2m-1)p}\}| 1\leq p \leq k\}$, $t_1$ must be able to compute all the messages from the sources in $S_1$, proceeding similar to Lemma \[le1\] we get $\frac{r}{l}\leq\frac{k}{n}$.
Let $y$ be a positive rational number such that $yk$ and $yn$ are integers. We have the following lemma:
\[l1\] If $\mathcal{N}_{||}$ has a $(yk,yn)$ fractional linear network code solution, then $\mathcal{N}$ has a $(yk,ykn)$ fractional linear network code solution.
This is because, what can be sent from sources to the terminals in one unit of time in $\mathcal{N}_{||}$, the same can be sent in $k$ units of time in $\mathcal{N}$.
We now present the main result of this paper.
\[thm2\] For any positive integers $k$, $n$, $y\geq 2$, and $w$, there exists a network which has a $(yk,yn)$ fractional linear network code solution, but has no $(wk,wn)$ fractional linear network code solution for all $w<y$.
Consider $m=yk$ in $\mathcal{N}$. Say $\mathcal{N}_{||}$ has a $(wk,wn)$ fractional linear network code solution where $w<y$. Then, from Lemma \[l1\], $\mathcal{N}$ has a $(wk,wkn)$ fractional linear network code solution. Therefore, from Theorem \[thm1\], $m$ divides $wk$. However, since $m=yk$ and $wk$ is less than $yk$, $m$ cannot divide $wk$. This is a contradiction to the assumption that $\mathcal{N}_{||}$ has a $(wk,wn)$ fractional linear network code solution for $w<y$.
We now show that $\mathcal{N}_{||}$ has a $(yk,yn)$ fractional linear solution. Note that $\mathcal{N}$ has a $(yk,ykn)$ fractional linear network code solution. Theorem \[thm1\] ensures the existence of such a network code solution. Moreover, there is a $(yk,ykn)$ routing solution for $\mathcal{N}$ (see the proof of Theorem \[thm1\] in Section \[sec4\]). Therefore, trivially, $\mathcal{N}_{||}$ has a $(yk,yn)$ routing solution. This completes the proof.
Proof of Theorem \[thm1\] {#sec4}
=========================
Our proof relies on a prior result on the connection between fractional linear network code solution and representable polymatroids. For the self-containment of the paper, we first provide a brief background material. For this, the definitions of a discrete polymatroid, and a discrete polymatroidal network are given in the following. These definitions are reproduced from [@sundar]. We also reproduce a prior result from [@sundar] which will be used in the proof of Theorem \[thm1\]. We note the definitions and the result from [@sundar] are specialised to the network used in this paper.
A discrete polymatroid is a set of vectors with some specific properties. Consider a set $G = \{1,2,\ldots,g\}$. Say $\mathbb{Z}_{\geq 0}$ denote the set of all positive integers. Also let $\mathbb{Z}_{\geq 0}^g$ be the set of all $g$ length vectors whose components belong to $\mathbb{Z}_{\geq 0}$. If $x$ is a vector then let $x(i)$ denote the $i^{th}$ component of $x$. A function $\rho : 2^G \rightarrow \mathbb{Z}_{\geq 0}$ qualifies to be a rank function if the following three conditions are satisfied:\
$(1)$ $\rho(\phi) = 0 $\
$(2)$ if $A\subseteq B\subseteq G$, then $\rho(A)\leq \rho(B)$\
$(3)$ $\rho(A) + \rho(B) \geq \rho(A\cup B) + \rho(A\cap B)$ for $\forall A,B \subseteq G$.\
A discrete polymatroid $\mathbb{D}$ with rank function $\rho$ is defined as $\mathbb{D} = \{x \in \mathbb{Z}_{\geq 0}^g | \sum_{i\in A} x(i) \leq \rho(A), \forall A \subseteq G\}$. $\mathbb{D}$ is said to be representable over $\mathbb{F}_q$ if there exist vector subspaces $V_1,V_2,\ldots,V_g$ of a vector space $E$ over $\mathbb{F}_q$ such that for any $A\subseteq G$, $dim(\sum_{i\in A} V_i) = \rho(A)$.\
We now state the conditions for $\mathcal{N}$ to be an $(r,l)$ discrete polymatroidal network. Recall that $X_{ij}$ is the messages generated by the source $s_{ij}\in S$. Say $X = \{X_{ij} | 1\leq i\leq m, 1\leq j\leq mn \}$. Let $Y$ be the collection of all messages carried by all the edges of the network. For any node $v$, let $Y_{In(v)}$ and $Y_{Out(v)}$ denote the set of messages carried the edges in $In(v)$ and $Out(v)$ respectively.\
The network $\mathcal{N}$ is an $(r_1,r_2,\ldots,r_{m^2n};l)$-discrete polymatroidal network with respect to the discrete polymatroid $\mathbb{D}$ if there exists a map $f: \{X \cup Y\} \rightarrow G$ which satisfies the following conditions:\
$(1)$ $f$ is one-to-one on the elements of $X$\
$(2)$ $\rho(f(X_{ij})) = r_{(i-1)mn+j}$ and $\rho(f(Y_e)) \leq l$ for any $X_{ij}\in X$, $Y_e\in Y$\
$(3)$ $\sum r_i\epsilon_{ig} \in \mathbb{D}$ where $\epsilon_{ig}\in \mathbb{Z}_{\geq 0}^g$, $\epsilon_{ig}(i) = 1$ and $\epsilon_{ig}(j) = 0$ if $j\neq i$ for $1\leq i\leq m^2n$\
$(4)$ For any node $v$, $\rho(f(Y_{In(v)})) = \rho(f(Y_{In(v)}\cup Y_{Out(v)}))$. The network $\mathcal{N}$ is an $(r,l)$-discrete polymatroidal network if it is an $(r_1,r_2,\ldots,r_{m^2n};l)$-discrete polymatroidal network for $r_1,r_2,\ldots,r_{m^2n} = r$.\
The following theorem is reproduced with slightly different notations from [@sundar].
A network has an $(r_1,r_2,\ldots,r_{m^2n};l)$ fractional linear network code solution over $\mathbb{F}_q$, if and only if it is an $(r_1,r_2,\ldots,r_{m^2n};l)$-discrete polymatroidal with respect to a discrete polymatroid $\mathbb{D}$ representable over $\mathbb{F}_q$.
An $(r,l)$ fractional linear network code solution is an $(r_1,r_2,\ldots,r_{m^2n};l)$ fractional linear network code solution for $r_1,r_2,\ldots,r_{m^2n} = r$.\
*Proof of Theorem \[thm1\]:* We now show that the network $\mathcal{N}$ has a $(d,dn)$ fractional linear network code solution if and only if $d$ is an integer multiple of $m$. We first prove the ‘only if’ part by showing that the network $\mathcal{N}$ is a $(d;dn)$-discrete polymatroidal network only if $m$ divides $d$. We then prove the ‘if’ part by presenting an $(m,mn)$ fractional linear network code solution.
Let $\mathtt{g} = \rho\circ f$. Also let $\mathbb{Z}_n = \{0,1,\ldots,n-1\}$. Define the set $U_i = \{a + 1 + (i-1)n\; | a\in \mathbb{Z}_n \}$ and the set $C_{ij} = \{X_{ia} | a\in U_j\}$ for $1\leq i,j\leq m$.
\[clai\] For any $X_{Q_i}\subseteq \{X_{i1},X_{i2},\ldots ,X_{i(mn)}\}$,
[l]{} (Y\_[11]{},X\_[Q\_[1]{}]{}) + (Y\_[22]{},X\_[Q\_2]{}) + + (Y\_[mm]{},X\_[Q\_m]{})\
= (Y\_[11]{},X\_[Q\_1]{},Y\_[22]{},X\_[Q\_2]{},…,Y\_[mm]{},X\_[Q\_m]{}).
Since none of the vectors in $X_{Q_i}$ and $Y_{ii}$ are dependent on any vector in $X_{Q_j}$ and $Y_{jj}$ for $i\neq j$, $1\leq i,j\leq m$, we have $H(Y_{ii},X_{Q_i}|Y_{jj},X_{Q_j}) = H(Y_{ii},X_{Q_i})$. Hence, $H(Y_{ii},X_{Q_i}) + H(Y_{jj},X_{Q_j}) = H(Y_{ii},X_{Q_i},Y_{jj},X_{Q_j})$. Similarly the following equation must also be true:
[l]{} H(Y\_[11]{},X\_[Q\_[1]{}]{}) + H(Y\_[22]{},X\_[Q\_2]{}) + + H(Y\_[mm]{},X\_[Q\_m]{})\
= H(Y\_[11]{},X\_[Q\_1]{},Y\_[22]{},X\_[Q\_2]{},…,Y\_[mm]{},X\_[Q\_m]{})\[ie1\]
Since $\mathtt{g}(\,)$ is a rank function of a discrete polymatroid, the equation (\[ie1\]) remains valid if $H(\,)$ is replaced by $\mathtt{g}(\,)$ [@dougherty].
\[cla1\] Let $X_{P_i}\!\subset \{X_{i1},X_{i2},\ldots ,X_{i(mn)}\}$, and $|X_{P_i}|=n$ for $1\leq i\leq m$. The following set of inequalities hold. $$\begin{gathered}
\mathtt{g}(Y_{11},X_{P_{1}}) + \mathtt{g}(Y_{22},X_{P_2}) + \cdots + \mathtt{g}(Y_{mm},X_{P_m})\\ \leq (2m-1)nd \label{mq1}\end{gathered}$$
We show the proof for the case when $X_{P_i} = C_{i1}$ for $1\leq i \leq m$. The rest of the cases can also be proved similarly.
[l]{} (Y\_[11]{},X\_[P\_[1]{}]{}) + (Y\_[22]{},X\_[P\_2]{}) + + (Y\_[mm]{},X\_[P\_m]{})\
= (Y\_[11]{},X\_[P\_1]{},Y\_[22]{},X\_[P\_2]{},…,Y\_[mm]{},X\_[P\_m]{})\[nq1\]\
(Y\_[11]{},X\_[P\_1]{},Y\_[22]{},X\_[P\_2]{},…,Y\_[mm]{},X\_[P\_m]{},\
Z\_[(m+1,1)]{},…,Z\_[(2m-1,1)]{})\
= (Y\_[11]{}…,Y\_[mm]{},Z\_[(m+1,1)]{},…,Z\_[(2m-1,1)]{})\[nq2\]\
mdn + (m-1)dn\
= (2m-1)dn
Equation (\[nq1\]) comes from Claim \[clai\]. Equation (\[nq2\]) is true because the terminal $t_1$ computes all the source symbols in the set $\{\cup_{1\leq i\leq m} C_{i1}\}$ from the vectors in the set $\{Y_{11}\ldots ,Y_{mm},Z_{(m+1,1)},\ldots,Z_{(2m-1,1)}\}$. Note that for all configuration of $X_{P_1},X_{P_2},\ldots ,X_{P_m}$ there is a terminal which demands all the sources contained in these sets. So the rest of the inequalities can be proved analogously.
\[cla2\] For $1\leq i\leq m$ the set $\{X_{i1},X_{i2},\ldots ,X_{i(mn)}\}$ is arbitrarily partitioned into $m$ mutually disjoint sets each containing $n$ elements. Denote the $j^{th}$ set as $X_{U_{ij}}$. The following inequalities hold true. $$\begin{gathered}
\text{For } 1\leq i\leq m,\\
\mathtt{g}(Y_{ii},X_{U_{i1}}) + \mathtt{g}(Y_{ii},X_{U_{i2}}) + \cdots + \mathtt{g}(Y_{ii},X_{U_{im}}) \\ \geq (2m-1)nd \label{claeq}\end{gathered}$$
To prove this result we first show that $\mathtt{g}(Y_{ii}) = \mathtt{g}(Y_{ij}) = nd$ for $1\leq i\leq m$ and $(m+1)\leq j\leq (2m-1)$.
[ll]{} m\^2nd&\
=(X\_[11]{},X\_[12]{},…,X\_[1(mq)]{},&…,X\_[m(mn)]{})\
(X\_[11]{},X\_[12]{},…,X\_[1(mn)]{},&…,X\_[m(mn)]{},Y\_[11]{},…,Y\_[mm]{},\
&Y\_[1(m+1)]{},…,Y\_[m(2m-1)]{})\
= (Y\_[11]{},…,Y\_[mm]{},Y\_[1(m+1)]{}&,…,Y\_[m(2m-1)]{})\[nq3\]\
(Y\_[11]{}) + + (Y\_[mm]{}) + &(Y\_[1(m+1)]{}) + + (Y\_[m(2m-1)]{})\
mnd + m(m-1)nd&\[nq4\]\
= m\^2nd&
Equation (\[nq3\]) is true as there exists no source symbol which is not demanded by any terminal. Equation (\[nq4\]) comes from the fact that $\mathtt{g}(Y_{ii}) \leq nd$ and $\mathtt{g}(Y_{ij}) \leq nd$ for $1\leq i\leq m$ and . So $\mathtt{g}(Y_{11}) + \cdots + \mathtt{g}(Y_{mm}) + \mathtt{g}(Y_{1(m+1)}) + \cdots + \mathtt{g}(Y_{m(2m-1)}) = m^2nd$. This implies $\mathtt{g}(Y_{ii}) = \mathtt{g}(Y_{ij}) = nd$.\
We now prove equation (\[claeq\]) when $X_{U_{ij}} = C_{ij}$ for $1\leq j\leq m$.
[l]{} (Y\_[ii]{},C\_[i1]{}) + (Y\_[ii]{},C\_[i2]{}) + + (Y\_[ii]{},C\_[im]{})\
(Y\_[ii]{}, C\_[i1]{}, C\_[i2]{},…,C\_[im]{}) + (m-1)Y\_[ii]{}\
= (C\_[i1]{}, C\_[i2]{},…,C\_[im]{}) + (m-1)Y\_[ii]{}\[hq1\]\
= mnd + (m-1)nd \[hq2\]\
= (2m-1)nd
The inequality in equation (\[hq1\]) is obtained by using the n-way submodularity given in [@rasala]. Equation (\[hq2\]) is true because $Y_{ii}$ can be computed from $C_{i1}, C_{i2},\ldots ,C_{im}$. For all other possible configurations of the sets $X_{U_{i1}},X_{U_{i2}},\ldots ,X_{U_{im}}$ the equations can be proved similarly.
\[cla3\] If $X_{R}\subset \{X_{i1},X_{i2},\ldots,X_{i(mn)}\}$ and $|X_{R}| = n$, then for $1\leq i\leq m$, $\mathtt{g}(Y_{ii},X_{R}) \leq \frac{(2m-1)nd}{m}$.
We show the proof of $\mathtt{g}(Y_{ii},X_{R}) \leq \frac{(2m-1)nd}{m}$ for $X_{R} = \{X_{m1},X_{m2},\ldots,X_{mn}\}$. For other possibilities of $X_{R}$ the claim can be proved similarly. Consider the following $m$ inequalities from equation (\[mq1\]) obtained by taking $X_{P_1} \in \{C_{11},C_{12},\ldots,C_{1m}\}$, and $X_{P_i} = C_{i1}$ for $2\leq i\leq m$.
[l]{} (Y\_[11]{},C\_[11]{}) [+]{} (Y\_[22]{},C\_[21]{}) [+]{} (Y\_[mm]{},C\_[m1]{}) (2m-1)nd\
(Y\_[11]{},C\_[12]{}) [+]{} (Y\_[22]{},C\_[21]{}) [+]{} (Y\_[mm]{},C\_[m1]{}) (2m-1)nd\
\
(Y\_[11]{},C\_[1m]{}) [+]{} (Y\_[22]{},C\_[21]{}) [+]{} [+]{} (Y\_[mm]{},C\_[m1]{}) (2m-1)nd
By summing up these $m$ equations, we get: $$\begin{gathered}
\mathtt{g}(Y_{11},C_{11}) + \mathtt{g}(Y_{11},C_{12}) + \cdots + \mathtt{g}(Y_{11},C_{1m}) + \\m\{\mathtt{g}(Y_{22},C_{21}) + \mathtt{g}(Y_{33},C_{31}) + \cdots + \mathtt{g}(Y_{mm},C_{m1})\}\\ \leq m(2m-1)nd \label{www}\end{gathered}$$ From Claim \[cla2\], we have $\mathtt{g}(Y_{11},C_{11}) + \mathtt{g}(Y_{11},C_{12}) + \cdots + \mathtt{g}(Y_{11},C_{1m}) \geq (2m-1)nd$. Substituting this in equation (\[www\]), we get: $$\begin{gathered}
m\{\mathtt{g}(Y_{22},C_{21}) + \mathtt{g}(Y_{33},C_{31}) + \cdots + \mathtt{g}(Y_{mm},C_{m1}) \} \\ \leq m(2m-1)nd - (2m-1)nd\end{gathered}$$ Similarly, considering the inequalities from equation (\[mq1\]) obtained by taking $X_{P_1} \in \{C_{11},C_{12},\ldots,C_{1m}\}$, $X_{P_2} = C_{22}$, and $X_{P_i} = C_{i1}$ for $3\leq i\leq m$; and then summing the $m$ equations, we can get the following: $$\begin{gathered}
m\{\mathtt{g}(Y_{22},C_{22}) + \mathtt{g}(Y_{33},C_{31}) + \cdots + \mathtt{g}(Y_{mm},C_{m1}) \} \\ \leq m(2m-1)nd - (2m-1)nd\end{gathered}$$ In the same manner it can be seen that for $1\leq i\leq m$, $$\begin{gathered}
m\{\mathtt{g}(Y_{22},C_{2i}) + \mathtt{g}(Y_{33},C_{31}) + \cdots + \mathtt{g}(Y_{mm},C_{m1}) \} \\ \leq m(2m-1)nd - (2m-1)nd \label{mu1}\end{gathered}$$ Note that in these $m$ equations, there is no term involving $\mathtt{g}(Y_{11},X_{P_1})$ for any $X_{P_1}$. Summing the $m$ equations in equation (\[mu1\]) we get: $$\begin{gathered}
m\{\mathtt{g}(Y_{22},C_{21}) + \mathtt{g}(Y_{22},C_{22}) + \cdots + \mathtt{g}(Y_{22},C_{2m}) \} + \\m^2\{\mathtt{g}(Y_{33},C_{31}) + \cdots + \mathtt{g}(Y_{mm},C_{m1}) \} \\ \leq m^2(2m-1)nd - m(2m-1)nd \label{www1}\end{gathered}$$ From equation (\[claeq\]), we have $\mathtt{g}(Y_{22},C_{21}) + \mathtt{g}(Y_{22},C_{22}) + \cdots + \mathtt{g}(Y_{22},C_{2m}) \geq (2m-1)nd$. Substituting this in equation (\[www1\]) we get: $$\begin{gathered}
m^2\{\mathtt{g}(Y_{33},C_{31}) + \cdots + \mathtt{g}(Y_{mm},C_{m1}) \} \\ \leq m^2(2m-1)nd - 2m(2m-1)nd\end{gathered}$$ Note again that the last equation has no terms involving either $\mathtt{g}(Y_{11},X_{P_1})$ or $\mathtt{g}(Y_{22},X_{P_2})$ for any $X_{P_1}$ and $X_{P_2}$. In this way after eliminating all terms involving $\mathtt{g}(Y_{ii},X_{P_i})$ for $1\leq i\leq m-1$, we get the following equation:
[l]{} m\^[m-1]{}(Y\_[mm]{},C\_[m1]{})\
m\^[m-1]{}(2m-1)nd - (m-1)m\^[m-2]{}(2m-1)nd\
(Y\_[mm]{},C\_[m1]{})
Now noting that $X_{R} = \{X_{m1},X_{m2},\ldots,X_{mn}\} = C_{m1}$,
[l]{} (Y\_[mm]{},X\_[R]{})
\[cla4\] For $1\leq i\leq m$, if $X_{V_i} \subset \{ X_{i1},X_{i2},\ldots,X_{i(mn)}\}$ and $|X_{V_i}| = y$, then for $1\leq y \leq n$, $\mathtt{g}(Y_{ii},X_{V_i}) \leq \frac{(nm + ym -y)d}{m}$.
From Claim (\[cla3\]) it can be seen that the result is true for $y=n$. We show that if the result is true for an arbitrary value of $y = z$ $(2\leq z\leq n)$, it is also true for $y = z-1$. The proof for the case when $X_{V_i} = \{X_{i1},X_{i2},\ldots,X_{i(z-1)} \}$ is given below assuming that for $\forall y\geq z$, $\mathtt{g}(Y_{ii},X_{V_i}) \leq \frac{(nm + ym -y)d}{m}$ when $|X_{V_i}| = y$. For the rest of the possibilities of $X_{V_i}$ when $|X_{V_i}| = z-1$, the proof is similar.
[l]{} (Y\_[ii]{},X\_[V\_i]{},X\_[iz]{}) + (Y\_[ii]{},X\_[V\_i]{},X\_[i(z+1)]{}) + +\
(Y\_[ii]{},X\_[V\_i]{},X\_[i(mn)]{}) (Y\_[ii]{},X\_[V\_i]{},X\_[iz]{},X\_[i(z+1)]{},…,X\_[i(mn)]{})\
+ (mn-z) (Y\_[ii]{},X\_[V\_i]{})\[a11\]\
(mn[-]{}z[+]{}1) mnd [+]{} (mn[-]{}z) (Y\_[ii]{},X\_[V\_i]{})\
(mn[-]{}z) + - mnd\
(mn[-]{}z) (Y\_[ii]{},X\_[V\_i]{})\
(mn[-]{}z) +\
- mnd (mn[-]{}z) (Y\_[ii]{},X\_[V\_i]{})\
(mn[-]{}z)\
- (mn[-]{}z) (Y\_[ii]{},X\_[V\_i]{})\
(mn[-]{}z) + (mn[-]{}z)\
- (mn[-]{}z) (Y\_[ii]{},X\_[V\_i]{})\
(mn[-]{}z) (mn[-]{}z) (Y\_[ii]{},X\_[V\_i]{})\
(Y\_[ii]{},X\_[V\_i]{})
Equation (\[a11\]) comes from the n-way submodularity formula given in [@rasala].
For $y=1$ in Claim (\[cla4\]), we get: $\mathtt{g}(Y_{ii},X_{ij}) \leq \frac{(nm + m - 1)d}{m}$ for $1\leq i\leq m$, $1\leq j\leq mq$.\
Now using the n-way submodularity again,
[l]{} (Y\_[ii]{},X\_[i1]{}) + (Y\_[ii]{},X\_[i2]{}) + + (Y\_[ii]{},X\_[i(mn)]{})\
(Y\_[ii]{},X\_[i1]{},X\_[i2]{},…,X\_[i(mn)]{}) + (mn-1)(Y\_[ii]{})\
(Y\_[ii]{},X\_[i1]{}) + (Y\_[ii]{},X\_[i2]{}) + + (Y\_[ii]{},X\_[i(mn)]{})\
mnd + (mn-1)nd = (nm + m -1)nd \[a22\]
Since there are $mn$ terms in the right hand side of equation (\[a22\]) and each term is less than or equal to $\frac{(nm + m - 1)d}{m}$, it must be that $\mathtt{g}(Y_{ii},X_{ij}) = \frac{(nm + m - 1)d}{m}$. Now note that $gcd(nm+m-1,m) = gcd(-1,m) = gcd(m-1,m) = gcd(m,1) = 1$. Since by definition, the rank function of a discrete polymatroid $\mathtt{g}(\,)$ is always an integer, $m$ must divide $d$ for $\mathtt{g}(Y_{ii},X_{ij})$ to be an integer.
We now show that the network $\mathcal{N}$ has an $(m,mn)$ fractional linear network code solution over any finite field by presenting an $(m,mn)$ routing solution. Note that an $(m,mn)$ routing solution is a special case of an $(m,mn)$ fractional linear network code solution. For any vector $X_{ij}$, $1\leq i\leq m$, $1\leq j\leq mn$, the first component is carried by $e_{ii}$, and the $p^{th}$ component for $2\leq p\leq m$ is carried by $e_{i(m-1+p)}$. Now the demands of all the terminals can be met by routing the appropriate symbols from the node $v_{i}$ for $1\leq i\leq 2m-1$. For example, the demands of the terminal $t_1$ gets fulfilled upon receiving the appropriate $n$ symbols of all vectors $Y_{ii}$ and $Y_{ij}$ for $1\leq i\leq m$, $m+1\leq j\leq 2m-1$.
Conclusion
==========
For non-multicast networks, it was shown that there exists networks which has an $(m,m)$ $(m\geq 2)$ vector linear network code solution, but has no $(w,w)$ vector linear network code solution for $w<m$. In this paper, we have generalized this result to show that for any positive integers $k$ and $n$, there exists a network which has no $(wk,wn)$ fractional linear network code solution for any $w<m$, but has an $(mk,mn)$ $(m\geq 2)$ fractional linear network code solution.
[18]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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|
---
address:
- '$^{1}$ Department of Physics, DDU College, University of Delhi, New Delhi, India '
- '$^{2}$Department of Physics, Birla Institute of Technology and Science, Pilani, Hyderabad Campus, Hyderabad - 500078, India'
author:
- 'Vijay Bhatt$^{1}$ Pradeep K. Jha$^{1}$ and Aranya B. Bhattacherjee$^{2}$'
title: 'Effect of second-order nonlinearity on quantum coherent oscillations in a quantum dot embedded in a doubly resonant-semiconductor micro-cavity'
---
Introduction
============
The recent developments in quantum technology physics have shown tremendous progress in the storage,processing and transfer of quantum information using quantum bits (Qubits) [@bou; @mon; @nie]. Quantum coherence which is a necessary requirement for realistic quantum communication system is extremely fragile and can be destroyed by interaction with the environment. Semiconductor quantum dots (QDs) embedded in micro-cavity have recently emerged as an attractive candidate for the implementation of quantum computing platforms [@pelli; @loss; @imam; @biola; @miran]. Instead of the usual two-level real atoms, excitons in the QDs are considered as an alternative two level systems characterized by strong exciton-phonon interactions [@hamea; @heitz; @turck; @beso]. For practical implementation of quantum information processing based on QDs, it is important to minimize the influence of lattice vibrations which tends to destroy their coherence. Thus it is important to take into account exciton-phonon interactions in the study of quantum-dot cavity system. Experimental observation of vacuum Rabi Oscillations in atomic [@brune] as well as in solid state systems [@reith; @khitro; @henn] provides evidence for strong coupling regime in micro cavity systems. Thus QDs embedded in semiconductor micro-cavity have emerged as an exciting platform to study cavity QED [@yama; @li; @wang].
Recently proposals have been put forward to use nano structured photonic nanocavities made of $\chi^{(2)}$ nonlinear materials as prospectives devices for application in quantum information processing, quantum logic gates and all optical switches [@arka; @Fryett].One of the main aims of working with such systems is to have a scalable integrated quantum photonic technology with the probability to work at telecommunication wavelengths. In this paper, we seek to theoretically study the quantum oscillations in a coherently driven quantum dot-cavity system in the presence of a $\chi^{(2)}$ nonlinear substrate and strong exciton-phonon interactions.
Theoretical Model
=================
The system considered here consist of an optical semiconductor microcavity supporting two field modes through nonlinear interaction $g_{nl}$. This nonlinear interaction is provided by the $\chi^{(2)}$ nonlinear substrate introduced into the microcavity as shown in Fig.1. In addition, a quantum dot (QD) is also embedded in the system which interacts with both the optical modes. Due to the nonlinear interaction process, one of the optical mode has two photons at the fundamental frequency while the second mode has a single photon at the second harmonic frequency. In order to ensure phase matching between the two modes, a non-zero spatial overlap between the cavity modes exists [@boyd; @rivoire]. The semiconductor microcavity considered here can be fabricated using distributed Bragg reflectors(DBR). The light field that is pumped into the cavity is confined in the x-direction by the DBR while the air guiding dielectric provides confinement in the y-z plane [@ali; @gudat; @bhatt]. A $\chi^{(2)}$ nonlinear substrate can be deposited on the GaAs cavity according to known experimental techniques [@arka].
\
The embedded semiconductor QD is assumed to be a simple two-level system which consists of the electronic ground state $ \lvert 1\rangle $ and the lowest-energy electron-hole (exciton) state $\lvert 1\rangle $. We consider that this QD via the exciton interacts with both the fundamental mode of frequency $\omega_{a}$ and second harmonic mode of frequency $\omega_{b}= 2 \omega_{a}$. The two level QD can be characterized by the pseudo spin-1/2 operators $\sigma_{\pm}$,$\sigma_{z}$ and the fundamental and second harmonic modes are characterized by the annihilation and creation operators $a (b)$ and $a^{\dagger}$ $(b^{\dagger})$ respectively with Bose commutation relation \[$a$,$a^{\dagger}$\]=1 (\[$b$,$b^{\dagger}$\]=1). In this model, we also consider strong coupling of exciton with bulk acoustic phonons. The total Hamiltonian for this coupled photon-exciton-phonon system in the dipole and rotating wave approximation is written as,
$$\begin{aligned}
H &=& \hbar\omega_{a}a^{\dagger}a + \hbar\omega_{b}b^{\dagger}b +\hbar\omega_{ex}\sigma_{z}+ \hbar g_{a}(\sigma_{-}a^{\dagger} +\sigma_{+}a) + \hbar g_{b}(\sigma_{-}b^{\dagger} + \sigma_{+}b) \nonumber \\
&+&\hbar g_{nl}\left(b (a^{\dagger})^2 + b^{\dagger}(a)^2\right) + \hbar\sum_{q}{\omega_q(b_q^{\dagger}b_q)} + \hbar \sigma_{z}\sum_{q}{M_{q}(b_q^{\dagger} + b_{q})}
\end{aligned}$$
The first two terms denote the energy of the fundamental and second harmonic mode respectively. The third term is the exciton energy with $\omega_{ex}$ as the exciton frequency. The fourth and fifth terms are the exciton-fundamental mode photon and exciton - second harmonic photon interaction with coupling constants $g_{a}$ and $g_{b}$ respectively. The sixth term is the nonlinear interaction between the fundamental and second harmonic mode with coupling constant $g_{nl}$ which can be expressed as
$$\hbar g_{nl}=\epsilon_{0}\left( \frac{\hbar\omega_{a}}{\epsilon_{0}\epsilon_{r}}\right) ^{3/2} \frac{\chi^{(2)}}{\sqrt{V_{r}}}$$
where, $\frac{1}{\sqrt{V_{r}}}$=$\int_{NL}\alpha(r)^{3} dr.$ Note that, the phase matching condition is implicit in the assumption of perfect overlap between the cavity modes. A perfect overlap between the cavity modes ($\alpha_{a}(r)=\alpha_{b}(r)=\alpha(r)$), $\alpha_{a}(r)$ and $\alpha_{b}(r)$ are the normalized field profiles of the cavity modes such that integration over the whole volume is unity.i.e; $\int\left|\alpha(r)\right|^{2}$ dr =1
The seventh term denotes the energy of the $q^{th}$ phonon mode having frequency $\omega_{q}$ and creation (annihilation) operator $b_q^{\dagger}(b_q)$. The last term is the exciton-phonon interaction characterized by the matrix element $M_{q}$. For the sake of simplicity, when the temperature is low($T < 50K$) [@beso; @wilson] we assume that the off-diagonal exciton-phonon interactions are negligible. For an InGaAs quantum dot, the energy separation is about 65MeV from the ground state transition.
Now first we apply canonical transformation to the Hamiltonian (1) [@mahan]
$$H^{'}= \exp{(S)} H \exp{(-S)},$$
where the generator is - $$S=\left( S_{z}+\frac{1}{2}\right )\sum_{q}\frac{M_q}{w_q}(b_q^{\dagger} + b_q) .$$
The transformed Hamiltonian is given by-
$$H^{'}=H_{0}^{'}+ H_{I}^{'} ,$$
where; $$\begin{aligned}
H_{0}^{'}&=&\omega_{a}a^{\dagger}a + \omega_{b}b^{\dagger}b + (\omega_{ex}-\Delta)\left(S_{z}+\frac{1}{2}\right)+ \sum_{q}\omega_{q}b_q^{\dagger}b_q ,\end{aligned}$$
$$H_{I}^{'}=g_a[\sigma_{+}aX^{\dagger}+\sigma{-}a^{\dagger}X] + g_b[\sigma_{+}bX^{\dagger}+\sigma{-}b^{\dagger}X] + g_{nl}[b(a^{\dagger})^2+b^{\dagger}(a)^2] ,$$
where $$\Delta=\sum_{q}\frac{M_{q}^2}{w_q} ,$$
$$X=exp\left[-\sum_{q}\frac{M_q}{w_q}(b_q^{\dagger}-b_q)\right] ,$$
$$X^{\dagger}=X^{-1} .$$
We will now work in the interaction picture with $H_{0}^{'}.$ The Hamiltonian in the interaction picture is evaluated as,
It is given by, $$H_{int}=e^{iH_{0}^{'}t}H_{I}^{'}e^{-iH_{0}^{'}t}$$
Using $$exp\left[ i\sum_{q}\omega_{q}b_q^{\dagger}b_qt\right] X exp\left[ -i\sum_{q}\omega_{q}b_q^{\dagger}b_qt\right] =exp\left[ -i\sum_{q}\frac{M_q}{w_q}(b_q^{\dagger}e^{i\omega_qt}-b_qe^{-iw_qt})\right] ,$$
we have $$\begin{aligned}
H_{int}&=&g_{a}\left[ \sigma_{+} a X^{\dagger}(t)e^{i\delta_a t} + \sigma_{-} a^{\dagger} X(t)e^{-i\delta_a t}\right] + g_{b}\left[ \sigma_{+} b X^{\dagger}(t)e^{i\delta_b t} + \sigma_{-} b^{\dagger} X(t)e^{-i\delta_b t}\right] \nonumber \\
&+&g_{nl}\left[ b(a^\dagger)^2 + b^{\dagger}(a)^2\right] \end{aligned}$$
Where; $\delta_a =\omega_{ex}-\omega_{a}-\Delta$ , $\delta_b =\omega_{ex}-2\omega_{a}-\Delta$
And $X(t) = \exp\left[-\sum_{q}\frac{M_{q}}{\omega_{q}}\left( b_{q}^{\dagger}e^{i \omega_{q}t} - b_{q}e^{-i \omega_{q}t}\right) \right]$
Now we proceed to solve the equation of motion for $\arrowvert \varPsi(t) \rangle$ ; i.e.
$i\frac{d \arrowvert \varPsi(t)\rangle}{dt}= H_{int}\arrowvert \varPsi(t) \rangle$
In general , the State vector $\arrowvert \varPsi(t)\rangle$ is a linear combination of states $\arrowvert 1, m, n \rangle$ $\arrowvert ph \rangle$ and $\arrowvert 2, m, n \rangle$ $\arrowvert ph \rangle$. Here $\arrowvert 2, m, n \rangle$ is the state in which the Quantum dot is in excited state. i.e. $\arrowvert 1, m, n \rangle$ is ground state. In the excited state Exciton are present.
As we are using the Interaction Picture, we use the slowly varying amplitude $C_{1, m ,n ,ph(t)}$ and $C_{2,m,n, ph(t)}$. The State vector is therefore:-
$$\arrowvert \varPsi(t)\rangle =\sum_{m,n}\left[ C_{1, m ,n ,ph(t)} \arrowvert 1, m, n \rangle \arrowvert ph \rangle + C_{2, m ,n ,ph(t)} \arrowvert 2, m, n \rangle \arrowvert ph \rangle\right]$$
( Note that, ’a’ operator will act on state n and ’b’ operator on m )
The Interaction Hamiltonian (13) Can cause transitions between the states as follows-
$\arrowvert 1, m+1, n \rangle \leftrightarrow \arrowvert 2, m, n \rangle$
$\arrowvert 1, m, n+1 \rangle \leftrightarrow \arrowvert 2, m, n \rangle$
$\arrowvert 1, m, n \rangle \leftrightarrow \arrowvert 1, m+2, n-1 \rangle$
$\arrowvert 1, m-2, n+1 \rangle \leftrightarrow \arrowvert 1, m, n \rangle$
$\arrowvert 2, m, n \rangle \leftrightarrow \arrowvert 2, m+2, n-1 \rangle$
$\arrowvert 2, m, n \rangle \leftrightarrow \arrowvert 2, m-2, n+1 \rangle$
\
We can write the Hamiltonian as-
$$H\arrowvert \varPsi \rangle = H_{a}\arrowvert \varPsi \rangle + H_{b}\arrowvert \varPsi \rangle + H_{nl}\arrowvert \varPsi \rangle$$
The equations of motion for the Probability Amplitudes are obtained by first substituting for $\arrowvert \varPsi(t) \rangle$ and $H_{eff}$ from equation (18) and equation (16) in equation (17),
We obtain the following linear Coupled Equations-
$$i\dot{C}_{2,m+2,n}(t)=g_{nl}C_{2,m,n+1}(t)\sqrt{n+1}\sqrt{m+1}\sqrt{m+2}$$
$$i\dot{C}_{2,m,n+1}(t)=g_{nl}C_{2,m+2,n}(t)\sqrt{n+1}\sqrt{m+1}\sqrt{m+2}$$
$$i\dot{C}_{1,m,n+1}(t)=g_{b} e^{-\frac{\lambda}{2}} \sqrt{n+1}e^{-i \delta_b t}C_{2,m,n}(t) + g_{nl}C_{1,m+2,n}(t)\sqrt{n+1}\sqrt{m+1}\sqrt{m+2}$$
$$i\dot{C}_{2,m,n}(t)=g_{a} e^{-\frac{\lambda}{2}} \sqrt{m+1}e^{i \delta_a t}C_{1,m+1,n}(t) + g_{b}C_{1,m,n+1}(t)\sqrt{n+1}e^{-\frac{\lambda}{2}}e^{i \delta_a t}$$
$$i\dot{C}_{1,m+1,n}(t)=g_{a} \sqrt{m+1}e^{-\frac{\lambda}{2}}e^{-i \delta_at}C_{2,m,n}(t)$$
$$i\dot{C}_{1,m+2,n}(t)=g_{nl}C_{1,m,n+1}(t)\sqrt{n+1}\sqrt{m+1}\sqrt{m+2}$$
Where, $\lambda = \sum_{q}( \frac{M_{q}}{\omega_{q}} )^2$ is the Huang-Rhys factor which corresponds to the exciton-phonon interactions. It can be determine by the experiment. [@zhu]
Results
=======
The coupled set of equations (20)-(25) above can be solved exactly subject to certain initial conditions. Initially the quantum dot is in the excited state $\arrowvert 2 \rangle$. (i.e, with the presence of the exciton). We have presented here result, after separating these equation into real and imaginary parts (See Appendix-A) and showing the results for Probability amplitude for $C_{2,m,n}$ (i.e excited state) with respect to time.
\
In figure-(3,4,5) we are using parameters $g_{nl}$ , $\delta_{a}$ , $\delta_{b}$ and $\lambda$ , where $g_{nl}$ is nonlinear coupling factor, $\delta_{a}$ is detuning for cavity mode ’a’ , $\delta_{b}$ is detuning for cavity mode ’b’. and $\lambda$ is Huang-Rhys factor.
From figure-(3), we can see that the system is undergoing Quantum Rabi Oscillations. In figure (4), if we change the value of detuning for cavity mode ’a’ (i.e $\delta_{a}$) to 0.2 and rest of parameters remain same . We see that Probability of finding the particle in excited state is decreasing by multiple of factor 2.
\
Now for figure (5), when we change the value of Nonlinear coupling factor $g_{nl}$ to 0.5 , We see that a periodic oscillating wave of larger amplitude is generating after two wave of lesser amplitude. It means for a couple of time, the probability of finding the particle in excited state is half and exist between 0.3 and 0.5 in the graph. Time duration is increasing for photons to exist in excited state.
\
It should be noted that the couplings of exciton and photons will not become too large so that the Q.Dot cavity system is not in the strong - coupling regime. The value of Huang-Rhys factor For CdSe quantum dots is $\lambda$ = 1 [@turck], For InAs/GaAs quantum dots $\lambda$ = 0.015 [@heitz] and for other semiconductor quantum dots such as GaAs [@arka2] and for InGaAs [@huang], $\lambda$ is even more small.
Conclusion
==========
In conclusion, applying Quantum treatment we have discussed the influence of strong exciton-phonon interaction on the quantum Rabi o scillations in a coherently driven quantum dot in a high-Q double -mode cavity in the presence of nonlinear substance. It is found that the coherent oscillations dressed by quantum lattice fluctuations can persist with the coupling constant $g_{a}e^{-\frac{\lambda}{2}}$ and $g_{b}e^{-\frac{\lambda}{2}}$ for two cavity modes. We have investigated that nonlinear substrate affect the Rabi oscillations. When we decrease the nonlinear coupling factor, we see that the number of photons in the higher quantum state exist for a longer time. In this way nonlinear coupling factor increasing the life time of photons to remain in exited state. Our result also indicate that even at the zero temperature, the strong exciton-phonon interactions still affect the quantum coherent oscillation. The interaction of exciton and phonon useful in many applications such as indistinguishable photon generation. It is shown by the result that even at the Zero temperature, the strong exciton- phonon interactions still affect the quantum coherent oscillation significantly and nonlinear coupling factor affecting the oscillation.
Thus we proposed a new way of quantum lattice fluctuation on quantum coherence oscillation in the presence of nonlinear coupling factor for Quantum dot-cavity system. This will make new plateform for more theoritical and practical applications in QED.
**P.K Jha** is thankful to **SERB-Department of Science and Technology, New Delhi** for the financial support. **Aranya B. Bhattacherjee** is grateful to **BITS Pilani, Hyderabad campus** for the facilities to carry out this research.
[99]{}
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****\
For separating equation (20)-(25) we let -
$C_{2,m+2,n}= a_{1} + i a_{2}$ , $C_{2,m,n+1}= b_{1} + i b_{2}$ , $C_{1,m,n+1}= c_{1} + i c_{2}$ , $C_{2,m,n}= d_{1} + i d_{2}$ ,
$C_{1,m+2,n}= e_{1} + i e_{2}$, $C_{1,m+1,n}= f_{1} + i f_{2}$
And we get the following equations after separating real and imaginary parts and comparing them.
$$\tag{A1}
\dot{a_1}= \sqrt{n+1}\sqrt{m+1}\sqrt{m+2} g_{nl} b_{2},$$
$$\tag{A2}
\dot{a_2}= -\sqrt{n+1}\sqrt{m+1}\sqrt{m+2} g_{nl} b_{1},$$
$$\tag{A3}
\dot{b_1}= \sqrt{n+1}\sqrt{m+1}\sqrt{m+2} g_{nl} a_{2},$$
$$\tag{A4}
\dot{b_2}= -\sqrt{n+1}\sqrt{m+1}\sqrt{m+2} g_{nl} a_{1},$$
$$\tag{A5}
\dot{c_1}=\sqrt{n+1}\cos{\delta_{b}t} g_{b} e^{-\lambda/2}d_{2} - \sqrt{n+1}\sin{\delta_{b}t} g_{b} e^{-\lambda/2}d_{1} + g_{nl} A e_{2},$$
$$\tag{A6}
\dot{c_2}= -\sqrt{n+1}\cos{\delta_{b}t} g_{b} e^{-\lambda/2}d_{1} - \sqrt{n+1}\sin{\delta_{b}t} g_{b} e^{-\lambda/2}d_{2} - g_{nl} A e_{1},$$
$$\begin{aligned}
&\;\dot{d_1}= \sqrt{m+1}\cos{\delta_{a}t} g_{a} e^{-\lambda/2}f_{2} + \sqrt{m+1}\sin{\delta_{a}t} g_{a} e^{-\lambda/2}f_{1}\nonumber\\
+&\;\sqrt{n+1}\cos{\delta_{b}t} g_{b} e^{-\lambda/2}c_{2} + \sqrt{n+1}\sin{\delta_{b}t} g_{b} e^{-\lambda/2}c_{1},\tag{A7}
\end{aligned}$$
$$\begin{aligned}
&\;\dot{d_2}= -\sqrt{m+1}\cos{\delta_{a}t} g_{a} e^{-\lambda/2}f_{1} + \sqrt{m+1}\sin{\delta_{a}t} g_{a} e^{-\lambda/2}f_{2}\nonumber\\
-&\;\sqrt{n+1}\cos{\delta_{b}t} g_{b} e^{-\lambda/2}c_{1} + \sqrt{n+1}\sin{\delta_{b}t} g_{b} e^{-\lambda/2}c_{2},\tag{A8}
\end{aligned}$$
$$\tag{A9}
\dot{e_1}=\sqrt{n+1}\sqrt{m+1}\sqrt{m+2} g_{nl} c_{2},$$
$$\tag{A10}
\dot{e_2}= -\sqrt{n+1}\sqrt{m+1}\sqrt{m+2} g_{nl} c_{1},$$
$$\tag{A11}
\dot{f_1}= -\sqrt{m+1}\sin{\delta_{a}t} g_{a} e^{-\lambda/2}d_{1} + \sqrt{m+1}\cos{\delta_{a}t} g_{a} e^{-\lambda/2}d_{2},$$
$$\tag{A12}
\dot{f_2}= -\sqrt{m+1}\cos{\delta_{a}t} g_{a} e^{-\lambda/2}d_{1} - \sqrt{m+1}\sin{\delta_{a}t} g_{a} e^{-\lambda/2}d_{2}.$$
For finding the Probability in excited state $\arrowvert 2, m, n \rangle$ $\arrowvert ph \rangle$ we calculate $d_{1}^2 + d_{2}^2$.
|
---
abstract: 'The search for extraterrestrial communication has mainly focused on microwave photons since the 1950s. We compare other high speed information carriers to photons, such as electrons, protons, and neutrinos, gravitational waves, inscribed matter, and artificial megastructures such as occulters. The performance card includes the speed of exchange, information per energy and machine sizes, lensing performance, cost, and complexity. In fast point-to-point communications, photons are superior to other carriers by orders of magnitude. Sending probes with inscribed matter requires less energy, but has higher latency. For isotropic beacons with low data rates, our current technological level is insufficient to determine the best choice. We discuss cases where our initial assumptions do not apply, and describe the required properties of hypothetical particles to win over photons.'
address: 'Sonneberg Observatory, Sternwartestr. 32, 96515 Sonneberg, Germany'
author:
- Michael Hippke
bibliography:
- 'references\_elsevier.bib'
title: Benchmarking information carriers
---
Introduction
============
Information carriers for interstellar communication can be electromagnetic waves (photons), other particles such as neutrinos or protons, gravitational waves and inscribed matter (probes). The search for extraterrestrial intelligence (SETI), which is in reality mainly a search for communications, focuses heavily on photons, particularly at microwave and optical frequencies . For photons, the search space is 9-dimensional: three of space, two of each polarization, intensity, modulation, frequency, and time. For other carriers, the search space has not been described in much detail yet. For example, Neutrinos come without polarization, but as particle/antiparticle and have three flavors (electron, muon, tau). A consolidated review of advantages and disadvantages for all carriers is missing in the literature accessible to us. In this paper, we will review all known possibilities for communication and benchmark them against photons. The academic literature indicates a rising and significant dissatisfaction with the orthodox SETI approach, which focuses almost exclusively on radio observations [@2011JBIS...64..156B; @2014ApJ...792...27W; @2014ApJ...792...26W]. Therefore, it is important to consider all alternative communication methods.
While we do not know the motivations and goals of other civilizations, if they exist, we here assume the universal value of exchanging more information rather than less. Other preferences which appear natural to us are the speed of exchange (a preference to obtain information earlier than later), the amount of energy required (obtaining as much information as possible for a finite amount of resources), and the ease of use (although this is difficult to measure). These aspects were already identified in the Cyclops Report, often dubbed the “SETI bible” [@1971asee.nasa.....O]. We will relax these constraints in section \[irrational\_aliens\].
In this work, we are largely agnostic to ever-changing technological cost functions, and focus on the physical optimum. We systematically compare all known communication methods with regards to their energy efficiency and data rates over pc to kpc distances. We benchmark them against photons, which have a capacity (in units of bits per energy) of [@2017arXiv171105761H]
$$\label{eq_photon}
C_{\gamma} =
\left(\frac{d}{1\,{\rm pc}} \right)^{-2}
\left(\frac{\lambda}{1\,{\rm nm}} \right)^{-1}
\left(\frac{D}{1\,{\rm m}} \right)^{4}
\,\,(\rm {bits\,J^{-1}})$$
where $d$ is the distance, $\lambda$ is the photon wavelength, $D=D_{\rm t}=D_{\rm r}$ are equal-sized transmitter and receiver apertures, and a conservative information efficiency of one bit per photon is assumed [@2017arXiv171205682H; @2018arXiv180106218H]. This estimate is valid to within 10% neglecting losses and noise.
Artificial occulters and other physical objects are treated in section \[occulter\], followed by charged (section \[charged\_particles\]), short-lifetime (section \[short\_particles\]) and massive (section \[massive\_particles\]) particles; gravitational waves (section \[gravitational\_waves\]) and neutrinos (section \[neutrinos\]).
Artificial occulters, inscribed matter, and other physical objects {#occulter}
==================================================================
For an information exchange, one can distinguish between scenarios where the transmitter controls only the energy output (category I, e.g., modulating the energy output of a Cepheid variable star), the directivity (category II, e.g., an occulting screen in front of a star), or both (category III, e.g., a laser).
Artificial occulters {#arti}
--------------------
Exotic examples for category II communications are artificial megastructures occulting a star, such as the geometrical objects described by @2005ApJ...627..534A. The advantage is that the energy source is already available, broadband, and powerful. Directivity is defined by the orbits of the structures, and limited by the fixed size of the star, which produces a wide (few degree for planet-sized structures at au distances) cone of visibility targeting $\mathcal{O}(10^6)$ stars within a few hundred pc. We may assume that curious civilizations would monitor nearby stars for (natural) exoplanet transits, and thus detect the non-spherical transiting structures. The obvious disadvantages are the need to build space-based megastructures with sizes $10^3 \dots 10^5$km, and the low information content. Encoding schemes could use transit timing and -duration variations and shape modifications, but are limited to a few bits per transit by the small number of these modes and stellar noise.
Megaengineering constructions may be intended for many purposes, such as shielding swarms [@2016AcAau.129..438C], and their visibility as communication beacons might be, deliberate or not, a side effect.
A modification of the transit idea, falling into category I, is the use of artificial light sent from Earth during a “planet Earth transit” which modifies the observed spectral transit signature of our planet [@2016MNRAS.459.1233K].
Other large structures described in the literature are (partial) @1960Sci...131.1667D spheres or some of their modifications such as @1987brig.iafcR....S thrusters, spherical arc mirrors to use the impulse from a star’s radiation pressure [@2013JBIS...66..144F], or starshades to reduce the irradiation from evolving stars [@2017MNRAS.469.4455G]. Such objects are not primarily built for communication and thus neglected here.
Similarly, other bright natural sources might be used as beacons, such as pulsars [@2003IJAsB...2..231E] or supernovae . Very advanced civilizations may modify the pulsation cycles of Cepheid variable stars as beacons to transmit all-call information throughout the galaxy and beyond [category I, @2012ConPh..53..113L; @2015ApJ...798...42H].
Some sources might appear as laser or microwave communication (category III), but could in fact be artificial directed energy to push light sails; these would appear as transient events with flux densities of Jy and durations of tens of seconds at 100 pc [@2015ApJ...811L..20G; @2017ApJ...837L..23L].
Inscribed matter
----------------
Sending a physical artifact can be the most energy-efficient choice [@2004Natur.431...47R], because it can be done at almost arbitrarily low velocities, and thus low energies. Also, an artifact can arrive at the destination in total, in contrast to a beam which is wider than the receiver in all realistic cases, so that most energy is lost. The obvious disadvantage is the large communication delay, e.g. sending a probe at a (relatively fast) $0.01\,c$ takes 438 years to the nearest star. We argue that inscribed matter on board of exploration probes could make sense, e.g. holding a galactic library of knowledge to be discovered by others, including a tutorial for communication. Then, the first contact would ease the need for fast communication, as reading the library could keep the impatient Aliens busy while waiting for the first “answer” from the stars. An in-depth analysis is given in @2017arXiv171210262H, which finds the capacity of the inscribed matter channel as
$$\label{eq_matter}
C_{\rm matter} = \eta\, S\, L^{-1}\, v^{-2} \,\,(\rm {bits\,J^{-1}}).$$
where $\eta$ is the propulsion efficiency factor for launch and deceleration, $S_{\rm base}$ is the information density per unit mass, and $L$ is the relativistic Lorentz factor. For $S_{\rm base}=10^{23}$bitsg$^{-1}$ and $v=0.1\,c$, we have an energy efficiency of $\approx10^{11}$bits per Joule. A photon channel requires large apertures to match this efficiency due to diffraction losses. For example, $d=1.3\,$pc requires $D = 1\,$km apertures at X-ray energy (1000km at $\lambda=1\,\mu$m). For a constant wavelength, apertures must increase with distance as $D \propto d^{1/2}$, making probes more attractive for large amounts of data over large distances. For low probe speeds ($v \ll 0.01\,$c), equivalent aperture sizes are implausibly high. Inscribed matter is more energy efficient by many orders of magnitude in this regime of slow probes.
Apart from a communication link between two distant species, an inscribed matter artefact can be placed in a stellar system, to be found at a later date [@2012AcAau..78..121C]. This scenario was popularized by Arthur C. Clarke’s moon monolith [@Clarke1953]. Searches for artefacts inside our solar system have been suggested [@1960Natur.186..670B; @1995ASPC...74..425P; @2004IAUS..213..487T; @2012AcAau..72...15H], as well as for starships [@Martin1980]. Possible locations include in geocentric, selenocentric, Earth-Moon libration, Earth-Moon halo orbits [@1980Icar...42..442F; @1983Icar...53..453V; @1983Icar...55..337F], the Kuiper belt [@2012AsBio..12..290L], and even “footprints of alien technology on Earth” [@2012AcAau..73..250D].
Assessment
----------
Leveraging available power sources as all-sky beacons appears attractive, but directivity requires building megastructures (e.g., a large Fresnel lens, occulters etc.). Such is physically possible, but the energy needed in the build process might be better spent in actual communication, e.g. using lasers. Structures such as small (planet-sized) occulters can not leverage the isotropic radiation; they only harvest a small fraction of the starlight into modulated beacons.
Throughout the paper we have made the assumption that the main purpose of a machine is communication, and all related costs (Table \[table1\]) are attributed to it. This may be false, as many buildings on Earth, such as skyscapers, serve double duty as office buildings and antenna carriers. Determining the actual communication costs, without the commercial part, is not trivial, and will be relevant for future SETI efforts with maturing technology [@Harrison2009].
On grand scale, even the cosmic microwave background might provide an opportunity to send a message to all occupants in the universe [@2006MPLA...21.1495H], although the information content might be limited to less than 1000 (unchangeable) bits [@2005physics..11135S].
We concede that it is crucial to keep an open mind for unknown phenomena [@2017arXiv170805318W], perhaps serving as beacons [@2016ApJ...816...17W]. However we can not see an attractive, yet superior, communication option in the schemes described in the literature when it comes to targeted communication with high data rates.
[lcccccccccc]{}
\
Carrier & MeV/c$^2$ & Lifetime (s) & Velocity ($c$) & Charged & & Extinction & Lensing & Difficulty & &\
Photon & 0 & stable & 1 & no & $10^{-4}$ & 0.001 & good & low & medium & medium\
Neutrino & $\approx 0.001$ & oscillations & $\lessapprox 1$ & no & $\textcolor{red}{10^6}$ & $\approx 0$ & very good & medium & & medium\
Electron & 0.51 & $>10^{36}$ & & & & medium & medium& & medium & medium\
Proton & 938.27 & $>10^{51}$ & & & & & very good & low & medium & medium\
Neutron & 939.56 & & & no & & & very good & low & medium & medium\
Muon & 105.66 & & & & & & very good & low& & medium\
Tau & 1776.82 & & & & & & very good & low& & medium\
Higgs & $1.25\times10^5$ & & & no & & $\approx 0$ & very good & & &\
Inscribed matter & $>0$ & $>10^{13}$ & $0 \dots 1$ & no & $\approx 0$ & $\approx 0$ & no & low& & zero\
Occulter & $>0$ & $>10^{13}$ & 1 & possible & & 0.001 & unlikely & low & & very low\
Gravitational wave & 0 & stable & 1 & no & & 0 & low &&&\
Axion & $\approx10^{-9}$ & stable & $\lessapprox 1$ & no & $10^{-4}$ & $\approx$ 0 & good & & unclear & unclear\
Tachyon & unclear & stable & $>1$ & no & unclear & unclear & unclear & & unclear & unclear\
All values for $E={\rm keV}$, transmitter aperture 1m, distance 1pc. Masses and lifetimes from @Mohr2006. Text in indicates problematic properties.
Charged and massive particles
=============================
This section covers all particles except neutrinos and photons.
Charged particles {#charged_particles}
-----------------
Particles with an electric charge, such as an electron (positron), proton, muon or tau, are deflected by magnetic fields and absorbed by the interstellar medium. It is expected that these particles lose any correlation with their original direction as they traverse through interstellar magnetic fields. In reality, small-scale anisotropies have been observed [@2017arXiv170803005T] which are believed to originate from structures in the heliomagnetic field, turbulence in galactic magnetic fields, and non-diffusive propagation [@2017PrPNP..94..184A]. Despite these anisotropies, it is highly unlikely that even perfect knowledge of the interstellar magnetic fields would allow for communication channels that are stable over useful timescales. Thus, we argue that charged particles are not usable for interstellar communication based on known physics.
Particles with short lifetimes {#short_particles}
------------------------------
Some particles like the muon or tau decay on very short timescales and are thus unsuitable. For example, muons with a lifetime of $10^{-6}$s (Table \[table1\]) decay within seconds even for high energies (PeV, Lorentz factor $\gamma \approx 10^6$). For a limited range to the nearest stars, a particle lifetime of a few years is required. For uncharged particles, this is only possible with Neutrons, as their half-life at $\mathcal{O}(100)$TeV, $\gamma \approx 10^5$ is $\approx 2.8$yrs. Such neutrons are easy to detect using atmospheric Cherenkov detectors [@2006ApJ...636..777A] and have low cosmic ray background (i.e. noise). Using muons (taus) for interstellar communication with year-long travel times would require extreme particle energies of $\mathcal{O}(10^{4})$EeV ($\mathcal{O}(10^{12})$EeV for taus), making these choices extremely energy inefficient. These energies are above Planck energy $E_{\rm P}=\sqrt{\hbar c^5/G}$ and thus physically implausible, because such high energy particles can not be produced, and would directly collapse into a black hole [@1995hep.ph...10364C].
Massive particles {#massive_particles}
-----------------
Some particles are heavy which makes it costly to accelerate them (proton, neutron, muon, tau). The Higgs boson has the highest mass of all known elementary particles ($125\,$GeV/c$^2$) and decays after $10^{-22}$s. Interstellar communication would thus require (unphysical) particle energies of $\mathcal{O}(10^{22})$EeV; substituting one Higgs would allow for sending $\mathcal{O}(10^{36})$keV photons instead, making this choice irrational.
Hypothetical particles {#hypo}
----------------------
It is sometimes argued in the hallways of astronomy departments that we “just have to tune into the right band” and – voilà – will be connected to the galactic gossip channel with the latest and greatest news about the Princess of Betelgeuse’s[^1] tragic death (R. H., priv. comm.). The problem with this argument is that isotropic radiation, filling the entire galaxy with information-carrying particles of any kind, would require a prohibitively large amount of energy due to the minimum requirement of energy per bit of information, $kT\ln2$ [@1998RSPSA.454..305L]. When it can not be isotropic, it must instead be targeted. So far, we have not detected any emission in the most trivial bands such as microwaves or optical. But could it be some yet unknown particle?
A candidate carrier for such a novel communication channel is the axion, a hypothetical particle postulated by @1977PhRvD..16.1791P [@1977PhRvL..38.1440P] to resolve the strong CP problem in quantum chromodynamics. The axion, if it exists, would have a low mass of order $10^{-5}$eV, which is negligible compared to the energy used for communication purposes at $\mu$m wavelength and below (eV energy and above). Axions, like neutrinos, do not interact much with matter, which makes extinction negligible, but detection difficult. The advantage with axions is that the signals at the input and output would be electromagnetic waves, so that existing modulation technology could be used. Despite the lack of strong interactions with matter, axions might be detected via their coupling to the electromagnetic field [@1985PhRvD..32.2988S]. A shielded, cooled, sharp resonator tuned to the right frequency could work as a receiver; given that the correct frequency is known to great (order Hz) accuracy [@2007PhRvD..76k1701S]. If the frequency is unknown, the receiver would need to sample the frequency space, or fall back to some magic frequency, which we can not yet guess given the unknown nature of axions.
Among the more “radical” speculations are objects such as stable remnants of the process of Hawking evaporation of black holes [@1993PhRvD..47..540B; @2005JHEP...10..053K]. If such objects exist, they might be excellent information repositories, with the highest possible information density, thus presenting a microscopic analogue of the inscribed matter.
The advantage of the axion over the photon is solely in its lower extinction. This is a small advantage, given that there are many photon wavelengths with low ($<1$%) extinction over pc, and even kpc, distances [@2017arXiv170603795H]. The added complexity in axion transmission and reception seems hardly worth the effort, unless one wants to deliberately set an entrance barrier for the primitive civilizations.
We note that there is a zoo of other hypothetical particles which could be discussed in a similar way, e.g. “hidden” photons [@2009EL.....8710010J], or the Neutralino . The discussion on such particles will be carried out in section \[we\_know\_nothing\].
Assessment {#acc}
----------
The only realistic candidate from this category is the neutron. The major issue for communication with neutrons is energy efficiency. Known physics only offers the focusing of such particle beams in accelerators. The beam width produced by an accelerator scales as [@2009JInst...4T5001I]
$$\theta_{\rm beam} = \frac{1}{\gamma} \approx \frac{10^{-4}}{E_{\mu} {\rm [TeV]}}$$
where $\gamma$ is the relativistic boost factor of a muon, and $E_{\mu}$ is its energy. The beam angle at GeV (TeV, PeV) energies is $6^{\circ}$ (21arcsec, 21mas). For comparison, the opening angle of diffraction-limited optics is $\theta_{\rm optics} = 1.22 \lambda / D_{\rm t}$. For $D_{\rm t}=1$m, $\theta_{\rm beam} = \theta_{\rm optics} $ at $\lambda=82$nm (15 eV), a difference of $7\times10^{10}$ in energy for the same beam width. In other words, focusing particles into a beam requires $7\times10^{10}$ more energy in a particle accelerator (using TeV particles) compared to a meter-sized mirror (with 15eV photons).
Practical examples for beam angles can be found from keV [@2012NaPho...6..308T; @2015PhPl...22b3106T], MeV [@PhysRevLett.110.155003; @2014OptL...39.4132L; @2017arXiv170508637S] to GeV energies, for a variety of particles, including neutrinos [@2008AcPPB..39.2943S].
An additional limit for high energy particles arises near the Greisen-Zatsepin-Kuzmin limit [@1966PhRvL..16..748G; @1966JETPL...4...78Z] at energies $>10^{19}\,$eV by slowing-interactions of cosmic ray photons with the microwave background radiation, which has been observationally confirmed [@2008PhRvL.101f1101A].
We conclude that high energy neutrons are energy inefficient for interstellar communication.
Gravitational waves {#gravitational_waves}
===================
Production and transmission
---------------------------
Gravitational waves (GW) are produced by asymmetric acceleration. In a binary system, the GW frequency is twice the orbital frequency, and the amplitude is determined by the change of the mass distribution. It is trivially possible to produce (low amplitude) GWs, in fact any movement of our bodies emits gravitons isotropically. It is however difficult to form a beam with this radiation, and to produce high amplitude GWs.
To our knowledge, the idea to use GWs for communication was first substantiated by @1977PoAn...10...39P and @1980toky.iafcQ....S who describe the need to focus the waves, and the authors suggest to use the “gravitational fields of massive objects with spherical symmetry”, i.e. lensing.
The advantage of GW signals is the complete transparency of the entire universe within out cosmological event horizon, similar to Neutrinos.
While the velocity of gravitational waves was historically debated to be superluminal [@1998PhLA..250....1V; @2001astro.ph..6350D], recent observations of a binary neutron star coalescence by LIGO show coincident signals from the gravitational waves and gamma-rays within $\sim1.7\,$s, indicating that the velocity of GWs is equal, or very close to, the speed of light [@2017ApJ...848L..12A].
Beacons
-------
GW detectors observe the intensity, or amplitude, which falls off as $1/d$ with distance $d$ whereas the flux of an isotropic electromagnetic source drops as $1/d^2$. Without the penalty of the inverse square law, gravitational waves have an advantage for observation (and thus communication) over large distances. They are preferable beacons for cases where the free-space loss of photons exceeds the efficiency penalty (or difficulty) of GW production. For sources (transmitters, t) of the same power, $P_{\rm t, \gamma}=P_{\rm t, GW}$, the receiver (r) flux ratio is $P_{\rm r, \gamma} / P_{\rm r, GW} = 1/d$. For identical receiver sensitivity, photons are preferable as long as $P_{\rm t, \gamma} > d P_{\rm t, GW}$. In other words, at $100\times$ greater distance, the photon power needs to be $100\times$ larger to compensate for its free space loss.
A BH merger emits $10^{47}$J as gravitational waves, as observed by LIGO [@2016PhRvL.116f1102A]. For comparison, a type Ia supernova has an energy release of $10^{42}$J in photons and $10^{40}$J as GWs [@2016MNRAS.461.3296N]. Therefore, in this category of most powerful beacons, GW are superior in detectability over Mpc and Gpc distances. The advantage is reduced because of increased technical difficulty for GW detection.
There are considerable issues with the artificial creation of high-amplitude high-frequency GWs. The large masses and velocity changes require large energies for the acceleration and deceleration of the bodies. Signals can only be injected through such changes, and the data rate will be slow because of mass inertia. Such a communication scheme appears energetically wasteful.
Lensing
-------
Gravitational lensing occurs in the same way for GWs as it does for photons. An important difference is that the commonly used geometrical optics approximation holds only as long as the wavelength is much smaller than the Schwarzschild radius of the lens mass, so that diffraction is small [@2010PhRvL.105y1101S]. This is the case for IR photons where $\lambda=1\,\mu$m and $r_g = 2GM_{\odot} /c^2 \approx 2,950$m is the Schwarzschild radius of the sun. However, GWs with $\lambda < r_g$ have frequencies $>10^5$Hz, so that artificial GWs for lensing need to have $>10^5$Hz.
Astrophysical GW sources are BH mergers with frequencies of $1 \dots 10^3$Hz, binaries with $10^{-4} \dots 1$Hz, and a small stochastic background of lower ($10^{-10} \dots 10^{-6}$Hz) frequencies. No natural sources are expected to exist for frequencies $>10^3$Hz.
### Backward foci
Lensing is typically described from the observational side, where the distance between the lens and the observer is typically larger, or of the same order, compared to the distance between the source and the lens.
Gravitational lensing of GWs increases the energy flux by a magnification factor $\mu > 0$, and the strain amplitude is amplified by $\sqrt{\mu}$ [@2017PhRvD..95d4011D]. The gain has a maximum on the axis $$\begin{aligned}
\label{eq_3}
\mu_{max}=4\pi\frac{r_g}{\lambda} \approx 12.57\end{aligned}$$
for $\lambda \approx r_g$ which is low compared to short wavelength lensing of particles [e.g., $10^9$ for IR photons, @2017arXiv170605570H].
More suitable lenses would be more massive. At the upper end, the center of our galaxy is the supermassive black hole, Sgr A\* [@2002Natur.419..694S] with a mass of $M_{\rm bh}=4.02\pm0.2\times10^6\,M_{\odot}$ at a distance from earth of $R_{\rm bh}=7.86\pm0.18$kpc [@2016ApJ...830...17B]. We can calculate its Schwarzschild radius as $r_{\rm g}=2\,GM_{\rm bh}/c^2\approx1.21\times10^{10}\,{\rm m}\approx17.1\,R_{\odot}\approx0.08\,{\rm au}$, and the apparent size as seen from earth as $\theta_{\rm bh}=3600\times180\times2\,r_{\rm g} / \pi R_{\rm bh} \approx 20.2\,\mu$as. A detector close the the BH has a useful gain of $10^4$ for frequencies of $10^5$Hz following eq. \[eq\_3\].
### Forward foci
The maximum BH forward gain for an isotropic radiator occurs if it is placed near the event horizon, where a substantial fraction of the flux is focused. Such a forward lensing has been suggested in the literature, “A star can produce a forward point focus of extreme magnification. A Schwarzschild black hole has an infinity of forward and backward foci, where there are two types of forward line foci and one kind of backward conical foci.” [@1991LNP...390..299V]. Using the forward focus (for IR lasers) was also suggested by @Jackson2015.
In this scenario, the distance between the source and the lens is much smaller than the distance between the lens and the observer. The problem for an object near the Schwarzschild radius is the high gravity of $F=GM/r^2\approx 4 \times 10^6$gee, implausibly high for advanced games of cosmic snooker producing GWs [@2010Natur.466..406B]. After all, the objects need to remain stationary (and can not be in orbit) to keep the alignment between source, lens and receiver.
Gravity decreases to 1gee at a distance of $621\,r_{\rm g}=49.3\,{\rm au}$. At this distance, only a small part of isotropic flux ends up in the Einstein ring, making the forward focus of the BH unattractive.
Encoding and noise
------------------
In the framework of quantum field theory, the graviton is a hypothetical elementary particle that mediates the force of gravitation. Usable dimensions are frequency and amplitude of the GW through the graviton spin; no polarization or charge is expected.
With astrophysical sources typically occurring at frequencies from $10^{-3}\dots10^3$Hz, no natural sources are reasonably observable through lensing which starts at $>10^5$Hz.
Assessment {#assessment-1}
----------
GWs are energetically inefficient for communication, except perhaps over the very largest distances. Strong GW lensing requires high masses $>10^6\,M_{\odot}$.
Neutrinos
=========
The author Stanislaw Lem envisioned Neutrino communication in his 1968 masterpiece “His Master’s Voice” [@lem1999his]. It was first mentioned in the scientific literature, in passing, by @1972Sci...177..163A. In the same decade, it was discussed extensively [@1977Sci...198..295S; @1977PoAn...10...39P; @1979CosSe...1....2P], but was long perceived as “so difficult that an advanced civilization may purposely choose such a system in order to find and communicate only with ETCs at their own level of development.” [@1979AcAau...6..213S]. Interstellar usage scenarios include clock synchronization [@1994QJRAS..35..321L], directed beam communication [@2008AcPPB..39.2943S; @2009PhLB..671...15L] and exotic scenarios such as using neutrinos to modify the periods of Cepheid variable stars as Morse-code like beacons [@2008arXiv0809.0339L; @2010NuPhA.844..248P; @2012ConPh..53..113L; @2015ApJ...798...42H].
Production
----------
On earth, neutrinos are artificially produced in fission reactors, nuclear bombs, and in accelerators.
### Fission reactors
In reactors, about 4.5% of fission energy is radiated away as antineutrino radiation with a peak (maximum) energy around 4MeV (10MeV). For interstellar communication purposes, this flux is sufficiently high (45 MW out of a GW reactor), but the emission is wide-angle [of order steradian, @2000PhRvD..61a2001A], impossible to modulate on short (sub-second) timescales, and the low energy makes detection difficult (section \[sub:neutrino\_detection\]).
### Nuclear bombs (and supernovae)
Nuclear bombs produce antineutrinos from the fission process, and both neutrinos and antineutrinos in case of a fusion stage. These isotropic flashes occur within a short time; 99.99% of the energy is released in $8\times10^{-8}$s. A large (100 MT) fusion bomb has an energy release of $4\times10^{17}$J (or 4kg mass equivalent), of which $\approx5$% or $2\times10^{16}\,{\rm J}\approx10^{38}\,{\rm eV}$ is in neutrinos. With a spectral peak of $\approx10$MeV, this translates into a flash of $10^{31}$ particles. With isotropic radiation, the flux at a distance to the nearest star (1.3pc) is $6\times10^{-10}$m$^{-2}$. Due to the low cross-section of MeV neutrinos, a planet-size detector would be required to detect this flash. To make a more easily detectable neutrino flash, the flux would need to be much larger. Supernovae emit of order $10^{46}$J ($10^{29}$kg mass equivalent) in neutrinos, a number sufficiently large to be detectable over kpc distance with small detectors [@1987Natur.326..135B; @2009APh....31..163P]. For comparison, the energy release of SNe is much smaller in photons ($10^{42}$J) and GWs ($10^{40}$J) [@2016MNRAS.461.3296N].
### Accelerators
The first accelerator-based neutrino beam used a proton beam hitting a beryllium target, producing pions, which decayed into GeV neutrinos [@1962PhRvL...9...36D]. This principle is still used in modern accelerators.
The ultimate (and not yet built) neutrino beam would be a “Neutrino Factory”, generating neutrinos by the decay of muons stored in a particle accelerator. Muons decay after $2.2\,\mu$s (ms at GeV energies) into a muon neutrino and an electron anti-neutrino. In this short time, the muons must be made, collimated, and accelerated; a very challenging technical problem [@2013arXiv1308.0494D; @2015arXiv150201647D]. These concepts are large (km size) machines with low ($\ll 1\%$) efficiency, comparable in immaturity to the first lasers in the 1960s. Physical efficiency limits are speculated to be of order 0.1% to 10% [@2009PhLB..671...15L; @2010PhLB..692..268H].
Extinction and detection {#sub:neutrino_detection}
------------------------
The minimal cross-section of neutrinos [@2012RvMP...84.1307F] is both a blessing and a curse. On the one hand, interstellar extinction is negligible in all circumstances. On the other hand, for low-energy (MeV) neutrinos, the mean free path is more than a light year of lead, requiring implausibly large detectors if a relevant fraction of particles shall be retrieved. The cross-section peaks for energies near 6.3PeV due to the @1960PhRv..118..316G resonance. At this energy, the detection fraction in $1\,{\rm km}^3$ of water is 1%.
Communication demonstration
---------------------------
Real-world neutrino communication has been demonstrated recently at the Fermilab through 240m of solid rock, at a data rate of 0.1bits/s [@2012MPLA...2750077S]. The neutrino beam at the Main Injector (NuMI) is one of the most intense neutrino beams worldwide, producing an arcmin beam peaking near 3.2GeV of mostly muon neutrinos, with a wall-plug power of 400kW [@2005physics...8001K]. The small distance between transmitter and receiver allowed for negligible free-space loss due to beam angle widening. Still, the low detector efficiency (section \[sub:neutrino\_detection\]) results in a low data rate, showing that significant improvements in neutrino beams and detectors are required for a real world application.
The technology is potentially useful for submarine communication [@2010PhLB..692..268H]. To establish even a low data rate communication between base and submarine, $10^{14}$ muon neutrinos per second at 150 GeV are required. Even with a theoretical transmitter efficiency of 10%, a wall-plug power of 65MW would be required.
Focusing {#neutrino_focusing}
--------
An ideal neutrino producing accelerator is based on muons, and thus its beam width scales as in section \[acc\]:
$$\theta_{\rm beam} = \frac{1}{\gamma} \approx \frac{10^{-4}}{E_{\mu} {\rm [TeV]}}$$
where $\gamma$ is the relativistic boost factor of a muon, and $E_{\mu}$ is its energy [@2009JInst...4T5001I]. The beam angle at GeV (TeV, PeV) energies is $6^{\circ}$ (21arcsec, 21mas). For PeV Neutrinos, we can approximate the capacity (in units of bits per energy) as
$$\label{eq_neutrino}
C_{\nu} \approx
10^{-10}
\left(\frac{d}{1\,{\rm pc}} \right)^{-2}
\left(\frac{V}{100\,{\rm km}^3} \right)
\,\,(\rm {bits\,J^{-1}})$$
where $d$ is the distance and $V$ is the water/ice detector volume. Estimates valid for all energies can not be given in closed form, because the cross section of Neutrinos is a complicated function of energy [@2012RvMP...84.1307F]. For example, Neutrinos at GeV instead of PeV energy require a collector volume larger by three orders of magnitude, in order to achieve the same collection fraction.
Even large detectors can not compensate for the focusing disadvantage of Neutrinos. For comparison, the opening angle of diffraction-limited optics is $\theta_{\rm optics} = 1.22 \lambda / D_{\rm t}$. For $D_{\rm t}=1$m, $\theta_{\rm beam} = \theta_{\rm optics} $ at $\lambda=82$nm (15 eV), a difference of $7\times10^{10}$ in energy for the same beam width. In other words, focusing neutrinos into a beam requires $7\times10^{10}$ more energy in a particle accelerator compared to a meter-sized mirror (using photons).
Capacity and encoding
---------------------
The capacity for neutrino communication (in bits per neutrino) can be calculated in the same way as for other particles. The usable dimensions are time of arrival, energy, and particle/antiparticle; there is no polarization. Instead, neutrinos come in three flavors (electron, muon, tau), but these cannot be used for encoding due to oscillations [@1998PhRvL..81.1562F]. The probability of neutrino oscillations follows a function of the ratio $L/E$ where $L$ is the distance traveled and $E$ is the energy. The distance-energy for a probability of order unity is $\approx1,000\,{\rm km\,GeV}^{-1}$, so that even for high-energy neutrinos (6.3PeV), oscillations randomize the flavors for $L>42$au, approximately the distance to Pluto.
Gravitational lensing
---------------------
Gravitational lensing of neutrinos follows the same laws as for photons, where the bending angle is inversely proportional to the impact parameter $b>R_{\odot}$ of a light ray with respect to the lens. This inverse bending angle is called astigmatism and produces a caustic focal line.
### Focal length
The difference to photons is that neutrinos (and GWs) also pass *through* the sun, resulting in a shorter minimum focal length of $z_{\rm 0,\nu}=23.5\pm0.1$au, about the distance of Uranus [@2000PhRvD..61h3001D; @2008ApJ...685.1297P], compared to $z_{\rm 0,\gamma}=R_{\odot}^2/2\,r_g \approx 546$au for photons, where $r_g = 2\,GM_{\odot} /c^2 \approx 2,950$m is the Schwarzschild radius of the sun [@1964PhRv..133..835L; @1979Sci...205.1133E].
### Aperture size
Classical lensing collects the photons from the very thin Einstein ring surrounding the star, whose width is equal to the receiver size, and whose circumference is a circle with an impact parameter $b=\sqrt{z/z_{\rm 0}}$ where $z$ is the heliocentric distance. Then, the area of collected light is $A_{\gamma}=\pi ((b + w) ^2 - b^2)$. For a meter-sized detector in the lens plane, the corresponding classical aperture is 74.6km [@2017arXiv170605570H; @2017arXiv170305783T].
In contrast, a transparent gravitational lens has an effective aperture of a fraction the stellar size, as most of the rays pass through the star. Based on solar density gradient models, different estimates of the effective lens radius have been calculated, ranging from $R_{\rm lens}=0.024\,R_{\odot}\approx16,700\,{\rm km}$ [@2008ApJ...685.1297P] to $R_{\rm lens}=0.17\,R_{\odot}\approx118,300\,{\rm km}$ [@2000PhRvD..61h3001D; @2005ApJ...628.1081N]. In realistic cases, $A_{\nu} \gg A_{\gamma}$. However, the Neutrino aperture is fixed and independent of the detector size, in contrast to the photon case.
### Size of the caustic
The caustic focal line of classical lensing extends unbroadened towards infinity. Transparent lensing, in contrast, produces a focal point, followed by an extending cone [@1981ApJ...244L...1B; @2008ApJ...685.1297P]. The minimum point spread function width in the focal plane (at a distance of 23.5au) is dominated by irregularities inside the sun, namely convection cells (contributing less than 50m), oblateness (1m), and spots (few m). The total effect has been estimated to about $1 \dots 50$m [@2000PhRvD..61h3001D] and more detailed modeling seems necessary before the deployment of a real detector. Of course no model is perfect, and the use of a real neutrino lens detector would inversely teach us a lot about the sun’s interior; perhaps worth a mission on its own.
### Detector mass equivalent
For a lens radius of $R_{\rm lens}=16,700 \dots 118,300$km which gets compressed in the image plane to a minimum (diameter) of $R_{\rm image}=0.025 \dots 0.005$km, we can calculate the corresponding mass gain as $R_{\rm lens}^2 / R_{\rm image}^2 \approx 5\times10^{11} \dots 6\times10^{16}$. Therefore, placing a receiver mass of one ton in the image plane will yield as many neutrino detections as having $10^{11} \dots 10^{16}$ tons in the receiver on earth.
### Transmission
The transmission probability for neutrinos of different energies and flavors through the SGL has been studied in detail by @1999hep.ph...10510E. There is essentially a cutoff near 100GeV, so that higher energy neutrinos are absorbed inside the sun (and re-radiated into arbitrary directions), while those of lower energy pass through almost unaffected.
### Station keeping
To observe a single source continuously, the detector (mass) must be kept stationary along the axis of source, sun, and detector. Therefore, the detector can not rotate around the sun, and must instead counter its (small) gravitational pull $F$ to keep in place,
$$F=\frac{G M_{\odot} m}{z_{\rm 0}^2}\approx0.01\,{\rm N.}$$
for $m=1,000$kg and $z_{\rm 0}=23.5$au. This value is small as even the smallest rocket engines have kN thrust, so that station keeping appears trivial. The location must be kept within meter accuracy, due to the small size of the caustic.
### Resolution
The magnification of the SGL is very high and can be calculated from geometrical optics. The image is smaller than the object by $R/z$, with $R$ as the distance to the object. For $z=23.5$au and our closest neighbor Alpha Centauri, $R=1.3$pc, we get $R/z=10^4$. Thus, an earth-sized (12,756km) source at this distance would appear with a size of 1km in the image plane. A meter-sized telescope in the image plane would resolve an area of 10km$^2$. Consequently, nearby astrophysical sources such as stars appear spatially resolved (and only part of the flux is collected). For imaging of extended objects, a scanning flight would be required. For interstellar communication purposes, transmitter apertures must be smaller than $D_{\rm r}R/z$. With such a high resolution, only one source can be observed at a time. To target a new object, the detector must be slewed.
### Point spread function
The relation between the relativistic momentum and the wavelength $\lambda$ is defined by the Broglie equation, $\lambda=hc/E$ where $h$ is Planck’s constant. For example, a GeV neutrino will have a wavelength of $10^{-15}$m, much ($10^5\times$) smaller than the size of an atom. In the image plane of the solar gravitational lens, the point spread function (PSF) width for $\lambda=10^{-15}$m is [@2017arXiv170406824T their Eq. 142]
$$\rho\approx 4.5~\Big(\frac{\lambda}{1\,\mu{\rm m}}\Big)\frac{b}{R_\odot}~{\rm cm} \approx 10^{-11}\,{\rm m}$$
which is much smaller than the image spread caused by the lens imperfections and can therefore be neglected.
Assessment {#assessment-2}
----------
Neutrinos are not ideal as beacons. Even large fusion bombs are insufficient to be detectable from the nearest stellar systems due to the low cross-section of the emitted MeV neutrinos. The larger fluxes of Supernovae are detectable; and one might imagine mid-sized fusion events (e.g. a moon-mass, $10^{22}$kg) to be detectable over kpc distance, but this appears extremely wasteful.
Gravitational lensing is attractive if the placement of large masses ($>1,000$kg) in the outer solar system is cheap. Precisely, the mass gain is $10^{11}$ to $10^{16}$ which can be used to calculate cost efficiency compared to a large planet-based neutrino detector, e.g. using water or ice. A disadvantage for a lens collector is the directivity. The resolution is so high that only one distant object can be observed at a time, while a classical (e.g. cubic, spherical) detector is sensitive to incoming neutrinos from all directions.
Regarding directed energy: The need for large accelerators, large receivers and low efficiency keeps neutrinos in the field of a “difficult” technology. Speculative future advances, such as (femto-)technology to manipulate nuclear matter [@bolonkin2009femtotechnology], may allow for more efficient neutrino capture, and thus higher detection fractions. It is presently unclear if there is a physically plausible solution to get in the 50% (or even 1%) efficiency range available for photons. Neutrino communication also suffers from the focusing efficiency issue: their beam width is $10^{10}$ times wider as for photons of the same energy. Finally, the large detectors disqualify neutrinos for the use on board of small probes.
Relaxing constraints {#irrational_aliens}
====================
Our initial assumptions of what ET values might be incorrect. We had assumed that ET favours more information over less, fast over slow, has energy limits, and wants to build less machinery rather than more, all else equal. We will now relax these constraints one by one and re-evaluate our analysis.
Little information is enough
----------------------------
Although humans are a curious species, an advanced civilization might be bored of factual communication with others. To exchange just a few bits (“I’m here!”), a beacon is sufficient. Isotropic microwave beacons are the most energy-efficient direct emitters, but are still very costly if they shall be seen over large distances [@2010AsBio..10..491B; @2010AsBio..10..475B]. Modulating a pulsator can be much easier at the same visibility (section \[arti\]). In this scenario, we should focus on a deep all-sky survey at GHz frequencies, as well as look for strange stellar-like object, such as Boyajian’s star [@2016MNRAS.457.3988B; @2016ApJ...829L...3W]. With sufficiently good instruments, the need for active beacons vanishes. Smaller and smaller macroengineering objects become remotely visible, such as “Clarke Exobelts”[@2018ApJ...855..110S], industrial air pollution [@2014ApJ...792L...7L], climate change, or catastrophic nuclear wars [@2016IJAsB..15..333S]. If such signals are detectable as by-catches of regular observations, the cost of (beacon style) communication is zero, making it the most efficient approach.
Preference of complicated technology
------------------------------------
Perhaps as an entry-barrier for lesser civilizations, ET might favor more complicated technology.
If this is to be done at the cost of higher energy usage, or at the cost of complicated (more expensive) machinery, an obvious choice would be $\gamma$-rays, Neutrinos or massive particles such as Neutrons.
At the cost of slower communication (time of arrival), one would send inscribed matter in the form of probes, or perhaps bullets, which need to be found and/or decelerated by the target civilization. Depending on how small such an object is, we would find it inside our solar system only at advanced stages of exploration.
Relaxed energy limits
---------------------
If energy efficiency is not considered important, wider beams are an obvious choice, as they have the advantage of more civilizations being potentially located inside the cone. In this lighthouse scenario, low photon energy communication, i.e. microwaves, are clearly favored because they maximize the information per unit area. Tight beams with wasteful energy usage would use heavy particles such as Neutrons.
Unimportant Time delay
----------------------
One could argue that entities in advanced civilizations live much longer, e.g. through hibernation, biological modifications, mind-uploading into interchangeable substrates such as silicon, or other trans-humanist mechanisms.
If time delay is not important, the obvious choice would be to send inscribed matter. This can be the most energy-efficient choice [@2004Natur.431...47R; @2017arXiv171210262H], because it can be done at almost arbitrarily low velocities ($\approx 20$kms$^{-1}$), and thus low energies. Also, an artifact can arrive at the destination in total, in contrast to a beam which is wider than the receiver in all realistic cases.
Major gaps in our knowledge of physics {#we_know_nothing}
--------------------------------------
Perhaps there exist yet unknown particles whose spec sheets exceed that of photons. While we can not know their nature, we can state the advantages we require them to posses in order to exceed photons. They must be either cheaper (in terms of energy requirements per beam width), faster, more robust, have more encoding modes, simpler to use, or a combination of these.
It appears implausible that yet unknown particles exist which are easier to use than photons or classical matter.
Regarding robustness, there are already known particles which have excellent extinction performance, e.g. high-energy photons or Neutrinos. Axions (section \[hypo\]), so they exist, would have zero extinction. However, they are more difficult to receive than photons, because their frequency needs to be known exactly. It also appears that these can only be received mono-chromatically (because the cavity is tuned to only one frequency), reducing the number of modes, and thus the number of bits per photon in any encoding scheme. While the total effect is unclear without knowing their exact properties, axions appear not to be superior over photons, and certainly not superior by more than a factor of a few.
For quicker communication, tachyonic particles would travel faster than light [@1967PhRv..159.1089F], and therefore be preferred over photons, all else equal. However, faster than light signals are generally assumed to violate causality and are therefore unlikely to exist, although hypothetical speculations remain [@2011arXiv1112.4714C].
It is also difficult to imagine particles which are preferred because of a higher number of encoding modes. The number of modes has a logarithmic influence on the data rate and is therefore only of marginal benefit [@2017arXiv171205682H; @2018arXiv180106218H].
Furthermore, we can exclude all hypothetical particles with high masses [e.g., the Neutralino with $>300$GeV, @2010pdmo.book..142E], because it is too costly in terms of required energy to accelerate these compared to mass-less photons.
Overall, it appears unlikely that any hypothetical particles exceeds keV photon performance by more than a factor of a few. Unless our understanding of physics has fundamental gaps, photons are the rational choice for most communications.
Multiple incorrect assumptions
------------------------------
Combining two or more of these arguments yields arbitrary results. For example, if advanced ET favors energy inefficiency *and* little information, it would be reasonable to create SNe. An approach combining slow data rate *and* expensive machinery could be to build artificial occulters.
Conclusion
==========
We have benchmarked all known information carriers against photons, including electrons, protons, neutrinos, inscribed matter (probes), gravitational waves and artificial megastructures such as occulters. We compared the speed of exchange, information per energy and machine sizes, lensing performance, and complexity.
Isotropic beacons with low data rates can not be constrained with our current level of understanding, and many options appear similarly attractive.
In point-to-point communications, we assumed that ET favours more information over less, fast over slow, simplicity over complexity, and has energy limits. We explored the consequences when dropping these assumptions one by one. If the assumptions hold, then photons are superior to other carries by orders of magnitude. If speed is not crucial, sending probes with inscribed matter would be preferred.
References
==========
[^1]: The M2 Iab red supergiant Betelgeuse (*$\alpha\,$Orionis*) is expected to explode in a supernova in $<10^6$ years [@2013EAS....60...17M; @2016ApJ...819....7D]. At a distance of $222 \substack{+48\\-38}\,$pc [@2017AJ....154...11H] it might well have exploded as of now, without us knowing yet.
|
---
abstract: |
The mean ground state occupation number and condensate fluctuations of interacting and non-interacting Bose gases confined in a harmonic trap are considered by using a canonical ensemble approach. To obtain the mean ground state occupation number and the condensate fluctuations, an analytical description for the probability distribution function of the condensate is provided directly starting from the analysis of the partition function of the system. For the ideal Bose gas, the probability distribution function is found to be a Gaussian one for the case of the harmonic trap. For the interacting Bose gas, using a unified approach the condensate fluctuations are calculated based on the lowest-order perturbation method and on Bogoliubov theory. It is found that the condensate fluctuations based on the lowest-order perturbation theory follow the law $\left\langle \delta ^{2}N_{{\bf {0}}%
}\right\rangle \sim N$, while the fluctuations based on Bogoliubov theory behave as $N^{4/3}$.
address:
- '$^{1}$Department of Applied Physics, Zhejiang University of Technology, Hangzhou, 310032, China'
- '$^{2}$Zhijiang College, Zhejiang University of Technology, Hangzhou, 310012, China '
- |
$^{3}$Department of Physics, East China Normal University, Shanghai, 200062,\
China
author:
- 'Hongwei Xiong$^{1,2}$, Shujuan Liu$^{1}$, Guoxiang Huang$^{3}$, and Zaixin Xu$^{3}$'
title: Canonical Statistics of Trapped Ideal and Interacting Bose Gases
---
Introduction
============
The experimental achievement of Bose-Einstein condensation (BEC) in dilute alkali atoms [@ALK], spin-polarized hydrogen [@MIT] and recently in metastable helium [@HEL] has enormously stimulated the theoretical research [@RMP; @PARKIN; @LEG] on the ultracold bosons. Among the several intriguing questions on the statistical properties of trapped interacting Bose gases, the problem of condensate fluctuations $\left\langle \delta
^{2}N_{{\bf {0}}}\right\rangle $ of the mean ground state occupation number $%
\left\langle N_{{\bf {0}}}\right\rangle $ is of central importance. Apart from the intrinsic theoretical interest, it is foreseeable that such fluctuations will become experimentally testable in the near future [@NEAR]. On the other hand, the calculations of $\left\langle \delta ^{2}N_{%
{\bf 0}}\right\rangle $ are crucial to investigate the phase collapse time of the condensate [@JAV; @WRI].
It is well known that within a grand canonical ensemble the fluctuations of the condensate are given by $\left\langle \delta ^{2}N_{{\bf {0}}%
}\right\rangle =N_{{\bf {0}}}\left( N_{{\bf {0}}}+1\right) $, implying that $%
\delta N_{{\bf {0}}}$ becomes of order $N$ when the temperature approaches zero. To avoid this sort of unphysically large condensate fluctuations, a canonical (or a microcanonical) ensemble has to be used to investigate the fluctuations of the condensate. On the other hand, because in the experiment the trapped atoms are cooled continuously from the surrounding, the system can be taken as being in contact with a heat bath but the total number of particles in the system is conserved. Thus it is necessary to use the canonical ensemble to investigate the statistical properties of the trapped weakly interacting Bose gas.
Within the canonical as well as the microcanonical ensembles, the condensate fluctuations have been studied systematically in the case of an ideal Bose gas in a box [@HAUGE; @FUJI; @ZIF; @BOR; @WIL], and in the presence of a harmonic trap [@WIL; @POL; @GAJ; @GRO; @NAVEZ; @BAL; @HOL1; @HOL2]. Recently, the question of how interatomic interactions affect the condensate fluctuations has been an object of several theoretical investigations [@GIO; @IDZ; @ILLU; @MEI; @KOC; @JAK]. Idziaszek [*et al.*]{} [@IDZ] investigated the condensate fluctuations of interacting Bose gases using the lowest-order perturbation theory and a two-gas model, while Giorgini [*et al.*]{} [@GIO] addressed this problem within a traditional particle-number-nonconserving Bogoliubov approach. Recently, Kocharovsky [*et al.*]{} [@KOC] supported and extended the results of the work of Giorgini [*et al.*]{} [@GIO] using a particle-number-conserving operator formalism.
Although the condensate fluctuations are thoroughly investigated in Ref.[@GIO; @IDZ; @ILLU; @MEI; @KOC], to best our knowledge up to now an analytical description of the probability distribution function for the interacting Bose gas directly from the microscopic statistics of the system has not been given. Note that as soon as the probability distribution function of the system is obtained, it is straightforward to get the mean ground state occupation number and the condensate fluctuations. The purpose of the present work is an attempt to provide such an analytical description of the probability distribution function of interacting and non-interacting Bose gases based on the analysis of the partition function of the system.
We shall investigate in this paper the condensate fluctuations of interacting and non-interacting Bose gases confined in a harmonic trap. The analytical probability distribution function of the condensate will be given directly from the partition function of the system using a canonical ensemble approach. For an ideal Bose gas, we find that the probability distribution of the condensate is a Gaussian function. In particular, our method can be easily extended to discuss the probability distribution function for a weakly interacting Bose gas. A unified way is given to calculate the condensate fluctuations from the lowest-order perturbation theory and from Bogoliubov theory. We find that different methods of approximation for the interacting Bose gas give quite different predictions concerning the condensate fluctuations. We show that the fluctuations based on the lowest-order perturbation theory follow the law $\left\langle \delta ^{2}N_{{\bf {0}}}\right\rangle
\sim N$, while the fluctuations based on the Bogoliubov theory behave as $N^{4/3} $.
The paper is organized as follows. Sec. II is devoted to outline the canonical ensemble, which is developed to discuss the probability distribution function of Bose gases. In Sec. III we investigate the condensate fluctuations of the ideal Bose gas confined in a harmonic trap. In Sec. IV the condensate fluctuations of the weakly interacting Bose gas are calculated based on the lowest order perturbation theory. In Sec. V the condensate fluctuations due to collective excitations are obtained based on Bogoliubov theory. Finally, Sec. VI contains a discussion and summary of our results.
Fluctuations and Mean Ground State Occupation Number of the Condensate in the Canonical Ensemble
================================================================================================
According to the canonical ensemble, the partition function of the system with $N$ trapped interacting bosons is given by
$${Z\left[ N\right] =\sum_{\Sigma _{{\bf {n}}}N_{{\bf {n}}}=N}\exp \left[
-\beta \left( \Sigma _{{\bf {n}}}N_{{\bf {n}}}\varepsilon _{{\bf {n}}%
}+E_{int}\right) \right]}, \label{par1}$$
where $N_{{\bf n}}$ and $\varepsilon _{{\bf n}}$ are occupation number and energy level of the state ${\bf
{n}}=\{n_{x},n_{y},n_{z}\}$, respectively. $\beta =1/k_{B}T$ and $\{n_{x},n_{y},n_{z}\}$ are non-negative integers. $E_{int}$ is the interaction energy of the system. For convenience, by separating out the ground state ${\bf {n}}={\bf 0}$ from the state ${\bf {n}}\neq {\bf 0}$, we have
$${Z\left[ N\right] =\sum_{N_{{\bf 0}}=0}^{N}\left\{ \exp \left[ -\beta \left(
E_{{\bf 0}}+E_{int}\right) \right] Z_{0}\left( N,N_{{\bf 0}}\right) \right\}}%
, \label{par2}$$
where $Z_{0}\left( N,N_{{\bf 0}}\right)$ stands for the partition function of a fictitious system comprising $N-N_{{\bf
0}}$ trapped ideal non-condensed bosons:
$${Z_{0}\left( N,N_{{\bf 0}}\right) =\sum_{\sum_{{\bf {n}}\neq {\bf 0}}N_{{\bf
{n}}}=N-N_{{\bf 0}}}\exp \left[ -\beta \sum_{{\bf {n}\neq 0}}N_{{\bf {n}}%
}\varepsilon _{{\bf {n}}}\right] .} \label{II-function-1}$$
Assuming $A_{0}\left( N,N_{{\bf 0}}\right) $ is the free energy of the fictitious system, we have
$${A_{0}(N,N_{{\bf 0}})=-k_{B}T\ln Z_{0}(N,N_{{\bf 0}}).} \label{free-energy}$$
The calculation of the free energy $A_{0}\left( N,N_{{\bf
{0}}}\right) $ is nontrivial because there is a requirement that the number of non-condensed bosons is $N-N_{{\bf {0}}}$ in the summation of the partition function $%
Z_{0}\left( N,N_{{\bf {0}}}\right) $. Using the saddle-point method developed by Darwin and Fowler [@DAR], it is straightforward to obtain a useful relation between the free energy $A_{0}\left( N,N_{{\bf {0}}%
}\right) $ and the fugacity $z_{0}$ of the fictitious $N-N_{{\bf 0}}$ non-interacting bosons
$${-\beta \frac{\partial }{\partial N_{{\bf {0}}}}A_{0}\left( N,N_{{\bf {0}}%
}\right) =\ln z_{0},} \label{relation1}$$
where the fugacity $z_{0}$ is determined by
$${N_{{\bf {0}}}=N-\sum_{{\bf {n}}\neq {\bf {0}}}\frac{1}{\exp \left[ {\beta }%
\varepsilon _{{\bf {n}}}\right] z_{0}^{-1}-1}.} \label{relation2}$$
We have given a simple derivation of Eqs. (\[relation1\]) and (\[relation2\]) in the Appendix.
Using the free energy $A_{0}\left( N,N_{{\bf {0}}}\right) $, the partition function of the system becomes
$${Z\left[ N\right] =\sum_{N_{{\bf 0}}=0}^{N}\exp \left[ q\left( N,N_{{\bf 0}%
}\right) \right]}, \label{par3}$$
where
$${q\left( N,N_{{\bf 0}}\right) =-\beta \left( E_{{\bf 0}}+E_{int}\right)
-\beta A_{0}\left( N,N_{{\bf 0}}\right).} \label{qqq}$$
It is obvious that $(1/Z\left[ N\right])\exp \left[
q\left( N,N_{{\bf {0}}}\right) \right]$ represents the probability finding $N_{{\bf 0}}$ atoms in the condensate.
To obtain the probability distribution function of the system, let us first investigate the largest term in the sum of the partition function $%
Z\left[ N\right]$. Assume the number of the condensed atoms is $N_{{\bf 0}%
}^{p}$ in the largest term of the partition function. The largest term $\exp \left[ q\left( N,N_{{\bf {0}}}^{p}\right) \right] $ is determined by requiring that $\frac{\partial }{\partial N_{{\bf {0}}}}q\left( N,N_{{\bf {%
0}}}\right) |_{N_{{\bf {0}}}=N_{{\bf {0}}}^{p}}=0$, [*i.e.*]{},
$${-\beta \frac{\partial }{\partial N_{{\bf 0}}^{p}}\left( E_{{\bf 0}%
}+E_{int}\right) -\beta \frac{\partial }{\partial N_{{\bf 0}}^{p}}A_{0}(N,N_{%
{\bf 0}}^{p})=0}. \label{main1}$$
Using Eq. (\[relation1\]) we obtain
$${\ln z_{0}^{p}=\beta \frac{\partial }{\partial N_{{\bf 0}}^{p}}\left( E_{%
{\bf 0}}+E_{int}\right).} \label{z0p}$$
In addition, from Eq. (\[relation2\]), the most probable value $%
N_{{\bf 0}}^{p}$ is determined by
$${N_{{\bf {0}}}^{p}=N-\sum_{{\bf {n}}\neq {\bf {0}}}\frac{1}{\exp \left[
\beta \varepsilon _{{\bf {n}}}\right] \left( z_{0}^{p}\right) ^{-1}-1}.}
\label{non1}$$
In the case of an ideal Bose gas, from Eq. (\[z0p\]) one obtains $\ln z_{0}^{p}=\beta \varepsilon _{{\bf {0}}}$. Thus $N_{{\bf {0}}%
}^{p}$ is the same as the mean ground state occupation number obtained by using a grand canonical ensemble approach. For sufficiently large $N$, the sum $\sum_{N_{%
{\bf {0}}}=0}^{N}$ in (\[par3\]) may be replaced by the largest term, since the error omitted in doing so is statistically negligible. In this situation, Eq. (\[non1\]) shows the equivalence between the canonical ensemble and the grand canonical ensemble for large $N$.
The other terms in the partition function (\[par3\]) will contribute to the fluctuations of the condensate, and lead to the deviation of $\left\langle N_{{\bf {0}}%
}\right\rangle $ from the most probable value $N_{{\bf 0}}^{p}$. If $N_{%
{\bf 0}}\neq N_{{\bf 0}}^{p}$, we have $\frac{\partial }{\partial N_{{\bf 0}}}%
q\left( N,N_{{\bf 0}}\right)\neq 0$. Assuming
$${\frac{\partial }{\partial N_{{\bf 0}}}q\left( N,N_{{\bf 0}}\right) = \alpha
\left( N,N_{{\bf 0}}\right),} \label{q-alpha}$$
from Eqs. (\[relation1\]) and (\[qqq\]), we obtain
$${\ln z_{0}=\beta \frac{\partial }{\partial N_{{\bf {0}}}}\left( E_{{\bf {0}}%
}+E_{int}\right) +\alpha \left( N,N_{{\bf {0}}}\right).} \label{z00}$$
By Eqs. (\[relation2\]) and (\[z00\]), we have
$${
N_{{\bf 0}}=N-
}$$
$${\sum_{{\bf n\neq 0}}\frac{1}{\exp \left[ \beta \varepsilon _{%
{\bf n}}\right] \exp \lbrack -\beta {\frac{\partial }{\partial N_{{\bf {0}}}}%
\left( E_{{\bf {0}}}+E_{int}\right) -\alpha \left( N,N_{{\bf {0}}}\right) }%
\rbrack -1}.} \label{main3}$$
Combining Eqs. (\[non1\]) and (\[main3\]), we get the following equation for determining $\alpha \left( N,N_{{\bf
{0}}}\right) $
$${N_{{\bf {0}}}\vspace{1pt}-N_{{\bf {0}}}^{p}=\sum_{{\bf n}\neq {\bf 0}}\frac{%
1}{\exp \left[ \beta \varepsilon _{{\bf n}}\right] \exp \lbrack -\beta {%
\frac{\partial }{\partial N_{{\bf {0}}}^{p}}\left( E_{{\bf {0}}%
}+E_{int}\right) }\rbrack -1}}$$
$${-\sum_{{\bf n}\neq {\bf 0}}\frac{1}{\exp \left[ \beta \varepsilon _{{\bf n}%
}\right] \exp \lbrack -\beta {\frac{\partial }{\partial N_{{\bf {0}}}}\left(
E_{{\bf {0}}}+E_{int}\right) -\alpha \left( N,N_{{\bf {0}}}\right) }\rbrack
-1}.} \label{alpha}$$
Once we know $E_{{\bf {0}}}$ and $E_{int}$ of the system, it is straightforward to obtain $\alpha \left( N,N_{{\bf {0}}}\right) $ from Eq. (\[alpha\]). Using $\alpha \left( N,N_{{\bf {0}}}\right) $, one can obtain the probability distribution function of the system.
From Eq. (\[q-alpha\]), we obtain the following result for $q\left( N,N_{%
{\bf {0}}}\right) $ $${q\left( N,N_{{\bf {0}}}\right) =\int_{N_{{\bf {0}}}^{p}}^{N_{{\bf {0}}%
}}\alpha \left( N,N_{{\bf {0}}}\right) dN_{{\bf {0}}}+q\left( N,N_{{\bf {0}}%
}^{p}\right).} \label{qalpha2}$$ Thus the partition function of the system becomes
$${Z\left[ N\right] =\sum_{N_{{\bf {0}}}=0}^{N}\left\{ \exp \left[ q\left(
N,N_{{\bf {0}}}^{p}\right) \right] G\left( N,N_{{\bf {0}}}\right) \right\},}
\label{par-alpha}$$
where
$${G\left( N,N_{{\bf {0}}}\right) =\exp \left[ \int_{N_{{\bf {0}}}^{p}}^{N_{%
{\bf {0}}}}\alpha \left( N,N_{{\bf {0}}}\right) dN_{{\bf {0}}}\right] .}
\label{ideal-dis}$$
Assuming $P\left( N_{{\bf {0}}}|N\right) $ is the probability to find $N_{{\bf {0}}}$ atoms in the condensate, $G\left( N,N_{{\bf {0}}%
}\right) $ represents the ratio $\frac{P\left( N_{{\bf {0}}}|N\right) }{%
P\left( N_{{\bf {0}}}^{p}|N\right) }$, [*i.e.*]{}, the relative probability to find $N_{{\bf 0}}$ atoms in the condensate. From Eq. (\[ideal-dis\]), the normalized probability distribution function is given by
$${G}_{n}{\left( N,N_{{\bf {0}}}\right) =A\exp \left[ \int_{N_{{\bf {0}}%
}^{p}}^{N_{{\bf {0}}}}\alpha \left( N,N_{{\bf {0}}}\right) dN_{{\bf {0}}%
}\right] ,} \label{norm-di}$$ where $A$ is a normalization constant and is given by the condition $A\int
G(N,N_{{\bf {0}}})dN_{{\bf {0}}}=1$.
As soon as we know $G\left( N,N_{{\bf {0}}}\right) $, the statistical properties of the system can be clearly described. From Eqs. (\[par-alpha\]) and (\[ideal-dis\]) one obtains the mean ground state occupation number $%
\left\langle N_{{\bf {0}}}\right\rangle $ and fluctuations $\left\langle \delta ^{2}N_{{\bf {0}}}\right\rangle $ in the canonical ensemble:
$${\left\langle N_{{\bf {0}}}\right\rangle =\frac{\sum_{N_{{\bf {0}}}=0}^{N}N_{%
{\bf {0}}}\exp \left[ q\left( N,N_{{\bf {0}}}\right) \right] }{\sum_{N_{{\bf
{0}}}=0}^{N}\exp \left[ q\left( N,N_{{\bf {0}}}\right) \right] }=\frac{%
\sum_{N_{{\bf {0}}}=0}^{N}N_{{\bf {0}}}G\left( N,N_{{\bf {0}}}\right) }{%
\sum_{N_{{\bf {0}}}=0}^{N}G\left( N,N_{{\bf {0}}}\right) }}
\label{mean-ideal}$$
$${
\left\langle \delta^{2} N_{{\bf {0}}}\right\rangle =\left\langle N_{{\bf {0}%
}}^{2}\right\rangle -\left\langle N_{{\bf {0}}}\right\rangle ^{2}=
}$$
$${\frac{%
\sum_{N_{{\bf {0}}}=0}^{N}N_{{\bf {0}}}^{2} G\left( N,N_{{\bf {0}}}\right) }{%
\sum_{N_{{\bf {0}}}=0}^{N} G\left( N,N_{{\bf {0}}}\right) }-\left[ \frac{%
\sum_{N_{{\bf {0}}}=0}^{N}N_{{\bf {0}}} G\left( N,N_{{\bf {0}}}\right) }{%
\sum_{N_{{\bf {0}}}=0}^{N} G\left( N,N_{{\bf {0}}}\right) }\right] ^{2}.}
\label{fluc-ideal}$$
Starting from Eqs. (\[mean-ideal\]) and (\[fluc-ideal\]), one can calculate the mean ground state occupation number and fluctuations for ideal and interacting Bose gases.
Ideal Bose Gases
================
We now study the condensate fluctuations of the system with $N$ non-interacting bosons trapped in an external potential. The potential is a harmonic one with the form
$$V_{ext}\left( {\bf {r}}\right) =\frac{m}{2}\left( \omega
_{x}^{2}x^{2}+\omega _{y}^{2}y^{2}+\omega _{z}^{2}z^{2}\right) {,}
\label{potential}$$
where $m$ is the mass of atoms, $\omega _{x}$, $\omega _{y}$, and $%
\omega _{z}$ are frequencies of the trap along three coordinate-axis directions. The single-particle energy level has the form
$${\varepsilon _{{\bf {n}}}=\left( n_{x}+\frac{1}{2}\right) \hbar \omega
_{x}+\left( n_{y}+\frac{1}{2}\right) \hbar \omega _{y}+\left( n_{z}+\frac{1}{%
2}\right) \hbar \omega _{z}.} \label{energy-trap}$$
From Eq. (\[non1\]) one can get easily the most probable value $N_{{\bf {0}}}^{p}$, which reads
$${N_{{\bf {0}}}^{p}=N-N\left( \frac{T}{T_{c}^{0}}\right) ^{3}-\frac{3%
\overline{\omega }\zeta \left( 2\right) }{2\omega _{ho}\left[ \zeta \left(
3\right) \right] ^{2/3}}\left( \frac{T}{T_{c}^{0}}\right) ^{2}N^{2/3},}
\label{ideal-most}$$
where $T_{c}^{0}=\frac{\hbar \omega _{ho}}{k_{B}}\left( \frac{N}{%
\zeta \left( 3\right) }\right) ^{1/3}$ is the critical temperature of the ideal Bose gas in the thermodynamic limit. $\overline{\omega
}=\left( \omega _{x}+\omega _{y}+\omega _{z}\right) /3$ and $\omega _{ho}=\left( \omega _{x}\omega _{y}\omega _{z}\right)
^{1/3}$ are arithmetic and geometric averages of the oscillator frequencies, respectively. When obtaining (\[ideal-most\]) we have used the following expression of the density of states [@FIN]
$${\rho \left( E\right) =\frac{1}{2}\frac{E^{2}}{\left( \hbar \omega
_{ho}\right) ^{3}}+\frac{3\overline{\omega }E}{2\omega _{ho}\left( \hbar
\omega _{ho}\right) ^{2}}.} \label{density-state}$$
On the basis of the same density of states a detailed study of the critical temperature and the ground state occupation number was given recently in [@MULKEN].
In a thermodynamic equilibrium the deviation from the most probable value $N_{{\bf {0}}}^{p}$ is small, therefore we can use the approximation $%
\exp \left[ -\alpha \left( N,N_{{\bf {0}}}\right) \right] \approx
1-\alpha \left( N,N_{{\bf {0}}}\right) $. From Eq. (\[alpha\]) and the single-particle energy level (\[energy-trap\]) we find the result for $\alpha \left( N,N_{{\bf {0}}%
}\right) $
$${\alpha \left( N,N_{0}\right) =-\frac{\zeta \left( 3\right) \left(
N_{0}-N_{0}^{p}\right) }{\zeta \left( 2\right) N\left( T/T_{c}^{0}\right)
^{3}}.} \label{alpha-har}$$
When obtaining $\alpha \left( N,N_{{\bf {0}}}\right) $ we have used the expansion $g_{3}\left( 1+\delta \right) \approx \zeta
\left( 3\right) +\zeta \left( 2\right) \delta $ [@ROB], where $g_{3}\left( z\right) $ belongs to the class of functions $g_{\alpha }\left( z\right) =\sum_{n=1}^{\infty }z^{n}/n^{\alpha
}$ and $\zeta \left( n\right) $ is Riemann $\zeta $ function. From (\[ideal-dis\]) and (\[alpha-har\]) we obtain the normalized probability distribution function of the harmonically trapped ideal Bose gas
$${G_{ideal}\left( N,N_{0}\right) =A}_{ideal}{\exp \left[ -\frac{\zeta \left(
3\right) \left( N_{0}-N_{0}^{p}\right) ^{2}}{2\zeta \left( 2\right) N\left(
T/T_{c}^{0}\right) ^{3}}\right] ,} \label{dis-har}$$
where $A_{ideal}$ is a normalization constant. It is interesting to note that the expression (\[dis-har\]) is a Gaussian distribution function. From the formulas (\[mean-ideal\]), (\[fluc-ideal\]), (\[ideal-most\]), and (\[dis-har\]) we can obtain $\left\langle N_{{\bf
{0}}}\right\rangle $ and $\left\langle \delta ^{2}N_{{\bf
{0}}}\right\rangle $ for the ideal Bose gas.
In Fig. 1(a) and Fig. 1(b) we plot $\left\langle N_{{\bf {0}}}\right\rangle /N$ as a function of temperature for $N=200$ and $N=10^{3}$ ideal bosons confined in an isotropic harmonic trap. The dashed line displays $%
\left\langle N_{{\bf {0}}}\right\rangle /N$ in the thermodynamic limit, while the solid line displays $\left\langle N_{{\bf
{0}}}\right\rangle /N$ within the grand canonical ensemble (or $N_{{\bf {0}}}^{p}$ within the canonical ensemble). The dotted line displays $\left\langle N_{{\bf {0}}}\right\rangle
/N$ within the canonical ensemble. When $N>10^{3}$, $\left\langle N_{{\bf {0}}%
}\right\rangle /N$ from the canonical ensemble agrees well with that from the grand canonical ensemble. Obviously, in the case of $N\rightarrow \infty $, $%
\left\langle N_{{\bf {0}}}\right\rangle /N$ obtained from the canonical ensemble coincides with that from the grand canonical ensemble.
From the formulas (\[mean-ideal\]) and (\[fluc-ideal\]) and the results (\[ideal-most\]) and (\[dis-har\]) we can obtain the condensate fluctuations of the ideal Bose gas. In Fig. 2 we plot the numerical result of $\delta N_{{\bf {0}}}=\sqrt{%
\left\langle \delta ^{2}N_{{\bf {0}}}\right\rangle }$ (solid line) for $%
N=10^{3}$ ideal bosons confined in an isotropic harmonic potential. The dashed line displays the result of Holthaus [*et al.*]{} [@HOL2], where the saddle-point method is developed to avoid the failure of the standard saddle-point approximation below the onset of BEC.
In Fig. 2 the dotted line displays the result given in Refs.[@POL; @GIO]. Our results coincide with those of Refs.[@POL; @GIO] when $T/T_{c}^{0}$ is smaller than $%
T_{m}/T_{c}^{0}$, which corresponds to the maximum fluctuations $\left\langle
\delta ^{2}N_{{\bf 0}}\right\rangle _{\max }$. In fact, when $%
T/T_{c}^{0}<T_{m}/T_{c}^{0}$, from (\[mean-ideal\]), (\[fluc-ideal\]) and (\[dis-har\]), we obtain the analytical result for the condensate fluctuations:
$${\left\langle \delta^{2} N_{{\bf 0}}\right\rangle =\frac{\pi
^{2}}{6\zeta \left( 3\right) }N\left( \frac{T}{T_{c}^{0}}\right)
^{3},} \label{analytical}$$
which recovers the result given in Refs.[@POL; @GIO]. This shows the validity of the probability distribution function (\[dis-har\]) for studying the statistical properties of the system. At the critical temperature, however, our results give
$${\left\langle \delta ^{2}N_{{\bf 0}}\right\rangle
|_{T=T_{c}}=\left( 1-\frac{2}{\pi} \right) \frac{\pi ^{2}N}{6\zeta
\left( 3\right) },} \label{critical-fluc}$$
which is much smaller than the result of Ref.[@GIO]. This difference is apprehensible because the analysis of Giorgini [*et al.*]{} [@GIO] holds in the canonical ensemble except near and above $T_{c}^{0}$, while our result holds also for the temperature near $T_{c}^0$. Near the critical temperature, our result (solid line) agrees with that of Holthaus [*et al.*]{} [@HOL2]. The results given by (\[analytical\]) and (\[critical-fluc\]) show a normal behavior of the condensate fluctuations for the harmonically trapped ideal Bose gas.
The fluctuations of the condensate can also be evaluated at $T=0$. In the case of $T\rightarrow 0$, from (\[dis-har\]) we get $G\left( N,N_{%
{\bf {0}}}\right) =A_{ideal}$ if $N_{{\bf {0}}}=N$, while $G\left( N,N_{{\bf {0}}%
}\right) \rightarrow 0$ when $N_{{\bf {0}}}\neq N$. Therefore, we obtain $%
\left\langle N_{{\bf {0}}}\right\rangle \rightarrow N$ and $\left\langle
\delta ^{2}N_{{\bf {0}}}\right\rangle \rightarrow 0$ when $T\rightarrow 0$.
Note that our results are reliable although the disputable saddle-point method is used to investigate the fluctuations of the condensate. It is well known that the applicability of the saddle-point approximation for the condensed Bose gas has been the subject of a long debate [@ZIF; @DEB]. Recently, the analysis given in Ref.[@GRO] showed that the fluctuations are overestimated, and do not appear to vanish properly with temperature using the conventional saddle-point method. Our discussions on the condensate fluctuations are reasonable because of two reasons: (i) As proved given in Ref.[@HOL2], the most probable value Eq. (\[non1\]) for the non-interacting Bose gas is still correct, even when carefully dealing with the failure of the standard saddle-point method below the critical temperature. (ii) In the usual statistical method $\left\langle N_{{\bf {0}}}\right\rangle $ and $%
\left\langle \delta ^{2}N_{{\bf {0}}}\right\rangle $ are obtained through the first and second partial derivatives of partition function, respectively. When the saddle-point approximation is used to calculate the partition function of the system, the error will be overestimated in the second partial derivative of the partition function. Thus one can not obtain correct condensate fluctuations using usual method. However, in our approach here what we used is the reliable result given by Eqs. (\[non1\]) and (\[main3\]). The probability distribution function of the ground state occupation number can be obtained directly from Eqs. (\[non1\]) and (\[main3\]), without resorting to the second partial derivative of the partition function. $\left\langle N_{{\bf {0}}}\right\rangle $ and $%
\left\langle \delta ^{2}N_{{\bf {0}}}\right\rangle $ are obtained from the probability distribution function in our approach. The correct description of $%
\delta N_{{\bf {0}}}$ near zero temperature and critical temperature also shows the validity of our method. Thus our approach has provided in some sense a simple method recovering the applicability of the saddle-point method through the calculations of the probability distribution function of the system.
Interacting Bose gases Based on the Lowest Order Perturbation Theory
=====================================================================
Below the critical temperature, Bose-Einstein condensation results in a sharp enhancement of the density in the central region of the trap. This makes the interacting effect between atoms be much more important than above $T_{c}$. The correction to the condensate fraction and critical temperature due to the interatomic interaction has been discussed within grand canonical ensemble [@GIO3; @NAR3; @BER; @LIU] and canonical ensemble[@XIONG; @MULKEN]. In this section we investigate the role of interaction on the condensate fluctuations of a weakly interacting Bose gas.
Using the lowest-order perturbation theory the interaction energy of the system takes the form
$${E_{int}=2g\int n_{0}\left( {\bf {r}}\right) n_{T}\left( {\bf {r}}\right)
d^{3}{\bf {r}}+ g\int n_{T}^{2}\left( {\bf {r}}\right) d^{3}{\bf {r},}}
\label{inter-energy}$$
where $g=4\pi \hbar ^{2}a/m$ is the coupling constant fixed by the s-wave scattering length $a$. $n_{0}\left( {\bf {r}}\right) $ and $%
n_{T}\left( {\bf {r}}\right) $ are the density distributions of the condensate and normal gas, respectively.
Below the critical temperature, by Thomas-Fermi approximation the density distribution of the condensate reads
$${n_{0}\left( {\bf {r}}\right) =\frac{\mu -V_{ext}\left( {\bf {r}}\right) }{g}%
,} \label{density-condensate}$$
where $\mu $ is the chemical potential of the system. The temperature dependence of the chemical potential is then fixed by the number of atoms in the condensate
$${\mu (N_{{\bf {0}}},T)=\frac{\hbar \omega _{ho}}{2}\left( \frac{15N_{{\bf {0}%
}}a}{a_{ho}}\right) ^{2/5},} \label{chemical}$$
where $a_{ho}=\left( \hbar /m\omega _{ho}\right) ^{1/2}$ is the harmonic oscillator length. Moreover, since $\mu =\partial E_{{\bf {0}}%
}/\partial N_{{\bf {0}}}$, the energy per particle in the condensate turns out to be
$${\varepsilon _{{\bf {0}}}^{TF}=E_{{\bf {0}}}/N_{{\bf {0}}}=\frac{5}{7}\mu
(N_{{\bf {0}}},T).} \label{TF-energy}$$
As a first-order approximation, omitting the interaction between condensed and non-condensed atoms, the partition function of the system is given by
$${Z_{int}\left[ N\right] =\sum_{N_{{\bf {0}}}=0}^{N} \left\{ \exp \left[
-\beta N_{{\bf {0}}}\varepsilon _{{\bf {0}}}^{TF}\right] Z_{0}\left( N,N_{%
{\bf {0}}}\right) \right\}.} \label{par-hf}$$
From Eq. (\[non1\]) the most probable value reads
$${N_{{\bf {0}}}^{p}=N-\sum_{{\bf {n}}\neq {\bf {0}}}\frac{1}{\exp \left[
\beta \left( \varepsilon _{{\bf {n}}}-\mu (N_{{\bf 0}}^{p},T)\right) \right]
-1}.} \label{most-hf-1}$$
Using the density of states, [*i.e.*]{}, (\[density-state\]), one obtains the result for the most probable value $N_{{\bf {0}}}^{p}$
$${
N_{{\bf {0}}}^{p}=N\left( 1-t^{3}\right) -
}$$
$${\frac{\zeta \left( 2\right) }{%
\zeta \left( 3\right) } \frac{\mu ( N_{{\bf {0}}}^{p},T) t^{3}N}{k_{B}T}-
\frac{3\overline{\omega }\zeta \left( 2\right) }{2\omega _{ho} \left[\zeta
\left( 3\right)\right] ^{2/3}}t^{2}N^{2/3},} \label{most-hf}$$
where $t=T/T_{c}^{0}$ is the reduced temperature. Introducing the scaling parameter $\eta$ [@RMP]
$${\eta =\frac{\mu (T=0)}{k_{B}T_{c}^{0}}=1.57\left( \frac{N^{1/6}a}{a_{ho}}%
\right) ^{2/5},} \label{eta}$$
(\[most-hf\]) becomes
$${
N_{{\bf {0}}}^{p}=N\left( 1-t^{3}\right) -
}$$
$${\frac{\zeta \left( 2\right) }{%
\zeta \left( 3\right) } \eta Nt^{2}\left( \frac{N_{{\bf {0}}}^{p}}{N}\right)
^{2/5}- \frac{3\overline{\omega }\zeta \left( 2\right) } {2\omega
_{ho}\left[\zeta \left( 3\right) \right]^{2/3}}t^{2}N^{2/3}.}
\label{most-hf-tra}$$
Note that the corrections due to the interatomic interaction and finite number of particle of the system can be obtained simultaneously when (\[most-hf-tra\]) is used to calculate $\left\langle N_{\bf{0}}\right\rangle $ and $\left\langle \delta ^{2}N_{\bf{0}}\right\rangle $ of the system. The second term on the right hand side of (\[most-hf-tra\]) accounts for the correction of the interaction effect. The correction due to the interatomic interaction coincides with the lowest-order thermal depletion obtained in the grand canonical ensemble approach [@RMP].
For other $N_{{\bf {0}}}$, assuming $\frac{\partial }{\partial N_{{\bf {0}}}}%
q\left( N,N_{{\bf {0}}}\right) =\alpha \left( N,N_{{\bf
{0}}}\right) $, we get
$${N_{{\bf {0}}}=N-\sum_{{\bf {n}}\neq {\bf {0}}}\frac{1}{\exp \left[ \beta
\left( \varepsilon _{{\bf {n}}}-\mu (N_{{\bf 0}},T)\right) \right] \exp
\left[ -\alpha \left( N,N_{{\bf {0}}}\right) \right] -1}.}
\label{alpha-hf-1}$$
Combining Eqs. (\[most-hf-1\]) and (\[alpha-hf-1\]), one obtains the result for $\alpha \left( N,N_{{\bf {0}}}\right)
$:
$${\alpha \left( N,N_{{\bf {0}}}\right) =-\frac{\zeta \left( 3\right) \left(
N_{{\bf {0}}}-N_{{\bf {0}}}^{p}\right) }{\zeta \left( 2\right) Nt^{3}}+\frac{%
\mu ( N_{{\bf {0}}}^{p},T) -\mu ( N_{{\bf {0}}},T) }{k_{B}T}.}
\label{alpha-hf}$$
The probability distribution function of the interacting Bose gas is then
$${
G_{int}
\left( N,N_{{\bf {0}}}\right)
=A_{int}{\exp \left[ \int_{N_{{\bf {0}%
}}^{p}}^{N_{{\bf {0}}}}\alpha \left( N,N_{{\bf {0}}}\right) dN_{{\bf {0}}%
}\right] =}
}$$
$$\frac{{{A}_{int}}}{A_{ideal}}{G_{ideal}\left( N,N_{{\bf {0}}%
}\right) R_{int}\left( N,N_{{\bf {0}}}\right) ,} \label{dis-hf}$$
where $A_{int}$ is a normalization constant. $G_{ideal}\left( N,N_{%
{\bf {0}}}\right) $ is the probability distribution function given by the formula (\[dis-har\]) for the ideal harmonically trapped Bose gas. The correction $%
R_{int}\left( N,N_{{\bf {0}}}\right) $ originating from the interatomic interaction takes the form
$${
R_{int}\left( N,N_{{\bf {0}}}\right) =
\exp \left\{
\frac{\hbar \omega _{ho}
}{2k_{B}T}\left( \frac{15a}{a_{ho}}\right) ^{2/5}\times
\right.
}$$
$${\left. \left[ \left( N_{{\bf {0}}
}^{p}\right) ^{2/5}\left( N_{{\bf {0}}}-N_{{\bf {0}}}^{p}\right)- \frac{5}{7}
\left( N_{{\bf {0}}}^{7/5}-\left( N_{{\bf {0}}}^{p}\right) ^{7/5}\right)
\right] \right\}.} \label{r-hf}$$
Note that $G_{int}\left( N,N_{{\bf {0}}}\right) $ is not a Gaussian distribution function because of the existence of the non-Gaussian factor $R_{int}\left( N,N_{{\bf {0}}}\right) $.
From (\[fluc-ideal\]) and (\[dis-hf\]) we can obtain the numerical result of $\left\langle \delta ^{2}N_{{\bf
{0}}}\right\rangle _{int}$. In Fig. 3 we have provided the numerical result of $\delta N_{{\bf {0}}}$ for $N=1000$ interacting bosons confined in an isotropic harmonic trap with $%
a/a_{ho}=10^{-4}$ and $a/a_{ho}=10^{-3}$, respectively. The crossover from the interacting to the non-interacting Bose gases is clearly shown. From Fig. 3 we find that the repulsive interaction between atoms results in a decrease of the condensate fluctuations. For an attractive interaction, we anticipate that the corrections between atoms result in an increase for the condensate fluctuations.
The interaction between condensed and non-condensed atoms gives high-order correction to the thermodynamic properties of the system. Near the critical temperature, [*i.e.*]{}, when $N_{{\bf 0}}a/a_{ho}<<1$ [@RMP], we have $n_{0}\left( {\bf {r}}\right) =N_{{\bf 0}}\left( \frac{m\omega _{ho}^{2}}{%
\pi \hbar }\right) ^{3/2}e^{-m\left( \omega _{x}x^{2}+\omega
_{y}y^{2}+\omega _{z}z^{2}\right) /\hbar }$. In addition, we can adopt the semiclassical approximation for the normal gas [@RMP], [*i.e.*]{}, $n_{T}\left( {\bf {%
r}}\right) =\lambda _{T}^{-3}g_{3/2}\left( e^{-\beta V_{ext}\left( {\bf {r}}%
\right) }\right) $ with $\lambda _{T}=\left[ 2\pi \hbar
^{2}/\left( mk_{B}T\right) \right] ^{1/2}$ being the thermal wavelength. From Eqs. (\[non1\]) and (\[inter-energy\]), it is straightforward to obtain the most probable value $N_{{\bf
0}}^{p}$ near the critical temperature:
$${\frac{N_{{\bf 0}}^{p}}{N} =\frac{1-t^{3} -\frac{\zeta \left( 2\right) }{%
\zeta \left( 3\right) }\left[ 2-\frac{S}{\zeta \left( 3/2\right) }\right]
\theta t^{7/2} -\frac{3\zeta \left( 2\right) }{2 \left [\zeta \left(
3\right) \right ]^{2/3}} \frac{\overline{\omega }}{\omega _{ho}}t^{2}N^{-1/3}%
} {1+\frac{\zeta \left( 2\right) \theta N^{1/2}t^{2}} {\left
[\zeta \left( 3\right) \right ]^{1/2}\zeta \left( 3/2\right) }}. }
\label{near}$$
where $S=\sum_{i,j=1}^{\infty }{1}/{\zeta \left( 3\right) \left[
ij\left( i+j\right) \right] ^{3/2}}$. When obtaining (\[near\]) we have introduced a scaling parameter $\theta ={%
gn_{T}\left( {\bf {r}}={\bf 0},T_{c}^{0}\right) }/{k_{B}T_{c}^{0}}=2.02\frac{%
a}{a_{ho}}N^{1/6}$. $\theta$ can also be written in the form of $\theta=0.65\eta^{5/2}$. By setting $N_{{\bf 0}}^{p}=0$, from (\[near\]) we obtain the shift of the critical temperature:
$$\frac{\delta T_{c}^{0}}{T_{c}^{0}}= -1.65\frac{a}{a_{ho}}N^{1/6}-\frac{\zeta
\left( 2\right) } {2 \left [\zeta \left( 3\right) \right ]^{2/3}}\frac{%
\overline{\omega }}{\omega _{ho}}N^{-1/3}. \label{shift}$$
The first term on the right hand side of (\[shift\]) is the shift due to the interatomic interaction. It agrees with the results based on the local density approximation [@GIO3] obtained by using the grand-canonical ensemble approach. The second term in (\[shift\]) gives exactly the usual results due to effects of the finite number of particles [@RMP]. Thus in our approach, within the canonical ensemble the corrections due to the effects of the finite particle number and the interatomic interactions can be obtained simultaneously.
Below the critical temperature, the most probable value is given by
$${
\frac{N_{{\bf 0}}^{p}}{N} =1-t^{3} - \frac{3\overline{
\omega }\zeta \left( 2\right) }{2\omega _{ho} \left [\zeta \left( 3\right)
\right ]^{2/3}}t^{2}N^{-1/3}
}$$
$${- \frac{\zeta \left( 2\right) }{\zeta
\left( 3\right) }t^{3}\left[ \frac{\eta \xi ^{2/5}}{t}+1.49\frac{\eta t^{2}}{
\xi ^{2/5}}F\left( w\right) +0.14\eta ^{5/2}t^{1/2}\right]
,} \label{below1}$$
where $w=\left( \eta \xi ^{2/5}/t\right) ^{1/2}$. $F\left( w\right) $ is defined by
$${F\left( w\right) =0.53\left( 1-0.5e^{-0.23w^{3}}-0.5e^{-1.51w^{3}}\right) }.
\label{below2}$$
Omitting the high-order terms of the parameter $\eta $, the expression (\[below1\]) gives exactly the lowest-order correction of (\[most-hf-tra\]).
From (\[inter-energy\]) we can obtain the probability distribution function of the condensate when the interaction between condensed and non-condensed atoms is considered. Combining with the most probable value, one obtains $\left\langle N_{{\bf {0}}}\right\rangle $ and $\delta N_{{\bf {0%
}}}$ of the interacting Bose gases. In Fig. 4 the experimental parameter by Ensher [*et al.*]{} [@ENS] is used to plot $\left\langle N_{{\bf {0}}}\right\rangle /N$ within the canonical ensemble. Our results (solid line) agree well with the conclusion of Ref.[@GIO3] (circles) where semiclassical approximation is used in the frame of the grand-canonical ensemble. We can also obtain the numerical results for $\delta N_{{\bf {0}}}$ in the presence of the interaction between condensed and non-condensed atoms. The numerical result of $\delta N_{{\bf {0}}}$ is displayed in Fig. 3 with $a/a_{ho}=10^{-4}$ (circles) and $a/a_{ho}=10^{-3}$ (squares), respectively. Our calculations show that the repulsive interaction between the condensed and non-condensed atoms lowers the condensate fluctuations further.
Interacting Bose gases based on Bogoliubov theory
=================================================
Condensate fluctuations due to collective excitations have been recently investigated by Giorgini [*et al.*]{} [@GIO] within the traditional particle-number-nonconserving Bogoliubov approach. In Ref.[@GIO] the fluctuations from collective excitations are shown to follow the law $\left\langle \delta ^{2}N_{{\bf
{0}}}\right\rangle \sim N^{4/3}$. In this section, the Bogoliubov theory will be developed based on our canonical statistics to discuss the condensate fluctuations originating from collective excitations. According to the Bogoliubov theory [@BOG; @GIO2], the total number of particles out of the condensate is given by
$${N_{T}=\sum_{nl\neq 0}N_{nl}=\sum_{nl\neq 0}(u_{nl}^{2}+v_{nl}^{2})f_{nl}.}
\label{bthermal}$$
The real quantities $u_{nl}$ and $v_{nl}$ satisfy the relations
$${u_{nl}^{2}+v_{nl}^{2}=\frac{\left[ \left( \varepsilon
_{nl}^{B}\right) ^{2}+g^{2}n_{0}^{2}\right] ^{1/2}}{2\varepsilon
_{nl}^{B}}}, \label{uv-1}$$
$${u_{nl}v_{nl}=-\frac{gn_{0}}{2\varepsilon _{nl}^{B}},} \label{uv-2}$$
where $f_{nl}$ is the number of the collective excitations excited in the system at the thermal equilibrium
$${f_{nl}=\frac{1}{\exp \left[ \beta \varepsilon _{nl}^{B}\right] -1}}.
\label{bdistribution}$$
In addition, the energy of the collective excitations entering Eqs. (\[uv-1\]) and (\[uv-2\]) is given by the dispersion law [@STR]
$${\varepsilon _{nl}^{B}=\hbar \omega _{ho}\left( 2n^{2}+2nl+3n+l\right)
^{1/2}.} \label{benergy}$$
These phonon-like collective excitations are in excellent agreement with the measurement of experiments. The dispersion law (\[benergy\]) is valid if the conditions $N_{{\bf {0}}}a/a_{ho}>>1$ and $%
\varepsilon _{nl}<<\mu $ are satisfied. The contribution to the condensate fluctuations due to these discrete low energy modes are important because $f_{nl},u_{nl}^{2}+v_{nl}^{2},u_{nl}v_{nl}\propto 1/\sqrt{%
2n^{2}+2nl+3n+l}$ at low excitation energies.
In Eq. (\[bthermal\]) $N_{nl}$ can be regarded as the effective occupation number of non-condensed atoms, while
$${N_{nl}^{B}=\frac{N_{nl}}{u_{nl}^{2}+v_{nl}^{2}}=f_{nl}} \label{bogo}$$
is the occupation number of the collective excitations. From the form of $f_{nl}$ one can construct the partition function of the collective excitations in the frame of canonical ensemble
$${Z_{B}=\sum_{\left\{nl\right\} }\exp \left[ -\beta \sum_{nl}
N_{nl}^{B}\varepsilon _{nl}^{B}\right].} \label{bpar1}$$
From Eq. (\[bogo\]) $Z_{B}$ becomes
$${Z_{B}=\sum_{\left\{ \Sigma N_{nl}=N\right\} } \exp \left[ -\beta
\sum_{nl}N_{nl}\varepsilon _{nl}^{eff}\right],} \label{bpar2}$$
where $\varepsilon _{nl}^{eff}= \varepsilon
_{nl}^{B}/\left( u_{nl}^{2}+v_{nl}^{2}\right) $ can be taken as an effective energy level of the thermal atoms. In this case $Z_{B}$ is the partition function of a fictitious boson system, which is composed of $N$ non-interacting Bosons whose energy level is determined by $\varepsilon _{nl}^{eff}$. From (\[bpar2\]) the most probable value $N_{{\bf {0}}}^{p}$ is given by
$${N_{{\bf {0}}}^{p}=N-\sum_{nl\neq 0}\frac{1}{\exp \left[ \beta \left(
\varepsilon _{nl}^{eff}-\varepsilon _{nl=0}^{eff}\right) \right] -1}.}
\label{bmost}$$
It is obvious that the occupation number of low $n,l$ in Eq. (\[bmost\]) coincides with that of Eq. (\[bthermal\]). Other $N_{{\bf {0}}}$ is determined by
$${N_{{\bf {0}}}=N-\sum_{nl\neq 0}\frac{1}{\exp \left[ \beta \left(
\varepsilon _{nl}^{eff}-\varepsilon _{nl=0}^{eff}\right) \right] \exp \left[
-\alpha \left( N,N_{{\bf {0}}}\right) \right] -1}.} \label{bother}$$
From Eqs. (\[bmost\]) and (\[bother\]) we obtain
$${\alpha \left( N,N_{{\bf {0}}}\right) \approx -}\frac{N_{{\bf {0}}}-N_{{\bf {%
0}}}^{p}}{\sum_{nl\neq 0}\left( u_{nl}^{2}+v_{nl}^{2}\right)
^{2}f_{nl}^{2}}. \label{balpha}$$
When getting (\[balpha\]) we have used the approximation $f_{nl}\approx k_{B}T/\varepsilon _{nl}^{B}$ for low energy collective excitations. Thus the probability distribution function of the condensate is given by
$${G_{B}\left( N,N_{{\bf {0}}}\right) =A}_{B}{\exp \left[ -\frac{\left( N_{%
{\bf {0}}}-N_{{\bf {0}}}^{p}\right) ^{2}}{2\sum_{nl\neq 0}\left(
u_{nl}^{2}+v_{nl}^{2}\right) ^{2}f_{nl}^{2}}\right] ,} \label{bgauss-dis}$$
where $A_{B}$ is a normalization constant. Therefore, the condensate fluctuations due to the collective excitations reads
$${
\left\langle \delta^{2} N_{{\bf {0}}}\right\rangle_{collective}=
}$$
$${ \frac{%
\sum_{N_{{\bf {0}}}=0}^{N}N_{{\bf {0}}}^{2}G_{B}\left( N,N_{{\bf {0}}%
}\right) }{\sum_{N_{{\bf {0}}}=0}^{N}G_{B}\left( N,N_{{\bf {0}}}\right) }-
\left[ \frac{\sum_{N_{{\bf {0}}}=0}^{N}N_{{\bf {0}}}G_{B}\left( N,N_{{\bf {0}%
}}\right) }{\sum_{N_{{\bf {0}}}=0}^{N}G_{B}\left( N,N_{{\bf {0}}}\right) }%
\right] ^{2}.} \label{f-b}$$
Eqs. (\[bgauss-dis\]) and (\[f-b\]) provide the formulas for calculating the condensate fluctuations originating from the collective excitations.
Below the temperature $T_{m}$ which corresponds to the maximum fluctuations, we obtain the analytical result for the condensate fluctuations
$${
\left\langle \delta ^{2}N_{0}\right\rangle_{collective}=\frac{\pi ^{2}}{%
12\zeta \left( 2\right) }B\left( \frac{ma^{2}k_{B}T_{c}}{\hbar ^{2}}\right)
^{2/5}N^{4/3}=
}$$
$${ \frac{1}{2}B\left( \frac{ma^{2}k_{B}T_{c}}{\hbar ^{2}}\right)
^{2/5}N^{4/3},} \label{b-below}$$
where $B$ is a dimensionless parameter, which is the same as that obtained in Ref.[@GIO]. Note that compared with the result obtained by Ref.[@GIO], the coefficient in (\[b-below\]) differs by a factor $\frac{1}{2}$. The expression (\[b-below\]) shows clearly that the condensate fluctuations due to the collective excitations are anomalous, [*i.e.*]{}, proportional to $N^{4/3}$. Note that $G_{B}\left( N,N_{{\bf
{0}}}\right) $ is a Gaussian distribution function, the anomalous behavior of the condensate fluctuations comes from the factor $2\sum_{nl\neq 0}\left(
u_{nl}^{2}+v_{nl}^{2}\right) ^{2}f_{nl}^{2}$, which is proportional to $%
N^{4/3}$.
At the critical temperature the probability distribution is given by $G_{B}\left( T=T_{c}\right) =\exp \left[ -N_{%
{\bf {0}}}^{2}/\gamma \right]$, where $\gamma =2\sum _{nl}\left(
u_{nl}^{2}+v_{nl}^{2}\right) ^{2}f_{nl}^{2}$. In this case, we obtains the analytical result of the condensate fluctuations
$${\left\langle \delta ^{2}N_{0}\right\rangle |_{T=T_{c}}=0.18\gamma=
0.18B\left( \frac{ma^{2}k_{B}T_{c}}{\hbar ^{2}}\right) ^{2/5}N^{4/3}.}
\label{bnear-trap}$$
From (\[b-below\]) and (\[bnear-trap\]) we find that the behavior of the condensate fluctuations based on the Bogoliubov theory is rather different from that of the lowest-order perturbation theory.
Discussion and conclusion
=========================
In this paper, a canonical ensemble approach has been developed to investigate the mean ground state occupation number and condensate fluctuations for interacting and non-interacting Bose gases. Different from the conventional methods, the analytical probability distribution function of the condensate has been obtained directly from the partition function of the system. Based on the probability distribution function, we have calculated the thermodynamic properties of the Bose gas, such as the condensate fraction and the fluctuations. Through the calculations of the probability distribution function, we have provided a simple method to recover the applicability of the saddle-point method for studying the condensate fluctuations. In fact, the theory of the improved saddle-point method developed in this work can be applied straightforwardly to consider the condensate fluctuations in other physical systems, such as the interacting Bose gas confined in a box [@xiong1], the interacting Bose gas in low-dimensions, etc.. The probability distribution function can also be used to discuss other interesting problems, such as the phase diffusion of the condensate.
For the harmonically trapped interacting Bose gas, we found that different approximations for weakly interacting Bose gases give quite different theoretical predictions concerning the condensate fluctuations. In our opinion the lowest-order perturbation theory gives in some sense the condensate fluctuations due to normal thermal atoms, while the Bogoliubov theory gives the condensate fluctuations originating from the collective excitations. The contributions to the condensate fluctuations due to the collective excitations mainly come from the low energy modes, and it is obvious that the condensate fluctuations based on the lowest-order perturbation theory miss the contributions coming from the collective excitations. Considering the fact that the contributions due to low energy thermal atoms in the lowest order perturbation theory is relatively small, the overall condensate fluctuations may be written as
$${\left\langle \delta ^{2}N_{{\bf {0}}}\right\rangle _{all}=\left\langle
\delta ^{2}N_{{\bf {0}}}\right\rangle _{int}+\left\langle \delta ^{2}N_{{\bf
{0}}}\right\rangle _{collective},} \label{all-fluctuation}$$
where $\left\langle \delta ^{2}N_{{\bf
{0}}}\right\rangle _{int}$ and $\left\langle \delta ^{2}N_{{\bf
{0}}}\right\rangle _{collective}$ are condensate fluctuations due to the normal thermal atoms and the collective excitations, respectively.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by the Science Foundation of Zhijiang College, Zhejiang University of Technology and Natural Science Foundation of Zhejiang Province. G. X. Huang was supported by the National Natural Science Foundation of China, the Trans-Century Training Programme Foundation for the Talents and the University Key Teacher Foundation of Chinese Ministry of Education. S. J. Liu and H. W. Xiong thank Professors G. S. Jia and J. F. Shen for their enormous encouragement.
Appendix {#appendix .unnumbered}
========
In this appendix, the method of saddle-point integration described by Darwin and Fowler [@DAR] is used to investigate the partition function of the fictitious $N-N_{{\bf {0}}}$ non-interacting bosons. The partition function of the fictitious system is given by
$${Z_{0}\left( N_{T}\right) =\sum_{\sum_{{\bf {n}}\neq {\bf 0}}N_{{\bf {n}}%
}=N_{T}}\exp \left[ -\beta \sum_{{\bf {n}\neq 0}}N_{{\bf {n}}}\varepsilon _{%
{\bf {n}}}\right] ,} \label{a-function-1}$$
where $N_{T}=N-N_{{\bf {0}}}$ is the number of particles out of the condensate.
Because of the restriction $\sum_{{\bf {n}\neq 0}}N_{{\bf {n}}}=N_{T}$ in the summation of Eq. (\[a-function-1\]), $Z_{0}\left( N_{T}\right) $ can not be explicitly evaluated. To proceed we define a generating function for $%
Z_{0}\left( N_{T}\right) $ in the following manner. For any complex number $%
z $, we take
$${G_{0}\left( T,z\right) =\sum_{N_{T}=0}^{\infty }z^{N_{T}}Z_{0}\left(
N_{T}\right).} \label{a-generate-define}$$
The generating function can be evaluated easily. The result of $%
G_{0}\left( T,z\right) $ is given by
$${G_{0}\left( T,z\right) =\prod_{{\bf {n}}\neq {\bf 0}}\frac{1}{1-z\exp
\left[ -\beta \varepsilon _{{\bf {n}}}\right] }.} \label{a-generate}$$
To obtain $Z_{0}\left( N_{T}\right) $ we note that by definition $%
Z_{0}\left( N_{T}\right) $ is the coefficient of $z^{N_{T}}$ in the expansion of $G_{0}\left( T,z\right) $ in powers of $z$. Therefore we have
$${Z_{0}\left( N_{T}\right) =\frac{1}{2\pi i}\oint dz\frac{G_{0}\left(
T,z\right) }{z^{N_{T}+1}},} \label{a-relation}$$
where the contour of integration is a closed path in the complex $%
z $ plane about $z=0$. Let $g\left( z\right) $ be defined by
$${\exp \left[ g\left( z\right) \right] =\frac{G_{0}\left( T,z\right) }{%
z^{N_{T}+1}},} \label{a-g-function}$$
then $Z_{0}\left( N_{T}\right) $ becomes
$${Z_{0}\left( N_{T}\right) =\frac{1}{2\pi i}\oint dz\exp \left[ g\left(
z\right) \right].} \label{a-z-gfunction}$$
The saddle point $z_{0}$ is determined by
$${\frac{\partial g\left( z_{0}\right) }{\partial z_{0}}=0.} \label{a-saddle}$$
From Eq. (\[a-g-function\]) we obtain
$${N_{T}=z_{0}\frac{\partial }{\partial z_{0}}\ln G_{0}\left( T,z_{0}\right)
-1.} \label{a-ttt}$$
By Eq. (\[a-generate\]), one gets
$${N_{T}=\sum_{{\bf {n}}\neq {\bf 0}}\frac{1}{\exp \left[ \beta \varepsilon _{%
{\bf {n}}}\right] z_{0}^{-1}-1}.} \label{a-thermal}$$
Noting that Eq. (\[a-thermal\]) is exactly the equation to determine the number of condensed atoms within the grand canonical ensemble, the saddle point $z_{0}$ can be also regarded as the fugacity of the fictitious $N-N_{{\bf {0}}}$ non-interacting bosons.
Expanding the integrand of Eq. (\[a-z-gfunction\]) about $z=z_{0}$, we have
$${
Z_{0}\left( N_{T}\right) =\frac{1}{2\pi i}\oint dz\exp \left[ g\left(
z_{0}\right) +\right.
}$$
$${\left. \frac{1}{2}\left( z-z_{0}\right) ^{2}\frac{\partial ^{2}}{%
\partial z_{0}^{2}}g\left( z_{0}\right) +\cdots \right],}
\label{a-z-expansion}$$
where
$${\frac{\partial ^{2}}{\partial z_{0}^{2}}g\left( z_{0}\right) =\frac{%
G_{0}^{\prime \prime} \left( T,z_{0}\right) }{G_{0}\left( T,z_{0}\right) }-%
\frac{N_{T}^{2}-N_{T}}{z_{0}^{2}}.} \label{a-g-second}$$
By putting $z-z_{0}=iy$ we obtain
$${Z_{0}\left( N_{T}\right) \approx \frac{\exp \left[ g\left( z_{0}\right)
\right] }{2\pi }\int_{-\infty }^{\infty }\exp \left[ -\frac{1}{2}\frac{%
\partial ^{2}}{\partial z_{0}^{2}}g\left( z_{0}\right) y^{2}\right] dy.}
\label{a-z-finial1}$$
Thus we have
$${Z_{0}\left( N_{T}\right) =\frac{G_{0}\left( T,z_{0}\right) }{%
z_{0}^{N_{T}+1} \left[ 2\pi g^{\prime \prime}\left( z_{0}\right) \right]
^{1/2}}.} \label{a-z-finial}$$
With these results the free energy $A_{0}\left( N,N_{{\bf
{0}}}\right) $ of the fictitious system is then given by
$${
A_{0}\left( N,N_{{\bf {0}}}\right) = -k_{B}T\left\{ \ln G_{0}\left(
T,z_{0}\right) -N_{T}\ln z_{0}\right.
}$$
$${\left .-\ln z_{0}-\frac{1}{2}\ln \left[ 2\pi
g^{\prime \prime}\left( z_{0}\right) \right] \right\}.} \label{a-free-1}$$
In the case of $N_{T}>>1$, the last two terms in Eq. (\[a-free-1\]) can be omitted. Therefore
$${A_{0}\left( N,N_{{\bf {0}}}\right) =-k_{B}T\left[ \ln G_{0}\left(
T,z_{0}\right) -N_{T}\ln z_{0}\right].} \label{a-free-2}$$
From Eq. (\[a-generate\]), we obtain the relation between $%
A_{0}\left( N,N_{{\bf {0}}}\right) $ and $z_{0}$ of the fictitious system:
$${-\beta \frac{\partial }{\partial N_{{\bf {0}}}}A_{0}\left( N,N_{{\bf {0}}%
}\right) =\ln z_{0}.} \label{a-free-relation}$$
Eqs. (\[a-thermal\]) and (\[a-free-relation\]) are useful relations used in the text.
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|
---
bibliography:
- 'neurips.bib'
title: |
Supplementary Material\
README: REpresentation learning by fairness-Aware Disentangling MEthod
---
Appendix A: Composition of UTK Face dataset in our experiments
==============================================================
In our experiments, we compose train, validation, and test sets of UTK Face dataset [@zhifei2017cvpr] as shown in Table \[table:appendixA\]. Firstly, we compose a skewed dataset since it is not challenging to perform fair classification when representation is learned on a balanced dataset. For example, we set more Caucasians (E=1) to be male (G=1) than female (G=0) but more other ethnicities (E=0) to be female than male. On the other hand, we compose balanced validation and test sets since skewed validation/test sets give an advantage to specific biased models in evaluation.
Appendix B: Comparison of standard accuracy and *equalized accuracy*
====================================================================
To demonstrate a contribution of *equalized accuracy* more clearly, we assume a skewed test dataset and two extremely biased classification results of model A and B as shown in Table \[table:appendixB\]. In this circumstance, the standard accuracy is calculated as follows: model A = $\frac{4}{10}\times90\% + \frac{1}{10}\times10\% + \frac{1}{10}\times10\% + \frac{4}{10}\times90\% = \textbf{74\%}$, model B =$\frac{4}{10}\times10\% + \frac{1}{10}\times90\% + \frac{1}{10}\times90\% + \frac{4}{10}\times10\% = \textbf{26\%}$. However, it is difficult to claim that model A performs better than model B even if the accuracy of model A is much higher. The result is due simply to the skewed test dataset and model A has the same accuracy with model B as 50% on a fully balanced test dataset. On the other hand, *equalized accuracy* is formulated as follows: and calculated as follows: model A = $\frac{1}{4}[90\% +10\% +10\% +90\%]= \textbf{50\%}$, model B = $\frac{1}{4}[10\% +90\% +90\% +10\%]= \textbf{50\%}$. Therefore, *equalized accuracy* measures a performance independent of the distribution of a test dataset and has same effect as a balanced test dataset.
Appendix C: The detailed structures of networks
===============================================
Table \[table:appendixC\_2\] and Table \[table:appendixC\] show the detailed structures of VAE, discriminator, and classifiers. All models [@vae; @Higgins2017betaVAELB; @pmlr-v80-kim18b; @pmlr-v97-creager19a] used in our experiments have the same structures as above for fair evaluation.
Appendix D: Visualization of disentangled representation
========================================================
To validate that the subspaces of our model are effectively disentangled, we visualize them using t-SNE [@t-sne]. Firstly, we learn VAE [@vae] and FD-VAE on CelebA dataset [@liu2015faceattributes]. In this process, we set the target and protected attributes to attractive and male attributes, respectively. Next, we use the learned models to get representation of all data in the test set. In addition, we separate the representation into three subspaces, which are used as data points in t-SNE (for VAE, we evenly divide it into three spaces that have the same dimensions with our subspaces). Figure \[fig:appendixD\] shows the t-SNE visualization on the subspaces. The blue, green, and orange points denote TAL, PAL, and MAL in our model, respectively, and each divided space in VAE. It shows that TAL, PAL, and MAL of our model are effectively disentangled.
Appendix E: Computing infrastructure and implementation details
===============================================================
We conduct all experiments on a single GTX 1080Ti GPU and develop all models [@vae; @Higgins2017betaVAELB; @pmlr-v80-kim18b; @pmlr-v97-creager19a] using PyTorch. In representation learning, the models are trained for about 120 epoch on CelebA dataset and 80 epoch on UTK Face dataset. The learning rate is set to $10^{-4}$ on both datasets. In downstream task learning, the models are trained for about 30 epoch and the learning rate is set to $10^{-6}$ on both datasets. The batch size is fixed at 256 in all steps.
|
---
abstract: 'We present stellar and dark matter (DM) density profiles for a sample of seven massive, relaxed galaxy clusters derived from strong and weak gravitational lensing and resolved stellar kinematic observations within the centrally-located brightest cluster galaxies (BCGs). In [Paper I]{} of the series, we demonstrated that the *total* density profile derived from these data, which span three decades in radius, is consistent with numerical *DM-only* simulations at radii $\gtrsim 5-10$ kpc, despite the significant contribution of stellar material in the core. Here we decompose the inner mass profiles of these clusters into stellar and dark components. Parametrizing the DM density profile as a power law $\rho_{\textrm{DM}} \propto r^{-\beta}$ on small scales, we find a mean slope $\langle \beta \rangle = 0.50 \pm 0.10~\textrm{(random)}~{}^{+0.14}_{-0.13}~\textrm{(systematic)}$. Alternatively, cored Navarro–Frenk–White (NFW) profiles with $\langle \log r_{\textrm{core}}/\textrm{kpc}\rangle = 1.14 \pm 0.13 ^{+0.14}_{-0.22}$ provide an equally good description. These density profiles are significantly shallower than canonical NFW models at radii $\lesssim 30$ kpc, comparable to the effective radii of the BCGs. The inner DM profile is correlated with the distribution of stars in the BCG, suggesting a connection between the inner halo and the assembly of stars in the central galaxy. The stellar mass-to-light ratio inferred from lensing and stellar dynamics is consistent with that inferred using stellar population synthesis models if a Salpeter initial mass function is adopted. We compare these results to theories describing the interaction between baryons and DM in cluster cores, including adiabatic contraction models and the possible effects of galaxy mergers and active galactic nucleus feedback, and evaluate possible signatures of alternative DM candidates.'
author:
- 'Andrew B. Newman, Tommaso Treu, Richard S. Ellis, and David J. Sand'
bibliography:
- 'paper2.bib'
title: |
The Density Profiles of Massive, Relaxed Galaxy Clusters.\
II. Separating Luminous and Dark Matter in Cluster Cores
---
Introduction {#sec:intro}
============
The internal structure of dark matter (DM) halos is a key prediction of the cold dark matter (CDM) paradigm. Numerical simulations following the detailed structure of collisionless CDM halos [e.g., @NFW96; @Ghigna00; @Diemand05; @Graham06; @Gao12] generically produce a central density cusp with $\rho_{\textrm{DM}} \sim r^{-1}$, characteristic of the Navarro–Frenk–White [NFW; @NFW96] form, probably becoming slightly shallower on very small scales [e.g., @Navarro10]. On the hand, simulations are only beginning to make predictions for DM halos that include baryons, which could profoundly reshape their host halos. The structure of real DM halos thus contains important information about galaxy formation, but there is currently no theoretical consensus on the magnitude or even sign of these baryonic effects, particularly over a wide range in mass. Additionally, the microphysics of the unknown DM particle could become important in the densest regions, and the inner structure of DM halos may therefore provide valuable indirect clues to its nature [e.g., @Spergel00; @Abazajian01; @Kaplinghat05; @Peter10].
Given the current uncertainty, observations are clearly in a good position to guide theoretical efforts. However, measurements of DM mass profiles are extremely challenging and are usually limited by confusion with baryons, the small dynamic range of the observations, and degeneracies that are inherent to individual mass probes (e.g., velocity anisotropy). Clusters of galaxies are promising locations to make progress. Accurate mass measures are available via many independent observational probes, especially gravitational lensing and X-ray emission (see @Allen11 [@Kneib11] for reviews, and references in [Paper I]{}). As we have shown [@N09; @N11], combining stellar kinematics with strong and weak gravitational lensing yields constraints over three decades in radius. This is comparable to the best simulations and is thus suitable for detailed comparison of the DM profile shape if the baryonic mass can be constrained. On small scales in relaxed clusters, the latter is dominated by stars in the central brightest cluster galaxy (BCG).
@Sand02 [@Sand04] demonstrated the utility of combining resolved stellar kinematics with strong lensing to constrain two-component mass models, i.e., the BCG stars and DM halo separately (see @MiraldaEscude95 [@Natarajan96]). @Sand04 studied six clusters and inferred a mean $\langle \beta \rangle = 0.52 \pm 0.05$, where $\rho_{\textrm{DM}} \propto r^{-\beta}$, significantly shallower than an NFW cusp having $\beta = 1$. They further noted possible variation in $\beta$ from cluster to cluster. @Sand08 improved on this analysis for two clusters (MS2137 and A383) by relaxing the assumption of axial symmetry in the lensing analysis, instead conducting a full two-dimensional study. They found this did not alter their earlier findings, but noted that the inferred DM slope $\beta$ is sensitive to the adopted scale radius $r_s$, which could only be constrained by additional mass probes at larger radii. This was implemented by @N09 in A611 through the addition of weak lensing data. In @N11, we further extended the methodology in A383 by constraining the role of projection effects (i.e., line-of-sight (l.o.s.) ellipticity; @Gavazzi05) via a comparison of X-ray and lensing data. We also presented a radially-extended velocity dispersion profile measured in a very deep spectroscopic observation. In both A611 and A383, we confirmed a shallow inner DM cusp with $\beta < 0.3$ (68% CL) and $\beta = 0.59^{+0.30}_{-0.35}$, respectively.[^1]
In [Paper I]{} of the present series, we presented strong and weak lensing and stellar kinematic data for a sample of seven massive ($M_{200} = 0.4 - 2 \times 10^{15} {\,\textrm{M}_\sun}$), relaxed galaxy clusters at $z = 0.19 - 0.31$. This built upon our earlier papers by enlarging the sample of clusters with the highest-quality data: weak lensing measured using deep multi-color imaging, primarily from the Subaru telescope, extended stellar kinematic profiles in the BCG obtained primarily at the Keck telescopes, and multiply-imaged sources located in *Hubble Space Telescope* (*HST*) imaging (25 strongly lensed sources in total, of which 21 have spectroscopic redshifts). We showed that the *total* inner density profile is remarkably well-described by numerical simulations containing *only CDM* at radii $\gtrsim 5-10$ kpc, despite the significant contribution of stellar mass on these scales.
Here we extend [Paper I]{} by dissecting the stellar and DM contributions, using improved versions of the techniques developed in our earlier papers. We first show how the mass content of the BCGs can be constrained using information from the entire sample. We then isolate the DM density profiles and quantify their behavior on small scales. We show that the DM profiles become shallower than NFW models within $\approx 30$ kpc, roughly the typical effective radius of the BCGs. Furthermore, the inner DM density profiles exhibit likely variation from cluster to cluster, and this variation is correlated with the properties of the BCG. Finally, we interpret our results in the context of the recent theoretical literature, focusing on the interactions between baryons and DM in galaxy clusters and the possibility that cores in galaxy clusters are imprints of DM particle physics.
{width="0.9\linewidth"}
Throughout we adopt a $\Lambda$CDM cosmology with $\Omega_m = 0.3$, $\Omega_{\Lambda} = 0.7$, and $H_0 = 70$ km s${}^{-1}$ Mpc${}^{-1}$. Error bars and upper limits encompass the 68% confidence interval. When pairs of errors are quoted, they refer to the random and systematic components, respectively.
Data and Modeling {#sec:datamodel}
=================
Whereas the *total* density profiles were studied in [Paper I]{}, the goal of this paper is to use our two-component fits to separate the stellar and dark mass contributions in the cluster cores. All aspects of the data and modeling were discussed extensively in [Paper I]{} (Section 7). Here we provide a summary of the features most relevant for this paper. Firstly, the stellar mass in the BCG is modeled based on fits to the surface luminosity in *HST* imaging. A uniform stellar mass-to-light ratio ${\Upsilon_*}$ is assumed within each BCG. As discussed in [Paper I]{} (Sections 5.1 and 9.3), this is justified by the mild or null color gradients observed over the relevant radial interval. Non-BCG cluster galaxies – relevant as perturbations in the strong lens model – are included via scaling relations based on the fundamental plane ([Paper I]{}, Section 7).
Secondly, the cluster-scale smooth DM halo is parameterized using either a generalized NFW (gNFW) model $$\rho_{\textrm{DM}}(r) = \frac{\rho_s}{(r/r_s)^{\beta} (1 + r/r_s)^{3-\beta}} \label{eqn:gnfw}$$ or a cored NFW (cNFW) model with $$\rho_{\textrm{DM}}(r) = \frac{b \rho_s}{(1 + b r/r_s)(1+r/r_s)^2}. \label{eqn:cnfw}$$ Both models have the same large scale behavior ($\rho_{\textrm{DM}}\propto r^{-3}$), but the gNFW model contains a central power-law cusp with $d \log \rho_{\textrm{DM}} / d \log r \rightarrow -\beta$ as $r \rightarrow 0$, while the cNFW model asymptotes to a constant-density core within a characteristic radius $r_{\textrm{core}}=r_s/b$. Both models contain the NFW profile in the limits $\beta = 1$ and $r_{\textrm{core}} \rightarrow 0$ and therefore allow us to explore deviations from canonical CDM halos in the central regions. Broad, uninformative priors are placed on the halo parameters ($\rho_s, r_s,$ and $\beta$ or $b$) and ${\Upsilon_*}$ ([Paper I]{}, Table 7).
Based on the close alignment between the optical centers of the BCGs and both the X-ray centroids (typically separated by $\simeq 3$ kpc, comparable to the measurement errors) and the lensing-derived centers of mass, we fix the center of the halo to that of the BCG ([Paper I]{}, Sections 2 and 7.3). Furthermore, mass estimates derived from lensing agree well with independent X-ray observations, which constrains the l.o.s. ellipticity of the halo to be mild in all clusters except A383 [see Paper I, Section 8 and @N11].
We do not specifically distinguish the hot gas in the intracluster medium (ICM), which is thus implicitly included in the halo in our models. Since the distribution of the ICM is similar to that of the halo and comprises only a $\simeq 10\%$ mass fraction [e.g., @Allen04], subtracting the ICM to isolate the DM has very little effect on the *slope* of the density profile ($\Delta d \log \rho / d \log r \lesssim 0.05$; @N09 [@SommerLarsen10]), which is the main focus of this paper.
These models are constrained by three data sets. Firstly, the mass on scales of $\simeq 100$ kpc to 3 Mpc is constrained using gravitational shear (weak lensing) measured in deep, multi-color images primarily from the Subaru telescope. Secondly, the angular positions and redshifts of background galaxies that are strongly lensed by the clusters precisely constrain the mass from $\simeq 20$ to $100$ kpc, varying from cluster to cluster. In total we located 25 multiply-imaged sources, of which 21 have spectroscopic redshifts (7 were first presented in [Paper I]{}). Finally, the most unique aspect of our analysis is the inclusion of spatially resolved stellar kinematics within the BCGs. These measures are derived from long-slit spectra primarily obtained at the Keck telescopes. The stellar kinematic data typically extend to $R \approx 10-20$ kpc and display a very homogenous shape that rises with radius, indicating a rising total mass-to-light ratio as expected at the centers of massive clusters. As demonstrated in [Paper I]{} (Section 9), the mass models provide good fits to the full range of data.
For the purpose of distinguishing dark and stellar mass, the most important physical assumptions are that stellar mass follows light, as justified above, and that the DM halo is adequately described by a gNFW- or cNFW-like profile. The precise parametric form is not as critical as the presumption that the DM density turns over smoothly at small radii – either to a power-law cusp in the gNFW case, or to a constant density in the cNFW models – without a sharp upturn on small scales. This is reasonable: by design these profiles describe pure CDM halos in the appropriate limits, and although the effects of adding baryons are uncertain, adiabatic contraction prescriptions [@Gnedin04; @Gnedin11] predict DM profiles that are well-fit by gNFW models over the relevant range of radii when applied to halos and BCGs representative of our sample. Due to the density and radial extent of observational constraints (extended stellar kinematic profiles, strongly lensed galaxies usually at multiple redshifts, weak lensing), we emphasize that we are able to consider quite general families of DM halos in each cluster.
Separating Luminous and Dark Mass: The Role of the Stellar Mass-to-Light Ratio
==============================================================================
![Probability densities for $\log \alpha_{\textrm{SPS}}$, which parameterizes the stellar mass-to-light ratio ${\Upsilon_{*\textrm{V}}}$ relative to SPS models (Equation \[eqn:alphaSPS\]), are shown for each cluster (thin lines) and jointly for the entire sample (thick). The dotted curve shows the effective prior, composed of the flat prior on $\log {\Upsilon_{*\textrm{V}}}$ convolved with a Gaussian uncertainty on ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ of $\sigma_{\textrm{SPS}}=0.07$ dex. Arrows indicate the effect of adopting mildly anisotropic orbits with $\beta_{\textrm{aniso}} = \pm 0.2$.\[fig:alphaIMF\]](MLjoint_dual){width="0.75\linewidth"}
In individual clusters there is a degeneracy between the stellar mass-to-light ratio ${\Upsilon_{*\textrm{V}}}= M_*/L_{\textrm{V}}$ and the inner DM slope. This is illustrated in Figure \[fig:degen\], which shows results for the mass models summarized in Section \[sec:datamodel\] and derived in [Paper I]{} (Section 9). This degeneracy is expected, since stellar mass in the BCG can be traded against DM. Owing to the multiplicity of constraints described above, particularly kinematic measurements at small radii in the stellar-dominated regime, the model degeneracy is not complete, and each cluster does carry information on both ${\Upsilon_*}$ and $\beta$ or $b$.
It is already evident in Figure \[fig:degen\] that most of the clusters in our sample prefer a DM inner slope that is shallower than an NFW profile (i.e., $\beta < 1$), consistent with our previous findings [@Sand02; @Sand04; @Sand08; @N09; @N11]. However, it is also clear that the precision of the constraints on the inner slope could be increased if additional information regarding ${\Upsilon_*}$ is available. Indeed, most clusters are consistent with a wide range of ${\Upsilon_*}$ when viewed in isolation, due to the uncertainty arising from the degeneracy described above. Furthermore, the figure suggests a possible variation from cluster to cluster in the DM inner slope, but this conclusion may be contingent upon substantial variations in ${\Upsilon_*}$ as well. We do not have strong a priori expectations about the possible variation from cluster to cluster in the DM inner slope, particularly recalling the uncertain role of baryons in theoretical predictions. There are, however, several strong reasons to believe that the true physical variation in ${\Upsilon_*}$ within our sample is small.
Firstly, Figure \[fig:degen\] shows estimates of the stellar mass-to-light ratio ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ derived by fitting stellar population synthesis (SPS) models to the broadband colors of the BCGs (see [Paper I]{}, Section 5.2). Currently, SPS models cannot predict absolute masses more accurately than a factor of $\simeq 2$, primarily due to the unknown stellar initial mass function (IMF), which we discuss further in Section \[sec:MLcompare\]. On the other hand, *relative* stellar masses are more robust, especially within a homogeneous galaxy population. As the bottom right panel of Figure \[fig:degen\] demonstrates, the range in ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ within our sample at a fixed IMF is small. Assuming a @Chabrier03 IMF, the median $\langle {\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}\rangle = 2.2$; the full range is only $1.80-2.32$, and the rms scatter is $9\%$.[^2]
Secondly, the rms dispersion in the absolute luminosities $L_{\textrm{V}}$ of the BCGs in our sample is only 0.1 dex. This small variation is consistent with previous studies of BCGs as “standard candles” with uniform luminosities and colors [e.g., @Sandage72; @Postman95; @Collins98; @Bernardi07]. Finally, the environments of the BCGs are the same: by construction they are all central galaxies in massive clusters, and their central velocity dispersions are comparable. It would be very surprising if this uniformity in luminosity and ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$, which are thought to derive from a similar assembly history, were the result of a conspiracy that masks larger variations in stellar mass. Instead, based on these physical similarities, it is very likely that the BCGs in our sample have similar stellar masses and ${\Upsilon_{*\textrm{V}}}$. As we discuss in Section \[sec:MLcompare\], this is further supported by recent, independent studies.
With the well-motivated assumption that the BCGs in our sample have a similar ${\Upsilon_{*\textrm{V}}}$, we can use the full sample of seven clusters to *jointly* constrain its value, thereby improving the precision and robustness of our measurements of the DM profile. Before embarking on this, we consider how to handle the small variations in ${\Upsilon_{*\textrm{V}}}$ that we do anticipate, despite the overall similarity. The sample spans a redshift range $z = 0.19 - 0.31$, so some mild passive evolution is expected. Additionally, the BCGs with the lowest ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ estimates show optical emission lines and far-infrared photometry indicative of ongoing star formation (although it involves a small fraction of the stellar mass; see [Paper I]{}). These BCGs reside in the cool core clusters, consistent with earlier studies [@Bildfell08; @Loubser09; @Sanderson09a].
Therefore, a more precise technique is to define ${\Upsilon_*}$ for each cluster *relative to the SPS measurement*: $$\log \alpha_{\textrm{SPS}} = \log {\Upsilon_{*\textrm{V}}}/ {\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}.
\label{eqn:alphaSPS}$$ We can then use the full cluster sample to constrain $\langle \log \alpha_{\textrm{SPS}} \rangle$, which parameterizes a common, systematic offset from photometrically-derived stellar mass-to-light ratios. As describe in Section \[sec:MLcompare\], the most probable source for large systematic offsets is an IMF that differs from that assumed in the SPS models: in this case, that of Chabrier. However, our analysis does *not* depend on the physical origin of the offset, only that is it common among our BCGs. Since the variation in ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ is small compared to the range of ${\Upsilon_{*\textrm{V}}}$ explored in our fits (25% versus a factor of 5.3), this approach is not radically different from assuming a common ${\Upsilon_{*\textrm{V}}}$. However, it improves on that assumption by making use of SPS models to adjust for small differences in ${\Upsilon_{*\textrm{V}}}$ arising from age and dust, while making no assumption on the validity of their absolute mass scale.
Constraining the stellar mass scale {#sec:massscale}
-----------------------------------
Figure \[fig:alphaIMF\] shows the probability distribution for $\log \alpha_{\textrm{SPS}}$ derived in each cluster. The uncertainty in $\log \alpha_{\textrm{SPS}}$ arises from two sources: that in the ${\Upsilon_{*\textrm{V}}}$ derived from dynamics and lensing, and the uncertainty in ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ arising from random photometric errors. In [Paper I]{} we estimated the latter as $\sigma_{\textrm{SPS}} = 0.07$ dex. Thus, the probability distributions for $\log \alpha_{\textrm{SPS}}$ are derived by broadening those for $\log {\Upsilon_{*\textrm{V}}}$ by a Gaussian with a dispersion of $\sigma_{\textrm{SPS}}$.
We have argued that there are strong a priori reasons to expect that $\alpha_{\textrm{SPS}}$ is uniform across our sample of BCGs. Using the probability distributions in Figure \[fig:alphaIMF\], we can ask whether the lensing and kinematic data are indeed consistent with this assumption. One way to quantify this is to suppose that the true distribution of $\log \alpha_{\textrm{SPS}}$ is Gaussian and infer its intrinsic dispersion $\sigma_{\log \alpha}$. The formalism for inferring the probability distribution $P(\sigma_{\log \alpha})$ was discussed in [Paper I]{} (Section 9, and see @Bolton12). The preference for non-zero intrinsic scatter can then be assessed by $$\Delta P = \sqrt{2 \ln[P(\sigma_{\log \alpha} = \sigma_{\textrm{peak}}) / P(\sigma_{\log \alpha} = 0)]},
\label{eqn:DeltaP}$$ where $\sigma_{\textrm{peak}}$ is the location of the maximum of $P(\sigma_{\log \alpha})$. For a Gaussian distribution, $\Delta P$ is the number of standard deviations from the mean. We find $\Delta P = 0.85$, i.e., a $<1\sigma$ preference for intrinsic scatter. Thus, the lensing and kinematic data are consistent with (although they alone cannot prove) our assumption that there is little intrinsic variation in $\alpha_{\textrm{SPS}}$ within our sample.
With the physically-motivated assumption that $\alpha_{\textrm{SPS}}$ is the same for each BCG, we can constrain its common value simply by multiplying the seven independent probability distributions. The results are shown by the thick curves in Figure \[fig:alphaIMF\]. Very similar values of $\log \alpha_{\textrm{SPS}}=0.28\pm 0.05$ and $0.26\pm 0.05$ are derived using the gNFW and cNFW models, respectively, demonstrating that these results do not strongly depend on the exact halo model. Given the closeness of these results, in the following analysis we adopt $\log \alpha_{\textrm{SPS}} = 0.27 \pm 0.05$. Despite marginalizing over fairly general parameterizations of the DM profile, we are able to obtain informative results due to the high density of observational constraints and the sample size.
Taking the median $\langle {\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}\rangle = 2.2$, we find that $\log \alpha_{\textrm{SPS}} = 0.27$ corresponds to ${\Upsilon_{*\textrm{V}}}= 4.1$. In Section \[sec:betasys\], we describe sources of systematic uncertainty leading to a final estimate $\log \alpha_{\textrm{SPS}} = 0.27 \pm 0.05 {}^{+0.10}_{-0.16}$. In Section \[sec:MLcompare\], we discuss the physical implications of this result and compare to the recent literature on the stellar mass-to-light ratio and IMF in early-type galaxies.
{width="\linewidth"}
The Inner Dark Matter Density Profile
=====================================
We now turn to the the inner DM density profiles. In our earlier papers, we studied the inner DM density slope $\beta$ by marginalizing over the uncertainty in ${\Upsilon_*}$ separately in each cluster. With the benefit of a larger sample with improved data, we have now combined constraints from seven clusters to arrive at a joint measurement of the stellar mass scale $\alpha_{\textrm{SPS}}$ (Section \[sec:massscale\]). Incorporating this information, we can now conduct our analysis in a more physically consistent way that recognizes the homogeneity of the BCGs, as well as further reducing the remaining degeneracies between dark and stellar mass.
Technically, we implement the joint constraint on $\log \alpha_{\textrm{SPS}}$ via importance sampling [e.g., @Lewis02], reweighting the Markov chain samples to effectively convert our flat prior on $\log \alpha_{\textrm{SPS}}$ to a Gaussian with mean $\langle \log \alpha_{\textrm{SPS}} \rangle = 0.27$ and dispersion $\sigma = (\sigma_{\alpha}^2 + \sigma_{\textrm{SPS}}^2)^{1/2} = 0.09$. Here $\sigma_{\alpha} = 0.05$ dex is the uncertainty in $\langle \log \alpha_{\textrm{SPS}} \rangle$, and $\sigma_{\textrm{SPS}} = 0.07$ dex is the random error in ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ for each BCG. The latter accounts for the fact that $\alpha_{\textrm{SPS}}$ refers to a systematic offset from SPS-based mass estimates, but random errors due to photometric noise remain in each cluster.[^3]
Dark and stellar mass profiles
------------------------------
Figure \[fig:dens\] shows the resulting spherically-averaged density profiles for the DM halo, BCG stars, and their sum. The results based on gNFW and cNFW models are again quite similar, showing that the choice of parameterization does not strongly affect the derived density profiles. We do not detect an overall preference for one model over the other: the ratio of the total Bayesian evidence is consistent with unity.[^4] The black line segment in each panel spans $r/r_{200} = 0.003 - 0.03$, which is the interval over which the total density slope $\gamma_{\textrm{tot}}$ was defined in [Paper I]{}. Its slope $r^{-1.13}$ is the average measured in CDM-only cluster simulations from the Phoenix project [@Gao12]. As quantified in [Paper I]{}, the stars and DM sum to produce a slope very close to CDM-only simulations over this interval.
Now we can see that both stars and DM contribute significantly to the mass in this regime: stars dominate the density in the inner radius, while virtually all the mass is DM at the outer radius. This demonstrates a tight coordination between the inner DM profile and the distribution of stars: the NFW-like density slope is not a property of the DM halo or the BCG alone, but of their sum. As noted in [Paper I]{}, at yet smaller radii $r \lesssim 5-10$ kpc where stars are dominant – well within the mean effective radius $\langle R_e \rangle = 30$ kpc – the total density profile generally steepens.
As expected if the total density is NFW-like, the DM profiles become shallower only on scales where the BCG contributes significantly, roughly within $R_e$. As we describe in Section \[sec:compare\], our results thus do not conflict with other studies that claim the DM alone follows an NFW profile but are confined to $r \gtrsim R_e$. The stellar mass density in our models reaches that of the DM at a median radius of $\langle r \rangle = 7$ kpc. In terms of enclosed mass, equality occurs at $\langle r \rangle = 12$ kpc. Within 5 kpc the median DM fraction is $\langle f_{\textrm{DM}}\rangle=25\%$, similar to massive field ellipticals [e.g., @Auger10b], but within their three-dimensional half-light radii $r_h$ the BCGs are far more DM-dominated: $\langle f_{\textrm{DM}} \rangle=80\%$.
[lcc]{} MS2137 & $0.65^{+0.23}_{-0.30}$ & $0.45^{+0.38}_{-0.48}$\
A963 & $0.50^{+0.27}_{-0.30}$ & $0.87^{+0.61}_{-0.71}$\
A383 & $0.37^{+0.25}_{-0.23}$ & $0.37^{+0.72}_{-0.64}$\
A611 & $0.79^{+0.14}_{-0.19}$ & $0.47^{+0.39}_{-0.50}$\
A2537 & $0.23^{+0.18}_{-0.16}$ & $1.67^{+0.24}_{-0.23}$\
A2667 & $0.42^{+0.23}_{-0.25}$ & $1.29^{+0.49}_{-0.49}$\
A2390 & $0.82^{+0.13}_{-0.18}$ & $0.30^{+0.53}_{-0.34}$\
**All clusters** & $0.50\pm0.13$ & $1.14\pm0.13$\
$\beta_{\textrm{aniso}} = +0.2$ & $0.38^{+0.09}_{-0.07}$ & $1.11^{+0.14}_{-0.10} $\
$\beta_{\textrm{aniso}} = -0.2$ & $0.64^{+0.05}_{-0.09}$ & $0.96^{+0.24}_{-0.11} $\
Separate $\alpha_{\textrm{SPS}}$ & $0.62\pm0.14$ & $1.09^{+0.12}_{-0.21}$
Inner DM density slopes and core radii {#sec:innerslopes}
--------------------------------------
{width="0.45\linewidth"} {width="0.45\linewidth"}
Figure \[fig:DMparams\] shows the probability distributions for $\beta$ (gNFW) and $r_{\textrm{core}}$ (cNFW) obtained by marginalizing over the other parameters, again weighting the samples to incorporate our joint constraint on $\alpha_{\textrm{SPS}}$. Results for the individual clusters are listed in Table \[tab:beta\]. Every cluster prefers $\beta < 1$, i.e., an inner slope shallower than an NFW model. Thick black lines shows constraints on the mean: $\langle \beta \rangle = 0.50 \pm 0.13$ and $\langle \log r_{\textrm{core}}/\textrm{kpc} \rangle = 1.14 \pm 0.13$; the method for deriving these is outlined in the Appendix. We note that while the typical $r_{\textrm{core}} \approx 14$ kpc is small, the cNFW profile turns over rather slowly at small radii. Thus, while $r_{\textrm{core}}$ is the radius where the density falls to half of the corresponding NFW profile, significant deviations extend to $r \simeq (3-4) r_{\textrm{core}}$.
![Correlation between the size of the BCG and the inner DM profile. **Top:** Gray points show the total density slope $\gamma_{\textrm{tot}}$ presented in [Paper I]{}; this is measured over $r/r_{200} = 0.003-0.03$ and is not an asymptotic slope. The dashed horizontal line shows the mean slope measured in CDM-only cluster simulations [@Gao12] over the same interval. Colored points denote the asymptotic DM density slope $\beta$ measured in the gNFW models. Dotted lines show least-squares linear fits. The Spearman rank correlation coefficient $\rho$ and the corresponding two-sided $P_0$-value are listed. **Bottom:** The core radii $r_{\textrm{core}}$ of the cNFW models are shown, again indicating a correlation with $R_e$.\[fig:bcg\_halo\]](bcg_halo){width="0.9\linewidth"}
We can also ask whether there is evidence for intrinsic variation in the inner DM profiles. This can be quantified by assuming that the parent distributions of $\beta$ and $\log r_{\textrm{core}}$ are Gaussian, and using the method described in Section \[sec:massscale\] to infer its dispersion. We find some evidence for intrinsic scatter with $\sigma_{\beta} = 0.22^{+0.15}_{-0.11}$ and $\sigma_{\log r_{\textrm{core}}} = 0.57^{+0.33}_{-0.21}$. Its statistical significance can be assessed with the $\Delta P$ statistic (Equation \[eqn:DeltaP\]): we derive $\Delta P = 1.5$ and 2.6 for $\beta$ and $\log r_{\textrm{core}}$, respectively. This indicates a $\simeq 2\sigma$ preference for the presence of intrinsic scatter in the inner DM profile shape. While we have focused on relaxed clusters, we expect this variation would increase if a broader sample of clusters that includes recent mergers were considered.
A possible physical origin of this scatter is illustrated in Figure \[fig:bcg\_halo\]. Gray points in the top panel show the total density slope $\gamma_{\textrm{tot}}$. As described in [Paper I]{}, these show mild scatter around the mean slope measured in CDM-only simulations [dashed line, @Gao12] over the same radial interval ($r/r_{200}=0.003-0.03$). Here we see signs of a correlation with the size of the BCG, with more extended BCGs corresponding to shallower total slopes. The effect on the DM slope (colored points) appears stronger: larger BCGs are hosted by clusters with shallower DM slopes $\beta$, or equivalently larger core radii $r_{\textrm{core}}$ (bottom panel). Such a correlation is necessary for the dark and stellar mass to combine to a similar total density profile. The significance can be assessed using the Spearman rank correlation test. We find a probabilities $P_0 = 0.18$ and 0.07 of obtaining an equally strong correlation between $R_e$ and $\beta$ or $r_{\textrm{core}}$, respectively, in the null hypothesis of uncorrelated data (see caption to Figure \[fig:bcg\_halo\]).
Figure \[fig:bcg\_halo\] suggests that the DM profile in the cluster core is connected to the build-up of stars in the BCG. We return to this point in Section \[sec:discussion\] and discuss physical scenarios that may explain this. Although the correlations with $R_e$ are most convincing, they are not unique: we find correlations between $\beta$ or $r_{\textrm{core}}$ and the stellar mass or luminosity with nearly equal statistical significance. There is no sign of a correlation with the virial mass $M_{200}$ ($\rho = 0.11$ and 0.04 for the gNFW and cNFW models; see caption to Figure \[fig:bcg\_halo\]).[^5]
We emphasize that it is preferable to compare directly to the physical density profiles (Figure \[fig:dens\]) when possible, rather than only marginalized distributions for $\beta$. These results do not imply, for example, that a CDM density profile should be modified simply by maintaining the same $r_s$ and changing $\beta = 1$ to $\beta = 0.5$. Rather, $r_s$ also shifts in our fits such that significant changes in $\rho_{\textrm{DM}}$ are kept within $r \lesssim 30$ kpc. This degeneracy is simply a result of the gNFW parameterization.
Systematic uncertainties {#sec:betasys}
------------------------
A full discussion of the systematic uncertainties affecting our analysis was presented in [Paper I]{}, Section 9.3 (see also @Sand04). In the following, we review the most important effects and estimate their impact on $\alpha_{\textrm{SPS}}$ and the inner DM halo parameters $\beta$ and $b$.
One of the main sources of systematic uncertainty is our use of spherical dynamical models based on isotropic velocity dispersion tensors. As discussed in [Paper I]{} (Section 9.3), this is a good approximation for luminous, non-rotating giant ellipticals in their central regions [e.g., @Gerhard01; @Cappellari07]. Nonetheless, individual galaxies can exhibit mild anisotropy with $|\beta_{\textrm{aniso}}| = |1 - \sigma_{\theta}^2 / \sigma_r^2| \approx 0.2$, and the population as a whole also may be slightly radially biased. To estimate the impact this has on our analysis, we repeated the dynamical analysis taking a constant anisotropy parameter $\beta_{\textrm{aniso}} = \pm 0.2$. Arrows in Figure \[fig:alphaIMF\] show that individual clusters may shift by $\Delta \log {\Upsilon_*}= -0.16$ ($\beta_{\textrm{aniso}} = +0.2$) or $\Delta \log {\Upsilon_*}= +0.10$ ($\beta_{\textrm{aniso}} = -0.2$). Since this bias may be correlated among the BCGs, we consider these as systematic uncertainties in the mean: $\langle \log \alpha_{\textrm{SPS}} \rangle = 0.27 \pm 0.05 {}^{+0.10}_{-0.16}$. We note that the effects of anisotropy are larger here than for studies of field elliptical lenses [e.g., @Auger10], since the latter do not resolve kinematics well within $R_e$ where the impact of anisotropy on the l.o.s. velocity dispersion is largest.
Uncertainties in the orbital distribution have a milder effect on the parameters describing inner DM profile. If we adopt the same prior in $\langle \log \alpha_{\textrm{SPS}} \rangle$, taking $\beta_{\textrm{aniso}} = \pm 0.2$ leads to systematic shifts of $\Delta \langle \beta \rangle = \pm 0.13$ and $\Delta \langle \log r_{\textrm{core}} \rangle \approx -0.18$ (Table \[tab:beta\]). If we instead shift the prior on $\langle \log \alpha_{\textrm{SPS}} \rangle$ to match the results obtained with the corresponding $\beta_{\textrm{aniso}}$, we find $\Delta \langle \beta \rangle = +0.11, -0.02$ and $\Delta \langle \log r_{\textrm{core}} \rangle = -0.21, +0.08$. Based on these results, we estimate systematic uncertainties of $\Delta \langle\beta\rangle = \pm 0.13$ and $\Delta \langle \log r_{\textrm{core}} \rangle = -0.2, +0.1$ due to the orbital anisotropy.
We note that the clusters with the lowest inferred $\alpha_{\textrm{SPS}}$ in Figure \[fig:alphaIMF\] (MS2137 and A611) are those with the highest halo concentration parameters ([Paper I]{}, Section 10). These clusters have NFW-like *total* density profiles down to unusually small radii, with very weak steeping on small scales. In view of the similarity of $\alpha_{\textrm{SPS}}$ among the other five clusters and the agreement with independent results discussed in Section \[sec:MLcompare\], a likely explanation is that some of the stellar mass is effectively counted in the halo when ${\Upsilon_*}$ is allowed to vary freely from cluster to cluster. Nevertheless, omitting MS2137 and A611 would shift $\langle\log \alpha_{\textrm{SPS}}\rangle$ by only $+0.02$. In this respect our results are encouragingly robust. This highlights the utility of the ensemble of clusters as a robust constraint on ${\Upsilon_*}$.
L.o.s. ellipticity in the cluster halo can complicate the coupling of lensing and dynamical mass measurements, since lensing measures the mass contained in cylinders, while dynamical and X-ray measurements nearly measure the spherically averaged mass distribution. The close agreement between lensing- and X-ray-based mass measurements shows that this is not a major effect in our sample; the only exception is A383, in which the l.o.s. shape is explicitly accounted for in our analysis ([Paper I]{}, Section 8.1, and @N11). Specifically, the mean ratio of spherical mass measures $\langle M_{\textrm{X}} / M_{\textrm{lens}}\rangle = 1.1$ at $r \simeq 60$ kpc, the typical Einstein radius in our sample ([Paper I]{}, Section 8). This could be explained by a mean elongation of the cluster halos along the l.o.s. with ellipticity $\langle q - 1 \rangle \approx 0.1 - 0.2$ (although, as described in [Paper I]{}, $\langle M_{\textrm{X}} / M_{\textrm{lens}}\rangle$ and thus $\langle q \rangle$ are actually consistent with unity within the systematic uncertainties). Based on our study of A383, we estimate that a mean l.o.s. ellipticity of this magnitude would cause systematic shifts of $\Delta \langle \beta \rangle \approx 0.06$ and $\Delta \langle \log r_{\textrm{core}} \rangle \approx -0.1$.
Combining the effects of l.o.s. ellipticity and orbital anisotropy in quadrature, we arrive at final measurements $\langle \beta \rangle = 0.50 \pm 0.13 {}^{+0.14}_{-0.13}$ and $\langle \log r_{\textrm{core}} \rangle = 1.14 \pm 0.13 {}^{+0.14}_{-0.22}$ including random and systematic error estimates. Naturally, variations in orbital anisotropy or l.o.s. ellipticity could cause larger shifts on a cluster-by-cluster basis. Such effects could decrease the intrinsic scatter in $\beta$ and $r_{\textrm{core}}$ that we infer, but they would have to be correlated with the size or mass of the BCG (Figure \[fig:bcg\_halo\]). While we have argued that our method of deriving a common value of $\alpha_{\textrm{SPS}}$ is superior, we note that marginalizing over ${\Upsilon_{*\textrm{V}}}$ separately in each cluster as in our earlier papers would shift the mean $\langle \beta \rangle$ by $<1\sigma$ (see “Separate $\alpha_{\textrm{SPS}}$” Table \[tab:beta\]).
In Paper I we evaluated the effect of varying the positional uncertainty $\sigma_{\textrm{pos}}$ in the strong lensing analysis. In the context of this paper, we find a mean shift of $\Delta \beta = -0.1$ when taking $\sigma_{\textrm{pos}} = 0\farcs3$ rather than our fiducial $\sigma_{\textrm{pos}} = 0\farcs5$, while $\Delta \beta = +0.1$ when $\sigma_{\textrm{pos}} = 1\farcs0$ (although this choice is strongly disfavored by the Bayesian evidence; see Section 7.2 of Paper I). There is no significant dependence of $\log \alpha_{\textrm{SPS}}$ on $\sigma_{\textrm{pos}}$.
Finally, we recall evidence presented in [Paper I]{} that A2537 is a possible l.o.s. merger. Such an alignment could produce a spuriously shallow DM profile in a lensing analysis, and A2537 indeed has the shallowest slope in our sample. However, Figure \[fig:bcg\_halo\] provides another explanation: A2537 has the second-largest BCG in the sample. Thus, it does not appear that our results for A2537 are exceptional. Nevertheless, recognizing its unique nature in our sample, we note that excluding A2537 yields $\langle \beta \rangle = 0.69^{+0.10}_{-0.14}$ and $\log r_{\textrm{core}} = 0.59^{+0.26}_{-0.37}$, which does not change our main conclusions.
Comparison to previous results {#sec:compare}
==============================
Stellar mass-to-light ratio {#sec:MLcompare}
---------------------------
These results on the inner DM profile are informed by the common stellar mass normalization that we infer, so it is important to compare this result to other measurements to assess its reliability (see also @Cappellari12c for a recent review). As shown in Section \[sec:massscale\], we find $\log \alpha_{\textrm{SPS}} = 0.27 \pm 0.05$ for isotropic orbits, with a corresponding ${\Upsilon_{*\textrm{V}}}= 4.1 \pm 0.5$ and ${\Upsilon_{*\textrm{B}}}= 5.3 \pm 0.6$ at the median ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ and ${\Upsilon_{*\textrm{B}}^{\textrm{SPS}}}$. When comparing mass-to-light ratios at different redshifts, it is essential to account for luminosity evolution. Where necessary, we evolve samples as $d \log {\Upsilon_{*\textrm{V}}}/ dz = -0.64$ [@Treu01]. We note that the $\simeq0.05$ dex systematic uncertainty in ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ ([Paper I]{}, Section 5.2) is relevant only for the interpretation of ${\Upsilon_{*\textrm{V}}}$ in terms of stellar populations, but it does *not* affect the stellar mass and so has no effect on the derived mass profiles.
Discussion of ${\Upsilon_*}$ is often tied to the IMF. This is because the unknown IMF is the dominant source of uncertainty in the absolute mass scale for SPS models, especially for old galaxies [e.g., @Bell01; @Bundy05; @Cappellari06; @Auger09; @Grillo09; @Stott10]. If interpreted as a difference in IMF, our measured $\alpha_{\textrm{SPS}}$ indicates a normalization consistent with that of the @Salpeter55 IMF, which has $\log M_{*,\textrm{Salp}}/M_{*,\textrm{Chab}} = 0.25$ when extended over $0.1-100 {\,\textrm{M}_\sun}$.[^6]
Several other studies have used lensing and stellar dynamics to probe massive field and group ellipticals. [@Auger10] study the SLACS samples of early-type lenses using strong and weak lensing and stellar kinematics [see also @Gavazzi07; @Treu10]. Assuming an NFW halo, they infer $\log \alpha_{\textrm{SPS}} = 0.28 \pm 0.03$ at $M_* = 10^{11} {\,\textrm{M}_\sun}$.[^7] Assuming an adiabatically-contracted halo lowers this value by $0.11-0.14$, i.e., still heavier than a Chabrier IMF. They infer an intrinsic scatter of $<0.09$ dex in $\log \alpha_{\textrm{SPS}}$ within their sample of $\sigma \gtrsim 200$ km s${}^{-1}$ lenses [@Treu10]. @Lagattuta10 study ellipticals at slightly higher redshift using strong and weak lensing. Evolving their ${\Upsilon_*}$ from $\langle z \rangle \approx 0.6$ to our $\langle z\rangle=0.25$ yields ${\Upsilon_{*\textrm{V}}}= 4.7 \pm 0.7$, consistent with our results. Both of these works assume an NFW halo and a mass–concentration relation that follows theoretical expectations (i.e., a one-parameter halo). Our models include much more general halos, and the BCGs are much more DM-dominated. Thus, the uncertainty in ${\Upsilon_*}$ on an object-by-object basis is larger; nonetheless, the ensemble averages agree well. @Sonnenfeld12 studied a rare early-type lens that presents two Einstein rings, which allowed them also to relax assumptions on the DM profile. They find $\alpha_{\textrm{SPS}} = 0.30 \pm 0.09$ in our notation (see also @Spiniello11). @Zitrin09 took advantage of the unusually flat surface density profile in the lensing cluster MACS J1149.5+2223 ($z = 0.544$), which offers a clean subtraction of the dark halo to isolate the mass of the BCG. They estimate ${\Upsilon_{*\textrm{B}}}\approx 4.5 \pm 1$ ($\approx 7 \pm 2$ if evolved to our $\langle z \rangle = 0.25$).
Other studies have used integral field spectroscopy to construct detailed dynamical models of local ellipticals. @Cappellari12 [@Cappellari12c; @Cappellari12b] discuss the ATLAS${}^{\textrm{3D}}$ sample of early-type galaxies. At the highest velocity dispersions present, they infer $\log \alpha_{\textrm{SPS}} = 0.25$ (@Cappellari12c, Figure 9, converted to our definition of $\alpha_{\textrm{SPS}}$). Interestingly, there appears to be little or no intrinsic scatter in $\alpha_{\textrm{SPS}}$ at $\sigma_e \gtrsim 250$ km s${}^{-1}$, nearly at the lower limit of our BCGs, although only a handful of such objects are present in their sample. Along with the tightness of the $M/L - \sigma_e$ relation at high $\sigma_e$ [@Cappellari12b], this supports our claim that $\alpha_{\textrm{SPS}}$ should be nearly constant within our sample of BCGs. @McConnell11 studied the BCG of A2162 using long-slit kinematics and integral field spectroscopy with adaptive optics, finding ${\Upsilon_{*\textrm{R}}}= 4.6^{+0.3}_{-0.7}$ in their “maximum halo” solution. For comparison, our result evolved to $z = 0$ is ${\Upsilon_{*\textrm{R}}}= 4.1 \pm 0.5$.
Finally, the IMF in early-type galaxies has recently been studied using detailed spectral synthesis models that take advantage of surface gravity-sensitive stellar absorption lines. In very high-quality spectra, these constrain the abundance of low-mass dwarfs that contribute much to the stellar mass but very little to the integrated light. Although the degree of scatter remains unclear, these studies suggest that a Salpeter-like IMF – or possibly even heavier – is typical in high-dispersion ellipticals [@vanDokkum10; @vanDokkum12; @Conroy12; @Smith12]. In summary, our measurements are consistent with a variety of other recent works indicating a heavy (Salpeter-like) ${\Upsilon_*}$ in massive early-type galaxies. Encouragingly, studies based on completely independent techniques are beginning to converge on the same results.
The total inner density slope
-----------------------------
When comparing results on the inner density profiles of clusters, it is essential to understand the *radial range* that is being fit and whether the *total* density profile or that of the *dark matter* is being considered. This distinction is most important at radii $\lesssim 30$ kpc where the BCG contributes noticeably to the total mass. In [Paper I]{} we showed that the *total* density profiles in our sample are consistent with CDM-only simulations down to $r \simeq 5-10$ kpc. The mean total density slope $\langle \gamma_{\textrm{tot}} \rangle = 1.16 \pm 0.05 {}^{+0.05}_{-0.07}$ was precisely measured over $r/r_{200} = 0.003-0.03$ and found to be consistent with collisionless CDM-only simulations, which have $\langle \gamma_{\textrm{tot}} \rangle = 1.13$ ([Paper I]{}, Section 9). Note that $\gamma_{\textrm{tot}}$ is measured over a specific radial interval and is distinct from the asymptotic inner slopes of gNFW models, which we denote $\beta_{\textrm{tot}}$ and $\beta_{\textrm{DM}}$ in the following.
Most observational studies have focused on the total density profile. @Umetsu11 stacked density profiles for four clusters with high-quality lensing data and found that $\beta_{\textrm{tot}} = 0.89^{+0.27}_{-0.39}$, with the inner 40 kpc/$h$ excluded from their fit. @Morandi11 measured $\beta_{\textrm{tot}} = 0.90 \pm 0.05$ in A1689, excluding the inner 30 kpc, and @Coe10 also found that the total mass distribution is NFW-like. Using imaging from the CLASH survey [@CLASH], @Umetsu12_J1206 and @Zitrin11 derived $\beta_{\textrm{tot}} = 0.96^{+0.31}_{-0.49}$ (their “method 7”) and $\beta_{\textrm{tot}} = 1.08 \pm 0.07$ in MACS J1206.2-0847 and A383, respectively. These lensing results are consistent with our claims that the *total* density profile is NFW-like at $r \gtrsim 5-10$ kpc.
@Morandi12 use lensing and X-ray data to derive a total slope $\beta_{\textrm{tot}} = 1.02 \pm 0.06$ in A383 and contrast this with our earlier finding that $\beta_{\textrm{DM}} = 0.59^{+0.30}_{-0.35}$ in the same cluster [@N11].[^8] These results are not inconsistent. Figure \[fig:dens\] shows that the DM profile we infer in A383 becomes shallower than an NFW model only at $r \lesssim 30$ kpc. These scales are excluded by Morandi & Limousin in their fits precisely because of the uncertainty in the BCG stellar mass that we have addressed using stellar kinematics. At $r \gtrsim 30$ kpc the total density profile in our models – nearly equal to that of the DM – is NFW-like.
The dark matter inner density slope {#sec:dmresults}
-----------------------------------
Among the main scientific goals of studying the inner regions of clusters are testing predictions of the collisionless CDM paradigm, and understanding the formation of the central galaxy and its impact on the DM halo. Thus, although precise and robust measurements of the total density profile are very valuable, for these goals it is clearly important to understand how much of this mass is DM and how much is baryonic. Over the past decade, we have been developing tools to perform this separation [@Sand02; @Sand04; @Sand08; @N09; @N11]. The history of this progress was described in Section \[sec:intro\].
[@Sand04] measured a mean $\langle \beta_{\textrm{tot}} \rangle = 0.52 \pm 0.05$ in a sample of six clusters. We have improved on this earlier work in many ways: through the use of elliptical lens models, the addition of weak lensing data, the incorporation of multiple strongly lensed sources (usually located at different redshifts), the comparison with X-ray results to quantify l.o.s. effects, the deeper spectroscopic observations of the BCGs that have yielded more precise and radially-extended kinematic profiles, and through joint constraints on the stellar mass scale $\alpha_{\textrm{SPS}}$. This work has essentially confirmed our initial findings, with the present value $\langle \beta_{\textrm{DM}} \rangle = 0.50 \pm 0.10 {}^{+0.14}_{-0.13}$ consistent with @Sand04. (The smaller error bars quoted in the latter work are due to the more restrictive model assumptions, particularly a fixed scale radius $r_s$.)
Four of the clusters in the present sample have been previously studied in our earlier papers. In general our results for MS2137 and A963 are consistent with @Sand04 [@Sand08] within their uncertainties, although the present measurements supercede earlier ones due to the improvements described above. Our analysis of A383 is consistent with @N11. The results presented here for A611, on the other hand, are significantly different from @N09: we find $\beta = 0.79^{+0.14}_{-0.19}$, rather than $\beta < 0.3$ (68% confidence). This is attributable to two changes in the data: a revised spectroscopic redshift for a multiply imaged galaxy, and improved stellar kinematic measurements (see Sections 4.4 and 6.4 in [Paper I]{}).
As we have shown, it is difficult to separate the BCG and DM profiles with lensing alone due to the low density (or lack) of constraints near the center. Only in clusters with exceptional lensing configurations is this feasible. An interesting such case is A1703, which presents an unusual quad image close to the BCG. @Limousin08 and @Richard09 performed a two-component fit – a gNFW halo and BCG stars following light, as in this work – and derive $\beta_{\textrm{DM}} = 0.92^{+0.05}_{-0.04}$. (See @Oguri09 for a consistent result with a much larger error bar.) This may not be inconsistent with our findings, since two clusters in our sample prefer a similar slope (A611 and A2390, see Figure \[fig:DMparams\]), and there may be scatter from cluster to cluster.[^9] @Zitrin10 found that the *total* density profile in A1703 is well-fit by an NFW model.
X-ray studies of two nearby clusters (A2589 and A2029) have also shown that the total density follows an NFW profile down to $\approx 0.002-0.01 r_{\textrm{vir}}$ [@Lewis03; @Zappacosta06]. The latter authors noted that for any reasonable ${\Upsilon_*}$, this implies a shallower DM profile in the central regions where the stellar mass is significant. Their finding agrees well with our work, which has quantified the split between stars and DM. [@Schmidt07] studied a large sample of distant X-ray clusters. By assuming a typical BCG stellar mass, they estimated $\langle\beta_{\textrm{DM}}\rangle = 0.88 \pm 0.29$ (95% CL). Often the inner $\simeq 40$ kpc must be excluded from their analysis, making a direct comparison difficult.
![**Top:** *Total* density profiles, including baryons and DM, for our sample are overlaid on CDM-only simulations of massive clusters [@Gao12 dashed line, with grey band indicating the full range of the simulated clusters; see [Paper I]{}, Section 10]. The dot-dashed line shows a system in which an NFW halo with concentration $c_{200} = 4.5$ is altered using the modified adiabatic contraction model of @Gnedin11. Parameters of $A_0 = 1.5, w_0 = 0.85$ were used, with the BCG described by a @Jaffe83 profile with scale length $r_J / r_{200} = 0.02$ and mass fraction $M_* / M_{200} = 0.002$, which are representative of our sample. The radial extent of the data is indicated at the bottom of the panel. **Bottom:** As in the top panel, but showing DM only. (The Phoenix simulations thus do not change.) Note that CDM halos match the observed total density profiles better than those of DM alone. The inclusion of halo contraction (dot-dashed line) only exacerbates the difference with the mean observed DM slope (thick black segment).\[fig:gaoplot\]](gaoplot_rvir_total_paper2 "fig:"){width="0.98\linewidth"}\
![**Top:** *Total* density profiles, including baryons and DM, for our sample are overlaid on CDM-only simulations of massive clusters [@Gao12 dashed line, with grey band indicating the full range of the simulated clusters; see [Paper I]{}, Section 10]. The dot-dashed line shows a system in which an NFW halo with concentration $c_{200} = 4.5$ is altered using the modified adiabatic contraction model of @Gnedin11. Parameters of $A_0 = 1.5, w_0 = 0.85$ were used, with the BCG described by a @Jaffe83 profile with scale length $r_J / r_{200} = 0.02$ and mass fraction $M_* / M_{200} = 0.002$, which are representative of our sample. The radial extent of the data is indicated at the bottom of the panel. **Bottom:** As in the top panel, but showing DM only. (The Phoenix simulations thus do not change.) Note that CDM halos match the observed total density profiles better than those of DM alone. The inclusion of halo contraction (dot-dashed line) only exacerbates the difference with the mean observed DM slope (thick black segment).\[fig:gaoplot\]](gaoplot_rvir_dm_paper2 "fig:"){width="0.98\linewidth"}
Discussion and Conclusions {#sec:discussion}
==========================
By combining strong lensing, weak lensing, and stellar kinematic observations that extend from $\simeq 3$ kpc to beyond the virial radius, thus spanning the baryon- to DM-dominated regimes, we constrained flexible, physically motivated models of the dark and stellar mass distributions in seven massive, relaxed galaxy clusters. As discussed extensively in [Paper I]{}, the density profiles of stars and DM sum to produce a slope close to *CDM-only* simulations, at least outside the very central $\approx 5-10$ kpc where stars strongly dominate the mass. In this paper we isolated the dark and stellar density profiles to quantify the behavior of the DM on small scales, finding a mean asymptotic inner power law slope of $\langle \beta \rangle = 0.50 \pm 0.13 {}^{+0.14}_{-0.13}$, or equivalently a mean DM core radius $\langle \log r_{\textrm{core}} \rangle = 1.14 \pm 0.13 {}^{+0.14}_{-0.22}$. We also presented evidence for possible variation in the inner DM profile from cluster to cluster, which correlates with the size and mass of the BCG (Figure \[fig:bcg\_halo\]). This suggests a connection between the DM profile in cluster cores and the assembly of stars in the BCG.
The conclusion that the inner DM profile is shallower than that of pure CDM halos is fully consistent with our previous claims [@Sand02; @Sand04; @Sand08; @N09; @N11]. We have improved upon these earlier works by collecting improved data for a larger sample of clusters and refining our analysis techniques, as discussed in Section \[sec:dmresults\]. A particular advance enabled by this enlarged, improved sample was a joint constraint on the stellar mass-to-light ratio ${\Upsilon_*}$ of the BCGs in our sample, which we found to be elevated by $\langle \log \alpha_{\textrm{SPS}} \rangle = 0.27 \pm 0.05 {}^{+0.10}_{-0.16}$ dex relative to fits to SPS models that assume a Chabrier IMF. Our measurements are instead consistent with a Salpeter IMF (or any equivalently “heavy” IMF). As reviewed in Section \[sec:MLcompare\], this agrees with recent, independent studies of massive, early-type galaxies based on lensing, dynamics, and detailed spectroscopy [@Treu10; @Auger10; @Cappellari12; @Cappellari12c; @vanDokkum10; @vanDokkum12; @Conroy12]. These rapid developments in our understanding of stellar populations promise significant advances in disentangling the distributions of dark and baryonic mass across a range of systems.
Figure \[fig:gaoplot\] compares our measurements to high-resolution CDM cluster simulations from the Phoenix project [@Gao12], clearly demonstrating these DM-only simulations are a better match to the total density profile than that of the DM alone. In assessing the role of baryons on their host halos, much of the focus of the theoretical literature has been on the halo contraction [e.g., @Blumenthal86; @Gnedin04; @Gnedin11] expected to result from a central dissipative build-up of baryons. The dot-dashed lines in Figure \[fig:gaoplot\] show the effect of applying the modified adiabatic contraction model of @Gnedin11 to an NFW halo and BCG with parameters typical of our sample and of the Phoenix simulations (see details in caption). As expected, the DM profile steepens (bottom panel), only worsening the disagreement with our observations. It has been argued that this increase in central DM density from adiabatic contraction will boost the gamma-ray flux from DM annihilation in clusters [@Ando12].[^10] However, our results suggest that adiabatic contraction is not the main process that sets the density profile and that the net effect on the halo is actually opposite to predictions. (As discussed in [Paper I]{}, this does not necessarily imply that the same theory cannot make valid predictions at the galaxy scale, where the star formation efficiency and assembly history are very different.)
A possible formation scenario is that early, dissipative star formation in the main BCG progenitor creates a steep total density slope in the inner $\simeq 5-10$ kpc, where stars dominate the mass. This size scale is indeed similar to the observed sizes of very massive galaxies at $z \gtrsim 2.5$ [e.g., @Trujillo06; @vanDokkum08; @Newman12]. Subsequent assembly of the extended stellar envelope of the BCG – thought to be dominated by low-mass, dry accretion of satellites [e.g., @Naab09; @Laporte12] – then mostly replaces the DM already in place with satellite material, roughly maintaining the density.
Controlled simulations have indeed shown that dynamical friction between infalling satellites and the DM halo can heat the cusp and reduce the central DM density [e.g., @ElZant01; @ElZant04; @Nipoti04; @Jardel09; @Cole11]. This process is dissipationless, since the orbital energy lost by the satellites is transferred to the halo, and thus contrasts with the AC picture, in which the baryons’ energy is radiated away [@Lackner10]. A connection between the assembled stellar mass and the central DM density is naturally expected. Indeed, @Nipoti04 find an anti-correlation between the amount of stellar mass assembled in the BCG and the inner DM density slope $\beta$, similar to our observations (Figure \[fig:bcg\_halo\]; we note the satellites in their simulations included no DM). @DelPopolo12 discusses a similar anti-correlation arising in their analytic models for the same physical reason, with higher central baryon fractions corresponding to shallower DM density cusps. The strength of the dynamical friction effect depends on the density of the satellites and their resistance to stripping. @Laporte12 showed that when a stellar mass–size relation in line with $z \gtrsim 2$ observations is imposed in their simulations (offset by $3-5\times$ in size from the local relation), the central DM cusp is flattened to $\beta \simeq 0.3 - 0.7$, comparable to our observations. It is important to realize that numerical experiments investigating this dynamical effect have generally lacked a fully realistic and consistent treatment of the satellites, so improved simulations are needed. Nonetheless, the current results are promising.
Until the last few years, full hydrodynamical, cosmological cluster simulations that include cooling, star formation, and feedback did not produce shallow DM cusps or cores, which probably reflected overcooling effects. [@Mead10] and @Martizzi12 showed that the inclusion of active galactic nucleus (AGN) feedback greatly improves this situation (see discussion and references in [Paper I]{}, Section 10) and may also play a key role in lowering the central DM density. Understanding the impact that gas cooling, dynamical friction from stellar “clumps,” and AGN feedback have on the small-scale DM distribution is an important avenue for future simulations, and the data we have presented provide strong constraints.
In addition to the effect of baryons on the halo, various DM particle scenarios have also been proposed to reduce tension between CDM and observations on small scales, including the “missing satellites” problem and evidence for central DM cores or shallow cusps (for a recent review, see @Primack09). These include warm sterile neutrinos at the $\sim\textrm{keV}$ scale [e.g., @Abazajian01; @Boyarsky09; @Maccio12; @Menci12], “fuzzy” CDM composed from an ultralight scalar particle [@Hu00; @Woo09], DM produced from early decays [@Kaplinghat05], and DM that itself decays with a long timescale [@Peter10], among many other possibilities. The goal is to preserve the large-scale successes of CDM, while allowing for modifications at higher densities where the detailed properties of the DM particle might manifest. A scenario for which halo density profiles has been worked out in detail is a self-interacting DM particle [@Spergel00; @Yoshida00; @Dave01]. @Rocha12 and @Peter12 showed that a cross-section $\sigma \sim 0.1$ cm${}^2$ g${}^{-1}$ can produce $\approx 20$ kpc cores in clusters without violating any current constraints, e.g., from the asphericity of cluster cores or the Bullet Cluster [@Randall08]. Only the dense central regions of the halo are affected, where scattering can occur within a Hubble time.
These $\approx 20$ kpc core sizes are intriguingly similar to our observations. On the other hand, they are also very similar to the scale of the baryons, i.e., the size of the BCG. It is unclear why the total density profile should then match the shape expected of collisionless CDM. In these scenarios, the core size arises from the microphysics of the DM particle and presumably should not “know” about the size of the central galaxy (Figure \[fig:bcg\_halo\]), for example. Thus, observations of clusters alone cannot provide unambiguous support for alternative DM theories. Global comparisons across a wide range of mass scales (for instance, a cross-section that also produces correct core sizes and densities in dwarf galaxies) remain an essential test for attempts to explain low central halo densities in terms of the DM particle.
It is a pleasure to acknowledge helpful conversations with Annika Peter. We thank Liang Gao for providing the Phoenix simulation results. The anonymous referee is thanked for a helpful report. R.S.E. acknowledges financial support from DOE grant DE-SC0001101. Research support by the Packard Foundation is gratefully acknowledged by T.T. The authors recognize and acknowledge the cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
In Section \[sec:innerslopes\], we described how posterior probability distributions $P(\beta)$ and $P(\log r_{\textrm{core}})$ are derived for each cluster by weighting the samples in the Markov chains derived in [Paper I]{}. The weights $$w = \frac{1}{\sqrt{2\pi} \sigma} \exp\left[-\frac{1}{2} \left(\frac{\log \alpha_{\textrm{SPS}} - \langle \log \alpha_{\textrm{SPS}} \rangle}{\sigma}\right)^2\right]
\label{eqn:weights}$$ effectively convert a flat prior on $\log \alpha_{\textrm{SPS}}$ ([Paper I]{}, Section 7) to a Gaussian with mean $\langle \log \alpha_{\textrm{SPS}} \rangle = 0.27$ and a dispersion $\sigma = (\sigma_{\alpha}^2 + \sigma_{\textrm{SPS}}^2)^{1/2}$. This dispersion accounts for two sources of error: the uncertainty $\sigma_{\alpha}=0.05$ dex in the global systematic offset $\langle \log \alpha_{\textrm{SPS}} \rangle$ from SPS estimates ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$, and the random photometric uncertainty $\sigma_{\textrm{SPS}}=0.07$ dex in ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ for each cluster. The first uncertainty is correlated across the entire sample, while the second is not.
Therefore, to obtain constrains on the mean $\langle \beta \rangle$ and $\langle \log r_{\textrm{core}} \rangle$, the probability distributions derived for each cluster in this manner cannot simply be multiplied, since they are not independent. Instead, we calculate the posterior probability of $\langle \beta \rangle$ as $$P(\langle \beta \rangle) \propto \int P(\langle \beta \rangle | \log \alpha_{\textrm{SPS}}) P(\log \alpha_{\textrm{SPS}})\,d\alpha_{\textrm{SPS}}.$$ Here $P(\langle \beta \rangle | \log \alpha_{\textrm{SPS}})$ is the posterior distribution of $\langle \beta \rangle$ at a fixed value of $\log \alpha_{\textrm{SPS}}$. It is obtained by multiplying the probability densities $P(\beta | \log \alpha_{\textrm{SPS}})$ for the seven clusters in our sample, which are each computed with Gaussian weights centered at the fixed value of $\log \alpha_{\textrm{SPS}}$ and a dispersion $\sigma_{\textrm{SPS}}$ (i.e., $\sigma = \sigma_{\textrm{SPS}}$ in Equation \[eqn:weights\]; we now account for only the random photometric errors in ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ since $\log \alpha_{\textrm{SPS}}$ is fixed). $P(\log \alpha_{\textrm{SPS}})$, which represents our constraint on the common stellar mass scale, is simply a Gaussian with mean $\langle \log \alpha_{\textrm{SPS}} \rangle = 0.27$ and dispersion $\sigma_{\alpha} = 0.05$ dex, as derived in Section \[sec:massscale\] for isotropic orbits.
We estimate the intrinsic scatter in $\beta$ (Section \[sec:innerslopes\]) using the posterior probability densities $P(\beta | \log \alpha_{\textrm{SPS}}=0.27)$ for each cluster. That is, we evaluate the cluster-to-cluster scatter in $\beta$ at a fixed value of $\log \alpha_{\textrm{SPS}}$. All of the above comments apply equally to our study of the cNFW models, simply replacing $\beta$ by $\log r_{\textrm{core}}$.
[^1]: In the present paper we present a significantly revised measurement for A611; see Section \[sec:dmresults\].
[^2]: Throughout, $L_{\textrm{V}}$ and ${\Upsilon_{*\textrm{V}}}$ refer to the observed luminosity, including any internal reddening from dust within the BCG. If we removed the reddening to obtain the intrinsic $L_{\textrm{V}}$ and ${\Upsilon_{*\textrm{V}}}$ of the stellar populations, their scatter would increase. (Reddening is indicated only in cool core clusters hosting some current star formation.) However, the SPS *stellar mass* estimates, which are significant for our analysis, are much more robust.
[^3]: This estimate of $\sigma_{\textrm{SPS}}$ may be conservative, given that the dispersion in ${\Upsilon_{*\textrm{V}}^{\textrm{SPS}}}$ measurements among the BCGs is smaller, and $\chi^2/\textrm{dof} \leq 1$ in the SPS model fits. Thus, in practice we are likely allowing for some mild intrinsic variation in $\alpha_{\textrm{SPS}}$.
[^4]: In Paper I we found that the evidence ratio mildly favored the cNFW models when taking a uniform prior on $\log \alpha_{\textrm{SPS}}$. When the joint constraint derived in this paper is taken as a prior, the evidence ratio is consistent with unity ($\ln E_{\textrm{gNFW}} / E_{\textrm{cNFW}} = -0.8 \pm 3.2$).
[^5]: Interestingly, the reverse seems to hold for $\gamma_{\textrm{tot}}$: there is no sign of a correlation with the stellar mass or luminosity, but a possible correlation with $M_{200}$ ($\rho = -0.68$, $P_0 = 0.09$). The latter may simply be because the radial range over which $\gamma_{\textrm{tot}}$ is measured is proportional to $r_{200}$.
[^6]: While @Salpeter55 did not measure the mass function down to $0.1 {\,\textrm{M}_\sun}$, this is the common meaning of a “Salpeter” IMF in extragalactic studies.
[^7]: Their $\alpha_{\textrm{IMF}}$ is defined relative to a Salpeter IMF and so differs from our definition by 0.25 dex.
[^8]: The present measurement of $\beta$ in A383 (Table \[tab:beta\]) is slightly shallower, but consistent with, @N11 due to our new joint constraint on $\alpha_{\textrm{SPS}}$.
[^9]: @Limousin08 imposed a tight prior on the BCG stellar mass derived from SPS fits, but did not consider uncertainty from the IMF. Their SPS estimates are quite high: ${\Upsilon_{*\textrm{B}}^{\textrm{SPS}}}\approx 11$, whereas we find ${\Upsilon_{*\textrm{B}}^{\textrm{SPS}}}= 3.0$ from fitting the SDSS photometry to this BCG, also using a Chabrier IMF. Adjusting the latter to our preferred $\alpha_{\textrm{SPS}} = 0.27$ yields ${\Upsilon_{*\textrm{B}}}= 5.7$, which agrees with the estimate ${\Upsilon_{*\textrm{B}}}\approx 6$ by @Zitrin10 in this cluster.
[^10]: In any case, the highly uncertain contribution from subhalos may dominate this signal [e.g., @Gao12susy].
|
[ ]{}
Introduction {#sec:introduction}
============
The concept of one-dimensional current-carrying edge channels in the quantum Hall regime has been the subject of detailed experimental and theoretical studies in recent years. It is now widely accepted as a basic ingredient for the understanding of the quantum Hall effect [@Halperin82; @Haug93; @MacDonald84; @Buttiker85; @Buttiker88; @Chklovskii92; @Chklovskii93]. In the edge state picture, the resistance quantization is attributed to current transport in quasi one-dimensional edge channels along the sample boundaries, assuming negligible backscattering between opposing edges of the two-dimensional electron gas (2DEG) due to vanishing bulk conductivity [@Buttiker85; @Buttiker88]. Edge channels are formed at the intersection of Landau-levels with the Fermi-energy, where the presence of unoccupied states allows for current transport. Theoretical treatments, such as the compressible/incompressible liquid picture [@Chklovskii92; @Chklovskii93] have led to a better understanding of the energetic structure of the 2DEG-edge. Various efforts have been made to experimentally investigate the details of edge-reconstruction and edge channel transport by means of tunneling- [@Zhitenev95; @Hwang93] or capacitance-spectroscopy [@Takaoka94]. At the same time advances in far infrared- [@Merz93; @Lorke96; @Hirakawa96], time-resolved transport- [@Zhitenev93] and edge-magnetoplasmon-spectroscopy [@Talyanskii92], as well as SET-measurements [@Wei98] have given further insight in the edge structure of the 2DEG. However, the experimental investigation of the potential profile at the sample boundaries remains a challenging task.
Here we report on the development and application of a new sample geometry, which renders possible a direct measurement of the charge transfer between adjacent edge channels at the sample boundaries. The combination of a quasi-Corbino [@Oto99] topology and the cross-gate technique [@Haug88; @Washburn88] has a number of advantages, which make it a versatile tool for studying edge channel transport: Requiring no sophisticated sample fabrication such as high-resolution lithography or cleaved-edge overgrowth [@Hilke01], it offers true 4-probe measurements in both, the linear and the non-linear transport regime. Furthermore it can “dissect” the edge channel structure, i.e. for $n$ edge channels, anyone of the $n-1$ gaps can individually be addressed. This is done by separately contacting single edge channels, selectively populating them [@vanWees89; @Komiyama89] and bringing them into a controlled interaction. We present experimental data obtained using this geometry which allows us to determine the equilibration-length in the linear regime and get information on the energetic edge structure (i.e. the spin- and Landau-gaps) by non-linear I–V-spectroscopy.
This report is organized as follows: Section \[sec:geometry\] gives a detailed description of the sample geometry, and the general concept behind the experiment. Also the application of the Landauer-Büttiker formalism [@Buttiker85; @Buttiker88] to the given topology is briefly outlined. Experimental results for different filling factor combinations, temperatures and interaction-lengths are presented in Section \[sec:experimental\]. These will be discussed in terms of the edge-reconstruction picture and evaluated with regard to the spin- and Landau-gap structure.
Device geometry {#sec:geometry}
===============
Concept and realization {#sec:concept}
-----------------------
The samples were fabricated from a molecular beam epitactically-grown GaAs/AlGaAs heterostructure, consisting of a 1 $\mu$m GaAs buffer layer, 20 nm undoped Al$_{0.3}$Ga$_{0.7}$As, 25 nm silicon-doped Al$_{0.3}$Ga$_{0.7}$As covered by a 45 nm superlattice and cap-layer. The mobility at liquid Helium temperature was 800 000 cm$^{2}$/Vs and the carrier density 3.7 $\cdot 10^{11}
$cm$^{-2}$. All structures were patterned by standard photolithography. The mesa was defined by wet chemical etching. The gate-electrode consists of 5 nm thermally evaporated NiCr (1:1). Ohmic contacts with typical resistances of 200 $\Omega$ at 30 mK are provided by alloyed AuGe/Ni/AuGe (88:12) pads.
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For the measurements reported here a total of 4 samples with slightly different geometries, fabricated from the same heterostructure, were investigated. The device geometry is given in Fig. \[fig1\], where the ring-shaped mesa is shown by the thick outline and the gate-electrode is indicated by the hatched area. In a quantizing magnetic field and for zero gate voltage, the present quasi-Corbino geometry has two sets of edge channels, called “inner" and “outer" edge states in the following. The shape and location of the gate is chosen such that, by application of a suitable negative bias, the inner edge channels can be redirected to run along the outer edge in the gate-gap region. This situation is shown in Fig. \[fig1\] for equilibrium conditions, i.e. without an external applied current. Here, the filling factors have been adjusted to $\nu=2$ in the ungated regions and $g=1$ under the gate. In the gate-gap there are two adjacent channels running in parallel. One of them is reflected by the gate potential, and connected only to the inner Ohmic contacts. The other, outer edge channel continues to run along the etched boundary, even under the gate-electrode, and is therefore connected only to the contacts positioned along the outer mesa edge. When the bulk region of the 2DEG under the gate is in its insulating state (integer filling factor and sufficiently low temperatures) current transport between inner and outer Ohmic contacts is possible only by charge equilibration among neighboring channels in the gate-gap region. The present geometry therefore allows for a direct investigation — in 2- or 4-probe geometries — of transport across the incompressible strip that separates the compressible edge channels within the gate-gap.
=1
Figure \[fig2\] illustrates the versatility of the present geometry. For any given integer filling factor $\nu$ all incompressible strips can be probed by adjusting the gate voltage to the appropriate integer filling factor $g$ in the gated region. When spin-splitting is resolved and the filling factors are adjusted to $\nu=4$ and $g=1$, the outermost spin-induced incompressible strip can be probed, whereas for $\nu=4$ and $g=2$ the wider, Landau-gap-induced strip between the second and third edge channel can be investigated. For clarity, in Fig. \[fig2\] only the four contacts used as current leads or voltage probes in the experiment are shown. Furthermore, only edge channels running by the interaction region (i.e. the gate gap) are drawn. In the evaluation of the experimental data it is of course necessary to take into account the complete sample geometry, as it will be discussed in the following section.
Application of the Landauer-Büttiker formalism to the experimental setup {#sec:LBapplic}
------------------------------------------------------------------------
Despite the apparent complexity of the device geometry, a relation between the measured resistances and equilibration based on inter-edge-channel transport (expressed in terms of an effective transmission-coefficient $T$ across the incompressible strip) can be derived in a straightforward manner. In the following, we will treat the case of a 4-probe measurement, restricting the discussion to the leads 3, 5, 7, 8. It is easy to show that the presence of the unused (floating) contacts 1, 2, 4, 6 neither changes any of the considerations below nor the obtained experimental results. Based upon the absence of backscattering across the bulk and the conservation of current, a Landauer-Büttiker-type multichannel-multiprobe formula can be derived (see Appendix A) to give the following results for the four different contact configurations for measuring the 4-probe resistance between inner and outer edge channels:
$$\begin{aligned}
R_{73,85}&=&\frac{h}{e^{2}}\left[\frac{1}{T}-\frac{\nu+g }{\nu
g}\right]\nonumber\\
R_{75,83}&=&\frac{h}{e^{2}}\left[\frac{1}{T}-\frac{1}{g}\right]\nonumber\\
R_{85,73}&=&\frac{h}{e^{2}}\left[\frac{1}{T}\right]\nonumber\\
R_{83,75}&=&\frac{h}{e^{2}}\left[\frac{1}{T}-\frac{1}{\nu}\right]
\label{eq:Req}\end{aligned}$$
Here $R_{ij,kl}$ is the resistance measured between the current leads $i$ and $j$ and voltage probes $k$ and $l$. Note that exchanging current and voltage probes changes the value of the resistance because of the chirality of edge states ($R_{ij,kl}(B)
= R_{kl,ij}(-B) \neq R_{kl,ij}(B)$). The measurements can therefore be used to confirm the direction of electron-drift, respectively the orientation of the magnetic field in the experimental setup.
Combining equation \[eq:Req\] with the relation between $T$ and the equilibration-length [@Muller92] $l_{eq}$ gives for the interaction of two edge channels
$$R_{85,73}=\frac{h}{e^{2}}\left[1+exp\left(\frac{-2d}{l_{eq}}\right)\right]^{-1}
\label{eq:leq}$$
with $d$ being the interaction-length, i.e. the gate-gap width. This means that the present topology makes it possible to determine the equilibration-length between neighboring edge channels by a single resistance measurement.
For elevated temperatures or macroscopic interaction-lengths, full equilibration is expected. Then the transmission-coefficient can be derived from a summation over all participating edge states and contacts:
$$T=\sum_{i=1}^{g}\sum_{j=g+1}^{\nu}1/ \nu=(\nu-g)\cdot g/ \nu .
\label{eq:Tfull}$$
In this case all the resistances $R_{ij,kl}$ are readily calculated and can be compared with the experimental values.
Finally, it should be mentioned that at no point in the derivation of equation \[eq:Req\] the transmission-coefficient is assumed to be independent of the electrochemical potential difference across the incompressible strip. Equilibration between edge states can therefore be studied as a function of bias. As will be shown in Section \[sec:weak\], this I–V-spectroscopy can be used not only in the linear transport regime ($T$ = const.), but also to investigate the local energetic structure of the edge region.
Experimental {#sec:experimental}
============
Equilibration across the Landau-gap at liquid helium-temperature {#sec:fulleq}
----------------------------------------------------------------
Figure \[fig3\] shows current-voltage curves obtained for a sample with 32 $\mu$m gate-gap width, a filling factor combination $\nu=4$, $g=2$ and a temperature of 4.2 K. The inset displays the corresponding, simplified configuration of edge channels (see also the discussion of Fig. \[fig2\]). As expected from equation \[eq:Req\], each configuration gives a different 4-point resistance (dashed lines indicate experimental curves). For comparison, the solid lines show the resistances for complete equilibration ($T = 1$), calculated as discussed in Section \[sec:LBapplic\][@byline].
=0.9
The very good agreement between the calculated resistances for fully equilibrated transport and the measurements at 4.2 K indicates that at this temperature the equilibration-length is much shorter than the gate-gap width of $d=32 \mu$m. Furthermore, the fact that equations \[eq:Req\] and \[eq:Tfull\] can so well account for the experimental data shows that the discussed Landauer-Büttiker picture is valid and in particular that transport through the bulk of the 2DEG is negligible. The absence of bulk leakage at 4.2 K for even filling factors is also confirmed by a direct determination of the bulk conductivity, adjusting the filling factors to $\nu=2$, $g=2$, and investigation of a reference-sample without a gate-gap [@Wildfeuer99]. Also the influence of non-ideal contact properties proves to be irrelevant.
Weak coupling among edge states at 30 mK {#sec:weak}
----------------------------------------
When the samples are cooled down to 30 mK in the mixing chamber of a $^{3}$He$^{4}$He dilution-refrigerator, the resistance at low bias increases dramatically. Even for an interaction-length of more than 30 $\mu$m, resistances of around 10 M$\Omega$ are observed, corresponding to macroscopic equilibration-lengths (see section \[sec:evaluation\]).
Figure \[fig4\] shows I–V-traces of a sample with 5 $\mu$m gate-gap width, obtained for the four different contact configurations and filling factors $\nu=4$ in the ungated areas and $g=2$ under the gate-electrode. As indicated in the inset, this filling factor combination corresponds to equilibration across the Landau-gap.
=0.97
The current-voltage characteristics are strongly nonlinear and asymmetric. Regarding the low positive bias region in Fig. \[fig4\], which corresponds to a decrease of the electrochemical potential of the two outermost edge channels in the gate-gap, the occurrence of an onset-voltage is observable for all contact configurations. At biases exceeding this voltage, the experimental curves have an almost constant differential resistance. For negative applied biases, no precise identification of an onset-voltage is possible. Here, in contrast to what is observed for positive biases, the I–V-traces are not exactly reproducible from cooling to cooling and differ qualitatively from sample to sample.
At a positive current of about 10 nA, the slope of the traces shown in Fig. \[fig4\] decreases drastically and the differential resistances drop to values close to those obtained for complete equilibration. We interpret this step in the I–V-characteristic as a novel type of breakdown-mechanism in the transport between adjacent edge channels, which is not related to the usual, complete breakdown of the quantum Hall effect. The latter is governed by transport through the bulk of the 2DEG, which for macroscopic samples (like the ones investigated here with lateral dimensions of a few millimeters), is only observed at much higher currents [@Shashkin94]. In particular, no step at small voltages is found for a reference-sample with gate-gap widths zero, where only the breakdown through the bulk can be seen, however at much higher voltages. As discussed above, already at 4.2 K and even integer filling factors the 2D bulk is in its insulating state and no indications of breakdown are observed in the range of biases and voltages considered here (see Fig. \[fig3\]). Also, bad Ohmic contacts cannot account for the step in the I–V-characteristic as the number of samples and contact combinations investigated make this explanation highly unlikely. Furthermore, in the applied 4-terminal configuration, contact resistances should not appreciably affect the results as long as the input-impedance of the experimental setup is sufficiently high.
Qualitative interpretation in terms of the incompressible/compressible liquid model {#sec:Shk}
-----------------------------------------------------------------------------------
In this section we will develop a first interpretation of the experimental observations shown in Fig. \[fig4\], which is based on the breakdown across the incompressible strip [@Chklovskii93] between adjacent, but differently biased edge channels. According to this picture it can also be understood, why the onset-voltage of about 4.4 mV, roughly corresponds to the Landau-gap $\hbar\omega_{c}$ for the given magnetic field $B=3.89$ T. Furthermore, the proposed model accounts for the asymmetry between positive and negative current directions seen e.g. in Fig. \[fig4\] and tries to explain why, at sufficiently large currents the linear parts of the observed traces have almost the same slope as the fully equilibrated resistance curves (see the solid lines in Fig. \[fig3\] and the dash-dotted lines in Fig. \[fig6\], obtained from equations \[eq:Req\] and \[eq:leq\]).
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The measured I–V-spectra in Fig. \[fig4\] resemble those of a backward diode [@Sze81]. Indeed, in the picture of edge-state-reconstruction developed by Chklovskii, Shklovskii et al. [@Chklovskii92] (taking into account screening-effects among electrons), transport across an incompressible edge-strip can be described in a manner comparable to the transport across the depletion-layer in a $p-n$ backward-diode. This is illustrated in Fig. \[fig5\] (b)–(d), where we have sketched the edge state reconstruction for $\nu= 2$, $g=1$. Also shown is the course of edge channels in the gate-gap region for this filling factor combination (Fig. \[fig5\] (a)). The mesa edge is positioned on the left hand side, shaded areas indicate compressible liquids, light regions incompressible strips and the bulk. The gate is drawn in black. Arrows indicate the direction of electron drift in the compressible liquid strips.
Figures \[fig5\] (b)–(d) show the reconstruction of the edge potential for the section indicated by the dashed line in (a). Occupied states beneath the Fermi-energy are symbolized by full circles, open circles represent unoccupied states, states at the Fermi-energy are shown as half-filled circles. In the configuration outlined in (b) no bias is applied between inner and outer contacts and no current will flow between the separately contacted edge states.
We now consider the case that one of the inner contacts is grounded and a positive voltage is applied to an outer contact. In this situation, the outer edge channel is shifted downwards in energy with respect to the inner one. For a very low bias, this aligns the occupied states in the inner channel with the energy-gap in the outer channel so that ideally no current is expected to flow. This gives reason for the appearance of the high resistance region at very small positive bias in complete analogy to the case of a backward-diode. Only when a high enough bias is applied so that the topmost occupied inner edge state becomes aligned with the first unoccupied outer edge state, transfer of electrons between the edge channels becomes energetically allowed and equilibration is readily achieved as depicted in Fig. \[fig5\] (c). This explains the sudden breakdown of the high differential resistance state when a positive voltage $\Delta
U_{oi}=+E_{g}/e$ is applied between inner and outer contacts. Here, $E_{g}$ is the relevant energy-gap, which, for the case $\nu=2$, $g=1$ is expected to be the spin-gap $g^{*}\mu_{B}B$, with $g^{*}$ being the effective Landé-factor.
=0.97
For a negative bias, on the other hand, the first unoccupied inner edge strip is energetically separated from the outer occupied states by roughly the Landau-gap $\hbar \omega_{c}$ (see Fig. \[fig5\] (d)). Therefore, a pronounced asymmetry is expected in the I–V-spectra, as it is observed experimentally for $\nu=2$, $g=1$ (Fig. \[fig6\]).
Here, at positive voltages exceeding 1 mV, almost full equilibration is established, whereas in the negative bias regime the onset of equilibration with a higher transmission coefficient can be detected only for $|U|
> 10$ mV. It should also be pointed out, that for a negative bias no true energy-gap opens up, since there are always unoccupied levels (arrow in Fig. \[fig5\] (d)) at the quasi-Fermi-level of the outer edge channels $E_{F}^{o}$. Therefore, no clear step-like features are observed for negative biases.
Of course this model can directly be transferred to the case of equilibration across the Landau-gap ($E_{g} = \hbar \omega_{c}$), as e.g. depicted in Fig. \[fig4\], where the step-like features are far more pronounced and the asymmetry is still observable.
Evaluation of the I–V-spectra {#sec:evaluation}
-----------------------------
The characteristic onset-voltage for equilibration across the Landau-gap is determined by linear extrapolation of the two different curve branches in the positive voltage region. For the data shown in Fig. \[fig4\], we obtain $E_{g} = 4.4$ meV. A comparison with $\hbar \omega_{c}$ for $B = 3.89$ T and an effective electron-mass of $m^{*} = 0.067 m_{e}$ shows a discrepancy of 2–3 meV. Similar deviations were observed for any of the four investigated samples and both filling factor combinations $\nu = 4$, $g = 3$ and $\nu = 4$, $g = 2$.
=0.97
Figure \[fig7\] shows the magnetic field dependence of the energy-gap determined for three different gate-gap widths. While the filling factor $g=2$ under the gate is held constant, the filling factor $\nu$ in the ungated areas varies with the strength of the magnetic field. Note that even though at non-integer filling factors the ungated bulk region are in a conducting state, [@Chklovskii93] I–V-spectroscopy across the incompressible strip is still possible. The solid line indicates the calculated values of $\hbar \omega_{c}$. For $\nu \leq 4$ we obtain a linear dependence for the samples with 2 and 5 $\mu$m gate-gap widths, but again the above mentioned deviation from the theoretical value of the Landau-gap is observed. A linearization of the curve slopes — where possible — allows to determine the effective mass $m^{*}$. The obtained values, as given in Table \[tab1\], are in good accordance with the standard value of 0.067 $m_{e}$ for bulk GaAs.
---------------- --------------- --
gate-gap width $m^{*}/m_{e}$
2 $\mu$m 0.067
5 $\mu$m 0.069
30 $\mu$m 0.060
---------------- --------------- --
: Effective masses $m^{*}$ in units of the free electron mass as calculated from the slopes of the linear fits to the data shown in Fig. \[fig7\]. \[tab1\]
The deviation from $\hbar \omega_{c}$ of roughly 2 meV can partly be understood in terms of the edge-state model discussed in section \[sec:Shk\], as the effective energy-gap width is reduced by the spin-splitting energy $g^{*} \mu_{B} B$. The Landau-level broadening might be a further cause for the onset of equilibration at biases less than $\hbar \omega_{c}/e$. The breakdown of the adiabatic transport regime at energy values smaller but comparable to the Landau-level spacing was also reported by Komiyama et al.[@Komiyama92; @Machida96] for samples with a cross-gate geometry. They proposed a self-consistent reconstruction of edge states to describe the observed nonlinear behaviour, finding $\hbar \omega_{c}/2$ to be the critical potential-difference for nonequilibrium population of edge channels.
The evaluation of the data shown in Fig. \[fig6\] gives an effective spin-gap of 0.57 meV. This corresponds to $g^{*} \approx 1.3$, about three times the bulk GaAs $g$-factor. Because of the smallness of the spin-gap it can only be evaluated from the present data with a large margin of error. A survey of values obtained for different gate-gap widths and filling factors (cf. Fig. \[fig2\], top row) shows a large scatter, with values from $g^{*} = 0.27$ to $g^{*} = 1.8$ [@Wuertz01]. A more thorough investigation of the parameter-space with further high-resolution measurements is therefore necessary.
Assuming a constant equililibration-rate per unit-length, the equilibration-length $l_{eq}$ can be deduced from the linear part of the I–V-curves at low positive voltages. This was done e.g. for the filling factor combination $\nu = 4$, $g = 2$. Making use of equation \[eq:Req\] and \[eq:leq\] we obtain $l_{eq}=$250 $\mu$m for 5 $\mu$m gate-gap width and 30 mK temperature. This corresponds to a transmission-coefficient of T = 0.072. Similar macroscopic equilibration-lengths have been reported by several authors using different approaches [@Muller92; @Alphenaar90; @Komiyama92; @Hirai95]. Depending on the gate-gap width we found equilibration-lengths up to the order of magnitude of the sample size. Surprisingly the equilibration-length also reveals a dependence on the choice of contacts, which is not compatible with the above discussed Landauer-Büttiker-type formulas.
conclusion
==========
In summary, we have realized a pseudo-Corbino geometry, suitable to obtain information on the energetic structure of the edge potential in the quantum Hall regime by means of simple I–V-spectroscopy. We directly observed non-linear transport across different energy-gaps and determined the transmission-coefficient for weak coupling among separately contacted edge channels. Interpreting our results in terms of the Landauer-Büttiker formalism [@Buttiker85; @Buttiker88] and the edge-reconstruction picture developed by Chklovskii et al. [@Chklovskii92; @Chklovskii93], we show that the described geometry enables us to shift the electrochemical potentials of single edge states with respect to each other. Transport across the incompressible liquid-strip can thereby be studied as a function of the bias applied between inner and outer contacts. Identifying the onset-voltage for inter-edge-channel current flow with the energy-gap, we found for equilibration across the Landau-gap a clear and reproducible deviation from the value of $\hbar \omega_{c}$. On the other hand, the dependence of the energy-gap on the magnetic field agrees well with the effective mass of GaAs. Furthermore, from the observed spin-gap, an effective $g$–factor can be determined, so far, however, only with a large margin of error.
As mentioned, our new spectroscopic technique has addressed a number of interesting questions regarding transport across incompressible liquid-strips and the energetic structure of the edge potential. Subject of future investigations with the proposed sample geometry will — besides continuing the study of non-linear transport — surely be the edge structure in the fractional quantum Hall regime as well as the influence of spin-flip processes on equilibration.
We wish to thank J.P. Kotthaus for his constant support of this work and gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft, SPP “Quantum Hall Systems", under grant LO 705/1-1. We also gratefully acknowledge help during the experiments and discussions with A.A. Shashkin. The part of the work performed in Russia was supported by RFBR grant 00-02-17294, INTAS YSF002, the programs “Nanostructures" and “Statistical Physics" from the Russian Ministry of Sciences.
Referring to the geometry shown in Fig. \[fig1\] we calculate as an example the 4-point resistance $R_{85,73}$. Again the previously applied nomenclature is used:
------- --- -----------------------------------------------------
$\nu$ : filling factor in the ungated regions of the sample
$g$ : filling factor in the gated regions
------- --- -----------------------------------------------------
From the application of the Landauer-Büttiker-multichannel-multiprobe formula [@Buttiker88] to the present geometry we derive the following “current-statistics" for the 8 shown contacts:
$$\begin{aligned}
I_{1}&=&-\frac{e^{2}}{h}\nu \left(U_{1}-U_{2}\right)\nonumber\\
I_{2}&=&-\frac{e^{2}}{h}\left[\nu U_{2}-(\nu - g) U_{1}-g
U_{8}\right]\nonumber\\
I_{3}&=&-\frac{e^{2}}{h} \nu
\left(U_{3}-U_{4}\right)\nonumber \\ I_{4}&=&-\frac{e^{2}}{h} \nu
\left(U_{4}-U_{5}\right)\nonumber\\ I_{5}&=&-\frac{e^{2}}{h} \nu
\left(U_{5}-U_{6}\right)\nonumber\\
I_{6}&=&-\frac{e^{2}}{h}\left[\nu U_{6}-g U_{3}- T_{36}
U_{3}-T_{76} U_{7} \right]\nonumber\\ I_{7}&=&-\frac{e^{2}}{h} g
\left(U_{7}-U_{1}\right)\nonumber\\
I_{8}&=&-\frac{e^{2}}{h}\left[g U_{8}-T_{78} U_{7}-T_{38} U_{3}
\right)\nonumber\end{aligned}$$
with $T_{ij}$ being the transmission-coefficient for charge-transfer from contact $i$ to contact $j$. Taking into account charge conservation in the gate-gap region and assuming at each contact the same number of incoming and outgoing edge channels we further obtain:
$$\begin{aligned}
T &=& T_{38}= T_{76}\nonumber\\ g &=& T_{78}+ T_{38}\nonumber\\
\nu &=& g + T_{36}+ T_{76}\nonumber\end{aligned}$$
Regarding now the 4-point resistance $R_{85,73}$, i.e. current transport between the contacts 8 and 5, with contacts 7 and 3 being used as voltage probes, we receive a surprisingly simple relation. (The currents at unused — respectively floating — contacts are vanishing.)
$$\begin{aligned}
I_{1}&=& 0 \Rightarrow U_{1}= U_{2}\nonumber\\ I_{2}&=& 0
\Rightarrow U_{1}= U_{8}\nonumber\\ I_{3}&=& 0 \Rightarrow U_{3}=
U_{4}\nonumber\\I_{4}&=& 0 \Rightarrow U_{4}=
U_{5}\nonumber\\I_{6}&=& 0 \nonumber\\ I_{7}&=& 0 \Rightarrow
U_{1}= U_{7}\nonumber\end{aligned}$$
$$\begin{aligned}
R_{85,73}\cdot \frac{e^{2}}{h}&=&\frac{U_{3}-
U_{7}}{I_{8}}\nonumber\\&=&\frac{U_{7}-U_{3}}{g U_{8}- \left(g -
T_{38}\right) U_{7} - T_{38} U_{3}}\nonumber\\&=& \frac{U_{7} -
U_{3}}{T_{38}\left(U_{7} - U_{3} \right)} =
\frac{1}{T_{38}}\nonumber\\&=& \frac{1}{T}\nonumber\end{aligned}$$
B. I. Halperin, Phys. Rev. B [**25**]{}, 2185 (1982). R. J. Haug, Semicond. Sci. Technol. [**8**]{}, 131-153 (1993). A. H. MacDonald, P. Strěda, Phys. Rev. B [**29**]{}, 1616 (1984). M. Büttiker, Y. Imry, R. Landauer and S. Pinhas, Phys. Rev. B [**31**]{}, 6207 (1985). M. Büttiker, Phys. Rev. B [**38**]{}, 9375 (1988). D. B. Chklovskii, B. I. Shklovskii and L. I. Glazman, Phys. Rev. B [**46**]{}, 4026 (1992). D. B. Chklovskii, K. A. Matveev and B. I. Shklovskii, Phys. Rev. B [**47**]{}, 12605 (1993). N. B. Zhitenev, R. J. Haug, K. v. Klitzing and K. Eberl, Phys. Rev. B [**51**]{}, 17820 (1995). S. W. Hwang, D. C. Tsui and M. Shayegan, Phys. Rev. B [**48**]{}, 8161 (1993). S. Takaoka, K. Oto, H. Kurimoto and K. Murase, K. Gamo and S. Nishi, Phys. Rev. Lett. [**72**]{}, 3080 (1994). R. Merz, F. Keilmann, R. J. Haug and K. Ploog, Phys. Rev. Lett. [**70**]{}, 651 (1993). A. Lorke, J. P. Kotthaus, J. H. English and A. C. Gossard, Phys. Rev. B [**53**]{}, 1054 (1996). K. Hirakawa et al., The Physics of Semiconductors, M. Scheffler and R. Zimmermann, Ed. World Scientific, Singapore, 1996, p. 2543. N. B. Zhitenev, R. J. Haug, K. v. Klitzing and K. Eberl, Phys. Rev. Lett. [**71**]{}, 2292 (1993). V. I. Talyanskii, A. V. Polisski, D. D. Arnone, M. Pepper, C. G. Smith, D. A. Ritchie, J. E. Frost and G. A. C. Jones , Phys. Rev. B [**46**]{}, 12427 (1992). Y. Y. Wei, J. Weis, K. v. Klitzing and K. Eberl, Phys. Rev. Lett. [**81**]{}, 1674 (1998). K. Oto, K. Arai, S. Uno, S. Takaoka and K. Murase, Microelectronic Engineering [**47**]{}, 27 (1999). R. J. Haug, A. H. MacDonald, P. Streda and K. von Klitzing, Phys. Rev. Lett. [**61**]{}, 2797 (1988). S. Washburn, A. B. Fowler, H. Schmid and D. Kern, Phys. Rev. Lett. [**61**]{}, 2801 (1988). M. Hilke, D. C. Tsui, M. Grayson, L. N. Pfeiffer and K. W. West, cond-mat/0104553 (2001). B. J. van Wees, E. M. M. Willems, L. P. Kouwenhoven, C. J. P. M. Harmans, J. G. Williamson, C. T. Foxon and J. J. Harris, Phys. Rev. B [**39**]{}, 8066 (1989). S. Komiyama, H. Hirai, S. Sasa and S. Hiyamizu, Phys. Rev. B [**40**]{}, 12566 (1989). G. Müller, D. Weiss, A. V. Khaetskii, K. v. Klitzing, S. Koch et al., Phys. Rev. B [**45**]{}, 3932 (1992). The data shown in Fig. \[fig3\] were taken on a sample with a slightly different geometry, having two separate inner etched regions. In this case, the fully equilibrated resistances in units of $h/e^2$ are 1/4, 1/2, 3/4, and 1 for the configurations $R_{73,85}$, $R_{83,75}$, $R_{75,83}$ and $R_{85,73}$, respectively. R. Wildfeuer, diploma thesis LMU Munich, unpublished (1999). A. A. Shashkin, A. J. Kent, P. A. Harrison, L. Eaves and M. Henini, Phys. Rev. B [**49**]{}, 5379 (1994). S. M. Sze, Physics of Semiconductor Devices, J. Wiley & Sons, 2nd Ed. (1981). S. Komiyama, H. Hirai, M. Ohsawa, Y. Matsuda, S. Sasa and T. Fujii, Phys. Rev. B [**45**]{}, 11085 (1992). T. Machida, H. Hirai, S. Komiyama, T. Osada and Y. Shiraki, Phys. Rev. B [**54**]{}, 14261 (1996). A. Würtz, diploma thesis LMU Munich, unpublished (2001). B. W. Alphenaar, P. L. McEuen, R. G. Wheeler and R. N. Sacks, Phys. Rev. Lett. [**64**]{}, 677 (1990). H. Hirai, S. Komiyama, S. Fukatsu, T. Osada, Y. Shiraki and H. Toyoshima, Phys. Rev. B [**52**]{}, 11159 (1995).
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---
abstract: 'Given a compact Kähler manifold, the Infinitesimal Torelli problem asks whether the differential of the period map of a Kuranishi family is injective. Unlike the classical Torelli theorem for curves, there is a negative answer for example for hyperelliptic curves of genus greater than $2$. Nevertheless the Infinitesimal Torelli Theorem holds for many other classes of manifolds. We will prove it for smooth hypersurfaces in simple abelian varieties with sufficiently high self-intersection giving an effective bound on a result by Green in this particular case.'
author:
- Patrick Bloß
bibliography:
- 'The\_Infinitesimal\_Torelli\_Theorem\_for\_hypersurfaces\_in\_abelian\_varieties\_-\_Patrick\_Bloss.bib'
title: The Infinitesimal Torelli Theorem for hypersurfaces in abelian varieties
---
Introduction
============
Consider a family of compact Kähler manifolds $\phi\colon \mathcal{X} \to B$, i.e. a proper holomorphic submersion of complex manifolds with Kähler fibres. Denote by $X_b$ the fibre $\phi^{-1}(b)$ of $\phi$ over $b\in B$ and fix $0 \in B$. Write $X=X_0$. Ehresmann’s theorem ensures that in some neighborhood $U$ of $0$ there are well defined isomorphisms of the cohomology groups $H^k(X_b,{\mathbb{Z}}) \cong H^k(X_0,{\mathbb{Z}})$. These will in general not preserve the Hodge structure so it makes sense to consider the *period map*. For given $k$ and $p$ the $p$-th piece of the period map with respect to the $k$-th cohomology group is defined by $$\mathcal{P}^{p,k}\colon U \to \mathrm{Grass}(b^{p,k},H^k(X,{\mathbb{C}})), b \mapsto F^pH^k(X_b,{\mathbb{C}})$$ where $F^pH^k(X_b,{\mathbb{C}})$ denotes the $p$-th step of the Hodge filtration and $b^{p,k}=\dim F^pH^k(X_b,{\mathbb{C}})$ (note that this is independent of $b$ as all $X_b$ have the same Hodge numbers). Griffiths showed that this map is holomorphic so we can consider its differential $$d\mathcal{P}^{p,k}\colon T_{B,0} \to {\mathrm{Hom}}(F^p H^k(X,{\mathbb{C}}),H^k(X,{\mathbb{C}})/F^p H^k(X,{\mathbb{C}})).$$ Furthermore he showed that $d\mathcal{P}^{p,k}$ is the composition of the Kodaira-Spencer map $T_{B,0} \to H^1(X,T_X)$ with the map $$H^1(X,T_X) \to {\mathrm{Hom}}(H^{k-p}(X,\Omega_X^p),H^{k-p+1}(X,\Omega_X^{p-1}))$$ given by the cup product and the interior product. Now we say that the Infinitesimal Torelli Theorem (ITT in the following) holds for a compact Kähler manifold $X$ if the period map $\mathcal{P}^{n,n}$ of a Kuranishi family of $X$ is an immersion. Since the Kodaira-Spencer map is an isomorphism for a Kuranishi family we need to show injectivity of the map $$H^1(X,T_X) \to {\mathrm{Hom}}(H^0(X,\omega_X),H^1(X,\Omega_X^{n-1})).$$ For a curve $C$ it follows easily from a classical result by Noether that the ITT holds if and only if $C$ has genus $g(C)=2$ or $g(C)> 2$ and $C$ is non-hyperelliptic. That is to say that in this case very ampleness of the canonical sheaf is a sufficient condition. For surfaces, however, Garra and Zucconi show that for any $n \geq 5$ there exists a generically smooth $n+9$ dimensional irreducible component of the moduli space of algebraic surfaces such that for a general element of it the ITT fails (see [@MR2413341]). Thus finding classes of objects that satisfy the ITT is still an open problem. Reider proves it for surfaces of irregularity at least $5$ with globally generated cotangent bundle that satisfy some additional conditions (see [@MR929539]).
The ITT has been shown by Griffiths to hold for hypersurfaces in projective space. Green then generalized this to *sufficiently ample* hypersurfaces in an arbitrary smooth projective variety $Y$. By sufficiently ample he means that there exists an ample line bundle $L_0$ such that it holds for all smooth hypersurfaces $X\subset Y$ with $\mathcal{O}_Y(X) {\otimes}L_0^{-1}$ ample. He does however not give an effective bound on how ample $L_0$ has to be.
We consider specifically the case where $Y=A$ is an abelian variety. Our main result is the following.
\[thm\_main\] Let $X$ be a smooth hypersurface in a $g$-dimensional simple abelian variety $A$. If $h^0(A,\mathcal{O}_A(X))>\left(\frac{g}{g-1}\right)^g \cdot g!$, then the Infinitesimal Torelli Theorem holds for $X$.
Following Green’s method one can see that for a hypersurface $X \subset A$ in a $g$-dimensional variety, if $L \coloneqq \mathcal{O}_A(X)$ is ample, the ITT holds if the mutliplication map $$H^0(A,L) {\otimes}H^0(A,L^{g-1}) \to H^0(A,L^g)$$ is surjective. We will discuss this in more detail in Section \[section\_green\]. Closely related to this is the notion of projective normality. An ample line bundle $L$ on an abelian variety $A$ is very ample and defines a projectively normal embedding if the mutliplication map $$H^0(A,L) {\otimes}H^0(A,L) \to H^0(A,L^2)$$ is surjective. By an inductive argument it is easy to see that in this case the ITT holds as well. It is well known that $L$ defines a projectively normal embedding if $L=M^n$ with $n \geq 3$ or $n=2$ and some additional condition on the basepoints of $L$ is satisfied. Projective normality for primitive line bundles is fully understood for abelian surfaces (see for example [@MR3656291]). For the higher dimensional case Hwang and To give a bound for projective normality to hold for a very general polarized abelian variety in terms of the self-intersection of the line bundle (see [@MR2964474]). Finally Iyer gives a bound for higher dimensional simple abelian varieties (see [@MR1974682]). In Section \[section\_main\_thm\] we will use this approach to prove the following theorem.
\[thm\_main\_2\] Let $L$ be a line bundle on a simple abelian variety $A$ of dimension $g$. Fix $n \in {\mathbb{N}}$. If $h^0(A,L)>\left(\frac{n+1}{n}\right)^g \cdot g!$ then the multiplication map $$\mu_n\colon H^0(A,L) {\otimes}H^0(A,L^n) \rightarrow H^0(A,L^{n+1})$$ is surjective.
Theorem \[thm\_main\] is then a corollary of this.
For non-simple abelian threefolds [@2018arXiv180308780L] gives numerical conditions for projective normality, taking into account all possible abelian subvarieties.
Green’s Approach {#section_green}
================
Let $X$ be a smooth ample hypersurface in an arbitary smooth projective variety $Y$ of dimension $d$. Let $L \coloneqq \mathcal{O}_Y(X)$. There is a short exact sequence $$0 \to N_X^{\vee}\to \Omega_Y^1 {\otimes}\mathcal{O}_X \to \Omega_X^1 \to 0$$ where $N_X$ denotes the normal bundle of $X$ in $Y$. For any $p\geq1$ this gives a long exact sequence $$0 \to S^p N_X^{\vee}\to \ldots \to \Omega_Y^{p-1} {\otimes}N_X^{\vee}\to \Omega_Y^p {\otimes}\mathcal{O}_X \to \Omega_X^1 \to 0.$$ Green then obtains a spectral sequence abutting to zero from which he ultimately deduces (under the assumption that $L$ is sufficiently ample) the following commutative diagram $$\label{cd_1}
\begin{tikzcd}
H^0(X,\omega_X) \arrow{r} {\otimes}H^1(X,\Omega_X^{d-1})^{\vee}&
H^1(X,T_X)^{\vee}\\
H^0(X,\omega_X) {\otimes}H^0(X,L|_X^{d-1} {\otimes}\omega_X) \arrow{r} \arrow{u} &
H^0(X,L|_X^{d-1} {\otimes}\omega_X^2) \arrow[twoheadrightarrow]{u} \\
H^0(Y,L {\otimes}\omega_Y) {\otimes}H^0(Y,L^d {\otimes}\omega_Y) \arrow{r} \arrow[twoheadrightarrow]{u} &
H^0(Y,L^{d+1} {\otimes}\omega_Y^2) \arrow[twoheadrightarrow]{u}
\end{tikzcd}$$ The two vertical maps on the bottom are simply restriction maps. Their surjectivity is obtained from the vanishing of certain cohomology groups. The vertical map on the top right comes from a quotient map and is thus surjective as well. Finally the map on the top is the dual of $d\mathcal{P}^{d,d}$ so the ITT holds if the mutliplication map on the bottom is surjective. Note that the ITT may still hold if this map fails to be surjective.
Consider the product $Y \times Y$ and denote by $\pi_i\colon Y \times Y \to Y$ for $i=1,2$ the projection maps to the two factors. Furthermore let $\Delta = \{(y,y) \ | \ y \in Y \}$ be the diagonal in $Y \times Y$ and let $\mathcal{I}_{\Delta/Y}$ denote its ideal sheaf. Under the assumption that $L$ is sufficiently ample Green then deduces the surjectivity of the map on the bottom of diagram (\[cd\_1\]) from the vanishing of $H^1(Y \times Y, \mathcal{I}_{\Delta/Y} {\otimes}\pi_1^*L {\otimes}\pi_2^*L^d)$.
Now consider a hypersurface $X$ in a $g$-dimensional abelian variety $A$ and let $L=\mathcal{O}_A(X)$.
If $L$ is ample then the surjectivity of the multiplication map $$H^0(A,L) {\otimes}H^0(A,L^{g-1}) \to H^0(A,L^g)$$ implies the ITT.
Using the fact that the cotangent bundle of an abelian variety is trivial and that for an ample line bundle $L$ we have $H^i(A,L) = 0$ for $i>0$, it is easy to check that in each instance in Green’s proof where $L$ is required to be sufficiently ample, ampleness is enough.
However, $L$ simply being ample is not sufficient to ensure the vanishing of $H^1(A \times A, \mathcal{I}_{\Delta/A} {\otimes}\pi_1^*L {\otimes}\pi_2^*L^{g-1})$. We will study more generally the surjectivity of the multipliction maps $$\mu_n\colon H^0(A,L) {\otimes}H^0(A,L^n) \to H^0(A,L^{n+1})$$ for $n \in {\mathbb{N}}$.
Surjectivity of multiplication maps
===================================
A concept related to the surjectivity of the multiplication maps $\mu_n$ is projective normality. It can be defined for any projective variety and thus in particular for abelian varieties. We use the definitions given in [@MR2062673].
A projective variety $Y \subset {\mathbb{P}}^N$ is called *projectively normal* in ${\mathbb{P}}^N$ if its homogeneous coordinate ring is an integrally closed domain. A line bundle $L \rightarrow Y$ is called *normally generated* if it is very ample and $Y$ is projectively normal under the associated projective embedding.
We can relate projective normality and the surjectivity of $\mu_n$. An ample line bundle $L$ on a projective variety $Y$ is normally generated if and only if the mutliplication map $H^0(Y,L) {\otimes}H^0(Y,L^n) \rightarrow H^0(Y,L^{n+1})$ is surjective for every $n \geq 1$ (see [@MR2062673 Lemma 7.3.2]).
This works for any projective variety but for abelian varieties surjectivity of $\mu_n$ implies surjectivity of $\mu_m$ for all $m \geq n$ (see for example [@MR1974682]). In particular we have that surjectivity of $\mu_1$ is equivalent to projective normality and that projective normality implies the ITT.
It is well known that a line bundle $L=M^n$ with $n \geq 3$ is normally generated and a line bundle $L=M^2$ is normally generated if and only if some additional assumption on the basepoints of $L$ holds. If we only care about surjectivity of $\mu_{g-1}$ the assumption on basepoints can be dropped at least when $g \geq 3$.
Recall that for a line bundle $L$ on an abelian variety $A = V/\Lambda$ the first Chern class $c_1(L)$ defines a hermitian form on $V$ whose imaginary part $E$ is integer valued on $\Lambda$. The elementary divisor theorem ensures that there is a basis of $\Lambda$ such that $E$ is given by the matrix $\begin{pmatrix}
0 & D \\
-D & 0
\end{pmatrix}$ where $D=\mathrm{diag}(d_1,\ldots,d_g)$ with integers $d_i \geq 0$ satisfying $d_i|d_{i+1}$ for $i=1,\ldots g-1$. The vector $(d_1,\ldots,d_g)$ is called the type of the line bundle $L$. If $L$ is ample we have $h^0(A,L) = d_1\cdots d_g$. Since $c_1(L^n)=nc_1(L)$ for any $n \in {\mathbb{N}}$ by the above discussion the ITT holds for any smooth divisor in the linear system of a line bundle of type $(d_1, \ldots, d_g)$ with $d_1 \geq 2$.
The question remains what happens for primitive line bundles, i.e. line bundles of type $(1,d_2,\ldots,d_g)$. Note that by the Riemann-Roch theorem we have $h^0(A,L) = (L^g)/g!$ so any numerical condition can be equivalently expressed in terms of the number of sections of $L$ or the top intersection number.
For polarized abelian surfaces projective normality is fully understood. By [@Laz] and [@MR2076454], if $(A,L)$ is a polarized abelian surface with $L$ very ample and of type $(1,d)$, then $L$ defines a projectively normal embedding if and only if $d>6$. Lazarsfeld’s paper is hard to find but [@MR3656291] summarizes the main points. We already know that the ITT fails exactly on the locus of hyperelliptic curves with genus greater than $2$. By [@MR3968899 Theorem 2.8] for any smooth hyperelliptic curve $C$ embedded in an abelian surface $A$ the genus $g(C)$ is $2,3,4$ or $5$ and $A$ is polarized of type $(1,g(C)-1)$. So smooth hypersurfaces of type $(1,5)$ and $(1,6)$ do not define projectively normal embeddings but satisfy the ITT. In the case that $g(C) = 2$ the ITT holds. By the above, $A$ is then principally polarized. The multiplication map $\mu$ cannot be surjective for purely dimensional reasons. This is however not a contradiction, as failure of $\mu$ to be surjective does not imply failure of the ITT.
For higher dimensional polarized abelian varieties Hwang and To show that a general polarized $g$-dimensional abelian variety with $h^0(A,L) \geq \frac{8^g}{2}\cdot \frac{g^g}{g!}$ is projectively normal (see [@MR2964474]). If we only want $\mu_{g-1}$ to be surjective we can in fact generalize the methods used in their proof to obtain a better bound.
We would prefer a different more explicit condition that we can check. Recall that an abelian variety $A$ is called *simple* if the only abelian subvarieties are $\{0\}$ and $A$ itself. Iyer proves the following theorem.
Let $L$ be an ample line bundle on a $g$-dimensional simple abelian variety $A$. If $h^0(A,L) > 2^g \cdot g!$, then $L$ gives a projectivelynormal embedding.
Asymptotically this bound is worse than the one in [@MR2964474]. It does give a better bound up to $g = 23$. However the main reason we prefer this is that simplicity is a more conrete condition to work with.
This already gives us a sufficient condition for the ITT to hold but we can relax it. We cannot remove the condition that $A$ be simple by making the numerical condition on the global sections of $L$ stronger, even if we only try to prove surjectivity of $\mu_{g-1}$. In fact an analogous statement for any abelian variety cannot hold. Consider the abelian variety $X=C \times A$ where $(C,\mathcal{O}_C(2p))$ is a $(2)$-polarized elliptic curve and $(A,L)$ is a $(g-1)$-dimensional polarized abelian variety with polarization of type $(d_2,\ldots,d_g)$ where all $d_i$ are odd, e.g. a third power of a principal polarization. Now $X$ carries the product polarization $\mathcal{O}_C(2p)\boxtimes L$ which must be primitive because $\gcd(2,3) = 1$ but it cannot be normally generated as the restriction to $C$ is only basepoint free but not very ample. One would expect that for each abelian subvariety $B$ a numerical condition on the sections of the restriction $L|_B$ implying projective normality can be derived. Indeed, in the case of abelian threefolds Lozovanu proves the following theorem.
Let $(A,L)$ be a polarized abelian threefold such that $h^0(A,L) > 78$. Assume the following conditions:
1. For any abelian surface $S \subseteq A$ one has $h^0(S,L|_S) > 4$.
2. For any elliptic curve $E \subseteq A$ one has $h^0(E,L|_E) > 4$.
Then $L$ gives a projectively normal embedding of $A$.
Note that he actually proves a more general result about $(A,L)$ satisfying the property $(N_p)$, however $(N_0)$ corresponds to projective normality.
Proof of the main theorem {#section_main_thm}
=========================
At the heart of the proof of Theorem \[thm\_main\_2\] is the following lemma. This is the only place where $A$ needs to be simple.
\[prop\_iyer\] Let $L$ be an ample line bundle on a $g$-dimensional simple abelian variety $A$. Let $G$ be a finite subgroup with $|G| > h^0(A,L)\cdot g!$. Then the image of $G$ under the rational map $\phi_L\colon A \to {\mathbb{P}}(H^0(A,L))^{\vee}$ generates ${\mathbb{P}}(H^0(A,L))^{\vee}$.
Before going into the proof of Theorem \[thm\_main\_2\] we recall some basic facts about polarized abelian varieties. Let $A = V/\Lambda$ be an abelian variety. A line bundle $L$ on $A$ induces a morphism $$\begin{aligned}
\psi_L\colon & A \to {\mathrm{Pic}}^0(A) \\
& a \mapsto t_a^*L {\otimes}L^{-1}.
\end{aligned}$$ Denote its kernel by $K(L)$. If $L$ is ample $\psi_L$ is an isogeny so that $K(L)$ is finite. A decomposition $\Lambda = \Lambda_1 \oplus \Lambda_2$ is a *decomposition for $L$* if $\Lambda_1$ and $\Lambda_2$ are maximally isotropic with respect to the alternating form $\mathrm{Im} \ c_1(L)$. A decomposition $V = V_1 \oplus V_2$ is called a *decomposition for $L$* if the induced decomposition $(V_1 {\cap}\Lambda) \oplus (V_2 {\cap}\Lambda)$ is a decomposition for $L$. Such a decomposition induces a decomposition $K(L) = K(L)_1 \oplus K(L)_2$.
In the following let $(B,M)$ be a principally polarized abelian variety with $\theta \in H^0(B,M)$ the unique (up to a scalar) section. Write $B=V/\Lambda$ and let $\Lambda = \Lambda_1 \oplus \Lambda_2$ be a decomposition for $M$. Fix $n \in {\mathbb{N}}$. There is a natural action on $H^0(B,M^n)$ by the theta group $\mathcal{G}(M^n) = \{(b,\varphi) \ | \ b \in K(M^n), \ \varphi\colon t_b^*M^n \overset{\cong}{\to} M^n\}$. We can choose compatible isomorphisms $\varphi_b\colon t_b^*M^n \to M^n$ for $b \in K(M^m)_1$ so that for any $b,b' \in K(M^n)_1$ we have $\varphi_b(t_{b'}^*\varphi_{b'}) = \varphi_{b'}(t_b^*\varphi_b)$. That means that the action of $\mathcal{G}(M^n)$ induces an action of $K(M^n)_1$. For our purpose we want to find a section $\widetilde{\theta} \in H^0(B,M^n)$ that is invariant under this action. Consider the isogeny $$\begin{aligned}
\varphi\colon B \to B'= B/K(M^n)_1
\end{aligned}$$ and let $M'$ be a line bundle on $B'$ such that $\varphi^*M' = M^n$. Since $M'$ is a principal polarization there is a unique (again up to a scalar) section $\theta' \in H^0(B',M')$. We can take $\widetilde{\theta} = \varphi^*\theta'$ since clearly for any $\lambda \in K(M^n)_1$ we have $t_\lambda^*\widetilde{\theta} = t_\lambda^* \varphi^* \theta' = \varphi^* t_\lambda^* \theta' = \varphi^* \theta' = \widetilde{\theta}$ for any $\lambda \in K(M^n)_1$. Abusing notation a little we will also write $\theta$ and $\widetilde{\theta}$ for the associated theta divisors.
Using the Theorem of the Square we see that for any $b \in B$ $$\begin{aligned}
t^*_{nb}M {\otimes}t^*_{-b}M^n & \cong t_b^* M^n {\otimes}M^{-n+1} {\otimes}t_{-b}^*M^n \\
& \cong M^n {\otimes}M^n {\otimes}M^{-n+1} \\
& \cong M^{n+1}
\end{aligned}$$ so the divisor $t^*_{nb}\theta + t^*_{-b}\widetilde{\theta}$ is an element of the linear system $|(n+1)\theta|$ thus we have a morphism $$\begin{aligned}
\phi\colon & B \rightarrow |(n+1)\theta| \\
& b \mapsto t^*_{nb}\theta + t^*_{-b}\widetilde{\theta}.
\end{aligned}$$ The following proposition is a generalization of a result by Wirtinger that can be found in [@MR0379510 p. 335].
\[prop\_1\] For any $n \in {\mathbb{N}}$ there is a nondegenerate bilinear form $\eta\colon H^0(B,M^{n+1}) {\otimes}H^0(B,M^{n+1}) \rightarrow {\mathbb{C}}$ inducing the isomorphism $$\eta'\colon {\mathbb{P}}(H^0(B,M^{n+1})^{\vee}) \overset{\cong}{\longrightarrow} {\mathbb{P}}(H^0(B,M^{n+1})) = |(n+1)\theta|$$ such that $$\begin{tikzcd}
\ & {\mathbb{P}}(H^0(B,M^{n+1})^{\vee}) \arrow{dd}{\eta'} \\
B \arrow{ur}[swap]{\phi_{M^{n+1}}} \arrow{dr}{\phi} & \ \\
\ & \ |(n+1)\theta|
\end{tikzcd}$$ commutes.
Consider the morphism $$\begin{aligned}
s\colon B \times B & \to B \times B \\
(x,y) & \mapsto (x+ny,x-y).
\end{aligned}$$ We now have an isomorphism $$s^*(p_1^*M {\otimes}p_2^*M^n) \cong p_1^*M^{n+1} {\otimes}p_2^*M^{n(n+1)}$$ To see this using the Appel-Humbert Theorem it suffices to compare the first Chern class and the semicharacters of both line bundles (see [@MR2062673 Lemma 7.1.1] for the case $n=1$). For any $m \in {\mathbb{N}}$ and $\alpha \in K(M^m)_1$ we will write $\theta_\alpha^m = \varphi_\alpha(t_\alpha^*\theta^m)$ with $\varphi_\alpha \colon t_\alpha^*M^m \to M^m$ the compatibly chosen isomorphisms from before so that $\{\theta_\alpha^m \ | \ \alpha \in K(M^m)_1 \}$ defines a basis for $H^0(B,M^m)$. Now we can write $$\begin{aligned}
\label{eq_1}
s^*(p_1^*\theta {\otimes}p_2^*\tilde{\theta}) = \sum_{\substack{\alpha \in K(M^{n+1})_1 \\ \beta \in K(M^{n(n+1)})_1}}
{c_{\alpha\beta} p_1^*\theta_\alpha^{n+1} {\otimes}p_2^*\theta_\beta^{n(n+1)}}
\end{aligned}$$ We want to obtain dependencies between the coefficients $c_{\alpha\beta}$ to see that they are determined by a square matrix which we will use to define $\eta$. Consider the pullback of equation (\[eq\_1\]) by $t_{(0,-\gamma)}$ with $\gamma \in K(M^n)_1$. On the left hand side, since $$\begin{aligned}
s(t_{(0,-\gamma)}(x,y)) &= s(x,y-\gamma) \\
&= (x+n(y-\gamma),x-(y-\gamma))\\
&= (x+ny-n\gamma,x-y+\gamma) \\
&=t_{(0,\gamma)}(s(x,y))
\end{aligned}$$ we get $$\begin{aligned}
t_{(0,-\gamma)}^*s^*(p_1^*\theta {\otimes}p_2^*\tilde{\theta})
& = s^*t_{(0,\gamma)}^*(p_1^*\theta {\otimes}p_2^*\tilde{\theta}) \\
& = s^*(p_1^*\theta {\otimes}p_2^*t_\gamma^*\tilde{\theta}) \\
& = s^*(p_1^*\theta {\otimes}p_2^*\widetilde{\theta}).
\end{aligned}$$ Here, we obtain the last line because we chose $\widetilde{\theta}$ such that it is invariant under translation by $\gamma \in K(M^n)_1$. On the right hand side we have $$\begin{aligned}
& t_{(0,-\gamma)}^*\left(\sum_{\substack{\alpha \in K(M^{n+1})_1 \\ \beta \in K(M^{n(n+1)})_1}}
{c_{\alpha\beta} p_1^*\theta_\alpha^{n+1} {\otimes}p_2^*\theta_\beta^{n(n+1)}} \right) \\
= & \sum_{\substack{\alpha \in K(M^{n+1})_1 \\ \beta \in K(M^{n(n+1)})_1}}
{c_{\alpha\beta} p_1^*\theta_\alpha^{n+1} {\otimes}p_2^*t_{-\gamma}^*\theta_\beta^{n(n+1)}}
\end{aligned}$$ The pullbacks on the right hand side permute the basis elements, comparing coefficients gives $c_{\alpha\beta} = c_{\alpha,\beta-\gamma}$.
Now because $\gcd(n,n+1)=1$, the exact sequence $$0 \to K(M^n)_1 \to K(M^{n(n+1)})_1 \to K(M^{n+1})_1 \to 0$$ splits and thus $K(M^{n(n+1)})_1 \cong K(M^n)_1 \oplus K(M^{n+1})_1$. Therefore for any $\beta \in K(M^{n(n+1)})_1$ there is exactly one $\gamma \in K(M^n)_1$ such that $\beta-\gamma \in K(M^{n+1})_1$, namely $\gamma$ is the $n$-torsion part of $\beta$. Ultimately this means that we can choose representatives $\alpha,\beta \in K(M^{n+1})_1$ so that the matrix $(c_{\alpha\beta})$ is determined by $\alpha, \beta \in K(M^{n+1})_1$.
We still need to show that $\det(c_{\alpha\beta}) \neq 0$. If the determinant were zero, the element $s^*(p_1^*\theta {\otimes}p_2^*\tilde{\theta})$ would be contained in a proper subspace $W_1 {\otimes}W_2$ with $W_1 \subsetneq H^0(B,M^{n+1})$ of $H^0(B,M^{n+1}) {\otimes}H^0(B,M^{n(n+1)})$. However, translation by an element $b \in K(M^n)$ acts on $H^0(B,M^{n+1})$ and since $K(M^{n+1}) \subset K(M^{n(n+1)})$ there is an action of $\Delta(B_{n+1}) = \{(b,b) \ | \ b \in B_{n+1}\}$ on $H^0(B,M^{n+1}) {\otimes}H^0(B,M^{n(n+1)})$. The element $s^*(p_1^*\theta {\otimes}p_2^*\widetilde{\theta})$ is invariant under this action and since the action on $H^0(B,M^{n+1})$ is irreducible it cannot lie in such a proper subspace. We conclude that $\det(c_{\alpha\beta}) \neq 0$ so $\eta(\theta_\alpha^{n+1},\theta_\beta^{n+1}) \coloneqq c_{\alpha\beta}$ defines the desired form $\eta$.
The equation (\[eq\_1\]) can be expressed as $$\theta(u+nv)\tilde{\theta}(u-v) = \sum_{\alpha,\beta \in K(M^{n+1})_1}
{c_{\alpha\beta} \theta_\alpha^{n+1}(u) \theta_\beta^{n(n+1)}(v)} \text{ for any } u,v \in B.$$ For each $v \in B$ this implies that $u$ is in the support of the divisor $t_{nv}^* \theta + t_{-v}^* \widetilde{\theta}$ if and only if it is a zero of $\sum{c_{\alpha\beta}\theta_\beta^{n(n+1)}(v)\theta_\alpha^{n+1}}$ which gives that $\phi(v)=\eta'(\phi_M(v))$.
With this we can prove the following theorem.
Let $L$ be a line bundle on a simple abelian variety $A$ of dimension $g$. Fix $n \in {\mathbb{N}}$. If $h^0(A,L)>\left(\frac{n+1}{n}\right)^g \cdot g!$ then the multiplication map $$\mu_n\colon H^0(A,L) {\otimes}H^0(A,L^n) \rightarrow H^0(A,L^{n+1})$$ is surjective.
Choose a maximal isotropic subgroup with respect to the Weil form, say $H=K(L)_1$ and cosider the isogeny $$\pi\colon A \to B=A/H.$$ There is a principal polarization $M$ on $B$ such that $\pi^*M = L$. The character group $\widehat{H} \coloneqq {\mathrm{Hom}}(H,{\mathbb{C}}^*)$ is a subgroup of ${\mathrm{Pic}}^0(B)$ so a character $\alpha \in \widehat{H}$ corresponds to a degree $0$ line bundle on $B$ also denoted by $\alpha$. We have a decomposition $\pi_*\mathcal{O}_A = \bigoplus_{\alpha \in \widehat{H}}\alpha$. This gives us $$\begin{aligned}
\pi_* L & = \pi_*(\mathcal{O}_A {\otimes}L) \\
& = \pi_*(\mathcal{O}_A {\otimes}\pi^*M) \\
& = \pi_* \mathcal{O}_A {\otimes}M \hspace{3cm} \text{(projection formula)} \\
& = \bigoplus_{\alpha \in \widehat{H}} M {\otimes}\alpha.
\end{aligned}$$ More generally, for any $m \in {\mathbb{N}}$, $\pi_* L^m = \bigoplus_{\alpha \in \widehat{H}} M^m {\otimes}\alpha$. Consequently $$H^0(A,L^m) \cong \bigoplus_{\alpha \in \widehat{H}} H^0(B,M^m {\otimes}\alpha).$$ for any $m \in {\mathbb{N}}$. However, given a power of $L$ we take the larger subgroup $K(L^n)_1$ and get a finer decomposition. We will do that specifically for the second factor of $\mu_n$. Analogously to before, let $G=K(L^n)_1$ and consider the isogeny $$\pi'\colon A \to B' = A/G.$$ Once again $B'$ is principally polarized say with polarization $M'$ and $L^n = \pi^*M'$. With the same arguments as above we can decompose $$H^0(A,L^n) \cong \bigoplus_{\alpha \in \widehat{G}} H^0(B,M' {\otimes}\alpha).$$ Due to our choices of subgroups $H=K(L)_1=nK(L^n)_1$ is a subgroup of $G$ so that these decompositions are compatible. This gives $\widehat{H} = \psi_M(\pi(K(L)_2))$ and $\widehat{G} = \psi_{M'}(\pi'(K(L^n)_2))$. The following diagram summarizes the situation $$\label{cd_9}
\begin{tikzcd}
A \arrow{r}{\pi} \arrow{d}{\psi_L} \arrow[rr, bend left, "\pi'"] & B \arrow{r}{\varphi} \arrow{d}{\psi_M} & B' \arrow{d}{\psi_{M'}} \\
{\mathrm{Pic}}^0(A) & {\mathrm{Pic}}^0(B) \arrow{l}{\pi^*} & {\mathrm{Pic}}^0(B') \arrow{l}{\varphi^*}.
\end{tikzcd}$$ Note that the second square does not commute but that we have instead $\varphi^* \circ \psi_{M'} \circ \varphi = n \cdot \psi_M$.
Now we can write our multiplication map as $$\mu_n\colon \bigoplus_{\alpha \in \widehat{H}, \beta \in \widehat{G}} H^0(B,M {\otimes}\alpha) {\otimes}H^0(B',M' {\otimes}\beta) \overset{1{\otimes}\varphi^*}{\to}
\bigoplus_{\gamma \in \widehat{H}} H^0(B,M^{n+1} {\otimes}\gamma).$$ We can decompose $\mu_n = \bigoplus_{\gamma \in \widehat{H}}\mu_{n,\gamma}$ with $$\mu_{n,\gamma}\colon \bigoplus_{\beta \in \widehat{G}} H^0(B,M{\otimes}\gamma {\otimes}\varphi^*\beta) {\otimes}H^0(B',M' {\otimes}\beta^{-1}) \to H^0(B,M^{n+1} {\otimes}\gamma).$$ Now since $\psi_M$ is an isomorphism we can take $H' \coloneqq \psi_M^{-1}(\widehat{H}) = \pi(K(L)_2)$, $G'\coloneqq \psi_{M'}^{-1}(\widehat{G}) = \pi'(K(L^n)_2)$ and $\widetilde{G} \coloneqq \varphi^{-1}(G') {\cap}\pi(K(L^n)_2)$ Taking $c \in H'$ such that $\gamma = \psi_M((n+1)c) = \psi_{M^{n+1}}(c)$ and writing out the definitions of $\psi_M$ and $\psi_{M'}$, we obtain
$$\mu_{n,\gamma}\colon \bigoplus_{\substack{b' \in G', b \in \widetilde{G} \\ \varphi(b) = b'}} H^0(B,t_{(n+1)c+nb}^*M) {\otimes}H^0(B',t_{-b'}^*M') \to H^0(B,t_c^*M^{n+1}).$$ The difference between this and the proof in [@MR1974682] is that we are now taking the sum over the much larger group $G'$. Let $\theta$ be the unique theta divisor of $|M|$ and $\widetilde{\theta} \in |M^{n+1}|$ the pullback along $\varphi$ of the unique theta divisor $\theta'$ in $|M'|$. We see that $\mu_{n,\gamma}$ is surjective if the linear system $|t_c^*M^{n+1}|$ is generated by divisors of the form $t_{(n+1)c+nb}^*\theta+t_{-b}^*\widetilde{\theta} = t_c^*( t_{n(c+b)}^*\theta+t_{-(c+b)}^*\widetilde{\theta})$ with $b \in \widetilde{G}$. By Proposition \[prop\_1\] it is thus surjective if the image of $\widetilde{G}$ under $\phi_c\coloneqq t_c^* \circ\phi$ generates $|t_c^*M^{n+1}|$ or equivalently if the image of $\widetilde{G}$ under $\phi$ generates $|M^{n+1}|$. Now by assumption we have $|\widetilde{G}| = h^0(A,L^n) = n^g\cdot h^0(A,L)>(n+1)^g\cdot g! = h^0(B,M^{n+1})\cdot g!$ and thus we can apply Proposition \[prop\_iyer\] to finish the proof.
Setting $n = g-1$ and using the method discussed in Section \[section\_green\] we obtain Theorem \[thm\_main\] as a corollary.
\[corollary\_ITT\] Let $X$ be a smooth hypersurface in a $g$-dimensional simple abelian variety $A$. If $h^0(A,\mathcal{O}_A(X))>\left(\frac{g}{g-1}\right)^g \cdot g!$, then the Infinitesimal Torelli Theorem holds for $X$.
For the case $g=2$ this is exactly the same as in [@MR1974682]. However for higher dimensions our result directly improves the bound. For $g=3$ for example, $h^0(A,\mathcal{O}_A(X))>20$ is a sufficient condition for a hypersurface on a simple abelian variety to satisfy the ITT as we have seen above whereas to show that it gives a projectively normal embedding we need $h^0(A,\mathcal{O}_A(X))>48$.
Let $S\subset A$ be a smooth complex projective surface that embeds into its Albanese $A$ as a hypersurface. If $S$ has geometric genus $p_g > 22$ and $A$ is simple then the ITT holds for $S$.
Consider the exact sequence $$0 \to \mathcal{O}_A \to \mathcal{O}_A(S) \to \mathcal{O}_S(S) \to 0$$ By adjunction we have $\omega_S \cong \mathcal{O}_S(S)$ so taking cohomology and comparing dimensions gives $$h^0(\mathcal{O}_A(S)) = p_g + 1 - 3 > 20$$ so we can apply Corollary \[corollary\_ITT\].
In [@MR929539] the ITT is proved for surfaces of irregularity greater than or equal to $5$ under the assumption that $\Omega_S^1$ is globally generated and that some other conditions hold. In our case $S$ has irregularity $3$ and we have to assume that the Albanese morphism $a:S \to A$ is an embedding which does in fact imply that $\Omega_S^1$ is globally generated. An interesting question would be if our approach can still be used to show the ITT when $\Omega_S^1$ is globally generated but $a$ is not an embedding.
|
---
abstract: |
A two-valued fitness landscape is introduced for the classical Eigen’s quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single or sharply peaked landscape. A general, non permutation invariant quasispecies model is studied, therefore the dimension of the problem is $2^N\times
2^N$, where $N$ is the sequence length. It is shown that if the fitness function is equal to $w+s$ on a $G$-orbit $A$ and is equal to $w$ elsewhere, then the mean population fitness can be found as the largest root of an algebraic equation of degree at most $N+1$. Here $G$ is an arbitrary isometry group acting on the metric space of sequences of zeroes and ones of the length $N$ with the Hamming distance. An explicit form of this exact algebraic equation is given in terms of the spherical growth function of the $G$-orbit $A$. Sufficient conditions for the so-called error threshold for sequences of orbits are given. Motivated by the analysis of the two-valued fitness landscapes an abstract generalization of Eigen’s model is introduced such that the sequences are identified with the points of a finite metric space $X$ together with a group of isometries acting transitively on $X$. In particular, a simplicial analogue of the original quasispecies model is discussed, which can be considered as a mathematical model of the switching of the antigenic variants for some bacteria.
#### Keywords:
Eigen’s quasispecies model, single peaked landscape, mean population fitness, regular polytope, finite metric space, isometry group
#### AMS Subject Classification:
15A18; 92D15; 92D25
author:
- |
Yuri S. Semenov$^{1}$, Artem S. Novozhilov$^{{2},}$[^1]\
*$^\textrm{\emph{1}}$Applied Mathematics–1, Moscow State University of Railway Engineering,*\
*Moscow 127994, Russia*\
*$^\textrm{\emph{2}}$Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA*
title: '[On Eigen’s quasispecies model, two-valued fitness landscapes, and isometry groups acting on finite metric spaces]{}'
---
Introduction. Classical quasispecies model
==========================================
A great deal of research on the border between mathematics and biology was spurred by Eigen’s *quasispecies model*, formulated in 1971 in [@eigen1971sma]. This model was suggested to describe the replication of prebiotic macromolecules in order to study various aspects of the problem of the origin of life. Independently, an equivalent model was suggested to study the change of frequencies of different genotypes in haploid multi-allele populations under the evolutionary forces of selection and mutation. Standard references to review the classical and recent developments are [@baake1999; @eigen1988mqs; @jainkrug2007; @wilke2005quasispecies; @schuster2012evolution]. We also refer to the introductory sections in [@bratus2013linear; @semenov2014; @semenov2015] for more details on various issues in the quasispecies theory. In the present work we are mostly concerned with some specific mathematical developments about the model, which can also describe various systems in population biology or chemical kinetics.
We start with formulating the model. We assume that we deal with a population of sequences of the fixed length $N$. Each sequence is composed of zeroes and ones, hence $l:=2^N$ being the total number of different types of sequences. The sequences can reproduce and mutate to each other. We also assume that the reproduction events occur at discrete time moments, and sequence $k$ produces $w_k$ offspring on average with the probabilities $q_{jk}$, where $q_{jk}$ is the probability to produce sequence $j$ by the parent of type $k$. Therefore, $q_{kk}$ is the probability of the error-free reproduction, and $\sum_{j}q_{jk}=1$. Let ${{\boldsymbol{p}}}\in{\mathbf{R}}^l$, ${{\boldsymbol{p}}}^\top=(p_0,\ldots,p_{l-1})$ be the vector of frequencies of different types of sequences at the selection-mutation equilibrium. Then it follows from the basic theory (e.g., [@burger2000mathematical]) that ${{\boldsymbol{p}}}$ can be found as the positive normalized eigenvector of the matrix ${{\boldsymbol{QW}}}$ corresponding to the dominant eigenvalue $\lambda$, i.e., $$\label{i:1}
{{\boldsymbol{QWp}}}=\lambda {{\boldsymbol{p}}}.$$ Here ${{\boldsymbol{W}}}=\operatorname{diag}(w_0,\ldots,w_{l-1})$ is the matrix describing the *fitness landscape* (note that we count the indices from 0), and ${{\boldsymbol{Q}}}=(q_{jk})_{l\times l}$ is the mutation matrix, which is stochastic by definition. At the equilibrium the dominant eigenvalue $\lambda$ is equal to the mean population fitness $\lambda=\overline{w}:=\sum_j w_jp_j$, and the vector ${{\boldsymbol{p}}}$ was called the *quasispecies* by Eigen and his co-authors.
The basic mathematical problem, given ${{\boldsymbol{W}}}$ and ${{\boldsymbol{Q}}}$, is to determine $\overline{w}$ and ${{\boldsymbol{p}}}$. This problem turned out to be very nontrivial and required an introduction of intricate methods of statistical physics, careful numerical procedures, and non-elementary mathematical analysis to achieve a partial progress (much more detail can be found in [@bratus2013linear; @semenov2014; @semenov2015] and references therein). No general analytical solution exists. Moreover, even numerically, there are important obstacles to find $\overline{w}$ and/or ${{\boldsymbol{p}}}$, most serious of which is the dimensionality of the problem, recall that the matrices have the dimensions $l\times l=2^N\times 2^N$, where $N$ is the sequence length. One particular solution to the problem of dimensionality is to consider very special fitness landscapes, such that the average number of offspring is determined not by the sequence type (which is the ordered list of ones and zeroes) but by the sequence composition (i.e., by the numbers of ones and zeroes in the sequence). Such fitness landscapes are sometimes called *symmetric* or *permutation invariant* and allow to reduce the dimension of the problem from $2^N\times 2^N$ to $(N+1)\times (N+1)$. This worked especially well for the so-called *single peaked fitness landscape* defined by $${{\boldsymbol{W}}}=\operatorname{diag}(w+s,w,\ldots,w),\quad w\geq 0,\,s>0,$$ see [@galluccio1997exact; @semenov2014; @swetina1982self] for additional details. Moreover, most limiting procedures when the sequence length $N$ tends to $\infty$, were applied to the models with the permutation invariant fitness landscapes, e.g., [@Baake2007; @saakian2006ese; @semenov2015]. At the same time it is clear that the assumption that the fitness landscape is permutation invariant should be relaxed at least in some specific biological situations.
Our first goal in this manuscript is to present an efficient method to reduce the dimensionality of the mathematical problem from $2^N$ to $N+1$ for some specific fitness landscapes that generalize the single peaked landscape but are not permutation invariant *sensu* the definition given above. These fitness landscapes still possess a great deal of symmetry but are much more flexible for assigning the fitness values compared to the permutation invariant landscapes. Second, by carefully analyzing the obtained algebraic equation for $\overline{w}$ we are able to give a precise mathematical definition of the threshold-like behavior, which is observed in some quasispecies models [@wilke2005quasispecies]. We present sufficient conditions for the model to demonstrate such behavior. The language of the group theory allows us to recast the conditions for the error threshold to occur into the geometric picture of sequences of orbits in the underlying metric space under the action of a given group. Third, motivated by these considerations, we introduce an abstract generalized Eigen quasispecies problem, give several specific examples, and briefly analyze a simplicial analogue of the original quasispecies model. Despite a high level of abstraction of the introduced model, even the simplest mathematical construction describes biologically realistic systems, in particular, the switching of the antigenic variants for some bacteria[^2].
Notation. The reduced problem {#sec:1}
=============================
In this section we introduce the required notation and also list several facts necessary for our exposition, additional details can be found in [@semenov2014]. Recall that $N$ denotes the sequence length and $l=2^N$.
Let $A$ be a non-empty fixed subset of indices: $A\subseteq\{0,1,\dots,l-1\}$. For some fixed $w\ge 0$, $s>0$ we consider the two-valued fitness landscapes of the form $$w_k=\left\{
\begin{array}{r}
w+s,\quad k\in A\;,\\
w,\quad k\notin A\;.\\
\end{array}
\right.$$ Thus, the diagonal matrix ${{{\boldsymbol{W}}}}$ of fitnesses can be represented as $$\label{eq:1}
{{{\boldsymbol{W}}}}=w {{{\boldsymbol{I}}}}+s{{{\boldsymbol{E}}}}_A=w {{{\boldsymbol{I}}}}+s
\sum_{a\in A}{{{\boldsymbol{E}}}}_a,$$ ${{\boldsymbol{ I}}}$ being the identity matrix and ${{{\boldsymbol{E}}}}_a$ being the elementary matrix with the only one nontrivial entry $e_{aa}=1$ on the main diagonal.
Consider the eigenvalue problem $$\label{eq:2}
{{{\boldsymbol{QW}}}} {{\boldsymbol{p}}}=\overline{w}\, {{\boldsymbol{p}}},$$ where ${{\boldsymbol{p}}}^{\top}=(p_0,\dots,p_{l-1})\in {\mathbf{R}}^l$ is the eigenvector of the matrix $ {{{\boldsymbol{QW}}}}$ corresponding to the leading (dominant) eigenvalue $\lambda=\overline{w}$. The vector ${{\boldsymbol{p}}}$ is normalized such that $$\label{eq:3}
\sum_{k=0}^{l-1}p_k=1,\quad p_k> 0\,,$$ and hence the equality $$\label{eq:4}
\sum_{k=0}^{l-1}w_kp_k=w+s\sum_{a\in A}p_a= \overline{w}$$ holds.
For the following we make an additional assumption that the mutations at different sites of the sequences are independent and the fidelity (i.e., the probability of the error-free reproduction) per site per replication is given by the same constant $0\leq q\leq 1$ for each site. Then $$q_{jk}=q^{N-H_{jk}}(1-q)^{H_{jk}}, \quad j,k=0,\ldots,l-1$$ defines the mutation matrix ${{\boldsymbol{Q}}}$. Here $H_{jk}$ is the standard Hamming distance between sequences $j$ and $k$ (i.e., the number of sites at which sequences $j$ and $k$ are different). Note that now both the leading eigenvalue $\lambda$ and the quasispecies ${{\boldsymbol{p}}}$ depend on the fitness landscape and, most importantly, on the mutation fidelity $q$, hence we sometimes denote $\overline{w}=\overline{w}(q)$ and ${{\boldsymbol{p}}}={{\boldsymbol{p}}}(q)$.
Using the special structure of the mutation matrix ${{\boldsymbol{Q}}}$, it can be shown (see, e.g, [@semenov2014]) that there exists a non-degenerate matrix ${{\boldsymbol{T}}}$ such that $${{{\boldsymbol{T}}}}^{-1}=\frac{1}{l}{{{\boldsymbol{T}}}},\quad{{{\boldsymbol{T}}}}^{-1}{{{\boldsymbol{QT}}}}=\frac{1}{l}{{{\boldsymbol{ TQT}}}}=:{{\boldsymbol{D}}},$$ with $${{\boldsymbol{D}}}=\operatorname{diag}\bigl(1,\dots,(2q-1)^{H_{j}},\dots,(2q-1)^{H_{l-1}}\bigr),$$ where $H_{j}$ is the Hamming norm of the sequence $j$, i.e., the number of ones in this sequence, $H_j:=H_{0j}$; we are using the lexicographical ordering of indices, hence, e.g., $H_0=0$ and $H_{l-1}=N$. Moreover, explicitly matrix ${{\boldsymbol{T}}}$ is given through the recursive procedure $${{\boldsymbol{T}}}={{\boldsymbol{T}}}_{N},\quad {{\boldsymbol{T}}}_{k}={{\boldsymbol{T}}}_1\otimes {{\boldsymbol{T}}}_{k-1},\quad k=2,3,\ldots,N,$$ and $${{\boldsymbol{T}}}_1=\begin{bmatrix}
1 & 1 \\
1 & -1 \\
\end{bmatrix}.$$ Here $\otimes$ denotes the Kronecker product (e.g., [@laub2005matrix]).
We write down the indices $a,b$, where $0\le a\le l-1$, $0\le b\le l-1$, in the binary representation: $$a=\alpha_0+\alpha_12+\dots+\alpha_{N-1}2^{N-1}=[\alpha_0,\,\alpha_1,\,\dots\,,\,\alpha_{N-1}]\,,\quad
\alpha_k\in\{0,1\}\,,$$ $$b=\beta_0+\beta_12+\dots+\beta_{N-1}2^{N-1}=[\beta_0,\,\beta_1,\,\dots\,,\,\beta_{N-1}]\,,\quad
\beta_k\in\{0,1\}\,.$$
One additional property of ${{\boldsymbol{T}}}$ that we will require in the following is given by the following lemma.
\[l:1\] Let ${{{\boldsymbol{T}}}}=(t_{ab})_{l\times l}$ be the transition matrix defined above, $P_a(z)=\sum\limits_{k=0}^{l-1}t_{ka}z^k $ be the generating polynomial of the $a$-th column of ${{\boldsymbol{T}}}$. Then $$\label{eq:5}
P_a(z)=\sum\limits_{k=0}^{l-1}t_{ka}z^k=\prod_{i=0}^{N-1}\left(1+(-1)^{\alpha_i}z^{2^i}\right)\;.$$ Moreover, $$\label{eq:6}
t_{ab}=t_{ba}=(-1)^{\left<a,b\right>}\;,\quad
\left<a,b\right>:=\sum\limits_{k=0}^{N-1}\alpha_k\beta_k\bmod 2.$$
We prove the formulas and by the induction on $N$. Indeed, according to the definition of ${{\boldsymbol{T}}}$ we have the Kronecker product $${{{\boldsymbol{T}}}}={{{\boldsymbol{T}}}}_{N}={{{\boldsymbol{T}}}}_1\otimes {{{\boldsymbol{T}}}}_{N-1}=\left[\begin{array}{rr}1&1\\1&-1\end{array}\right]\otimes
{{{\boldsymbol{T}}}}_{N-1}=\left[\begin{array}{rr}{{{\boldsymbol{T}}}}_{N-1}&{{{\boldsymbol{T}}}}_{N-1}\\{{{\boldsymbol{T}}}}_{N-1}&-{{{\boldsymbol{T}}}}_{N-1}\end{array}\right].$$ Let us represent $a=a_N=a_{N-1}+\alpha_{N-1} 2^{N-1}$. The block form of ${{{\boldsymbol{T}}}}={{{\boldsymbol{T}}}}_{N}$ implies the equality $P_a(z)=P_{a_{N-1}}(z)(1+(-1)^{\alpha_{N-1}}z)$ and by induction.
Consider again the representations $a=a_N=a_{N-1}+\alpha_{N-1}
2^{N-1}$, $b=b_N=b_{N-1}+\beta_{N-1}2^{N-1}$. It follows from the above form of the matrix ${{{\boldsymbol{T}}}}$ that $$t_{ab}=({{{\boldsymbol{T}}}}_N)_{ab} =({{{\boldsymbol{T}}}}_{N-1})_{a_{N-1}b_{N-1}}(-1)^{\alpha_{N-1}\beta_{N-1}}\;.$$ The induction on $N$ completes the proof of .
Let us now return to the problem . We have $${{{\boldsymbol{T}}}}^{-1}{{{\boldsymbol{ QT}}}} {{{\boldsymbol{T}}}}^{-1}{{{\boldsymbol{ WT}}}} {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}=\overline{w}\, {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}\;,$$ or, in view of , $$\label{eq:7}
{{{\boldsymbol{D}}}} \left(w{{{\boldsymbol{I}}}}+s\sum_{a\in A}{{{\boldsymbol{T}}}}^{-1}{{{\boldsymbol{E}}}}_a{{{\boldsymbol{T}}}}\right) {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}=\overline{w}\, {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}\;,$$ which yields, after some rearrangement, $$\label{eq:8}(\overline{w}{{{\boldsymbol{I}}}}-w{{{\boldsymbol{D}}}}) {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}=\frac{s}{l}\sum_{a\in A}{{{\boldsymbol{ DT}}}}{{{\boldsymbol{E}}}}_a{{\boldsymbol{p}}}\;.$$
Let ${{\boldsymbol{x}}}:={{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}$, ${{\boldsymbol{p}}}={{{\boldsymbol{T}}}}{{\boldsymbol{x}}}$. Then implies $$\label{eq:9}
{{\boldsymbol{x}}}=\frac{s}{l}\sum_{a\in A}(\overline{w}{{{\boldsymbol{I}}}}-w{{{\boldsymbol{D}}}})^{-1}
{{{\boldsymbol{ DT}}}}{{{\boldsymbol{E}}}}_a{{\boldsymbol{p}}}\;,$$ or, in coordinates, $$\label{eq:10}
x_k=\frac{s}{l}\sum_{a\in A}\frac{(2q-1)^{H_k}
t_{ka}p_a}{\overline{w}-w(2q-1)^{H_k}}\,,\quad
k=0,\dots,l-1.$$ Since ${{\boldsymbol{p}}}={{{\boldsymbol{T}}}}{{\boldsymbol{x}}}$, then we get $$\label{eq:11}
p_b=\sum_{k=0}^{l-1}t_{bk}x_k=\frac{s}{l}\sum_{a\in
A}\sum_{k=0}^{l-1}\frac{(2q-1)^{H_k}
t_{bk}t_{ak}}{\overline{w}-w(2q-1)^{H_k}}p_a\,.$$
Note that only the components $p_a$, where $a\in A$, are involved in the right-hand side of . We can omit the components $p_b$ for $b\notin A$ and obtain the “reduced” column-vector ${{\boldsymbol{p}}}_A=(p_a)$, $a\in A$. Considering only $a\in A$ we can rewrite as $$\label{eq:12}
{{\boldsymbol{p}}}_A={{\boldsymbol{M}}}{{\boldsymbol{p}}}_A\,,$$ where ${{\boldsymbol{M}}}=(m_{ab})_{r\times r}$ is the square matrix of the order $r=|A|$ with the entries $$\label{eq:13}
m_{ab}=m_{ba}=\frac{s}{l}\sum_{k=0}^{l-1}\frac{(2q-1)^{H_k}
t_{ak}t_{bk}}{\overline{w}-w(2q-1)^{H_k}}=\frac{s}{l}\sum_{k=0}^{l-1}\frac{(2q-1)^{H_k}
(-1)^{\left<a,k\right>+\left<b,k\right>}}{\overline{w}-w(2q-1)^{H_k}}\,$$ in view of and Lemma \[l:1\]. The equality means that the reduced vector ${{\boldsymbol{p}}}_A$ is an eigenvector of ${{\boldsymbol{M}}}$ corresponding to the eigenvalue $\lambda=1$.
We consider $\overline{w}$ in as a parameter. It follows from that $\overline{w}$ depends only on $p_a$, $a\in
A$, that is, on the reduced vector ${{\boldsymbol{p}}}_A$. The original eigenvector ${{\boldsymbol{p}}}$ can be reconstructed from ${{\boldsymbol{p}}}_A$ with the help of if ${{\boldsymbol{p}}}_A$ is known. Therefore, for the introduced special two-valued fitness landscapes, instead of the original problem , we can consider the problem to find the reduced eigenvector ${{\boldsymbol{p}}}_A$ satisfying and corresponding to the eigenvalue $\lambda=1$ of the matrix ${{\boldsymbol{M}}}$ defined in . Since defines $p_a,\,a\in A$ in terms of $\overline{w}$, then, finally, formula can be used to determine $\overline{w}$ implicitly in terms of the system parameters $w,s$ and $q$.
To conclude, we remark that the eigenvalue $\overline{w}$ can be also found from the equation $$\label{eq:15}\det({{{\boldsymbol{M}}}}-{{{\boldsymbol{I}}}})=0\,,$$ but in general it is not easier than to solve the original problem . For the single peaked landscapes (i.e., when $A$ consists of a single element) the corresponding equation was obtained and investigated in [@semenov2014]. In the next section we propose a different approach that can be further elaborated on for some special cases.
Equation for the leading eigenvalue $\overline{w}$ {#sec:2}
==================================================
In this section we show that, using the preliminary analysis from the previous section, it is possible to find an algebraic equation for the eigenvalue $\overline{w}$ under some additional symmetry requirements on the set $A$, and this equation is of degree at most $N+1$.
First we transform in the following way: $$\label{eq:16}
\begin{split}
m_{ab}&=\frac{s}{l}\sum_{k=0}^{l-1}\frac{(2q-1)^{H_k}t_{ak}t_{bk}}{\overline{w}-w(2q-1)^{H_k}}=\frac{s}{l\overline{w}}\sum_{k=0}^{l-1}\frac{(2q-1)^{H_k}t_{ak}t_{bk}}{1-\frac{w}{\overline{w}}(2q-1)^{H_k}}\\
&=\frac{s}{l\overline{w}}\sum_{k=0}^{l-1}t_{ak}t_{bk}\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}}\right)^c (2q-1)^{(c+1)H_k}=\frac{s}{l\overline{w}}\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}}\right)^c\sum_{k=0}^{l-1}(2q-1)^{(c+1)H_k}t_{ak}t_{bk}\;.
\end{split}$$
\[l:2\] We have the following factorization: $$\label{eq:17}\sum_{k=0}^{l-1}z^{H_k} t_{ak}t_{bk}=
(1-z)^{H_{ab}}(1+z)^{N-H_{ab}}\;.$$
It is straightforward to see that $\sum\limits_{k=0}^{l-1}z^{H_k}
t_{ak}t_{bk}= \sum\limits_{k=0}^{l-1}t_{ak}z^{H_k} t_{kb}$ is the entry $z_{ab}$ of the matrix $${{{\boldsymbol{Z}}}}:={{{\boldsymbol{T}}}}\operatorname{diag}(1,\dots,z^{H_k},\dots,z^N){{{\boldsymbol{T}}}}=2^N {{{\boldsymbol{T}}}}\operatorname{diag}(1,\dots,z^{H_k},\dots,z^N){{{\boldsymbol{T}}}}^{-1}.$$ It follows from the properties of ${{\boldsymbol{T}}}$ that $$z_{ab}=2^N(1-q)^{H_{ab}}q^{N-H_{ab}}=(1-z)^{H_{ab}}(1+z)^{N-H_{ab}},\quad
\mbox{where}\; q=\frac{1+z}{2}\;.$$ Thus, the lemma is proved.
Applying Lemma \[l:2\] to we get $$\label{eq:18}
m_{ab}=\frac{s}{l\overline{w}}
\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}}\right)^c
\left(1-(2q-1)^{c+1}\right)^{H_{ab}}\left(1+(2q-1)^{c+1}\right)^{N-H_{ab}}.$$
We have the following consequence of : $$\label{eq:19}
\sum_{b\in A} p_b= \sum_{b\in A}\sum_{a\in A} m_{ba}p_a=\sum_{a\in
A}p_a \sum_{b\in A} m_{ba}.$$
Now we introduce a *key* assumption that will allow us to simplify the analysis.
We assume that the sum $\sum\limits_{b\in A}m_{ab}$ does not depend on $a\in A$. Then it follows from that $\sum\limits_{b\in A}
m_{ba}=1$ for each $a\in A$. In view of it implies that $$\label{eq:20}
\frac{l\overline{w}}{s}=
\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}}\right)^c
\sum\limits_{b\in A}
\left(1-(2q-1)^{c+1}\right)^{H_{ab}}\left(1+(2q-1)^{c+1}\right)^{N-H_{ab}}\,,$$ if the inner sum does not depend on $a\in A$.
The main question is when our key assumption holds. We present some sufficient conditions for this, and hence for the equation .
We refer to the well known geometric interpretation of the metric space $V=V_N=\{0,1\}^N$ with the Hamming distance. Consider 1-skeleton of the $N$-dimensional cube $[0,1]^N$ with the set of vertices $V$. The vertices $a$ and $b$ are connected by the (unique) edge $e_{ab}$ if $H_{ab}=1$. The Hamming distance between vertices $u$ and $v$ is the length of a shortest path connecting these vertices, that is, the number of edges in this path. The set $V$, due to the binary representation $$a=\alpha_0+\alpha_1
2+\dots+\alpha_{N-1}2^{N-1}=[\alpha_0,\,\alpha_1,\,\dots\,,\,\alpha_{N-1}]\,,\quad
\alpha_k\in\{0,1\}\,,$$ can be identified with the set of indices $X=X_N=\{0,1,\dots,2^N-1\}$ with the Hamming distance. In what follows we will usually make no difference between metric spaces $V$ and $X$.
We note that the group ${\rm Iso}(X_N)$ of all isometries of $X_N$, acting on the set $X_N$, is also known as the *Weyl group* $W_N$ of order $2^NN!$ of the root system of type $B_N$ (or $C_N$, see, e.g., [@bourbaki22001lie]).
\[pr:1\] Let $G$ be a group that acts on the metric space $X$ by isometries *(*i.e., $G\leqslant{\rm Iso}(X)$*)* and let $A$ be a $G$-orbit. Then the equality holds.
Since $G$ acts transitively on $A$ and preserves the Hamming distance $H_{ab}$, the inner sum in does not depend on $a\in
A$.
Now we can state the following basic result.
\[corr:2:1\]Under the conditions of Proposition \[pr:1\] the eigenvalue $\overline{w}=\overline{w}(q)$ of is a root of an algebraic equation *(*with the coefficients depending on $q$*)* of degree at most $N+1$.
Consider the polynomial (see and Lemma \[l:2\]) $$\label{eq:21}F_A(z):=\frac{1}{2^N}\sum_{b\in A}(1-z)^{H_{ab}}(1+z)^{N-H_{ab}}\,.$$ We have $$\label{eq:22}
F_A(z)>0,\;-1<z\leq 1\,,\quad F_A(0)=\frac{|A|}{2^N}\;,\quad F_A(1)=1.$$ Since $0\leq H_{ab}\leq N$, we can rewrite as $$\label{eq:23}
F_A(z)=\frac{1}{2^N}\sum_{d=0}^N f_d\,(1-z)^{d}(1+z)^{N-d}
,$$ where $f_d=\#\{b\in A\,|\,H_{ab}=d\}$. Applying the binomial expansion to we get $$\label{eq:24}
F_A(z)=\sum_{d=0}^N h_d\,z^d\;,\quad
h_d=\frac{1}{2^N}\sum_{j=0}^d(-1)^j f_j{\binom{d}{j}}{\binom{N-d}{d-j}}.$$
With the introduced notation equation reads $$\label{eq:25}
\frac{\overline{w}}{s}=
\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}}\right)^c
F_A((2q-1)^{c+1})\,,$$ or $$\frac{\overline{w}}{s}= \sum_{d=0}^N h_d(2q-1)^d
\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}}\right)^c(2q-1)^{cd}=\sum_{d=0}^N
\frac{h_d(2q-1)^d\overline{w}}{\overline{w}-w(2q-1)^d}\,.$$ Finally, the last equation can be transformed into $$\label{eq:26}
\sum_{d=0}^N
\frac{h_d(2q-1)^d}{\overline{w}-w(2q-1)^d}=\frac{1}{s}$$ with the rational coefficients $h_d$ defined in .
We have the following corollary that allows us to reduce the number of computations in some special cases.
\[corr:2:3\] Let $\Gamma=\Gamma_N={\rm Iso}(X_N)$ be the group of all isometries [*(*]{}of order $2^N N!$[*)*]{} acting on the metric space $X=X_N=\{0,1,\dots,2^N-1\}$ with the Hamming distance and let $\gamma A$ be the image of $A$ under the [*(*]{}left[*)*]{} action of $\gamma\in \Gamma$. Then the equations , , , and are the same for $A$ and $\gamma A$.
The equations , , , and were obtained only on the ground of the metric properties of $A$. Note that $A$ is a $G$-orbit if and only if $\gamma A$ is a $\gamma
G\gamma^{-1}$-orbit. Thus, the (left-)acting group $G$ should be substituted by the conjugate $\gamma G\gamma^{-1}$.
The case $w=0$ corresponds to the lethal mutations. In particular, we have
If $w=0$ then we get the following polynomial expression for the leading eigenvalue, where $a\in
A$ may be chosen arbitrarily in the $G$-orbit $A$: $$\label{eq:27}
\overline{w}=s \sum\limits_{b\in A}
(1-q)^{H_{ab}}q^{N-H_{ab}}=sF_A(2q-1)=s\sum_{d=0}^N
h_d(2q-1)^d\,.$$
Examples and applications {#sec:3}
=========================
In this section we consider several simple examples of the two-valued fitness landscapes and apply the obtained equation for the leading eigenvalue. The examples we consider are mostly based on various subgroups of the symmetric group $S_N$.
The symmetric group $G=S_N$ acts on the metric space $X=X_N=\{0,1,\ldots,2^N-1\}$ with the Hamming distance by isometries. To wit, let $\sigma\in S_N$. Then $$\sigma(a)=\sigma[\alpha_0,\,\alpha_1,\ldots,\alpha_{N-1}]=
[\alpha_{\sigma^{-1}(0)},\,\alpha_{\sigma^{-1}(1)},\ldots,\alpha_{\sigma^{-1}(N-1)}],\quad
\alpha_k\in\{0,1\}\,.$$ Note that $G=S_N$ is a proper subgroup of $\Gamma={\rm Iso}(X_N)$. The latter is of the order $2^N N!$ and contains also the elements that correspond to reflections of the $N$-dimensional cube $[0,1]^N$, see, e.g., Example \[ex:3:2\] below.
The $S_N$-orbits are the subsets of $$A_p=\{a\in X\,|\,H_a=p\}\;,\quad p=0,1,\dots,N\,.$$
\[ex:3:1\] Recall that we defined the permutation invariant fitness landscape to be a diagonal matrix ${{\boldsymbol{W}}}$ such that the elements on the main diagonal are $w_j=w_{H_j},$ i.e., the fitness of the sequence $j$ depends only on the total number of ones in this sequence. To satisfy this definition the orbit for the two-valued fitness landscape must coincide with one of $A_p$ defined above.
We can consider only the case $2p\le N$. Indeed, let $\gamma(a)=a^*=l-1-a$ be the index conjugate to $a$. The conjugation $\gamma$ is an involution in $\Gamma$. The binary representation of $a^*$ differs at each position from that of $a$. Then $H_{a^*}=N-H_a$ and $a\in A_p\Leftrightarrow
a^*\in A_p^*=A_{N-p}$. In other words, according to Corollary \[corr:2:3\], the equations , , , and for $A_p$ and $A_p^*=A_{N-p}$ are the same.
To obtain an equation for $\overline{w}$ we will need an auxiliary
\[l:3:1\] For $a,b\in A=A_p$ the distance $H_{ab}$ is even. Moreover, for each $k=0,1,\dots,p\,$ $$\#\{b\in A_p\,|\,H_{ab}=2k\}={\binom{p}{k}}{\binom{N-p}{k}}.$$
If $H_a$ and $H_b$ have the same parity, in particular, coincide then $H_{ab}$ is even, hence $H_{ab}=2k$.
The binary representations $$a=[\alpha_0,\,\alpha_1,\,\dots\,,\,\alpha_{N-1}]\,,
\quad b=[\beta_0,\,\beta_1,\,\dots\,,\,\beta_{N-1}]\,,\quad
\alpha_j,\beta_j\in\{0,1\}\,$$ differ at exactly $2k$ positions. Thus, in order to obtain the binary representation of $b$ from that of $a$ we need to substitute exactly $k$ ones in $[\alpha_0,\,\alpha_1,\,\dots\,,\,\alpha_{N-1}]$ by zeroes and exactly $k$ zeroes in $[\alpha_0,\,\alpha_1,\,\dots\,,\,\alpha_{N-1}]$ by ones since the total number of ones in both representations of $a$, $b$ is equal to $p$, $H_a=H_b=p$. There are ${\binom{p}{k}}{\binom{N-p}{k}}$ variants of such substitutions.
Lemma \[l:3:1\] applied to yields $$\label{eq:28}
\frac{\overline{w}}{s}=
\frac{1}{2^N}\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}}\right)^c
\sum\limits_{k=0}^p {\binom{p}{k}}{\binom{N-p}{k}}\left(1-(2q-1)^{c+1}\right)^{2k}\left(1+(2q-1)^{c+1}\right)^{N-2k}.$$ We note that in we can disregard the restriction $2p\le N$. The polynomial $$\label{eq:29}F_{A_p}(z)=\frac{1}{2^N}\sum\limits_{k=0}^p
{\binom{p}{k}}{\binom{N-p}{k}}(1-z)^{2k}(1+z)^{N-2k}=\sum\limits_{d=0}^N h_dz^d$$ of degree $N$ satisfies the conditions . Moreover, $h_d=h_{N-d}$.
Therefore, we conclude that for the permutation invariant fitness landscapes we obtained the explicit equation for the leading eigenvalue $\overline{w}$ with $h_d$ defined in . While it is a common wisdom that the dimensionality of the quasispecies problem for the permutation invariant fitness landscapes can be reduced to $N+1$ from $2^N$, the explicit equation to determine the leading eigenvalue $\overline{w}$ is, to the best of our knowledge, new.
For the permutation invariant fitness landscapes arguably the most transparent and efficient way to analyze the problem is to invoke the so-called maximum principle [@Baake2007; @saakian2006ese; @wolff2009robustness], therefore, first several examples in this section should be mostly considered as an illustration of the suggested technique. Nevertheless, the results we present are exact, contrary to the approximate nature of the maximum principle, for which also some technical conditions on the fitness landscape must be imposed (without these conditions the maximum principle can lead to incorrect conclusions, e.g., [@semenov2015; @wolff2009robustness]). Our equations work for any fitness landscape and therefore are of interest on their own.
In what follows we consider several special cases of the previous example.
\[ex:3:1:1\] Let $p=0$ in the previous example. Then we deal with the classical single peaked fitness landscape. The equation for $\overline{w}$ was studied in great details in [@semenov2014]. We would like to mention that in view of Corollary \[corr:2:3\], since the group of isometries $\Gamma$ acts transitively on the set $X$ of indices then each equation for the single peaked landscape $A=\{a\}$ is the same. Consequently, for the leading eigenvalue we can consider the basic case $A_0=\{0\}$, that is, the single peak at $w_0=w+s$.
We can also consider the trivial group $G=\{1\}$ acting on $X$ in order to treat the same case.
The polynomial becomes $$F_{A_0}(z)=\frac{1}{2^N}(1+z)^N= \frac{1}{2^N}\sum\limits_{d=0}^N
{\binom{N}{d}}z^d.$$ Hence, the equation reads $$\label{eq:30}
\frac{1}{2^N} \sum_{d=0}^{N} {\binom{N}{d}}\frac{(2q-1)^{d}}{\overline{w}-w(2q-1)^{d}}=\frac{1}{s}\,.$$ A very similar expression for a slightly different model was obtained originally in [@galluccio1997exact].
\[ex:3:1:2\]In the previous notation let $A=A_1=\{1,2,4,8,\dots,2^{N-1}\}$. Now, for $a,b\in A$ $$H_{ab}=\left\{
\begin{array}{r}
0,\quad a=b\;,\\
2,\quad a\ne b\;.\\
\end{array}
\right.$$ The calculation of the polynomial yields $$F_{A_1}(z)=\frac{1}{2^N}\left((1+z)^N+(N-1)(1-z)^2(1+z)^{N-2}\right)=
\frac{1}{2^N}\sum\limits_{d=0}^N \frac{(N-2d)^2}{N}{\binom{N}{d}}z^d.$$ Hence transforms into $$\label{eq:31}
\frac{1}{2^N}\sum_{d=0}^{N} \frac{(N-2d)^2}{N}{\binom{N}{d}}\frac{(2q-1)^{d}}{\overline{w}-w(2q-1)^{d}}=\frac{1}{s}\,.$$
\[ex:3:1:3\] Let $N=2n$ be an even number and let $A=A_n$. Applying Lemma \[l:3:1\] to we find $$\label{eq:32}
\frac{\overline{w}}{s}=
\frac{1}{2^N}\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}}\right)^c
\sum\limits_{k=0}^n {\binom{n}{k}}^2\left(1-(2q-1)^{c+1}\right)^{2k}\left(1+(2q-1)^{c+1}\right)^{2n-2k}.$$
\[ex:3:2\] Consider the set $A=\{a,a^*\}$, where, as before, $a^*$ is the conjugate index, $a^*=l-1-a$. Let $G=\{1,g\}$ be the group of order 2 whose nontrivial element (involution) $g$ maps each $a\in X$ to the conjugate $a^*$. Thus, the set $A=\{a,a^*\}$ is a $G$-orbit. In view of Proposition \[pr:1\], since $H_{aa}=0$ and $H_{aa^*}=N$ then the equation reads $$\label{eq:34}
\frac{\overline{w}}{s}=
\frac{1}{2^N}\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}}\right)^c
\left(
\left(1-(2q-1)^{c+1}\right)^{N}+\left(1+(2q-1)^{c+1}\right)^{N}\right)\,.$$ In this case the polynomial takes the form ($\lfloor\,\cdot\,\rfloor$ stands for the integer part): $$F_{A}(z)=\frac{1}{2^N}((1+z)^N+(1-z)^N)= \frac{1}{2^{N-1}}
\sum_{d=0}^{ \lfloor N/2\rfloor} {\binom{N}{2d}}z^d\;.$$ Hence the equation is of degree $\lfloor N/2\rfloor+1$: $$\label{eq:35}
\frac{1}{2^{N-1}}\sum_{d=0}^{ \lfloor N/2 \rfloor} {\binom{N}{2d}}\frac{(2q-1)^{2d}}{\overline{w}-w(2q-1)^{2d}}=\frac{1}{s}\,.$$
\[ex:3:3\] According to a well known theorem of Cayley each (finite) group $G$ is a permutation group (which acts on itself by, for instance, left shifts). It follows that each finite group $G$ can be embedded into symmetric group $S_{n}$, $n=|G|$. Since $S_n$ acts on the set of indices $X=X_n$ then we can find many $G$-orbits restricting on $G$ the canonical action of $S_n$ on $X_n$ (see the beginning of this section). Moreover, since there are standard embeddings $S_n\to S_{n+1}\to S_{n+2}\to\dots$ there is no problem to construct the action of any finite group $G$ on the set $X_N$ for $N\ge n$. This gives us a virtually unlimited list of the two-valued fitness landscapes, which are not permutation invariant.
For instance, let $$G=Q_8=\{\pm 1,\,\pm i,\,\pm j,\,\pm
k\,|\,i^2=j^2=k^2=-1\,,\;ij=k,\,jk=i,ki=j\}$$ ($-1$ commutes with each element of $Q_8$) be the classical quaternion group of order 8. The embedding $Q_8\to S_8$ is chosen so that $i\to
(0213)(4657)$, $j\to(0415)(2736)$.
Consider a $G$-orbit, say, $$A=\{7,11,13,14,112,176,208,224\}\subset X_N\,,\quad N\geq 8.$$ Direct calculations yield the polynomial $$\label{eq:36}
F_{A,N}(z)=\frac{1}{2^N}((1+z)^N+3(1-z)^2(1+z)^{N-2}+4(1-z)^6(1+z)^{N-6})\,.$$ For $N=8$ we have $$\label{eq:36a}
F_{A,8}(z)=\frac{1}{2^8}((1+z)^8+3(1-z)^2(1+z)^6+4(1-z)^6(1+z)^2)=$$$$=
\frac{1}{64}(2+z+14z^2+15z^3+15z^5+14z^6+z^7+2z^8)\,.$$ Finally, we obtain the following form of $$\label{eq:37}
\sum_{d=0}^8
\frac{R_d(2q-1)^d}{\overline{w}-w(2q-1)^d}=\frac{64}{s}\;,$$ where $R_0=R_8=2$, $R_1=R_7=1$, $R_2=R_6=14$, $R_3=R_5=15$, $R_4=0$.
Examples of calculating $\overline{w}$ are given in Fig. \[fig:1\], where the case $N=8$ was also checked numerically using the full matrix ${{\boldsymbol{QW}}}$.
![The leading eigenvalue $\overline{w}$ depending on the fidelity $q$ for the two-valued fitness landscape with $w=2,\,s=5$ and the set $A$ as in Example \[ex:3:3\]. $(a)$ $N=8$ (this case was also checked numerically, using the full matrix ${{\boldsymbol{QW}}}$), $(b)$ $N=50$.[]{data-label="fig:1"}](fig1.eps){width="95.00000%"}
\[ex:3:8\]If $w=0$ the calculations can be significantly simplified (see ). Moreover, we can find not only the polynomial expression for the leading eigenvalue provided $A$ is a $G$-orbit, $G\leqslant{\rm Iso}(X)$, but we can find the eigenvector ${{\boldsymbol{p}}}$ (the quasispecies distribution) as well.
On substituting $w=0$ into we obtain $$\label{eq:38}
{{{\boldsymbol{W}}}}=s {{{\boldsymbol{E}}}}_A=s \sum_{a\in A}{{{\boldsymbol{E}}}}_a.$$ Here ${{{\boldsymbol{E}}}}_A$ is the diagonal matrix corresponding to the projection on the orbit $A$, ${{{\boldsymbol{E}}}}_A^2={{{\boldsymbol{E}}}}_A$. The problem can be transformed now as follows: $$\label{eq:39}
s{{{\boldsymbol{Q}}}} {{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}}=\overline{w} \left({{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}}+({{\boldsymbol{p}}}-{{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}})\right)\;\quad(=\overline{w}\,{{\boldsymbol{p}}}).$$ Multiplying from the left by ${{{\boldsymbol{E}}}}_A$ and taking into account that ${{{\boldsymbol{E}}}}_A$ is a projection matrix we find $$\label{eq:40}
{{{\boldsymbol{E}}}}_A{{{\boldsymbol{Q}}}} {{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}}=\frac{\overline{w}}{s}\, {{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}}.$$ Note that if we omit zeroes in ${{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}}$, we obtain the reduced vector ${{\boldsymbol{p}}}_A$ introduced in Section \[sec:1\].
Direct calculations and Lemma \[l:2\] show that if we take $$\label{eq:41}
{{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}}=\theta(0,\ldots,0,1,0,\dots,0,1,0,\ldots),\quad
\theta>0,$$ where the ones stand only for the indices $a\in A$, we get $$\label{eq:42}
{{{\boldsymbol{E}}}}_A{{{\boldsymbol{Q}}}} {{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}}=F_A(2q-1)\, {{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}}.$$ Let us compare and . In view of we conclude that the vector ${{\boldsymbol{p}}}$ satisfying is a solution of the problem provided $\overline{w}=sF_A(2q-1)$ (possibly not unique).
The equality implies, regardless of $\theta$, that $$\label{eq:43}
{{\boldsymbol{p}}}=\frac{s}{\overline{w}}{{{\boldsymbol{Q}}}} {{{\boldsymbol{E}}}}_a{{\boldsymbol{p}}}=\frac{1}{F_A(2q-1)}\,{{{\boldsymbol{Q}}}} {{{\boldsymbol{E}}}}_a{{\boldsymbol{p}}},$$ where ${{{\boldsymbol{E}}}}_a{{\boldsymbol{p}}}$ has the form . The normalizing factor $\theta$ should be chosen in such a way that holds. Thus, in coordinates we have $$p_k=\frac{1}{|A|\cdot F_A(2q-1)}\sum\limits_{b\in
A}(1-q)^{H_{kb}}q^{N-H_{kb}},\quad k=0,\ldots,2^N-1,$$ or $$\label{eq:44}
p_k=\frac{1}{|A|}\frac{\sum\limits_{b\in
A}(1-q)^{H_{kb}}q^{N-H_{kb}}}{\sum\limits_{b\in
A}(1-q)^{H_{ab}}q^{N-H_{ab}}}\,,\quad k=0,\ldots,2^N-1,\quad a\in
A.$$ The expressions imply that the distribution ${{\boldsymbol{p}}}$ is constant for any fixed $q$ on the $G$-orbits in the set of indices $A$.
Using the discussed approach, for Example \[ex:3:1\] and $w=0$ we obtain from that $$\label{eq:45}
\overline{w}=s \sum\limits_{k=0}^p {\binom{p}{k}}\binom{N-p}{k}(1-q)^{2k}q^{N-2k}.$$ In Example \[ex:3:1:3\] ($N=2n$) we find $$\label{eq:48}
\overline{w}= s \sum\limits_{k=0}^n {\binom{n}{k}}^2(1-q)^{2k}q^{2n-2k}\approx \frac{s}{\sqrt{\pi n}}
\frac{1}{\sqrt{1-(2q-1)^{2}}}\,,\quad\mbox{when}\;\;n\gg
1.$$ In Example \[ex:3:2\] we have $$\label{eq:49}
\overline{w}=\frac{s}{2^{N-1}}\sum_{d=0}^{ \lfloor N/2\rfloor} {\binom{N}{2d}}(2q-1)^{2d}=s(q^N+(1-q)^N).$$ Other examples can be treated similarly.
The infinite sequence limit $N\to\infty$ {#sec:5}
========================================
In Corollary \[corr:2:1\] we obtained the algebraic equation of degree at most $N+1$ for the leading eigenvalue $\overline{w}=\overline{w}(q)$. The advantage of having a polynomial equation of degree $N+1$ notwithstanding, solving becomes complicated as $N\to\infty$. Moreover, it is well known that at least for some fitness landscapes (including the classical single peaked fitness landscape) the phenomenon of the *error threshold* is observed: there exists a critical mutation rate $q$, after which the quasispecies distribution ${{\boldsymbol{p}}}$ becomes uniform. This phenomenon is usually identified with a non-analytical behavior of the limiting eigenvalue $\overline{w}$ when $N\to\infty$, a general idea can be grasped from Fig. \[fig:1\]b, where it is seen that there exists a corner point on the graph of the function $\overline{w}$.
In this section we propose several steps to rigorously define and analyze this kind of behavior in terms of sequences of orbits $A_n$ that determine our two-valued fitness landscapes. First, we find some bounds for the function $\overline{w}$ provided $0.5\leq q\leq 1$. Next, we restrict our attention at the special class of sequences $(A_n)_{n=n_0}^\infty$, which we call *admissible* and of the *moderate growth* (here $n_0$ is a sufficiently large natural number). Finally, among all those admissible sequences of the moderate growth we identify the ones that demonstrate some kind of non-uniform convergence for the corresponding sequence of eigenvalues $(\overline{w}^{(n)})_{n=n_0}^\infty$.
Lower and upper bounds on $\overline{w}(q)$ {#sec:5:1}
-------------------------------------------
First we note that for our purposes it is easier to deal with the series rather than . We also make the following substitutions $$\label{eq:51}
w=us\;,\quad \overline{w}=\overline{u}s.$$ Then turns into $$\label{eq:52}
\overline{u}=
\sum_{c=0}^{\infty}\left(\frac{u}{\overline{u}}\right)^c
F_A((2q-1)^{c+1}),$$ where the polynomial $F_A(z)$, defined in , can be represented in the form .
From Example \[ex:3:8\] we have that $sF_A(2q-1)=\overline{w}(q)$ is the leading eigenvalue if $w=0$. It was proved in [@semenov2014] that $\overline{w}(q)$ increases on the segment $0.5\leq q\leq 1$. Therefore, on this segment we have the non-increasing sequence (for any fixed $q$) $$\label{eq:53}
F_A(2q-1)\geq F_A((2q-1)^2)\geq\dots\geq F_A((2q-1)^{c})\geq
F_A((2q-1)^{c+1})\geq\dots >0,$$ since $F_A((2q-1)^{c})>0$ according to . Hence, $$\label{eq:54}
\overline{u}=
\sum_{c=0}^{\infty}\left(\frac{u}{\overline{u}}\right)^c
F_A((2q-1)^{c+1})\leq
F_A(2q-1)\sum_{c=0}^{\infty}\left(\frac{u}{\overline{u}}\right)^c=
\frac{F_A((2q-1))\,\overline{u}}{\overline{u}-u}\,.$$ It follows that $ \overline{u}\leq u+ F_A(2q-1)$, or $$\overline{w}(q)\leq w+sF_A(2q-1)=:\overline{w}_{up,1}(q).$$
A second upper bound can be obtained as follows: $$\begin{aligned}
\overline{u}&=
\sum_{c=0}^{\infty}\left(\frac{u}{\overline{u}}\right)^c
F_A((2q-1)^{c+1})=F_A(2q-1)+
\sum_{c=1}^{\infty}\left(\frac{u}{\overline{u}}\right)^c
F_A((2q-1)^{c+1})\\
&\leq F_A(2q-1)+F_A((2q-1)^2)\sum_{c=1}^{\infty}\left(\frac{u}{\overline{u}}\right)^c=
F_A(2q-1)+\frac{uF_A((2q-1)^2)}{\overline{u}-u}\,.\end{aligned}$$ Solving the quadratic inequality we get $$\overline{u}\leq\frac{u+F_A(2q-1)+\sqrt{(u+F_A(2q-1))^2-4u(F_A(2q-1)-F_A((2q-1)^2))}}{2}\,,$$ or, $$\label{eq:55}
\overline{w}(q)\leq\frac{\overline{w}_{up,1}(q)+
\sqrt{\overline{w}^2_{up,1}(q)-4w(sF_A(2q-1)-sF_A((2q-1)^2))}}{2}=:\overline{w}_{up,2}(q).$$
In view of $\overline{w}_{up,2}(q)\leq\overline{w}_{up,1}(q)$.
To obtain a lower bound on $\overline{w}(q)$ we use the approach applied in [@semenov2014]. Since $\overline{w}(q)$ increases on the segment $0.5\leq q\leq 1$ therefore $$\label{eq:56}
\overline{w}(q)\geq\overline{w}(0.5)=w+\frac{s|A|}{2^N}\,.$$
By the definition of $$\begin{aligned}
F_A((2q-1)^{c+1})&=\sum_{b\in A}\left(\frac{1-(2q-1)^{c+1}}{2}\right)^{H_{ab}}
\left(\frac{1+(2q-1)^{c+1}}{2}\right)^{N-H_{ab}}\\
&\geq \left(\frac{1+(2q-1)^{c+1}}{2}\right)^{N}\geq
\left(\frac{1+(2q-1)}{2}\right)^{(c+1)N}=q^{(c+1)N}\,,\end{aligned}$$ since $a\in A$, $H_{aa}=0$ and the function $f(t)=t^{c+1}$ is convex (downward) on the segment $[0,1]$.
Now from $$\label{eq:57}
\overline{u}\geq
\sum_{c=0}^{\infty}\left(\frac{u}{\overline{u}}\right)^c
q^{(c+1)N}=\frac{\overline{u}\,q^N}{\overline{u}-uq^N}\,,\quad\mbox{or}\;\;
\overline{u}\geq(u+1)q^N\,,\quad\mbox{or}\;\; \overline{w}\geq
(w+s)q^N.$$ Combining and yields $$\label{eq:58}
\overline{w}(q)\geq \max\left(w+\frac{s|A|}{2^N},
(w+s)\,q^N\right)=:\overline{w}_{low}(q)\,.$$ Thus we have proved
For the leading eigenvalue $\overline{w}(q)$ of in the case of the two-valued fitness landscape we have $$\overline{w}_{low}(q)\leq w(q)\leq \overline{w}_{up,2}(q),\quad 0.5\leq q\leq 1,$$ where $\overline{w}_{low}(q)$ is given by , and $\overline{w}_{up,2}(q)$ is given by .
A numerical example with the obtained bounds is given in Figure \[fig:2\].
![The lower and upper bounds on the leading eigenvalue $\overline{w}$ in the case of the quaternion landscape (Example \[ex:3:3\]), $(a)$ $N=8$, $(b)$ $N=50$.[]{data-label="fig:2"}](fig2.eps){width="95.00000%"}
Admissible sequences of orbits
------------------------------
To make a progress in analyzing the limit behavior of our system when $N\to\infty$ we introduce in this subsection two definitions in terms of which this behavior will be described.
From the previous subsection, we see that the curve $\overline{w}=\overline{w}_{low}(q)$ has a corner point on $[0.5,1]$, which we denote $q_\ast$: $$\label{eq:59}
q_*=q_*^{(N)}=\sqrt[N]{\frac{w+s|A|2^{-N}}{w+s}}=\sqrt[N]{\frac{u+|A|2^{-N}}{u+1}}=\sqrt[N]{\frac{\overline{w}(0.5)}{\overline{w}(1)}}\,.$$ The function $\overline{w}_{low}(q)$ is constant for $0.5\leq
q\leq q_*$ and increases for $q_*<q\leq 1$ (see Figure \[fig:2\]). It was shown in [@semenov2014] that for the single peak landscapes ($|A|=1$) the lower bound $\overline{w}_{low}(q)$ provides a close approximation for $\overline{w}(q)$ for sufficiently large $N$. Our goal is to generalize these results on the case of the two-valued fitness landscapes.
From this point on we shall use $n$ as the index, which tends to infinity. In most cases it actually coincides with the sequence length $N$, albeit not always, hence the choice of notation.
One of the main underlying questions concerning the quasispecies model and especially its infinite sequence limit, is how actually the fitness landscape is scaled when $N\to\infty$. In most works in literature a continuous limit is used, which basically narrows the pull of the allowed fitness landscapes to the ones which have, given this continuous limit, a limit fitness function, which must be also continuous (e.g., [@Baake2007; @saakian2006ese]). Here we take a different approach by specifying sequences of orbits $(A_n)_{n=n_0}^\infty$, on which the fitness landscape is defined. The sequences that are of interest to us will be called *admissible*.
Suppose that for any $n\geq n_0$ a sequence of $G_n$-orbits $A_n\in X_n$ is given, where $G_n\leqslant{\rm Iso}\, (X_n)$. When $n\to \infty$ the group ${\rm Iso}\, (X_n)$ will be always viewed as a subgroup of ${\rm Iso}\, (X_{n+1})$. More precisely, let $g\in {\rm Iso}\, (X_n)$ be a fixed isometry and let $a\in
X_{n+1}$ be represented as $a=a_{n}+\alpha_{n} 2^{n}$ where $a_{n},\alpha_{n}\in X_n$. Then $g$, viewed as an element of ${\rm
Iso}\, (X_{n+1})$, maps $a\in X_{n+1}$ to $g(a):=g(a_{n})+g(\alpha_{n}) 2^{n}$. In other words, ${\rm Iso}\,
(X_n)$ as a subgroup of ${\rm Iso}\, (X_{n+1})$ is acting on the “upper” hyperface $V_n\times\{1\}$ of the cube $V_{n+1}=\{0,1\}^{n+1}=V_n\times\{0,1\}$ in the same way as it acts on the “lower” hyperface $V_n\times\{0\}\cong V_n$. Thus, we have the ascending chain $${\rm Iso}\, (X_{n_0})<\ldots< {\rm Iso}\, (X_{n})<{\rm Iso}\,
(X_{n+1})<\ldots$$ and the corresponding ascending chain of symmetric subgroups $$S_{n_0}<\ldots< S_{n}<S_{n+1}<\ldots\,.$$
For a fixed $w\geq 0$ consider a sequence of landscapes $({{\boldsymbol{w}}}^{(n)})_{n\geq n_0}$ such that $w_k^{(n)}=w+s$ if $k\in A_n$ and $w_k^{(n)}=w$ otherwise. The sequence $(A_n)_{n=n_0}^\infty$ and the parameters $w$, $s$, and $u=w/s$ define the corresponding family of leading eigenvalues $\overline{w}^{(n)}=\overline{w}^{(n)}(q)$, which are solutions of , and the family $\overline{u}^{(n)}=\overline{u}^{(n)}(q)$, such that $\overline{u}^{(n)}=\overline{w}^{(n)}/s$.
In [@semenov2014] it was proved that for any $n\geq n_0$ the function $\overline{u}^{(n)}(q)$ has the following properties:
1. The function $\overline{u}^{(n)}(q)$ increases on the segment $[0.5,1]$ and is convex (downward) there.
2. $\overline{u}^{(n)}(0.5)=u+\displaystyle{\frac{|A_n|}{2^n}},\quad \overline{u}^{(n)}(1)=u+1$.
\[d:5:1\] A sequence $(A_n)_{n=n_0}^\infty$ of $G_n$-orbits is called admissible if the corresponding sequence of values of polynomials $F_{A_n}(2q-1)$ in is non-increasing for each $q\in [0.5,1]$: $$\label{eq:61}
F_{A_n}(2q-1)=\sum_{d=0}^n f^{(n)}_{d}\,(1-q)^dq^{n-d}\geq
\sum_{d=0}^{n+1}
f^{(n+1)}_{d}\,(1-q)^dq^{n+1-d}=F_{A_{n+1}}(2q-1)\,,\;n\geq
n_0.$$
A sequence $(A_n)_{n=n_0}^\infty$ of $G_n$-orbits is called a sequence of the moderate growth if $$\label{eq:62}
\lim_{n\to\infty}\frac{|A_n|}{2^n}=0\,,\quad\mbox{or}\quad |A_n|=o(2^n),\quad n\to\infty\,.$$
To show that our definitions make sense we state
\[pr:ad\]In all the examples of Section \[sec:3\] the corresponding sequences of orbits are admissible and of the moderate growth.
See Appendix \[ap:1\].
Consider a sequence $(A_n)_{n=n_0}^\infty$ of $G_n$-orbits. Our next aim is to investigate what happens with the corresponding family $(\overline{u}^{(n)})_{n=n_0}^\infty$ as $n\to\infty$.
\[pr:5:1\] If $(A_n)_{n=n_0}^\infty$ is an admissible sequence of $G_n$-orbits then for each fixed $q\in[0.5,1]$ the sequence $(\overline{u}^{(n)}(q))_{n=n_0}^\infty$ is a non-increasing sequence as $n\to\infty$. If, additionally, $(A_n)_{n=n_0}^\infty$ is a sequence of the moderate growth then $\lim\limits_{n\to\infty}\overline{u}^{(n)}(0.5)=u$ and $\lim\limits_{n\to\infty}\overline{u}^{(n)}(1)=u+1$.
The second assertion follows directly from Property 2 of $\overline{u}^{(n)}(q)$ above. Let us proof the first one. The equation for $u\ne 0$ can be rewritten in the form $$\label{eq:69}
u=\frac{u}{\overline{u}^{(n)}(q)}\overline{u}^{(n)}(q)=
\sum_{c=0}^{\infty}\left(\frac{u}{\overline{u}^{(n)}(q)}\right)^{c+1}
F_{A_n}((2q-1)^{c+1})=\sum_{m=1}^{\infty}\left(\frac{u}{\overline{u}^{(n)}(q)}\right)^{m}
F_{A_n}((2q-1)^m)\,.$$
It follows from Definition \[d:5:1\] that at each fixed point $q\in[0.5,1]$ the sequence of positive coefficients $(F_{A_n}((2q-1)^{c+1}))_{n\geq n_0}$ is non-increasing for any $c+1\in
{\mathbf{N}}$. But the left-hand side $u$ of is constant. This implies that $(\overline{u}^{(n)}(q))_{n\geq n_0}$ must be a non-increasing sequence for each $q\in[0.5,1]$.
Hence we can conclude that the curve $\overline{u}=\overline{u}^{(n+1)}(q)$ always passes [*under*]{} the curve $\overline{u}=\overline{u}^{(n)}(q)$ in the rectangle $\{0.5\leq q\leq 1\,,\;u\leq\overline{u}\leq u+1\}$ if $(A_n)_{n=n_0}^\infty$ is an admissible sequence of $G_n$-orbits, see Figure \[fig:3\].
Proposition \[pr:5:1\] and Property 1 of $\overline{u}^{(n)}(q)$ yield
If $(A_n)_{n=n_0}^\infty$ is an admissible sequence of $G_n$-orbits of the moderate growth then for any fixed $\varepsilon\in(0,1]$ there exists $N_0\in {\mathbf{N}}$ such that for any $n\geq N_0$ the curve $\overline{u}=\overline{u}^{(n)}(q)$ intersects the line $\overline{u}=u+\varepsilon$ at a unique point $q^{(n)}(\varepsilon,u)\in(0.5,1]$.
Note that by virtue of , , and the value $q^{(n)}(\varepsilon,u)$ from the previous corollary can be found from one of the following equations $$\label{eq:70}
u+\varepsilon=
\sum_{c=0}^{\infty}\left(\frac{u}{u+\varepsilon}\right)^{c}
F_{A_n}((2q-1)^{c+1})\,,$$ or, $$\label{eq:728}
\sum_{d=0}^N \frac{h_d(2q-1)^d}{u+\varepsilon-u(2q-1)^d}=1.$$
Another almost immediate result is given in the following
If $(A_n)_{n\geq N_0}$ is an admissible sequence of $G_n$-orbits of the moderate growth then for fixed $(\varepsilon,u)$ the sequence $(q^{(n)}(\varepsilon,u))_{n\geq N_0}$ is non-decreasing as $n\to\infty$ and the inequality $$\label{eq:72}
q^{(n)}(\varepsilon,0)\leq q^{(n)}(\varepsilon,u)\leq
\sqrt[n]{\frac{u+\varepsilon}{u+1}}\leq
1-\frac{1-\varepsilon}{n(u+1)}\,$$ holds.
The upper bound (see Section \[sec:5:1\]) $\overline{u}=u^{(n)}_{up,1}(q)= u+ F_{A_n}(2q-1)$ gives rise to the lower bound in since the equation $u+\varepsilon=u+
F_{A_n}(2q-1)$ is equivalent to when $u=0$. The lower bound $\overline{u}=(u+1)q^n$ provides the upper bound in .
Since $\overline{u}=(u+1)q^n$ is convex downward if $q\in[0.5,1]$ and $\overline{u}=u+1-n(u+1)(1-q)$ is the equation of the tangent at $q=1$ to the curve $\overline{u}=(u+1)q^n$ then we get the last inequality in . Note that the curve $\overline{u}=\overline{u}^{(n)}(q)$ has the same tangent at $q=1$ (see, for instance, [@semenov2014]).
The obtained results are illustrated in Figure \[fig:3\].
![The curves in the coordinates $q,\overline{u}$ defined by (from top to bottom): $\overline{u}=u+F_{A_n}(2q-1),\,\overline{u}=\overline{u}^n(q),\,\overline{u}=\overline{u}^{n+1}(q),\,\overline{u}=(u+1)q^{n+1}$. The points of intersections of these curves with the dotted line $\overline{u}=u+\varepsilon$ define the values $q^{(n)}(\varepsilon,0),\,q^{(n)}(\varepsilon,u),\,q^{(n+1)}(\varepsilon,u),\,1-\frac{1-\varepsilon}{(n+1)(u+1)}$ respectively, see also .[]{data-label="fig:3"}](fig3.eps){width="50.00000%"}
Threshold-like behavior
-----------------------
In this subsection we define rigorously what we call the threshold-like behavior and provide sufficient conditions for the sequences of admissible orbits to possess this kind of behavior. The main conclusion, which can be stated in a form of a conjecture, emphasizes the role of geometry for the threshold-like behavior to occur. Loosely speaking, if the admissible sequence of orbits “looks like a point” asymptotically, i.e., basically indistinguishable from the single peaked landscape in the infinite length limit, then the threshold-like behavior is observed. We conjecture, as numerical experiments show, that the opposite is true: If asymptotically the admissible sequence of orbits is different from a point, then there exits no threshold-like behavior.
Let us introduce the notation $$\label{eq:73}
q^{(n)}_*(\varepsilon,u)=\sqrt[n]{\frac{u+\varepsilon}{u+1}}\,,$$ from where $$\label{eq:74}
\lim_{n\to\infty}n(1-q^{(n)}_*(\varepsilon,u))=\log\frac{u+1}{u+\varepsilon}\,.$$ It follows that for a fixed $u>0$ $$\label{eq:75}
\lim_{\varepsilon\downarrow 0}\lim_{n\to\infty}n(1-q^{(n)}_*(\varepsilon,u))=\log\frac{u+1}{u}\,.$$
It is known (e.g., [@semenov2014]) that for the single peaked landscape the curve $\overline{u}=\overline{u}^{(n)}(q)$ passes very close to the lower bound $\overline{u}=\max\{u,(u+1)q^n\}$ in such a way that $$q^{(n)}(\varepsilon,u)=q^{(n)}_*(\varepsilon,u)-o\left(\frac{1}{n}\right)=
\sqrt[n]{\frac{u+\varepsilon}{u+1}}-o\left(\frac{1}{n}\right)$$ as $n\to\infty$ (from we have the inequality $q^{(n)}(\varepsilon,u)\leq q^{(n)}_*(\varepsilon,u)$).
Our next aim is to investigate what happens with the curve $\overline{u}=\overline{u}^{(n)}(q)$ as $n\to\infty$. It is more conveniently done in coordinates $x$, $L$, defined by $$\label{eq:76}q=1-\frac{x}{n}\,,\quad 0\leq x\leq \frac{n}{2}\,,\quad \overline{u}=(u+1)e^{-L}\,,\quad 0\leq L\leq
\log\frac{u+1}{u}\,.$$ We will assume that $u>0$ in . Hence the curve $\overline{u}=\overline{u}^{(n)}(q)$ transforms into the curve $$\label{eq:77}
L_n(x)=\log(u+1)-\log\overline{u}^{(n)}\left(
1-\frac{x}{n}\right).$$ Note that $L_n(0)=0$ since ${u}^{(n)}(1)=u+1$ for any $n$.
\[def:5:3\] We say that an admissible sequence $(A_n)_{n\geq n_0}$ of the moderate growth, or, equivalently, the family $(\overline{u}^{(n)})_{n\geq n_0}$ possesses the threshold-like behavior on the segment $[0.5,1]$ if for each fixed $x\geq 0$ and the corresponding functions $L_n(x)$ it is true that $$\label{eq:78}
\lim_{n\to\infty} L_n(x)=L(x)=\begin{cases}x,&0\leq x<\log \frac{u+1}{u},\\
\log\frac{u+1}{u}\,,&x\geq \log\frac{u+1}{u}\,.
\end{cases}$$
The definition above is illustrated in Fig. \[fig:4\].
![The limit function $L(x)$ in Definition \[def:5:3\] of the threshold-like behavior[]{data-label="fig:4"}](fig4.eps){width="55.00000%"}
If the threshold-like behavior is present in the two-valued fitness landscape, then the following formula provides an approximation for the threshold mutation rate $q_*^{(n)}(u)$, $n\gg 1$: $$\label{eq:79}
q^{(n)}_*(u)\approx
1-\frac{1}{n}\log\frac{u+1}{u}=1-\frac{1}{n}\log\frac{w+s}{w}\,,$$ which, of course, coincides with the classical estimate for the error threshold for the single peaked landscape [@eigen1988mqs; @semenov2014].
If the sequence of continuous functions $(\overline{u}^{(n)})_{n\geq n_0}$ has the threshold-like behavior then it converges not uniformly on $[0.5,1]$, as $n\to\infty,$ to the discontinuous function $\psi(q)$ such that $\psi(q)=u$ if $0.5\leq q<1$ and $\psi(1)=u+1$.
The following theorems and corollaries provide sufficient conditions under which an admissible sequence of orbits of the moderate growth shows the threshold-like behavior.
\[th:5:1\]In the above notation suppose that for $n\geq n_0$ an admissible sequence of $G_n$-orbits $A_n\subset
X_n$ *(*${G}_n\leqslant{\rm Iso}\, (X_n)$*)* of the moderate growth is given and $u>0$. Suppose also that for $n\geq
n_0$ the inequality $$\label{eq:80}
F_{A_n}(2q-1)\leq (2q-1)^{n/2}+M_n\,,\quad
0.5\leq q\leq 1 \,,$$ is satisfied for some constants $M_n$ such that $\lim\limits_{n\to \infty}M_n=0$. Then the sequence $(A_n)_{n\geq n_0}$ shows the threshold-like behavior on the segment $[0.5,1]$.
In view of equation , in coordinates $x$, $L$: $$(u+1)e^{-L_n(x)}=
\sum_{c=0}^{\infty}\left(\frac{u}{(u+1)e^{-L_n(x)}}\right)^c
F_{A_n}\left(\left(1-\frac{2x}{n}\right)^{c+1}\right),$$ therefore (putting $m=c+1$) $$\label{eq:81}
u= \sum_{m=1}^{\infty}\left(\frac{u}{u+1}\right)^m
e^{mL_n(x)}F_{A_n}\left(\left(1-\frac{2x}{n}\right)^{m}\right).$$
The lower bound , $\overline{u}\geq(u+1)q^n$, implies for $x\in\left[0,\log\frac{u+1}{u}\right)$, $n\gg 1$, $$\label{eq:82}
L_n(x)=\log\frac{u+1}{\overline{u}^{(n)}(1-\frac{x}{n})}\leq
-n\log\left(1-\frac{x}{n}\right).$$ Consequently, we have on $\left[0,\log\frac{u+1}{u}\right)$ $$\label{eq:83}
\limsup_{n\to\infty}
L_n(x)\leq -\lim_{n\to\infty}n\log\left(1-\frac{x}{n}\right)=
x.$$
On the other hand, the function $F_{A_n}(2q-1)$, as the leading eigenvalue for $u=0$ (see ), is increasing on the segment $[0.5, 1]$. In view of the inequality $1-t\leq e^{-t}$ we have $$F_{A_n}\left(\left(1-\frac{2x}{n}\right)^{m}\right)\leq
F_{A_n}\left(e^{-2mx/n}\right)\,.$$ Make the substitution $2q-1=e^{-2x/n}$ into , where $0\leq
x<+\infty$. Then the following inequality $$e^{mL_n(x)}F_{A_n}\left(\left(1-\frac{2x}{n}\right)^{m}\right)\leq
e^{mL_n(x)}F_{A_n}\left(e^{-2mx/n}\right)\leq
e^{m(L_n(x)-x)}+M_ne^{mL_n(x)}\,$$ holds. Hence, yields $$u\leq\sum_{m=1}^{\infty}\left(\frac{u}{u+1}\right)^m
e^{m(L_n(x)-x)}+M_n\sum_{m=1}^{\infty}\left(\frac{ue^{L_n(x)}}{u+1}\right)^m
\,.$$ In view of both progressions in the right-hand side converge for $x\in \left[0,\log\frac{u+1}{u}\right)$ and $n\gg 1$.
The simplification provides the inequality $$1\leq e^{L_n(x)-x}+
M_n\frac{e^{L_n(x)}(u+1-ue^{L_n(x)-x})}{(u+1)(u+1-ue^{L_n(x)})}\,.$$ Since $M_n\to 0$ and the inequality holds we get finally $
\liminf\limits_{n\to\infty} e^{L_n(x)-x}\geq 1\,$, or, $$\label{eq:84}
\liminf_{n\to\infty} L_n(x)\geq x\,.$$
It follows from , that $\lim\limits_{n\to\infty}L_n(x)=x$ if $0\leq x< \log\frac{u+1}{u}$. The increasing functions $L_n(x)$ cannot exceed the value $\log\frac{u+1}{u}$. Thus, the threshold-like behavior is observed.
\[corr:5:2\] The sequence of constant single peaked landscapes $A_n\equiv \{a\},\,n\geq n_0$ possesses the threshold-like behavior.
Condition of Theorem \[th:5:1\] reads as follows: the inequality $$\label{eq:85}q^n\leq (2q-1)^{n/2}+M_n,\quad
0.5\leq q\leq 1,$$ holds for some constants $M_n$ such that $\lim\limits_{n\to \infty}M_n=0$.
Let us show that the constants $$M_n=\frac{1}{n}\left(1-\frac{1}{n}\right)^{n-1}\leq\frac{1}{e(n-1)}$$ fit. Consider the auxiliary function $\varphi_n(q)=q^n- (2q-1)^{n/2}$. Its maximum $\mu_n$ on the segment $[0.5,1]$ is reached at some point $q_n<1$. This point is a root of the equation $$\varphi'_n(q)=nq^{n-1}-n (2q-1)^{n/2-1}=0\;, \quad
\mbox{or}\;\;(2q-1)^{n/2}=(2q-1)q^{n-1}\,.$$ Hence, by the definition of $\varphi_n(q)$, we get $$\mu_n=\varphi_n(q_n)=q_n^n-(2q_n-1)q_n^{n-1}=q_n^{n-1}(1-q_n)\,.$$ The function $M(t)=t^{n-1}(1-t)$ achieves its maximum on \[0,1\], which is equal to $M_n=\frac{1}{n}\left(1-\frac{1}{n}\right)^{n-1}$, at $t_n=1-\frac{1}{n}$. Hence, $\mu_n\leq M_n$ and the conditions of Theorem \[th:5:1\] hold.
Theorem \[th:5:1\] together with Corollary \[corr:5:2\] are the key results as the following theorem shows. We are convinced that the reason for the error threshold effect is geometric. More precisely, in view of the polynomial $F_{A_n}(2q-1)$ can be always represented in the form $$\label{eq:86}F_{A_n}(2q-1)=q^n+\sum_{k=1}^n f_k^{(n)}\,(1-q)^{k}q^{n-k},$$ where $f_k^{(n)}=\#\{b\in A_n\,|\,H_{ab}=k\}$. Thus, this polynomial can be viewed as a kind of the spherical growth function of the orbit $A_n$ with respect to an arbitrary fixed point $a\in A_n$.
\[th:5:2\] In the above notation assume that for any $n\geq n_0$ an admissible sequence of $G_n$-orbits $A_n\subset
X_n$ *(*$G_n\leqslant{\rm Iso}\,( X_n)$*)*, $n\geq n_0$ of the moderate growth is given and $u>0$. If either $$\label{eq:87}
\lim_{n\to\infty}\max_{q\in [0.5,1]}\sum_{k=1}^{\lfloor n/2\rfloor}
f_k^{(n)}\,(1-q)^{k}q^{n-k}=0\,,$$ or $$\label{eq:88}
\lim_{n\to\infty}\sum_{k=1}^{\lfloor n/2\rfloor}
f_k^{(n)}\,\left(\frac{k}{n}\right)^{k}\left(1-\frac{k}{n}\right)^{n-k}=0\,,$$ then the sequence $(A_n)_{n\geq n_0}$ possesses the threshold-like behavior.
The polynomial $P_k^{(n)}(q)=(1-q)^kq^{n-k}$ decreases on the segment \[0.5,1\] if $n<2k\leq 2n$ and achieves its maximal value $2^{-n}$ at $q=0.5$. If $1\leq k\leq \lfloor n/2\rfloor$ then the maximal value of $P_k^{(n)}(q)=(1-q)^kq^{n-k}$ on \[0.5,1\] is achieved at the point $q_k^{(n)}=1-\frac{k}{n}$ and is equal to $\left(\frac{k}{n}\right)^{k}\left(1-\frac{k}{n}\right)^{n-k}$.
Denote $F_n=\max\limits_{q\in
[0.5,1]}\sum_{k=1}^{\lfloor n/2\rfloor}f_k^{(n)}(1-q)^{k}q^{n-k}$. Then Corollary \[corr:5:2\], and together yield $$\label{eq:89}
\begin{split}
F_{A_n}(2q-1)&=q^n+\sum_{k=1}^n f_k^{(n)}(1-q)^{k}q^{n-k} \leq q^n+F_n+ \sum_{k=\lfloor n/2\rfloor+1}^n \frac{f_k^{(n)}}{2^n}\\
&\leq (2q-1)^{n/2}+\frac{1}{n}\left(1-\frac{1}{n}\right)^{n-1}+F_n+\frac{|A_n|}{2^n}=(2q-1)^{n/2}+o(1)\,,\quad n\to \infty.
\end{split}$$
On the other hand, if the equality holds we can substitute $\sum_{k=1}^{\lfloor n/2\rfloor}
f_k^{(n)}\,\left(\frac{k}{n}\right)^{k}\left(1-\frac{k}{n}\right)^{n-k}$ for $F_n$ since $F_n\leq \sum_{k=1}^{\lfloor n/2\rfloor}
f_k^{(n)}\left(\frac{k}{n}\right)^{k}\left(1-\frac{k}{n}\right)^{n-k}$. Hence, Theorem \[th:5:1\] implies that the sequence $(A_n)_{n\geq n_0}$ shows the threshold-like behavior.
The following sequences of orbits possess the threshold-like behavior:
- All the constant sequences $A_n\equiv A$ *(*see, for instance, Example \[ex:3:1:3\] of the quaternion landscape*)*;
- All the antipodal sequences $A_n=
\{a,a^*\}\subset X_n$ *(*see Example \[ex:3:2\]*)*;
- All the permutation invariant sequences $A_{p,n}$ where $ A_{p,n}=\{a\in X\,|\,H_a=p\}\;$, $
p=0,1,\dots,n\,$, and $p$ does not depend on $n\geq p$ *(*see Examples \[ex:3:1\] and \[ex:3:1:2\]*)*.
$(i)$ Let $A\subset X_{n_0}$. Since the orbit is fixed then for $n\geq n_0$ all the coefficients $f_k^{(n)}\equiv
f_k^{(n_0)}$ and $f_k^{(n)}\equiv 0$ when $k>n_0$. It follows that the assumption of Theorem \[th:5:2\] that $$\lim_{n\to\infty}\sum_{k=1}^{\lfloor n_0/2\rfloor}
f_k^{(n_0)}\,\left(\frac{k}{n}\right)^{k}\left(1-\frac{k}{n}\right)^{n-k}=0\,$$ holds since $k\leq \lfloor n_0/2\rfloor$ and, consequently, $\left(\frac{k}{n}\right)^{k}\to 0$ as $n\to\infty$ provided $\left(1-\frac{k}{n}\right)^{n-k}\leq 1$.
$(ii)$ In this case $F_{A_n}(q)=q^n+(1-q)^n$. Then $f_k^{(n)}=0$, $k=1,\dots, \lfloor n/2\rfloor$. Then both assumptions , hold.
$(iii)$ We may suppose that $n\ge 2p=n_0$. In view of $$F_{A_{p,n}}(q)=\sum\limits_{k=0}^p
{\binom{p}{k}}{\binom{n-p}{k}}(1-q)^{2k}q^{n-2k}\,.$$ Hence, since $p$ is fixed, ${\binom{p}{k}}<2^p$, $1-\frac{2k}{n}\leq
1$: $$\sum\limits_{k=1}^p {\binom{p}{k}}{\binom{n-p}{k}}\left(\frac{k}{n}\right)^{2k}\left(1-\frac{2k}{n}\right)^{n-2k}
\leq (2p)^{2p}\sum\limits_{k=1}^p \frac{1}{n^{2k}}\,{\binom{n-p}{k}}\leq (2p)^{2p}\sum\limits_{k=1}^p \frac{n^k}{k!\,n^{2k}}=o(1)$$ as $n\to\infty$. Consequently, the condition is satisfied.
\[remark:5\]In contrast, if $A_{2n}=A_{n,2n}$ is the sequence of the fitness landscapes in Example \[ex:3:1:3\] then the numerical calculations (Fig. \[fig:5\]) show that this sequence does not demonstrate the threshold-like behavior. The approximate formula provides a lower bound $$\max_{q\in [0.5,1]}\sum_{k=1}^{n}
{\binom{n}{k}}^2\,(1-q)^{2k}q^{2n-2k}\geq 0.183\approx
\frac{4r-1}{8r\sqrt{\pi r}}\,,\quad
r=-\frac{3}{4}W\left(-\frac{1}{3\sqrt[3]{\pi}}\right)\approx
1.7423,$$ where $W(z)$ is (a branch of) the Lambert $W$ function ($W(z)e^{W(z)}=z$). Hence, the sufficient conditions for the threshold-like behavior are not satisfied in this case.
![The leading eigenvalue $\overline{w}(q)$ versus the mutation rate $q$ in the case of the fitness landscape in Example \[ex:3:1:3\]. The sequence length $2n=50$ in the left panel and $2n=100$ in the right panel. Note the absence of the threshold-like behavior[]{data-label="fig:5"}](fig5.eps){width="95.00000%"}
A natural question to ask is whether the given sufficient conditions are also necessary for the threshold-like behavior. While at this point we do not have a full answer for this question, we can present a sufficient condition for the absence of the threshold like behavior of the sequence $(\overline{u}^{(n)})_{n\geq n_0}$ as $n\to\infty$. This sufficient condition shows in a way that the condition is “almost” necessary for the error threshold.
Suppose that there exist constants $\varepsilon>0$, $x>0$ such that for all $n\gg n_0$ the inequality $$\label{eq:90}
F_{A_n}(2q_n-1)\geq (u+1)(q_n^n+2\varepsilon)\,,\quad
q_n=1-\frac{x}{n}\,,$$ holds for sufficiently small $u>0$. Then the sequence $(A_n)_{n\geq n_0}$ possesses no threshold-like behavior.
We can assume that $x<\log\frac{u+1}{u}$ for sufficiently small $u>0$ and $q_n>0.5$ for sufficiently large $n$. In view of $$\overline{u}^{(n)}(q_n)=F_{A_n}(2q_n-1)+\sum_{c=1}^{\infty}\left(
\frac{u+1}{\overline{u}^{(n)}(q_n)}\right)^c
F_{A_n}\left((2q_n-1)^{c+1}\right)\geq F_{A_n}(2q_n-1)\geq
(u+1)(q_n^n+2\varepsilon)\,.$$ Hence, $$L_n(x)=\log(u+1)-\log\overline{u}^{(n)}(q_n)\leq-\log
(q_n^n+2\varepsilon)=-\log\left(\left(1-\frac{x}{n}\right)^n+2\varepsilon\right)<-\log(e^{-x}+\varepsilon).$$ for $n\gg n_0$. Consequently, $\limsup\limits_{n\to\infty}
L_n(x)\leq -\log(e^{-x}+\varepsilon)<x$.
Note that in Remark \[remark:5\] we can take $x=r\approx 1.7423$, $u\leq 0.1$, $\varepsilon=0.01$, $n\ge 4$.
General construction for the Eigen evolutionary problem {#sec:6}
=======================================================
The classical Eigen quasispecies model uses as the underlying geometry the $N$-dimensional hypercube. The distances between the vertices of this hypercube are measured by the number of edges connecting them. While this geometry has a transparent biological interpretation in terms of sequences composed of zeroes and ones, which can be identifies with, e.g., purine and pyrimidine, we feel that it is a natural generalization to consider an arbitrary isometry group acting on an abstract metric space to move to a next level of abstraction of the quasispecies model (a somewhat relevant discussion of the original Eigen model can be found in [@dress1988evolution; @rumschitzki1987spectral]). This section provides a concise description of such generalization. While we concentrate here on the mathematical development of the model, we would like to note that an abstract construction of a simplicial fitness landscape can be used to model real biological systems, in particular the switching of the antigenic variants of some bacteria [@avery2006microbial].
Groups of isometries and a generalized algebraic Eigen quasispecies problem {#sec:6:1}
---------------------------------------------------------------------------
The previous results, when we encode individuals of a population by the vertices of the binary cube $X=\{0,1\}^N$ equipped with the Hamming distance, can be generalized as follows. Let $(X,d)$ be a finite metric space. We will assume that the metric $d\colon X\times X
\longrightarrow {\mathbf{N}}_0$ is an [*integer*]{}-valued function.
Consider a group $\Gamma\leqslant{\rm Iso}(X)$ of isometries of $X$ and suppose that $\Gamma$ acts [*transitively*]{} on $X$, that is, $X$ is a single $\Gamma$-orbit (we use the notation for the left action). Since $\Gamma$ acts transitively on $X$ we can fix an arbitrary point $x_0\in X$ and consider the function $d_{x_0}\colon X\longrightarrow {\mathbf{N}}_0$ such that $d_{x_0}(x)=d(x,x_0)$. By definition, $$\operatorname{diam}(X):=\max\{d_{x_0}(x)\,|\,x\in X\}$$ is called the [*diameter*]{} of $X$. The number $N=\operatorname{diam}(X)$ does not depend on the choice of $x_0$.
Let us point out a few of important general geometric examples.
Let $\Gamma=W$ be the Weyl group of the root system $\Delta$ of a simple finite-dimensional Lie algebra ${\mathfrak g}$ over $\bf C$ acting on the Weyl chamber system $X$ (see [@bourbaki22001lie], chapter VI). For instance, if $\Delta$ is of type $A_N$ then $W\cong S_N$. The distance between two chambers $x$, $y$ is the minimal number of chamber walls we need to pass from $x$ to $y$. It is known (e.g., see [@bourbaki22001lie], chapters IV, V) that $d(x,y)$ is just the length of the unique element $w\in W$ such that $y=wx$ when $W$ is viewed as a reflection group generated by a set $S$ of reflections which correspond to fundamental roots (see more general Example 6.3 below.) The number $N=\operatorname{diam}(X)$ is known as the Coxeter number of $W$ and $|X|=|W|$.
On the other hand, the Weyl group $W\cong S_N$ of type $A_N$ acts on the $N$-dimensional regular simplex, the Weyl group $W$ of type $B_N$ (or $C_N$) acts on the $N$-dimensional cube since the root lattice is cubic in the latter case. Thus, we come to the next class of geometric examples.
Let $X=P^{(0)}$ be the the set of vertices of an $n$-dimensional regular polytope $P$ (see, e.g., [@coxeter1973regular] and Fig. \[fig:6\]), all edges of which have an integer length $e$, say, of a regular $m$-gon ($m\ge
2$) on the plane, of a tetrahedron, octahedron or icosahedron in the 3-dimensional space (see Fig. \[fig:6\] for some examples) and so on, equipped with the “edge” metric: the distance between $x$ and $y$ is the minimal number of edges of $P$ connecting $x$ and $y$ multiplied by $e$. For $n$-dimensional unit cube the edge metric is the same as the Hamming metric.
The group of all isometries $\Gamma={\rm Iso}(P)$ acts on $P$ and, consequently, on $X$. For instance, let $P$ be an icosahedron or dodecahedron. Then $\Gamma\cong A_5$ where $A_5<S_5$ is the alternating group of order 60.
![Examples of regular polytopes in dimension 3: tetrahedron (regular simplex), cube, octahedron[]{data-label="fig:6"}](fig6.eps){width="85.00000%"}
\[ex:6:3\] Let $G$ be a finite group generated by a set $S=S^{-1}$. The [*word*]{} metric $d=d_S$ on $G$ is defined as follows (see [@de2000topics], chapter IV for more details and examples): $d(g,h)=l(g^{-1}h)$ where $l(g^{-1}h)=l$ is the minimal number of generators $s\in S$ needed to represent $g^{-1}h$ as a product $s_1\dots s_l$. The word metric is invariant with respect to the action of $G$ on itself by left shifts $h\to gh$. Hence, we have the metric space $X=G$ and the transitive action of $\Gamma=G$ on $X$ by isometries.
More generally, for any subgroup $H\leqslant G$ we can define the metric space $X_H=\{gH\,|\,g\in G\}$ of the left cosets of $G$ by $H$. The group $G$ acts on $X_H$ by left shifts and $$d(gH,aH)=\min\{d(x,y)\,|\,x\in gH, y\in aH\}\,.$$
If $G$ acts transitively by isometries on a metric space $X$ then as a $G$-set $X$ is isomorphic to the set of left cosets $G/{\rm
St}_{\Gamma}(x_0)$, $x_0\in X$.
Let $p$ be any fixed prime, ${{\mathbf{Z}}}_p$ be the commutative ring of $p$-adic integers equipped with the standard $p$-adic metrics $d_p(x,y)=\|
x-y\|_p$. Consider the quotient rings $X_{n,p}={{\mathbf{Z}}}_p/p^{n}{{\mathbf{Z}}}_p$, $n\in {\mathbf{N}}$, with the scaled metric $\overline{d}_p(\overline{x},\overline{y})=p^{n-1}\|
x-y\|_p$ ($\overline{x}$ denotes the coset $x+p^{n}{{\mathbf{Z}}}_p$) on which the additive group $\Gamma=\Gamma_{n,p}=X_{n,p}$ acts 1-transitively by isometries $L_{\overline{\gamma}}:
\overline{x}\to \overline{\gamma} +\overline{x}$. Here $N_{n,p}=\operatorname{diam}(X_{n,p})=p^{n-1}$, $l_{n,p}=|X_{n,p}|=p^n$.
For $p=2$, $n=3$ we have 2-adic “cube” $X_{3,2}$ which is different from the binary cube with the Hamming metric.
Now consider a quadruple $(X,d,\Gamma, {{\boldsymbol{w}}})$ where $(X,d)$ is a finite metric space with integer distances between points of diameter $N$ and cardinality $l=|X|$, a group $\Gamma\leqslant{\rm
Iso} (X)$ is a fixed group and a [*fitness landscape*]{} ${{\boldsymbol{w}}}=(w_x)^\top$ is a vector-column of non-negative real numbers called [*fitnesses*]{} indexed by $x\in X$. The quadruple $(X,d,\Gamma, {{\boldsymbol{w}}})$ will be called [*homogeneous*]{} $\Gamma$-landscape. In other words, the sequences of the population are encoded by $x\in X$.
Consider also the diagonal matrix ${{{\boldsymbol{W}}}}={\operatorname{diag}}(w_x)$ of order $l$ called the [*fitness matrix*]{}, the symmetric distance matrix ${{{\boldsymbol{D}}}}=\bigl(d(x,y)\bigr)_{l\times l}$ with integer entries of the same order and the symmetric matrix ${{{\boldsymbol{Q}}}}=\left((1-q)^{d(x,y)} q^{N-d(x,y)}\right)_{l\times l}$ for $q\in
[0,1]$. Finally, we introduce the [*distance polynomial*]{} $$\label{6.1}
P_X(q)=\sum_{x\in X} (1-q)^{d(x,x_0)}q^{N-d(x,x_0)}\,,\quad
x_0\in X.$$
Since $\Gamma$ acts transitively on $X$ this polynomial is independent on the choice of $x_0\in X$ and is the sum of entries in each row (column) of ${{{\boldsymbol{Q}}}}$.
The following definition generalizes the classical Eigen quasispecies problem we dealt with in the previous sections.
The problem to find the leading eigenvalue $\overline{w}=\overline{w}(q)$ of the matrix $\frac{1}{P_X(q)}{{{\boldsymbol{ QW}}}}$ and the eigenvector ${{\boldsymbol{p}}}={{\boldsymbol{p}}}(q)$ satisfying $$\label{6.2}
{{{\boldsymbol{QWp}}}}=P_X(q)\overline{w}\,{{\boldsymbol{p}}},\;\quad p_x=p_x(q)\ge
0,\quad\sum_{x\in X} p_x(q)=1\,$$ will be called [*the generalized algebraic Eigen quasispecies problem*]{}.
Note that in ([\[6.2\]]{}) $$\label{6.3} \overline{w}=\sum_{x\in X} w_xp_x\,.$$ Due to the Perron–Frobenius theorem a solution of this problem always exists. Also note that the uniform distribution vector $$\label{6.4}
{{\boldsymbol{p}}}=\frac{1}{|X|}(1,\dots,1)^\top=\frac{1}{l}(1,\dots,1)^\top$$ provides a solution to (\[6.2\]) in the case of the constant fitnesses $w_x\equiv w>0$.
The problem (\[6.2\]) turns into the classical Eigen evolutionary problem for the $N$-dimensional binary cube $X=\{0,1\}^N$ with the Hamming metric and $\Gamma={\rm Iso}
(X)$ which was named in 1930 by A. Young a [*hyperoctahedral*]{} group. $\Gamma$ is isomorphic as an abstract group to the Weyl group of the root system of type $B_N$ or $C_N$ and is acting on the cube. In the classical case $P_X(q)\equiv 1$.
Consider also the following serial examples. If $X$ is the set of vertices of an $n$-dimensional regular simplex with all edges of unit length then $\Gamma={\rm Iso} (X)\cong S_{n+1}$, $N=\operatorname{diam}(X)=1$ and $l=|X|=n+1$. The distance polynomial is $$\label{6.5}
P_X(q)=q+n(1-q)\,.$$ If $X$ is the set of vertices of an $n$-dimensional hyperoctahedron with all edges of unit length then $\Gamma={\rm
Iso} (X)$ is again a hyperoctahedral group (the hyperoctahedron is the dual polytope to the cube), $N=\operatorname{diam}(X)=2$ and $l=|X|=2n$. The distance polynomial is $$\label{6.6}
P_X(q)=q^2+(2n-2)(1-q)q+(1-q)^2\,.$$
Properties of the distance polynomial
-------------------------------------
In the notation of Section \[sec:6:1\] consider the polynomial $P_X(q)=P_{X,d}(q)$. The polynomial $P_X(q)$ is strictly positive on $[0,1]$ (if $N$ is strictly equal to $\operatorname{diam}(X)$. If $N>\operatorname{diam}(X)$ then $P_X(0)=0$, such cases sometimes we will need to consider) and possesses the following properties:
1. $$\label{6.7}
P_X(1)=1,\quad P_X\left(\frac{1}{2}\right)=
\frac{|X|}{2^N}=\frac{l}{2^N}\;.$$
2. $$\label{6.8}
P_X(q)=\sum_{k=0}^N f_k\,(1-q)^k q^{N-k}\in \mathbf Z[q]\,,$$
where the non-negative integers $f_k=f_k(X):=\#\{x\in X\,|\,d(x,x_0)=k\}$ are the cardinalities of $d$-spheres in $X$ with the center at the fixed point $x_0$ and of radius $k$.
The polynomial $S_X(t)=\sum_{k=0}^N f_k
t^k$ is often called the [*spherical growth function*]{} of $(X,d)$. See, for instance, [@de2000topics], chapter VI for details and examples.
Suppose that we scaled the metric $d$ by a positive integer factor $e$. Let $P_{X,e\cdot d}(q)$ denote the new distance polynomial. Then $$\label{6.9} P_{X,e\cdot d}(q)=\sum_{x\in X}
(1-q)^{e\,d(x,x_0)}q^{e(N-d(x,x_0))}=\sum_{k=0}^N
f_k\,(1-q)^{e\,k} q^{e(N-k)}\,,\quad x_0\in X\,.$$ Since $q\in [0,1]$ we may assert that the sequence $\{P_{X,e\cdot
d}(q)\,|\,e\in {\mathbf{N}}\}$ is non-increasing at each fixed point $q\in[0,1]$.
Regular simplicial fitness landscapes
-------------------------------------
To give a specific example of the analysis of the generalized algebraic Eigen quasispecies problem we shall briefly consider two-valued fitness landscapes related to the set of vertices of the regular $n$-dimensional simplex $X$ with ${\rm Iso}(X)\cong S_{n+1}$. Here we follow the main lines of Section \[sec:1\].
Biologically, the simplicial fitness landscape means that we deal with a population of individuals such that any individual can mutate to any other individual with the same probability equal to $1-q$. Even such oversimplified construction can model a non-trivial biological system. Here, for example, if we consider “mutation” as a sudden discrete genetic (heritable) change then the simplicial geometry can describe, at a first approximation, the switching of the antigenic variants for some bacteria. These variants turns one into another with almost equal probabilities, whereas the corresponding fitnesses of different variants are defined by interactions with the host immune system (e.g., [@avery2006microbial]).
### General scheme
Let $X=\{0,1,\dots,n\}$ and $d(i,j)=1$ if $i\ne j$, $d(i,i)=0$. Hence, $X$ is a metric space with the trivial metric, $N=\operatorname{diam}(X)=1$ and $l=|X|=n+1$. The distance polynomial is defined by (\[6.5\]).
Let $A\subset\{0,1,\dots,n\}$. Consider the landscape $$w_k=\left\{
\begin{array}{r}
w+s,\quad k\in A\;,\\
w,\quad k\notin A\;.\\
\end{array}
\right.$$ The matrix ${{{\boldsymbol{W}}}}$ of fitnesses can be represented as follows $${\label{6.10}}
{{{\boldsymbol{W}}}}=w{{{\boldsymbol{I}}}}+s {{{\boldsymbol{E}}}}_A=w {{{\boldsymbol{I}}}}+s
\sum_{a\in A}{{{\boldsymbol{E}}}}_a,$$ ${{\boldsymbol{ I}}}$ being the identity matrix and ${{{\boldsymbol{E}}}}_a$ being the elementary matrix with the only one nontrivial entry $e_{aa}=1$ on the diagonal.
We want to solve the problem (\[6.2\]). The matrix ${{{\boldsymbol{Q}}}}=({{{\boldsymbol{Q}}}}_{ba})=(2q-1){{{\boldsymbol{I}}}}+(1-q){{{\boldsymbol{E}}}}$ where all the entries of ${{\boldsymbol{ E}}}$ are ones, that is $${{{\boldsymbol{Q}}}}_{ba}=\left\{
\begin{array}{cc}
1-q\,,& b\ne a\,,\\
q\,,&b=a\,.
\end{array}
\right.$$
It can be directly checked that $${\label{6.11}}
{{{\boldsymbol{D}}}}(n,q):={{{\boldsymbol{T}}}}^{-1} {{{\boldsymbol{QT}}}}=\operatorname{diag}(q+n(1-q),2q-1,\dots,2q-1),$$ where for the symmetric transition matrix ${{{\boldsymbol{T}}}}$ of order $n+1$ we have $${\label{6.12}}
{{{\boldsymbol{T}}}}=\left(\begin{array}{crrrr} 1&1&1&\dots&1\\
1&-1&0&\dots&0\\
1&0&-1&\dots&0\\
\vdots&\vdots&\vdots&\ddots&0\\
1&0&0&\dots&-1\\
\end{array}\right)\,,\quad {{{\boldsymbol{T}}}}^{-1}=\frac{1}{n+1}\left(\begin{array}{crrrr} 1&1&1&\dots&1\\
1&-n&1&\dots&1\\
1&1&-n&\dots&1\\
\vdots&\vdots&\vdots&\ddots&1\\
1&1&1&\dots&-n\\
\end{array}\right)\,.$$
The transformation of (\[6.2\]) yields $${{{\boldsymbol{T}}}}^{-1}{{{\boldsymbol{ QT}}}} {{{\boldsymbol{T}}}}^{-1}{{{\boldsymbol{ WT}}}} {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}=P_X(q)\overline{w}\, {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}=(q+n(1-q)) \overline{w}\,
{{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}.$$ or, in view of (\[6.11\]), $${\label{6.13}}
{{{\boldsymbol{D}}}}(n,q) \left(w{{{\boldsymbol{I}}}}+s\sum_{a\in A}{{{\boldsymbol{T}}}}^{-1}{{{\boldsymbol{E}}}}_a{{{\boldsymbol{T}}}}\right) {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}=P_X(q)\overline{w}\, {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}},$$ whence $${\label{6.14}}(P_X(q)\overline{w}{{{\boldsymbol{I}}}}-w{{{\boldsymbol{D}}}}(n,q)) {{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}=\sum_{a\in A} {{{\boldsymbol{D}}}}(n,q){{{\boldsymbol{T}}}}^{-1}{{{\boldsymbol{E}}}}_a{{\boldsymbol{p}}}.$$
Let ${{\boldsymbol{x}}}={{{\boldsymbol{T}}}}^{-1}{{\boldsymbol{p}}}$, ${{\boldsymbol{p}}}={{{\boldsymbol{T}}}}{{\boldsymbol{x}}}$. Then (\[6.14\]) implies $${\label{6.15}} {{\boldsymbol{x}}}=s\sum_{a\in A}(P_X(q)\overline{w}\,{{{\boldsymbol{I}}}}-{{{\boldsymbol{D}}}}(n,q)))^{-1}{{{\boldsymbol{D}}}}(n,q){{{\boldsymbol{T}}}}^{-1}{{{\boldsymbol{E}}}}_a{{\boldsymbol{p}}},$$ or, in coordinates, $${\label{6.16}}
x_k=s\sum_{a\in A}\frac{{{{\boldsymbol{D}}}}(n,q)_k
t^{(-1)}_{ka}p_a}{P_X(q)\overline{w}-w{{{\boldsymbol{D}}}}(n,q)_k},\quad
k=0,\ldots,n.$$
Since ${{\boldsymbol{p}}}={{{\boldsymbol{T}}}}{{\boldsymbol{x}}}$, then we get from (\[6.15\]) $${\label{6.17}}
{{\boldsymbol{p}}}=s\sum_{a\in A}{{{\boldsymbol{T}}}}(P_X(q)\overline{w}{{{\boldsymbol{I}}}}-{{{\boldsymbol{D}}}}(n,q))^{-1} { {{\boldsymbol{D}}}}(n,q){{{\boldsymbol{T}}}}^{-1}\,{{{\boldsymbol{E}}}}_a{{\boldsymbol{p}}}.$$
Only the components $p_a$, $a\in A$, are involved in the right-hand side of (\[6.17\]). By definition, ${{{\boldsymbol{E}}}}_A=\sum_{a\in A} {{{\boldsymbol{E}}}}_a$ and ${{{\boldsymbol{E}}}}_A$ is a projection matrix. Hence, we can multiply both sides of (\[6.17\]) by ${{{\boldsymbol{E}}}}_A$: $${\label{6.18}}
{{{\boldsymbol{E}}}}_A {{\boldsymbol{p}}}=s{{{\boldsymbol{E}}}}_A{{{\boldsymbol{T}}}}(P_X(q)\overline{w}{{{\boldsymbol{I}}}}-{{{\boldsymbol{D}}}}(n,q)))^{-1} {{{\boldsymbol{D}}}}(n,q){{{\boldsymbol{T}}}}^{-1}\,{{{\boldsymbol{E}}}}_A {{\boldsymbol{p}}}.$$
We can rewrite (\[6.18\]) as $${\label{6.19}}
{{\boldsymbol{p}}}_A={{{\boldsymbol{M}}}}{{\boldsymbol{p}}}_A,\quad {{\boldsymbol{p}}}_A={{{\boldsymbol{E}}}}_A {{\boldsymbol{p}}}\,,$$ where $${\label{6.20}}
\begin{split}
{{{\boldsymbol{M}}}}&=s{{{\boldsymbol{E}}}}_A \,{{{\boldsymbol{T}}}}(P_X(q)\overline{w}{{{\boldsymbol{I}}}}-{{{\boldsymbol{D}}}}(n,q)))^{-1}{{{\boldsymbol{D}}}}(n,q){{{\boldsymbol{T}}}}^{-1}\\
&=\frac{s}{\overline{w}P_X(q)}\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}P_X(q)}\right)^c{{{\boldsymbol{E}}}}_A \,{{{\boldsymbol{T}}}}{{{\boldsymbol{D}}}}(n,q)^{c+1}{{{\boldsymbol{T}}}}^{-1}\\
&=\frac{s}{\overline{w}P_X(q)}\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}P_X(q)}\right)^c{{{\boldsymbol{E}}}}_A \,{{{\boldsymbol{Q}}}}^{c+1}\,.
\end{split}$$
It follows that vector ${{\boldsymbol{p}}}_A$ is an eigenvector of ${{{\boldsymbol{M}}}}$ corresponding to the eigenvalue $\lambda=1$.
Consider $\overline{w}$ in (\[6.19\]), (\[6.20\]) as a parameter. It follows from (\[6.3\]) that $\overline{w}$ depends only on $p_a$, $a\in A$, that is, on the “reduced” vector ${{\boldsymbol{p}}}_A={{{\boldsymbol{E}}}}_A{{\boldsymbol{p}}}$. The original eigenvector ${{\boldsymbol{p}}}$ can be reconstructed from ${{\boldsymbol{p}}}_A$ with the help of (\[6.17\]). Thus, instead of the original problem we arrive to the reduced problem to find the eigenvector ${{\boldsymbol{p}}}_A$ satisfying (\[6.18\]) and corresponding to the eigenvalue $\lambda=1$ of the matrix ${{{\boldsymbol{M}}}}=(m_{ba})$ defined in (\[6.20\])). The parameter $\overline{w}=\overline{w}(q)$ satisfies the formula $${\label{6.21}}
\overline{w}=w+s\sum_{a\in A}p_a.$$
### Equation for the eigenvalue $\overline{w}$
In we have $${\label{6.22}}
{{{\boldsymbol{Q}}}}^{c+1}=(2q-1)^{c+1}{{{\boldsymbol{I}}}}+\sum_{m=1}^{c+1}{\binom{c+1}{m}}(n+1)^{m-1}(2q-1)^{c+1-m}(1-q)^m{{{\boldsymbol{E}}}}\,.$$
We apply directly the binomial expansion for the matrix ${{{\boldsymbol{Q}}}}^{c+1}=((2q-1){{{\boldsymbol{I}}}}+(1-q){{{\boldsymbol{E}}}})^{c+1}$. Since all the entries of ${{\boldsymbol{ E}}}$ are ones then ${{\boldsymbol{E}}}^2=(n+1){{\boldsymbol{E}}}$ and, consequently, ${{\boldsymbol{E}}}^m=(n+1)^{m-1}{{\boldsymbol{E}}}$.
The equality (\[6.19\]) implies that $${\label{6.23}}
\sum_{b\in A} p_b= \sum_{b\in A}\sum_{a\in A} m_{ba}p_a=\sum_{a\in
A}p_a \sum_{b\in A} m_{ba}.$$ Suppose that the sum $\sum\limits_{b\in A}m_{ba}$ does not depend on $a\in A$. Then it follows from (\[6.23\]) that $\sum\limits_{b\in A} m_{ba}=1$ for each $a\in A$. In view of (\[6.20\]) $${\label{6.24}}
\frac{\overline{w}P_X(q)}{s}=
\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}P_X(q)}\right)^c
\sum_{b\in A}({{{\boldsymbol{E}}}}_A \,{{{\boldsymbol{Q}}}}^{c+1})_{ba}\,,$$ if the inner sum does not depend on $a\in A$.
In the previous notation let $A$ be a subset of a simplicial metric space $X$. Then the equality $${\label{6.25}}
\frac{\overline{w}P_X(q)}{s}=
\!\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}P_X(q)}\right)^c
\left(\!(2q-1)^{c+1}+\frac{|A|}{n+1}\sum_{m=1}^{c+1}{\binom{c+1}{m}}(n+1)^{m}(2q-1)^{c+1-m}(1-q)^m\right)\,$$ holds.
Since ${\rm Iso}(X)\cong S_{n+1}$ acts $(n+1)$-transitively on $X$ we may assume that $A$ is the subset $\{0,1,\dots, |A|-1\}$ on which the cyclic subgroup $C_{|A|}=\left<(0,1,\dots, |A|-1)\right>$ is acting. Then we apply (\[6.22\]) to (\[6.24\]) .
The formula (\[6.25\]) can be simplified as follows. Recall (see (\[6.5\])) that $P_X(q)=q+n(1-q)$. The binomial expansion yields $$\sum_{m=1}^{c+1}{\binom {c+1}{m}}(n+1)^{m}(2q-1)^{c+1-m}(1-q)^m$$ $$=
((2q-1)+(n+1)(1-q))^{c+1}-(2q-1)^{c+1}=P_X(q)^{c+1}-(2q-1)^{c+1}\,.$$ Hence, (\[6.25\]) reads $${\label{6.26}}
\frac{\overline{w}P_X(q)}{s}=
\!\sum_{c=0}^{\infty}\left(\frac{w}{\overline{w}P_X(q)}\right)^c
\left(\!\left(1-\frac{|A|}{n+1}\right)(2q-1)^{c+1}+\frac{|A|}{n+1}P_X(q)^{c+1}\right).$$
Summing the geometric progressions we finally get $${\label{6.27}}
\frac{|A|}{(n+1)(\overline{u}-u)}+
\left(1-\frac{|A|}{n+1}\right)\frac{2q-1}{(q+n(1-q))\overline{u}-(2q-1)u}=1,\quad
u=\frac{w}{s}\,,\quad \overline{u}=\frac{\overline{w}}{s}\,.$$
Note that the equation (\[6.27\]) depends only on $|A|$ and the dimension $n$. It follows that (\[6.27\]) provides the eigenvalue of the two-valued fitness problem (\[6.2\]) for any subset $A\subset X$. Note also that the equation (\[6.27\]) turns into the equation of degree $2=N+1$ where $N= 1$ is the diameter of the simplex (compare with Corollary \[corr:2:1\]). We expect that for the hyperoctahedral landscapes we will get cubic equations since $N=2$ for a hyperoctahedron (with unit edges) in any dimension.
The solution to (\[6.27\]) is given by the following formula[^3] $${\label{6.28}}
\begin{split}
\overline{u}&=\frac{v(q)+\sqrt{v^2(q)-4(u+u^2)(2q-1)(q+n(1-q))}}{2(q+n(1-q))}\,,\\
v(q)&=(q+n(1-q))\left(u+\frac{|A|}{n+1}\right)+(2q-1)\left(u+1-\frac{|A|}{n+1}\right).
\end{split}$$
### Simplicial error threshold
In this subsection some results of Section \[sec:5\] are appropriated for the case of the simplicial landscapes.
Let $X_n$ be the set of vertices of an $n$-dimensional regular simplex with edges of unit length and let $(A_n)_{n=n_0}^\infty$, $A_n\subset
X_n$, be a sequence of subsets. Let $\overline{u}=\overline{u}^{(n)}(q)$ be the sequence of the corresponding eigenvalues (see (\[6.28\])). It can be checked that each function $\overline{u}^{(n)}$ is increasing on the segment $[0.5,1]$ and convex downward.
A sequence $(A_n)_{n\geq n_0}$ is called a sequence of the [*moderate growth*]{} if $${\label{6.29}}
\lim_{n\to \infty}\frac{|A_n|}{n+1}=0\;.$$
Let us denote $\alpha_n=\frac{|A_n|}{n+1}$. In view of (\[6.28\]) $\overline{u}^{(n)}(0.5)=u+\alpha_n\to u$ as $n\to\infty$ if the sequence $(A_n)_{n\geq n_0}$ is of the moderate growth. On the other hand, $\overline{u}^{(n)}(1)\equiv u+1$.
Consider new coordinates $x$, $L$ such that $${\label{6.30}}q=1-\frac{x}{n},\;0\leq x\leq \frac{n}{2},
\quad L=\frac{u+1}{\overline{u}}-1\,,
\;0\leq L\leq 1/u.$$ We assume that $u>0$ in (\[6.30\]). The curve $\overline{u}=\overline{u}^{(n)}(q)$ transforms into the curve $${\label{6.31}}
L=L_n(x)=\frac{u+1}{\overline{u}^{(n)}\left(
1-\frac{x}{n}\right)}-1.$$
\[def:6:1\] We say that a sequence $(A_n)_{n=n_0}^\infty$ of the moderate growth, or, equivalently, the family $(\overline{u}^{(n)})_{n=n_0}^\infty$ possesses the threshold-like behavior on the segment $[0.5,1]$ if for each fixed $x\geq 0$ and the corresponding functions $L_n(x)$ it it true that $${\label{6.32}}
\lim_{n\to\infty} L_n(x)=L(x)=\left\{\begin{array}{l}\vspace{3pt}
\;\displaystyle x,\quad 0\leq x< \frac{1}{u}\,,\\
\displaystyle \frac{1}{u},\quad x\geq \frac{1}{u}\;.
\end{array}
\right.$$
![Illustration to Definition \[def:6:1\]](fig7.eps){width="50.00000%"}
The following formula provides an approximation for the error threshold value (if exists) $q_*^{(n)}(u)$, $n\gg 1$: $$q^{(n)}_*(u)\approx 1-\frac{1}{nu}=1-\frac{s}{nw}\,.$$
In the above notation suppose that a sequence of subsets $A_n\subset X_n$ of the moderate growth is given and $u>0$. Then the sequence $(A_n)_{n=n_0}^\infty$ shows the threshold-like behavior on the segment $[0.5,1]$.
In view of the equation (\[6.27\]), in coordinates $x$, $L$: $${\label{6.33}}
\frac{\alpha_n(1+L_n(x))}{1-uL_n(x)}+
\frac{(1-\alpha_n)(1-2x/n)}{(1+x-x/n)\frac{1+u}{1+L_n(x)}-\left(1-2x/n\right)u}=1\;.$$
The existence of $\lim\limits_{n\to\infty}L_n(x)$ for a fixed $x$, $0\leq x<1/u$, can be proved with the help of lower and upper estimates. If $n\to\infty$ in (\[6.33\]) we get (since $\alpha_n=|A_n|/(n+1)\to 0$) $$\frac{1}{(1+x)\frac{1+u}{1+L(x)}-u}=1\,,$$ or $L(x)\equiv x$ on $[0,1/u)$. Since $L_n(x)$ increases with respect to $n$ and cannot exceed the value $1/u$ we obtain the desired result.
In a similar fashion other geometric examples can be analyzed.
Concluding remarks
==================
There are two main points to emphasize in order to conclude the presentation. First, in this text we put forward general, rigorous, and quite elementary methods to analyze two-valued fitness landscapes in the classical quasispecies model. While a great deal of analysis of this problem in the existing literature was inspired by the analogies with the famous Ising model of statistical physics, we show that direct methods of linear algebra allow gaining full understanding of the properties of the selection–mutation equilibrium in this model at least in some special cases.
Second, the language of the group theory gives us an opportunity to look at the phenomena associated with the quasispecies model from a more general and abstract point of view. In particular, the infamous error threshold can be looked at from the position of the external and internal metric properties of orbits. If the set of population sequences is enumerated by points of a finite metric space $X$ with integer-valued metric $d$ on which a group $\Gamma$ acts transitively by isometries then we can involve group theoretical and algebraic tools in order to obtain not very complicated solutions for the leading eigenvalue problem, at least in the special case of the two-valued fitness landscapes. Such a classical approach is in accordance with the well known F. Klein’s [*Erlangen program*]{}. We are convinced that this connection between mathematical biology, finite geometries, combinatorics and algebra confirms the importance of Eigen’s model from various viewpoints.
To reiterate, in the general case we consider a quadruple $(X,d,\Gamma, {{\boldsymbol{w}}})$ — [*homogeneous*]{} $\Gamma$-landscape — with the fitness function ${{\boldsymbol{w}}}\colon X\longrightarrow {\mathbf{R}}_{\geq 0}$. The information of the geometric properties of the underlying metric space $(X,d)$ is contained in the symmetric matrix ${{{\boldsymbol{Q}}}}=\bigl((1-q)^{d(x,y)} q^{N-d(x,y)}\bigr)$, $q\in
[0,1]$, where $N=\operatorname{diam}(X)$ is the diameter of $X$. The diameter $N$ as well as the cardinality $l=|X|$ are the two main numerical characteristics of the model $(X,d,\Gamma, {{\boldsymbol{w}}})$. The distance polynomial $P_X(q)$, which is the leading eigenvalue of the matrix ${{{\boldsymbol{Q}}}}$, plays the key role in the analysis. For the classical Eigen’s quasispecies model $X=\{0,1\}^N$ is the binary cube with the Hamming distance, $\operatorname{diam}(X)=N$, $l=|X|=2^N$ and $P_X(q)\equiv 1$.
Suppose that we have a subgroup $G\leqslant\Gamma$ which also acts on $X$ and the fitness function ${{\boldsymbol{w}}}$ is constant on the orbits of $G$-action. We saw in Sections \[sec:2\], \[sec:6\] that for the two-valued fitness functions ${{\boldsymbol{w}}}(A)=w+s$, ${{\boldsymbol{w}}}(X\setminus A)=w$, $A$ being any $G$-orbit, the degree of the equation on the leading eigenvalue can be reduced from $l$ to $N+1$. Although the solution of the leading eigenvalue problem appears in an implicit form we are able to obtain lower and upper bounds for it.
We can also consider sequences of metric spaces $X_n$ and orbits $A_n$ as $n\to\infty$. Usually we have a chain $$X_{n_0}\subset\dots \subset X_{n}\subset X_{n+1}\subset\dots
\subset \bigcup_{n} X_n=X_{\infty}\,.$$ The analysis presented in the main text allows to conjecture that the [*error threshold*]{}, i.e., non-analytical behavior of the leading eigenvalue $\overline{w}$ in the infinite sequence limit, occurs when the cardinalities $|A_n|$ grow not rapidly enough comparing with the growth of $|X_n|$. For instance, when $A_n\equiv A$, where $A$ contains a single point (the single peaked landscape), or is a fixed constant set (orbit) then the threshold-like behavior is observed. This topic is to be investigated in the general situation.
At the beginning of Section \[sec:6\] we pointed out the most interesting geometric examples of groups and metric spaces for which the generalized Eigen’s algebraic problem could be solved. Among them are the Weyl groups acting on the chamber systems (the [*reflection*]{} groups should be added) and groups of symmetry of regular polytopes. Example \[ex:6:3\] deals with all finite groups in general. It is very possible, and genuinely intriguing, that some infinite finitely generated groups (free groups, non-Euclidean crystallographic groups and others) can be included in the list of groups for the future research (see, for instance, [@de2000topics; @gromov1993]).
Proof of Proposition \[pr:ad\] {#ap:1}
==============================
The following three lemmas and corollary provide the full proof that all the examples in Section \[sec:3\] deal with admissible sequences of orbits of the moderate growth (Proposition \[pr:ad\]).
Let $A\subset X_{n_0}$ be a fixed $G$-orbit. Consider the constant sequences $A_n\equiv A$ and $G_n\equiv G$, $n\geq n_0$. Then the sequence $(A_n)_{n=n_0}^\infty$ is admissible.
Since the orbit is not changing as $n\to \infty$ then it follows from that $$F_{A_n}(2q-1)=q^{n-n_0}F_{A_{n_0}}(2q-1)\,,\quad q\in[0,1].$$ The polynomial $F_{A_n}(2q-1)>0$ and $q^{n-n_0}\geq q^{n+1-n_0}$ on $[0.5,1]$. Hence, holds.
Let $a_n\in X_n$, $a_n^*=2^n-1-a_n$, and $A_n=\{a_n,a_n^*\}$. Let $G_n=G=\{1,g\}$ be the group of order 2 such that $g(a)=a^*$ for any $a\in X_n$. Then the sequence $(A_n)_{n=n_0}^\infty$ is admissible.
In view of and $$F_{A_n}(2q-1)=q^n+(1-q)^n\geq
q^{n+1}+(1-q)^{n+1}=F_{A_{n+1}}(2q-1)\,,\quad q\in[0,1].$$
\[l:5:3\] Let $p$ be a fixed number, $n\geq n_0=2p$. Let $A_n=A_{n,p}=\{a\in X_n\,|\,H_a=p\}$. Then $(A_n)_{n=n_0}^\infty$ is an admissible sequence.
It follows from that $$\label{eq:63}
F_{A_n}(2q-1)=\sum\limits_{k=0}^p {\binom{p}{k}}{\binom{n-p}{k}}(1-q)^{2k}q^{n-2k}=:F_{n,p}(q)\;,\quad 0\leq p\leq n.$$ At the same time consider the polynomials $$\label{eq:64}
G_{n,p}(q):=\sum\limits_{k=1}^p {\binom{p}{k}}{\binom{n-p}{k-1}}(1-q)^{2k}q^{n+1-2k}.$$ By definition, $F_{0,0}(q)\equiv 1$, $G_{n,0}(q)=0$. Applying the binomial formulas ${\binom{n+1-p}{k}}={\binom{n-p}{k}}+{\binom{n-p}{k-1}}$ to and ${\binom{p}{k}}={\binom{p-1}{k}}+{\binom{p-1}{k-1}}$ to we get the following recursive relations: $$\label{eq:65}
F_{n+1,p}(q)=qF_{n,p}(q)+G_{n,p}(q)\,,\quad G_{n+1,p}(q)=(1-q)^2
F_{n,p-1}(q)+qG_{n,p-1}(q)\,.$$
When we substitute the left-hand-side of the second formula into the first one (with the change $n\to n-1$) and then iterate such substitutions we get $$\label{eq:66}
F_{n+1,p}(q)=qF_{n,p}(q)+(1-q)^2\sum_{j=1}^p
q^{j-1}F_{n-j,p-j}(q)\;.$$ In the same way the equality $$\label{eq:67}
G_{n+1,p}(q)=(1-q)^2\sum_{j=1}^p q^{j-1}F_{n+1-j,p-j}(q)$$ can be obtained.
Formulas imply also that $$F_{n,p}(q)-F_{n+1,p}(q)=(1-q)F_{n,p}(q)-G_{n,p}(q)=
(1-q)qF_{n-1,p}(q)+(1-q)G_{n-1,p}(q)-G_{n,p}(q)\,,$$ or $$\label{eq:68}
F_{n,p}(q)-F_{n+1,p}(q)=q\left((1-q)F_{n-1,p}(q)-G_{n,p}(q)\right)+
(1-q)\left(G_{n-1,p}(q)-G_{n,p}(q)\right)\,.$$ Our objective is to prove that $$F_{2p+k+1,p}(q)\leq
F_{2p+k,p}(q)\,,\qquad k\geq 0\,,\quad q\in[0,1]\,.$$ We will proceed by induction on $p$ and, for a fixed $p$, by induction on $k$.
First of all, the case $p=0$ is trivial since $F_{n,0}(q)=q^n$.
Let $p\ge 1$ be fixed and let $k=0$. Substituting $n=2p$ into we get $$F_{2p,p}(q)-F_{2p+1,p}(q)=q\left((1-q)F_{2p-1,p}(q)-G_{2p,p}(q)\right)+
(1-q)\left(G_{2p-1,p}(q)-G_{2p,p}(q)\right)\,.$$ Let us show that both summands in the right-hand side are nonnegative on $[0,1]$. On the one hand, by definition we have $F_{n,p}(q)=F_{n,n-p}(q)$. Then in view of $$\begin{aligned}
(1-q)F_{2p-1,p}(q)-G_{2p,p}(q)&=(1-q)F_{2p-1,p-1}(q)-G_{2p,p}(q)\\
&=(1-q)F_{2p-1,p-1}(q)-((1-q)^2F_{2p-1,p-1}(q)+qG_{2p-1,p-1}(q))\\
&=q(1-q)F_{2p-1,p-1}(q)-qG_{2p-1,p-1}(q)\\
&=q(1-q)F_{2p-1,p-1}(q)-q(F_{2p,p-1}-qF_{2p-1,p-1}(q))\\
&=q(F_{2p-1,p-1}(q)-F_{2p,p-1}(q))\geq 0\;,\quad q\in [0,1]\,,\end{aligned}$$ by the inductive hypothesis.
On the other hand, it follows from that $$G_{2p-1,p}(q)-G_{2p,p}(q)=(1-q)^2\sum_{j=1}^p
q^{j-1}(F_{2p-j,p-j}(q)-F_{2p+1-j,p-j}(q)) \geq 0$$ on $[0,1]$ by the same reasons. This finishes the proof for the case $k=0$.
Let $k\ge 1$. Then by virtue of we can assert that $$\begin{aligned}
&F_{2p+k,p}(q)-F_{2p+k+1,p}(q)=\\
&=q(F_{2p+k-1,p}(q)-F_{2p+k,p}(q))+(1-q)^2\sum_{j=1}^p
q^{j-1}(F_{2p+k-1-j,p-j}(q)-F_{2p+k-j,p-j}(q))\geq 0\end{aligned}$$ on $[0,1]$ by the inductive hypothesis. The lemma is proved.
Let $A_{2n}=A_{2n,n}=\{a\in X_{2n}\,|\,H_a=n\}$. Then $(A_{2n})_{n=n_0}^\infty$ is an admissible sequence.
In the notation of Lemma \[l:5:3\] let us prove that $F_{2n,n}(q)\leq F_{2n-2,n-1}(q)$ on $[0,1]$. From the first formula we can find the expressions $G_{n+1,p}(q)=F_{n+2,p}(q)-qF_{n+1,p}(q)$, $G_{n,p-1}(q)=F_{n+1,p-1}(q)-qF_{n,p-1}(q)$ and substitute them into the second one. The simplification yields $$F_{n+2,p}(q)=(1-2q)F_{n,p-1}(q)+qF_{n+1,p}(q)+qF_{n+1,p-1}(q)\,.$$ Consequently, choosing appropriate values for $n$, $p$ in this formula, we get $$F_{2n-2,n-1}(q)-F_{2n,n}(q)=q(F_{2n-2,n-1}(q)-F_{2n-1,n}(q))+
q(F_{2n-2,n-1}(q)-F_{2n-1,n-1}(q))\,.$$ But in view of $F_{n,p}(q)=F_{n,n-p}(q)$ for all $n$ and $p$, $0\leq p\leq n$. Hence $F_{2n-1,n}(q)=F_{2n-1,n-1}(q)$ and it follows from Lemma \[l:5:3\] that $$F_{2n-2,n-1}(q)-F_{2n,n}(q)=2q(F_{2n-2,n-1}(q)-F_{2n-1,n-1}(q))\geq
0\,$$ on the segment $[0,1]$.
#### Acknowledgements:
The YSS’s research is partially supported by the joint grant between the Russian Foundation for Basic Research (RFBR) and Taiwan National Council \#12-01-92004HHC-a and by RFBR grant \#13-01-00779. ASN’s research is supported in part by ND EPSCoR and NSF grant \#EPS-0814442. We thank Yuri Wolf from NCBI/NLM/NIH for a profitable discussion on the biological examples of the generalized Eigen’s models.
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[^1]: Corresponding author: [email protected]
[^2]: A somewhat modified version of this text appeared in Bull Math Bio, 78(5), 991–1038, 2016, which also includes a nontechnical discussion of the main results. The same discussion can be found also at
[^3]: This is a correct formula, unfortunately, in the published version the factor $(q+n(1-q))$ is missing
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---
abstract: 'Living organisms capitalize on their ability to predict their environment to maximize their available free energy, and invest this energy in turn to create new complex structures. Is there a preferred method by which this manipulation of structure should be done? Our intuition is “simpler is better,” but this is only a guiding principal. Here, we substantiate this claim through thermodynamic reasoning. We present a new framework for the manipulation of patterns—structured sequences of data—by predictive devices. We identify the dissipative costs and how they can be minimized by the choice of memory in these predictive devices. For pattern generation, we see that simpler is indeed better. However, contrary to intuition, when it comes to extracting work from a pattern, any device capable of making statistically accurate predictions can recover all available energy.'
author:
- 'Andrew J. P. Garner'
- Jayne Thompson
- Vlatko Vedral
- Mile Gu
title: 'When is simpler thermodynamically better?'
---
[**[Simpler is better—]{}**]{}an idea that has resonated with culture throughout history. Amongst many examples, the value of simplicity has been recognized in the minimalist movement in 1960s art and music, [*shibui*]{} in Japanese ceramics, productivism in post-revolutionary Russian, and the guiding principles of twenty-first century Apple product design [@Richmond12]. Our predilection for simplicity even permeates natural philosophy, as most-famously formalized by William of Ockham’s razor. This notion, “Everything should be made as simple as possible but no simpler,” [@NoSimpler] suggests that satisfying explanations for reality must avoid unnecessary complications.
The virtue of simplicity in art is subjective, but in science this can be quantified, provided that we explain what is meant both by “simpler” and by “better”. In thermodynamics, a subject whose original purpose relates to the design of steam-engines, “better” suggests efficiency and thrift (e.g. more power for less coal). This carries through to modern thermodynamics: the best approach for a given task minimizes the expenditure of a limited resource [@BrandaoHORS13] (e.g. work or tricky-to-create states).
Meanwhile in computational mechanics [@CrutchfieldY89; @ShaliziC01], we can formalize “simpler” in the context of pattern generation and manipulation. Everything we observe in the environment can be considered to be pattern—a temporal sequence of data that exhibits statistical structure. Much of science then deals with the construction of machines that take information from past observations, and use it to generate future statistics. Here, the [*simplest*]{} machine stores this information about the past using the minimal amount of internal structure.
![ \[fig:SimExtCycle\] [[**Cycle of pattern generation and extraction.**]{}]{} A tape moves from left to right. A generator expends work to write a pattern to the tape. The extractor then uses the pattern on this tape to extract work. To run cyclically, each device maintains [*prescient*]{} memory that keeps track of the pattern. In this article, we identify the dissipative work costs. We find that simpler is thermodynamically better [*only*]{} when it comes to generating patterns. ](SimExtCycle){width="42.50000%"}
To answer “When is simpler thermodynamically better?” we present a thermodynamic framework for pattern manipulation. We evaluate the steady-state operation of a full thermal cycle, consisting of a [*generator*]{} that work to accurately produce a pattern and an [*extractor*]{} that anticipates the pattern to exploit it for free energy (see \[fig:SimExtCycle\]). We examine the work exchange in these devices, and how they use internal memory to retain their predictive ability. We then identify dissipative work costs that worsen their thermodynamic performance. In this context, a [*simpler*]{} device uses less internal memory, and a [*better*]{} one wastes less work maintaining this memory.
Ockham’s razor is a guiding philosophical principle, rather than a cast-in-stone physical law; and so there is no a priori physical motivation to assume that using the smallest, simplest, memory is always thermodynamically better. By minimizing the dissipative work costs, we identify the best choice of internal memory. In the generator, using the simplest possible internal memory (the causal states associated with the pattern’s [*statistical complexity*]{} [@CrutchfieldY89; @ShaliziC01]) minimizes dissipation. For generating a pattern: simpler is thermodynamically better. For the extractor, defying our intuition, it transpires that the choice of memory has no thermodynamic consequence.
[**[Patterns are a resource.]{}**]{} Knowing a system’s internal state has thermodynamic consequence: this knowledge can be used to perform work (drive a mechanical task), as illustrated by the Szilárd engine thought experiment [@LeffR02]. A box has a single particle inside, on the left- or right-hand-side. A movable barrier inserted in the center acts as a piston that expands as the particle pushes against it. If an agent knows which side of the barrier the particle is on, she can couple the barrier (e.g. via a pulley) to raise a weight. As the piston expands it generates an amount of work ${k_\mathrm{B}}T \ln 2$ (at temperature $T$) by drawing in the same amount of heat from its surroundings.
Knowledge about patterns may also be exploited. Suppose an agent attempts to extract work from a series of Szilárd engines (labelled sequentially by index $t$), prepared such that the particle in engine $t$ is on the same side of the barrier as in engine $t\!-\!1$ with probability $p\neq\frac{1}{2}$. An agent unware of this pattern would only be able to correctly predict the particle’s location half of the time, and hence will extract less work than an agent who knows the pattern and couples her pulley accordingly. As such, the ability to predict grants thermodynamic advantage.
This is a manifestation of Maxwell’s dæmon—an apparently paradoxical conversion of heat into work that is only resolved by accepting that information is physical and hence subject to the laws of thermodynamics [@Bennett82; @Landauer96; @LeffR02; @ParrondoHS15]. For the single Szilárd engine, we must also account for the cost of resetting the agent’s memory about the particle’s location—this knowledge must be thought of as a resource. Likewise, since it is more thermodynamically useful that the sequence follows a pattern than be uncorrelated, the pattern itself must also be considered as a resource. Producing a pattern hence requires an investment of work. Moreover, any physical device that generates (or exploits) a pattern contains some memory about what has happened in the pattern so far, in order to accurately generate (or anticipate) upcoming parts of the pattern. Any thermodynamic costs of maintaining this internal memory must also be accounted for.
The quantitative link between information, entropy and heat dissipation is given by Landauer’s principle [@LeffR02]: the minimum work cost of any information-processing task is proportional to the change in information entropy[^1] (just like macroscopic thermodynamics, where the work required to slowly change between two states of the same internal energy is proportional to the change in thermodynamic entropy).
[**[The mathematics of patterns.]{}**]{} We approach this mathematically using the language of statistical complexity [@CrutchfieldY89; @ShaliziC01]. The outputs of a physical process are represented using a bi-infinite sequence of random variables $\ldots X^{t-1} X^t X^{t+1} \ldots$, where $X^t$ describes the random variable at time $t$. (In our example above, $X^t$ is whether engine $t$ has a particle on the left or the right). These are partitioned such that variables up to $X^t$ are in the past ${\cev{X}}^t$, and from $X^{t+1}$ onwards are in the future ${\vec{X}}^t$. A particular [*sequence*]{} is specified by the values that each random variable ultimately takes ($X^1\!=\!x_A$, $X^2\!=\!x_B$ etc.). A statistical description of the [*pattern*]{} is given by the probability distribution over these sequences ${\mathrm{P}\small({\cev{X}}^t,{\vec{X}}^t\small)}$.
When the outputs follow a pattern, the past and future are correlated. If a particular sequence ${\cev{x}}$ is observed in the past, the statistics of the future are given by ${\mathrm{P}\small({\vec{X}}^t \,|\, {\cev{X}}^t={\cev{x}}\small)}$. Information theory quantifies how useful this past knowledge is for predicting the future: the difference in the entropy with and without knowledge of the past, ${{H}\small({\vec{X}}^t\small)}-{{H}\small({\vec{X}}^t \,|\, {\cev{X}}^t\small)}$. This value is exactly the [*mutual information*]{} between the past and future ${{I}\small({\cev{X}}^t \,; {\vec{X}}^t\small)}$ (also known as the [*excess entropy*]{} $E$).
We make the simplifying assumption that the pattern is a [*stationary process*]{} [@ShaliziC01]– the statistics ${\mathrm{P}\small({\cev{X}}^t,{\vec{X}}^t\small)}$ are invariant under time translation. This does not mean that every output $x^t$ in the sequence is identical, or that the pattern is Markovian, but rather that the statistics of $X^{t+1}$ onwards, given a past sequence, have no explicit time dependence: ${\mathrm{P}\small({\vec{X}}^t \,|\, {\cev{X}}^t={\cev{x}}\small)} = {\mathrm{P}\small({\vec{X}}^{t'} \,|\, {\cev{X}}^{t'}={\cev{x}}\small)}$ for all $t$, $t'$ and ${\cev{x}}$. Hence, we can often omit the superscript $t$.
The notion of an agent is formalised by constructing a [*prescient device*]{}: a machine whose knowledge of future statistics is as good as is causally possible (i.e. without knowledge about the outcomes of random events in the future). It would be cumbersome to store the entire infinite past string in this device’s memory, since this requires an unbounded amount of storage. Instead, one uses a (many-to-one) function to map multiple past sequences to the device’s internal memory $R$, which will itself be a random variable over a finite alphabet. Two different pasts are only mapped to the same internal memory state if the future statistics conditional on them are the same. For the purpose of predicting future statistics, knowing $R$ is as useful as knowing the entire past and ${\mathrm{P}\small({\vec{X}} \,|\, R\small)} = {\mathrm{P}\small({\vec{X}} \,|\, {\cev{X}}\small)}$. Such internal states $R$ are known as [*prescient states*]{}.
Storing these states requires an amount of memory given by the entropy ${{H}\small(R\small)}$ over their probability distribution. We define states that use less memory, and hence have lower entropy, as [*simpler*]{}. The simplest prescient states requiring the least memory are known as [*causal states*]{} (written $S$). These correspond to the equivalence classes of the pasts that predict the same future statistics. As any given pattern has a unique set of causal states, the memory ${{H}\small(S\small)}$ needed to store these states quantifies how difficult it is to describe that pattern, and so is a [*statistical measure of complexity*]{} [@ShaliziC01]. Unlike Kolmogorov complexity [@Kolmogorov63] which concerns the generation of a specific string, the statistical complexity avoids classifying structureless but highly-random processes as complicated. Statistical complexity has been applied to analysing structure in diverse contexts, such as neural networks [@HaslingerKS10], financial markets [@ParkWLYJM07] and
The intuition bestowed upon us by Ockham’s razor suggests that in any realization of a predictive device, we should use these simplest causal states. In the remainder of this article, we shall see where this intuition holds, and substantiate the penalties incurred by deviation from it.
[**[A new framework for pattern manipulation.]{}**]{} The classical expressions of thermodynamic laws (e.g. Kelvin’s statement that a device cannot convert heat into work with no other effect [@Thomson51]) concern cyclic behaviour— processes that leave the system in a state allowing for repetition with the same thermodynamic consequence. Without a full cycle in mind, there is the danger that the thermal benefit of a process may come at the expense of consuming an unaccounted-for resource. A cycle does not require the [*microstate*]{} of the system to return to its original value. Consider a piston of gas expanding and compressing: it does not matter if the individual molecules have moved to new locations by the end of the cycle, as long as the important thermodynamic variables—the pressure and volume—return to their original values.
In the same way, we present a cycle of generating and consuming a pattern (see \[fig:SimExtCycle\]). Visualise the pattern as a series of symbols on a tape (each position labelled by $t$). These symbols are initially distributed according to uncorrelated high-entropy[^2] random variables $X_{{\rm dflt}}\ldots X_{{\rm dflt}}$. The tape is acted on by two machines, which are in contact with a thermal reservoir (i.e. heat bath at inverse temperature $\beta=\frac{1}{{k_\mathrm{B}}T}$) and a battery for storing free energy (e.g. a raising weight). The first machine, a [*generator*]{}, writes $k$ steps of the pattern onto the tape with an associated work cost. The second machine, an [*extractor*]{}, resets $k$ symbols on the patterned tape back to the uncorrelated default states, outputting work as it does so. An [*instantaneous*]{} device with $k=1$ acts on one symbol at a time; whereas a machine with a larger [*stride*]{} $k$ processes a larger section of the pattern at once.
Since the generator and extractor require accurate statistical knowledge of the future (the generator to know what it must produce next; the extractor to anticipate the upcoming inputs), they must both maintain prescient internal memory. As such they are realisations of prescient devices. We shall assume that these devices begin with their internal memories synchronized, such that the part of the pattern the generator is about to produce matches with the part of the pattern the extractor anticipates acting upon \[i.e. the machines have internal states $R^t = r$ and $\tilde{R}^t = r'$ respectively such that ${\mathrm{P}\small({\vec{X}} \,|\, R=r\small)} = {\mathrm{P}\small({\vec{X}} \,|\, \tilde{R}=r'\small)}$.\] To run continually in an cycle, the generator and extractor must remain synchronized after they have generated and extracted the same number of symbols. This ensures that their performance on the next part of the pattern remains the same.
Thus, after each device has processed $k$ symbols, the prescient state must be updated from $R^t$ to $R^{t+k}$. How this update is performed can be deterministic or otherwise. The behaviour is [*deterministic*]{} if given an initial memory state $R^t$ and the next $k$ symbols of the pattern $X^{t+1}\ldots X^{t+k}$, the final state $R^{t+k}$ is completely defined (i.e. the randomness is entirely confined to the outputs $X^{t+1}\ldots X^{t+k}$). The alternative is that there is some additional randomness in the final value of $R^{t+k}$. However, this randomness can only be between choices of $R^{t+k}$ that predict exactly the same future statistics of $X$, as otherwise the devices will cease to be prescient (and may desynchronize).
![ \[fig:UpdateTape\] [[**Writing a pattern to a tape.**]{}]{} The various choices of symbols on a tape can each be associated with a different energy level of a system (drawn as a black horizontal lines whose relative height indicates relative energy). The statistical state of the symbol is a probability distribution (drawn as grey bars) over these configurations. By changing the Hamiltonian of the tape whilst remaining in contact with a thermal reservoir, the statistics can be altered to ${\mathrm{P}\small(X^{t+1} \,|\, S^t\small)}$. At this point, the system is isolated from the heat reservoir and the Hamiltonian is adiabatically removed. The whole procedure requires an investment of work proportional to the reduction in the state’s entropy. ](HamilUpdate){width="46.00000%"}
[**[Investing work to generate a pattern.]{}**]{} For a given pattern, there is a family of generators, characterized by the prescient states $R^t$ used as internal memory, and by the the number of steps $k$ of the pattern that are generated at once. The work cost of generating a pattern consists of two contributions: one from changing the entropy of the tape, and one from updating the memory.
From Landauer’s principle, the minimal work investment required to write $k$ symbols of the pattern onto the tape (at inverse temperature $\beta = \frac{1}{{k_\mathrm{B}}T}$) is given by $$\label{eq:TapeCost}
\beta \, W_{\rm tape}^k = k \left[ {{H}\small(X_{\rm dflt}\small)} - {{H}\small(X^{t+1} \,|\, S^{t}\small)} \right],$$ \[See \[fig:UpdateTape\], and [*Appendix*]{}.\] This value relates to change in entropy of the tape itself, and so is an intrinsic property of the pattern rather than of the machine generating it. As such it can be expressed in terms of the causal states $S$ (which are unique for any given pattern [@ShaliziC01]) rather than the arbitrary device-dependent internal states $R$.
![ \[fig:UpdateState\] [[**Updating the generator’s memory.**]{}]{} (The deterministic case is shown.) A blank ancilla state $R_{\rm dflt}$ is [*updated*]{} at no cost to $R^{t+k}$, conditioned on the values of the initial internal state $R^{t}$ and outputs $X^{t+1}\ldots X^{t+k}$. The old state $R^{t}$ is then [*decorrelated*]{} from $R^{t+1}$ and $X^{t+1}\ldots X^{t+k}$, and the mutual information used to reduce the entropy of the internal state (transforming it into $\tilde{R}$). Finally $\tilde{R}$ is [*reset*]{} back to the blank ancilla state at work cost ${{H}\small(R^{t} \,|\, X^{t+1}\ldots X^{t+k} R^{t+k}\small)}$, so that the generator’s internal state is ready to produce the next part of the pattern. ](CausalStateUpdateK){width="44.00000%"}
The second contribution, the cost of updating the internal memory into $R^{t+k}$ (so that the generator is ready to produce the next part of the pattern) is given by $$\begin{aligned}
\beta W^k_{\rm diss} & = && {{H}\small(R^{t} \,|\, X^{t+1}\ldots X^{t+k} R^{t+k}\small)} \nonumber \\
& && - {{H}\small(R^{t+k} \,|\, R^t X^{t+1}\ldots X^{t+k}\small)}. \label{eq:cEntExplicit}\end{aligned}$$ \[See \[fig:UpdateState\] and [*Appendix*]{}.\] The first term is the cost of erasing the previous state $R^t$, offset by the mutual information that is contained within the new state of the memory $R^{t+k}$ and the patterned outputs $X^{t+1}\ldots X^{t+k}$. The second term (which is zero for deterministic updates) reflects the fact that if the update is indeterministic, according to Landauer’s principle we can recover a portion of the work cost associated with the memory’s change in entropy. However, by expanding the first term (details in [*Appendix*]{}), we arrive at $$\begin{aligned}
\beta W^k_{\rm diss}
& = && \hspace{-2em} {{H}\small(X^{t+1}\!\ldots\!X^{t+k} \,|\, R^t\small)} \nonumber \\
& && \hspace{-2em} - {{H}\small(X^{t+1}\!\ldots\!X^{t+k} \,|\, R^{t+k}\small)}, \label{eq:cEntNost}\end{aligned}$$ where the indeterministic term has been eliminated. Whatever work might have been gained by introducing randomness into $R^{t+k}$ is entirely cancelled out by the cost of resetting this randomness in the previous state $R^t$. Hence determinism does not in fact play a role in deciding the thermodynamic advantage of prescient memory.
Rather, we see that the dissipative cost is proportional to the difference between $R^t$’s predictive power to guess the next $k$ symbols, and $R^{t+k}$’s [*retrodictive*]{} power to guess the preceding $k$ symbols. Failure to predict increases the first entropy, and hence the amount of dissipation. On the other hand, failure to retrodict increases the second entropy and lessens the total dissipation.
What choice of prescient memory $R$ minimizes this cost? The first term ${{H}\small(X^{t+1}\!\ldots\!X^{t+k} \,|\, R^t\small)}$ is the same for [*all*]{} choices of prescient states, since they all must reduce the entropy of the future statistics by the largest possible amount. The second term ${{H}\small(X^{t+1}\!\ldots\!X^{t+k} \,|\, R^{t+k}\small)}$ takes its largest value when $R^{t+k}$ contains the least information about the preceding $k$ symbols. The causal state $S^{t+k}$ contain the least information about the past required to predict the future, so all alternative prescient states must contain this amount or more. As such, ${{H}\small(X^{t+1}\!\ldots\!X^{t+k} \,|\, R^{t+k}\small)}\!\leq\!{{H}\small(X^{t+1}\!\ldots\!X^{t+k} \,|\, S^{t+k}\small)}$. Out of all the possible prescient states $R$ to use as our internal memory, the simplest ones—the causal states $S$—hence dissipate the least heat. We thus conclude that for generating a pattern, [*simpler is better*]{}.
[**[Extracting work from a pattern.]{}**]{} Let us now evaluate how much work we can extract from a pattern, by considering the prescient [*extractor*]{}, This device is a sophisticated Szilárd engine [@LeffR02] that maintains an up-to-date internal memory (the prescient state $R^t$) allowing it to anticipate the upcoming $k$ symbols in the pattern. Work is extracted by converting $X^{t+1}\ldots X^{t+k}$ into uncorrelated output states $X_{\rm dflt} \ldots X_{\rm dflt}$.
In order to fully account for all the changes in entropy in our system, again we must consider both the tape on which the pattern is written and the internal memory of the extractor. We do this in a single step; the total average work released by taking $k$ symbols to the default state $X_{\rm dflt}\ldots X_{\rm dflt}$, and updating the internal memory from $R^t$ to $R^{t+k}$is given by $$\label{eq:TapeFreeEnergy}
\beta W^k_{\rm out} = k \left[ {{H}\small(X_{\rm dflt}\small)} - {{H}\small(X^{t+1} \,|\, S^t\small)} \right]$$ \[Proof in [*Appendix*]{}.\]
The value here is entirely proportional to the change in entropy of the $k$ symbols on the tape, and is the maximum amount we would expect from Landauer’s principle. The curious result here is that although we have accounted for the cost of updating, there is absolutely no dependence on the choice of internal memory $R^t$, and the scaling with the number of symbols processed $k$ is trivial. Unlike with the generator, it does not matter what sort of memory is used for extraction.
Moreover, we remark that there may always be dissipation in generation since even causal states can contain superfluous information about the past for some patterns . However, this is not the case for extraction, even though we would expect larger $R^t$ to perform worse since there should be more about the past to “clean up” when updating the internal state.
This apparent paradox may be explained by carefully considering the difference between the generator and the extractor. Crucially, the output pattern must be undisturbed at the end of the generation procedure but not for extraction. Thus, the difference is between a device that moves information, and a device that copies information. Moving information is a logically reversible process, whereas copying information is not—and it is [*logical irreversibility*]{} that lies behind dissipative costs in computation [@Bennett82]. At the end of the extraction, there is only one copy of this information (encoded in $R^{t+k}$), whereas in generation, the information is present both in the internal memory and in the tape. Thus the extractor can move the information from $X^{t+1}\ldots X^{t+k}$ into $R^{t+k}$, but the generator must copy it. If more information is stored in $R^t$, then more has to be copied. This subtle, but important, distinction reveals to us why updating the memory must dissipate heat in pattern generation, but not for pattern extraction.
[**[Relation to existing results.]{}**]{} We first observe that if we consider only updating the generator’s memory, and restrict ourselves to causal states and the limit $k\!\to\!\infty$, then the dissipative work cost of updating the generator’s memory \[\[eq:cEntNost\]\] converges on ${{I}\small({\cev{X}} \,; {\vec{X}}\small)} - {{H}\small(S\small)}$, recovering the result of Wiesner et al. [@WiesnerGRV12].
Secondly, we note that the dissipative cost \[\[eq:cEntNost\]\] may be re-expressed in terms of mutual information: $$\begin{aligned}
\beta W^k_{\rm diss} & = & \hspace{-0.5em} {{I}\small(X^{t+1}\!\ldots\!X^{t+k} \,; R^{t+k}\small)} - {{I}\small(X^{t+1}\!\ldots\!X^{t+k} \,; R^t\small)}. \label{eq:InfoNost}\end{aligned}$$ In the opposite, instantaneous, case $k=1$, this recovers a mathematical quantity similar to that introduced by Still et al. [@StillSBC12] as the [*useless instantaneous nostalgia*]{}. That this coincides with our result in certain limits, despite very different derivations, demonstrates the universality of the information-theoretic concepts underlying the thermal costs of prediction. In both cases, the need to clean up unnecessary information about a process’s past has been identified as a source of dissipation.
Our work extends this by examining the thermodynamics of a full cycle of pattern generation and extraction (see \[fig:SimExtCycle\]). By doing so, we have identified that penalty for storing more than is necessary depends very much on how the prediction is used. Any prescient extractor can recover the entire free energy of the pattern—such a device has no concept of useless nostalgia, and no dissipation. It is not unnecessary storage in and of itself that induces the dissipative cost, but rather logical irreversibility. Only when the pattern manipulation is irreversible (such as in our generator, or the system in [@StillSBC12]) does storing too much information incur a thermal penalty.
[**[The thermodynamics of patterns.]{}**]{} The two devices we have presented in this article are building blocks that can be combined into complex pattern manipulators. The simplest example is the cycle of generation followed by extraction at the same temperature (\[fig:SimExtCycle\]). This configuration could be considered as charging a battery that stores energy in the form of a pattern, to be later released by the extractor. Since the contributions from writing and consuming the pattern cancel out, the net cost is from the work dissipation whilst updating the generator’s memory. Using the simplest internal memory ensures that the least work is wasted.
An alternative configuration is for a generator and extractor to act in parallel on different patterns. Here, the extractor consumes one pattern, and uses the energy released to power a generator that writes a different pattern. This approximates the action of all living organisms: for example, a lion metabolizes the structure of an antelope (destroying it in the process), and uses the energy released to build more lion. Using the simplest internal memory for generation grants the advantage that less antelope needs to be consumed in order to produce the same amount of lion.
[**[Outlook.]{}**]{} In this article, we have treated the thermodynamics of patterns, and identified the dissipative work costs in a cycle of producing and consuming a pattern. For generation, simpler is thermodynamically better; the cost is minimized when using the simplest internal states possible: the causal states associated with the pattern’s statistical complexity. On the other hand, for extraction we found that any prescient device has the ability to recover the entire free energy of the pattern, because the internal memory could be updated in a logically reversible way. The answer to “When is simpler thermodynamically better?” is that simpler is better when it helps us avoid logical irreversibility.
Our discussion has thus far been restricted to the average behaviour of classical systems operating in the quasistatic regime. By taking the quasistatic limit, we do not account for additional dissipation that occurs when writing to the tape in finite time. The additional costs of this are highly implementation-dependent: typically, faster processes dissipate more heat. By considering only the average behaviour, we have avoided addressing the fluctuations in the work costs. Fluctuations are intrinsically related to dissipation and dissipative work [@Crooks99; @GomezMarinPvB08; @ParrondoHS15], but it is not yet clear what the associated fluctuation relation is for the dissipative work discussed here.
Recent research indicates that by using quantum states, less memory is required than the best classical alternative for the same predictive power [@GuWRV12]. In a random process, there is a limit to how distinct the futures may be: two different pasts may in some (but not all) cases lead to exactly the same future. There is hence no need to perfectly distinguish between the two internal states, but this is unavoidable when using classical memory. Quantum memory does not require this, and hence has a lower entropy for the same level of prescience as the classical case. It is therefore feasible that by storing less unnecessary information about the past, quantum machines could be simpler devices and hence dissipate less heat. This would suggest that there are certain pattern-processing tasks where the quantumness of a device could yield thermodynamic advantage. Could quantum pattern generators, through their increased simplicity, be better than the classical alternatives?
[**[Acknowledgements.]{}**]{} We are grateful for funding from the John Templeton Foundation Grant 53914 [*“Occam’s Quantum Mechanical Razor: Can Quantum theory admit the Simplest Understanding of Reality?”*]{}; the National Research Foundation; the Ministry of Education in Singapore, the Academic Research Fund Tier 3 MOE2012-T3-1-009; the National Basic Research Program of China Grants 2011CBA00300 and 2011CBA00302; the National Natural Science Foundation of China Grants 11450110058, 61033001 and 61361136003; the 1000 Talents Program of China; and the Oxford Martin School. We thank , , and for illuminating .
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APPENDIX {#app:Costs .unnumbered}
========
[**[Stationarity, synchronization and determinism.]{}**]{} If a process is [*stationary*]{}, it follows that ${\mathrm{P}\small(X^{t_1}\small)} = {\mathrm{P}\small(X^{t_2}\small)}$ for all $t_1$, $t_2$ and hence ${{H}\small(X^{t_1}\small)} = {{H}\small(X^{t_2}\small)}$ for all $t_1$, $t_2$. Any prescient states derived from this sequence then also satisfy ${{H}\small(R^{t_1}\small)} = {{H}\small(R^{t_2}\small)}$ for all $t_1$, $t_2$: the memory required to store the current position in the pattern does not change with time.
If we can [*synchronize*]{} our memory, it means that we can determine which prescient state we are in from a sufficiently long observation of the past (at least as many observations as the [*crypticity*]{} of the pattern [@MahoneyEC09]). Expressed in terms of conditional entropy, ${{H}\small(R^t \,|\, {\cev{X}}\small)} = 0$, which reflects no uncertainty in the state $R^t$ if the entire past is known.
Closely related is the concept of [*determinism*]{}, which is when the state of the internal memory at the end of any action is completely fixed by the initial state of the memory and the symbols observed in the pattern. Entropically expressed, ${{H}\small(R^{t+k} \,|\, R^t X^{t+1} \ldots X^{t+k}\small)} = 0$. Once a process has been synchronized to deterministic memory, it will remain synchronized. Causal states are automatically deterministic [@ShaliziC01].
[**[Work cost of writing a pattern to a tape.]{}**]{} Let us specifically examine the cost of writing $k$ parts of some pattern to a tape using a generator with prescient states $R$. Initially, all $k$ positions on the tape will be in their default uncorrelated state ${X_{\rm dflt}}\ldots {X_{\rm dflt}}$ with associated entropy $k {{H}\small(X_{\rm dflt}\small)}$. For an outside observer who is not aware of the particular choice of symbols written by the machine (but who knew the machine’s initial internal state) after the pattern has been written, the entropy of the tape will have changed to ${{H}\small(X^{t+1} \ldots X^{t+k} \,|\, R^t\small)}$. This value can be expanded to ${{H}\small(X^{t+1} \,|\, R^t\small)} + {{H}\small(X^{t+2} \,|\, R^t X^{t+1}\small)} + \ldots + {{H}\small(X^{t+k} \,|\, R^t X^{t+1} \ldots X^{t+k-1}\small)}$.
As $R$ is prescient, $R^{t+j}$ is exactly as useful as $R^t X^{t+1} \ldots X^{t+j}$ for predicting the future of $X$. Hence, the entropy of the tape can be re-written as ${{H}\small(X^{t+1} \,|\, R^t\small)} + {{H}\small(X^{t+2} \,|\, R^{t+1}\small)} + \ldots + {{H}\small(X^{t+k} \,|\, R^{t+k-1}\small)}$. Using stationarity, the entropy of the tape can then be rewritten as $k {{H}\small(X^{t+1} \,|\, R^t\small)}$. Finally, we note that all prescient states are as good as each other for predicting the future, and so we can replace ${{H}\small(X^{t+1} \,|\, R^t\small)}$ with the equivalent entropy conditioned on causal states ${{H}\small(X^{t+1} \,|\, S^t\small)}$. The work cost of writing this pattern is hence proportional to this change in entropy: $$\beta \, W_{\rm tape}^k = k \left( {{H}\small(X_{\rm dflt}\small)} - {{H}\small(X^{t+1} \,|\, S^{t}\small)} \right).$$
[**[Work cost of updating the generator’s memory.]{}**]{} The above generator must update its internal state from $R^t$ to $R^{t+k}$ before it can produce the next part of the pattern. As with the extractor, we admit an ancilla system of the same size as the internal memory $R$, which starts and ends in the pure state $R_{\rm dflt}$. (See \[fig:UpdateState\].) For deterministic memory ${{H}\small(R^{t+k} \,|\, R^t X^{t+1} \ldots X^{t+k}\small)} = 0$ and the ancilla can be set to state $R^{t+k}$ by a reversible operation with no associated work exchange. For indeterministic memory, the ancilla is slightly randomised and its entropy increases by ${{H}\small(R^{t+k} \,|\, R^t X^{t+1} \ldots X^{t+k}\small)}$ (as with the extractor). Let us consider the cost of resetting the previous internal state $R^t$ back to $R_{\rm dflt}$. Again, this does not cost the full entropy ${{H}\small(R^t\small)}$, since there may be some mutual information ${{I}\small(R^{t} \,; R^{t+k} X^{t+1} \ldots X^{t+k}\small)}$ between the old state, and the new state and tape. \[Constructively one could imagine this mutual information as being used to reduce the entropy of $R^t$ without work cost, transforming $R^t$ into a new state $\tilde{R^t}$ with entropy ${{H}\small(R^{t} \,|\, R^{t+k}, X^{t+1} \ldots X^{t+k}\small)}$.\] Remarking that, as before, the increased entropy of $R^t$ due to indeterminism is completely negated by the work that could have been extracted by randomizing the ancilla in the first place, we see that the work cost of resetting $R^t$ to $R_{\rm dflt}$ is $$\beta W_{\rm diss}^k = {{H}\small(R^{t} \,|\, X^{t+1} \ldots X^{t+k} R^{t+k}\small)}.$$
We first remark that in the limit of $k\to\infty$, the information ${{I}\small(R^{t} \,; X^{t+1} \ldots X^{t+k} R^{t+k} \small)} \to E$, the excess entropy, because $R$ is prescient and so contains all the information that could be known about the future. Hence the work dissipated converges on ${{H}\small(R^t\small)} - E$ in the limit of large $k$ (in agreement with [@WiesnerGRV12] for causal states).
Secondly, by expanding the joint entropy ${{H}\small(R^t X^{t+1}\ldots X^{t+k} R^{t+k}\small)}$ in two different ways, we can rewrite the dissipative entropy as
$$\begin{aligned}
{{H}\small(R^t \,|\, X^{t+1}\ldots X^{t+k} R^{t+k}\small)} & = & {{H}\small(R^t X^{t+1}\ldots X^{t+k} R^{t+k}\small)} - {{H}\small(R^{t+k}\small)} - {{H}\small(X^{t+1}\ldots X^{t+k} \,|\, R^{t+k}\small)} \nonumber \\
& = & {{H}\small(R^t\small)} + {{H}\small(X^{t+1}\ldots X^{t+k} \,|\, R^t\small)} + {{H}\small(R^{t+k} \,|\, R^t X^{t+1}\ldots X^{t+k}\small)} \nonumber \\
& & \hspace{1em} - {{H}\small(R^{t+k}\small)} - {{H}\small(X^{t+1}\ldots X^{t+k} \,|\, R^{t+k}\small)} \nonumber \\
& = & {{H}\small(X^{t+1}\ldots X^{t+k} \,|\, R^t\small)} - {{H}\small(X^{t+1}\ldots X^{t+k} \,|\, R^{t+k}\small)} \label{eq:app:cEntNost} \\
& = & {{I}\small(X^{t+1}\ldots X^{t+k} \,; R^{t+k}\small)} - {{I}\small(X^{t+1}\ldots X^{t+k} \,; R^t\small)} \label{eq:app:InfoNost} .\end{aligned}$$
\[We have used ${{H}\small(R^t\small)}\!=\!{{H}\small(R^{t+k}\small)}$ from stationarity; and since the term ${{H}\small(R^{t+k} \,|\, R^t X^{t+1}\ldots X^{t+k}\small)}$ arising from indeterminism can be perfectly cancelled out, we have omitted it from the expansion.\]
[**[Work extractable from a pattern.]{}**]{} Consider an extractor whose internal memory begins in state $R^t$. For ease of calculation, we admit an ancillary variable of the same size as the internal memory, that is initially in a pure state $R_{\rm dflt}$. If the device is deterministic, upon the input of $k$ values on the tape, the joint system can be put into the state $R^t X^{t+1} \ldots X^{t+k} R^{t+k}$ reversibly and hence without any work cost. If the device is not-deterministic, but still prescient for $X$, then the final $R^{t+k}$ has some additional randomness, constrainted by the fact $R^t$ is a fine-graining of the causal state. As such, the entropy of the ancilla is increased to ${{H}\small(R^{t+k} \,|\, S^{t+k}\small)}$. We can rewrite this value as ${{H}\small(R^{t+k} \,|\, R^t X^{t+1} \ldots X^{t+k}\small)}$, since the causal state is itself determined by $R^t X^{t+1} \ldots X^{t+k}$, and no other parts of the past are useful for determining $R^{t+k}$.
We can now evaluate the entropy of the previous internal state and the tape, noting its mutual information with the new internal state: ${{I}\small(R^t X^{t+1} \ldots X^{t+k} \,; R^{t+k}\small)} = {{H}\small(R^{t+k}\small)} - {{H}\small(R^{t+k} \,|\, R^t X^{t+1} \ldots X^{t+k}\small)}$. $$\begin{aligned}
{{H}\small(R^t X^{t+1} \ldots X^{t+k} \,|\, R^{t+k}\small)} & && \nonumber \\
& \hspace{-10em} = && \hspace{-9.5em} {{H}\small(R^t X^{t+1} \ldots X^{t+k}\small)} - {{I}\small(R^t X^{t+1} \ldots X^{t+k} \,; R^{t+k}\small)} \nonumber\\
& \hspace{-10em} = && \hspace{-9.5em} {{H}\small(R^t\small)} + {{H}\small(X^{t+1} \,|\, R^t\small)} + \ldots \nonumber \\
& \hspace{-10em} && \hspace{-9.5em} + {{H}\small(X^{t+k} \,|\, R^t X^{t+1}\ldots X^{t+k-1}\small)} \nonumber\\
& \hspace{-10em} && \hspace{-9.5em} - {{H}\small(R^{t+k}\small)} + {{H}\small(R^{t+k} \,|\, R^t X^{t+1}\ldots X^{t+k}\small)} \nonumber\\
& \hspace{-10em} = && \hspace{-9.5em} k {{H}\small(X^{t+1} \,|\, S^t\small)} + {{H}\small(R^{t+k} \,|\, R^t X^{t+1}\ldots X^{t+k}\small)}
\label{eq:ExtractorCostApp}\end{aligned}$$ We have used that for a stationary process ${{H}\small(R^{t+k}\small)} = {{H}\small(R^t\small)}$. We have also used prescience, such that conditioning the future of $X$ on $R^t X^{t+1} \ldots X^{t+j}$ is equivalent to conditioning it on the causal state $S^{t+j}$. Since the terms in $S^{t+j}$ are shifted in time with respect to each other, from stationarity they all represent the same value, and so each addend contributes the same entropy.
The extractor takes each value on the tape back to the default uncorrelated states $X_{\rm dflt}$, with entropy $k {{H}\small(X_{\rm dflt}\small)}$. The final state of the ancilla must also be pure and uncorrelated with the reset tape, such that the total final entropy of the tape-ancilla system is also $k {{H}\small(X_{\rm dflt}\small)}$. This corresponds to a change in entropy $k \left[ {{H}\small(X_{\rm dflt} - {{H}\small(X^{t+1} \,|\, S^t\small)} \small)} \right] - {{H}\small(R^{t+k} \,|\, R^t X^{t+1}\ldots X^{t+k}\small)}$. The final term, arising only when there is indeterminism, is equal in magnitude but opposite in sign to the amount of randomness we introduced to the tape from indeterminism in the first place. As such, these terms perfectly cancel out, and do not affect the final value of the work exchange. By Landauer’s principal, we hence see that the work extracted when resetting $k$ symbols on the tape and advancing the internal state from $R^t$ to $R^{t+k}$ is $$\beta W^k_{\rm out} = k \left[ {{H}\small(X^{t+1} \,|\, S^t\small)} - {{H}\small(X_{\rm dflt}\small)} \right].$$
[^1]: Classically, given by the [*Shannon entropy*]{} ${{H}\small(X\small)} = -\sum_i P(X=x_i) \log_2 P\left(X=x_i\right)$. Using a base 2 logarithm gives units of [*bits*]{}.
[^2]: The mathematical results presented will hold regardless of the particular statistics used for $X_{{\rm dflt}}$, so long as we use the same statistical states for the generator’s input and extractor’s output.
|
---
author:
- |
Bing-Long Chen and Xi-Ping Zhu\
Department of Mathematics\
Zhongshan University\
Guangzhou, P.R.China
date: 'May, 2005'
title: '**Uniqueness of the Ricci Flow on Complete Noncompact Manifolds**'
---
0.5cm
The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [@Ha1]. Later on, De Turck [@De] gave a simplified proof. In the later of 80’s, Shi [@Sh1] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.
Introduction
============
0.5cm
Let $(M^n, g_{ij})$ be a complete Riemannian (compact or noncompact) manifold. The Ricci flow $$\frac{\partial}{\partial t}g_{ij}(x,t)=-2R_{ij}(x,t), \ \ \ \ \ \
\ \ \ \text{for}\ x\in M^{n} \text{ and }\ t\geq 0, \eqno{(1.1)}$$ with $g_{ij}(x,0)=g_{ij}(x)$, is a weakly parabolic system on metrics. This evolution system was introduced by Hamilton in [@Ha1]. Now it has proved to be powerful in the research of differential geometry and lower dimensional topology (see for example Hamilton’s works [@Ha1], [@Ha2], [@Ha3], [@Ha6] and the recent works of Perelman [@P1], [@P2]). The first matter for the Ricci flow (1.1) is the short time existence and uniqueness of the solutions. When the manifold $M^{n}$ is compact, Hamilton proved in [@Ha1] that the Ricci flow (1.1) has a unique solution for a short time. So the problem has been well settled on compact manifolds. In [@De], De Turck introduced an elegant trick to give a simplified proof. Later on, Shi [@Sh1] extended the short time existence result to noncompact manifolds. More precisely, Shi [@Sh1] proved that if $(M^n, g_{ij})$ is complete noncompact with bounded curvature, then the Ricci flow (1.1) has a solution with bounded curvature on a short time interval. In this paper, we will deal with the uniqueness of the Ricci flow on complete noncompact manifolds.
The uniqueness of the Ricci flow is important in the theory of the Ricci flow with surgery (see for example [@P1], [@P2] and [@CZ1]). When we consider the Ricci flow on a compact manifold, the Ricci flow will generally develop singularities in finite time. In the theory of the Ricci flow with surgery, one eliminates the singularities by Hamilton’s geometric surgeries (cut off the high curvature part and glue back a standard cap, then run the Ricci flow again). An important question in this theory is to control the curvature of the glued cap after surgery. The uniqueness theorem of the Ricci flow insures that the solution on glued cap is sufficiently close to a (complete noncompact) standard solution, which is the evolution of capped round cylinder. Then we can apply the estimate of the standard solutions [@P2] and [@CZ1] to get the desired control on curvature. The employing of the uniqueness theorem is essential. So even if we consider the Ricci flow on compact manifolds, we still have to encounter the problem of uniqueness on noncompact manifolds.
It is well-known that the uniqueness of the solution of a parabolic system on a complete noncompact manifold does not always hold if one does not impose any growth condition of the solutions. For example, even the simplest linear heat equation on $\mathbb{R}$ with zero as initial data has a nontrivial solution which grows faster than $e^{a|x|^{2}}$ for any $a>0$ whenever $t>0$. This says, for the standard linear heat equation, the most growth rate for the uniqueness is $e^{a|x|^{2}}$. Note that in a K$\ddot{a}$hler manifold, the Ricci curvature is given by $$R_{i\bar{j}} = -\frac{\partial^2}{\partial z^i \partial
\bar{z}^j}log det (g_{k\bar{l}}).$$ Thus the reasonable growth rate that we can expect for the uniqueness of the Ricci flow is the solution with bounded curvature.
In this paper, we will prove the following uniqueness theorem of the Ricci flow.
Let $(M^n,g_{ij}(x))$ be a complete noncompact Riemannian manifold of dimension $n$ with bounded curvature. Let ${g}_{ij}(x,t)$ and $\bar{g}_{ij}(x,t)$ be two solutions to the Ricci flow on $M^n\times[0,T]$ with the same $g_{ij}(x)$ as initial data and with bounded curvatures. Then $g_{ij}(x,t)= \bar{g}_{ij}(x,t)$ for all $(x,t)\in M^{n}\times [0,T]$.
Since the Ricci flow is not a strictly parabolic system, our argument will apply the De Turck trick. This is to consider the composition of the Ricci flow with a family of diffeomorphisms generated by the harmonic map flow. By pulling back the Ricci flow by this family of diffeomorphisms, the evolution equations become strictly parabolic. In order to use the uniqueness theorem of a strict parabolic system on a noncompact manifold, we have to overcome two difficulties. The first one is to establish a short time existence for the harmonic map flow between noncompact manifolds. The second one is to get a priori estimates for the harmonic map flow so that after pulling backs, the solutions to the strictly parabolic system still satisfy suitable growth conditions. To the best of our knowledge, one can only get short time existence of harmonic map flow by imposing negative curvature or convex condition on the target manifolds (see for example, [@ES] and [@DL]) or by simply assuming the image of initial data lying in a compact domain on the target manifold (see for example [@LT]). In [@CZ1], we observed that the condition of injectivity radius bounded from below ensures certain uniform (local) convexity and showed that this is sufficient to give the short time existence and the a priori estimates for the harmonic map flow. Thus in [@CZ1], we obtained the uniqueness under an additional assumption that the initial metric has a positive lower bound on injectivity radius. The main purpose of this paper is to remove this additional assumption. Note from [@CLY] or [@CGT] that the injectivity radius of the initial manifold decays at worst exponentially. This allows us to conformally straighten the initial manifold at infinity. Our idea is to study the evolution equations coming from the composition of the Ricci flow and harmonic map flow, as well as a conformal change.
This new approach has the advantage of transforming the Ricci flow equation to a strictly parabolic system on a manifold with uniform geometry at infinity. We expect that it could also give new short time existence for the Ricci flow without assuming the boundedness of the curvature of the initial metric. As a direct consequence, we have the following result.
Suppose $(M^{n},g_{ij}(x))$ is a complete Riemannian manifold, and suppose $g_{ij}(x,t)$ is a solution to the Ricci flow with bounded curvature on $M^{n}\times[0,T]$ and with $g_{ij}(x)$ as initial data. If $G$ is the isometry group of $(M^{n},g_{ij}(x))$, then $G$ remains to be an isometric subgroup of $(M^{n},g_{ij}(x,t))$ for each $t\in[0,T]$.
This paper is organized as follows. In Section 2, we study the harmonic map flow coupled with the Ricci flow. In Section 3, we study the Ricci-De Turck flow and prove the uniqueness theorem.
We are grateful to Professor S. T. Yau for many helpful discussions and encouragement. The second author is partially supported by the IMS of The Chinese University of Hong Kong and the first author is supported by FANEDD 200216 and NSFC 10401042.
0.8cm
Harmonic map flow coupled with the Ricci flow
=============================================
0.5cm Let $(M^{n},g_{ij}(x))$ and $(N^{m},h_{ij}(y))$ be two Riemannian manifolds, $f:M^{n}\rightarrow N^{m}$ be a map. The harmonic map flow is the following evolution equation for maps from $M^n$ to $N^m$, $$\tag{2.1}
\left\{
\begin{split}
\quad \frac{\partial}{\partial t}f(x,t)&=\triangle f(x,t), \ \ \ \mbox{ for }x \in M^n, t>0, \\
f(x,0)&= f(x), \ \ \ \ \ \mbox{ for }x \in M^n ,
\end{split}
\right.$$ where $\triangle$ is defined by using the metric $g_{ij}(x)$ and $h_{\alpha\beta}(y)$ as follows $$\triangle
f^{\alpha}(x,t)=g^{ij}(x)\nabla_{i}\nabla_{j}f^{\alpha}(x,t),
\ \ \ \ \ \ \ \ \ \$$ and $$\nabla_{i}\nabla_{j}f^{\alpha}=\frac{\partial^{2}f^{\alpha}}{\partial
x^{i}\partial
x^{j}}-\Gamma^{k}_{ij}\frac{\partial f^{\alpha}}{\partial
x^{k}}+\Gamma^{\alpha}_{\beta\gamma}\frac{\partial f^{\beta}}{\partial x^{i}}
\frac{\partial f^{\gamma}}{\partial x^{j}}. \eqno(2.2)$$ Here we use $\{x^{i}\}$ and $\{y^{\alpha}\}$ to denote the local coordinates of $M^{n}$ and $N^{m}$ respectively, $\Gamma^{k}_{ij}$ and $\Gamma^{\alpha}_{\beta\gamma}$ the corresponding Christoffel symbols of $g_{ij}$ and $h_{\alpha\beta}$.
Let $g_{ij}(x,t)$ be a complete smooth solutions of the Ricci flow with $g_{ij}(x)$ as initial data, then the harmonic map flow coupled with Ricci flow is the following equation $$\left\{
\begin{split}
\quad \frac{\partial}{\partial t}f(x,t)&=\triangle_{t} f(x,t), \ \ \ \mbox{ for }x \in M^n, t>0, \\
f(x,0)&= f(x), \ \ \ \ \ \mbox{ for }x \in M^n ,
\end{split}
\right.$$ where $\triangle_{t}$ is defined as above by using the metric $g_{ij}(x,t)$ and $h_{\alpha\beta}(y)$.
Suppose $g_{ij}(x,t)$ is a solution to the Ricci flow on $
M^{n}\times [0,T]$ with bounded curvature $$|Rm|(x,t)\leq k_{0}$$ for all $(x,t)\in M^{n}\times[0,T]$. Let $(N^{n},h_{\alpha\beta})=(M^{n},g_{ij}(\cdot,T))$ be the target manifold. The purpose of this section is to prove the following theorem
There exists $0<T_0<T$, depending only on $k_{0}$, $T$ and $n$ such that the harmonic map flow coupled with the Ricci flow $$\tag{2.3}
\left\{
\begin{split}
\quad \frac{\partial}{\partial t}F(x,t)&=\triangle_{t} F(x,t), \\
F(\cdot,0)&= identity, \ \ \ \ \
\end{split}
\right.$$ has a solution on $M^{n}\times[0,T_0]$ satisfying the following estimates $$\begin{array}{lr}
|\nabla F|\leq \tilde{C}_{1},\\
|\nabla^{k} F|\leq \tilde{C}_{k} t^{-\frac{k-2}{2}},\ \ \ \ \
\mbox{for all k }\ \geq 2,
\end{array}\eqno{(2.4)}$$ for some constants $\tilde{C}_{k}$ depending only on $k_{0}$, $T$, $k$ and $n$.
The proof will occupy the rest of this section.
Expanding base and target metrics at infinity
---------------------------------------------
We will construct appropriate auxiliary functions on $M^{n}$ and $N^{n}$ and do conformal deformations for the base and the target metrics. Firstly, we construct the function on $(N^{n},h_{\alpha\beta})$. The function can be obtained by solve certain equations [@ScY] or smoothing certain functions by convolution [@GW].
Fix $p\in N^{n}$. Then for any $a\geqslant 1$, there exists a $C^{\infty}$ nonnegative function $\varphi_{a}$ on $N^{n}$ such that $$\tag{2.5}
\left\{
\begin{split}
\varphi_{a}(y)&\equiv 0 \ \ \ &&\text{on}\ \ \ \ B(p,a), \\
d(y,p)&\leqslant\varphi_{a}(y)\leqslant C_{0}d(y,p) \ \ \ &&\text{on}\ \ \ \ N^{n}\backslash B(p,2a),\\
|\nabla^{k}\varphi_{a}| &\leqslant C_{k} \ \ \ && \text{on}\ \ \ \
N^{n},\ \ \ \ \text {for}\ k\geqslant1,
\end{split}
\right.$$ where $C_{k}$, $i=0, 1, 2, \cdots,$ are constants depending only on $k_{0}$ and $T$; the distance $d(y,p)$, the covariant derivatives $\nabla^{k}\varphi_{a}$ and the norms $|\nabla^{k}\varphi_{a}|$ are computed by using the metric $h_{\alpha\beta}$.
Proof. Let $\xi$ be a smooth nonnegative increasing function on $\mathbb{R}$ such that $\xi(s)=0$ for $s\in
(-\infty,\frac{5}{4}]$, and $\xi=1$ for $s\in[\frac{7}{4},\infty)$. For each $y\in N^{n}$, by averaging the functions $\xi(\frac{d(p,y)}{a})$ and $d(p,y)$ over a suitable ball of the tangent space $T_{y}N^{n}$ (see for example [@GW]), we obtain two smooth functions $\xi_{a}$ and $\rho$. Notice that $(N^{n},h_{\alpha\beta})=(M^{n},g_{ij}(\cdot,T))$, thus all the covariant derivatives of the curvatures of $h_{\alpha\beta}$ are bounded by using Shi’s gradient estimates [@Sh1]. Then $\varphi_{a}=C\xi_{a}\rho $, for some constant $C$ depending only on $k_{0}$ and $T$, is the desired function.
$\hfill\#$
Recall from [@CLY] and [@CGT] that on a complete manifold with bounded curvature, the injectivity radius decays at worst exponentially; more precisely, there exists a constant $\tilde{C}(n)>0$ depending only on the dimension, and there exists a constant $\delta>0$ depending on $n$, $k_{0}$ and the injectivity radius at $p$ such that $$inj(N^{n},h_{\alpha\beta},y)\geqslant \delta
e^{-\tilde{C}(n)\sqrt{k_{0}}d(y,p)}. \eqno(2.6)$$
Fix $a\geqslant1$, let $\varphi^{a}=4\tilde{C}(n)\sqrt{k_{0}}\varphi_{a}$ and set $$h^{a}_{\alpha\beta}=e^{\varphi^{a}}h_{\alpha\beta}. \eqno(2.7)$$ Clearly, $h^{a}_{\alpha\beta}=h_{\alpha\beta}$ on $B(p,a)$. Note that $(N^{n},h_{\alpha\beta})=(M^{n},g_{ij}(\cdot,T))$, so the function $\varphi_{a}$ is also a function on $M^{n}$. Let $$g^{a}_{ij}(x,t)=e^{\varphi^{a}}g_{ij}(x,t) \eqno(2.8)$$ be the new family of metrics on $M^{n}$. Instead of (2.3), we will consider a new harmonic map flow $$\left\{
\begin{split}
\quad \frac{\partial}{\partial t}\overset{a}{F}(x,t)&=\overset{a}{\triangle}_{t} \overset{a}{F}(x,t), \\
\overset{a}{F}(\cdot,0)&= identity, \ \ \ \ \
\end{split}
\right.\eqno(2.3)_{a}$$ where $\overset{a}{\triangle}_{t}\overset{a}{F}$ is defined by using the metric $g^{a}_{ij}(x,t)$ and $h^{a}_{\alpha\beta}(y)$.
Before we can solve $(2.3)_{a}$, we have to discuss the geometry of the new metrics $h^{a}_{\alpha\beta}(y)$ and $g^{a}_{ij}(x,t)$. Let us first compute the curvature and its covariant derivatives and injectivity radius of $(N^{n},h^a_{\alpha\beta})$ as follows.
By a direct computation, we get $$\tag{2.9}
\begin{split}
\overset{a}{R}_{\alpha\beta\gamma\delta}=&e^{\varphi^{a}}
R_{\alpha\beta\gamma\delta}+\frac{e^{\varphi^{a}}}{4}\{|\nabla
\varphi^{a}|^{2}(h_{\alpha\delta}h_{\beta\gamma}-h_{\alpha\gamma}
h_{\beta\delta})\\&+(2\nabla_{\alpha}\nabla_{\delta}\varphi^{a}-
\nabla_{\alpha}\varphi^{a}\nabla_{\delta}\varphi^{a})h_{\beta\gamma}
+(2\nabla_{\beta}\nabla_{\gamma}\varphi^{a}-
\nabla_{\beta}\varphi^{a}\nabla_{\gamma}\varphi^{a})h_{\alpha\delta}\\&-(2\nabla_{\beta}\nabla_{\delta}\varphi^{a}-
\nabla_{\beta}\varphi^{a}\nabla_{\delta}\varphi^{a})h_{\alpha\gamma}-(2\nabla_{\alpha}\nabla_{\gamma}\varphi^{a}-
\nabla_{\alpha}\varphi^{a}\nabla_{\gamma}\varphi^{a})h_{\beta\delta}\}
\end{split}$$ where $\overset{a}{R}_{\alpha\beta\gamma\delta}$ is the curvature of $h^{a}_{\alpha\beta}$, $\nabla_{\alpha}\varphi^{a}$,$\nabla_{\alpha}\nabla_{\delta}\varphi^{a}$ and $|\nabla_{\alpha}\varphi^{a}|$ are computed by the metric $h_{\alpha\beta}$. Therefore, by combining with (2.5), we have $$\tag{2.10}
\begin{split}
|\overset{a}{R}_{m}|_{h^{a}}&\leqslant
e^{-\varphi^{a}}(k_{0}+C(n)(C_{2}+C_{1}^{2}))\\
&<\infty.
\end{split}$$
For higher derivatives, we rewrite (2.9) in a simple form $$\overset{a}{R}_{m}=e^{\varphi^{a}}\{R_{m}+\nabla\varphi^{a}\ast\nabla\varphi^{a}\ast
h^{2}\ast h^{-1}+\nabla^{2}\varphi^{a}\ast h\}$$ where we use $A\ast B$ to express some linear combinations of tensors formed by contractions of tensor product of $A$ and $B$. Note that $$\begin{split}
\overset{a}{\Gamma^{\alpha}_{\beta\gamma}}-\Gamma^{\alpha}_{\beta\gamma}
&=\frac{1}{2}[\nabla_{\beta}\varphi^{a}\delta^{\alpha}_{\gamma}+\nabla_{\gamma}\varphi^{a}
\delta^{\alpha}_{\beta}-h^{\alpha\eta}h_{\beta\gamma}\nabla_{\eta}\varphi^{a}]\\
&=(\nabla\varphi^{a}\ast h\ast h^{-1})^{\alpha}_{\beta\gamma},
\end{split}$$ so by induction, we have $$\tag{2.11}
\begin{split}
\overset{a}{\nabla^{k}}\overset{a}{R}_{m}&=\nabla\overset{a}{{\nabla}^{k-1}}
\overset{a}{R_{m}}+(\overset{a}{\Gamma}-\Gamma)\ast\overset{a}{\nabla^{k-1}}\overset{a}{R_{m}}\\
&=e^{\varphi^{a}}\{\sum^{k}_{l=0}\nabla^{l}R_{m}\ast\sum_{i_{1}+\cdots+i_{p}=k-l}\nabla^{i_{1}}\varphi^{a}\ast
\cdots\ast
\nabla^{i_{p}}\varphi^{a}+\sum_{i_{1}+\cdots+i_{p}=k+2}\nabla^{i_{1}}\varphi^{a}\ast
\cdots\ast \nabla^{i_{p}}\varphi^{a}\},
\end{split}$$ where we denote $\nabla^{0}\varphi^{a}=1$. By combining with (2.5) and gradient estimate of Shi [@Sh1], we get $$\tag{2.12}
\begin{split}
|\overset{a}{\nabla^{k}}\overset{a}{R}_{m}|_{h^{a}}&\leqslant
e^{-\frac{k+2}{2}\varphi^{a}}C(n,k_{0},k,C_{1},\cdots,C_{k+2})(\sum^{k}_{l=0}|\nabla^{l}R_{m}|+1)\\
&\leqslant C(n,k_{0},T,k)e^{-\frac{k+2}{2}\varphi^{a}}\\
&\leqslant C(n,k_{0},T,k).
\end{split}$$
For the injectivity radius of $h^{a}_{\alpha\beta}$, we know from (2.5) and (2.7) that for any $y\in N^{n}\backslash B(p,2a+1)$, $$\overset{a}{B}(y,1)\supset
B(y,e^{-2\tilde{C}(n)\sqrt{k_{0}}(\varphi_{a}+C_{1})})$$ and $$\tag{2.13}
\begin{split}
Vol_{h^{a}}(\overset{a}{B}(y,1))&=\int_{\overset{a}{B}(y,1)}(e^{4\tilde{C}(n)\sqrt{k_{0}}\varphi_{a}})^{\frac{n}{2}}
\\
&\geqslant
e^{2\tilde{C}(n)\sqrt{k_{0}}(\varphi_{a}-C_{1})}Vol_{h}(B(y,e^{-2\tilde{C}(n)\sqrt{k_{0}}(\varphi_{a}+C_{1})}))
\end{split}$$ where we denote by $\overset{a}{B}(y,1)$ the ball centered at $y$ and of radius 1 with respect to metric $h^{a}_{\alpha\beta}$, and $Vol_{h^{a}}(\overset{a}{B}(y,1))$ its volume.
Since $$\varphi_{a}(y)\geqslant d(y,p),$$ for $y\in
N^{n}\backslash B(p,2a+1)$, there holds $$e^{-2\tilde{C}(n)\sqrt{k_{0}}(\varphi_{a}+C_{1})}\leqslant \delta
e^{-\tilde{C}(n)\sqrt{k_{0}}d(y,p)} \eqno(2.14)$$ for $y\in N^{n}\backslash
B(p,2a+1+|\frac{log\delta^{-1}}{\tilde{C}(n)\sqrt{k_{0}}}|)$. By (2.6), (2.10), (2.13), (2.14) and volume comparison theorem, we have $$\tag{2.13}
\begin{split}
Vol_{h^{a}}(\overset{a}{B}(y,1))&\geqslant c(n,k_{0})
e^{2n\tilde{C}(n)\sqrt{k_{0}}(\varphi_{a}-C_{1})}(e^{-2\tilde{C}(n)\sqrt{k_{0}}(\varphi_{a}+C_{1})})^{n}
\\
&\geqslant c(n,k_{0}),
\end{split}$$ By combining this with the local injectivity radius estimate in [@CLY] or [@CGT], we get $$inj(N^{n},h^{a},y)\geqslant \tilde{C}(n,k_{0})>0, \ \ \ \mbox{for}
\ \ y\in N^{n}\backslash
B(p,2a+1+|\frac{log\delta^{-1}}{\tilde{C}(n)\sqrt{k_{0}}}|).$$ Consequently, we have proved the following lemma.
There exists a sequence of constants $\bar{C_{0}}$, $
\bar{C_{1}}$, $ \cdots$, with the following property. For all $a\geqslant 1$, there exists $i_{a}>0$, such that the metrics $h^{a}_{\alpha\beta}=e^{\varphi^{a}}h_{\alpha\beta}$ on $N^{n}$ satisfy $$\tag{2.15}
\begin{split}
|\overset{a}{\nabla^{k}}\overset{a}{R}_{m}|_{h^{a}}&\leqslant \bar{C}_{k}e^{-\frac{k+2}{2}\varphi^{a}}\leqslant \bar{C}_{k}\\
inj(N^{n},h^{a}_{\alpha\beta})&\geqslant i_{a}>0
\end{split}$$ for $k=0, 1, \cdots.$
$\hfill\#$
We next estimate the curvature and the its covariant derivatives of $g^{a}_{ij}(x,t)=e^{\varphi^{a}}g_{ij}(x,t)$.
By the Ricci flow equation, we have $$\tag{2.16}
\begin{split}
\Gamma^{l}_{ij}(\cdot,T)-\Gamma^{l}_{ij}(\cdot,t)&=\int_{t}^{T}(g^{-1}\ast\nabla
Ric)(\cdot,s)ds,\\
\nabla^{k}_{g(\cdot,T)}(\Gamma(\cdot,T)-\Gamma(\cdot,t))&=\int_{t}^{T}\sum^{k}_{l=0}\nabla^{k+1-l}Ric
\ast
\sum_{i_1+1+\cdots+i_{p}+1=l}\nabla^{i_1}_{g(\cdot,T)}(\Gamma(\cdot,s)-\Gamma(\cdot,T))\\&
\ast\cdots\ast\nabla^{i_p}_{g(\cdot,T)}(\Gamma(\cdot,s)-\Gamma(\cdot,T))\ast
g^{k}\ast g^{-(k+1)}(\cdot,s)ds.
\end{split}$$ By combining with the gradient estimates of Shi [@Sh1] and induction on $k$, we have $$\tag{2.17}
\left\{
\begin{split}
|\Gamma(\cdot,T)-\Gamma(\cdot,t)|&\leqslant
C(n,k_{0},T)\int_{t}^{T}\frac{1}{\sqrt{s}}ds,\\
|\nabla_{g(\cdot,T)}(\Gamma(\cdot,T)-\Gamma(\cdot,t))|&\leqslant
C(n,k_{0},T)(1+|log t|),\\
|\nabla^{k}_{g(\cdot,T)}(\Gamma(\cdot,T)-\Gamma(\cdot,t))|&\leqslant
C(n,k_{0},T,k)t^{-\frac{k-1}{2}}, \ \ \ \mbox{ for } k\geq2.\\
\end{split}
\right.$$ Since $$\nabla^{k}_{g(\cdot,t)}\varphi^{a}=\sum^{k-1}_{l=0}\nabla_{g(\cdot,T)}^{k-l}\varphi^{a}\ast
\sum_{i_1+1+\cdots+i_p+1=l}\nabla^{i_1}_{g(\cdot,T)}(\Gamma(\cdot,t)-\Gamma(\cdot,T))
\ast\cdots\ast\nabla^{i_p}_{g(\cdot,T)}(\Gamma(\cdot,t)-\Gamma(\cdot,T))$$ for $k\geqslant1$, the combination with (2.17) and (2.5) gives $$\tag{2.18}
\left\{
\begin{split}
&|\nabla_{g(\cdot,t)}\varphi^{a}|+|\nabla_{g(\cdot,t)}^{2}\varphi^{a}|\leqslant
C(n,k_{0},T),\\
&|\nabla_{g(\cdot,t)}^{3}\varphi^{a}|\leqslant
C(n,k_{0},T)(1+|log t|),\\
&|\nabla^{k}_{g(\cdot,t)}\varphi^{a}|\leqslant
C(n,k_{0},T,k)t^{-\frac{k-3}{2}}, \ \ \ \ \text{for}\ \ k\geqslant4.\\
\end{split}
\right.$$ Then by combining (2.11) and (2.18), the curvature and the covariant derivatives of $g^{a}(\cdot,t)$ can be estimated as follows $$\begin{split}
|\overset{a}{\nabla}^{k}\overset{a}{R}_{m}|_{g^{a}(\cdot,t)}\leqslant
C(n,k_{0},T,k) e^{-\frac{k+2}{2}\varphi^{a}}t^{-\frac{k}{2}}, \ \ \ \ \text{for}\ \ k\geqslant 0.\\
\end{split}$$ Summing up, the above estimates give the following
There exists a sequence of constants $\bar{k_{0}}$, $
\bar{k_{1}}$, $ \cdots$, with the following property. For all $a\geqslant 1$, the metrics $g^{a}_{ij}(\cdot,t)=e^{\varphi^{a}}g_{ij}(\cdot,t)$ on $M^{n}$ satisfy $$\tag{2.19}
\begin{split}
|\overset{a}{\nabla^{l}}\overset{a}{R}_{m}|_{g^{a}(\cdot,t)}
&\leqslant \bar{k}_{l}
e^{-\frac{k+2}{2}\varphi^{a}}t^{-\frac{l}{2}}, \ \ \ \text{for}\ \
\ l\geqslant 0.
\end{split}$$ on $M^{n}\times [0,T]$.
$\hfill\#$
We remark that the fact that the curvatures of $h^{a}_{\alpha\beta}$ and $g^{a}_{ij}(\cdot,t)$ are uniformly bounded (independent of $a$) is essential in our argument. While the injectivity radius bound $i_a$ may depend on $a$.
For the new family of metrics $g^{a}_{ij}(\cdot,t)$, we have the following lemma.
$$\begin{aligned}
&\frac{\partial }{\partial
t}g^{a}_{ij}=e^{\varphi^{a}}(-2\overset{a}{R}_{ij}+(\overset{a}{\nabla}^{2}\varphi^{a}+\overset{a}{\nabla}\varphi^{a}\ast
\overset{a}{\nabla}\varphi^{a})\ast
\overset{a}{g}\ast(\overset{a}{g})^{-1}),\\
&\frac{\partial}{\partial t}\overset{a}{\Gamma}^{k}_{ij}=
e^{\varphi^{a}}(\overset{a}{g})^{-1}\ast
\overset{a}{\nabla}\overset{a}{Ric}+e^{\varphi^{a}}(\overset{a}{g})^{-2}\ast\overset{a}{g}\ast
(\overset{a}{Ric}\ast \nabla
\varphi^{a}+\overset{a}{\nabla}^{3}\varphi^{a})\\&\ \
+e^{\varphi^{a}}(\overset{a}{g})^{-3}\ast(\overset{a}{g})^{2}\ast
[(\overset{a}{\nabla}\overset{a}{\varphi})^{3}+\overset{a}{\nabla}^{3}\varphi^{a}],\end{aligned}$$
$$\tag{2.20}
\begin{split}
e^{\frac{\varphi^{a}}{2}}|\overset{a}{\nabla}{\varphi}^{a}|_{g^{a}(\cdot,t)}
+e^{\varphi^{a}}|\overset{a}{\nabla}^{2}_{g^{a}(\cdot,t)}\varphi^{a}|_{g^{a}(\cdot,t)}&\leqslant
C(n,k_{0},T),\\
e^{\frac{3}{2}\varphi^{a}}|\overset{a}{\nabla}^{3}_{g^{a}(\cdot,t)}\varphi^{a}|_{g^{a}(\cdot,t)}&\leqslant
C(n,k_{0},T)(1+|\log t|),\\
e^{\frac{k}{2}\varphi^{a}}|\overset{a}{\nabla}^{k}_{g^{a}(\cdot,t)}\varphi^{a}|_{g^{a}(\cdot,t)}&\leqslant
C(n,k_{0},T,k)\frac{1}{t^{\frac{k-3}{2}}},\ \ \mbox{for}\ \
k\geqslant4.
\end{split}$$
Proof. Note that $$\begin{split}
\overset{a}{\Gamma}-\Gamma&=g\ast g^{-1}\ast \nabla\varphi^{a}\\
\overset{a}{\nabla^{2}}\varphi^{a}&=\nabla^{2}\varphi^{a}+(\overset{a}{\Gamma}-\Gamma)\ast
\nabla \varphi^{a}\\
\overset{a}{\nabla^{k}}\varphi^{a}&=\sum_{i_1+\cdots+i_p=k}
g^{k-1}\ast (g^{-1})^{k-1}\ast
\nabla^{i_1}\varphi^{a}\ast\cdots\ast\nabla^{i_p}\varphi^{a}
\end{split}$$ where the summation is taken over all indices $i_j>0$. By combining this with (2.18), we get the desired estimates for $|\overset{a}{\nabla^{k}}_{g^{a}(\cdot,t)}\varphi^{a}|_{g^{a}(\cdot,t)}$. One the other hand, since $$\overset{a}{R}_{ij}=R_{ij}+(\overset{a}{\nabla^{2}}\varphi^{a}+\nabla\varphi^{a}\ast\nabla\varphi^{a})\ast
g\ast g^{-1},$$ it follows that $$\begin{split}
\overset{a}{\nabla}_{i}\overset{a}{R}_{jl}&=\nabla_{i}R_{jl}+g^{a}\ast
{g^{a}}^{-1}\ast(\overset{a}{Ric}\ast\nabla\varphi^{a}+\overset{a}{\nabla^{3}}\varphi^{a})\\
&\ \ +{(g^{a})}^{2}\ast {(g^{a})}^{-2}\ast
(\overset{a}{\nabla^{2}}\varphi^{a}\ast\overset{a}{\nabla}\varphi^{a}+(\overset{a}{\nabla}\varphi^{a})^{3}).
\end{split}$$ By combining this with $$\begin{split}
\frac{\partial}{\partial
t}\overset{a}{\Gamma}^{k}_{ij}&=\frac{\partial}{\partial
t}{\Gamma}^{k}_{ij}+\frac{\partial}{\partial t}(g^{-1}\ast g\ast
\nabla \varphi^{a})\\
&=-g^{kl}(\nabla_{i}R_{jl}+\nabla_{j}R_{li}-\nabla_{l}R_{ij})+g\ast
g^{-2}\ast Ric \ast \nabla\varphi^{a},
\end{split}$$ we have proved the lemma.
$\hfill\#$
Modified harmonic map flow
--------------------------
The purpose of this subsection is to solve the equation $(2.3)_{a}$. More precisely, we will prove the following theorem
There exists $0<T_1<T$, depending only on $k_{0}$, $T$ and $n$ such that for all $a\geqslant1$ the modified harmonic map flow flow coupled with the Ricci flow $$\left\{
\begin{split}
\quad \frac{\partial}{\partial t}\overset{a}{F}(x,t)&=\overset{a}{\triangle}_{t} \overset{a}{F}(x,t) \\
\overset{a}{F}(\cdot,0)&= identity \ \ \ \ \
\end{split}
\right.\eqno(2.3)_{a}$$ has a solution on $M^{n}\times[0,T_0]$ satisfying the following estimates $$\begin{array}{lr}
|\overset{a}{\nabla}\overset{a}{F}|\leq C(n,k_{0},T),\\
|{\overset{a}{\nabla}}^{k}\overset{a}{ F}|\leq C(n,k_{0},T,k) t^{-\frac{k-2}{2}},\ \ \ \ \
\mbox{for all k }\ \geq 2,
\end{array}\eqno{(2.21)}$$ for some constants $C(n,k_{0},T,k)$ depending only on $n$, $k_{0}$, $T$, and $k$ but independent of $a$.
Note that $\overset{a}{F}$ is viewed as a map from $(M^{n},g^{a}_{ij}(x,t))$ and $(N^{n},h^{a}_{\alpha\beta}(y))$, all the covariant derivatives and the norms in Theorem 2.6 are computed with respect $g^{a}_{ij}(x,t)$ and $h^{a}_{\alpha\beta}(y)$. We begin with a easier short time existence of $(2.3)_{a}$ where the short time interval may depend on $a$.
### Short time existence of the modified harmonic map flows
We consider $(2.3)_{a}$ with general initial data.
Let $f$ be a smooth map from $M^{n}$ to $N^{n}$ with $$E_{0}=\sup_{x\in
M^{n}}|\overset{a}{\nabla}f|_{g^{a}_{ij}(\cdot,0),h^{a}_{\alpha\beta}}(x)+\sup_{x\in
M^{n}}|\overset{a}{\nabla^{2}}f|_{g^{a}_{ij}(\cdot,0),h^{a}_{\alpha\beta}}(x)<\infty.$$ Then there exists a $\delta_{0}>0$ such that the initial problem $$\left\{
\begin{split}
\quad \frac{\partial}{\partial t}\overset{a}{F}(x,t)&=\overset{a}{\triangle}_{t} \overset{a}{F}(x,t), \\
\overset{a}{F}(x,0)&= f(x), \ \ \ \ \
\end{split}
\right.\eqno{(2.3)_{a}}^{\prime}$$ has a smooth solution on $M^{n}\times[0,\delta_{0}]$ satisfying the following estimates $$\sup_{(x,t)\in
M^{n}\times[0,\delta_0]}|\overset{a}{\nabla}\overset{a}{F}|_{g^{a}_{ij}(\cdot,0),h^{a}_{\alpha\beta}}(x,t)+\sup_{(x,t)\in
M^{n}\times[0,\delta_0]}|\overset{a}{\nabla^{2}}\overset{a}{F}|_{g^{a}_{ij}(\cdot,0),h^{a}_{\alpha\beta}}(x,t)\leqslant
C(n,k_{0},T,a,E_{0}),$$ $$\sup_{(x,t)\in
M^{n}\times[0,\delta_0]}|\overset{a}{\nabla^{k}}\overset{a}{F}|_{g^{a}_{ij}(\cdot,0),h^{a}_{\alpha\beta}}(x,t)\leqslant
\frac{C(n,k_{0},T,k,a)}{t^{\frac{k-2}{2}}}, \eqno(2.22)$$ for $k
\geq 3.$
We will prove the theorem by solving the corresponding initial-boundary value problem on a sequence of exhausted bounded domains $D_1\subseteq D_2 \subseteq \cdots $ with smooth boundaries and $D_j\supseteq B_{g^a(\cdot,0)}^{a}(P,j+1)$ : $$\left\{
\begin{split}
\quad \frac{\partial}{\partial t}\overset{a}{F^{j}}(x,t)&=\overset{a}{\triangle}_{t} \overset{a}{F^{j}}(x,t),
\ \ \mbox{ for }x \in D_{j} \mbox{ and } t>0,\\
F^{j}(x,0)&= f(x) \ \ \ \ \ \mbox{ for }x \in D_{j} ,\\
\overset{a}{F^{j}}(x,t)&= f(x) \ \ \ \ \ \ \ \ \mbox{ for } x\in \partial
D_{j},
\end{split}
\right.\eqno(2.23)$$ and $\overset{a}{F}$ will be obtained as the limit of a convergent subsequence of $\overset{a}{F^{j}}$ as $j\rightarrow
\infty$. Here $P$ is a fixed point on $M^n$ and $B_{g^a(\cdot,0)}^{a}(P,j+1)$ is the geodesic ball centered at $P$ of radius $j+1$ with respect to the metric $g^a_{ij}(\cdot,0)$
The following lemma gives the zero-order estimate of $\overset{a}{F^{j}}$.
There exist positive constants $0<T_2<T$ and $C>0$ such that for any $j$, if (2.23) has a smooth solution $\overset{a}{F^{j}} $ on $\bar{D_{j}}\times[0,T_3]$ with $T_3\leq T_2$, then we have $$d_{(N^{n},h^{a})}(f(x),\overset{a}{F^{j}}(x,t))\leq C\sqrt{t}, \eqno(2.24)$$ for any $(x,t)\in
D_{j}\times[0,T_3]$.
Proof. For simplicity, we drop the superscripts $a$ and $j$ of $\overset{a}{F^{j}}$. Note that the distance function $d_{(N^{n},h^{a})}(y_{1},y_{2})$ can be regarded as a function on $N^{n}\times
N^{n}$. Set $\psi(y_{1},y_{2})=\frac{1}{2}d^2_{(N^{n},h^{a})}(y_{1},y_{2})$ and $\rho(x,t)=\psi(f(x),F(x,t))$. Then $\psi(x,t)$ is smooth when $\psi<\frac{1}{2}i_{a}^{2}$. Now we compute the equation of $\rho(x,t)$: $$(\frac{\partial}{\partial
t}-\overset{a}{\triangle_{t}})\rho=-d_{h^{a}}(f(x),F(x,t))
\frac{\partial d_{h^{a}}}{\partial {y_{1}}^{\alpha}}\overset{a}
{\triangle_{t}}f^{\alpha}-Hess(\psi)(X_{i},X_{j})(g^{a})^{ij}\eqno(2.25)$$ where the vector fields $X_{i}$, $i=1, 2, \cdots, n$, in local coordinates $(y_{1}^{\alpha},y_{2}^{\beta})$ on $N^{n}\times N^{n}$ are defined as follows $$X_{i}=\frac{\partial f^{\alpha}}{\partial
x^{i}}\frac{\partial}{\partial y_{1}^{\alpha}}+\frac{\partial
F^{\beta}}{\partial x^{i}}\frac{\partial}{\partial y_{2}^{\beta}}.$$
To handle the first term on the right hand side of (2.25), we use $$\overset{a}{\Gamma^{k}_{ij}}(x,t)-\overset{a}{\Gamma^{k}_{ij}}(x,0)=
\Gamma^{k}_{ij}(x,t)-{\Gamma}^{k}_{ij}(x,0)+g(\cdot,t)\ast
g^{-1}(\cdot,t)\ast \nabla \varphi^{a}+g(\cdot,0)\ast
g^{-1}(\cdot,0)\ast \nabla \varphi^{a},$$ to conclude that $$|\overset{a}{\triangle_{t}}f|_{g^{a}(\cdot,t),h^{a}}\leqslant C(n,k_{0},T)E_{0}.$$ Recall from Lemma 2.3 that the curvature of the metric $h^a_{\alpha\beta}$ is bounded by $\bar{C_0}$. We claim that if $d_{h^{a}}(f(x),F(x,t))\leq \min\{\frac{i_a}{4},\frac{\pi}{4\sqrt{\bar{C_{0}}}}\}$, then $$Hess(\psi)(X_{i},X_{j})(g^{a})^{ij}\geq
\frac{1}{2}|\overset{a}{\nabla}\overset{a}{F}|_{g^{a},h^{a}}^{2}-C \eqno(2.26)$$ where $C=C(E_0,\bar{C_0})$ depends only on $E_0$ and $\bar{C_0}$.
Indeed, recall the computation of $Hess(\psi)$ in [@ScY1]. For any $(u,v)\in
D=\{(u,v): (u,v)\in N^{n}\times
N^{n},d_{N^{n}}(u,v)<\min\{\frac{i_{a}}{2},\frac{\pi}{2\sqrt{\bar{C_0}}}\}\}$, let $\gamma_{uv}$ be the minimal geodesic from $u$ to $v$ and $e_{1}\in T_{u}N^{n}$ be the tangent vector to $\gamma_{uv}$ at $u$. Then $e_{1}(u,v)$ defines a smooth vector field on $D$. Let $\{e_{i}\}$ be an orthonormal basis for $T_{u}N^{n}$ which depends $u$ smoothly. By parallel translation of $\{e_{i}\}$ along $\gamma$, we define $\{\bar{e}_{i}\}$ an orthonormal basis for $T_{v}N^{n}$. Thus $\{e_{1},\cdots e_{n},\bar{e}_{1},\cdots \bar{e}_{n}\}$ is a local frame on $D$. Then For any $X=X^{(1)}+X^{(2)}\in T_{(u,v)}D$, where $$X^{(1)}=\sum_{i=1}^{n}\xi_{i}e_{i}, \ \ \mbox{and}\ \
X^{(2)}=\sum_{i=1}^{n}\eta_{i}\bar{e}_{i},$$ by the formula (16) in [@ScY1], $$\begin{aligned}
Hess(\psi)(X,X)&=&\sum_{i=1}^{n}(\xi_{i}-\eta_i)^{2}
+\int_{0}^{r}t\langle\nabla_{e_{1}}V,\nabla_{e_{1}}V\rangle
+\int_{0}^{r}t\langle\nabla_{\bar{e}_{1}}V,\nabla_{\bar{e}_{1}}V\rangle\\
&&-\int_{0}^{r}t\langle
R(e_{1},V)V,e_{1}\rangle-\int_{0}^{r}t\langle
R(\bar{e}_{1},V)V,\bar{e}_{1}\rangle\end{aligned}$$ where $V$ is a Jacobi field on geodesic $\sigma$ (connecting $(v,v)$ to $(u,v)$) and $\bar{\sigma}$ (connecting $(u,u)$ to $(u,v)$) with $X$ as the boundary values, where $X$ is extended to be a local vector field by letting its coefficients with respect to $\{e_{1},\cdots e_{n},\bar{e}_{1},\cdots \bar{e}_{n}\}$ be constant(see [@ScY1]). By the Jacobi equation, $|V|$, $|\nabla_{e_{1}}V|$ and $|\nabla_{\bar{e}_{1}}V|$ are bounded. Thus we have $$|Hess(\psi)|_{h^{a}}\leq C(i_a,\bar{C_0})$$ under the assumption of the claim. So the mixed term $\frac{\partial^{2}\psi}{\partial y^{\alpha}_{1}\partial y^{\beta}_{2}}f^{\alpha}_{i}F^{\beta}_{j}(g^{a})^{ij}$ in $Hess(\psi)(X_i,X_j)(g^{a})^{ij}$ can be bounded by $C(E_0,\bar{C_0})E_0|\overset{a}{\nabla}\overset{a}{F}|_{g^{a},h^{a}}$. On the other hand, the Hessian comparison theorem for the points which are not in the cut locus gives $$\begin{aligned}
\frac{\partial \psi}{\partial {y_{2}}^{\alpha}\partial
{y_{2}}^{\beta}}
-(\overset{a}{\Gamma^{\gamma}_{\alpha\beta}}\circ \overset{a}{F})\frac{\partial \psi}{\partial
y_{2}^{\gamma}}\geq \frac{\pi}{4}{h}^{a}_{\alpha\beta},\\
\frac{\partial \psi}{\partial {y_{1}}^{\alpha}\partial
{y_{1}}^{\beta}}
-(\overset{a}{\Gamma^{\gamma}_{\alpha\beta}}\circ f)\frac{\partial \psi}{\partial
y_{1}^{\gamma}}\geq \frac{\pi}{4}{h}^{a}_{\alpha\beta}.
\end{aligned}$$ Thus the claim follows.
Let $$T^{\prime}_2=\max \{t\leq T: \sup\limits_{D} {d}_{h^{a}}(f(x),F(x,t))\leq
\min\{i_a,\frac{\pi}{4\sqrt{\bar{C_0}}}\}\}.$$ If $\overset{a}{F}(x,t)$ is a smooth solution of (2.23) on $\bar{D}\times[0,T_3]$ with $T_3\leqslant T^{\prime}_{2}$, by (2.25) and (2.26), we get $$\tag{2.27} \begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle_{t}})\rho\leq -\frac{1}{2}|\overset{a}{\nabla}\overset{a}{F}|_{g^{a},h^{a}}^{2}
+C\sqrt{\rho}
+C
\end{split}$$ on $D\times[0,T_3]$, for some constant $C$ depending on $E_0$, $i_{a}$ and $\bar{C_{0}}$. Note that the initial and boundary values of $\rho$ are zero, so by the maximum principle, we get $$d_{h^{a}}(f(x),\overset{a}{F}(x,t))\leq C\sqrt{t}.$$ This implies $$T^{\prime}_2\geq
\min\{\frac{\min\{i_{a},\frac{\pi}{4\sqrt{\bar{C_{0}}}}\}^{2}}{C^{2}},T_3\}.$$ Hence the lemma holds with $$T_2=\min\{\frac{\min\{i_{a},\frac{\pi}{4\sqrt{\bar{C_{0}}}}\}^{2}}{C^{2}},T\}.$$ .
$\hfill\#$
After we have the zero order estimate (2.24), we now apply the standard parabolic equation theory to get the following short time existence for (2.23).
There exists a positive constant $T_3 \leq T_2$ depending only on the dimension $n, a, T_2$ and $C$ in Lemma 2.8 such that for each $j$, the initial-boundary value problem (2.23) has a smooth solution $\overset{a}{F^{j}}$ on $\bar{D_{j}}\times[0,T_3]$.
$\underline{\mbox{\textbf{Proof}}}$. For an arbitrarily fixed point $x_0$ in $\bar{D_j}$, choose normal coordinates $\{x^i\}$ and $\{y^\alpha \}$ on $(M^{n},g^{a}(\cdot,0))$ and $(N^{n},h^{a})$ around $x_0$ and $f(x_0)$ respectively. The equation (2.23) can be written as $$\frac
{\partial {y^{\alpha}}}{\partial t} (x,t) = (g^{a})^{ij}(x,t)
\{\frac{\partial^{2}y^{\alpha}}{\partial
x^{i}\partial
x^{j}}-\overset{a}{\Gamma^{k}_{ij}}(x,t)\frac{\partial y^{\alpha}}{\partial
x^{k}}
+\overset{a}{\Gamma^{\alpha}_{\beta\gamma}}(y^1(x,t), \cdots, y^n(x,t))\frac{\partial y^{\beta}}{\partial x^{i}}
\frac{\partial y^{\gamma}}{\partial x^{j}}\}.\eqno(2.28)$$ Note that $\overset{a}{\Gamma^{\alpha}_{\beta\gamma}}(f(x_0))=0$. By applying (2.24) and a result of Hamilton (Corollary (4.12) in [@Ha4]), we know that the coefficients of the quadratic terms on the RHS of (2.28) can be as small as we like provided $T_3>0$ sufficiently small (independent of $x_0$ and $j$).
Now for fixed $j$, we consider the corresponding parabolic system of the difference of the map $\overset{a}{F^j}$ and $f(x)$. Clearly the coefficients of the quadratic terms of the gradients are also very small. Thus, whenever (2.23) has a solution on a time interval $[0,T'_3]$ with $T'_3 \leq T_3$, we can argue exactly as in the proof of Theorem 6.1 in Chapter VII of the book [@LSU] to bound the norm of $\overset{a}{\nabla}
\overset{a}{F^{j}}$ over $\bar{D_j}\times[0,T'_3]$ by a constant depending only on the $L^{\infty}$ bound of $\overset{a}{F}$ in (2.23), the map $f(x)$, the domain $D_j$, and the metrics $g^{a}_{ij}(\cdot,t)$ and $h^{a}_{\alpha
\beta}$ over the domain $D_{j+1}$. Hence by the same argument as in the proof of Theorem 7.1 in Chapter VII of the book [@LSU], we deduce that the initial-boundary value problem (2.23) has a smooth solution $\overset{a}{F^{j}}$ on $\bar{D_{j}}\times[0,T_3]$.
$\hfill\#$
Unfortunately, the gradient estimates of $\overset{a}{F^{j}}$ in the proof of the above lemma depend also on the domain $D_j$. In order to get a convergent subsequence of $\overset{a}{F^{j}}$, we have to estimate the covariant derivatives of $\overset{a}{F^{j}}$ uniformly in each compact subsets. Before we proceed, we need some preliminary estimates.
The covariant derivatives of $\overset{a}{F^{j}}$ satisfy the following equations $$\tag{2.29} \begin{split}
\frac{\partial}{\partial t}\overset{a}{\nabla}\overset{a}{F^{j}}
&=\overset{a}{\triangle}_{t} \overset{a}{\nabla}\overset{a}{F^{j}}+
\overset{a}{Ric}(M^{n})\ast\overset{a}{\nabla}\overset{a}{F^{j}}
+\overset{a}{R_{N}}\ast(\overset{a}{\nabla}\overset{a}{F^{j}})^{3}, \\
\frac{\partial}{\partial t}\overset{a}{\nabla^{k}}\overset{a}{F^{j}}
&=\overset{a}{\triangle}_{t} \overset{a}{\nabla^{k}}\overset{a}{F^{j}}+
\sum_{l=0}^{k-1}\overset{a}{\nabla^{l}}[(\overset{a}{R_M}+\overset{a}{R_{N}}\ast
(\overset{a}{\nabla}\overset{a}{F^{j}})^{2}+e^{\varphi^{a}}\overset{a}{R_M}
+\overset{a}{\nabla^{2}}e^{\varphi^{a}})\ast\overset{a}{\nabla^{k-l}}\overset{a}{F^{j}}],
\end{split}$$ where $\overset{a}{\nabla^{l}}(A\ast B)$ represents the linear combinations of $\overset{a}{\nabla^{l}}A\ast
B$,$\overset{a}{\nabla^{l-1}}A\ast
\overset{a}{\nabla}B$, $\cdots$, $A\ast
\overset{a}{\nabla^{l}}B$, and $\overset{a}{\nabla^{2}}e^{\varphi^{a}}=e^{\varphi^{a}}(\overset{a}{\nabla^{2}}
\varphi^{a}+\overset{a}{\nabla}\varphi^{a}\ast \overset{a}{\nabla}\varphi^{a}).$
$\underline{\mbox{\textbf{Proof}}}$ For $k=1$, by direct computation and Ricci formula, we have $$\frac{\partial}{\partial
t}\overset{a}{\nabla_{i}}\overset{a}{F^{\alpha}}=\overset{a}{\triangle_{t}}\overset{a}{\nabla_{i}}\overset{a}{F^{\alpha}}
-\overset{a}{R^{l}_{i}}\overset{a}{\nabla_{l}}\overset{a}{F^{\alpha}}+\overset{a}{R}^{\alpha}_{\beta\delta\gamma}
\overset{a}{\nabla_{i}}\overset{a}{F^{\beta}}\overset{a}{\nabla_{k}}\overset{a}{F^{\delta}}\overset{a}{\nabla_{l}}\overset{a}{F^{\gamma}}(
g^{a})^{kl}.$$ For $k\geqslant2$, by Ricci formula, it follows $$\overset{a}{\nabla}\overset{a}{\triangle}\overset{a}{\nabla^{k-1}}\overset{a}{F^{j}}=\overset{a}{\triangle}
\overset{a}{\nabla^{k}}\overset{a}{F^{j}}+\overset{a}{\nabla}
[(\overset{a}{R_M}+\overset{a}{R_N}\ast(\overset{a}{\nabla}\overset{a}{F^{j}})^{2})\ast
\overset{a}{\nabla^{k-1}}\overset{a}{F^{j}}].$$ Recall from (2.20) that $$\frac{\partial}{\partial
t}\overset{a}{\Gamma^{i}_{jk}}=\overset{a}{\nabla}(e^{\varphi^{a}}\overset{a}{R_M}+\overset{a}{\nabla^{2}}e^{\varphi^{a}}).$$ Then we have $$\begin{split}
\frac{\partial}{\partial t}\overset{a}{\nabla^{k}}\overset{a}{F^{j}}
-\overset{a}{\triangle}_{t}
\overset{a}{\nabla^{k}}\overset{a}{F^{j}}=&\overset{a}{\nabla}[(\frac
{\partial}{\partial t}-\overset{a}{\triangle_{t}})\overset{a}{\nabla^{k-1}}
\overset{a}{F^{j}}]+\overset{a}{\nabla}(e^{\varphi^{a}}\overset{a}{R_M}
+\overset{a}{\nabla^{2}}e^{\varphi^{a}})
\ast\overset{a}{\nabla^{k-1}}\overset{a}{F^{j}}\\&\ \
+\overset{a}{R_N}\ast\overset{a}{\nabla}\overset{a}
{F^{j}}\ast\overset{a}{\nabla^{2}}\overset{a}{F^{j}}\ast
\overset{a}{\nabla^{k-1}}\overset{a}{F^{j}}+\overset{a}
{\nabla}[(\overset{a}{R_M}+\overset{a}{R_N}\ast(\overset{a}
{\nabla}\overset{a}{F^{j}})^{2})\ast \overset{a}{\nabla^{k-1}}\overset{a}{F^{j}}]\\
=& \overset{a}{\nabla}[(\frac{\partial}{\partial t}
-\overset{a}{\triangle})\overset{a}{\nabla^{k-1}}
\overset{a}{F^{j}}]+\overset{a}{\nabla}\{
(\overset{a}{R_M}+\overset{a}{R_{N}}\ast
(\overset{a}{\nabla}\overset{a}{F^{j}})^{2}+(e^{\varphi^{a}}\overset{a}{R_M}
+\overset{a}{\nabla^{2}}e^{\varphi^{a}})\ast\overset{a}{\nabla^{k-1}}\overset{a}{F^{j}}\}\\
=&
\sum_{l=0}^{k-1}\overset{a}{\nabla^{l}}[(\overset{a}{R_M}+\overset{a}{R_{N}}\ast
(\overset{a}{\nabla}\overset{a}{F^{j}})^{2}+e^{\varphi^{a}}\overset{a}{R_M}
+\overset{a}{\nabla^{2}}e^{\varphi^{a}})\ast\overset{a}{\nabla^{k-l}}\overset{a}{F^{j}}].
\end{split}$$ This proves the lemma.
$\hfill\#$
For each $k>0$, let $\xi_k$ be a smooth non-increasing function from $(-\infty,+\infty)$ to $[0,1]$ so that $\xi_k(s)=1$ for $s\in (-\infty,\frac{1}{2}+\frac{1}{2^{k+1}}]$, and $\xi_k(s)=0$ for $s\in[\frac{1}{2}+\frac{1}{2^k})$; moreover for any $\epsilon>0$ there exists a universal $C_{k,\epsilon}>0$ such that $$|\xi^{\prime}_k(s)|+|\xi^{\prime\prime}_k(s)|\leq
C_{k,\epsilon}{\xi_k(s)}^{1-\epsilon}.$$
There exists a positive constant $T_4$, $0<T_4\leq T_3$ independent of $j$ such that for any geodesic ball $B_{g^{a}(\cdot,0)}(x_0,\delta)\subset D_j$, there is a constant $C=C(a,\delta,E_0,\bar{C_0},\bar{k_0})$ such that the smooth solution of (2.23) satisfies $$|\overset{a}{\nabla}\overset{a}{
F^{j}}|_{g^{a}(\cdot,t),h^{a}}\leqslant C$$ on $B_{g^{a}(\cdot,0)}(x_0,\frac{3\delta}{4})\times [0,T_4]$.
$\underline{\mbox{\textbf{Proof.}}}$ We compute the equation of $|\overset{a}{\nabla}\overset{a}{
F^{j}}|^{2}_{g^{a}(\cdot,t),h^{a}}.$ For simplicity, we drop the superscript $j$. By (2.20), we have $$\tag{2.30} \begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
&=\langle \overset{a}{Ric}(M^{n})\ast\overset{a}
{\nabla}\overset{a}{F}
+\overset{a}{R_{N}}\ast(\overset{a}{\nabla}\overset{a}{F})^{3},
\overset{a}{\nabla}\overset{a}{F}\rangle_{g^{a},h^{a}}
-2|\overset{a}{\nabla^{2}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}\\
&\ \ +e^{\varphi^{a}}
(\overset{a}{Ric(M^{n})}+\overset{a}{\nabla^{2}}
\varphi^{a}+
\overset{a}{\nabla}\varphi^{a}\ast\overset{a}
{\nabla}\varphi^{a})
\ast\overset{a}{\nabla}
\overset{a}{F}\ast\overset{a}{\nabla}\overset{a}{F}\\&\leqslant
-2|\overset{a}{\nabla^{2}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C(n,k_0,T)|\overset{a}{\nabla}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C(n)\bar{C_0}
|\overset{a}{\nabla}\overset{a}{F}|^{4}_{g^{a}(\cdot,t),h^{a}}.
\end{split}$$ Setting $$\rho_{A}(x,t)=(d_{h^{a}}^{2}(f(x),F(x,t))+A)|\overset{a}{\nabla}\overset{a}{
F}|^{2}_{g^{a}(\cdot,t),h^{a}}$$ where $A$ is determined later, and combining with (2.27) and (2.24), we have $$\begin{aligned}
\frac{\partial}{\partial
t}\rho_{A}&\leqslant&\overset{a}{\triangle}\rho_A-2|\overset{a}{\nabla^{2}}
\overset{a}{F}|^{2}_{g^{a},h^{a}}(d_{h^{a}}^{2}(f(x),\overset{a}{F}(x,t))+A)-
|\overset{a}{\nabla}\overset{a}{ F}|^{4}_{g^{a},h^{a}}\\&
&+C(n)\bar{C_0}(d^{2}_{h^{a}}(f(x),\overset{a}{F}(x,t))+A)|\overset{a}{\nabla}
\overset{a}{F}|^{4}_{g^{a},h^{a}}+C|\overset{a}{\nabla}\overset{a}{F}|^{2}_{g^{a},h^{a}}+C(n,k_0,T)\rho_A
\\& &+2|\nabla
d^{2}_{h^{a}}(f(x),\overset{a}{F}(x,t))|_{g^{a}} |\nabla
|\overset{a}{\nabla}\overset{a}{
F}|^{2}_{g^{a}(\cdot,t),h^{a}}|_{g^{a}}.
\end{aligned}$$ Since $$\begin{aligned}
|\nabla
d^{2}_{h^{a}}(f(x),\overset{a}{F}(x,t))|_{g^{a}}&\leqslant& 2
d_{h^{a}}(f(x),\overset{a}{F}(x,t))(|\overset{a}{\nabla}\overset{a}{F}|_{g^{a},h^{a}}+
|\overset{a}{\nabla}f|_{g^{a},h^{a}})\\
&\leqslant&
C\sqrt{t}+C\sqrt{t}|\overset{a}{\nabla}\overset{a}{F}|_{g^{a},h^{a}},\\
|\nabla
|\overset{a}{\nabla}\overset{a}{
F}|^{2}_{g^{a}(\cdot,t),h^{a}}|_{g^{a}}&\leqslant& 2|\overset{a}{\nabla^{2}}\overset{a}{F}|
_{g^{a},h^{a}}|\overset{a}{\nabla}\overset{a}{F}|_{g^{a},h^{a}},
\end{aligned}$$ by choosing $T_4=\min\{T_3,\frac{1}{4C(n)\bar{C_0}C^{2}}\}$, $A=\frac{1}{4C(n)\bar{C_0}}$, and applying Cauchy-Schwartz inequality, we have $$(\frac{\partial}{\partial
t}-\overset{a}{\triangle})\rho_{A}\leqslant-(C(n)\bar{C_0})\rho_A^{2}+C.$$ Here and in the following we denote by $C$ various constants depending only on $n$, $k_0$, $T$, $E_0$ and $a$.
We compute the equation of $u=\xi_{1}(\frac{d_{g^{a}(\cdot,0)}(x_0,\cdot)}{\delta})\rho_{A}$ at the smooth points of $d_{g^{a}(\cdot,0)}(x_0,\cdot)$, $$(\frac{\partial}{\partial t}-\overset{a}{\triangle})u\leqslant
C\xi_{1}-(C(n)\bar{C_0})\rho_{A}^{2}\xi_{1}-2(g^{a})^{ij}\nabla_{i}\xi_{1}\nabla_{j}\rho_{A}
+(-\xi_{1}^{\prime}\frac{\overset{a}{\triangle}d_{g^{a}(\cdot,0)}(x_0,\cdot)}
{\delta}+e^{nk_0T}\frac{|\xi_{1}^{\prime\prime}|}{\delta^{2}})\rho_{A}.$$ By the Hessian comparison theorem and the fact that $-\xi^{\prime}_{1}\geqslant0$, we have $$\begin{aligned}
\overset{a}{\nabla_{i}}\overset{a}{\nabla_{j}}d_{g^{a}(\cdot,0)}&\leqslant&
\overset{a}{\nabla_{i}^{0}}\overset{a}{\nabla_{j}^{0}}d_{g^{a}(\cdot,0)}
+(\overset{a}{\Gamma}(\cdot,0)-\overset{a}{\Gamma}(\cdot,t))\ast
\nabla d_{g^{a}(\cdot,0)}\\
&
\leqslant&(\frac{1+\bar{k_0}d_{g^{a}(\cdot,0)}}{d_{g^{a}(\cdot,0)}}+C)g^{a}_{ij}(\cdot,0),\\
-\xi_{1}^{\prime}\overset{a}{\triangle}d_{g^{a}(\cdot,0)}&\leqslant&\frac{C|\xi_{1}^{\prime}|}{\delta}.\end{aligned}$$ These two inequalities hold on the whole manifold in the sense of support functions. Thus for any $x_{1}\in M^{n}$, there is a function $h_{x_{1}}$ which is smooth on a neighborhood of $x_{1}$ with $h_{x_{1}}(\cdot)\geqslant d_{g^{a}(\cdot,0)}(x_0,\cdot)$, $h_{x_{1}}(x_{1})= d_{g^{a}(\cdot,0)}(x_0,x_{1})$ and
$$-\xi_{1}^{\prime}\overset{a}{\triangle}h_{x_{1}}\mid_{x_{1}}\leqslant2\frac{C|\xi_{1}^{\prime}|}{\delta}.$$ Indeed, $h_{x_{1}}$ can be chosen to having the form $d_{g^{a}(\cdot,0)}(q,\cdot)+d_{g^{a}(\cdot,0)}(q,x_{0})$ for some $q$, so we may require $|\overset{a}{\nabla}h_{x_{1}}|_{g^{a}(\cdot,0)}\leqslant1$. Let $(x_{1},t_0)$ be the maximum point of $u$ over $M^{n}\times[0,T_{4}]$. If $t_{0}=0$, then $\xi_{1}\rho_{A}\leqslant E_{0}$. Assume $t_{0}>0$. At the point $(x_{1},t_{0})$, we have $\frac{\partial}{\partial
t}(\xi_{1}\rho_{A})(x_{1},t_{0})\geqslant0$. If $x_{1}$ does not lie on the cut locus of $x_{0}$, then $$\begin{aligned}
0&\leqslant&-C(n)\bar{C_0}\rho_{A}^{2}\xi_{1}+\frac{1}{\delta^{2}}
(e^{nk_0T}\frac{|\xi_{1}^{\prime}|^{2}}{\xi_{1}}+2C(|\xi_{1}^{\prime}|+|\xi_{1}^{\prime\prime}|))\rho_{A}+C\xi_{1}\\
&\leqslant
&-C(n)\bar{C_0}\rho_{A}^{2}\xi_{1}+\frac{C}{\delta^{2}}\sqrt{\xi_{1}}\rho_{A}+C\xi_{1}\\
&\leqslant&
-C(n)\bar{C_0}\rho_{A}^{2}\xi_{1}+\frac{C}{\delta^{4}}\\
&\leqslant&
-C(n)\bar{C_0}(\rho_{A}\xi_{1})^{2}+\frac{C}{\delta^{4}}.\end{aligned}$$ We get $$\xi_{1}\rho_{A}\leqslant\max\{E_{0},\sqrt{\frac{C}{C(n)\bar{C_{0}}\delta^{4}}}\}$$ for all $(x,t)\in B_{g^{a}(\cdot,0)}(x_0,\delta)\times[0,T_4]$. If $x_{1}$ lies on the cut locus of $x_{0}$, then by applying the standard support function technique (see for example [@ScY]), the above maximum principle argument still works. So by the definition of $\xi_{1}$ and $\rho_{A}$, we have
$$|\overset{a}{\nabla}\overset{a}{
F^{j}}|_{g^{a}(\cdot,t),h^{a}}\leqslant \frac{C}{\delta}$$ on $B_{g^{a}(\cdot,0)}(x_0,\frac{3\delta}{4})\times [0,T_4]$. The proof of the lemma is completed.
$\hfill\#$
The next lemma estimates the higher derivatives in terms of the bound of $|\overset{a}{\nabla}\overset{a}{
F^{j}}|_{g^{a}(\cdot,t),h^{a}}$.
Let $\overset{a}{F}$ be a smooth solution of equation $$(\frac{\partial}{\partial
t}-\overset{a}{\triangle})\overset{a}{F}=0$$ on $B_{g^{a}(\cdot,0)}(x_{0},\delta)\times[0,\bar{T}]$, with $\bar{T}\leqslant T$. Suppose $$\tag{2.31}
\begin{split}
\sup_{(x,t)\in
B_{g^{a}(\cdot,0)}(x_0,\frac{3\delta}{4})\times[0,\bar{T}]}|\overset{a}
{\nabla}\overset{a}{F}|_{g^{a}_{ij}(\cdot,0),h^{a}_{\alpha\beta}}(x,t)&\leqslant
E_{1},\\ \mbox{and } \sup_{x\in
B_{g^{a}(\cdot,0)}(x_0,\frac{3\delta}{4})}
|\overset{a}{\nabla^{2}}\overset{a}{F}|_{g^{a}_{ij}(\cdot,0),h^{a}_{\alpha\beta}}(x,0)&\leqslant
E_{1}.
\end{split}$$ Then for any $k\geqslant2$, there exists a positive constant $C=C(k,E_{1},\delta,k_{0},T)>0$ such that $$|\overset{a}{\nabla^{k}} \overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}\leq C t^{-\frac{k-2}{2}}
\eqno(2.32)$$ on $B_{g^a(\cdot,0)}(x_{0},\frac{\delta}{2})\times [0,\bar{T}]$.
Proof. The proof is using the Bernstein trick. We assume $\delta<1$ without loss of generality. For $k=2$, from (2.15), (2.19), (2.20) and (2.29), we have $$\tag{2.33} \begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla^{2}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
&=\langle \sum_{l=0}^{1}\overset{a}{\nabla^{l}}[(\overset{a}{R_M}+\overset{a}{R_{N}}\ast
(\overset{a}{\nabla}\overset{a}{F^{j}})^{2}+e^{\varphi^{a}}\overset{a}{R_M}
+\overset{a}{\nabla^{2}}e^{\varphi^{a}})\ast\overset{a}{\nabla^{2-l}}\overset{a}{F}],\overset{a}{\nabla^{2}}
\overset{a}{F}\rangle_{g^{a},h^{a}}\\&\ \ \ \
-2|\overset{a}{\nabla^{3}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}} +e^{\varphi^{a}}
(\overset{a}{Ric(M^{n})}+\overset{a}{\nabla^{2}}
\varphi^{a}+
\overset{a}{\nabla}\varphi^{a}\ast\overset{a}
{\nabla}\varphi^{a})
\ast(\overset{a}{\nabla^{2}}
\overset{a}{F})^{2}\\&\leqslant
-2|\overset{a}{\nabla^{3}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C|\overset{a}{\nabla^{2}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+\frac{C}{\sqrt{t}}
|\overset{a}{\nabla^{2}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}.
\end{split}$$ In this lemma, we use $C$ to denote various constants depending only on $E_1$, $k_0$, $T$, $k$ and $\delta$. Note that by (2.30) and (2.33), we have $$\begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
&\leqslant
-2|\overset{a}{\nabla^{2}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C,\\
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla^{2}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}
&\leqslant
C|\overset{a}{\nabla^{2}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}+\frac{C}{\sqrt{t}}.
\end{split}$$ So by setting $$v=|\overset{a}{\nabla^{2}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}-2C\sqrt{t}+2C\sqrt{T}+|\overset{a}{\nabla}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}},$$ we have $$\begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})v
&\leqslant
-2|\overset{a}{\nabla^{2}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
+C|\overset{a}{\nabla^{2}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}+C\\
&\leqslant -v^{2}+C.
\end{split}$$ Since at $t=0$, $$v\leqslant2C\sqrt{T}+E_{1}+E_{1}^{2}$$ on $B_{g^{a}(\cdot,0)}(x_{0},\frac{3\delta}{4})$, we apply the maximum principle as in Lemma 2.11 to get $$\xi_{2}(\frac{d_{g^{a}(\cdot,0)}(x_0,\cdot)}{\delta})v\leqslant C$$ on $B_{g^{a}(\cdot,0)}(x_{0},\frac{3\delta}{4})\times[0,\bar{T}]$. This implies $$|\overset{a}{\nabla^{2}} \overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}\leq C$$ on $B_{g^{a}(\cdot,0)}(x_{0},(\frac{1}{2}+\frac{1}{2^{3}})\delta)\times[0,\bar{T}]$.
Now we estimate the third-order derivatives. From Shi’s gradient estimate [@Sh1], the estimate of $ |\overset{a}{\nabla^{2}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}\leq C $ and (2.15), (2.19), (2.20) and (2.29), we have: $$\tag{2.34} \begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla^{3}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
&=\langle \sum_{l=0}^{2}\overset{a}{\nabla^{l}}[(\overset{a}{R_M}+\overset{a}{R_{N}}\ast
(\overset{a}{\nabla}\overset{a}{F})^{2}+e^{\varphi^{a}}\overset{a}{R_M}
+\overset{a}{\nabla^{2}}e^{\varphi^{a}})\ast\overset{a}{\nabla^{3-l}}\overset{a}{F}],\overset{a}{\nabla^{3}}
\overset{a}{F}\rangle_{g^{a},h^{a}}\\&\ \ \ \
-2|\overset{a}{\nabla^{4}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}} +e^{\varphi^{a}}
(\overset{a}{Ric(M^{n})}+\overset{a}{\nabla^{2}}
\varphi^{a}+
\overset{a}{\nabla}\varphi^{a}\ast\overset{a}
{\nabla}\varphi^{a})
\ast(\overset{a}{\nabla^{3}}
\overset{a}{F})^{2}\\&\leqslant
-2|\overset{a}{\nabla^{4}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C|\overset{a}{\nabla^{3}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+\frac{C}{t}
|\overset{a}{\nabla^{3}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}.
\end{split}$$ on $B_{g^{a}(\cdot,0)}(x_{0},(\frac{1}{2}+\frac{1}{8})\delta)\times[0,\bar{T}]$. Here we used the estimates $|\overset{a}{\nabla^{4}}e^{\varphi^{a}}|_{g^{a}}\leqslant \frac{C}{\sqrt{t}}$, $|\overset{a}{\nabla^{3}}e^{\varphi^{a}}|_{g^{a}}\leqslant C(1+|\log t|)$, and $|\overset{a}{\nabla^{2}}\overset{a}{R_m}|\leqslant\frac{C}{t}$.
By (2.33), it follows $$(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla^{2}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
\leqslant
-2|\overset{a}{\nabla^{3}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+\frac{C}{\sqrt{t}} \eqno(2.35)$$ on $B_{g^{a}(\cdot,0)}(x_{0},(\frac{1}{2}+\frac{1}{8})\delta)\times[0,\bar{T}]$. Let $v=(|\overset{a}{\nabla^{2}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+A)|\overset{a}{\nabla^{3}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}$, where $A=100 \sup\limits_{B_{g^{a}(\cdot,0)}(x_0,(\frac{1}{2}+\frac{1}{2^{3}})\delta)\times[0,\bar{T}]}|\overset{a}
{\nabla^{2}}\overset{a}{F}|_{g^{a}_{ij}(\cdot,t),h^{a}_{\alpha\beta}}(x,t)+C.$ By a direct computation, it follows $$\begin{aligned}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle})v&\leq&|\overset{a}{\nabla^{3}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}(-2|\overset{a}{\nabla^{3}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+\frac{C}{\sqrt{t}})+(|\overset{a}{\nabla^{3}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+A)\\& &\times( -2|\overset{a}{\nabla^{4}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C|\overset{a}{\nabla^{3}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+\frac{C}{t}
|\overset{a}{\nabla^{3}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}})\\& &
+
8|\overset{a}{\nabla^{2}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}
|\overset{a}{\nabla^{3}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
|\overset{a}{\nabla^{4}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}.\end{aligned}$$ Since $$8|\overset{a}{\nabla^{2}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}
|\overset{a}{\nabla^{3}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
|\overset{a}{\nabla^{4}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}\leqslant
-|\overset{a}{\nabla^{3}}\overset{a}{F}|^{4}_{g^{a}(\cdot,t),h^{a}}+16
|\overset{a}{\nabla^{4}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
|\overset{a}{\nabla^{2}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}},$$ we deduce $$(\frac{\partial}{\partial
t}-\overset{a}{\triangle})v\leqslant-|\overset{a}{\nabla^{3}}\overset{a}{F}|^{4}_{g^{a}(\cdot,t),h^{a}}+
\frac{C}{t}|\overset{a}{\nabla^{3}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}
+\frac{C}{\sqrt{t}}|\overset{a}{\nabla^{3}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}$$ and $$\begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle})(tv)&\leqslant
v-t|\overset{a}{\nabla^{3}}\overset{a}{F}|^{4}_{g^{a}(\cdot,t),h^{a}}+
C|\overset{a}{\nabla^{3}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}
+C\sqrt{t}|\overset{a}{\nabla^{3}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}\\
&\leqslant-\frac{1}{t}\{t^{2}
|\overset{a}{\nabla^{3}}\overset{a}{F}|^{4}_{g^{a}(\cdot,t),h^{a}}-C\sqrt{t}(\sqrt{t}
|\overset{a}{\nabla^{3}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}})-tv-C\sqrt{t}(t
|\overset{a}{\nabla^{3}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}})\}\\
&\leqslant-\frac{1}{t}\{\frac{(tv)^{2}}{10^{5}C^{2}}-C\}.
\end{split}$$ So at the maximum point of $\xi_{3}(\frac{d_{g^{a}(\cdot,0)}(x_{0},\cdot)}{\delta})(tv)$, applying the maximum principle as in Lemma 2.11, we have $$\begin{split}0&\leqslant-\frac{1}{t}\{\frac{\xi_{3}(tv)^{2}}{10^{5}C^{2}}-C\xi_{3}\}
+C(\frac{|\xi_{3}^{\prime}|^{2}}{\xi_{3}}+|\xi_{3}^{\prime\prime}|)(tv)\\
&\leqslant-\frac{1}{t}\{\frac{\xi_{3}(tv)^{2}}{10^{5}C^{2}}-C\xi_{3}
-Ct\sqrt{\xi_{3}}(tv)\}\\
&\leqslant-\frac{1}{t}\{\frac{\xi_{3}(tv)^{2}}{10^{6}C^{2}}-C^{4}\},\\
\end{split}$$ which gives $$\xi_{3}(tv)\leqslant\sqrt{10^{6}C^{6}}.$$ Thus by the definition of $v$ and $\xi_{3}$, we get
$$|\overset{a}{\nabla^{3}} \overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}\leq C t^{-\frac{1}{2}}$$ on $B_{0}(x_{0},(\frac{1}{2}+\frac{1}{2^{4}})\delta)\times [0,\bar{T}]$.
Now we estimate the higher derivatives by induction. Suppose we have proved that
$$|\overset{a}{\nabla^{l}} \overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}\leq C t^{-\frac{l-2}{2}}
\ \ \ \ \ \ \mbox{ for }
l=3, \cdots, k-1$$ on $B_{0}(x_{0},(\frac{1}{2}+\frac{1}{2^{k}})\delta)\times [0,\bar{T}]$. By (2.29), we have $$\begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla^{k}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
&=\langle \sum_{l=0}^{k-1}\overset{a}{\nabla^{l}}[(\overset{a}{R_M}+\overset{a}{R_{N}}\ast
(\overset{a}{\nabla}\overset{a}{F})^{2}+e^{\varphi^{a}}\overset{a}{R_M}
+\overset{a}{\nabla^{2}}e^{\varphi^{a}})\ast\overset{a}{\nabla^{k-l}}\overset{a}{F}],\overset{a}{\nabla^{3}}
\overset{a}{F}\rangle_{g^{a},h^{a}}\\&\ \ \ \
-2|\overset{a}{\nabla^{k+1}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}} +e^{\varphi^{a}}
(\overset{a}{Ric(M^{n})}+\overset{a}{\nabla^{2}}
\varphi^{a}+
\overset{a}{\nabla}\varphi^{a}\ast\overset{a}
{\nabla}\varphi^{a})
\ast(\overset{a}{\nabla^{k}}
\overset{a}{F})^{2}\\&\leqslant
-2|\overset{a}{\nabla^{k+1}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C|\overset{a}{\nabla^{k}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}} +C(n) \sum_{l=1}^{k-1}|\overset{a}{\nabla^{l}}[\overset{a}{R_M}+\overset{a}{R_{N}}\ast
(\overset{a}{\nabla}\overset{a}{F})^{2}\\&\ \ +e^{\varphi^{a}}\overset{a}{R_M}
+\overset{a}{\nabla^{2}}e^{\varphi^{a}}]|_{g^{a},h^{a}}|\overset{a}{\nabla^{k-l}}\overset{a}{F}|_{g^{a},h^{a}}
|\overset{a}{\nabla^{k}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}.
\end{split}$$ By the induction hypothesis, the local derivative estimate of Shi, and (2.15), (2.19) and (2.20), it follows $$\begin{aligned}
\sum_{l=0}^{k-1}|\overset{a}{\nabla^{l}}\overset{a}{R_{M}}|_{g
^{a}}|\overset{a}{\nabla^{k-l}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}&\leqslant &\frac{C}{
t^{\frac{k-1}{2}}},\\
\sum_{l=0}^{k-1}|\overset{a}{\nabla^{l}}[\overset{a}{R_{N}}\ast(\overset{a}{\nabla}\overset{a}{F})^{2}]|_{g
^{a}}|\overset{a}{\nabla^{k-l}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}&\leqslant &\frac{C}{
t^{\frac{k-1}{2}}}+C|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}},\\
\sum_{l=0}^{k-1}|\overset{a}{\nabla^{l+2}}e^{\varphi^{a}}|_{g
^{a}}|\overset{a}{\nabla^{k-l}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}&\leqslant &\frac{C}{
t^{\frac{k-2}{2}}},\\
\sum_{l=0}^{k-1}|\overset{a}{\nabla^{l}}\overset{a}{e^{\varphi^{a}}R_{M}}|_{g
^{a}}|\overset{a}{\nabla^{k-l}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}&\leqslant &\frac{C}{
t^{\frac{k-1}{2}}}.\\\end{aligned}$$ This gives $$\begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla^{k}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
&\leqslant
-2|\overset{a}{\nabla^{k+1}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C|\overset{a}{\nabla^{k}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}} +\frac{C}{t^{\frac{k-1}{2}}}
|\overset{a}{\nabla^{k}}\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}},\\
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}
&\leqslant
C|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}
+\frac{C}{t^{\frac{k-1}{2}}},
\\
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla^{k-1}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
&\leqslant
-2|\overset{a}{\nabla^{k}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+
+\frac{C}{t^{k-\frac{5}{2}}}.
\end{split}$$ Let $\varepsilon=\frac{2(k-3)}{k-2}-1$, then $0\leq\varepsilon<1$ for $k\geq 4$. It is clear that $$(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}^{1+\varepsilon}
\leqslant
C|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}^{1+\varepsilon} +\frac{C}{t^{\frac{k-1}{2}}}|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}^{\varepsilon},$$ and $$\begin{aligned}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})(|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}^{1+\varepsilon}+|\overset{a}{\nabla^{k-1}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}})
&\leqslant&
-2|\overset{a}{\nabla^{k}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
C|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}^{1+\varepsilon} \\& &+\frac{C}{t^{\frac{k-1}{2}}}|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}^{\varepsilon}+\frac{C}{t^{k-\frac{5}{2}}},\end{aligned}$$ on $B_{g^{a}(\cdot,0)}(x_{0},(\frac{1}{2}+\frac{1}{2^{k}})\delta)\times[0,\bar{T}]$.
Let $$v=t^{k-3}(|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}^{1+\varepsilon}+|\overset{a}{\nabla^{k-1}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}).$$ Then we have $$\begin{aligned}
(\frac{\partial}{\partial t}-\overset{a}{\triangle})v&\leqslant&
(k-3)\frac{v}{t}+ t^{k-3}(-|\overset{a}{\nabla^{k}}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
+\frac{C}{t^{\frac{k-1}{2}}}|\overset{a}{\nabla^{k}}
\overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}^{\varepsilon}+\frac{C}{t^{k-\frac{5}{2}}}\\ &\leqslant&
-\frac{1}{t}\{v^{\frac{2}{1+\varepsilon}}
-C\sqrt{t}v^{\frac{\varepsilon}{1+\varepsilon}}-C\sqrt{t}\}\\
&\leqslant&-\frac{1}{t}\{v^{\frac{2}{1+\varepsilon}}
-C\}\end{aligned}$$ on $B_{g^{a}(\cdot,0)}(x_{0},(\frac{1}{2}+\frac{1}{2^{k}})\delta)\times[0,\bar{T}]$. Similarly, at the maximum point of $\xi_{k}(\frac{d_{g^{a}(\cdot,0)}(x_{0},\cdot)}{\delta})v$, we have $$\begin{split}0&\leqslant-\frac{1}{t}\{\xi_{k}v^{\frac{2}{1+\varepsilon}}-C\xi_{k}\}
+C(\frac{|\xi_{k}^{\prime}|^{2}}{\xi_{k}}+|\xi_{k}^{\prime\prime}|)v\\
&\leqslant-\frac{1}{t}\{\xi_{k}v^{\frac{2}{1+\varepsilon}}-C\xi_{k}^{\frac{1+\varepsilon}{2}}v-C\}\\
&\leqslant-\frac{1}{t}\{\frac{1}{2}\xi_{k}v^{\frac{2}{1+\varepsilon}}-C\}\\
&\leqslant-\frac{1}{t}\{\frac{1}{2}(\xi_{k}v)^{\frac{2}{1+\varepsilon}}-C\}.
\end{split}$$ since $\frac{2}{1+\varepsilon}>1$. So we proved the $k$-th order estimate $$|\overset{a}{\nabla^{k}} \overset{a}{F}|_{g^{a}(\cdot,t),h^{a}}\leq C t^{-\frac{k-2}{2}}$$ on $B_{g^{a}(\cdot,0)}(x_{0},(\frac{1}{2}+\frac{1}{2^{k+1}})\delta)\times [0,\bar{T}]$. This completes the proof of the lemma.
$\hfill\#$
Now we are ready to prove Theorem 2.7.
0.5cm $\underline{\mbox{\textbf{Proof of Theorem 2.7.}}}$ 0.3cm
Since $D_j\supseteq B_{g^{a}(\cdot,0)}(P,j+1),$ by choosing $\delta=1$ and $\bar{T}=T_4$ in Lemma 2.11 and Lemma 2.12, we get a convergent subsequence of $\overset{a}{F^{j}}$ (as $j\rightarrow\infty$) on $B_{g^{a}(\cdot,0)}(P,j)\times [0,T_4]$. Denote the limit by $\overset{a}{F}$ (as $j\rightarrow\infty$). Then $\overset{a}{F}$ is the desired solution of $(2.3)_{a}^{\prime}$ with estimates (2.22).
Finally we prove a uniqueness theorem for the solutions of $(2.3)_{a}^{\prime}$ with estimates (2.22).
Let $\overset{a}{F}$ and $\overset{a}{\bar{F}}$ be two solutions of the intial problem $(2.3)_{a}^{\prime}$ on $[0,\bar{T}]$, $\bar{T}\leqslant T$, with estimates (2.22). Then $\overset{a}{F}=\overset{a}{\bar{F}}$ on $[0,\bar{T}]$.
$\underline{\mbox{\textbf{Proof }}}$ Set $\psi(y_{1},y_{2})=\frac{1}{2}d^2_{(N^{n},h^{a})}(y_{1},y_{2})$ and $\rho(x,t)=\psi(\overset{a}{F}(x,t),\overset{a}{\bar{F}}(x,t))$. Then $\psi(x,t)$ is smooth when $\psi<\frac{1}{2}i_{a}^{2}$. Now by the same calculation as in Lemma 2.8, we have: $$(\frac{\partial}{\partial
t}-\overset{a}{\triangle_{t}})\rho=-Hess(\psi)(X_{i},X_{j})(g^{a})^{ij}$$ where the vector fields $X_{i}$, $i=1, 2, \cdots, n$, in local coordinates $(y_{1}^{\alpha},y_{2}^{\beta})$ on $N^{n}\times N^{n}$ are defined as follows $$X_{i}=\frac{\partial \overset{a}{F^{\alpha}}}{\partial
x^{i}}\frac{\partial}{\partial y_{1}^{\alpha}}+\frac{\partial
\overset{a}{\bar{F}^{\beta}}}{\partial
x^{i}}\frac{\partial}{\partial y_{2}^{\beta}}.$$ By the estimates (2.22), we know that there is a constant $0<\bar{T^{\prime}}\leqslant\bar{T}$ such that there holds $$\rho<\min\{\frac{i_{a}^{2}}{8},\frac{\pi^{2}}{8\bar{C_0}}\}.$$ on $M^n\times [0,\bar{T^{\prime}}]$.
Similarly as in the proof of Lemma 2.8. By using the computation of $Hess(\psi)$ in [@ScY1] (the formula (16) in [@ScY1]), for any $(u,v)\in
D=\{(u,v): (u,v)\in N^{n}\times N^{n} \mbox{ with }
d_{N^{n}}(u,v)<\min\{\frac{i_{a}}{2},\frac{\pi}{2\sqrt{\bar{C_0}}}\}\}$, and any $X \in T_{(u,v)}D$, $$\begin{aligned}
Hess(\psi)(X,X)&\geqslant &-\int_{0}^{r}t\langle
R(e_{1},V)V,e_{1}\rangle-\int_{0}^{r}t\langle
R(\bar{e}_{1},V)V,\bar{e}_{1}\rangle\end{aligned}$$ where $V$ is a Jacobi field on geodesic $\sigma$ (connecting $(v,v)$ to $(u,v)$) and $\bar{\sigma}$ (connecting $(u,u)$ to $(u,v)$) with $X$ as the boundary values. Since $|\overset{a}{\nabla} F|_{g^{a},h^{a}}$ and $|\overset{a}{\nabla}\bar{F}|_{g^{a},h^{a}}$ are bounded, we know from above formula that $$Hess(\psi)(X_{i},X_{j})(g^{a})^{ij}\geqslant -C\rho$$ on $M^{n} \times[0,\bar{T^{\prime}}]$. Thus we have $$(\frac{\partial}{\partial
t}-\overset{a}{\triangle_{t}})\rho \leqslant C\rho$$ on $M^{n} \times[0,\bar{T^{\prime}}]$. By the maximum principle, it follows that $\rho=0$ on $M^{n} \times[0,\bar{T^{\prime}}]$. Then the lemma follows by continuity method.
$\hfill\#$
### Proof of theorem 2.6 and Theorem 2.1
0.5cm $\underline{\mbox{\textbf{Proof of Theorem
2.6.}}}$0.3cm
Let us check the initial data. Now $f=identity$, so $$\tag{2.36}
\begin{split}|\overset{a}{\nabla}f|^{2}_{g^{a}(\cdot,0),h^{a}}&=
g^{ij}(\cdot,0)g_{ij}(\cdot,T)\\
&\leqslant ne^{2nk_0T}
\end{split}$$ $$\tag{2.37}
\begin{split}|\overset{a}{\nabla^{2}}f|^{2}_{g^{a}(\cdot,0),h^{a}}
&=|\overset{a}{\Gamma^{k}_{ij}}(\cdot,0)-\overset{a}{\Gamma^{k}_{ij}}(\cdot,T)|_{g^{a}(\cdot,0),h^{a}}
\\
&\leqslant
C(n,k_0,T)\int_{0}^{T}e^{\varphi^{a}}(|\overset{a}{\nabla}\overset{a}{R}_{M}|_{g^{a}(\cdot,t)}+
|\overset{a}{R}_{M}\ast\overset{a}{\nabla}\varphi^{a}|_{g^{a}(\cdot,t)})\\&
\ \ \
+|\overset{a}{\nabla}\varphi^{a}|_{g^{a}(\cdot,t)}|\overset{a}{\nabla^{2}}\varphi^{a}|_{g^{a}(\cdot,t)}+
|\overset{a}{\nabla^{3}}\varphi^{a}|_{g^{a}(\cdot,t)})dt\\
&\leqslant C(n,k_0,T)\int_{0}^{T}\frac{1}{\sqrt{t}}+|\log t|dt\\
& \leqslant C(n,k_0,T).
\end{split}$$ By applying Theorem 2.7, we know that there is $\delta_0>0$ such that $(2.3)_a$ has a smooth solution $\overset{a}{F}$ on $M^{n}\times[0,\delta_0]$ with estimates (2.22). In views of Lemma 2.12 and Lemma 2.13, in order to prove Theorem 2.6, we only need to bound $|\overset{a}{\nabla}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}$ uniformly on a uniformly interval $[0,T_1]$ with $T_1$ independent of $a$. To this end, let $$\begin{split}
\tilde{T}=\sup\{\tilde{T_0}\mid &\tilde{T_0}\leqslant T, (2.3)_a
\text{ has a smooth solution on } M^{n}\times [0,\tilde{T_0}]\\
&\text{with}
\sup_{M^{n}\times [0,\tilde{T_0}]}|\overset{a}{\nabla}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
<\infty\},
\end{split}$$ We will estimate $\tilde{T}$ from below.
We come back to the equation (2.30) of $|\overset{a}{\nabla}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}$, where there holds $$\begin{split}
(\frac{\partial}{\partial
t}-\overset{a}{\triangle}_{t})|\overset{a}{\nabla}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
\leqslant
-2|\overset{a}{\nabla^{2}}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C_1(n,k_0,T)|\overset{a}{\nabla}
\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+C_2(n,k_0,T)
|\overset{a}{\nabla}\overset{a}{F}|^{4}_{g^{a}(\cdot,t),h^{a}}
\end{split}$$ on $ M^{n}\times [0,\tilde{T}]$. We remark that $\overset{a}{F}$ is defined on a complete manifold with bounded curvature and $\sup_{M^{n}\times
[0,\tilde{T_0}]}|\overset{a}{\nabla}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}
<\infty$, for each $\tilde{T_0}<\tilde{T}$. So by applying the maximum principle on complete manifolds, we have $$\frac{d^{+}}{d
t}(\sup_{M^{n}}|\overset{a}{\nabla}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}})\leqslant C_1(n,k_0,T)
\sup_{M^{n}}|\overset{a}{\nabla}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}+
C_2(n,k_0,T)\sup_{M^{n}}|\overset{a}{\nabla}\overset{a}{F}|^{4}_{g^{a}(\cdot,t),h^{a}}$$ where $\frac{d^{+}}{d
t}$ is the upper right derivative defined by $$\frac{d^{+}}{d
t} u=\limsup_{\triangle t\searrow 0}\frac{u(t+\triangle t)-u(t)}{\triangle
t}.$$ By combining with (2.36), we have $$\sup_{M^{n}\times
[0,\tilde{T_0}]}|\overset{a}{\nabla}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}\leqslant
2n e^{2nk_0T},$$ provided $\tilde{T_0}\leqslant \min\{T,
\frac{\log 2}{C_1(n,k_0,T)+2ne^{2nk_0T}C_2(n,k_0,T)}\}$.
By Lemma 2.12 and Lemma 2.13 and Theorem 2.7, the solution $\overset{a}{F}$ exists smoothly until $|\overset{a}{\nabla}\overset{a}{F}|^{2}_{g^{a}(\cdot,t),h^{a}}$ blows up, so we know $\tilde{T}\geqslant \min\{T, \frac{\log
2}{C_1(n,k_0,T)+2ne^{2nk_0T}C_2(n,k_0,T)}\}$. By choosing $T_1=\min\{T, \frac{\log
2}{C_1(n,k_0,T)+2ne^{2nk_0T}C_2(n,k_0,T)}\}$, Theorem 2.6 follows.
$\hfill\#$
0.5cm $\underline{\mbox{\textbf{Proof of Theorem 2.1.}}}$ 0.3cm
Note that $\varphi^{a}=0$ on $B_{g(\cdot,T)}(P,a)$, and $g^{a}_{ij}(x,t)=e^{\varphi^{a}}g_{ij}(x,t)$, $h^{a}_{\alpha\beta}(y)=e^{\varphi^{a}}h_{\alpha\beta}$. It follows that $$g^{a}_{ij}(x,t)=g_{ij}(x,t) \ \ \ \text{on} \
B_{g(\cdot,T)}(P,a),$$ $$h^{a}_{\alpha\beta}(y)=h_{\alpha\beta}(y) \ \ \ \text{on} \
B_{g(\cdot,T)}(P,a).$$ By Theorem 2.6 and estimates (2.21) and letting $a\rightarrow
\infty$, the solutions $\overset{a}{F}$ of $(2.3)_{a}$ on $M^{n}\times[0,T_1]$ have a convergent subsequence so that the limit is a solution of (2.3) with the estimates (2.4).
$\hfill\#$ 0.8cm
The uniqueness of the Ricci flow
=================================
0.5cm
Preliminary estimates for the Ricci-De Turck flow
-------------------------------------------------
Let $F(x,t)$ be a solution to (2.3) in Theorem 2.1 on $M^{n}\times[0,T_0]$. Let $\tilde{g}_{ij}(x,t)=h_{\alpha\beta}(F(x,t))\frac{\partial F^{\alpha}}{\partial x^{i}}
\frac{\partial F^{\beta}}{\partial x^{j}}$ be the one-parameter family of pulled back metrics $F^{*}{h}$. We will estimate $g_{ij}(x,t)$ in terms of $\tilde{g}_{ij}(x,t)$.
There exists a constant $0<T_5\leq T_0$ depending only on $k_0$ and $T$ such that for all $(x,t)\in M^{n}\times
[0,T_5]$, we have $$\tag{3.1}
\begin{split}
\frac{1}{C(n,k_0,T)}\tilde{g}_{ij}(x,t)\leq g_{ij}(x,t)&\leq
C(n,k_0,T)\tilde{g}_{ij}(x,t)\\
|\tilde{\nabla}^{k}g|_{\tilde{g}}\leq \frac{C(n,k_0,T,k)}{t^{\frac{k-1}{2}}}
\end{split}$$ for $k=1,2,\cdots$
Proof. We first consider the zero-order estimate of $g_{ij}(x,t)$. The estimate $|\nabla F|^{2}=\tilde{g}_{ij}g^{ij}\leq C$ in (2.4) implies $\tilde{g}_{ij}(x,t)\leq Cg_{ij}(x,t)$. For the reverse inequality, we compute the equation of $\tilde{g}_{ij}(x,t)$: $$\begin{aligned}
\frac{\partial}{\partial t}\tilde{g}_{ij}&=&\triangle
\tilde{g}_{ij}-2R_{ik}F^{\alpha}_{l}F^{\beta}_{j}h_{\alpha\beta}g^{kl}
+2R_{\alpha\beta\gamma\delta}F^{\alpha}_{i}F^{\beta}_{k}
F^{\gamma}_{j}F^{\delta}_{l}g^{kl}-2h_{\alpha\beta}F^{\alpha}_{k,i}F^{\beta}_{l,j}g^{kl}\\
&\geq&\triangle \tilde{g}_{ij}-
2R_{ik}\tilde{g}_{jl}g^{kl}-2k_0|\nabla
F|^{2}g_{ij}-2|\nabla^{2}F|^{2}g_{ij}\\
&\geq&\triangle
\tilde{g}_{ij}-2R_{ik}\tilde{g}_{jl}g^{kl}-C(n,k_0,T) g_{ij},\end{aligned}$$ by (2.4). Combining this with the Ricci flow equation gives $$\begin{aligned}
(\frac{\partial}{\partial t}-\triangle)(\tilde{g}_{ij}+
C(n,k_0,T)t
g_{ij}-\frac{1}{2ne^{2nk_0T}}g_{ij})\geqslant-2R_{ik}(\tilde{g}_{lj}+
C(n,k_0,T)t
g_{lj}-\frac{1}{2ne^{2nk_0T}}g_{lj})g^{kl}.\end{aligned}$$ Note that at $t=0$, $$(\tilde{g}_{ij}+
C(n,k_0,T)t
g_{ij}-\frac{1}{2ne^{2nk_0T}}g_{ij})\mid_{t=0}=g_{ij}(\cdot,T)-\frac{1}{2ne^{2nk_0T}}g_{ij}(\cdot,0)>0.$$ By applying the maximum principle to above equation, we obtain $$\tilde{g}_{ij}+ C(n,k_0,T)t
g_{ij}-\frac{1}{2ne^{2nk_0T}}g_{ij}>0$$ on $M^{n}\times[0,T_0]$. Let $T_5=\min\{T_0,\frac{1}{4ne^{2nk_0T}C(n,k_0,T)}\}$. Then we have $$\tilde{g}_{ij}\geq \frac{1}{4ne^{2nk_0T}} g_{ij}, \ \ \ \text{on}\ \ M^{n}\times[0,T_5].$$ This gives the zero-order estimate of $g_{ij}(x,t)$.
For the first order derivative of $g_{ij}$, we compute $$\begin{aligned}
\tilde{\nabla}_{k}g_{ij}=(\tilde{\nabla}_{k}-\nabla_{k})g_{ij}
=(\Gamma^{l}_{ki}-\tilde{\Gamma}^{l}_{ki})g_{lj}+(\Gamma^{l}_{kj}-\tilde{\Gamma}^{l}_{kj})g_{li}\end{aligned}$$ and $$\begin{aligned}
|(\Gamma^{l}_{ki}-\tilde{\Gamma}^{l}_{ki})|^{2}_{\tilde{g}}&=
&|(\Gamma^{p}_{ki}-\tilde{\Gamma}^{p}_{ki})\tilde{g}_{lp}|^{2}_{\tilde{g}}\\
&=&|\nabla_k\nabla_iF^{\alpha}\frac{\partial F^{\beta}}{\partial
x^{l}}h_{\alpha\beta}|_{\tilde{g}}\\
&\leqslant& C(n,k_0,T)|\nabla_k\nabla_iF^{\alpha}\frac{\partial
F^{\beta}}{\partial
x^{l}}h_{\alpha\beta}|_{g}\\
&\leqslant& C(n,k_0,T)|\nabla^{2}F|_{g,h}|\nabla F|_{g,h}\\
&\leqslant& C(n,k_0,T).\end{aligned}$$ This gives the first order estimate.
For higher order estimates, we prove it by induction. Suppose we have showed $$|\tilde{\nabla^{l}}g|_{\tilde{g}}\leqslant\frac{C}{t^{\frac{l-1}{2}}}
\ \ \ \ \text{for}\ \ l=1,2,\cdots,k-1,$$ $$|\tilde{\nabla^{l}}(\Gamma-\tilde{\Gamma})|_{\tilde{g}}\leqslant\frac{C}{t^{\frac{l}{2}}}
\ \ \ \ \text{for}\ \ l=0,1,\cdots,k-2.$$ Since by induction $$\begin{aligned}
|\tilde{\nabla}^{k-1}(\Gamma-\tilde{\Gamma})|_{\tilde{g}}&=&
|\tilde{\nabla}^{k-1}[(\Gamma-\tilde{\Gamma})\ast\tilde{g}]|_{\tilde{g}}\\
&=&|\sum_{j=0}^{k-1}\nabla^{k-1-j}[(\Gamma-\tilde{\Gamma})\ast\tilde{g}]\ast\sum_{i_1+1+\cdots+i_q+1=j
}\tilde{\nabla}^{i_1}(\Gamma-\tilde{\Gamma})\ast\cdots\ast\tilde{\nabla}^{i_q}(\Gamma-\tilde{\Gamma})|_{\tilde{g}}\\
&\leqslant& C(n,k_0,T)
\sum_{j=0}^{k-1}|\nabla^{k-1-j}(\nabla^{2}F\ast\nabla F)|_{g,h}\\
& & \times\sum_{i_1+1+\cdots+i_q+1=j}|\tilde
{\nabla}^{i_1}(\Gamma-\tilde{\Gamma})|_{\tilde{g}}\cdots |\tilde
{\nabla}^{i_q}(\Gamma-\tilde{\Gamma})|_{\tilde{g}}\\
&\leqslant&
C(n,k_0,T,k)(\frac{1}{t^{\frac{k-1-j}{2}}}\frac{1}{t^{\frac{j-2}{2}}}+\frac{1}{t^{\frac{k-1}{2}}})\\
&\leqslant& \frac{C(n,k_0,T,k)}{t^{\frac{k-1}{2}}}\end{aligned}$$ and $$\begin{aligned}
\tilde{\nabla}^{k}g&=&\tilde{\nabla}^{k-1}((\Gamma-\tilde{\Gamma})\ast
g
)\\&=&\sum_{i=0}^{k-1}\tilde{\nabla^{i}}(\Gamma-\tilde{\Gamma})\ast\tilde{\nabla}^{k-1-i}g,\end{aligned}$$ then we have $$|\tilde{\nabla}^{k}g|_{\tilde{g}}\leqslant
\frac{C}{t^{\frac{k-1}{2}}}.$$ This completes the induction argument and the proposition is proved.
$\hfill\#$
Let $F(x,t)$ be the solution of (2.3) in Theorem 2.1. Then $F(\cdot,t)$ are diffeomorphisms for all $t\in [0,T_5]$; moreover, there exists a constant $C(n,k_0,T)>0$ depending only on $n$, $k_0$ and $T$ such that $$d_{h}(F(x_1,t),F(x_2,t))\geqslant e^{-C(n,k_0,T)}d_{h}(x_1,x_2)$$ for all $x_1,x_2\in M^{n}$, $t\in[0,T_{5}]$.
Proof. Note that $$\frac{1}{C}\tilde{g}_{ij}(x,t)\leq g_{ij}(x,t)\leq
C\tilde{g}_{ij}(x,t)$$ implies that $F$ are local diffeomorphisms. So we only need to prove that $F(\cdot,t)$ is injective. Suppose not. Then there exist two points $x_1\neq x_2$, such that $ F(x_1,t)= F(x_2,t)$, for some $t_0\in(0,T_5]$. Assume $t_0>0$ be the first time so that $ F(x_1,t)= F(x_2,t)$. Choose small $\delta>0$, such that there exist a neighborhood $\tilde{O}$ of $F(x_1,t_0)$ and a neighborhood $O$ of $x_1$ such that $F^{-1}(\cdot,t)$ is a diffeomorphism from $\tilde{O}$ to $O$ for all $t\in[t_0-\delta,t_0]$, moreover, letting $\tilde{\gamma_t}$ be a shortest geodesic( parametrized by arc length) on the target $(N^{n},h_{\alpha\beta})$ connecting $F(x_1,t)$ and $F(x_2,t)$, we require $\tilde{\gamma}\in \tilde{O}$ for $t\in[t_0-\delta,t_0]$. We compute $$\begin{aligned}
\frac{\partial}{\partial t}d_{h}(F(x_1,t),F(x_2,t))&=&\langle
V,{\tilde{\gamma}}^{\prime}(l)\rangle_{h}-\langle
V,{\tilde{\gamma}}^{\prime}(0)\rangle_{h}
\end{aligned}$$ where $\tilde{\gamma}(0)=F(x_1,t)$ , $\tilde{\gamma}(l)=F(x_2,t)$, and $V^{\alpha}=\triangle
F^{\alpha}$. Now we pull back everything by $F^{-1}$ to $O$, $$\begin{aligned}
\frac{\partial}{\partial t}d_{h}(F(x_1,t),F(x_2,t))&=&\langle
P_{-\tilde{\gamma}}V-V,{\gamma}^{\prime}(0)\rangle_{F^{*}h}\\
&\geq&- \sup_{x\in F^{-1}\tilde{\gamma}}|\tilde{\nabla}V|(x,t) d_{h}(F(x_1,t),F(x_2,t))
\end{aligned}$$ where $P_{\tilde{\gamma}}$ is the parallel translation along $F^{-1}\tilde{\gamma}$ using the metric $F^{*}h$. By (2.4), $$\begin{aligned}
|\tilde{\nabla}_{k} V^{l}|_{\tilde{g}}&=&|\tilde{\nabla}_{k}(\triangle
F^{\alpha}\frac{\partial F^{\alpha}}{\partial x^{l} }h_{\alpha\beta})|_{\tilde{g}}\\&\leqslant&
|\nabla_{k}(\triangle
F^{\alpha}\frac{\partial F^{\alpha}}{\partial x^{l} }h_{\alpha\beta})|_{\tilde{g}}+
C|\Gamma-\tilde{\Gamma}|_{\tilde{g}}|\nabla^{2}F|_{g,h}|\nabla F|_{g,h}\\
&\leqslant&
C(n,k_0,T)(|\nabla^{3}F|_{g,h}|\nabla F|_{g,h}+|\nabla^{2}F|^{2}_{g,h}+|\nabla^{2}F|^{2}_{g,h}|\nabla F|_{g,h})\\
&\leqslant& \frac{C(n,k_0,T)}{\sqrt{t}}.
\end{aligned}$$ It follows that we have $$d_{h}(F(x_1,t),F(x_2,t))\leqslant e^{C(\sqrt{t_0}-\sqrt{t_0-\delta})
}d_{h}(F(x_1,t_0),F(x_2,t_0))=0,$$ for $t\in [t_0-\delta,t_0]$, which contradicts with the choice of $t_0$. So $F(\cdot,t)$ are diffeomorphisms.
By choosing $\tilde{O}=N^{n}$, $O=M^{n}$, the above computation also gives $$d_{h}(F(x_1,t),F(x_2,t))\geqslant e^{-C(n,k_0,T)}d_{h}(x_1,x_2).$$ The proof of the proposition is completed.
$\hfill\#$
Ricci-De Turck flow
-------------------
From the previous section, we know that the harmonic map flow coupled with Ricci flow (2.3)with identity as initial data has a short time solution $F(x,t)$ on $M^{n}\times[0,T_5]$, which remains being a diffeomorphism with good estimates (2.4). Let ${(F^{-1})}^{*}g$ be one-parameter family of pulled back metrics on the target $(N^{n},h_{\alpha\beta})$. Denote $g_{\alpha\beta}(y,t)$. Then $g_{\alpha\beta}(y,t)$ satisfies the so called Ricci-De Turck flow: $$\frac{\partial}{\partial
t}g_{\alpha\beta}(y,t)=-2R_{\alpha\beta}(y,t)+\nabla_{\alpha}
V_{\beta}+\nabla_{\beta} V_{\alpha} \eqno(3.3)$$ where $V^{\alpha}=g^{\beta\gamma}(\Gamma^{\alpha}_{\beta\gamma}(g)-{\Gamma}^{\alpha}_{\beta\gamma}(h))$, $\Gamma^{\alpha}_{\beta\gamma}(g)$ and ${\Gamma}^{\alpha}_{\beta\gamma}(h)$ are the Christoffel symbols of the metrics $g_{\alpha\beta}(y,t)$ and $h_{\alpha\beta}(y)$ respectively.
By (3.1) of Proposition 3.1, we already have the following estimates for $g_{\alpha\beta}(y,t)$
$$\frac{1}{C(n,k_{0},T)}h_{\alpha\beta}(y)\leqq g_{\alpha\beta}(y,t) \leqq C(n,k_{0},T) h_{\alpha\beta}(y)$$ $$|{\nabla^{k}_{h}}g|_{h}\leqq
\frac{C(n,k_{0},T,k)}{t^{\frac{k-1}{2}}}.\eqno(3.4)$$ on $N^{n}\times[0,T_5]$.
Let $g_{ij}(x,t)$ and $\bar{g}_{ij}(x,t)$ be two solutions to the Ricci flow with bounded curvature and with the same initial value as assumed in Theorem 1.1. We solve the corresponding harmonic map flow with same target $(N^{n},h_{\alpha\beta})=(M^{n},g_{ij}(\cdot,T))$ by $$\tag{3.5}
\left\{
\begin{split}
\quad \frac{\partial}{\partial t}F(x,t)&=\triangle F(x,t), \\
F(\cdot,0)&= identity,
\end{split}
\right.$$ and $$\tag{3.6}
\left\{
\begin{split}
\quad \frac{\partial}{\partial t}\bar{F}(x,t)&=\bar{\triangle}\bar{F}(x,t), \\
\bar{F}(\cdot,0)&=identity,
\end{split}
\right.$$ respectively. Then we obtained two solutions $F(x,t)$ and $\bar{F}(x,t)$ on $M^{n}\times[0,T_5]$. It is clear that $\bar{F}(x,t)$ still satisfies (2.4), Proposition 3.1 and Proposition 3.2. Let $\bar{g}_{\alpha\beta}(y,t)={(\bar{F}^{-1})}^{*}\bar{g}(y,t)$, then $\bar{g}_{\alpha\beta}(y,t)$ still satisfies (3.4). Now we have two solutions $g_{\alpha\beta}(y,t)$ and $\bar{g}_{\alpha\beta}(y,t)$ to the Ricci De-Turck flow with same initial data and with good estimates (3.4).
There holds $$g_{\alpha\beta}(y,t)=\bar{g}_{\alpha\beta}(y,t)$$ on $N^{n}\times [0,T_5]$.
Proof. We can write the Ricci-De Turck flow (3.3) by using the fixed metric $h_{\alpha\beta}(y)$ in the following form (see [@Sh1]): $$\tag{3.7}
\begin{split}
\quad \frac{\partial}{\partial t}g_{\alpha\beta}=&g^{\gamma\delta}\tilde{\nabla}_{\gamma}\tilde{\nabla}_{\delta}g_{\alpha\beta}-
g^{\gamma\delta}g_{\alpha\xi}\tilde{g}^{\xi\eta}\tilde{R}_{\beta\gamma\eta\delta}
-g^{\gamma\delta}g_{\beta\xi}\tilde{g}^{\xi\eta}\tilde{R}_{\alpha\gamma\eta\delta}+\frac{1}{2}g^{\gamma\delta}g^{\xi\eta}
(\tilde{\nabla}_{\alpha}g_{\xi\gamma}\tilde{\nabla}_{\beta}g_{\eta\delta}\\&+
2 \tilde{\nabla}_{\gamma}g_{\beta\xi}\tilde{\nabla}_{\eta}g_{\alpha\delta}-2\tilde{\nabla}_
{\gamma}g_{\beta\xi}\tilde{\nabla}_{\delta}g_{\alpha\eta}-2\tilde{\nabla}_{\beta}
g_{\xi\gamma}\tilde{\nabla}_{\delta}g_{\alpha\eta}
-2 \tilde{\nabla}_{\alpha}g_{\xi\gamma}\tilde{\nabla}_{\delta}g_{\beta\eta})
\end{split}$$ where $\tilde{g}_{\alpha\beta}=h_{\alpha\beta},$ $\tilde{\nabla}$ and $\tilde{R}$ are the covariant derivative and the curvature of $\tilde{g}_{\alpha\beta}$. Note that $\bar{g}_{\alpha\beta}$ also satisfies (3.7), then the difference $g_{\alpha\beta}-\bar{g}_{\alpha\beta}$ satisfies the following equation: $$\tag{3.8}
\begin{split}
\frac{\partial}{\partial
t}(g-\bar{g})=&g^{\gamma\delta}\tilde{\nabla}_{\gamma}\tilde{\nabla}_{\delta}(g-\bar{g})
+g^{-1}\ast\bar{g}^{-1}\ast\tilde{\nabla}^{2}\bar{g}\ast(\bar{g}-g)\\
&+ \bar{g}^{-1}\ast\tilde{g}^{-1}\ast\tilde{Rm}\ast(g-\bar{g})+g^{-1}\ast\bar{g}^{-1}\ast{g}\ast\tilde{g}^{-1}\ast\tilde{Rm}\ast(g-\bar{g})\\
& +g^{-1}\ast
g^{-1}\ast\bar{g}^{-1}\ast\tilde{\nabla}g\ast\tilde{\nabla}g\ast(g-\bar{g})+g^{-1}\ast
\bar{g}^{-1}\ast\bar{g}^{-1}\ast\tilde{\nabla}g\ast\tilde{\nabla}g\ast(g-\bar{g})\\&
+
\bar{g}^{-1}\ast\bar{g}^{-1}\ast\tilde{\nabla}g\ast\tilde{\nabla}(g-\bar{g})+\bar{g}^{-1}\ast\bar{g}^{-1}\ast\tilde{\nabla}\bar{g}\ast\tilde{\nabla}(g-\bar{g})
\end{split}$$ since $g^{\alpha\beta}-\bar{g}^{\alpha\beta}=g^{\alpha\xi}\bar{g}^{\eta\beta}(\bar{g}_{\eta\xi}-g_{\eta\xi})$. Let $$|g-\bar{g}|^{2}=\tilde{g}^{\alpha\gamma}\tilde{g}^{\beta\delta}
(g_{\alpha\beta}-\bar{g}_{\alpha\beta})(g_{\gamma\delta}-\bar{g}_{\gamma\delta}).$$ It follows from (3.8) that: $$\begin{aligned}
(\frac{\partial}{\partial
t}-g^{\gamma\delta}\tilde{\nabla}_{\gamma}\tilde{\nabla}_{\delta})|g-\bar{g}|^{2}
&\leqslant&-2g^{\xi\eta}\tilde{g}^{\alpha\gamma}\tilde{g}^{\beta\delta}
(\tilde{\nabla}_{\xi}g_{\alpha\beta}-\tilde{\nabla}_{\xi}\bar{g}_{\alpha\beta})
(\tilde{\nabla}_{\eta}g_{\gamma\delta}-\tilde{\nabla}_{\eta}\bar{g}_{\gamma\delta})\\&
& +100[|\tilde{Rm}|(1+|g||g^{-1}|)|\bar{g}^{-1}|
+|\tilde{\nabla}^{2}\bar{g}||\bar{g}^{-1}||g^{-1}|\\& & \ \ \ \ \ +|\tilde{\nabla}g|^{2}
(|\bar{g}^{-1}|^{2}|g^{-1}|+|\bar{g}^{-1}||g^{-1}\|^{2})]|g-\bar{g}|^{2}\\&
& +100|\bar{g}^{-1}|^{2}(|\tilde{\nabla}g|+|\tilde{\nabla}\bar{g}|)|\tilde{\nabla}(g-\bar{g})||g-\bar{g}|
\end{aligned}$$ where all the norms are computed with the metric $\tilde{g}=h$. By Cauchy-Schwartz inequality and (3.4), we have $$\tag{3.9}
\begin{split}
(\frac{\partial}{\partial
t}-g^{\gamma\delta}\tilde{\nabla}_{\gamma}\tilde{\nabla}_{\delta})|g-\bar{g}|^{2}
\leqslant&-2g^{\xi\eta}\tilde{g}^{\alpha\gamma}\tilde{g}^{\beta\delta}
(\tilde{\nabla}_{\xi}g_{\alpha\beta}-\tilde{\nabla}_{\xi}\bar{g}_{\alpha\beta})
(\tilde{\nabla}_{\eta}g_{\gamma\delta}-\tilde{\nabla}_{\eta}\bar{g}_{\gamma\delta})\\
&
+\frac{C}{\sqrt{t}}|g-\bar{g}|^{2}+C|\tilde{\nabla}(g-\bar{g})||g-\bar{g}|\\ \leqslant&
\frac{C}{\sqrt{t}}|g-\bar{g}|^{2}
\end{split}$$ on $N^{n}\times [0,T_5]$.
Let $\varphi_1$ be the nonnegative function in Lemma 2.2 with $a=1$, then $$\begin{aligned}
\frac{1}{C}(1+\tilde{d}(y,p))&\leqslant& \varphi_1(y)\leqslant C_0\tilde{d}(y,p)\ \ \ \text{on} \ N^{n}\backslash B(P,2),\\
|\tilde{\nabla}\varphi_1|&+&|\tilde{\nabla}^{2}\varphi_1|\leqslant C, \ \ \ \ \text{on} \
N^{n}.
\end{aligned}$$ For any fixed $t$ and any $\varepsilon>0$, consider the maximum of $|g-\bar{g}|^{2}-\varepsilon
\varphi$. Clearly, the maximum is achieved at some point $P_{\varepsilon}^{t}$ and there hold $$\begin{aligned}
|g-\bar{g}|^{2}(P_{\varepsilon}^{t})&\geqslant&
|g-\bar{g}|^{2}(y)-\varepsilon\varphi(y),\\
|\tilde{\nabla}|g-\bar{g}|^{2}|(P_{\varepsilon}^{t})\leqslant
C\varepsilon
&,&\tilde{\nabla}_{\alpha}\tilde{\nabla}_{\beta}|g-\bar{g}|^{2}
(P_{\varepsilon}^{t})\leqslant
C\varepsilon \tilde{g}_{\alpha\beta}(P_{\varepsilon}^{t})
\end{aligned}$$ for all $y\in N^{n}$. This gives $$\tag{3.10}
\begin{split}
\limsup_{\varepsilon\rightharpoonup0}|g-\bar{g}|^{2}
(P_{\varepsilon}^{t})&=\sup|g-\bar{g}|^{2}\\
g^{\alpha\beta}\tilde{\nabla}_{\alpha}\tilde{\nabla}_
{\beta}|g-\bar{g}|^{2}(P_{\varepsilon}^{t})&\leqslant
C\varepsilon
\end{split}$$ by the equivalence of $g$ and $\tilde{g}$.
Define a function $$|g-\bar{g}|^{2}_{max}(t)=\sup_{y\in N^{n}}|g-\bar{g}|^{2}(y,t).$$ By (3.9), and (3.10), we have $$\begin{aligned}
\frac{d^{+}}{d t}|g-\bar{g}|^{2}_{max}(t)&\leqslant&
\frac{C}{\sqrt{t}}|g-\bar{g}|^{2}_{max}(t)\end{aligned}$$ and then $$|g-\bar{g}|^{2}_{max}(t)\leqslant
e^{C\sqrt{T}}|g-\bar{g}|^{2}_{max}(0)=0$$ Therefore the proof of the Proposition 3.3 is completed.
$\hfill\#$
Proof of the main theorem
-------------------------
Let $g_{ij}(x,t)$ and $\bar{g}_{ij}(x,t)$ be two solutions to the Ricci flow (1.1) with bounded curvature and with the same initial data. We solve the corresponding harmonic map flow (3.5) and (3.6) with the same target $(N^{n},h_{\alpha\beta})=(M^{n},g_{ij}(\cdot,T))$ respectively. We obtain two solutions $F(x,t)$ and $\bar{F}(x,t)$ which are diffeomorphisms for $t\in[0,T_5]$, where $T_5>0$ depends only on $n, k_0, T$. Then ${(F^{-1})}^{*}g$ and ${(\bar{F}^{-1})}^{*}\bar{g}$ are two solutions to the Ricci-De Turck flow with the same initial value. It follows from Proposition 3.3 that $${(F^{-1})}^{*}g={(\bar{F}^{-1})}^{*}\bar{g},$$ on $N^{n}\times[0,T_5]$. So in order to prove $g_{ij}(x,t)\equiv\bar{g}_{ij}(x,t)$, we only need to show $F\equiv\bar{F}$. Let $$\begin{aligned}
V^{\alpha}(y,t)&=&g^{\beta\gamma}(\tilde{\Gamma}^{\alpha}_
{\beta\gamma}-{\Gamma}^{\alpha}_{\beta\gamma})=-(\triangle
F\circ F^{-1})^{\alpha}\\
\bar{V}^{\alpha}(y,t)&=&\bar{g}^{\beta\gamma}(\tilde{\Gamma}
^{\alpha}_{\beta\gamma}-\bar{\Gamma}^{\alpha}_{\beta\gamma})=-
(\bar{\triangle}\bar{F}\circ
\bar{F}^{-1})^{\alpha}.
\end{aligned}$$ be two one-parameter family of vector fields on $N^{n}$, where ${g}_{\alpha\beta}(y,t)=({({F}^{-1})}^{*}{g})_
{\alpha\beta}(y,t)$ and $\bar{g}_{\alpha\beta}(y,t)=({(\bar{F}^{-1})}^{*}\bar{g})_
{\alpha\beta}(y,t)$. By Proposition 3.3, we have $g_{\alpha\beta}(y,t)=\bar{g}_
{\alpha\beta}(y,t)$, thus the vector fields $V\equiv \bar{V}$ on the target $N^{n}$. Therefore, $F$ and $\bar{F}$ satisfy the same ODE equation with the same initial value: $$\begin{aligned}
\frac{\partial}{\partial t}F&=&V\circ F,\\
F(\cdot,0)&=&identity,\end{aligned}$$ and $$\begin{aligned}
\frac{\partial}{\partial t}\bar{F}&=&V\circ \bar{F},\\
\bar{F}(\cdot,0)&=&identity,
\end{aligned}$$ By the same calculation as in the proof of Proposition 3.2, we have $$\begin{aligned}
\frac{\partial}{\partial t}{d}_{N^{n}}(F(x,t),\bar{F}(x,t))
&\leqslant& \sup_{y\in N^{n}}|\tilde{\nabla}V|(y,t)
{d}_{N^{n}}(F(x,t),\tilde{F}(x,t))\\
&\leqslant& \frac{C}{\sqrt{t}}
{d}_{N^{n}}(F(x,t),\tilde{F}(x,t)).
\end{aligned}$$ This gives $${d}_{N^{n}}(F(x,t),\bar{F}(x,t))\leqslant
e^{C\sqrt{T}} {d}_{N^{n}}(F(x,0),\bar{F}(x,0))=0,$$ which concludes that $$F(x,t)\equiv\bar{F(x,t)}.$$ Thus $g(x,t)=\bar{g}(x,t)$, for all $x\in M^{n}$ and $t\in [0,T_{5}]$ and for some $T_{1}>0$. Clearly, we can extend the interval $[0,T_{1}]$ to the whole $[0,T]$ by continuity method.
Therefore we complete the proof of the Theorem 1.1.
$\hfill\#$
Finally, Corollary 1.2 is a direct consequence of Theorem 1.1. Indeed, since $G$ is the isometry group of $g_{ij}(x,0)$, then for any $\sigma\in G$, $\sigma^{*}g(\cdot,t)$ is still a solution to the Ricci flow with bounded curvature and $\sigma^{*}g(\cdot,t)\mid_{t=0}=\sigma^{*}g(\cdot,0)=g(\cdot,0).$ By applying Theorem 1.1, we have $\sigma^{*}g(\cdot,t)=g(\cdot,t)$, $\forall t\in[0,T]$. So the corollary follows.
$\hfill\#$
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|
---
abstract: 'We propose a straightforward and efficient procedure to perform dynamical mean-field (DMFT) calculations on the top of the static mean-field LDA+U approximation. Starting from self-consistent LDA+U ground state we included multiplet transitions using the Hubbard-I approximation, which yields a very good agreement with experimental photoelectron spectra of $\delta$-Pu, Am, and their selected compounds.'
author:
- 'Alexander Shick, Jindrich Kolorenč, Ladislav Havela, Václav Drchal, and Thomas Gouder'
title: 'Multiplet effects in the electronic structure of $\delta$-Pu, Am and their compounds'
---
Introduction
============
Electronic, magnetic and superconducting properties of actinide elements recently attracted significant interest and attention in the condensed matter physics. Most intriguing are the phenomena at the localization threshold of the $5f$ series, which is crossed between Pu and Am, where the electron-electron correlations play a prominent role [@Kotliar; @grivenau].
During last few years, electronic structure calculations of Pu and Am based on the conventional band theoretical methods (the local density or generalized gradient expantion approximations LDA/GGA to the density functional theory) could not explain essental experimental data. While, the LDA/GGA band structure calculations predict a local magnetic moment (ordered or disordered) to form on the Pu [@soderlind] and Am [@soderlind2] atoms, none of them were seen in the experiment [@lashley]. Also, the same papers attempted to evaluate the photoemission spectra (PES) and electronic specific heat in Pu and Am making use of single-particle LDA/GGA densities of states (DOS), incorrectly assuming weak electron correlation character of 5$f$ systems at the borderline between the localized, nonbonding, behaviour and the bonding situation of electronic bands.
It was shown recently that the [*around-mean-field*]{} (AMF)-LSDA+U correlated band theory gives non-magnetic ground state for Pu [@shick05] and Am [@shick06]. Also, the equilibrium volumes and bulk moduli for $\delta$-Pu and [*fcc-*]{}Am are calculated in a good agreement with experiment. However, there is a clear disagreement between the AMF-LDA+U calculated DOS and PES, questioning the validity of this approximation. The DMFT calculations [@EPLP] using the (AMF)-LSDA+U form of interacting Hamiltonian demonstrated that experimental PES and high $\gamma$ coefficient of the electronic specific heat in $\delta$-Pu and its selected compounds [@PLK05; @PKL06] originate from the excitations, and not from the ground state DOS alone. While accounting for dynamical fluctuations, the DMFT of Ref. [@EPLP] does not take into account atomic-like excitations which can play an important role in Pu and Am.
In this Letter, we develop a DMFT based computational scheme based on multi-orbital Hubbard-I approximation (HIA) [@LK99; @Dai03; @Dai05] including the spin-orbit coupling (SOC) which explicitly accounts for the atomic-like multiplet transition excitations in Pu, Am and their compounds. Starting from the non-magnetic ground state calculated with the static mean-field AMF-LSDA+U approximation, we obtain excitation spectra of Pu and Am in surprisingly good agreement with PES, in support of the atomic-like origin of the electronic excitations in these materials.
Methodology
===========
We start with the multi-band Hubbard Hamiltonian [@LK99] $H = H^0 + H^{\rm int} $, where $$\begin{aligned}
\label{eq:1ph} H^0 = \sum_{i,j} \sum_{\alpha, \beta} H^0_{i \alpha,
j \beta}
c^{\dagger}_{i \alpha} c_{j \beta}
= \sum_{\bf k} \sum_{\alpha, \beta} H^0_{\alpha, \beta} ({\bf k})
c^{\dagger}_{\alpha}({\bf k}) c_{\beta}({\bf k}) \, ,\end{aligned}$$ is the one-particle Hamiltonian found from ab initio electronic structure calculations for a periodic crystal. The indices $i,j$ label the lattice sites, $\alpha = (\ell m \sigma)$ denote the spinorbitals, and ${\bf k}$ is the k-vector from the first Brillouin zone. It is assumed that the electron-electron correlations between s, p, and d electrons are well described within the density functional theory, while the correlations between the f electrons have to be considered separately by introducing the interaction Hamiltonian $$\begin{aligned}
\label{eq:hint} H^{\rm int} = \frac{1}{2} \sum_i
\sum_{m_1,m_2,m_3,m_4}
\sum_{\sigma,\sigma'}
\langle m_1,m_2|V_{i}^{ee}|m_3,m_4 \rangle
c_{i m_1 \sigma}^{\dagger} c_{i m_2 \sigma'}^{\dagger}
c_{i m_4 \sigma'} c_{i m_3 \sigma} \, .\end{aligned}$$ The $V^{ee}$ is an effective on-site Coulomb interaction [@LK99] expressed in terms of the Slater integrals $F_k$ and the spherical harmonic ${|lm \rangle}$. The corresponding one-particle Green function $$\begin{aligned}
\label{eq:1gf} G({\bf k},z) = \Big( z + \mu - H^0 ({\bf k}) - \Sigma({\bf k},z) \Big)^{-1}\end{aligned}$$ is expressed via $H^0$ and the one-particle selfenergy $\Sigma({\bf k},z)$ which contains the electron-electron correlations, where $z$ is a (complex) energy with respect to the chemical potential $\mu$. The interactions (\[eq:hint\]) act only in the subspace of f-states. Consequently, the selfenergy $\Sigma({\bf k},z)$ is nonzero only in the subspace of the f-states. The simplest mean-field approximation (often called L(S)DA+U) [@LAZ95] neglects the ${\bf k}$- and energy-dependence of $\Sigma$ replacing it by the on-site potential $V_{+U}$. For a given set of spin-orbitals $|m \sigma \rangle$ the potential reads: $$\begin{aligned}
[V_{+U}]^{\sigma}_{mm'} = \sum_{p,q,\sigma'} \Big( \langle
m,p|V^{ee}|m',q \rangle - \langle m,p|V^{ee}|q,m' \rangle
\delta_{\sigma,\sigma '} \Big) n^{\sigma'}_{p,q} \, ,
\label{eq:4}\end{aligned}$$ where $n^{\sigma}_{mm'}$ is the local orbital occupation matrix of the orbitals $|m \sigma \rangle$. It was shown in Ref. [@SLP99] that the Kohn-Sham equation with the potential Eq.(\[eq:4\]) can be obtained by making use of variational minimization of the LDA+U total energy functional (i.e. of the expectation value of the multiband Hubbard Hamiltonian) in a way similar to the conventional density functional theory [@PZ81].
In what follows we use the local approximation for the selfenergy, i.e., we assume that it is site-diagonal and therefore independent of ${\bf k}$. Then we can employ the “impurity" method of Ref. [@LK99]. We first obtain a local Green function integrating $G({\bf k},z)$, Eq.(\[eq:1gf\]), over the Brillouin zone $$\begin{aligned}
G(z) = \frac{1}{V_{BZ}} \int_{BZ}d{\bf k}
\Big( z + \mu - H^0 ({\bf k}) - \Sigma(z) \Big)^{-1}
\label{eq:gff}\end{aligned}$$ and define a “bath" Green function (the so-called Weiss field) ${\cal G}_{0}(z)$ $$\begin{aligned}
{\cal G}_{0}(z) = \Big( G^{-1}(z) + \Sigma(z) \Big)^{-1} \, .
\label{eq:6}\end{aligned}$$ All ${\cal G_0}$, $G$, and $\Sigma$ in Eqs. (\[eq:gff\], \[eq:6\]) are matrices in the subspace of the $|m \sigma \rangle$ f-orbitals.
The DMFT self-consistency condition is now formulated by equating $G
(z)$, Eq.(\[eq:gff\]), to the Green function $\tilde{G}(z)$ of a single-impurity Anderson model (SIAM) [@PWN97] $$\begin{aligned}
\label{eq:8} \tilde{G}(z) = \Big( z + \mu - \epsilon_0 -
\tilde{\Delta}(z) -
\tilde{\Sigma}(z) \Big)^{-1}\end{aligned}$$ where $\tilde{\Delta}(z)$ is the effective hybridization function and $\tilde{\Sigma}(z)$ is the SIAM self-energy. We write the Eq.(\[eq:8\]) in the form of the Eq.(\[eq:6\]) $$\begin{aligned}
\Big( z + \mu - \epsilon_0 - \tilde{\Delta}(z) \Big)^{-1}
= \tilde{{\cal G}}_0(z)
= \Big( \tilde{G}^{-1}(z) + \tilde{\Sigma}(z) \Big)^{-1}
\label{eq:9}\end{aligned}$$ from which it follows that $\tilde{\cal G}_0(z)$ has the meaning of the SIAM “bath" Green function. The iterative procedure to solve the periodic lattice problem in the DMFT approximation is now formulated in a usual way [@LK99]: starting with single particle Hamiltonian $H^0({\bf k})$ and a guess for local $\Sigma$, the local Green function is calculated using the Eq.(\[eq:gff\]) and the “bath" Green function is calculated from Eq.(\[eq:6\]); the SIAM is solved for this “bath" and a new local $\Sigma$ is calculated from Eq.(\[eq:8\]), which is inserted back into Eq.(\[eq:gff\]).
The LDA+U procedure can be viewed in the same way. There is no need to apply the full DMFT iterative procedure described above and to solve the SIAM with the “bath" from Eq.(\[eq:6\]). The self-energy $\Sigma(z)$ is now approximated by a static potential $V_{+U}$ from Eq.(\[eq:4\]), and the well-known relation between the Green function, Eq.(\[eq:6\]), and the local orbital occupation matrix $n^{\sigma}_{mm'} \, = \, -\pi^{-1}{\rm Im}
\int^{\mu} {\rm d} E \, G(E)^{\sigma}_{mm'}$ is used. In addition, the charge- and spin-densities needed to construct the single-particle Hamiltonian $H_0({\bf k})$ in Eq.(\[eq:gff\]) are calculated self-consistently. We emphasize that LDA+U approximation is generically connected with the LDA+DMFT procedure.
Hubbard-I approximation
=======================
Here we attempt to build a computational scheme based on self-consistent static mean-field LDA+U ground state that will allow us to access the correlated electron excitations. We specifically choose the [*around-mean-field*]{} version of LSDA+U (AMF-LDA+U) which was shown to describe correctly the non-magnetic ground state properties of $\delta$-Pu [@shick05], fcc-Am, Pu-Am alloys [@shick06], and selected Pu-compounds [@PLK05]. We extend our previous works [@shick05; @shick06] towards the DMFT to account for the multiplet transitions which are necessary for a correct description of PE excitation spectra. We use the multiorbital HIA which is suitable for incorporating the multiplet transitions in the electronic structure, as it is explicitly based on the exact diagonalization of an isolated atomic-like f-shell.
Further, we restrict our formulation to the paramagnetic phase, and we closely follow the procedure described in [@LK99]. We construct the atomic Hamiltonian including the spin-orbit coupling (SOC): $$\begin{aligned}
H^{\rm at} = \sum_{m_1,m_2}^{\sigma, \sigma'}
\xi ({\bf l} \cdot {\bf s})_{m_1 m_2}^{\sigma \; \; \sigma'}
c_{m_1 \sigma}^{\dagger}c_{m_2 \sigma'}
+ \frac{1}{2} \sum_{m_{1}...m_{4}}^{\sigma, \sigma'} \langle
m_1 m_2|V^{ee}|m_3 m_4 \rangle c_{m_1 \sigma}^{\dagger} c_{m_2
\sigma'}^{\dagger} c_{m_4 \sigma'} c_{m_3 \sigma} \, , \label{eq:10}\end{aligned}$$ and perform exact diagonalization of $H^{\rm at} |\nu \rangle =
E_\nu |\nu\rangle $ to obtain all possible eigenvalues $E_\nu$ and eigenvectors $|\nu \rangle$.
The HIA “chemical potential" $\mu_{H}$ is then calculated as $$\langle n \rangle = \frac{1}{Z} {\rm Tr} \Big[ N \exp(-\beta [H^{\rm
at}
- \mu_H \hat{N}]) \Big]
\label{eq:11}$$ for a given number of particles $\langle n \rangle$. Here, $\beta$ is the inverse temperature and $Z$ is the partition function.
Finally, the atomic Green function is calculated as follows: $$\begin{aligned}
[G^{\rm at}(z)]_{m_1 m_2}^{\sigma \; \; \sigma'}(z) &=& \frac{1}{Z}
\, \sum_{\nu,\mu} \frac{\langle \mu|c_{m_1 \sigma}|\nu \rangle
\langle \nu|c_{m_2 \sigma'}^{\dagger}|\mu \rangle} {z + (E_\mu -
\mu_H N_{\mu}) - (E_\nu - \mu_H N_{\nu})}
\nonumber \\
&& \times [\exp(-\beta (E_\nu - \mu_H N_\nu))
+ \exp(-\beta (E_\mu - \mu_H N_\mu))]
\label{eq:12}\end{aligned}$$ and the atomic self-energy is evaluated as: $$\begin{aligned}
\label{eq:13} [\Sigma_{H}(z)]^{\sigma \; \; \sigma'}_{mm'} =
z \delta_{m_1 m_2} \delta_{\sigma \sigma'} - \xi ({\bf l} \cdot
{\bf s})_{m_1 m_2}^{\sigma \; \; \sigma'} - \Big[ \Big(G^{\rm
at}(z)\Big)^{-1}\Big]_{m_1 m_2}^{\sigma \; \; \sigma'} \, .\end{aligned}$$
Assuming that the self-consistent LDA+U calculations are performed, the LDA+U Green function is evaluated as $$\begin{aligned}
G_{+U}(z) = \frac{1}{V_{\rm BZ}} \int_{\rm BZ}d \, {\bf k}
\Big(z+\mu - H_0({\bf k}) - V_{+U} \Big)^{-1} \, . \label{eq:14}\end{aligned}$$ In Eq.(\[eq:14\]) we took into account the presence of SOC for both $H_0({\bf{k}})$ and $\hat{V}_{+U}$ as described in Ref. [@shick05]. We used LDA+U eigenvalues and eigenfunctions calculated in the full-potential LAPW basis [@SLP99; @shick05] to construct the on-site spin-orbital Green function matrix, Eq.(\[eq:14\]).
We evaluate the static “bath" ${\cal G}_{0}(z)$ from Eq.(\[eq:6\]) and find the hybridization function $\Delta(z)$ together with $( \epsilon_0 - \mu)$ which determines the energy of “impurity" level with respect to the solid potential: $$\begin{aligned}
\Big( z + \mu - \epsilon_{+U} - \xi ({\bf l} \cdot {\bf s}) -
\Delta(z) \Big)^{-1} = {\cal G}^0_{+U}(z)
= \Big( G_{+U}^{-1}(z) + V_{+U}(z) \Big)^{-1} \, . \label{eq:15}\end{aligned}$$ Here we added the SOC explicitly and assumed the paramagnetic case.
We point on a difference in a physical meaning of $\epsilon_0$ in the original SIAM and in the auxiliary SIAM used in LDA+DMFT (LDA+U): in the former, it labels the position of the non-interacting f(d)-level so that the chemical potential is determined self-consistently for a given $\Delta(z)$; in the latter, the chemical potential $\mu$ is determined by a periodical crystal - basically by the Green function of Eq.(\[eq:gff\]) - so that the auxiliary $\epsilon_0$ accommodates all the electrostatic shifts between correlated electrons and the potential of the solid.
Now we can formulate a simple approximate procedure to solve the DMFT Eqs.(\[eq:gff\],\[eq:6\],\[eq:8\]) with Hubbard-I self-energy, Eq.(\[eq:13\]). We assume that the self-consistent static mean-field LDA+U already gives a correct number of particles and hybridization. Using the LDA+U Green function, Eq.(\[eq:10\]), and the potential, Eq.(\[eq:4\]), we evaluate the Weiss field ${\cal G}_{+U}^0(z)$. Then we insert the HIA self-energy, calculated for the same number of correlated electrons as given by LDA+U, into this “bath", and calculate the new Green function $$\begin{aligned}
G(z) = \Big( [{\cal G}_{0}^{+U}(z)]^{-1} + (\epsilon_{+U} -
\epsilon_{H}) - \Sigma_{H}(z) \Big)^{-1} \, , \label{eq:16}\end{aligned}$$ where $(\epsilon_{+U} - \epsilon_{H})$ is chosen so as to ensure that $n \; = \; \pi^{-1} {\rm Im} \int^{\mu} d E {\rm Tr}[ G(E)]$ is equal to a given number of correlated electrons [@shift].
Valence band photoemission spectra
==================================
Starting from self-consistent AMF-LSDA+U ground state solutions for ${\delta}$-Pu [@shick05], and *fcc*-Am [@shick06], and making use of the corresponding eigenvalues and eigenfunctions we evaluate the $G_{+U}(z)$ Green function given by Eq.(\[eq:14\]) and the ${\cal G}_{0}^{+U}(z)$ “bath" Green Function (Eq.(\[eq:15\])) [@details]. In HIA calculations Eq.(\[eq:10\] - \[eq:13\]) we used the commonly accepted values of SOC constants $\xi$ = 0.3 eV (Pu) and 0.34 eV (Am). To find $\mu_H$ Eq.(\[eq:11\]), we choose the AMF-LDA+U values of $\langle
n \rangle$, namely, $n_f$=6.0 for Am and $n_f$=5.4 for Pu [@details2]. The self-energy Eq.(\[eq:13\]) was calculated along the real axis for $z=E-E_{\rm F} + i \delta$, where $\delta$ = 63 meV [@Svane], and $\beta$ was varied from 100 to 1000 eV$^{-1}$. We found no sizable effect due to the variation of $\beta$ for a given $\mu_H$ in the resulting spectral density Eq.(\[eq:16\]).
In Fig.1a we show the f-projected DOS (fDOS) from AMF-LDA+U together with spectral density calculated from Eq.(\[eq:16\]). For [*fcc*]{}-Am, the well localised fDOS peak at -4 eV transforms to the multiplet of excited state transitions $f^6 \rightarrow f^5$ below the Fermi energy, and fDOS-manifold around +2 eV to $f^6 \rightarrow
f^7$ multiplet transitions. Although there is no doubt that the 5f multiplets must dominate the experimental valence-band spectra of Am-based systems, individual lines are not resolved (except for partly resolved features in the spectrum of Am metal), and the position of the 5f intensity in the energy spectrum is the main indicator of the agreement with calculations. In this sense, the calculated Am spectral density is in a good agreement with PES [@gouder05]. It also agrees with similar calculations [@Svane] [@note]. Furthermore, we performed the calculations of AmN and AmSb using the same $\{U,J\}$ set of values as for elemental Am. The AMF-LDA+U yields AmN as an indirect gap semiconductor and AmSb as a semi-metal. The HIA spectral densities for Am $f$ manifolds in AmN and AmSb are shown in Fig.1b,c and are in good agreement with PES experiments [@gouder05].
Experimental valence-band spectra of $\delta$-Pu and several other Pu systems exhibit three narrow features within 1 eV below $E_F$, the most distinct one very close to $E_F$ being accompanied by a weaker feature at 0.5 eV and another one at 0.8-0.9 eV. Their general occurrence and invariability of characteristic energies practically excludes any relation to individual features in density of electronic states. Instead, a relation to final state multiplets has been suggested [@gouder00; @havela02] among other possible explanations . Similar to Am, the link to atomic multiplet is corroborated by the present calculations also for Pu. For $\delta$-Pu shown in Fig.2a, the AMF-LDA+U fDOS manifold at around -1 eV transforms into a set of multiplet transitions with high value of spectral density at $E_F$. Similarly to the Pu metal, the PES exhibits the three most intense peaks for a broad class of Pu compounds. The calculations performed for PuTe (see Fig. 2b) indeed demonstrate the three-peak pattern similar to Pu, in agreement with experiment [@durakiewicz]. Also we point out a good agreement between our calculations and recent DMFT calculations of Ref. [@EPLP], as well as those of Svane for PuSe [@Svane]. One should note that the high spectral intensity at the Fermi level is explaining enhanced values of the $\gamma$-coefficient of electronic specific heat, observed in $\delta$-Pu and other Pu systems. As a conclusion, we have shown that the three narrow features observed in the valence band spectra of Pu and majority of Pu compounds can be identified with the most intense atomic excitations (multiplets), calculated using the LDA+U and the multi-orbital HIA calculations. The calculations explain that the atomic excitations can be observed even if the 5f states are not fully localized as in $\delta$-Pu, and the atomic character fixes the characteristic energies (not intensities) such that similar features are found in spectra of diverse Pu systems. A reasonable agreement with experiment is found also for Am and its compounds calculated on the same footing.
The research was carried out as a part of research programs AVOZ10100520 of the Academy of Sciences, MSM 0021620834 of the Ministry of Education of Czech Republic. Financial support was provided by the Grant Agency of the Academy of Sciences (Project A100100530), the Grant Agency of Czech Republic (Projects 202/04/1005 and 202/04/1103), and by the action COST P16 (Project OC144, financed by the Czech Ministry of Education). We gratefully acknowledge valuable discussions with V. Janiš, P. Oppeneer, and A. Lichtenstein.
[99]{} S.Y. Savrasov , G. Kotliar E. Abrahams, [*Nature*]{} 410 (London), 793 (2001). J.-C. Griveau, J. Rebizant, G.H. Lander, and G. Kotliar, Phys. Rev. Lett. [**94**]{}, 097002 (2005). P. Söderlind and B. Sadigh, Phys. Rev. Lett. [**92**]{}, 185702 (2004) and references therein. P. Söderlind and A. Landa, Phys. Rev. B [**72**]{}, 024109 (2005) and refernces therein. Lashley J.C., Lawson A., McQueeney R.J., Lander G.H., Phys. Rev. B [**72**]{}, 054416 (2005). Also, it is well established experimentally that the ground state of Am is a $f^6$ $J=0$ singlet. A. B. Shick, V. Drchal, and L. Havela, Europhys. Lett. [**69**]{}, 588 (2005). A. B. Shick, L. Havela, J. Kolorenč, T. Gouder, and P. Oppeneer, Phys. Rev. B [**73**]{}, 104415 (2006). L. V. Pourovskii, M. I. Katsnelson, A. I. Lichtenstein, L. Havela, T. Gouder, F. Wastin, A. B. Shick, V. Drchal, and G. H. Lander, Europhys. Lett. [**75**]{}, 479 (2006).
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Also, we used the experimental values of $\delta$-Pu and *fcc*-Am lattice constats. The Am atom Coulomb $U$ = 4 eV, and the intra-atomic exchange parameter $J$=0.75 eV (corresponding Slater integrals $F_0$=4.0 eV, $F_2$= 8.93967 eV $F_4$= 5.97302 eV, and $F_6$= 4.41907 eV), and for the Pu atom $U$ = 4 eV, $J$=0.7 eV corresponding Slater integrals $F_0$=4.0 eV, $F_2$= 8.34369 eV $F_4$= 5.57482 eV, and $F_6$= 4.12446 eV) were used.
We found that the Eq.(\[eq:11\]) gives practically the same values for $\mu_H$ for inverse temperature $\beta = 1/ (k_B T)$ in the range of $30-50$ eV$^{-1}$ as the procedure of Ref. [@LK99] based on the summation over Matsubara frequencies (see Eq.(\[eq:12\]) of Ref. [@LK99]).
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The experimental spectra were shifted upwards for clarity.
|
---
abstract: 'Adaptive behavior emerges through a dynamic interaction between cognitive agents and changing environmental demands. The investigation of information processing underlying adaptive behavior relies on controlled experimental settings in which individuals are asked to accomplish demanding tasks whereby a hidden state or an abstract rule has to be learned dynamically. Although performance in such tasks is regularly considered as a proxy for measuring high-level cognitive processes, the standard approach consists in summarizing response patterns by simple heuristic scoring measures. With this work, we propose and validate a new computational Bayesian model accounting for individual performance in the established Wisconsin Card Sorting Test. We embed the new model within the mathematical framework of Bayesian Brain Theory, according to which beliefs about the hidden environmental states are dynamically updated following the logic of Bayesian inference. Our computational model maps distinct cognitive processes into separable, neurobiologically plausible, information-theoretic constructs underlying observed response patterns. We assess model identification and expressiveness in accounting for meaningful human performance through extensive simulation studies. We further apply the model to real behavioral data in order to highlight the utility of the proposed model in recovering cognitive dynamics at an individual level. Practical and theoretical implications of our computational modelling approach for clinical and cognitive neuroscience research are finally discussed, as well as potential future improvements.'
author:
- |
Marco D’Alessandro\
Department of Psychology and Cognitive Science\
University of Trento\
Corso Bettini, 84, 38068 Rovereto\
`[email protected]` Stefan T. Radev\
Institute of Psychology\
Heidelberg University\
Hauptstr. 47-51, 69117 Heidelberg\
`[email protected]` Andreas Voss\
Institute of Psychology\
Heidelberg University\
Hauptstr. 47-51, 69117 Heidelberg\
`[email protected]` Luigi Lombardi\
Department of Psychology and Cognitive Science\
University of Trento\
Corso Bettini, 84, 38068 Rovereto\
`[email protected]`
bibliography:
- 'ref.bib'
title: 'A Bayesian brain model of adaptive behavior: An application to the Wisconsin Card Sorting Task '
---
Introduction
============
Computational models of cognition provide a way to formally describe and empirically account for mechanistic, process-based theories of adaptive cognitive functioning [@sun2009theoretical; @cooper1996systematic; @lee2014bayesian]. A foundational theoretical framework for describing functional characteristics of neurocognitive systems has recently emerged under the hood of Bayesian brain theories [@knill2004bayesian; @friston2010free]. Bayesian brain theories owe their name to their core assumption that neural computations resemble inference processes following the logic of Bayesian probability theory.
From a Bayesian perspective, cognitive agents exist in an uncertain environment and adaptive behavior emerges through a dynamic interaction between cognitive agents and environmental demands. In order to behave adaptively, cognitive agents must be sensitive to changes in their environment. More formally, they must generate and maintain internal probabilistic models of environmental states as external sensory information is gathered [@friston2005theory]. From these internal models, they derive beliefs about the causal structure of the environment and make predictions about future environmental states. Moreover, internal models form a basis for choosing future actions which can change the state of the environment and are, in turn, modified and refined by changes in the environment. As a result, internal beliefs and predictions are also updated to match the new model, according to principles of Bayesian inference [@friston2009predictive; @friston2010free; @buckley2017free].
The empirical assessment of adaptive functioning often relies on dynamic reinforcement learning (RL) tasks which require participants to adapt their behavior during the unfolding of the task. A typical RL task unfolds through multiple trials as participants observe certain environmental contingencies, take actions, and receive feedback based on their actions. Optimal performance in a RL experimental paradigm requires that agents infer the probabilistic model underlying the hidden environmental states. Since these models usually change as the task progresses, agents, in turn, need to adapt their inferred model, in order to take optimal actions.
In the present work we propose and validate a computational Bayesian model which accounts for the adaptive behavior of cognitive agents in reinforcement learning tasks. More precisely, we focus on the widely adopted Wisconsin Card Sorting Test (WCST; [@berg1948simple; @heaton1981wisconsin]) as a particular instance of such a task. The WCST is perhaps the most popular neuropsychological setting employed to measure set-shifting, cognitive flexibility and impulsive response modulation [@bishara2010sequential; @alvarez2006executive] and we consider it as a fundamental paradigm for investigating adaptive behavior from a Bayesian perspective.
The environment of the WCST consists of a target and a set of stimulus cards with geometric figures which vary according to three perceptual features. The WCST requires participants to infer the correct classification principle by trial and error using the examiner’s feedback. The feedback is thought to carry a positive or negative information signaling the agent whether the immediate action was appropriate or not. Modeling adaptive behavior in the WCST from a Bayesian perspective is straightforward, since observable actions emerge from the interaction between the internal probabilistic model of the agent and a set of discrete environmental states.
Performance in WCST and similar RL tasks [@bechara2001insensitivity; @frank2004carrot] is usually measured via a rough summary metric such as the number of correct/incorrect responses or pre-defined psychological scoring criteria (see for instance [@heaton1981wisconsin]). These metrics are then used to infer the underlying cognitive processes involved in the task. A major shortcoming of this approach is that it simply assumes the cognitive processes to be inferred without specifying an explicit *process model*. Moreover, summary measures do not utilize the full information present in the data, such as trial-by-trial fluctuations or various interesting agent-environment interactions. For this reason, crude scoring measures are often insufficient to disentangle the dynamics of the relevant cognitive (sub)processes involved in a RL task. Consequently, an entanglement between processes at the metric level can prevent us from answering interesting research questions about aspects of adaptive behavior.
In our view, a sound computational account for adaptive behavior in RL tasks needs to provide at least a quantitative measure of effective belief updating about the environmental states at each trial. This measure should be complemented by a measure of how feedback-related information influences behavior. The first measure should account for the integration of meaningful information. In other words, it should describe how prior beliefs about the current environmental state change after an observation has been made. The second measure should account for signaling the (im)probability of observing a certain environmental configuration (e.g., an (un)expected feedback given a response) [@schwartenbeck2016neural].
Indeed, recent studies suggest that the meaningful information content and the pure unexpectedness of an observation are processed differently at the neural level. Moreover, such disentanglement appears to be of crucial importance to the understanding of how new information influences adaptive behavior [@nour2018dopaminergic; @schwartenbeck2016neural; @o2013dissociable]. Inspired by these results and previous computational proposals [@koechlin2007information], we integrate these different information processing aspects into the current model from an information-theoretic perspective.
Our computational cognitive model draws heavily on the mathematical frameworks of Bayesian probability theory and information theory [@sayood2018information]. First, it provides a parsimonious description of observed data in the WCST via two neurocognitively meaningful parameters, which we dub *flexibility* and *information loss* (to be explained in the **Model** section). Moreover, it captures the main response patterns obtainable in the WCST via different parameter configurations. Second, we formulate a functional connection between cognitive parameters and underlying information processing mechanisms related to belief updating and prediction formation. We formalize and distinguish between *Bayesian surprise* and *Shannon surprise* as the main mechanisms for adaptive belief updating. Moreover, we introduce a third quantity, which we dub predictive *Entropy* and which quantifies an agent’s subjective uncertainty about the current internal model. Finally, we propose to measure these quantities on a trial-by-trial basis and use them as a proxy for formally representing the dynamic interplay between agents and environments.
The rest of the paper is organized as follows. First, the WCST is described in more detail and a mathematical representation of the new Bayesian computational model is provided. Afterwards, we explore its characteristics through simulations. We also present an application in which we apply a novel and powerful Bayesian deep neural network method [@radev2020bayesflow] for model evaluation and parameter estimation. We apply the model to a real behavioral data from an already published dataset. Finally, we discuss the results as well as the main strengths and limitations of the proposed model.
The Wisconsin Card Sorting Test
===============================
In a typical WCST, participants learn to pay attention and respond to relevant stimulus features, while ignoring irrelevant ones, as a function of experimental feedback. Individuals are asked to match a target card with one of four stimulus cards. Each card depicts geometric figures that vary in terms of three features, namely, color (red, green, blue, yellow), shape (triangle, star, cross, circle) and number of objects (1, 2, 3 and 4), according to a correct sorting rule on any given trial (see ).
[.99]{} ![Suppose that the current sorting rule is the feature shape. The target card in the first trial (left box) contains two blue triangles. A correct response requires that the agent matches the target card with the stimulus card containing the single triangle (arrow represents the correct choice), regardless of the features color and number. The same applies for the second trial (right box) in which matching the target card with the stimulus card containing three yellow crosses is the correct response.[]{data-label="fig:Fig.1"}](Fig1.png "fig:"){width="50.00000%"}
Each response in the WCST is followed by a feedback informing the participant if his/her response is correct or incorrect. After some fixed number of consecutive responses, the sorting rule is changed by the experimenter without warning, and participants are required to infer the new sorting rule. Clearly, the most adaptive response would be to explore the remaining possible rules. However, participants sometimes would persist responding according to the old rule and produce what is called a *perseverative response*.
The Model
=========
The core idea behind our computational framework is to encode the concept of *belief* into a generative probabilistic model of the environment. Belief updating then corresponds to recursive Bayesian updating of the internal model based on current and past interactions between the agent and its environment. Optimal or sub-optimal actions are selected according to a well specified or a misspecified internal model and, in turn, cause perceptible changes in the environment.
We assume that the cognitive agent aims to infer the *true hidden state* of the environment by processing and integrating sensory information from the environment. Within the context of the WCST, the hidden environmental states might change at a non-constant rate, so the agent needs to rely on environmental feedback and own actions to infer the current state. We assume that the agent maintains an internal probability distribution over the states at each individual trial of the WCST. The agent then updates this distribution upon making new observations. In particular, the hidden environmental states to be inferred are the three features, $s_t \in \{1,2,3\}$. The posterior probability of the states depends on an observation vector ${\boldsymbol}{x}_t=(a_t,f_t)$, which consists of the pair of agent’s response (actions) $a_t \in \{1,2,3,4\}$ and received feedback $f_t \in \{0,1\}$ in a given trial $t = 0,...,T$. The discrete response $a_t$ represents the stimulus card indicator being matched with a target card at trial $t$. We denote a sequence of observations as ${\boldsymbol}{x}_{0:t} = ({\boldsymbol}{x}_0,{\boldsymbol}{x}_1,...,{\boldsymbol}{x}_t) = ((a_0,f_0), (a_1,f_1),(a_2,f_2),...,(a_t,f_t))$ and set ${\boldsymbol}{x}_0 = \varnothing$ in order to indicate that there are no observations at the onset of the task. Thus, trial-by-trial belief updating is recursively computed according to Bayes’ rule:
$$p(s_t|{\boldsymbol}{x}_{0:t})=\frac{p({\boldsymbol}{x}_t|s_t,{\boldsymbol}{x}_{0:t-1})p(s_t|{\boldsymbol}{x}_{0:t-1})}{p({\boldsymbol}{x}_t|{\boldsymbol}{x}_{0:t-1})}$$
Accordingly, the agent’s posterior belief about the task-relevant features $s_t$ after observing a sequence of response-feedback pairs ${\boldsymbol}{x}_{0:t}$ is proportional to the product of the likelihood of observing a particular response-feedback pair and the agent’s prior belief about the task-relevant feature in the current trial. The likelihood of an observation is computed as follows:
$$p({\boldsymbol}{x}_t|s_t,{\boldsymbol}{x}_{0:t-1}) = \frac{f_t p(a_t|s_t=i) + (1-f_t)(1-p(a_t|s_t=i))}{f_t\sum_j p(a_t|s_t=j) + (1-f_t)\sum_j(1-p(a_t|s_t=j))}$$
and $p(a_t|s_t=i)$ indicates the probability of a matching between the target and the stimulus card assumed that the current feature is $i$. Here, we assume the likelihood of a current observation to be independent from previous observations without loss of generality, that is:
$$\begin{aligned}
p({\boldsymbol}{x}_t|s_t,{\boldsymbol}{x}_{0:t-1}) = p({\boldsymbol}{x}_t|s_t) \nonumber\end{aligned}$$
The prior belief for a given trial $t$ is computed based on the posterior belief generated in the previous trial, $p(s_{t-1}|{\boldsymbol}{x}_{0:t-1})$, and the agent’s belief about the probability of transitions between the hidden states, $p(s_t|s_{t-1})$. The prior belief can also be considered as a predictive probability over the hidden states. The predictive distribution for an upcoming trial $t$ is computed according to the Chapman-Kolmogorov equation:
$$p(s_{t+1}=j|{\boldsymbol}{x}_{0:t}) = \sum_{i=1}^3 p(s_{t+1}=j|s_{t}=i,\boldsymbol{\Gamma}(t))p(s_{t}=i|{\boldsymbol}{x}_{0:t})$$
where ${\boldsymbol}{\Gamma}(t)$ represents a stability matrix describing transitions between the states (to be explained shortly). Thus, the agent combines information from the updated belief (posterior distribution) and the belief about the transition properties of the environmental states to predict the most probable future state. The predictive distribution represents the internal model of the cognitive agent according to which actions are generated.
The stability matrix ${\boldsymbol}{\Gamma}(t)$ encodes the agent’s belief about the probability of states being stable or likely to change in the next trial. In other words, the stability matrix reflects the cognitive agent’s internal representation of the dynamic probabilistic model of the task environment. It is computed on each trial based on the response-feedback pair, ${\boldsymbol}{x}_t$, and a matching signal, ${\boldsymbol}{m}_t$, which are observed.
The matching signal ${\boldsymbol}{m}_t$ is a vector informing the cognitive agent which features are currently relevant (meaningful), such that $m^{(i)}_t=1$ when a positive feedback is associated with a response implying feature $s_t=i$, and $m^{(i)}_t=0$ otherwise. Note, that the matching signal is not a free parameter of the model, but is completely determined by the task contingencies. The matching signal vector allows the agent to compute the *state activation level* $\omega_t^{(i)} \in [0,1]$ for the hidden state $s_t=i$, which provides an internal measure of the (accumulated) evidence for each hidden state at trial $t$. Thus, the activation levels of the hidden states are represented by a vector ${\boldsymbol}{\omega}_t$. The stability matrix is a square and asymmetric matrix related to hidden state activation levels such that:
$$\renewcommand\arraystretch{1.5}
\boldsymbol{\Gamma}(t) =
\begin{bmatrix}
\omega_t^{(1)} & \frac{1}{2}(1-\omega_t^{(1)}) & \frac{1}{2}(1-\omega_t^{(1)}) \\
\frac{1}{2}(1-\omega_t^{(2)}) & \omega_t^{(2)} & \frac{1}{2}(1-\omega_t^{(2)}) \\
\frac{1}{2}(1-\omega_t^{(3)}) & \frac{1}{2}(1-\omega_t^{(3)}) & \omega_t^{(3)}
\end{bmatrix}$$
where the entries $\Gamma_{ii}(t)$ in the main diagonal represent the elements of the activation vector ${\boldsymbol}{\omega}_t$, and the non-diagonal elements are computed so as to ensure that rows sum to 1. The state activation vector is computed in each trial as follows:
$$\begin{bmatrix}
\omega_t^{(1)} \\
\omega_t^{(2)} \\
\omega_t^{(3)}
\end{bmatrix}
= f_t {\boldsymbol}{\omega}_{t-1}^{\delta}
\begin{bmatrix}
m_t^{(1)} \\
m_t^{(2)} \\
m_t^{(3)}
\end{bmatrix}
+ \lambda \left[(1-f_t) {\boldsymbol}{\omega}_{t-1}^{\delta}
\begin{bmatrix}
1-m_t^{(1)} \\
1-m_t^{(2)} \\
1-m_t^{(3)}
\end{bmatrix}
\right]
\begin{bmatrix}
\omega_{t-1}^{(1)} \\
\omega_{t-1}^{(2)} \\
\omega_{t-1}^{(3)}
\end{bmatrix}.$$
This equation reflects the idea that state activations are simultaneously affected by the observed feedback, $f_t$, and the matching signal vector, ${\boldsymbol}{m}_t$. However, the matching signal vector conveys different information based on the current feedback. Matching a target card with a stimulus card makes a feature (or a subset of features) informative for a specific state. The vector ${\boldsymbol}{m}_t$ contributes to increase (resp. decrease) the activation level of a state if the feature is informative for that state when a positive (resp. negative) feedback is received.
The parameter $\lambda \in [0,1]$ modulates the efficiency to disengage attention to a given state-activation configuration when a negative feedback is processed. We therefore term this parameter $\textit{flexibility}$. We also assume that information from the matching signal vector can degrade by slowing down the rate of evidence accumulation for the hidden states. This means that the matching signal vector can be re-scaled based on the current state activation level. The parameter $\delta \in [0,1]$ is introduced to achieve this re-scaling. When $\delta=0$, there is no re-scaling and updating of the state activation levels relies on the entire information conveyed by ${\boldsymbol}{m}_t$. On the other extreme, when $\delta=1$, several trials have to be accomplished before converging to a given configuration of the state activation levels. Equivalently, higher values of $\delta$ affect the entropy of the distribution over hidden states by decreasing the probability of sampling of the correct feature. We therefore refer to $\delta$ as *information loss*.
The free parameters $\lambda$ and $\delta$ are central to our computational model, since they regulate the rate at which the internal model converges to the true task environmental model. Eq. (5) can be expressed in compact notation as follows:
$${\boldsymbol}{\omega}_t = f_t {\boldsymbol}{\omega}_{t-1}^{\delta} {\boldsymbol}{m}_t + \lambda \left[ (1-f_t){\boldsymbol}{\omega}_{t-1}^{\delta}(1-{\boldsymbol}{m}_t) \right]{\boldsymbol}{\omega}_{t-1}$$
Note that the information loss parameter $\delta$ affects the amount of information that a cognitive agent acquires from environmental contingencies, irrespective of the type of feedback received. Global information loss thus affects the rate at which the divergence between the agent’s internal model and the true model is minimized. Figure 2 illustrates these ideas.
[.99]{} ![Suppose the correct sorting rule is the feature *shape*. The figure shows the rate of convergence of the predictive distributions to the true task environmental model. The predictive distributions at trial $t+1$ depends on the sorting action $a_t$ (first row) and the received feedback $f_t$ (second row). Two examples of updating a predictive distribution are shown: one in which information loss is high ($\delta=0.7$, third row), and one in which information loss is low ($\delta=0.3$, fifth row). High information loss slows down the convergence of the internal model to the true environmental model. The gray bar plots represent the predictive probability distribution over the rules from which an action is sampled at each trial. Dotted bars represent the updated predictive distribution after the feedback observation. For each scenario, trial-by-trial information-theoretic measures are shown.[]{data-label="fig:Fig.2"}](Fig2.png "fig:"){width="\textwidth"}
The probabilistic representation of adaptive behaviour provided by our Bayesian agent model allows us to quantify (latent) cognitive dynamics by means of meaningful information-theoretic measures. Information theory has, indeed, proven to be an effective and natural mathematical language to account for functional integration of structured cognitive processes and to relate them to brain activity [@koechlin2007information; @friston2017active; @collell2015brain; @strange2005information; @friston2003learning]. In particular, we are interested in three key measures, namely, *Bayesian surprise*, $\mathcal{B}_t$, *Shannon surprise*, $\mathcal{I}_t$, and *entropy*, $\mathcal{H}_t$. The subscript $t$ indicates that we can compute each quantity on a trial-by-trial basis. Each quantity is thought to reflect a specific interpretation in terms of separate neurocognitive processes. Bayesian surprise $\mathcal{B}_t$ quantifies the magnitude of the update from prior belief to posterior belief. Shannon surprise $\mathcal{I}_t$ quantifies the improbability of an observation given an agent’s prior expectation. Finally, entropy $\mathcal{H}_t$ measures the degree of epistemic uncertainty regarding the true environmental states. Such measures are thought to account for the ability of the agent to manage uncertainty as emerging as a function of competing behavioral affordances [@hirsh2012psychological]. We expect an efficient adaptive functioning system to attenuate uncertainty over environmental states (current features), by reducing the entropy of its internal probabilistic model.
Bayesian surprise can be computed as the Kullback–Leibler ($\mathbb{KL}$) divergence between prior and posterior beliefs about the environmental states. In our model representation, actions are sampled from predictive distributions which integrate information from the posterior belief about the hidden states and belief about their dynamics. The Bayesian surprise is then thought to account for the divergence between the predictive model for the current trial, and the updated predictive model for the upcoming trial. It is computed as follows:
$$\begin{split}
\mathcal{B}_t & = \mathbb{KL}[p(s_{t+1}|{\boldsymbol}{x}_{0:t}) || p(s_t|{\boldsymbol}{x}_{0:t-1})] \\
& = \sum_{i=1}^3 \left[ p(s_{t+1}=i|{\boldsymbol}{x}_{0:t})\log\left( \frac{p(s_{t+1}=i|{\boldsymbol}{x}_{0:t})}{p(s_t=i|{\boldsymbol}{x}_{0:t-1})} \right)\right]
\end{split}$$
The Shannon surprise of a current observation given a previous one is computed as follows:
$$\begin{split}
\mathcal{I}_t & =-\log p({\boldsymbol}{x}_t|{\boldsymbol}{x}_{0:t-1}) \\
& = -\log \sum_{i=1}^3 \left[ p({\boldsymbol}{x}_t|s_t=i)p(s_t=i|{\boldsymbol}{x}_{0:t-1}) \right]
\end{split}$$
Finally, the entropy is computed over the predictive distribution in order to account for the uncertainty in the internal model of the agent in trial $t$ as follows:
$$\begin{split}
\mathcal{H}_t
&= \mathbb{E}\left[-\log p(s_t|{\boldsymbol}{x}_{0:t-1})\right] \\
&= -\sum_{i=1}^3 p(s_t=i|{\boldsymbol}{x}_{0:t-1}) \log p(s_t=i|{\boldsymbol}{x}_{0:t-1})
\end{split}$$
Once the flexibility ($\lambda$) and information loss ($\delta$) parameters are recovered from data, the information-theoretic quantities can be easily computed and visualized for each trial of the WCST (see ). This allows to rephrase standard neurocognitive constructs in terms of measurable information-theoretic quantities. Moreover, the dynamics of these quantities, as well as their interactions, can be used for formulating and testing hypotheses about the neurcognitive underpinnings of adaptive behavior in a principled way, as discussed later in the paper.
Simulations
-----------
In this section we evaluate the expressiveness of the model by assessing its ability to reproduce meaningful behavioral patterns as a function of its two free parameters. We study how the generative model behaves when performing the WCST in a 2-factorial simulated Monte Carlo design where flexibility ($\lambda$) and information loss ($\delta$) are systematically varied.
In this simulation, the Heaton version of the task [@heaton1981wisconsin] is administered to the Bayesian cognitive agent. In this particular version, the sorting rule (true environmental state) changes after a fixed number of consecutive correct responses. In particular, when the agent correctly matches the target card in 10 consecutive trials, the sorting rule is automatically changed. The task ends after completing a maximum of 128 trials.
### Generative Model
The cognitive agent’s responses are generated at each time step (trial) by processing the experimental feedback. Its performance depends on the parameters governing the computation of the relevant quantities. The generative algorithm is outlined in **Algorithm 1**.
### Simulation 1: Clinical Assessment of the Bayesian Agent
Ideally, the qualitative performance of the Bayesian cognitive agent will resemble human performance. To this aim, we adopt a metric which is usually employed in clinical assessment of test results in neurological and psychiatric patients [@braff1991generalized; @zakzanis1998subcortical; @bechara2002decision; @landry2016meta]. Thus, agent performance is codified according to a neuropsychological criterion [@heaton1981wisconsin; @flashman1991note] which allows to classify responses into several response types. These response types provide the scoring measures for the test.
Here, we are interested in: 1) non-perseverative errors (E); 2) perseverative errors (PE); 3) number of trials to complete the first category (TFC); and 4) number of failures to maintain set (FMS). Perseverative errors occur when the agent applies a sorting rule which was valid before the rule has been changed. Usually, detecting a perseveration error is far from trivial, since several response configurations could be observed when individuals are required to shift a sorting rule after completing a category (see [@flashman1991note] for details). On the other hand, non-perseverative errors refer to all errors which do not fit the above description, or in other words, do not occur as a function of changing the sorting rule, such as casual errors.
The number of trials to complete the first category tells us how many trials the agent needs in order to achieve the first sorting principle, and can be seen as an index of conceptual ability [@anderson2010towards; @singh2017wisconsin]. Finally, a failure to maintain a set occurs when the agent fails to match cards according to the sorting rule after it can be determined that the agent has acquired the rule. A given sorting rule is assumed to be acquired when the individual correctly sorts at least five cards in a row [@heaton1981wisconsin; @figueroa2013failure]. Thus, a failure to maintain a set arises whenever a participant suddenly changes the sorting strategy in the absence of negative feedback. Failures to maintain a set are mostly attributed to distractibility. We compute this measure by counting the occurrences of first errors after the acquisition of a rule.
We run the generative model by varying flexibility across four levels, $\lambda \in \{0.3, 0.5, 0.7, 0.9\}$, and information loss across three levels, $\delta \in \{0.4, 0.7, 0.9\}$. We generate data from 150 synthetic cognitive agents per parameter combination and compute standard scoring measures for each of the agents simulated responses. Results from the simulation runs are depicted in .
[.99]{} ![Clinical scoring measures as functions of flexibility and information loss - simulated scenarios. Cells show the density of scoring measures for the levels of $\lambda$ across different levels of $\delta$. In particular, they show the distribution of non-perseverative errors (E), perseverative errors (PE), number of trials to complete the first category (TFC), number of failures to maintain set (FMS) obtained from 150 synthetic agent’s response simulations for each cell of the factorial design.[]{data-label="fig:Fig.3"}](Fig3.png "fig:"){width="\textwidth"}
The simulated performance of our Bayesian cognitive agent demonstrates that different parameter combinations capture different meaningful behavioral patterns. In other words, flexibility and information loss seem to interact in a theoretically meaningful way.
First, overall errors increase when flexibility decreases, which is reflected by the inverse relation between the number of casual, as well as perseverative, errors and the values of parameter $\lambda$. Moreover, this pattern is consistent across all the levels of parameter $\delta$. More precisely, information loss seems to contribute to the characterization of the casual and the perseverative components of the error in a different way. Perseverative errors are likely to occur after a sorting rule change and reflect the inability of the agent to use feedback to disengage attention from the currently attended feature. They therefore result from local cognitive dynamics conditioned on a particular stage of the task (e.g., after completing a series of correct responses).
Second, information loss does not interact with flexibility when perseverative errors are considered. This is due to the fact that high (resp. low) information loss affects general performance by yielding a dysfunctional response strategy which increases (resp. decreases) the probability of making an error at any stage of the task. The lack of such interaction provides evidence that our computational model can disentangle between error patterns due to perseveration and those due to general distractibility, according to neuropsychological scoring criteria.
However, in our framework, flexibility is allowed to yield more general and non-local cognitive dynamics as well. Indeed, $\lambda$ plays a role whenever belief updating is demanded as a function of negative feedback. An error classified as non-perseverative (e.g., casual error) by the scoring criteria might still be processed as a feedback-related evidence for belief updating. Consistently, the interaction between $\lambda$ and $\delta$ in accounting for causal errors shows that performance worsens when both flexibility and information loss become less optimal, and that such pattern becomes more pronounced for lower values of $\delta$.
On the other hand, a specific effect of information loss can be observed for the scoring measures related to slow information processing and distractibility. The number of trials to achieve the first category reflects the efficiency of the agent in arriving at the first true environmental model. Flexibility does not contribute meaningfully to the accumulation of errors before completing the first category for some levels of information loss. This is reflected by the fact that the mean number of trials increases as a function of $\delta$, and do not change across levels of $\lambda$ for low and mid values of $\delta$. A similar pattern applies for failures to maintain a set. Both scoring measures index a deceleration of the process of evidence accumulation for a specific environmental configuration, although the latter is a more exhaustive measures of dysfunctional adaptation.
Therefore, an interaction between parameters can be observed when information loss is high. A slow internal model convergence process increases the amount of errors due to improper rule sampling from the internal environmental model. However, internal model convergence also plays a role when a new category has to be accomplished after completing an older one. On the one hand, compromised flexibility increases the amount of errors due to inefficient feedback processing. This leads to longer trial windows needed to achieve the first category. On the other hand, when information loss is high, belief updating upon negative feedback is compromised due to high internal model uncertainty. At this point, the probability to err due to distractibility increases, as accounted by the failures to maintain a set measures.
Finally, the joint effect of $\delta$ and $\lambda$ for high levels of information loss suggests that the roles played by the two cognitive parameters in accounting for adaptive functioning can be entangled when neuropsychological scoring criteria are considered.
### Simulation 2: Information-theoretic Analysis of the Bayesian Agent
In the following, we explore a different simulation scenario in which information-theoretic measures are derived to assess performance of the Bayesian cognitive agent. In particular, we explore the functional relationship between cognitive parameters and the dynamics of the recovered information-theoretic measures by simulating observed responses by varying flexibility across three levels, $\lambda \in \{0.1,0.5,0.9\}$, and information loss across three levels, $\delta \in \{0.1,0.5,0.9\}$.
For this simulation scenario, we make no prior assumptions about sub-types of error classification. Instead, we investigate the dynamic interplay between Bayesian surprise, $\mathcal{B}_t$, Shannon surprise, $\mathcal{I}_t$, and entropy, $\mathcal{H}_t$ over the entire course of 128 trials in the WCST.
[.99]{} ![Information-theoretic measures varying as a function of flexibility $\lambda$ and information loss $\delta$ across 128 trials of the WCST. Optimal belief updating and uncertainty reduction are achieved with low information loss and high flexibility (first row, third column).[]{data-label="fig:Fig.4"}](Fig4.png "fig:"){width="\textwidth"}
depicts results from the nine simulation scenarios. Although an exhaustive discussion on cognitive dynamics should couple information-theoretic measures with patterns of correct and error responses, we focus solely on the information-theoretic time series for illustrative purposes. We refer to the **Application** section for a more detailed description of the relation between observed responses and estimated information-theoretic measures in the context of data from a real experiment.
Again, simulated performance of the Bayesian cognitive agent shows that different parameter combinations yield different patterns of cognitive dynamics. Observed spikes and their related magnitudes signal informative task events (e.g., unexpected negative feedback), as accounted by Shannon surprise, or belief updating, as accounted by Bayesian surprise. Finally, entropy encodes the epistemic uncertainty about the environmental model on a trial-by-trial basis.
In general, low information loss ensures optimal behavior by speeding up internal model convergence by decreasing the number of trials needed to minimize uncertainty about the environmental states. Low uncertainty reflects two main aspects of adaptive behavior. On the one hand, the probability that a response occurs due to sampling of improper rules decreases, allowing the agent to prevent random responses due to distractibility. On the other hand, model convergence entails a peaked Shannon surprise when a negative feedback occurs, due to the divergence between predicted and actual observations.
Flexibility plays a role in integrating feedback information in order to enable belief updating. The first row depicted in shows cognitive dynamics related to low information loss, across the levels of flexibility. As can be noticed, there is a positive relation between the magnitude of the Bayesian surprise and the level of flexibility, although unexpectedness yields approximately the same amount of signaling, as accounted by peaked Shannon surprise. From this perspective, surprise and belief updating can be considered functionally separable, where the first depends on the particular internal model probability configuration related to $\delta$, whilst the second depends on flexibility $\lambda$.
However, more interesting patterns can be observed when information loss increases. In particular, model convergence slows down and several trials are needed to minimize predictive model entropy. Casual errors might occur within trial windows characterized by high uncertainty, and interactions between entropy and Shannon surprise can be observes in such cases. In particular, Shannon surprise magnitude increases (resp. decreases) when model’s entropy decreases (resp. increases), that is, during the task phases in which the internal model has converged (resp. not converged). As a consequence, negative feedback could be classified as informative or uninformative, based on the uncertainty in the current internal model. This is reflected by the negative relation between entropy and Shannon surprise, as can be noticed by inspecting the graphs depicted in the third row of . Therefore, the magnitude of belief updating depends on the interplay between entropy and Shannon surprise, and can differ based on the values of the two measures in a particular task phase.
In sum, both simulation scenarios suggest that the simulated behavior of our generative model is in accord with theoretical expectations. Moreover, the flexibility and information loss parameters can account for a wide range of observed response patterns and inferred dynamics of information processing.
Model Identification
====================
In this section, we discuss the computational framework for recovering the parameters of our model from observed behavioral data. Parameter recovery is essential to inferring the cognitive dynamics underlying observed behavior in real-world applications of the model. This section is slightly more technical and can be skipped without significantly affecting the flow of the text.
Making our cognitive model suitable for application in real-world contexts entails estimating parameters from available data and accounting for uncertainty about parameter estimates. Indeed, uncertainty quantification turns out to be a fundamental and challenging goal when first-level quantities, that is, cognitive parameter estimates, are used to recover (second-level) information-theoretic measures of cognitive dynamics. The main difficulties arise when model complexity makes estimation and uncertainty quantification intractable at both analytical and numerical levels. For instance, in our case, probability distributions for the hidden model are generated at each trial, and the mapping between hidden states and responses changes depending on the structure of the task environment.
Identifying such a dynamic mapping is relatively easy from a generative perspective, but it becomes challenging, and almost impossible, when reverse engineering is required. Generally, this problem arises when no likelihood function relating model parameters to the data is available, or when the likelihood function is too complex to be evaluated [@sisson2011likelihood]. To overcome these limitations, we apply the recently developed *BayesFlow* method [@radev2020bayesflow]. *BayesFlow* is a powerful computational tool that allows to estimate parameters and quantify uncertainty in a unified probabilistic framework when inverting the generative model is intractable. The method is based on recent advances in deep probabilistic modeling and makes no assumptions about the shape of the true parameter posteriors. Thus, our ultimate goal becomes to approximate and analyze the joint posterior distribution over the model parameters. The posterior is given via an application of Bayes’ rule:
$$p({\boldsymbol{\theta}}|{\boldsymbol}{x}_{0:T},{\boldsymbol}{m}_{0:T}) = \frac{p({\boldsymbol}{x}_{0:T},{\boldsymbol}{m}_{0:T}|{\boldsymbol{\theta}})p({\boldsymbol{\theta}})}{\int p({\boldsymbol}{x}_{0:T},{\boldsymbol}{m}_{0:T}|{\boldsymbol{\theta}})p({\boldsymbol{\theta}})d{\boldsymbol{\theta}}}$$
where we set ${\boldsymbol{\theta}}= (\lambda, \delta)$ and stack all observations and matching signals into the vectors ${\boldsymbol}{x}_{0:T} = ({\boldsymbol}{x}_0, {\boldsymbol}{x}_1,...,{\boldsymbol}{x}_T)$ and ${\boldsymbol}{m}_{0:T} = ({\boldsymbol}{m}_0, {\boldsymbol}{m}_1,...,{\boldsymbol}{m}_T)$, respectively. The *BayesFlow* method uses simulations from the generative model to learn and clibrate a probabilistic mapping between data and parameters. First, it utilizes the fact that the data likelihood at each trial $t$ can be reparameterized as:
$${\boldsymbol}{x}_t \sim p({\boldsymbol}{x}_t|{\boldsymbol}{\theta},{\boldsymbol}{m}_t) \Longleftrightarrow {\boldsymbol}{x}_t = g({\boldsymbol}{\theta},{\boldsymbol}{m}_t,\xi) \textrm{ with } \xi \sim p(\xi)$$
with $g$ being the generative Bayesian cognitive model (**Algorithm 1**) and $\xi$ independent noise representing the non-deterministic relationship between data-generating parameters and generated data. Second, *BayesFlow* utilizes the fact that data can easily be simulated by repeatedly running $g$ with different ${\boldsymbol{\theta}}$ and thereby iteratively minimizes the divergence between the true posterior and an approximate posterior via an invertible neural network. This approach allows to obtain samples from the approximate joint posterior distribution of the cognitive parameters of interest, which can be further processed in order to extract meaningful statistics (e.g., posterior mean, maximum a posteriori).
At this point, we must ensure that our computational model can be reliably fit to data. To this purpose, the main requirement is that the parameters can be recovered accurately and uncertainty in estimates is well-calibrated.
To address such a requirement, we train the invertible network for 50 epochs which amount to 50000 backpropagation updates. We then validate performance on a separate validation set of 1000 simulated data sets with known different *ground truth* parameter values. Training the networks took less than a day on a single machine with an NVIDIA^^ GTX1060 graphics card. In contrast, obtaining full parameter posteriors from the entire validation set took approximately 1.78 seconds. In what follows, we describe and report all performance validation metrics.
To assess the accuracy of point estimates, we compute the root mean squared error (RMSE) and the coefficient of determination ($R^{2}$) between estimated and true parameter values. To assess the quality of the approximate posteriors, we compute a calibration error [@radev2020bayesflow] of the empirical coverage of each marginal posterior Finally, we implement simulation-based calibration (SBC, [@talts2018validating]) for visually detecting systematic biases in the approximate posteriors.
*Point Estimates*. Point estimates obtained by posterior means as well as corresponding RMSE and $R^{2}$ metrics are depicted in . Note, that point estimates do not have any special status in Bayesian inference, as they could be misleading depending on the shape of the posteriors. However, they are simple to interpret and useful for ease-of-comparison. We observe that pointwise recovery of $\lambda$ is better than that of $\delta$. This is mainly due to suboptimal pointwise recovery in the lower $(0,0.1)$ range of $\delta$. This pattern is evident in and is due to the fact that $\delta$ values in this range produce almost indistinguishable data patterns. Bootstrap estimates yielded an average RMSE of $0.155$ ($SD = 0.004$) and an average $R^{2}$ of $0.708$ ($SD = 0.015$) for the $\delta$ parameter. An average RMSE of $0.094$ ($SD = 0.002$) and an average $R^{2}$ of $0.895$ ($SD = 0.007$) were obtained for the $\lambda$ parameter. These results suggest good global pointwise recovery but also warrant the inspection of full posteriors, especially in the low ranges of $\delta$.
*Full Posteriors*. Average bootstrap calibration error was $0.011$ ($SD = 0.005$) for the marginal posterior of $\delta$ and $0.014$ ($SD = 0.007$) for the marginal posterior of $\lambda$. Calibration error is perhaps the most important metric here, as it measures potential under- or overconfidence across all confidence intervals of the approximate posterior (i.e., an $\alpha$-confidence interval should contain the true posterior with a probability of $\alpha$, for all $\alpha \in (0, 1)$). Thus, low calibration error indicates a faithful uncertainty representation of the approximate posteriors. Additionally, SBC-histograms are depicted in . As shown by [@talts2018validating], deviations from the uniformity of the rank statistic (also know as a PIT histogram) indicate systematic biases in the posterior estimates. A visual inspection of the histograms reveals that the posterior means slightly overestimate the true values of $\delta$. This corroborates the pattern seen in for the lower range of $\delta$.
Finally, depicts the full marginal posteriors on two validation sets. Even on these two data sets, we observe strikingly different posterior shapes. The marginal posterior of $\delta$ obtained from the first data set is slightly left-skewed and has its density concentrated over the $(0.8, 1.0)$ range. On the other hand, the marginal posterior of $\delta$ from the second data set is noticeably right-skewed and peaked across the lower range of the parameter. The marginal posteriors of $\lambda$ appear more symmetric and warrant the use of the posterior mean as a useful summary of the distribution. These two examples underline the importance of investigating full posterior distributions as a means to encode all relevant information about the parameters. Moreover, they demonstrate the advantage of imposing no distributional assumptions on the resulting posteriors, as their form and sharpness can vary widely depending on the concrete data set.
[.49]{} ![Parameter recovery results on validation data; **(a)** Posterior means vs. true parameter values; **(b)** Histograms of the rank statistic used for simulation-based calibration; **(c)** Example full posteriors for two validation data sets; (**d**) Example information-theoretic dynamics recovered from the parameter posteriors.[]{data-label="fig:Fig.5"}](Fig5a.png "fig:"){width="\textwidth"}
[.49]{} ![Parameter recovery results on validation data; **(a)** Posterior means vs. true parameter values; **(b)** Histograms of the rank statistic used for simulation-based calibration; **(c)** Example full posteriors for two validation data sets; (**d**) Example information-theoretic dynamics recovered from the parameter posteriors.[]{data-label="fig:Fig.5"}](Fig5b.png "fig:"){width="\textwidth"}
[.49]{} ![Parameter recovery results on validation data; **(a)** Posterior means vs. true parameter values; **(b)** Histograms of the rank statistic used for simulation-based calibration; **(c)** Example full posteriors for two validation data sets; (**d**) Example information-theoretic dynamics recovered from the parameter posteriors.[]{data-label="fig:Fig.5"}](Fig5c.png "fig:"){width="\textwidth"}
[.49]{} ![Parameter recovery results on validation data; **(a)** Posterior means vs. true parameter values; **(b)** Histograms of the rank statistic used for simulation-based calibration; **(c)** Example full posteriors for two validation data sets; (**d**) Example information-theoretic dynamics recovered from the parameter posteriors.[]{data-label="fig:Fig.5"}](Fig5d.png "fig:"){width="\textwidth"}
Application
===========
In this section we fit the Bayesian cognitive model to real clinical data. The aim of this application is to evaluate the ability of our computational framework to account for dysfunctional cognitive dynamics of information processing in psychiatric patients. To this aim, we estimate parameters at individual level from a group of participants from an already published dataset [@bechara2002decision].
Here, we focus on the estimation of the two relevant parameters $\lambda$ and $\delta$ from a participant’s observed response and feedback data. Our goal is to utilize the full information contained in the data and, further, quantify the uncertainty in parameter estimates.
The Data
--------
The dataset used in this application consists of responses collected by administering the Heaton version of the WCST to healthy and substance dependent individuals (SDIs). Participants in the study were adults ($>18$ years old) and gave their informed consent for inclusion which was approved by the appropriate human subject committee at the University of Iowa. SDIs were diagnosed as substance dependent based on the Structured Clinical Interview for DSM-IV criteria [@first1997structured].
For this application, we focus on SDI participants who achieved all 128 trials in the task. This is the only selection criterion employed, and is motivated by the aim to utilize a maximum amount of data for model identification. However, this decision is not necessitated by the estimation method, since several trial numbers can be used for parameter recovery. Thus, the resulting dataset consists consists of 10 SDIs.
Results
-------
We fit the Bayesian cognitive agent to data from each participant and obtain individual posterior distributions (see ) over the parameters. The advantage of modeling cognitive dynamics of individuals from a clinical population is that model predictions can be examined in light of available evidence about individual performances. SDIs are known to demonstrate inefficient conceptualization of the task and dysfunctional error-prone response strategies. This has been attributed to defective error monitoring and behavior modulation systems, which depend on cingulate and frontal brain regions functionality [@kubler2005cocaine; @willuhn2003topography]. Therefore, we expect our model to consistently capture such characteristics.
The recovered joint posteriors reveal a rather homogeneous pattern across SDI participants. Flexibility appears seriously impaired, as reflected by the low values of $\lambda$. The ability to efficiently achieve a suitable representation of the (task) environment also appears compromised due to abnormal information loss, as reflected by the high values of $\delta$. However, slight individual differences in the parameters can be observed.
[.99]{} ![Joint posteriors of the flexibility ($\lambda$) and information loss ($\delta$) parameters obtained from the sample of 10 patients. We observe low flexibility and high information loss across all patient. Darker colors represent regions of low posterior density; lighter colors represent regions of high posterior density.[]{data-label="fig:Fig.6"}](Fig6.png "fig:"){width="\textwidth"}
Parameter estimates suggest that error patterns produced by these individuals might be induced by a non-trivial interaction between cognitive sub-components.
Lower values of $\lambda$ imply that errors are likely to be produced by generating responses from an internal environmental model which is no longer valid. In other words, the agent is unable to rely on local feedback-related information in order to update beliefs about hidden states. On the other hand, higher values of $\delta$ reflect a general inefficiency of belief updating processes due to slow convergence to the optimal probabilistic environmental model.
From this perspective, Bayesian surprise $\mathcal{B}_t$ and Shannon surprise $\mathcal{I}_t$ might play different roles in regulating behavior based on different internal model probability configurations. These configurations are governed by the interplay between cognitive parameters.
For instance, it is often the case that psychiatric patients produce a noticeable amount of errors distributed sparsely across windows of trials. However, errors might be processed differently based on the status of the internal environmental states representation, as reflected by the entropy of the predictive model, $\mathcal{H}_t$. Thus, information-theoretic measures allow to describe cognitive dynamics on a trial-by-trial basis and, further, to disentangle the effect that different feedback-related information processing dynamics exert on adaptive behavior.
To further clarify these concepts, we investigate the reconstructed time series of information-theoretic quantities of an exemplary individual response pattern (Patient 7; ).
[.9]{} ![Recovered cognitive dynamics of patient 7. (**a**) Joint posterior of the flexibility and information loss parameters. The marginal posteriors indicate very low flexibility and very high information loss; (**b**) Time series of information-theoretic measures depicting belief updating and agent’s internal model uncertainty during the unfolding of the task. Labels C and E indicate correct and error responses.[]{data-label="fig:Fig.7"}](Fig7a.png "fig:"){width="\textwidth"}
[.99]{} ![Recovered cognitive dynamics of patient 7. (**a**) Joint posterior of the flexibility and information loss parameters. The marginal posteriors indicate very low flexibility and very high information loss; (**b**) Time series of information-theoretic measures depicting belief updating and agent’s internal model uncertainty during the unfolding of the task. Labels C and E indicate correct and error responses.[]{data-label="fig:Fig.7"}](Fig7b.png "fig:"){width="\textwidth"}
depicts the unfolding of cognitive dynamics across a subset of trials in the task. Information-theoretic measures are recovered by computing the posterior mean of parameters.
Processing unexpected observations is accounted by the quantification of surprise at observing a response-feedback pair which is inconsistent with the current internal model of the task environment. Negative feedback is maximally informative when errors occur after the internal model has converged to the true task model (grey area), or the entropy approaches zero (grey line). The Shannon surprise (orange line) is maximal when errors occur within trial windows in which the agent’s uncertainty about environmental states is minimal (orange areas).
However, internal model updates following an informative feedback are not optimally performed, which is reflected by very small Bayesian surprise (blue line). This is due to impaired flexibility, and reflects the fact that after internal model convergence, informative feedback is not processed adequately and the internal model becomes impervious to change.
Conversely, errors occurring when the agent is uncertain about the true environmental state carry no useful information for belief updating, since the system fails to conceive such errors as unexpected and informative. The information loss parameter plays a crucial role in characterizing this cognitive behavior. The slow convergence to the true environmental model, accompanied by the slow reduction of entropy in the predictive model, leads to a large number of trials required to achieve a good representation of the current task environment (white areas). Errors occurring within trial windows with large predictive model entropy (green area) do not affect subsequent behavior, and feedback is maximally uninformative.
The role that predictive (internal) model uncertainty plays in characterizing the way the agent processes feedback allows to disentangle sub-types of errors based on the information they convey for subsequent belief updating. From this perspective, error classification is entirely dependent on the status of the internal environmental model across task phases. Identifying such a dynamic latent process is therefore fundamental, since the error codification criterion evolves with respect to the internal information processing dynamics. Otherwise, the problem of inferring which errors are due to perseverance in maintaining an older (converged) internal model and which due to uncertainty about the true environmental state becomes intractable, or even impossible.
Discussion
==========
Investigating information processing related to changing environmental contingencies is fundamental to understanding adaptive behavior. For this purpose, cognitive scientists usually rely on controlled settings in which individuals are asked to accomplish (possibly) highly demanding tasks whose demands are assumed to resemble those of natural environments. Even in the most trivial cases, such as the WCST, optimal performance requires integrated and distributed neurocognitive processes. Moreover, these processes are unlikely to be isolated by simple scoring or aggregate performance measures.
In the current work, we developed and validated a new computational Bayesian model which maps distinct cognitive processes into separable information-theoretic constructs underlying observed adaptive behavior. We argue that these constructs could help describe and investigate the neurocognitive processes underlying adaptive behavior in a principled way.
In contrast to similar modeling approaches involving information-theoretic constructs [@o2013dissociable; @nour2018dopaminergic; @schwartenbeck2016neural], we adopt a powerful computational method for model identification. The method allows us to recover and quantify uncertainties in parameter estimates which is important for assessing the reliability of information-theoretic constructs in accounting for cognitive properties. In our case, uncertainty or identifiability of cognitive parameters is captured via a full joint posterior, and then a representative statistics of parameter posteriors (e.g., maximum a posteriori, posterior mean) can be used to derive the unfolding of information-theoretic quantities on a trial-by-trial basis.
Several computational models have been proposed to analyze performances in the WCST (and similar RL tasks), ranging from behavioral [@bishara2010sequential; @steinkemulti] to neural network models [@dehaene1991wisconsin; @amos2000computational; @levine1993methodological; @monchi2000neural]. These models aim to provide psychologically interpretable parameters or biologically inspired network structures, respectively, accounting for specific qualitative patterns of observed data. The main advantage of our Bayesian cognitive agent representation is that it provides both a cognitive and a measurement model which coexist within a substantiated theoretical framework.
Therefore, although our computational model is not a neural model, it might provide a suitable description of cognitive dynamics at a representational and computational level [@marr1982vision]. This description can then be related to neural functioning underlying adaptive behavioral. Indeed, there is some evidence to suggest that neural processes related to belief maintenance/updating and unexpectedness are crucial for performance in the WCST. In particular, brain circuits associated with cognitive control and belief formation, such as the parietal cortex and prefrontal regions, seem to share a functional basis with neural substrates involved in adaptive tasks [@nour2018dopaminergic]. Prefrontal regions appear to mediate the relation between feedback and belief updating [@lie2006using] and efficient functioning in such brain structures seems to be heavily dependent on dopaminergic neuromodulation [@ott2019dopamine]. Moreover, the dopaminergic system plays a role in the processing of salient and unexpected environmental stimuli, in learning based on error-related information, and in evaluating candidate actions [@nour2018dopaminergic; @daw2011model; @gershman2018successor]. Accordingly, dopaminergic system functioning has been put in relation with performance in the WCST [@hsieh2010correlation; @rybakowski2005association] and shown to be critical for the main executive components involved in the task, that is, cognitive flexibility and set-shifting [@bestmann2014role; @stelzel2010frontostriatal]. Further, neural activity in the anterior cingulate cortex (ACC) is increased when a negative feedback occurs in the context of the WCST [@lie2006using]. This finding corroborates the view that the ACC is part of an error-detection network which allocates attentional resources to prevent future errors. The ACC might play a crucial role in adaptive functioning by encoding error-related or, more generally, feedback-related information. Thus, it could facilitate the updating of internal environmental models [@rushworth2008choice].
Such neurobiological evidence suggests that brain networks involved in the WCST might endow adaptive behavior by accounting for maintaining/updating of an internal model of the environment and efficient processing of unexpected information. Is it noteworthy, that these processing aspects are incorporated into our computational framework. At this point, the empirical and theoretical potentials of the proposed computational framework for investigating adaptive functioning can be outlined.
*Model-Based Neuroscience*. Recent studies have pointed out the advantage of simultaneously modeling and analyzing neural and behavioral data within a joint modeling framework. In this way, the latter can be used to provide information for the former, as well as the other way around [@turner2017approaches; @turner2013bayesian; @forstmann2011reciprocal]. This involves the development of joint models which encode assumptions about the probabilistic relationships between neural and cognitive parameters.
Within our framework, the reconstruction of information-theoretic discrete time series yields a quantitative account of the agent’s internal processing of environmental information. Event-related cognitive measures of belief updating, epistemic uncertainty and surprise can be put in relation with neural measurements by explicitly providing a formal account of the statistical dependencies between neural and cognitive (information-theoretic) quantities. In this way, latent cognitive dynamics can be directly related to neural event-related measures (e.g., fMRI, EEG). Applications in which information-theoretic measures are treated as dependent variables in standard statistical analysis are also possible.
*Neurological Assessment*. Although neuroscientists have considered performance in the WCST as a proxy for measuring high-level cognitive processes, the usual approach to the analysis of human adaptive behavior consists in summarizing response patterns by simple heuristic scoring measures (e.g. occurrences of correct responses and sub-types of errors produced) and classification rules [@flashman1991note]. However, the theoretical utility of such a summary approach remains questionable. Indeed, adaptive behavior appears to depend on a complex and intricate interplay between multiple network structures [@barcelo2006task; @monchi2001wisconsin; @lie2006using; @barcelo1998non; @buchsbaum2005meta]. This posits a great challenge for disentangling high-level cognitive constructs at a model level and further investigating their relationship with neurobiological substrates. It appears that standard scoring measures might not be able to fulfil these tasks. Moreover, there is a pronounced lack of anatomical specificity in previous research concerning the neural and functional substrates of the WCST [@nyhus2009wisconsin].
Thus, there is a need for more sophisticated modeling approaches. For instance, disentangling errors due to perseverative processing of previously relevant environmental models from those due to uncertainty about task environmental states, is important and nontrivial. Sparse and distributed error patterns might depend on several internal model probability configurations. Such internal models are latent, and can only be uncovered through cognitive modeling. Therefore, information-based criteria to response (error) classification can enrich clinical evaluation beyond heuristically motivated criteria.
*Generalizability*. Another important advantage of the proposed computational framework is that it is not solely confined to the WCST. In fact, one can argue that the seventy-year old WCST does not provide the only or even the most suitable setting for extracting information about cognitive dynamics from general populations or maladaptive behavior in clinical populations. One can envision tasks which embody probabilistic (uncertain) or even chaotic environments (for instance with partially observable or unreliable feedback or partially observable states) and demand integrating information from different modalities [@o2013dissociable; @nour2018dopaminergic]. These settings might prove more suitable for investigating changes in uncertainty-related processing or cross-modal integration than deterministic and fully observable WCST-like settings. Note that, as it currently stands, our framework is directly extendable to these richer settings.
Despite these advantages, our proposed computational framework has some limitations. A first limitation might concern the fact that the new Bayesian cognitive model accounts for the main dynamics in adaptive tasks by relying on only two parameters. Although such a parsimonious proposal suffices to disentangle latent data-generating processes, a more exhaustive formal description of cognitive sub-components might be envisioned. However, model identification can become challenging is such a scenario, especially when sparse one-dimensional response data is used as a basis for parameter recovery.
Second, as it currently stands, model identification is optimal only when the entire sequence of 128 trials in the WCST is used. However, in the Heaton version, the task can end with only after several sorting rule changes. Using incomplete data appears suboptimal for parameter recovery and results in large uncertainty estimates and multimodal posteriors. Future research should focus on designing and employing more data-rich RL tasks which can provide a better starting point for recovering complex latent cognitive dynamics.
In conclusion, the proposed model can be considered as the basis for a (bio)psychometric tool for measuring the dynamics of cognitive processes under changing environmental demands. Furthermore, it can be seen as a step towards a theory-based framework for investigating the relation between such cognitive measures and their neural underpinnings. Further investigations are needed to refine the proposed computational model and systematically explore the advantages of the Bayesian brain theoretical framework for empirical research on high-level cognition.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Karin Prillinger and Luca D’Alessandro for reading the manuscript and providing useful suggestions which significantly improved the original text.
|
=1
Introduction {#sec:Introduction}
============
The tridiagonal $(N+1)\times(N+1)$ matrix of the following form $$\begin{gathered}
C_{N+1} = \left(
\begin{matrix}
0 & 1 & & & &\\
N & 0 & 2 & & &\\
& N-1 & 0 & 3 & & \\
& & \ddots & \ddots & \ddots & \\
& & & 2 & 0 & N \\
& & & & 1 & 0
\end{matrix}
\right)
\label{Kac}\end{gathered}$$ appears in the literature under several names: the Sylvester–Kac matrix, the Kac matrix, the Clement matrix, $\ldots$. It was already considered by Sylvester [@Sylvester], used by M. Kac in some of his seminal work [@Kac], by Clement as a test matrix for eigenvalue computations [@Clement], and continues to attract attention [@Bevilacqua; @Boros; @Taussky]. The main property of the matrix $C_{N+1}$ is that its eigenvalues are given explicitly by $$\begin{gathered}
-N, -N+2, -N+4, \dots, N-2, N.
\label{KacEig}\end{gathered}$$ Because of this simple property, $C_{N+1}$ is a standard test matrix for numerical eigenvalue computations, and part of some standard test matrix toolboxes (e.g., [@Higham]).
One of the outcomes of the current paper implies that $C_{N+1}$ has appealing two-parameter extensions. For odd dimensions, let us consider the following tridiagonal matrix $$\begin{gathered}
C_{2N+1}(\gamma,\delta) = \left(
\begin{matrix}
0 & 2\gamma+2 & & & & & &\\
2N & 0 & 2 & & & & &\\
& 2\delta+2N & 0 & 2\gamma+4 & & & &\\
& & 2N-2 & 0 & 4 & & &\\
& & &\ddots & \ddots & \ddots & &\\
& & & & 2\delta+4 & 0 & 2\gamma+2N & \\
& & & & & 2 & 0 & 2N \\
& & & & & & 2\delta+2 & 0
\end{matrix}
\right).
\label{Kac-odd}\end{gathered}$$ In the following, we shall sometimes use the term “two-diagonal” [@BI] for tridiagonal matrices with zero entries on the diagonal (not to be confused with a bidiagonal matrix, which has also two non-zero diagonals, but for a bidiagonal matrix the non-zero entries are on the main diagonal and either superdiagonal or the subdiagonal). So, just as $C_{2N+1}$ the matrix is two-diagonal, but the superdiagonal of $C_{2N+1}$, $$\begin{gathered}
[1, 2, 3, 4, \ldots, 2N-1, 2N]\end{gathered}$$ is replaced by $$\begin{gathered}
[2\gamma+2, 2, 2\gamma+4, 4, \ldots, 2\gamma+2N, 2N],\end{gathered}$$ and in the subdiagonal of $C_{2N+1}$, $$\begin{gathered}
[2N, 2N-1, 2N-2, \ldots, 3, 2, 1]\end{gathered}$$ the odd entries are replaced, leading to $$\begin{gathered}
[2N, 2\delta+2N, 2N-2, \ldots, 2\delta+4, 2, 2\delta+2].\end{gathered}$$ Clearly, for $\gamma=\delta=-\frac12$ the matrix $C_{2N+1}(\gamma,\delta)$ just reduces to $C_{2N+1}$. One of our results is that $C_{2N+1}(\gamma,\delta)$ has simple eigenvalues for general $\gamma$ and $\delta$, given by $$\begin{gathered}
0, \pm 2\sqrt{1(\gamma+\delta+2)}, \pm 2\sqrt{2(\gamma+\delta+3)}, \pm 2\sqrt{3(\gamma+\delta+4)}, \ldots,
\pm 2\sqrt{N(\gamma+\delta+N+1)}.\end{gathered}$$ This spectrum simplifies even further for $\delta=-\gamma-1$; in this case one gets back the eigenvalues .
For even dimensions, we have a similar result. Let $C_{2N}(\gamma,\delta)$ be the $(2N)\times(2N)$ tridiagonal matrix with zero diagonal, with superdiagonal $$\begin{gathered}
[2\gamma+2, 2, 2\gamma+4, 4, \ldots, 2N-2, 2\gamma+2N]\end{gathered}$$ and with subdiagonal $$\begin{gathered}
[2\delta+2N, 2N-2, 2\delta+2N-2, \ldots, 4, 2\delta+4, 2, 2\delta+2].\end{gathered}$$ Then $C_{2N}(\gamma,\delta)$ has simple eigenvalues for general $\gamma$ and $\delta$, given by[^1] $$\begin{gathered}
\pm 2\sqrt{(\gamma+1)(\delta+1)}, \pm 2\sqrt{(\gamma+2)(\delta+2)}, \ldots, \pm 2\sqrt{(\gamma+N)(\delta+N)}.
\label{Kac-even-Eig}\end{gathered}$$ This spectrum simplifies for $\delta=\gamma$, and obviously for $\gamma=\delta=-\frac12$ one gets back the eigenvalues since in that case $C_{2N}(\gamma,\delta)$ just reduces to $C_{2N}$.
What is the context here for these new tridiagonal matrices with simple eigenvalue properties? Well, remember that $C_{N+1}$ also appears as the simplest example of a family of Leonard pairs . In that context, this matrix is related to symmetric Krawtchouk polynomials [@Ismail; @Koekoek; @Suslov]. Indeed, let $K_n(x)\equiv K_n(x;\frac12,N)$, where $K_n(x;p,N)$ are the Krawtchouk polynomials [@Ismail; @Koekoek; @Suslov]. Then their recurrence relation [@Koekoek equation (9.11.3)] yields $$\begin{gathered}
n K_{n-1}(x) + (N-n) K_{n+1}(x) = (N-2x) K_n(x), \qquad n=0,1,\ldots,N.
\label{Kraw-recur}\end{gathered}$$ Writing this down for $x=0,1,\ldots,N$, and putting this in matrix form, shows indeed that the eigenvalues of $C_{N+1}$ (or rather, of its transpose $C_{N+1}^T$) are indeed given by . Moreover, it shows that the components of the $k$th eigenvector of $C_{N+1}^T$ are given by $K_n(k)$.
So we can identify the matrix $C_{N+1}$ with the Jacobi matrix of symmetric Krawtchouk polynomials, one of the families of finite and discrete hypergeometric orthogonal polynomials. The other matrices $C_{N}(\gamma,\delta)$ appearing in this introduction are not directly related to Jacobi matrices of a simple set of finite orthogonal polynomials. In this paper, however, we show how two sets of distinct dual Hahn polynomials [@Ismail; @Koekoek; @Suslov] can be combined in an appropriate way such that the eigenvalues of matrices like $C_N(\gamma,\delta)$ become apparent, and such that the eigenvector components are given in terms of these two dual Hahn polynomials. This process of combining two distinct sets is called “doubling”. We examine this not only for the case related to the matrix $C_N(\gamma,\delta)$, but stronger: we classify all possible ways in which two sets of dual Hahn polynomials can be combined in order to yield a two-diagonal “Jacobi matrix”. It turns out that there are exactly three ways in which dual Hahn polynomials can be “doubled” (for a precise formulation, see later). By the doubling procedure, one automatically gets the eigenvalues (and eigenvectors) of the corresponding two-diagonal matrix in explicit form.
This process of doubling and investigating the corresponding two-diagonal Jacobi matrix can be applied to other classes of orthogonal polynomials (with a finite and discrete support) as well. In this paper, we turn our attention also to Hahn and to Racah polynomials. The classification process becomes rather technical, however. Therefore, we have decided to present the proof of the complete classification only for dual Hahn polynomials (Section \[sec:dHahn\]). For Hahn polynomials (Section \[sec:Hahn\]) we give the final classification and corresponding two-diagonal matrices (but omit the proof), and for Racah polynomials we give the final classification and some examples of two-diagonal matrices in Appendix \[sec:Racah\].
We should also note that the two-diagonal matrices appearing as a result of the doubling process are symmetric. So matrices like do not appear directly but in their symmetrized form. Of course, as far as eigenvalues are concerned, this makes no difference (see Section \[sec:testmatrix\]).
The doubling process of the polynomials considered here also gives rise to “new” sets of orthogonal polynomials. One could argue whether the term “new” is appropriate, since they arise by combining two known sets. The peculiar property is however that the combined set has a common unique weight function. Moreover, we shall see that the support set of these doubled polynomials is interesting, see the examples in Section \[sec:orthpoly\]. In this section, we also interpret the doubling process in the framework of Christoffel–Geronimus transforms. It will be clear that from our doubling process, one can deduce for which Christoffel parameter the Christoffel transform of a Hahn, dual Hahn or Racah polynomial is again a Hahn, dual Hahn or Racah polynomial with shifted parameters.
In Section \[sec:testmatrix\] we reconsider the two-diagonal matrices that have appeared in the previous sections. It should be clear that we get several classes of two-diagonal matrices (with parameters) for which the eigenvalues (and eigenvectors) have an explicit and rather simple form. This section reviews such matrices as new and potentially interesting examples of eigenvalue test matrices.
In Section \[sec:oscillators\] we explore relations with other structures. Recall that in finite-dimensional representations of the Lie algebra $\su(2)$, with common generators $J_+$, $J_-$ and $J_0$, the matrix of also has a symmetric two-diagonal form. The new two-diagonal matrices appearing in this paper can be seen as representation matrices of deformations or extensions of $\su(2)$. We give the algebraic relations that follow from the “representation matrices” obtained here. The algebras are not studied in detail, but it is clear that they could be of interest on their own. The general algebras have two parameters, and we indicate how special cases with only one parameter are of importance for the construction of finite oscillator models.
Introductory example {#sec:Example}
====================
We start our analysis by the explanation of a known example taken from [@Stoilova2011]. For this example, we first recall the definition of Hahn and dual Hahn polynomials and some of the classical notations and properties.
The Hahn polynomial $Q_n(x;\alpha, \beta, N)$ [@Ismail; @Koekoek; @Suslov] of degree $n$, $n=0,1,\ldots,N$, in the variable $x$, with parameters $\alpha>-1$ and $\beta>-1$ (or $\alpha<-N$ and $\beta<-N$) is defined by [@Ismail; @Koekoek; @Suslov] $$\begin{gathered}
Q_n(x;\alpha,\beta,N) = {}_3F_2 \left( {\genfrac{}{}{0pt}{}{-n,n+\alpha+\beta+1,-x}{\alpha+1,-N}} ; 1 \right).
\label{defQ}\end{gathered}$$ Herein, the function $_3F_2$ is the generalized hypergeometric series [@Bailey; @Slater] $$\begin{gathered}
{}_3F_2 \left( {\genfrac{}{}{0pt}{}{a,b,c}{d,e}} ; z \right)=\sum_{k=0}^\infty \frac{(a)_k(b)_k(c)_k}{(d)_k(e)_k}\frac{z^k}{k!}.
\label{defF}\end{gathered}$$ In (\[defQ\]), the series is terminating because of the appearance of the negative integer $-n$ as a numerator parameter. Note that in (\[defF\]) we use the common notation for Pochhammer symbols [@Bailey; @Slater] $(a)_k=a(a+1)\cdots(a+k-1)$ for $k=1,2,\ldots$ and $(a)_0=1$. Hahn polynomials satisfy a (discrete) orthogonality relation [@Ismail; @Koekoek] $$\begin{gathered}
\sum_{x=0}^N w(x;\alpha, \beta,N) Q_n(x;\alpha, \beta, N) Q_{n'}(x;\alpha,\beta,N) = h_n(\alpha,\beta,N)\, \delta_{n,n'},
\label{orth-Q}\end{gathered}$$ where $$\begin{gathered}
w(x;\alpha, \beta,N) = \binom{\alpha+x}{x} \binom{N+\beta-x}{N-x}, \qquad x=0,1,\ldots,N, \\
h_n (\alpha,\beta,N)= \frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_n n!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_n N!}.\end{gathered}$$ We denote the orthonormal Hahn functions as follows $$\begin{gathered}
\tilde Q_n(x;\alpha,\beta,N) \equiv \frac{\sqrt{w(x;\alpha,\beta,N)}\, Q_n(x;\alpha,\beta,N)}{\sqrt{h_n(\alpha,\beta,N)}}.
$$ The Hahn polynomials satisfy the following recurrence relation [@Koekoek equation (9.5.3)] $$\begin{gathered}
\label{re}
\Lambda ( x) y_n(x) = A(n) y_{n+1}(x) - \bigl( A(n) + C(n)\bigr) y_n(x) + C(n) y_{n-1}(x)\end{gathered}$$ with $$\begin{gathered}
y_n(x) = Q_n(x;\alpha,\beta,N),\qquad \Lambda(x) = -x, \label{ABD}\\
A(n) = \frac{(n+\alpha+1)(n+\alpha+\beta+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)},\qquad
C(n) = \frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}.\nonumber\end{gathered}$$
Related to the Hahn polynomials are the dual Hahn polynomials: $R_n(\lambda(x);\gamma, \delta, N)$ of degree $n$, $n=0,1,\ldots,N$, in the variable $\lambda(x)=x(x+\gamma+\delta+1)$, with parameters $\gamma>-1$ and $\delta>-1$ (or $\gamma<-N$ and $\delta<-N$) which are defined similarly to [@Ismail; @Koekoek; @Suslov] $$\begin{gathered}
\label{DHahn}
R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2 \left( {\genfrac{}{}{0pt}{}{-x,x+\gamma+\delta+1,-n}{\gamma+1,-N}} ; 1 \right).\end{gathered}$$ As is well known, the (discrete) orthogonality relation of the dual Hahn polynomials is just the “dual” of $$\begin{gathered}
\sum_{x=0}^N \overline{w}(x;\gamma, \delta,N) R_n(\lambda(x);\gamma, \delta, N) R_{n'}(\lambda(x);\gamma,\delta,N) = \overline{h}_n(\gamma,\delta,N) \delta_{n,n'},
\label{orth-R}\end{gathered}$$ where $$\begin{gathered}
\overline{w}(x;\gamma, \delta,N) = \frac{(2x+\gamma+\delta+1)(\gamma+1)_x(-N)_x N!}{(-1)^x(x+\gamma+\delta+1)_{N+1}(\delta+1)_x x!},\\
\overline{h}_n (\gamma,\delta,N)= \left[\binom{\gamma+n}{n} \binom{N+\delta-n}{N-n}\right]^{-1} .\end{gathered}$$ Orthonormal dual Hahn functions are defined by $$\begin{gathered}
\tilde R_n(\lambda(x);\gamma,\delta,N) \equiv \frac{\sqrt{\overline{w}(x;\gamma,\delta,N)} R_n(\lambda(x);\gamma,\delta,N)}{\sqrt{\overline{h}_n(\gamma,\delta,N)}}.
\label{R-tilde}\end{gathered}$$ Dual Hahn polynomials also satisfy a recurrence relation of the form , with [@Koekoek equation (9.6.3)] $$\begin{gathered}
y_n(x) = R_n(\lambda ( x);\gamma,\delta,N),\qquad \Lambda(x)=\lambda(x) = x(x+\gamma+\delta+1),\nonumber\\
A(n) = (n+\gamma+1)(n-N),\qquad C(n) = n(n-\delta-N-1).
\label{AC}\end{gathered}$$
In [@Stoilova2011], the following difference equations involving two sets of Hahn polynomials were derived (for convenience we use the notation $Q_n(x)\equiv Q_n(x;\alpha,\beta+1,N)$ and $\hat Q_n(x)\equiv Q_n(x;\alpha+1,\beta,N)$): $$\begin{gathered}
(N+\beta+1-x) Q_n(x)-(N-x) Q_{n}(x+1) =\frac{(n+\alpha +1)(n+\beta+1)}{\alpha+1} \hat Q_n(x), \label{Q-rec1} \\
(x+1) \hat Q_n(x)-(\alpha +x+2) \hat Q_{n}(x+1) =-(\alpha+1) Q_n(x+1). \label{Q-rec2}\end{gathered}$$ Writing out these difference equations for $x=0,1,\ldots,N$, the resulting set of equations can easily be written in matrix form. For this matrix form, let us use the normalized version of the polynomials, and construct the following $(2N+2)\times (2N+2)$ matrix $U$ with elements $$\begin{gathered}
U_{2x,N-n} = U_{2x,N+n+1} = \frac{(-1)^x}{\sqrt{2}} \tilde Q_n(x;\alpha,\beta+1,N), \label{Ueven}\\
U_{2x+1,N-n} = -U_{2x+1,N+n+1} = -\frac{(-1)^x}{\sqrt{2}} \tilde Q_n(x;\alpha+1,\beta,N), \label{Uodd}\end{gathered}$$ where $x,n\in\{0,1,\ldots,N\}$. By construction, this matrix is orthogonal [@Stoilova2011]: the fact that the columns of $U$ are orthonormal follows from the orthogonality relation of the Hahn polynomials, and from the signs in the matrix $U$. Thus $U^TU=UU^T=I$, the identity matrix.
The normalized difference equations , for $x=0,1,\ldots,N$ can then be cast in matrix form. The coefficients in the left hand sides of , give rise to a tridiagonal $(2N+2)\times(2N+2)$-matrix of the form $$\begin{gathered}
\label{MK}
M= \left( \begin{matrix}
0 & M_0 & 0 & & \\
M_0 & 0 & M_1 & \ddots & \\
0 & M_1 & 0 & \ddots & 0 \\
&\ddots & \ddots & \ddots & M_{2N} \\
& & 0 & M_{2N} & 0
\end{matrix} \right),\end{gathered}$$ with $$\begin{gathered}
M_{2k}= \sqrt{(k+\alpha+1)(N+\beta+1-k)}, \qquad M_{2k+1}=\sqrt{(k+1)(N-k)}.
\label{M_k}
\end{gathered}$$ Suppose $\alpha>-1$, $\beta>-1$ or $\alpha<-N-1$, $\beta<-N-1$ and let $U$ be the orthogonal matrix determined in , . Then [@Stoilova2011] the columns of $U$ are the eigenvectors of $M$, i.e., $$\begin{gathered}
M U = U D,
\label{MUUD}\end{gathered}$$ where $D$ is a diagonal matrix containing the eigenvalues of $M$ $$\begin{gathered}
D= \operatorname{diag} (-\epsilon_N,\ldots,-\epsilon_1,-\epsilon_0,\epsilon_0,\epsilon_1,\ldots,\epsilon_{N}), \nonumber\\
\epsilon_{k}=\sqrt{(\alpha+k+1)(\beta+k+1)},\qquad k=0,1,\ldots,N. \label{epsilon}\end{gathered}$$
Note that the eigenvalues of the matrix $M$ are (up to a factor 2) the same as those of the matrix $C_{2N+2}(\alpha,\beta)$, the two-parameter extension of the Sylvester–Kac matrix. As we will further discuss in Section \[sec:testmatrix\], the above result proves that the eigenvalues of $C_{2N+2}(\alpha,\beta)$ are indeed given by . Even more: the orthonormal eigenvectors of $M$ are just the columns of $U$.
Another way of looking at is in terms of the dual Hahn polynomials. Interchanging $x$ and $n$ in the expressions , , we have $$\begin{gathered}
U_{2n,N-x} = U_{2n,N+x+1} = \frac{(-1)^n}{\sqrt{2}} \tilde R_n(\lambda(x);\alpha,\beta+1,N), \label{UevenR}\\
U_{2n+1,N-x} = -U_{2n+1,N+x+1} = -\frac{(-1)^n}{\sqrt{2}} \tilde R_n(\lambda(x);\alpha+1,\beta,N), \label{UoddR}\end{gathered}$$ where $x,n\in\{0,1,\ldots,N\}$. In this way, each row of the matrix $U$ consists of a dual Hahn polynomial of a certain degree, having different parameter values for even and odd rows. Now, the relation can be interpreted as a three-term recurrence relation with $M$ being the Jacobi matrix. Two sets of (dual) Hahn polynomials (with different parameters) are thus combined into a new set of polynomials such that the Jacobi matrix for this new set has a simple two-diagonal form, with simple eigenvalues. The pair of difference equations , involving two sets of Hahn polynomials then corresponds to the following relations involving the dual Hahn polynomials $R_n(x)\equiv R_{n}(\lambda(x);\gamma,\delta+1,N)$ and $\hat R_n(x)\equiv R_{n}(\lambda(x);\gamma+1,\delta,N)$: $$\begin{gathered}
\label{R-rec1}
(N+\delta+1-n)R_{n}(x)-(N-n) R_{n+1}(x) =\frac{(x+\gamma+1)(x+\delta+1)}{(\gamma+1)} \hat R_n(x),\\
(n+1) \hat R_n(x)- (n+\gamma+2) \hat R_{n+1}(x) = -(\gamma+1) R_{n+1}(x).\label{R-rec2}\end{gathered}$$ This is in fact a special case of the so-called Christoffel transform of a dual Hahn polynomial with its transformation parameter chosen specifically so that the result is again a dual Hahn polynomial (with different parameters). We will further elaborate on this in Section \[sec:orthpoly\].
This introductory example, taken from [@Stoilova2011], opens the following question: in how many ways can two sets of (dual) Hahn polynomials be combined such that the Jacobi matrix is two-diagonal? This will be answered in the following section.
Doubling dual Hahn polynomials: classification {#sec:dHahn}
==============================================
The essential relation in the previous example is the existence of a pair of “recurrence relations” , intertwining two types of dual Hahn polynomials (or equivalently a couple of difference equations , for two types of their duals, the Hahn polynomials). Let us therefore examine the existence of such relations in general. Say we have two types of dual Hahn polynomials with different parameter values for $\gamma$ and $\delta$ (and possibly $N$) denoted by $R_n(\lambda ( x);\gamma,\delta,N)$ and $R_{n}(\lambda (\hat x);\hat{\gamma},\hat{\delta},\hat{N})$, that are related in the following manner $$\begin{gathered}
\label{rel1}
a(n) R_n(\lambda ( x);\gamma,\delta,N) + b(n) R_{n+1}(\lambda ( x);\gamma,\delta,N) ={\hat{d}}(x) R_{n}(\lambda (\hat x);\hat{\gamma},\hat{\delta},\hat{N}) \\
\hat{a}(n) R_{n}(\lambda (\hat x);\hat{\gamma},\hat{\delta},\hat{N}) + \hat{b}(n) R_{n+1}(\lambda (\hat x);\hat{\gamma},\hat{\delta},\hat{N}) = d(x) R_{n+1}(\lambda ( x);\gamma,\delta,N) .
\label{rel2}\end{gathered}$$ If we want these relations to correspond to a matrix identity like , then it is indeed necessary that the (unknown) functions $a(n)$, $\hat a(n)$, $b(n)$ and $\hat b(n)$ are functions of $n$ and not of $x$, and that $d(x)$ and $\hat d(x)$ are functions of $x$ and not of $n$. Of course, the parameters $\gamma$, $\delta$, $N$, $\hat\gamma$, $\hat\delta$, $\hat N$ can appear in these functions.
In order to lift this technique also to other polynomials than just the dual Hahn polynomials, say we have the following relations between two sets of orthogonal polynomials of the same class, denoted by $y_{n}$ and $\hat{y}_{n}$, but with different parameter values $$\begin{gathered}
\label{yyh}
a(n) y_{n} + b(n) y_{n+1} ={\hat{d}}(x) \hat{y}_{n} ,\\
\hat{a}(n) \hat{y}_{n} + \hat{b}(n) \hat{y}_{n+1} = d(x) y_{n+1} ,
\label{hhy}\end{gathered}$$ where $a$, ${\hat{a}}$, $b$, ${\hat{b}}$ are independent of $x$ and $d$, ${\hat{d}}$ are independent of $n$. Although , are not actual recurrence relations since they involve both $y_{n}$ and $\hat{y}_{n}$, we will refer to a couple of such relations intertwining two types of orthogonal polynomials as “a pair of recurrence relations”.
When substituting in , we arrive at the following recurrence relation for $\hat y_{n}$ $$\begin{gathered}
\label{de1}
a(n) \bigl\lbrack \hat{a}(n-1) \hat{y}_{n-1} + \hat{b}(n-1) \hat{y}_{n}\bigr\rbrack + b(n) \bigl\lbrack \hat{a}(n) \hat{y}_{n} + \hat{b}(n) \hat{y}_{n+1}\bigr\rbrack =d(x) {\hat{d}}(x) \hat{y}_{n}.\end{gathered}$$ In the same manner, $\hat{y}_{n}$ can be eliminated to find a recurrence relation for $y_{n}$ $$\begin{gathered}
\label{deh1}
\hat{a}(n-1) \bigl\lbrack a(n-1) y_{n-1} + b(n-1) y_{n} \bigr\rbrack + \hat{b}(n-1) \bigl\lbrack a(n) y_{n} + b(n) y_{n+1}\bigr\rbrack = {\hat{d}}(x) d(x) y_{n} .\end{gathered}$$
Of course, the orthogonal polynomials $y_{n}$ already satisfy a three-term recurrence relation of the form . A comparison of the coefficients of $y_{n+1}$, $ y_{n}$, $y_{n-1}$ in , with the known coefficients given in leads to the following set of requirements for $a$, ${\hat{a}}$, $b$, ${\hat{b}}$, $d$, ${\hat{d}}$ $$\begin{gathered}
\label{Dh}
a(n) \hat{a}(n-1) ={\hat{C}}(n), \\
\label{D}
a(n-1) \hat{a}(n-1) =C(n), \\
\label{BDh}
a(n) \hat{b}(n-1) + \hat{a}(n) b(n) - d(x)\,{\hat{d}}(x) = - \bigl\lbrack \hat{\Lambda} ( x)+ {\hat{A}}(n) + {\hat{C}}(n)\bigr\rbrack, \\
\label{BD}
a(n) \hat{b}(n-1) + \hat{a}(n-1) b(n-1) - {\hat{d}}(x)d(x) = - \bigl\lbrack \Lambda ( x)+ A(n) + C(n)\bigr\rbrack, \\
\label{Bh}
b(n) \hat{b}(n) = {\hat{A}}(n), \\
\label{B}
b(n) \hat{b}(n-1) = A(n).\end{gathered}$$ After a slight rearrangement of terms in the requirements and , we arrive at two new equations where the left hand side is independent of $x$ while the right hand side is independent of $n$, namely, $$\begin{gathered}
\label{BDh2}
a(n) \hat{b}(n-1) + \hat{a}(n) b(n) + {\hat{A}}(n) + {\hat{C}}(n) = d(x){\hat{d}}(x)- \hat{\Lambda} ( x), \\
\label{BD2}
a(n) \hat{b}(n-1) + \hat{a}(n-1) b(n-1) + A(n) + C(n) = {\hat{d}}(x) d(x)- \Lambda ( x).\end{gathered}$$ Hence, the two sides must be independent of both $n$ and $x$. By means of – we can eliminate $A$, $\hat{A}$, $C$, $\hat{C}$ to find $$\begin{gathered}
a(n) \bigl\lbrack \hat{a}(n-1) + \hat{b}(n-1) \bigr\rbrack + b(n)\bigl\lbrack \hat{a}(n) +\hat{b}(n) \bigr\rbrack = d(x) {\hat{d}}(x)- \hat{\Lambda} ( x), \\
\hat{a}(n-1) \bigl\lbrack a(n-1) + b(n-1) \bigr\rbrack +\hat{b}(n-1) \bigl\lbrack a(n) + b(n)\bigr\rbrack = {\hat{d}}(x) d(x)- \Lambda ( x).
\end{gathered}$$ Moreover, subtracting one from the other yields $$\begin{gathered}
\Lambda ( x) - \hat{\Lambda} ( x)
= \hat{a}(n-1) \bigl\lbrack a(n)-a(n-1) - b(n-1) \bigr\rbrack
+ b(n) \bigl\lbrack \hat{a}(n) + \hat{b}(n) - \hat{b}(n-1) \bigr\rbrack. \!\!\!\label{LL}\end{gathered}$$
Now, for a given class of orthogonal polynomials with recurrence relation of the form , we determine all possible functions $a$, $\hat{a}$, $b$, $\hat{b}$, $d$, ${\hat{d}}$ satisfying the list of requirements –. Hereto, we proceed as follows
- From and we observe that, up to a multiplicative factor, $C(n)$ is split into two functions, $a(n-1)$ and $\hat{a}(n-1)$. When $a(n-1)$ is shifted by 1 in $n$ and multiplied again by $\hat{a}(n-1)$ we must arrive at $\hat{C}(n)$. Hence, $C$ and $\hat{C}$ consist of an identical part, and a part which differs by a shift of 1 in $n$. This observation gives a first list of possibilities for $a$ and $\hat{a}$.
- Similarly we find a list for $b$ and $\hat{b}$ by means of and .
- These possibilities are then to be compared with requirements and . From , and we get an expression for the product $d(x) {\hat{d}(x)}$. Finally, the set of remaining choices for $a$, ${\hat{a}}$, $b$, ${\hat{b}}$ are to be plugged in and in order to get $d$, ${\hat{d}}$ and to verify if these relations indeed hold.
The actual performance of the procedure just described is still quite long and tedious, when carried out for a fixed class of polynomials. In what follows we achieve this for the dual Hahn polynomials, which have the easiest recurrence relation, and it takes about three pages to present this. The reader who wishes to skip the details can advance to Theorem \[theo2\].
For dual Hahn polynomials, the data is given by $$\begin{gathered}
y_n = R_n(\lambda ( x);\gamma,\delta,N),\qquad \hat{y}_n = R_{n}\big(\lambda (\hat x);\hat{\gamma},\hat{\delta},\hat{N}\big), \qquad \Lambda(x) =\lambda ( x)= x(x+\gamma+\delta+1),
\\
A(n) = (n+\gamma+1)(n-N),\qquad
C(n) = n(n-\delta-N-1),\end{gathered}$$ and with similar expressions for $\hat\Lambda(x)$, $\hat A(n)$ and $\hat C(n)$ (with $x$, $\gamma$, $\delta$, $N$ replaced by $\hat x$, $\hat\gamma$, $\hat\delta$, $\hat N$). From , the following expression must be independent of $x$ $$\begin{gathered}
\Lambda ( x) - \hat{\Lambda} ( x) = x(x+\gamma+\delta+1) - \hat{x}\big(\hat{x}+\hat{\gamma}+\hat{\delta}+1\big).\end{gathered}$$ In order for the term in $x^2$ to disappear, we must have $\hat{x} = x + \xi$ which gives $$\begin{gathered}
x(x+\gamma+\delta+1) - ( x + \xi)( x + \xi+\hat{\gamma}+\hat{\delta}+1) = (\gamma+\delta-\hat{\gamma}-\hat{\delta} -2 \xi ) x - \xi( \xi+\hat{\gamma}+\hat{\delta}+1)\end{gathered}$$ and as we require the coefficient of $x$ to be zero we find the following condition for $\xi$ $$\begin{gathered}
\label{ab}
\gamma+\delta-(\hat{\gamma}+\hat{\delta}) = 2 \xi .\end{gathered}$$
From we see that we have four distinct possible combinations for $a(n-1)$ and $\hat{a}(n-1)$ $$\begin{aligned}
{3}
& a(n-1) = 1 c_a, \qquad && \hat{a}(n-1) = n(n-\delta-N-1) c_a^{-1}, & \tag{a1}\label{a1}\\
& a(n-1) = n c_a, \qquad && \hat{a}(n-1) = (n-\delta-N-1) c_a^{-1}, & \tag{a2}\label{a2}\\
& a(n-1) = (n-\delta-N-1) c_a, \qquad && \hat{a}(n-1) = n c_a^{-1}, & \tag{a3}\label{a3}\\
& a(n-1) = n(n-\delta-N-1) c_a, \qquad && \hat{a}(n-1) = 1 c_a^{-1}, &\tag{a4}\label{a4}\end{aligned}$$ with $c_a$ a factor. Combining this with we must have $$\begin{gathered}
a(n) \hat{a}(n-1) ={\hat{C}}(n) = n\big(n-\hat{\delta}-\hat{N}-1\big).\end{gathered}$$ This immediately implies that $c_a$ is independent of $n$, and – yield the following possibilities $$\begin{aligned}
& n(n-\delta-N-1) = n(n-\hat{\delta}-\hat{N}-1) \ && \implies \ \delta+N =\hat{\delta}+\hat{N}, & \tag{a1$'$}\label{a1'}\\
& (n+1) (n-\delta-N-1) = n(n-\hat{\delta}-\hat{N}-1)\!\!\! && \implies\ \delta+N+1 = 0 \, \land\, \hat{\delta}+\hat{N}+2=0,\!\!\!\!\!\!\!\! & \tag{a2$'$}\label{a2'}\\
& (n-\delta-N) n = n(n-\hat{\delta}-\hat{N}-1) \ && \implies \ \delta+N =\hat{\delta}+\hat{N}+1, & \tag{a3$'$}\label{a3'}\\
& (n+1)(n-\delta-N)= n(n-\hat{\delta}-\hat{N}-1) \ && \implies \ \delta+N = 0\, \land \, \hat{\delta}+\hat{N}+2 =0. & \tag{a4$'$}\label{a4'}\end{aligned}$$ Because of the restriction on $\delta$ the option is ineligible, leaving – as only viable options.
In a similar way, from we see that we have four possible combinations for $b(n)$ and $\hat{b}(n)$, $$\begin{aligned}
{3}
& b(n)= 1 c_b,\qquad & & \hat{b}(n-1) = (n+\gamma+1)(n-N) c_b^{-1}, & \tag{b1}\label{b1}\\
& b(n)= (n+\gamma+1) c_b,\qquad & & \hat{b}(n-1) = (n-N) c_b^{-1}, & \tag{b2}\label{b2}\\
& b(n)= (n-N)c_b,\qquad & & \hat{b}(n-1) = (n+\gamma+1) c_b^{-1}, &\tag{b3}\label{b3} \\
& b(n)= (n+\gamma+1)(n-N) c_b,\qquad & & \hat{b}(n-1) = 1 c_b^{-1}. &\tag{b4}\label{b4}\end{aligned}$$ Combining this with we must have $$\begin{gathered}
b(n) \hat{b}(n) ={\hat{A}}(n) = (n+\hat{\gamma}+1)(n-\hat{N})\end{gathered}$$ This implies that $c_b$ is independent of $n$ and moreover for – yields $$\begin{aligned}
& (n+\gamma+2)(n-N+1) = (n+\hat{\gamma}+1)(n-\hat{N}) && \implies \ \gamma+1 =\hat{\gamma} \, \land \, N-1 =\hat{N}, & \tag{b1$'$}\label{b1'}\\
& (n+\gamma+1)(n-N+1) = (n+\hat{\gamma}+1)(n-\hat{N}) && \implies \ \gamma =\hat{\gamma} \, \land \, N-1 =\hat{N}, & \tag{b2$'$}\label{b2'}\\
& (n+\gamma+2)(n-N) = (n+\hat{\gamma}+1)(n-\hat{N}) && \implies \ \gamma+1 =\hat{\gamma} \, \land \, N =\hat{N}, & \tag{b3$'$}\label{b3'}\\
& (n+\gamma+1)(n-N)= (n+\hat{\gamma}+1)(n-\hat{N}) && \implies \ \gamma =\hat{\gamma} \, \land \, N =\hat{N}. & \tag{b4$'$}\label{b4'}\end{aligned}$$ We thus have four viable options for $b$, $\hat{b}$ and three for $a$, $\hat{a}$, giving a total of 12 possible combinations, which we will systematically consider and treat.
[*Case* ]{}. Plugging in , we get $$\begin{gathered}
a(n) (n+\gamma+1)(n-N)\, c_b^{-1} + \hat{a}(n-1) c_b + (n+\gamma+1)(n-N)+
n(n-\delta-N-1)\\
\qquad{} = {\hat{d}}(x)d(x)- \Lambda ( x).\end{gathered}$$ As the right hand side is independent of $n$, so must be the left hand side. This eliminates options and for $a$, $\hat{a}$ as that would result in a third order term in $n$ which cannot vanish. On the other hand, yields $$\begin{gathered}
(n+\gamma+1)(n-N) \frac{c_a}{ c_b} + n(n-\delta-N-1)\frac{c_b}{ c_a} + (n+\gamma+1)(n-N)+
n(n-\delta-N-1) \\
\qquad{} = {\hat{d}}(x)d(x)- \Lambda ( x).\end{gathered}$$ This must be independent of $n$, so the coefficient of $n^2$ in the left hand side must vanish, hence ${c_a}/{ c_b} + {c_b}/{ c_a} +2=0$ or thus ${c_a}/{ c_b}=-1$. For this value of ${c_a}/{ c_b}$ the left hand side equals zero and is indeed independent of $n$. Note that this leaves one degree of freedom as only the ratio ${c_a}/{ c_b}$ is fixed. This is just a global scalar factor for and , also present in . Henceforth, for convenience, we set $c_a =1$ and $c_b=-1$.
The combined options and thus give a valid set of equations of the form and , and they correspond to the parameter values $$\begin{gathered}
\hat{\gamma} = \gamma+1,\qquad \hat{\delta}= \delta+1,\qquad \hat{N} = N-1.\end{gathered}$$ Moreover, by means of we find $\xi=-1$ and so $\hat{x} = x-1$. Finally, plugging these $a$, ${\hat{a}}$, $b$, ${\hat{b}}$ in and , and putting $n=0$ we find $$\begin{gathered}
R_{0}(\lambda(x);\gamma,\delta,N)- R_{1}(\lambda(x);\gamma,\delta,N) = \frac{x(x+\gamma+\delta+1)}{N(\gamma+1)} = \hat{d}(x)\end{gathered}$$ and similarly $d(x)= N(\gamma+1)$. Hence, for $R_n(x)\equiv R_n(\lambda(x);\gamma,\delta,N)$ and $\hat R_n(x)\equiv R_n(\lambda(x-1)$; $\gamma+1,\delta+1,N-1)$ we have the relations $$\begin{gathered}
R_{n}(x)- R_{n+1}(x) = \frac{x(x+\gamma+\delta+1)}{N(\gamma+1)} \hat R_{n}(x), \\
-(n+1)(N-n+\delta) \hat R_{n}(x)+(N-n-1)(n+\gamma+2) \hat R_{n+1}(x) = N(\gamma+1)R_{n+1}(x).\end{gathered}$$ Interchanging $x$ and $n$, these recurrence relations for dual Hahn polynomials are precisely the known actions of the forward and backward shift operator for Hahn polynomials [@Koekoek equations (9.5.6) and (9.5.8)].
[*Case* ]{}. Next, we consider the option for $b$, $\hat{b}$. Plugging in , we get $$\begin{gathered}
a(n) (n-N) c_b^{-1} + \hat{a}(n-1) (n+\gamma) c_b + (n+\gamma+1)(n-N)+
n(n-\delta-N-1)\\
\qquad{} = {\hat{d}}(x) d(x)- \Lambda ( x).\end{gathered}$$ Since the left hand side must be independent of $n$, option is ruled out. Also option is ruled out: using and $\delta+N+1=0$ (from ), the left hand side again cannot be independent of $n$. Only remains, giving $$\begin{gathered}
(n-\delta-N) (n-N)\,\frac{c_a}{ c_b} + n (n+\gamma) \frac{c_b}{ c_a} + (n+\gamma+1)(n-N)+
n(n-\delta-N-1)\\
\qquad{} = {\hat{d}}(x) d(x)- \Lambda ( x).\end{gathered}$$ In order for $n^2$ in the left hand side to vanish, we again require ${c_a}/{ c_b}=-1$. This gives $$\begin{gathered}
-N(N+\gamma+\delta+1) = {\hat{d}}(x) d(x)- \Lambda ( x),\end{gathered}$$ and we see that both sides are indeed independent of $n$.
The combined options and also give a valid set of equations of the form and , now corresponding to the parameter values $$\begin{gathered}
\hat{\gamma} = \gamma, \qquad \hat{\delta}= \delta,\qquad \hat{N} = N-1.\end{gathered}$$ Moreover, by means of we find $\xi=0$ and so $\hat{x} = x$. Putting again $n=0$ in and for these $a$, ${\hat{a}}$, $b$, ${\hat{b}}$ we find $$\begin{gathered}
\begin{split}
&(-\delta-N) R_0(\lambda(x);\gamma,\delta,N) - (\gamma+1) R_1(\lambda(x);\gamma,\delta,N) \\
&\qquad{}=
- \frac{(N-x)(x+\gamma+\delta+N+1)}{N} = \hat{d}(x)
\end{split}\end{gathered}$$ and similarly $d(x)= N$. The relations in question are then, for $R_n(x)\equiv R_n(\lambda(x);\gamma,\delta,N)$ and $\hat R_n(x)\equiv R_n(\lambda(x);\gamma,\delta,N-1)$ $$\begin{gathered}
(n-\delta-N) R_n(x) - (n+\gamma+1) R_{n+1}(x) =- \frac{(N-x)(x+\gamma+\delta+N+1)}{N} \hat R_{n}(x), \\
(n+1) \hat R_{n}(x) -(n-N+1) \hat R_{n+1}(x) = N R_{n+1}(x),\end{gathered}$$ which can be verified algebraically or by means of a computer algebra package.
[*Case* ]{}. The next option to consider is , for which becomes $$\begin{gathered}
a(n) (n+\gamma+1) c_b^{-1} + \hat{a}(n-1) (n-N-1) c_b + (n+\gamma+1)(n-N)+
n(n-\delta-N-1)\\
\qquad{} = {\hat{d}}(x) d(x)- \Lambda ( x).\end{gathered}$$ The independence of $n$ in the left hand side again rules out options and , while gives $$\begin{gathered}
(n-\delta-N) (n+\gamma+1) \frac{c_a}{ c_b} + n (n-N-1)\frac{c_b}{ c_a} + (n+\gamma+1)(n-N)+
n(n-\delta-N-1)\\
\qquad{} = {\hat{d}}(x) d(x)- \Lambda ( x).\end{gathered}$$ Also here, we require ${c_a}/{ c_b}=-1$ to arrive at a left hand side independent of $n$, namely $$\begin{gathered}
(\gamma+1)\delta = {\hat{d}}(x) d(x)- \Lambda ( x).\end{gathered}$$ The combined options and thus give a valid set of equations of the form and , and they correspond to the parameter values $$\begin{gathered}
\hat{\gamma} = \gamma+1, \qquad \hat{\delta}= \delta-1,\qquad \hat{N} = N;\end{gathered}$$ by means of we find $\xi=0$ and so $\hat{x} = x$. Finally, plugging these $a$, ${\hat{a}}$, $b$, ${\hat{b}}$ in and and putting $n=0$ we find $$\begin{gathered}
(-\delta-N)R_{0}(\lambda(x);\gamma,\delta,N)+N R_{1}(\lambda(x);\gamma,\delta,N) =- \frac{(x+\gamma+1)(x+\delta)}{(\gamma+1)} = \hat{d}(x)\end{gathered}$$ and similarly $d(x)= \gamma+1$.
Hence we have the relations, for $R_n(x)\equiv R_n(\lambda(x);\gamma,\delta,N)$ and $\hat R_n(x)\equiv R_n(\lambda(x);\gamma+1$, $\delta-1,N)$ $$\begin{gathered}
-(n-\delta-N)R_{n}(x)+(n-N) R_{n+1}(x) =\frac{(x+\gamma+1)(x+\delta)}{(\gamma+1)} \hat R_n(x),\\
-(n+1) \hat R_n(x)+ (n+\gamma+2) \hat R_{n+1}(x) = (\gamma+1) R_{n+1}(x).\end{gathered}$$ These can again be verified algebraically or by means of a computer algebra package. Note that these relations coincide with , from the previous section (up to a shift $\delta\rightarrow\delta+1$).
[*Case* ]{}. The final option for $b$, $\hat{b}$ does not correspond to a valid set of equations of the form and as the left hand side of can never be independent of $n$ for either options , or .
This completes the analysis in the case of dual Hahn polynomials, and we have the following result
\[theo2\] The only way to double dual Hahn polynomials, i.e., to combine two sets of dual Hahn polynomials such that they satisfy a pair of recurrence relations of the form , is one of the three cases:
[**dual Hahn I**]{}, $R_n(x)\equiv R_n(\lambda(x);\gamma,\delta,N)$ and $\hat R_n(x)\equiv R_n(\lambda(x-1);\gamma+1,\delta+1,N-1)$: $$\begin{gathered}
R_{n}(x)- R_{n+1}(x) = \frac{x(x+\gamma+\delta+1)}{N(\gamma+1)} \hat R_{n}(x), \\
-(n+1)(N-n+\delta) \hat R_{n}(x)+(N-n-1)(n+\gamma+2) \hat R_{n+1}(x) = N(\gamma+1)R_{n+1}(x).
\end{gathered}$$ [**dual Hahn II**]{}, $R_n(x)\equiv R_n(\lambda(x);\gamma,\delta,N)$ and $\hat R_n(x)\equiv R_n(\lambda(x);\gamma,\delta,N-1)$: $$\begin{gathered}
(n-\delta-N) R_n(x) - (n+\gamma+1) R_{n+1}(x) =- \frac{(N-x)(x+\gamma+\delta+N+1)}{N} \hat R_{n}(x), \\
(n+1) \hat R_{n}(x) -(n-N+1) \hat R_{n+1}(x) = N R_{n+1}(x).
\end{gathered}$$ [**dual Hahn III**]{}, $R_n(x)\equiv R_n(\lambda(x);\gamma,\delta,N)$ and $\hat R_n(x)\equiv R_n(\lambda(x);\gamma+1,\delta-1,N)$: $$\begin{gathered}
-(n-\delta-N)R_{n}(x)+(n-N) R_{n+1}(x) =\frac{(x+\gamma+1)(x+\delta)}{(\gamma+1)} \hat R_n(x),\\
-(n+1) \hat R_n(x)+ (n+\gamma+2) \hat R_{n+1}(x) = (\gamma+1) R_{n+1}(x).
\end{gathered}$$
By interchanging $x$ and $n$, each of the recurrence relations for dual Hahn polynomials in the previous theorem gives rise to a set of forward and backward shift operators for regular Hahn polynomials. The case [**dual Hahn I**]{} corresponds just to the known forward and backward shift operators for Hahn polynomials [@Koekoek]: $Q_n(x)\equiv Q_n(x;\alpha,\beta,N)$ and $\hat Q_n(x)\equiv Q_n(x;\alpha+1,\beta+1$, $N-1)$: $$\begin{gathered}
Q_{n}(x)- Q_{n}(x+1) = \frac{n(n+\alpha+\beta+1)}{N(\alpha+1)} \hat Q_{n-1}(x), \\
-(x+1)(N-x+\beta) \hat Q_{n-1}(x)+(N-x-1)(x+\alpha+2) \hat Q_{n-1}(x+1) \\
\qquad{} = N(\alpha+1)Q_{n}(x+1).\end{gathered}$$ The case [**dual Hahn III**]{} corresponds to our introductory example , (up to a shift $\beta\rightarrow\beta+1$), and appears already in [@Stoilova2011]. The case [**dual Hahn II**]{} yields a new set of relations (encountered recently in [@JSV2014 equations (16), (17)]), namely $Q_n(x)\equiv Q_n(x;\alpha,\beta,N)$ and $\hat Q_n(x)\equiv Q_n(x;\alpha,\beta,N-1)$: $$\begin{gathered}
(x-\beta-N) Q_n(x) - (x+\alpha+1) Q_n(x+1) =- \frac{(N-n)(n+\alpha+\beta+N+1)}{N} \hat Q_{n}(x), \\
(x+1) \hat Q_{n}(x) -(x-N+1) \hat Q_{n}(x+1) = N Q_n(x+1).\end{gathered}$$ The most important thing is, however, that we have classified the possible cases.
Because the sets of recurrence relations are of the form , , they can be cast in matrix form, like in , with a simple two-diagonal matrix. For the case [**dual Hahn I**]{}, note that the $N$-values of $R_n(x)$ and $\hat R_n(x)$ differ by 1, so the definition of the matrix $U$ (again in terms of the normalized version of the polynomials) requires a little bit more attention. The matrix $U$ is now of order $(2N+1)\times (2N+1)$ with matrix elements $$\begin{gathered}
U_{2n,N-x} = U_{2n,N+x} = \frac{(-1)^n}{\sqrt{2}} \tilde R_n(\lambda(x);\gamma,\delta,N), \qquad x=1,\ldots,N, \nonumber \\
U_{2n+1,N-x} = -U_{2n+1,N+x} = -\frac{(-1)^n}{\sqrt{2}} \tilde R_n(\lambda(x-1);\gamma+1,\delta+1,N-1), \qquad x=1,\ldots,N, \nonumber \\
U_{2n,N}= (-1)^n \tilde R_n(\lambda(0);\gamma,\delta,N),\qquad U_{2n+1,N}=0,\label{U-HahnI}\end{gathered}$$ where the row index of the matrix $U$ (denoted here by $2n$ or $2n+1$, depending on the parity of the index) also runs over the integers from 0 up to $2N$. This matrix $U$ is orthogonal: the orthogonality relation of the dual Hahn polynomials and the signs in the matrix $U$ imply that its rows are orthonormal. Thus $U^TU=UU^T=I$, the identity matrix. Then the recurrence relations for [**dual Hahn I**]{} of Theorem \[theo2\] are now reformulated in terms of a two-diagonal $(2N+1)\times(2N+1)$-matrix of the form $$\begin{gathered}
\label{MK-I}
M= \left( \begin{matrix}
0 & M_0 & 0 & & \\
M_0 & 0 & M_1 & \ddots & \\
0 & M_1 & 0 & \ddots & 0 \\
&\ddots & \ddots & \ddots & M_{2N-1} \\
& & 0 & M_{2N-1} & 0
\end{matrix} \right).\end{gathered}$$
Explicitly
\[prop3\] Suppose $\gamma>-1$, $\delta>-1$. Let $M$ be the two-diagonal matrix with $$\begin{gathered}
M_{2k}=\sqrt{(k+\gamma+1)(N-k)}, \qquad M_{2k+1}=\sqrt{(k+1)(N+\delta-k)},
\label{M_kI}\end{gathered}$$ and $U$ the orthogonal matrix determined in . Then the columns of $U$ are the eigenvectors of $M$, i.e., $M U = U D$, where $D$ is a diagonal matrix containing the eigenvalues of $M$ $$\begin{gathered}
D= \operatorname{diag} (-\epsilon_N,\ldots,-\epsilon_1,0,\epsilon_1,\ldots,\epsilon_{N}),\nonumber\\
\epsilon_{k}=\sqrt{k(k+\gamma+\delta+1)},\qquad k=1,\ldots,N. \label{324}\end{gathered}$$
Note that we have kept only the conditions under which the matrix $M$ is real. The other conditions for which the dual Hahn polynomials in can be normalized (namely $\gamma<-N$, $\delta<-N$) would give rise to imaginary values in . In such a case, the relation $MU=UD$ remains valid, and also $D$ would have imaginary values.
For the case [**dual Hahn II**]{}, the matrix $U$ is again of order $(2N+1)\times (2N+1)$ with matrix elements $$\begin{gathered}
U_{2n,x} = U_{2n,2N-x} = \frac{1}{\sqrt{2}} \tilde R_n(\lambda(x);\gamma,\delta,N), \qquad x=0,\ldots,N-1, \nonumber \\
U_{2n+1,x} = -U_{2n+1,2N-x} = -\frac{1}{\sqrt{2}} \tilde R_n(\lambda(x);\gamma,\delta,N-1), \qquad x=0,\ldots,N-1, \nonumber \\
U_{2n,N}= \tilde R_n(\lambda(N);\gamma,\delta,N),\qquad U_{2n+1,N}=0,\label{U-HahnII}\end{gathered}$$ where the row indices are as in . The orthogonality relation of the dual Hahn polynomials and the signs in the matrix $U$ imply that its rows are orthonormal, so $U^TU=UU^T=I$. The pair of recurrence relations for [**dual Hahn II**]{} of Theorem \[theo2\] yield
\[prop4\] Suppose $\gamma>-1$, $\delta>-1$. Let $M$ be a tridiagonal $(2N+1)\times(2N+1)$-matrix of the form with $$\begin{gathered}
M_{2k}= \sqrt{(N+\delta-k)(N-k)}, \qquad M_{2k+1}=\sqrt{(k+1)(k+\gamma+1)},
\label{M_kII}\end{gathered}$$ and $U$ the orthogonal matrix determined in . Then the columns of $U$ are the eigenvectors of $M$, i.e., $M U = U D$, where $D$ is a diagonal matrix containing the eigenvalues of $M$ $$\begin{gathered}
D= \operatorname{diag} (-\epsilon_N,\ldots,-\epsilon_1,0,\epsilon_1,\ldots,\epsilon_{N}), \\
\epsilon_{k}=\sqrt{k(\gamma+\delta+1+2N-k)},\qquad k=1,\ldots,N.\end{gathered}$$
Note that the order in which the normalized dual Hahn polynomials appear in the matrix $U$ is different for and . This is related to the indices of the polynomials in the relations of Theorem \[theo2\].
Finally, for the case [**dual Hahn III**]{}, the matrix $U$ is given by , and we recapitulate the results given at the end of the previous section, now in terms of the dual Hahn parameters $\gamma$ and $\delta$.
\[prop1\] Suppose $\gamma>-1$, $\delta>-1$ or $\gamma<-N-1$, $\delta<-N-1$. Let $M$ be the tridiagonal matrix with $$\begin{gathered}
M_{2k}= \sqrt{(k+\gamma+1)(N+\delta+1-k)}, \qquad M_{2k+1}=\sqrt{(k+1)(N-k)},
\label{M_kIII}
\end{gathered}$$ and $U$ the orthogonal matrix determined in , . Then the columns of $U$ are the eigenvectors of $M$, i.e., $M U = U D$, where $D$ is a diagonal matrix containing the eigenvalues of $M$ $$\begin{gathered}
D= \operatorname{diag} (-\epsilon_N,\ldots,-\epsilon_1,-\epsilon_0,\epsilon_0,\epsilon_1,\ldots,\epsilon_{N}),\\
\epsilon_{k}=\sqrt{(k+\gamma+1)(k+\delta+1)}, \qquad
k=0,1,\ldots,N.
\end{gathered}$$
To conclude for dual Hahn polynomials: there are three sets of recurrence relations of the form , . Each of the three cases gives rise to a two-diagonal matrix with simple and explicit eigenvalues, and eigenvectors given in terms of two sets of dual Hahn polynomials.
Doubling Hahn polynomials {#sec:Hahn}
=========================
The technique presented in the previous section can be applied to other types of discrete orthogonal polynomials with a finite spectrum. We have done this for Hahn polynomials. One level up in the hierarchy of orthogonal polynomials of hypergeometric type are the Racah polynomials. Also for Racah polynomials we have applied the technique, but here the description of the results becomes very technical. So we shall leave the results for Racah polynomials for Appendix \[sec:Racah\].
For Hahn polynomials the analysis is again straightforward but tedious, so let us skip the details of the computation and present just the final outcome here. Applying the technique described in –, with $y_n = Q_n(x;\alpha,\beta,N)$ and $\hat{y}_n = Q_{n}(\hat{x};\hat{\alpha},\hat{\beta},\hat{N})$ yields the following result.
\[theo5\] The only way to combine two sets of Hahn polynomials such that they satisfy a pair of recurrence relations of the form , is one of the four cases:\
[**Hahn I**]{}, $Q_n(x)\equiv Q_n(x;\alpha,\beta,N)$ and $\hat Q_n(x)\equiv Q_n(x;\alpha+1,\beta,N)$: $$\begin{gathered}
\frac{(n+\alpha+\beta+N+2)}{(2n+\alpha+\beta+2)} Q_n(x) - \frac{(N-n)}{(2n+\alpha+\beta+2)} Q_{n+1}(x)
= \frac{(\alpha+x+1)}{(\alpha+1)} \hat Q_n(x), \\
- \frac{(n+1)(n+\beta+1) }{(2n+\alpha+\beta+3)} \hat Q_n(x) + \frac{(n+\alpha+\beta+2)(n+\alpha+2)}{(2n+\alpha+\beta+3)} \hat Q_{n+1}(x)
= (\alpha+1) Q_{n+1}(x).\end{gathered}$$ [**Hahn II**]{}, $Q_n(x)\equiv Q_n(x;\alpha,\beta,N)$ and $\hat Q_n(x)\equiv Q_n(x-1;\alpha+1,\beta,N-1)$: $$\begin{gathered}
\frac{1}{(2n+\alpha+\beta+2)} Q_n(x) - \frac{1}{(2n+\alpha+\beta+2)} Q_{n+1}(x)
= \frac{x}{N(\alpha+1)} \hat Q_{n}(x), \\
- \frac{(n+1)(n+\beta+1) (n+\alpha+\beta+N+2)}{(2n+\alpha+\beta+3)} \hat Q_{n}(x) \\
\qquad{} + \frac{(n+\alpha+\beta+2)(N-n-1)(n+\alpha+2)}{(2n+\alpha+\beta+3)} \hat Q_{n+1}(x)
= N(\alpha+1) Q_{n+1}(x) .\end{gathered}$$ [**Hahn III**]{}, $Q_n(x)\equiv Q_n(x;\alpha,\beta,N)$ and $\hat Q_n(x)\equiv Q_n(x;\alpha,\beta+1,N)$: $$\begin{gathered}
\frac{(n+\beta+1 )(n+N+2+\alpha+\beta)}{(2n+\alpha+\beta+2)} Q_n(x) +\frac{(N-n)(n+\alpha+1)}{(2n+\alpha+\beta+2)} Q_{n+1}(x)\\
\qquad{} = (\beta+1+N-x) \hat Q_n(x), \\
\frac{(n+1)}{(2n+\alpha+\beta+3)} \hat Q_n(x) + \frac{(n+\alpha+\beta+2)}{(2n+\alpha+\beta+3)} \hat Q_{n+1}(x)
= Q_{n+1}(x) .\end{gathered}$$ [**Hahn IV**]{}, $Q_n(x)\equiv Q_n(x;\alpha,\beta,N)$ and $\hat Q_n(x)\equiv Q_n(x;\alpha,\beta+1,N-1)$: $$\begin{gathered}
\frac{(n+\beta+1)}{(2n+\alpha+\beta+2)} Q_n(x) +\frac{(n+\alpha+1)}{(2n+\alpha+\beta+2)} Q_{n+1}(x)
= \frac{(N-x)}{N} \hat Q_n(x) , \\
\frac{(n+1)(n+\alpha+\beta+N+2)}{(2n+\alpha+\beta+3)} \hat Q_n(x) + \frac{(N-n-1)(n+\alpha+\beta+2)}{(2n+\alpha+\beta+3)} \hat Q_{n+1}(x)
= N Q_{n+1}(x) .\end{gathered}$$
Note that when interchanging $x$ and $n$ the relations in [**Hahn II**]{} coincide with the known forward and backward shift operator relations for dual Hahn polynomials [@Koekoek equations (9.6.6) and (9.6.8)]. In the same way, the other cases yield new forward and backward shift operator relations for dual Hahn polynomials.
Since the recurrence relations are of the form , , they can be cast in matrix form with a two-diagonal matrix. We shall write the matrix elements again in terms of normalized polynomials. For the case [**Hahn I**]{}, the matrix $U$ of order $(2N+2)\times(2N+2)$, with elements $$\begin{gathered}
U_{2n,N-x} = U_{2n,N+x+1} = \frac{(-1)^n}{\sqrt{2}} \tilde Q_n(x;\alpha,\beta,N), \nonumber\\
U_{2n+1,N-x} = -U_{2n+1,N+x+1} = -\frac{(-1)^n}{\sqrt{2}} \tilde Q_n(x;\alpha+1,\beta,N) \label{dU-I}\end{gathered}$$ where $x,n\in\{0,1,\ldots,N\}$, is orthogonal, and the recurrence relations yield
\[prop6\] Suppose that $\gamma,\delta >-1$. Let $M$ be a tridiagonal $(2N+2)\times(2N+2)$-matrix of the form with $$\begin{gathered}
M_{2k} = \sqrt{ \frac{ (k+\alpha+1)(k+\alpha+\beta+1)(k+\alpha+\beta+2+N)}{(2k+\alpha+\beta+1)(2k+\alpha+\beta+2)}}, \nonumber \\
M_{2k+1} = \sqrt{ \frac{(k+\beta+1)(k+1)(N-k) }{(2k+\alpha+\beta+2)(2k+\alpha+\beta+3) } },
$$ and $U$ the orthogonal matrix determined in . Then the columns of $U$ are the eigenvectors of $M$, i.e., $M U = U D$, where $D$ is a diagonal matrix containing the eigenvalues of $M$ $$\begin{gathered}
D= \operatorname{diag} (-\epsilon_N,\ldots,-\epsilon_1,-\epsilon_0,\epsilon_0,\epsilon_1,\ldots,\epsilon_{N}),\nonumber\\
\epsilon_{k}=\sqrt{k+\alpha+1},\qquad k=0,1,\ldots,N.
\label{47}\end{gathered}$$
For the case [**Hahn II**]{}, the orthogonal matrix $U$ is of order $(2N+1)\times(2N+1)$, with elements $$\begin{gathered}
U_{2n,N-x} = U_{2n,N+x} = \frac{(-1)^n}{\sqrt{2}} \tilde Q_n(x;\alpha,\beta,N), \qquad x=1,\ldots,N, \nonumber \\
U_{2n+1,N-x} = -U_{2n+1,N+x} = -\frac{(-1)^n}{\sqrt{2}} \tilde Q_n(x-1;\alpha+1,\beta,N-1), \qquad x=1,\ldots,N, \label{dU-II} \\
U_{2n,N}= (-1)^n \tilde Q_n(0;\alpha,\beta,N),\qquad U_{2n+1,N}=0, \nonumber\end{gathered}$$ where the row indices are as in . The recurrence relations for [**Hahn II**]{} yield
\[prop7\] Suppose that $\alpha,\beta >-1$ or $\alpha,\beta<-N$. Let $M$ be a tridiagonal $(2N+1)\times(2N+1)$-matrix of the form with $$\begin{gathered}
M_{2k} = \sqrt{ \frac{ (k+\alpha+1)(k+\alpha+\beta+1)(N-k)}{(2k+\alpha+\beta+1)(2k+\alpha+\beta+2)}} ,\nonumber \\
M_{2k+1} = \sqrt{ \frac{(k+\beta+1)(k+\alpha+\beta+2+N)(k+1) }{(2k+\alpha+\beta+2)(2k+\alpha+\beta+3) } },
$$ and $U$ the orthogonal matrix determined in . Then the columns of $U$ are the eigenvectors of $M$, i.e., $M U = U D$, where $D$ is a diagonal matrix containing the eigenvalues of $M$: $$\begin{gathered}
D= \operatorname{diag} (-\epsilon_N,\ldots,-\epsilon_1,0,\epsilon_1,\ldots,\epsilon_{N}), \qquad
\epsilon_{k}=\sqrt{k},\qquad k=1,\ldots,N.\end{gathered}$$
Note that for both cases, the two-diagonal matrix $M$ becomes more complicated compared to the cases for dual Hahn polynomials, but the matrix $D$ of eigenvalues becomes simpler.
For the two remaining cases we need not give all details: the matrix $M$ for the case [**Hahn III**]{} is equal to the matrix $M$ for the case [**Hahn I**]{} with the replacement $\alpha \leftrightarrow \beta$, and so its eigenvalues are $\pm \sqrt{k+\beta+1}$, $k=0,1,\ldots,N$. And the matrix $M$ for the case [**Hahn IV**]{} is equal to the matrix $M$ for the case [**Hahn II**]{} with the same replacement $\alpha \leftrightarrow \beta$, so its eigenvalues are $0$ and $\pm \sqrt{k}$, $k=1,\ldots,N$.
Polynomial systems, Christoffel and Geronimus transforms {#sec:orthpoly}
========================================================
So far, we have only partially explained why the technique in the previous sections is referred to as “doubling” polynomials. It is indeed a fact that the combination of two sets of polynomials, each with different parameters, yields a new set of orthogonal polynomials. This can be compared to the well known situation of combining two sets of generalized Laguerre polynomials (both with different parameters $\alpha$ and $\alpha-1$) into the set of “generalized Hermite polynomials” [@Chihara]. There, for $\alpha>0$, one defines $$\begin{gathered}
P_{2n}(x) = \sqrt{\frac{n!}{(\alpha)_n}} L_n^{(\alpha-1)}\big(x^2\big),\qquad
P_{2n+1}(x) =\sqrt{\frac{n!}{(\alpha)_{n+1}}} x L_n^{(\alpha)}\big(x^2\big). \label{Lcase2}\end{gathered}$$ Then the orthogonality relation of Laguerre polynomials leads to the orthogonality of the polynomials : $$\begin{gathered}
\int_{-\infty}^{+\infty} w(x) P_n(x)P_{n'}(x)dx = \Gamma(\alpha) \delta_{n,n'}, $$ where $$\begin{gathered}
w(x) = e^{-x^2} |x|^{2\alpha-1}.
\label{w2}\end{gathered}$$ Note that the even polynomials are Laguerre polynomials in $x^2$ (for parameter $\alpha-1$), and the odd polynomials are Laguerre polynomials in $x^2$ (for parameter $\alpha$) multiplied by a factor $x$. The weight function is common for both types of polynomials. It is this phenomenon that appears here too in our doubling process of Hahn or dual Hahn polynomials.
From a more general point of view, this fits in the context of obtaining a new family of orthogonal polynomials starting from a set of orthogonal polynomials and its kernel partner related by a Christoffel transform [@Chihara; @Marcellan; @Vinet2012]. In a way, our classification determines for which Christoffel parameter $\nu$ (see [@Vinet2012] for the notation) the Christoffel transform of a Hahn, dual Hahn or Racah polynomial is again a Hahn, dual Hahn or Racah polynomial with possibly different parameters. This determines moreover quite explicitly the common weight function.
For a dual Hahn polynomial $R_n(x)\equiv R_n(\lambda(x);\gamma,\delta,N)$, with data given in , and a Christoffel parameter $\nu$ the kernel partner is given by the transform $$\begin{gathered}
\label{CT}
P_n(x) = \frac{R_{n+1}(x)-a_n R_n(x)}{\Lambda(x)-\Lambda(\nu)},\qquad a_n = \frac{R_{n+1}(\nu)}{R_n(\nu)}.\end{gathered}$$ Because of the recurrence relation and what is called the Geronimus transform the original polynomials can also be expressed in terms of the kernel partners. This is usually done for monic polynomials (see [@Vinet2012 equations (3.2) and (3.3)]), but it can be extended to non-monic dual Hahn polynomials as follows $$\begin{gathered}
\label{GT}
R_n(x) = A(n) P_n(x) - b_n P_{n-1}(x)\end{gathered}$$ where the coefficients $b_n$ are related to the recurrence relation as follows $$\begin{gathered}
b_na_{n-1} = C(n),\qquad A(n)a_n+b_n =A(n)+C(n)+\Lambda(\nu).
\label{baC}\end{gathered}$$
Our classification now shows that only for $\nu$ equal to one of the values $0$, $N$ or $-\delta$, the kernel partner $P_n(x)$ will again be a dual Hahn polynomial. Indeed, taking for example $\nu =0$ in we have $R_n(0) = 1$ and $$\begin{gathered}
P_n(x)= \frac{R_{n+1}(x)-R_n(x)}{\Lambda(x)} = \frac{-1}{N(\gamma+1)}R_{n}(\lambda ( x-1);{\gamma+1},{\delta+1},{N-1}),\end{gathered}$$ where we used the first relation of [**dual Hahn I**]{} to obtain again a dual Hahn polynomial. The reverse transform follows immediately from the second relation of [**dual Hahn I**]{}. Similarly, taking $\nu=N$ in we have $R_n(N) = {(-N-\delta)_{n}}/{(\gamma+1)_n}$ and $$\begin{gathered}
P_n(x)
= \frac{R_{n+1}(x)-R_n(x)(n-\delta-N)/(n+\gamma+1)}{(x-N)(x+N+\gamma+\delta+1)}
= \frac{-1}{N(n+\gamma+1)}R_n(\lambda(x);\gamma,\delta,N-1),\end{gathered}$$ which we obtained using the first relation of [**dual Hahn II**]{}. For the reverse transform we find, using the second relation of [**dual Hahn II**]{} with shifted $n\mapsto n-1$, $$\begin{gathered}
A(n) P_n(x) - b_n P_{n-1}(x) = \frac{-(n-N)}{N}R_n(\lambda(x);\gamma,\delta,N-1) + \frac{n}{N}R_{n-1}(\lambda(x);\gamma,\delta,N-1)\\
\hphantom{A(n) P_n(x) - b_n P_{n-1}(x)}{} = R_n(x).\end{gathered}$$ For the last case, taking $\nu=-\delta$ in we have $R_n(-\delta) = {(-N-\delta)_{n}}/{(-N)_n}$ and $$\begin{gathered}
P_n(x) = \frac{R_{n+1}(x)-R_n(x)(n-\delta-N)/(n-N)}{(x+\gamma+1)(x+\delta) } \\
\hphantom{P_n(x)}{}
= \frac{1}{(\gamma+1)(n-N)}R_n(\lambda(x);\gamma+1,\delta-1,N),\end{gathered}$$ which we obtained using the first relation of [**dual Hahn III**]{}. For the transform we have $$\begin{gathered}
A(n) P_n(x) - b_n P_{n-1}(x) = \frac{n+\gamma+1}{\gamma+1}R_n(\lambda(x);\gamma+1,\delta-1,N-1) \\
\hphantom{A(n) P_n(x) - b_n P_{n-1}(x) =}{} - \frac{n}{\gamma+1}R_{n-1}(\lambda(x);\gamma+1,\delta-1,N),\end{gathered}$$ which equals $R_n(x)$ by the second relation of [**dual Hahn III**]{}.
In a similar way, for the Hahn polynomials, putting $Q_n(x)\equiv Q_n(x;a,b,N)$, using the data in $$\begin{gathered}
P_n(x) = \frac{Q_{n+1}(x)-a_n Q_n(x)}{\Lambda(x)-\Lambda(\nu)},\qquad a_n = \frac{Q_{n+1}(\nu)}{Q_n(\nu)},\end{gathered}$$ and in , , the cases [**Hahn I**]{}, [**II**]{}, [**III**]{}, [**V**]{} correspond respectively to the choices $-\alpha-1$, $0$, $N+\beta+1$ and $N$ for $\nu$.
The task of determining for which Christoffel parameter $\nu$ the kernel partner of a dual Hahn polynomial is again of the same family is not trivial. It comes down to finding a pair of recurrence relations of the form , with coefficients related to $\nu$ as in . We have classified these for general coefficients, without a relation to $\nu$, and we observe that each solution indeed corresponds to a specific choice for $\nu$.
The transforms , give rise to new orthogonal systems, but in general there is no way of writing the common weight function. However, since here both sets are of the same family, we can actually do this. Let us begin with the dual Hahn polynomials, in particular the case [**dual Hahn I**]{}, for which the corresponding matrix $U$ is given in . They give rise to a new family of discrete orthogonal polynomials with the relation $MU=UD$ corresponding to their three term recurrence relation with Jacobi matrix $M$ . In general the support of the weight function is equal to the spectrum of the Jacobi matrix [@Berezanskii; @Klimyk2006; @Koelink2004; @Koelink1998]. After simplifying with the normalization factors , this leads to a discrete orthogonality of polynomials, with support equal to the eigenvalues of $M$ (so in this case, the support follows from ). Concretely, for the case under consideration, we have
\[prop10\] Let $\gamma>-1$, $\delta>-1$, and consider the $2N+1$ polynomials $$\begin{gathered}
{P}_{2n} (q) = \frac{(-1)^n}{\sqrt{2}} {R}_n\big(q^2;\gamma,\delta,N\big), \qquad n=0,1,\ldots,N,\nonumber\\
P_{2n+1} (q) = - \frac{(-1)^n}{\sqrt{2}}\frac{\sqrt{(n+\gamma+1)(N-n)}}{(\gamma+1)N} q {R}_n\big(q^2-\gamma-\delta-2;\gamma+1,\delta+1,N-1\big),\nonumber\\
\hphantom{P_{2n+1} (q) =}{} n=0,1,\ldots,N-1.
\end{gathered}$$ These polynomials satisfy the discrete orthogonality relation $$\begin{gathered}
\sum_{q\in S} \frac{(-1)^k (2k+\gamma +\delta +1) (\gamma+1)_k (-N)_k N! }{ (k+\gamma +\delta +1)_{N+1} (\delta+1)_k k! }
(1+\delta_{q,0}) P_n(q) P_{n'}(q) \nonumber\\
\qquad {}= \left[ \binom{\gamma+\lfloor n/2\rfloor }{\lfloor n/2\rfloor} \binom{\delta+N-\lfloor n/2\rfloor }{N-\lfloor n/2\rfloor} \right]^{-1}
\delta_{n,n'}
\label{511}
\end{gathered}$$ with $$\begin{gathered}
S=\big\{ 0,\pm\sqrt{k(k+\gamma+\delta+1)}, \ k=1,2,\ldots,N\big\}.
\end{gathered}$$
Note that for $q\in S$, $q^2=k(k+\gamma+\delta+1)\equiv \lambda(k)$, and the polynomial $P_{2n}(q)$ is of the form $R_n(\lambda(k);\gamma,\delta,N)$. In that case, $q^2-\gamma-\delta-2 = (k-1)((k-1)+(\gamma+1)+(\delta+1)+1) \equiv \lambda(k-1)$, so the polynomial $P_{2n+1}(q)$ is of the form $R_n(\hat \lambda(k-1);\gamma+1,\delta+1,N-1)$. The interpretation of the weight function in the left hand side of is as follows: each $q$ in the support $S$ is mapped to a $k$-value belonging to $\{0,1,\ldots,N\}$, and then the weight depends on this $k$-value.
Now we turn to the classification of Section \[sec:Hahn\], where the corresponding orthogonal matrices $U$ are given in terms of (normalized) Hahn polynomials. for the case [**Hahn I**]{}, the matrix $U$ is given in , and the spectrum of the matrix $M$ is given by . After simplifying the normalization factors, the orthogonality of the rows of $U$ gives rise to
\[prop8\] Let $\alpha>-1$, $\beta>-1$, and consider the $2N+2$ polynomials, $n=0,1,\ldots,N$, $$\begin{gathered}
{P}_{2n} (q) = \frac{(-1)^n}{\sqrt{2}} {Q}_n\big(q^2-\alpha-1;\alpha,\beta,N\big),\nonumber\\
P_{2n+1} (q) = - \frac{(-1)^n}{\sqrt{2}}\frac{1}{(\alpha+1)} \sqrt{\frac{(n+\alpha+1)(n+\alpha+\beta+1)(2n+2+\alpha+\beta)}{(n+N+\alpha+\beta+2)(2n+\alpha+\beta+1)}}\nonumber\\
\hphantom{P_{2n+1} (q) =}{} \times q {Q}_n\big(q^2-\alpha-1;\alpha+1,\beta,N\big).\end{gathered}$$ These polynomials satisfy the discrete orthogonality relation $$\begin{gathered}
\sum_{q\in S} \binom{ q^2-1}{q^2-\alpha-1} \binom{ N - q^2+\alpha+\beta +1}{N - q^2+\alpha+1} P_n(q) P_{n'}(q) =h_{\lfloor n/2\rfloor}(\alpha,\beta,N) \beta_{n,n'}\end{gathered}$$ with $$\begin{gathered}
S=\big\{ {-}\sqrt{N+\alpha+1}, -\sqrt{N+\alpha}, \ldots, -\sqrt{\alpha+1}, \sqrt{\alpha+1}, \ldots, \sqrt{N+\alpha},\sqrt{N+\alpha+1}\big\}\end{gathered}$$ and $$\begin{gathered}
h_{n}(\alpha,\beta,N) = \frac{(-1)^n (n+\alpha +\beta +1)_{N+1} (\beta+1)_n n! }{ (2n+\alpha +\beta +1) (\alpha+1)_n (-N)_n N! }.\end{gathered}$$
So $P_n(q)$ is a polynomial of degree $n$ in the variable $q$, of different type (with different parameters when expressed as a Hahn polynomial) depending on whether $n$ is even or $n$ is odd. The support points of the discrete orthogonality are given by $$\begin{gathered}
q = \pm\sqrt{k+ \alpha+1}, \qquad k=0,\dots, N.\end{gathered}$$
In the same way, the dual orthogonality for the case [**Hahn II**]{} gives rise to
\[prop9\] Let $\alpha>-1$, $\beta>-1$, and consider the $2N+1$ polynomials $$\begin{gathered}
{P}_{2n} (q) = \frac{(-1)^n}{\sqrt{2}} {Q}_n\big(q^2;\alpha,\beta,N\big), \qquad n=0,1,\ldots,N,\nonumber\\
P_{2n+1} (q) = - \frac{(-1)^n}{\sqrt{2}}\frac{1}{(\alpha+1)N} \sqrt{\frac{(N-n)(n+\alpha+1)(n+\alpha+\beta+1)(2n+\alpha+\beta+2)}
{(2n+\alpha+\beta+1)}} \nonumber\\
\hphantom{P_{2n+1} (q) =}{} \times q {Q}_n\big(q^2-1;\alpha+1,\beta,N-1\big), \qquad n=0,1,\ldots,N-1.\end{gathered}$$ These polynomials satisfy the discrete orthogonality relation $$\begin{gathered}
\sum_{q\in S} \binom{q^2+\alpha}{q^2} \binom{N - q^2+\beta}{N - q^2}
(1+\delta_{q,0}) P_n(q) P_{n'}(q) =h_{\lfloor n/2\rfloor}(\alpha,\beta,N) \beta_{n,n'}\end{gathered}$$ with $$\begin{gathered}
S=\big\{ {-}\sqrt{N}, -\sqrt{N-1}, \ldots, -1,0,1,\ldots, \sqrt{N-1},\sqrt{N}\big\}\end{gathered}$$ and $$\begin{gathered}
h_{n}(\alpha,\beta,N) = \frac{(-1)^n (n+\alpha +\beta +1)_{N+1} (\beta+1)_n n! }{ (2n+\alpha +\beta +1) (\alpha+1)_n (-N)_n N! }.\end{gathered}$$
The ideas described in the three propositions of this section should be clear. It would lead us too far to give also the explicit forms corresponding to the remaining cases. Let us just mention that also for these cases the support of the new polynomials coincides with the spectrum of the corresponding two-diagonal matrix $M$.
First application: eigenvalue test matrices {#sec:testmatrix}
===========================================
In Sections \[sec:dHahn\] and \[sec:Hahn\] we have encountered a number of symmetric two-diagonal matrices $M$ with explicit expressions for the eigenvectors and eigenvalues. In general, if one considers a two-diagonal matrix $A$ of size $(m+2)\times(m+2)$, $$\begin{gathered}
\label{A}
A= \left( \begin{matrix}
0 & b_0 & 0 & & \\
c_0 & 0 & b_1 & \ddots & \\
0 & c_1 & 0 & \ddots & 0 \\
&\ddots & \ddots & \ddots & b_{m} \\
& & 0 & c_{m} & 0
\end{matrix} \right),\end{gathered}$$ then it is clear that the characteristic polynomial depends on the products $b_ic_i$, $i=0,\ldots,m$, only, and not on $b_i$ and $c_i$ separately. So the same holds for the eigenvalues. Therefore, if all matrix elements $b_i$ and $c_i$ are positive, the eigenvalues of $A$ or of the related symmetric matrix $$\begin{gathered}
A'= \left( \begin{matrix}
0 & \sqrt{b_0c_0} & 0 & & \\
\sqrt{b_0c_0} & 0 & \sqrt{b_1c_1} & \ddots & \\
0 & \sqrt{b_1c_1} & 0 & \ddots & 0 \\
&\ddots & \ddots & \ddots & \sqrt{b_{m}c_m} \\
& & 0 & \sqrt{b_{m}c_m} & 0
\end{matrix} \right)\end{gathered}$$ are the same. The eigenvectors of $A'$ are those of $A$ after multiplication by a diagonal matrix (the diagonal matrix that is used in the similarity transformation from $A$ to $A'$).
For matrices of type , it is sufficient to denote them by their superdiagonal $[\mathbf{b}]=[b_0,\ldots,b_m]$ and their subdiagonal $[\mathbf{c}]=[c_0,\ldots,c_m]$. So the Sylvester–Kac matrix from the introduction is denoted by $$\begin{gathered}
[\mathbf{b}]=[1,2,\ldots,N],\qquad [\mathbf{c}]=[N,\ldots,2,1],\end{gathered}$$ with eigenvalues given by .
The importance of the Sylvester–Kac matrix as a test matrix for numerical eigenvalue routines has already been emphasized in the Introduction. In this context, it is also significant that the matrix itself has integer entries only (so there is no rounding error when represented on a digital computer), and that also the eigenvalues are integers. Of course, matrices with rational numbers as entries suffice as well, since one can always multiply the matrix by an appropriate integer factor.
Let us now systematically consider the two-diagonal matrices encountered in the classification process of doubling Hahn or dual Hahn polynomials. For the matrix of the [**dual Hahn I**]{} case, the corresponding non-symmetric form can be chosen as the two-diagonal matrix with $$\begin{gathered}
[\mathbf{b}]=[\gamma+1,1,\gamma+2,2,\ldots,\gamma+N,N], \nonumber\\
[\mathbf{c}]=[N,N+\delta,N-1,N-1+\delta,\ldots,1,\delta+1].
\label{Mat1}\end{gathered}$$ The eigenvalues are determined by Proposition \[prop3\] and given by $0$, $\pm\sqrt{k(k+\gamma+\delta+1)}$, $k=1,\ldots,N$. This is (up to a factor 2) the matrix mentioned in the Introduction. As test matrix, the choice $\gamma+\delta+1=0$ (leaving one free parameter) is interesting as it gives rise to integer eigenvalues. In Proposition \[prop3\] there is the initial condition $\gamma>-1$, $\delta>-1$. Clearly, if one is only dealing with eigenvalues, the condition for is just $\gamma+\delta+2\geq 0$. And when one substitutes $\delta=-\gamma-1$ in , there is no condition at all for the one-parameter family of matrices of the form .
For the [**dual Hahn II**]{} case, the matrix is given in Proposition \[prop4\], and its non-symmetric form can be taken as $$\begin{gathered}
[\mathbf{b}]=[\gamma+N,1,\gamma+N-1,2,\ldots,\gamma+1,N], \nonumber\\
[\mathbf{c}]=[N,\delta+1,N-1,\delta+2,\ldots,1,\delta+N].
\label{Mat2}\end{gathered}$$ The eigenvalues are given by $0$, $\pm\sqrt{k(\gamma+\delta+1+2N-k)}$, $k=1,\ldots,N$. There is no simple substitution that reduces these eigenvalues to integers.
For the [**dual Hahn III**]{} case, the matrix is given in Proposition \[prop1\], and its simplest non-symmetric form is $$\begin{gathered}
[\mathbf{b}]=[\gamma+1,1,\gamma+2,2,\ldots,\gamma+N,N,\gamma+N+1], \nonumber\\
[\mathbf{c}]=[\delta+N+1,N,\delta+N,N-1,\ldots,\delta+2,1,\delta+1].
\label{Mat3}\end{gathered}$$ The eigenvalues are given by , i.e., $\pm\sqrt{(\gamma+k+1)(\delta+k+1)}$, $k=0,\ldots,N$. Up to a factor 2, this is the third matrix mentioned in the Introduction. The substitution $\delta=\gamma$ leads to a one-parameter family of two-diagonal matrices with square-free eigenvalues. And in particular when moreover $\gamma$ is integer, all matrix entries and all eigenvalues are integers.
The two-diagonal matrices arising from the Hahn doubles or the Racah doubles can also be written in a square-free form of type . However, for these cases the entries in the two-diagonal matrices $M$ are already quite involved (see, e.g., Propositions \[prop6\], \[prop7\], \[propR1\] or \[propR2\]), and we shall not discuss them further in this context. The three examples given here, –, are already sufficiently interesting as extensions of the Sylvester–Kac matrix as potential eigenvalue test matrices.
Further applications: related algebraic structures\
and finite oscillator models {#sec:oscillators}
===================================================
The original example of a (dual) Hahn double, described here in Section \[sec:Example\], was encountered in the context of a finite oscillator model [@JSV2011]. In that context, there is also a related algebraic structure. In particular, the two-diagonal matrices $M$ of the form or are interpreted as representation matrices of an algebra, which can be seen as a deformation of the Lie algebra $\su(2)$. Once an algebraic formulation is clear, this structure can be used to model a finite oscillator. The close relationship comes from the fact that for the corresponding finite oscillator model the spectrum of the position operator coincides with the spectrum of the matrix $M$.
Therefore, it is worthwhile to examine the algebraic structures behind the current matrices $M$. We shall do this explicitly for the three double dual Hahn cases.
For the case [**dual Hahn I**]{}, we return to the form of the matrix $M$ given in or . For any positive integer $N$, let $J_+$ denote the lower-triangular tridiagonal $(2N+1)\times(2N+1)$ matrix given below, and let $J_-$ be its transpose $$\begin{gathered}
\label{J+}
J_+= 2 \left( \begin{matrix}
0 & 0 & & & \\
M_0 & 0 & 0 & & \\
0 & M_1 & 0 & 0 & \\
& 0 & M_2 & 0 & \ddots \\
& &\ddots & \ddots & \ddots
\end{matrix} \right),
\qquad
J_-= J_+^\dagger. \end{gathered}$$ Let us also define the common diagonal matrix $$\begin{gathered}
J_0=\operatorname{diag}(-N, -N+1, \ldots, N),
\label{J0}\end{gathered}$$ and the “parity matrix” $$\begin{gathered}
P=\operatorname{diag}(1,-1,1,-1,\ldots).
\label{P}\end{gathered}$$ Then it is easy to check that these matrices satisfy the following relations (as usual, $I$ denotes the identity matrix) $$\begin{gathered}
P^2=1, \qquad PJ_0=J_0P, \qquad PJ_\pm=-J_\pm P, \nonumber\\
[J_0, J_\pm] = \pm J_\pm, \nonumber\\
[J_+,J_-]=2 J_0 + 2 (\gamma+\delta+1)J_0P - (2N+1)(\gamma-\delta)P + (\gamma-\delta)I.
\label{alg-I}\end{gathered}$$ Especially the last equation is interesting. From the algebraic point of view, it introduces some two-parameter deformation or extension of $\su(2)$. When $\gamma=\delta=-1/2$, the equations coincide with the $\su(2)$ relations. Another important case is when $\delta=-\gamma-1$, leaving a one-parameter extension of $\su(2)$ without quadratic terms.
For the case [**dual Hahn II**]{}, the corresponding expressions of $J_+$, $J_-$, $J_0$ and $P$ are the same as above in –, but with $M_k$-values given by . As far as the algebraic relations are concerned, they are also given by but with the last relation replaced by $$\begin{gathered}
[J_+,J_-]=-2 J_0 + 2 (\gamma+\delta+2N+1)J_0P + (2N+1) (\gamma-\delta)P -(\gamma-\delta)I.
$$
For the case [**dual Hahn III**]{}, the size of the matrices changes to $(2N+2)\times(2N+2)$. For $J_+$ and $J_-$ one can use , with $M_k$-values given by . $P$ has the same expression , but for $J_0$ we need to take $$\begin{gathered}
J_0=\operatorname{diag} \left(-N-\frac12, -N+\frac12, \ldots, N+\frac12\right).
$$ With these expressions, the algebraic relations are given by but with the last relation replaced by $$\begin{gathered}
[J_+,J_-]=2 J_0 + 2 (\gamma-\delta)J_0P - ((2N+2)(\gamma+\delta+1)+(2\gamma+1)(2\delta+1))P\nonumber\\
\hphantom{[J_+,J_-]=}{} + (\gamma-\delta)I .
\label{alg-III}\end{gathered}$$
The structure of these algebras is related to the structure of the so-called algebra ${\cal H}$ of the dual $-1$ Hahn polynomials, see [@Genest2013; @Tsujimoto]. It is not hard to verify that the algebra ${\cal H}$, determined by [@Genest2013 equations (3.4)–(3.6)] or [@Genest2013 equations (6.2)–(6.4)], can be cast in the form (or vice versa). Indeed, starting from the form [@Genest2013 equations (6.2)–(6.4)] coming from dual $-1$ Hahn polynomials, we can take $$\begin{gathered}
J_0 = \widetilde{K_1}- \frac{\rho}{4}, \qquad J_+ = \widetilde{K_2} + \widetilde{K_3} ,\qquad J_- = \widetilde{K_2} - \widetilde{K_3},\end{gathered}$$ to get the same form as $$\begin{gathered}
P^2=1, \qquad PJ_0=J_0P, \qquad PJ_\pm=-J_\pm P, \nonumber\\
[J_0, J_\pm] = \pm J_\pm, \qquad
[J_+,J_-]=2 J_0 + 2 \nu J_0P +\frac{\sigma}{2}P + \frac{\rho}{2}I,
\label{nu-rho}\end{gathered}$$ where $\nu$, $\sigma$, $\rho$ depend on the parameters of the dual $-1$ Hahn polynomials $\alpha$, $\beta$, $N$ through [@Genest2013 equations (3.4)–(3.6)]. In our case, the algebraic relations are the same, but the dependence of the “structure constants” in on the parameters $\gamma$, $\delta$, $N$ of the dual Hahn polynomials is different.
As far as we can see, the doubling of dual Hahn polynomials as classified in this paper gives a set of polynomials that is similar but in general not the same as a set of dual $-1$ Hahn polynomials [@Tsujimoto] (except for specific values of parameters, e.g., $\delta=-\gamma-1$ does coincide with a specific dual $-1$ Hahn polynomial). For general parameters, the support of the weight function is different, the recurrence relations (or difference relations) are different, and the hypergeometric series expression is different.
The algebraic structures obtained here (or special cases thereof) can be of interest for the construction of finite oscillator models [@Atak-Suslov; @Atak2001; @Atak2005; @JSV2011]. Two familiar finite oscillator models fall within this framework: the model discussed in [@JSV2011] corresponds to with $\delta=\gamma$, and the one analysed in [@JSV2011b] to with $\delta=\gamma$. Observe that there are some other interesting special values. For example, the case with $\delta=-\gamma-1$ gives rise to an interesting algebra, and in particular also to a very simple spectrum . We intend to study the finite oscillator that is modeled by this case, and study in particular the corresponding finite Fourier transform; but this will be the topic of a separate paper.
Conclusion {#sec:conclusion}
==========
We have classified all pairs of recurrence relations for two types of dual Hahn polynomials (i.e., dual Hahn polynomials with different parameters), and refer to these as dual Hahn doubles. The analysis is quite straightforward, and the result is given in Theorem \[theo2\], yielding three cases. For each case, we have given the corresponding symmetric two-diagonal matrix $M$, its matrix of orthonormal eigenvectors $U$ and its eigenvalues in explicit form. The same classification has been obtained for Hahn polynomials and Racah polynomials.
The orthogonality of the matrix $U$ gives rise to new sets of orthogonal polynomials. These sets could in principle also be obtained from, for example, a set of dual Hahn polynomials and a certain Christoffel transform. In our approach, the possible cases where such a transform gives rise to a polynomial of the same type follow naturally, and also the explicit polynomials and their orthogonality relations arise automatically.
As an interesting secondary outcome, we obtain nice one-parameter and two-parameter extensions of the Sylvester–Kac matrix with explicit eigenvalue expressions. Such matrices can be of interest for testing numerical eigenvalue routines.
The first example of a (dual) Hahn double appeared in a finite oscillator model [@JSV2011]. For this model, the Hahn polynomials (or their duals) describe the discrete position wavefunction of the oscillator, and the two-diagonal matrix $M$ lies behind an underlying algebraic structure. Here, we have examined the algebraic relations corresponding to the three dual Hahn cases. It is clear that the analysis of finite oscillators for some of these cases is worth pursuing.
Appendix: doubling Racah polynomials {#sec:Racah}
====================================
The technique presented in Sections \[sec:dHahn\] and \[sec:Hahn\] is applied here for Racah polynomials.
Racah polynomials $R_n(\lambda(x);\alpha,\beta,\gamma,\delta)$ of degree $n$ ($n=0,1,\ldots,N$) in the variable $\lambda(x)=x(x+\gamma+\delta+1)$ are defined by [@Ismail; @Koekoek; @Suslov] $$\begin{gathered}
R_n(\lambda(x);\alpha,\beta,\gamma,\delta)= {}_4F_3 \left( {\genfrac{}{}{0pt}{}{-n,n+\alpha+\beta+1,-x,x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}} ; 1 \right),\end{gathered}$$ where one of the denominator parameters should be $-N$: $$\begin{gathered}
\label{-N}
\alpha+1=-N\qquad\hbox{or}\qquad \beta+\delta+1=-N \qquad\hbox{or}\qquad \gamma+1=-N.\end{gathered}$$ For the (discrete) orthogonality relation (depending on the choice of which parameter relates to $-N$) we refer to [@Koekoek equation (9.2.2)] or [@NIST Section 18.25]
Racah polynomials satisfy a recurrence relation of the form with $$\begin{gathered}
y_n(x) = R_n(\lambda(x);\alpha,\beta,\gamma,\delta), \qquad \Lambda(x) =\lambda(x) = x(x+\gamma+\delta+1), \nonumber \\
A(n) = \frac{(n+\alpha+1)(n+\alpha+\beta+1)(n+\gamma+1)(n+\beta+\delta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)},\nonumber\\
C(n) = \frac{n(n+\alpha+\beta-\gamma)(n+\alpha-\delta)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)} . \label{Racahdata}\end{gathered}$$
We have applied the technique described in –, with $y_n = R_n(\lambda(x);\alpha,\beta,\gamma,\delta) $ and $\hat{y}_n = R_{n}(\lambda (\hat x);\hat{\alpha},\hat{\beta},\hat{\gamma},\hat{\delta})$. The analysis is again straightforward but tedious, and the final outcome is
\[theoR\]The only way to combine two sets of Racah polynomials such that they satisfy difference relations of the form , is one of the four cases:\
[**Racah I**]{}, $R_n(x)\equiv R_n(\lambda(x);\alpha,\beta,\gamma,\delta)$ and $\hat R_n(x)\equiv R_n(\lambda(x);\alpha,\beta+1,\gamma+1,\delta-1)$: $$\begin{gathered}
\frac{(n+\beta+\delta+1)(n+\alpha+1)}{(2n+\alpha+\beta+2)} R_{n+1}(x) - \frac{(n-\delta+\alpha+1)(n+\beta+1)}{(2n+\alpha+\beta+2)} R_n(x)\\
\qquad{} = \frac{(x+\delta)(x+\gamma+1)}{\gamma+1} \hat R_n(x),\\
\frac{(n+\alpha+\beta+2)(n+\gamma+2)}{(2n+\alpha+\beta+3)} \hat R_{n+1}(x) -
\frac{(n+1)(n-\gamma+\alpha+\beta+1)}{(2n+\alpha+\beta+3)} \hat R_n(x) \\
\qquad{}
= (\gamma+1) R_{n+1}(x).\end{gathered}$$ [**Racah II**]{}, $R_n(x)\equiv R_n(\lambda(x);\alpha,\beta,\gamma,\delta)$ and $\hat R_n(x)\equiv R_n(\lambda(x);\alpha,\beta+1,\gamma,\delta)$: $$\begin{gathered}
\frac{(n+\gamma+1)(n+\alpha+1)}{(2n+\alpha+\beta+2)} R_{n+1}(x) - \frac{(n-\gamma+\alpha+\beta+1)(n+\beta+1)}{(2n+\alpha+\beta+2)} R_n(x)\\
\qquad{} = \frac{(x+\beta+\delta+1)(x+\gamma-\beta)}{\beta+\delta+1}\hat R_n(x),\\
\frac{(n+\beta+\delta+2)(n+\alpha+\beta+2)}{(2n+\alpha+\beta+3)} \hat R_{n+1}(x) -
\frac{(n+1)(n-\delta+\alpha+1)}{(2n+\alpha+\beta+3)} \hat R_n(x) \\
\qquad{} = (\beta+\delta+1) R_{n+1}(x).\end{gathered}$$ [**Racah III**]{}, $R_n(x)\equiv R_n(\lambda(x);\alpha,\beta,\gamma,\delta)$ and $\hat R_n(x)\equiv R_n(\lambda(x-1);\alpha+1,\beta,\gamma+1,\delta+1)$: $$\begin{gathered}
\frac{1}{(2n+\alpha+\beta+2)} R_{n+1}(x) - \frac{1}{(2n+\alpha+\beta+2)} R_n(x)
= \frac{x(x+\gamma+\delta+1)}{(\gamma+1)(\beta+\delta+1)(\alpha+1)} \hat R_{n}(x), \\
\frac{(n+\gamma+2)(n+\beta+\delta+2)(n+\alpha+2)(n+\alpha+\beta+2)}{(2n+\alpha+\beta+3)} \hat R_{n+1}(x) \\
\qquad \quad{}- \frac{(n+1)(n-\gamma+\alpha+\beta+1)(n-\delta+\alpha+1)(n+\beta+1)}{(2n+\alpha+\beta+3)} \hat R_{n}(x) \\
\qquad {} = (\gamma+1)(\beta+\delta+1)(\alpha+1) R_{n+1}(x).\end{gathered}$$ [**Racah IV**]{}, $R_n(x)\equiv R_n(\lambda(x);\alpha,\beta,\gamma,\delta)$ and $\hat R_n(x)\equiv R_n(\lambda(x);\alpha+1,\beta,\gamma,\delta)$: $$\begin{gathered}
\frac{ (n+\gamma+1)(n+\beta+\delta+1) }{(2n+\alpha+\beta+2)} R_{n+1}(x) - \frac{ (n-\gamma+\alpha+\beta+1)(x-\delta+\alpha+1)}{(2n+\alpha+\beta+2)} R_n(x)\\
\qquad{} = \frac{(x+\gamma+\delta-\alpha)(x+\alpha+1)}{(\alpha+1)} \hat R_{n}( x), \\
\frac{(n+\alpha+2)(n+\alpha+\beta+2)}{(2n+\alpha+\beta+3)} \hat R_{n+1}(x)
- \frac{(n+1)(n+\beta+1)}{(2n+\alpha+\beta+3)} \hat R_{n}(x)
= (\alpha+1) R_{n+1}(x).\end{gathered}$$
Note that after interchanging $n$ and $x$, and $\alpha\leftrightarrow\gamma$ and $\beta\leftrightarrow\delta$, the relations in [**Racah III**]{} coincide with the known forward and backward shift operator relations [@Koekoek equations (9.2.6) and (9.2.8)]. The relations in [**Racah I**]{} were already found in [@JSV2014 equations (5) and (6)].
In the context of Section \[sec:orthpoly\] it is worth noting that the above relations also correspond to Christoffel-Genonimus transforms. Taking $R_n(x)\equiv R_n(\lambda(x);\alpha,\beta,\gamma,\delta)$ in the relations –, with data given by , the above cases [**Racah I**]{}, [**II**]{}, [**III**]{}, [**IV**]{} correspond respectively to the choices $\nu=-\delta$, $\nu=\beta-\gamma$, $\nu=0$ and $\nu=-\alpha-1$.
For each of the four cases, one can translate the set of difference relations to a matrix identity of the form $MU=UD$. In fact, for each of the four cases, there are three subcases depending on the choice of $-N$ in . We shall not give all of these cases: they should be easy to construct for the reader who needs one. Let us just give an example or two.
Consider the case [**Racah I**]{} with $\alpha+1=-N$. It is convenient to perform the shift $\delta \rightarrow \delta+1$ in the two difference relations of Theorem \[theoR\]. The orthogonal matrix $U$ is of order $(2N+2)\times(2N+2)$, with elements $$\begin{gathered}
U_{2n,N-x} = U_{2n,N+x+1} = \frac{(-1)^n}{\sqrt{2}} \tilde R_n(\lambda(x);\alpha,\beta,\gamma,\delta+1), \nonumber\\
U_{2n+1,N-x} = -U_{2n+1,N+x+1} = -\frac{(-1)^n}{\sqrt{2}} \tilde R_n(\lambda(x);\alpha,\beta+1,\gamma+1,\delta), \label{R-I}\end{gathered}$$ where $\tilde R_n$ is the notation for a normalized Racah polynomial. Then, one has
\[propR1\] Suppose that $\gamma,\delta>-1$ and $\beta>N+\gamma$ or $\beta<-N-\delta-1$. Let $M$ be a tridiagonal $(2N+2)\times(2N+2)$-matrix of the form with $$\begin{gathered}
M_{2k} = \sqrt{ \frac{(N-\beta-k)(\gamma+1+k)(N+\delta+1-k)(k+\beta+1) }{(N-\beta-2k)(2k-N+1+\beta) }},\nonumber\\
M_{2k+1} = \sqrt{ \frac{(\gamma+N-\beta-k)(k+1)(N-k)(k+\beta+\delta+2) }{(N-\beta-2k-2)(2k-N+1+\beta) }},
$$ and $U$ the orthogonal matrix determined in . Then the columns of $U$ are the eigenvectors of $M$, i.e., $M U = U D$, where $D$ is a diagonal matrix containing the eigenvalues of $M$ $$\begin{gathered}
D= \operatorname{diag} (-\epsilon_N,\ldots,-\epsilon_1,-\epsilon_0,\epsilon_0,\epsilon_1,\ldots,\epsilon_{N}), \\
\epsilon_{k}=\sqrt{(k+\gamma+1)(k+\delta+1)},\qquad k=0,1,\ldots,N.\end{gathered}$$
As a second example, consider the case [**Racah III**]{} with $\alpha+1=-N$. The orthogonal matrix $U$ is now of order $(2N+1)\times(2N+1)$, with elements $$\begin{gathered}
U_{2n,N-x} = U_{2n,N+x} = \frac{(-1)^n}{\sqrt{2}} \tilde R_n(\lambda(x);\alpha,\beta,\gamma,\delta), \qquad n=1,\ldots,N, \nonumber \\
U_{2n+1,N-x-1} = -U_{2n+1,N+x+1} = -\frac{(-1)^n}{\sqrt{2}} \tilde R_n(\lambda(x);\alpha+1,\beta,\gamma+1,\delta+1),\nonumber\\
\hphantom{U_{2n+1,N-x-1} = -U_{2n+1,N+x+1} =}{} n=0,\ldots,N-1, \nonumber \\
U_{2n,N}= (-1)^n \tilde R_n(\lambda(0);\alpha,\beta,\gamma,\delta),\qquad U_{2n+1,N}=0.\label{R-III}\end{gathered}$$ Then, one has
\[propR2\] Suppose that $\gamma,\delta>-1$ and $\beta>N+\gamma$ or $\beta<-N-\delta$. Let $M$ be a tridiagonal $(2N+1)\times(2N+1)$-matrix of the form with $$\begin{gathered}
M_{2k} = \sqrt{ \frac{(k+\gamma+1)(-N+\beta+k)(N-k)(k+\beta+\delta+1) }{(N-\beta-2k)(N-\beta-2k-1) }},\nonumber\\
M_{2k+1} = \sqrt{ \frac{(\gamma+N-\beta-k)(k+1)(k+\beta+1)(k-\delta-N) }{(N-\beta-2k-2)(N-\beta-2k-1) }},
$$ and $U$ the orthogonal matrix determined in . Then the columns of $U$ are the eigenvectors of $M$, i.e., $M U = U D$, where $D$ is a diagonal matrix containing the eigenvalues of $M$ $$\begin{gathered}
D= \operatorname{diag} (-\epsilon_N,\ldots,-\epsilon_1,0,\epsilon_1,\ldots,\epsilon_{N}), \qquad
\epsilon_{k}=\sqrt{k(k+\gamma+\delta+1)},\qquad k=1,\ldots,N.\end{gathered}$$
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors wish to thank the referees for their insightful remarks and suggestions which helped to enhance the clarity of the matter covered here.
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[^1]: In this paper, we shall usually take $\gamma>-1$ and $\delta>-1$, yielding real eigenvalues for the matrices under consideration. But the expressions for the eigenvalues remain valid even when these conditions are not satisfied.
|
---
author:
- 'M. Dubé [^1], M. Rost'
- 'M. Alava'
title: 'Conserved Dynamics and Interface Roughening in Spontaneous Imbibition : A Critical Overview '
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#intro}
============
A considerable amount of effort has been spent in the field of kinetic roughening over the last two decades. Apart from technological interests in crystal growth, a good part of the fascination for this field comes from the possibility of describing many different types of interfaces by a few distinct universality classes, in terms of scaling exponents and scaling functions [@ref_gen; @Krug_97].
However, if the theoretical and numerical aspects of the field are extremely rich and varied, the experimental backing of these ideas is quite lacking. This is for instance the case in kinetic roughening in crystal growth due to the great variety of atomistic processes. Much of the experimental attention has rather concentrated on simpler one–dimensional “toy-systems”, in particular driven interfaces in random media. A definite advantage of these systems is that the interface configuration is directly observable in the experiment (while, it must often be deduced, from some probe-surface interaction, in crystal growth). Even these simple systems are challenging. They are most commonly grouped in universality classes described by the Kardar-Parisi-Zhang [@KPZ_86] (KPZ) or Edwards-Wilkinson [@EW_82] (EW) equations, with either quenched or thermal noise, depending on the driving regime. In the case of quenched noise the exact low-dimensional scaling behaviour is however still somewhat contradictory [@Leschhorn_96; @Nattermann_94; @Leschhorn_97; @Narayan_93; @Csahok_93]).
Examples of experimental studies in these systems are slow combustion fronts [@jyvkl_97] (for which KPZ scaling was recently demonstrated), shock fronts, fluid-gas interfaces in Hele-Shaw cells [@Rubio_89; @Horvath_91a; @He_92b; @Delker_96; @Dougherty_98], or in paper [@Buldyrev_92; @Barabasi_92; @Family_92; @Buldyrev_ph; @Amaral_94; @Horvath_95; @Kumar_96; @Kwon_96; @Zik_97] and fracture surfaces [@Bourev] such as one-dimensional fracture lines [@Kertesz_93; @Engoy_94], to name a few. In this paper, we concentrate on a particular system, namely the spontaneous imbibition of a porous medium by a liquid [@Buldyrev_92; @Barabasi_92; @Family_92; @Buldyrev_ph; @Amaral_94; @Horvath_95; @Kumar_96; @Kwon_96; @Zik_97]. Our goal is to critically review the experiments and theories that exist in the literature and to indicate a direction for further investigation and comprehensive understanding of imbibition front roughening. The key questions are (i) whether imbibition should present any kind of scaling behaviour at all, and (ii), if so, under which conditions can one of the – possibly several – scaling regimes become observable.
Our motivations in writing this paper are twofold. First, the fluid dynamics aspect of imbibition is itself very complex, and should not be ignored in any discussion of the statistical fluctuations of the interface. We feel that this has not been properly done so far. Secondly, we present in a companion paper the results of a line of investigation of spontaneous imbibition based on a phase field formalism [@Dube_2005]; the present paper is intented to lay down the aspects which we believe essentials to any models of imbibition.
We start in Section \[macroscopicfeatures\] by reviewing the macroscopic properties of imbibition and show that even the simple propagation of an imbibition front may have several dynamical regimes, depending on the design of the experiment. We also show that the macroscopic progression of the interface will have a very strong influence on the roughening process. Section \[previousmodels\] outlines the models that have been proposed for imbibition experiments as well as their predictions. In Section \[experimentalefforts\], we discuss the existing experimental data, and point along the way to several features missed by these experiments. Finally we conclude with a discussion and proposals concerning future experiments.
Macroscopic Features of Imbibition {#macroscopicfeatures}
==================================
Although extremely familiar to researchers working in the field of flow in porous media, the details of imbibition would seem to be relatively unknown to the statistical physics community. It is generally defined in reference to two-fluid flow in porous media, and corresponds to the displacement of the lesser wetting fluid by the more wetting one [@Scheidegger_57; @Sahimi_93]. Notice that this definition is irrespective of whether the flow of the fluids is spontaneous or induced (eg., by a pump). In this work, we will restrict ourselves to the case of [*spontaneous*]{} imbibition. The flow of the fluids is thus driven solely by capillary forces, with gravity and/or evaporation being the only external influences on the fluids’ motion.
Capillary Rise {#capillaryrise}
--------------
The simplest example of spontaneous imbibition is capillary rise: a part of a capillary tube, of radius $R$, is immersed into a reservoir, exposed to an ambient atmospheric pressure $P_0$. We assume that the fluid wets the capillary so that a meniscus, described by the surface $z-h(x,y,t)=0$ is formed. At equilibrium, this surface is characterised by a contact angle $\theta$, obtained from Young’s law [@Rowlinson_82] $$\gamma_{lg} \cos \theta = \gamma_{sg} - \gamma_{sl},$$ where $\gamma_{ij}$ is the surface tension between the phases $i$ and $j$, and $s$, $l$ and $g$ stand for solid, liquid and gas respectively. If the meniscus is in motion, the contact angle differs from its equilibrium value [@deGennes_85], but we neglect this effect here. The important point is that the curvature causes a pressure difference $\Delta P \equiv P_c = P_0 - P(z=h^-) = 2
\gamma_{lg} \cos \theta /R$ where $\gamma$ is the surface tension of the liquid-gas interface and $R/ \cos \theta$ is the curvature of the meniscus. We refer to $P_c$ as the capillary pressure.
The motion of the fluid, of density $\rho$ and viscosity $\eta$, can be treated within the assumption of Poiseuille flow [@Landau_fl], i.e. the full Navier-Stokes equation is replaced by the simpler Stokes equation for an incompressible fluid under the influence of the gravitational force $\rho {\bf g}$. As usual, the pressure field is found from Laplace’s equation $\nabla^2 P = 0$, but with boundary condition $P(z \! = \! 0) = P_0$ and $P(z \! = \! h) =
P_0 - P_c$. (We can neglect corrections to the pressure field close to the meniscus, if its height is a lot larger than the radius of the tube). This yields $ P(x,y,z) = P(z) = P_0 - P_c z/h(t)$, and it is straightforward to obtain the progression of the interface, $$\frac{ d h}{d t} = \frac{\kappa}{\eta} \rho g
\left( \frac{h_{eq}}{h(t)} - 1 \right)
\label{washburn}$$ a classical result derived by Washburn [@Washburn_21] and Rideal [@Rideal_22]. Washburn’s and Rideal’s equation includes the permeability of the tube $\kappa = R^2 / 8$ and the equilibrium height of the meniscus $h_{eq} = P_c / \rho g$. Studies of capillary rise including the inertial term of the Navier-Stokes equation [@Szekely_71] show that Eq. (\[washburn\]) is essentially correct, with the exception of very short times. Notice also that this equation of motion also neglects completely the problem of the actual motion of the contact line between the gas-solid and liquid phases [@deGennes_85].
The (transcendental) solution of Eq. (\[washburn\]) has the following asymptotic properties: Defining $h(t_0)$ as the initial height of the column and $\tau_{eq} \equiv h_{eq} \eta / \kappa (\rho
g)^2$ as an equilibration time, for low heights $h \ll h_{eq}$ (and also in the absence of gravity where $h_{eq} = \infty$) the rise is of the form $$h^2 (t) - h^2 (t_0) \; = \; \frac{\kappa P_c}{\eta} \; (t - t_0).
\label{t-half}$$ The equilibrium height is approached by $h$ exponentially $$h (t) \; \sim \; h_{eq} ( 1- e^{-t/\tau_{eq}}).
\label{rise-eq}$$
This was examined experimentally by Washburn [@Washburn_21] and Rideal [@Rideal_22] as well as numerous others, who found very good agreement between theory and experiment. Capillary rise in a liquid–liquid system was also examined by Mumley et al.[@Mumley_86]. They confirmed Washburn’s result in the case of perfect wetting, but reported discrepancies for systems with non-zero contact angle. In this case, the rise was [*slower*]{} that $t^{1/2}$, a behaviour attributed to the motion of the contact line itself [@deGennes_85].
Capillary Rise in Porous Media
------------------------------
With the basic capillary rise phenomenon understood, we can now turn to spontaneous imbibition in porous media. The flow of liquids in porous media is generally described in terms of Darcy’s equation [@Scheidegger_57; @Sahimi_93] $$\langle {\bf Q} \rangle = - \rho A \frac{\kappa}{\eta}
( {\mbox {\boldmath $\nabla$} } P - \rho {\bf g} ),
\label{darcy}$$ where $ \langle {\bf Q} \rangle$ is the average mass of fluid transported per unit time through the cross-section $A$ of the porous medium, and $\kappa$ and $\eta$ are the average permeability and viscosity respectively. Darcy’s equation arises from an averaging procedure of the porous medium and ignores all details on length scales smaller than the pores. If we assume the porous medium to be homogeneous, such that the permeability is a constant $\kappa_0$ independent of the fluid concentration, we can still solve Laplace’s equation for the pressure and obtain a coarse-grained pressure gradient, ${\mbox {\boldmath $\nabla$} } P = {\bf \hat{z}} P_c / h(t)$, with $P_c$ an effective capillary pressure, and $h$ the average height of the fluid column. This is of course valid only when the notion of an interface is itself well defined, i.e. it should not be too “fuzzy” up to macroscopic scales. Under these assumptions, and with the identification $\langle Q \rangle = \rho A \,dh /dt$, Darcy’s equation leads directly to the Washburn–Rideal result, Eq.(\[washburn\]) with a dynamical rise given by Eqs. (\[t-half\]) and (\[rise-eq\]).
This form of Darcy’s equation has been used to study many experiments of fluid propagation in porous media, including some fibrous materials (see below). The agreement is not always perfect – we shall come back to this in Section \[validity\]. At this point, a more complete system of equations [@Sahimi_93; @Hilfer_98] would only obscure the physics.
Evaporation
-----------
Considering the typical experimental setups for imbibition front propagation [@Amaral_94], evaporation effects should also be included to Washburn’s equation. As far as we are aware, no detailed studies of fluid motion through thin porous media including the effects of evaporation have been performed. A natural assumption is to introduce an evaporation rate proportional to the area of the fluid exposed to air, i.e. the net loss of fluid mass per unit time due to evaporation is $Q_e = - 2
{\cal E} \rho L h (t)$ where $L$ is the lateral width, and ${\cal E}$ is a phenomenological evaporation rate per unit area. Thus Eq.(\[washburn\]) is modified to $$\frac{ d h}{d t} = \frac{\kappa}{\eta} \left(
\frac{p_c}{h} - \rho g \right) - \epsilon h
\label{wash-grav-evap}$$ where $\epsilon = 2 {\cal E} \rho$ is the evaporation rate. This form obviously neglects changes in the concentration profile towards the surface, as well as the evaporation at the interface itself, which is both justified for thin media such as a sheet of paper.
An immediate consequence of Eq. (\[wash-grav-evap\]) is an equilibrium height depending on the evaporation rate $$h_{eq} = \frac{h_e^2}{2 h_g} \left( \left( 1 + \frac{4 h_g^2}{h_e^2}
\right)^{1/2} - 1 \right)$$ where $h_g = P_c /\rho g$ and $h_e = ( P_c \kappa / \epsilon \eta
)^{1/2}$ are respectively the equilibrium heights in the cases where only gravity or evaporation have a significant influence, with a crossover defined by $2 h_g \sim h_e$. When gravity can be neglected, the height $h(t)$ behaves as $$h^2 (t) = h^2 (t_0) \; e^{-2 \epsilon (t - t_0)}
+ h_e^2 \; \left( 1 - e^{-2 \epsilon (t - t_0)} \right)
\label{wash-evap}$$ Well below $h_e$ the rise still follows $h(t) \sim t^{1/2}$, and it again approaches exponentially the equilibrium height.
Addition of a Solution
----------------------
Many experiments on the roughness of the interface in an imbibition context have studied the behaviour of a dye, added to the pure fluid [@Buldyrev_92; @Amaral_94; @Kwon_96]. In the literature on porous media this is referred to as the hydrodynamic dispersion phenomenon [@Sahimi_93]. At the simplest level, it can be treated by the introduction of a dye concentration field $K({\bf x},t)$ advected by the macroscopic velocity field ${\bf v}$, $$\frac{\partial K}{\partial t} + {\bf v} \cdot
{\mbox {\boldmath $\nabla$} } K = D \nabla^2 K,
\label{dye_dynamics}$$ where $D$ is a diffusion constant (in some cases, it may be necessary to introduce a diffusion tensor $D_{ij}$ [@Sahimi_93]).
Two remarks on Eq. (\[dye\_dynamics\]) are in order. First, in everyday life, it is easily seen that the fluid front is always faster than the dye front. This clogging phenomenon, which may be due to a smaller permeability for the dye, may easily be incorporated phenomenologically by introducing a constant $\lambda < 1$ in the front of the convective term, i.e. $${\bf v} \cdot {\mbox {\boldmath $\nabla$} } K
\rightarrow \lambda {\bf v} \cdot {\mbox {\boldmath $\nabla$} } K.$$ The second point concerns the stopped interface. If the pinning of the interface, at a distance $h_e$ from the reservoir, is due to evaporation, there will nevertheless be a fluid motion, with approximate velocity $ v_e \sim \kappa P_c / \eta h_e $, in order to compensate the losses. This implies that more and more dye particles will be brought to the fluid interface, thus creating a band of high dye concentration, analogous to the coffee rings examined recently [@Deegan_97]. The increase of the width of this band with time can be used as an alternative way of measuring the relevant macroscopic parameters.
Validity of Macroscopic Description {#validity}
-----------------------------------
We now discuss the validity of the macroscopic results for the specific case of spontaneous imbibition in paper. Ordinary paper is made out of wood fibers and, in many cases, chemical additives and filler materials like talc and clay, arranged in a disordered structure. Not only is there a wide distribution of pore sizes, with a high effective tortuosity but the surface of the network is extremely uneven. This gives rise to all the standard problems in defining a static/dynamical contact angle for inhomogeneous substrates.
Darcy’s equation, Eq. (\[darcy\]), is relatively well established (to the point that it is even often referred to as a law) for general porous media. However, paper, as well as other fibrous materials present the peculiarity that the fiber structure may be modified by the contact with the liquid, a phenomenon known as swelling [@Bristow_71]. Cellulose fibers show a great affinity to water and can absorb large quantities in millisecond timescales, giving rise to concomitant changes in fiber volume and pore structure (this is however not the case for many organic fluids and oils). Thus one’s intuitive picture of fibers as capillary tubes is false: the pore structure is highly non-trivial and in some cases time–dependent. There are two different effects that play a role: the volume to be filled with liquid increases, and the flow resistance of the pore network changes.
There can also be an exchange of liquid between the inside of a fiber and the “surface” pores, which complicates enormously the fluid flow, since there are no well defined “structures” (either the pores, or the fibers) responsible for the capillary forces [@Aspler_87; @Pezron_95]. This is of course a very serious impediment to any kind of attempt at a “microscopic description” of the structure, say in terms of a percolation network [@Lenormand_88].
A more serious problem with water is the fact that it is not actually known whether a pressure balance argument is always valid. On short time scales (smaller than a few seconds), it has been proposed that front penetration would proceed in pores first with the establishment of a prewetting layer through [*diffusion*]{} [@Salminen_88], in which case the front position could advance as $t^k$ with $k$ varying up to unity. Experiments have demonstrated that clear, Washburn-like behavior can be obtained in conditions that amount to basically no interaction (swelling, prewetting) between the material and the penetrating liquid [@Gillespie_58]. Note that minute applications of chemicals, e.g. during paper manufacture, may induce drastic changes in the effective viscosity or surface tension of the invading liquid.
There are however some advantages to paper. The very high permeability of the fibers will reduce the formation of overhangs in the interface. Darcy’s equation cannot treat trapped air bubbles in bulk porous media [@Hilfer_98], which in paper only play a minor role: paper is thin and the pores should be connected well to the surface. Moreover spontaneous imbibition is slow and allows more time for the removal of overhangs and trapped bubbles.
In conclusion, one would believe that a well defined fluid-air interface should exist, and that Darcy’s equation should be valid on length scales larger than the interface width, provided that interactions between the liquid and the fibers are minimal. In that sense, a full hydrodynamical treatment of the problem may not be necessary (as noted in [@Dube_2005]). In many cases, it may also be necessary to introduce a concentration dependent permeability, i.e. $\kappa = \kappa (\rho)$ [@Gillespie_58]. In any case a Washburn approach may be a first step. In that respect, recent experimental work with paper and [*deionised*]{} water, showing front propagation consistent with Washburn’s equation is quite encouraging [@Zik_98].
Statistical Fluctuations of the Interface
-----------------------------------------
So far, we have considered a flat interface in a completely homogeneous medium. In general, the disorder in the paper causes fluctuations which will accumulate to roughen the advancing front. In the standard picture of kinetic roughening [@ref_gen; @Krug_97] the fluctuations of the interface are correlated up to a distance $\xi_{\|} (t)$, a lateral correlation length, increasing in time as $\xi_{\|} \sim t^{1/z}$ and described by the dynamical exponent $z$. The vertical extent of the interface fluctuations, its width $W$, is related to $\xi_\|$ through the roughness exponent $W \sim \xi_\|^\chi$. Thus the width increases with time as $W \sim t^\beta$ with $\beta = \chi / z$, until the interface fluctuations have saturated, ie., until $\xi_\| (t) = L$, which defines the crossover time $t_\times \sim L^z$.
The initial increase of the width and eventual saturation are comprised in a (Family–Vicsek) scaling form $$W (L,t) = L^{\chi} f (t/L^z)$$ with the scaling function $f(u) \sim u^{\beta}$ for $u \ll 1$ and $f(u) \sim const$ for $u \gg 1$.
The spatial height difference correlation function (of the $q$th moment) $$G_q ({\bf r},t) = \langle | h({\bf r}+{\bf r}',t)-h({\bf r}',t) |^q
\rangle^{1/q}$$ generally scales similarly to the total width, i.e. $G_q \sim r^\chi$ for small $r$, and approaching a constant as $r \to
\infty$. However, care has to be taken: In the case of [*anomalous*]{} scaling the height differences for fixed $|{\bf r}|$ do not saturate with time, and $G_q$ shows the [*local*]{} roughness exponent $\chi_{loc}$ [@Krug_97; @Lopez_97]. The height difference distribution for fixed $|{\bf r}|$ may also have a long tail, causing intermittent or “turbulent” jumps in the height configuration in which case each moment has a different exponent $\chi_q$, ie., the interface exhibits multiscaling [@Krug_94]. Also a temporal height difference correlation function $$C_q (t) = \langle | h({\bf r},t+t')-h({\bf r},t')-(\bar
h(t+t')-\bar h(t')) |^q \rangle^{1/q}_{{\bf r}}$$ can be considered (e.g. in [@Horvath_95]), which is of particular use in the case the system exhibits time–translational invariance.
To our knowledge none of the works studying interface fluctuations in imbibition have checked for anomalous (as determined by the higher moments of the correlation functions) types of scaling behaviour [@Buldyrev_92; @Barabasi_92; @Family_92; @Buldyrev_ph; @Amaral_94; @Horvath_95; @Kumar_96; @Kwon_96; @Zik_97], although this is in principle difficult, since an exact diagnosis for higher moments $q>2$ is severely hampered by large statistical fluctuations. These concepts are nevertheless important when the randomness of the medium is described with random-field disorder [@Dube_2005].
Another peculiarity of imbibition is that, apart from the standard kinetic lateral correlation length $\xi_\|(t)$, arising from the accumulating history of fluctuations, another lateral length scale is to be seen: local conservation of the fluid determines a lateral length $\xi_\times$ related to the average motion of the interface. Intuitively, it is clear that such a length scale must exist, since fluctuations ahead and behind the average position of the imbibition front have respectively a slower and faster instantaneous velocity. To make a quantitative argument, we introduce an effective surface tension $\gamma^*$, representing the energy cost of a curved air-liquid interface on a [*macroscopic*]{} scale [@Krug_91]. Any curvature at the interface – the typical size of fluctuations being $W$ vertically and $\xi_\times$ laterally – modifies the pressure by an amount $\Delta P = \gamma^* W / \xi_\times^2$ (the Laplace pressure effect). This should be compared with the difference in the pressure field across the same vertical distance $W$ due to the pressure gradient derived in Section \[capillaryrise\], namely $\Delta P = P_c W / H$, from which we obtain $$\xi_\times^2 \sim \gamma^* H / P_c,$$ relating the parallel length scale $\xi_\times$ to the height of the interface. Beyond this length scale we do not expect any correlated roughness of the interface, since those fluctuations would be suppressed by the overall gradient in the pressure field $P_c/ H$.
Let us recall once again that the interface continuously slows down [@note_4], without gravity nor evaporation $H \sim t^{1/2}$. Thus the slow increase of $\xi_\times \sim H^{1/2} \sim t^{1/4}$ with time leads to an increase of the width $W \sim t^\beta$ with $\beta =
\chi/4$. However, we stress that this increase is conceptually different from the increase of the lateral correlation length with a dynamic exponent $z$ in standard models of kinetic roughening.
Models of Imbibition {#previousmodels}
====================
In parallel with experimental work, many theoretical models of imbibition have been developed. These models fall either in discrete (cellular automata) or continuous classes. Most of the theoretical discussion on imbibition was done in terms of the DPD model, which was shown to belong to the same universality class as the quenched KPZ equation. Other types of models or continuum equations have also been proposed.
Cellular Automata Models
------------------------
The earliest theoretical investigations of imbibition phenomena were based on the [*Directed Percolation Depinning*]{} (DPD) model [@Buldyrev_92; @Barabasi_92; @Amaral_94]. It is a cellular automaton, with space discretized in cells which are either blocked or wettable, with the blocked cells being a fraction $p$. The particularity of the model is that overhangs in the interface are removed as soon as they occur, which thereby introduces anisotropy in the interface. Growth takes place by invasion of available cells until the interface comes across a percolating directed path of blocked cells. Clearly, the statistical properties of the interface are related to the properties of the percolating path. Close to the percolation threshold $p_c \simeq 0.47$, the directed percolating path is characterized by the parallel and perpendicular length scales $\xi_{\|} \sim | p- p_c|
^{-\nu_{\|}}$ and $\xi_{\bot} \sim | p- p_c| ^{-\nu_{\bot}}$, with values $\nu_{\|} \simeq 1.7$ and $\nu_{\bot} \simeq 1.09$. so the width $ W(L) \sim \xi_{\bot} \sim L^{\nu_{\bot}/\nu_{\|}}$, yielding $\chi \simeq 0.63$.
To treat evaporation, the DPD model was modified by the introduction of height dependent fraction of blocked cells, i.e. $p = p(h) \sim
h /h_0$ with an equilibrium height $h_0 \sim (\nabla p)^{-1}$ defined by $p(h_0)=p_c$ [@Amaral_94]. In this model, the interface is described by the standard DPD roughness exponent up to a length $l_{\times} \sim (\nabla
p)^{-\gamma/\chi_{DPD}}$ after which it saturates to a constant value $w_{sat} \sim ( \nabla p)^{-\gamma}$. The identification $w_{sat} \sim \xi_{\bot}$ then yields $\gamma =
\nu_{\bot} / (1+\nu_{\bot}) \sim 0.52$, a value comparable to the experimental results.
As already noted [@Horvath_95], there is a physical justification to the main feature of the DPD model, namely the erosion of overhangs. It is plausible that propagation of the fluid in a direction parallel to the reservoir, if it has a chance to occur, will be favoured over the motion of the fluid away from it, since this does not result in any change in the average capillary pressure. In that sense, any overhangs will eventually be removed, although the question of the different time scales involved in the dynamics is certainly extremely complex [@Dube_2005].
The DPD model in its simplest form then probably describes well the statistical properties of the pinned interface. It is however difficult to believe that this model can describe the whole dynamical motion of the interface, since it does not account for liquid conservation. It already fails by predicting a [*constant*]{} average interface velocity. As we saw in section \[macroscopicfeatures\], one of the principal features of imbibition is a constant slowing down of the front, a result intimately related to fluid conservation. It can be argued that the DPD refers specifically to the motion of the dye, but the rise of the dye front is necessarily bounded by the liquid-air interface, and the conservation law governing the motion of the fluid must be reflected on the motion of the dye particles. From that point of view, simply replacing ${\bf v} \sim t^{-1/2}$ in Eq. (\[dye\_dynamics\]) certainly would be incorrect.
It is interesting to show how the global behaviour of the modified DPD model may be related to the macroscopic Washburn behaviour, Eq. (\[wash-grav-evap\]). The density of [*open*]{} cells $1-p(h)$ can be thought of as an effective force acting on the interface, similar to the gradient in pressure on fluid motion. Since the Washburn Eq. is by essence a dissipative equation, we can associate, [*away from pinning*]{} $$p (h) \rightarrow
\frac{\kappa}{\eta} \left(
\frac{p_c}{h} - \rho g \right) - \epsilon h$$ The assumption of constant gradient thus effectively corresponds to an evaporation rate. It is also possible, at the macroscopic level, to include gravity, and to mimic the slowing down of the interface. Note however that we predict a pinning height $h_e \sim \epsilon^{-1/2}$ while the assumption of Amaral et al. implies $h_e \sim (\nabla p)^{-1}$, since they neglect the capillary driving term.
Another model of evaporation was introduced by Kumar and Jana [@Kumar_96]. The model is essentially similar to the DPD model, with the modification that each cell is not necessarily “full” or “empty”, but may contain many “subunits”. Evaporation is modeled by a loss of $n$ subunits in the transfer between cells. It is found that below a critical $n_{c1}$, the interface propagates indefinitely. Between this value and a second critical loss rate $n_{c2}$, they observe an interface behaviour similar to the one observed by Amaral et al. [@Amaral_94], but with value $\chi =
0.5$ and $\gamma = 3.0$. For $n > n_{c2}$, this scaling regime breaks down and the roughness exponent becomes $n$–dependent. These results were backed experimentally, but not in any consistent way.
It is rather difficult to believe that indefinite front propagation under evaporation (obtained for $n < n_{c1}$) is physically realistic, nor seems the way of including evaporation convincing. In this model, the concentration of fluid molecules decreases continuously from a maximum $N_0$ to a value of $0$ at the interface, which seems physically unrealistic.
We terminate by presenting invasion models based on the random field Ising model [@Martys_91], which predict a depinning transition, as well as a change of morphology of the interface at some length scale related to the capillary pressure. It seems however unsuited for spontaneous imbibition, because any advance of the front requires the increase of an externally applied pressure.
A variation of this idea, introduced by Sneppen [@Sneppen_92], allows invasion always at sites of lowest resistance. An interesting aspect of this model is temporal multiscaling [@SnepJen_93], due to avalanche motion [@Olami_94; @Leschhorn_94]. Avalanches are also present in the process of spontaneous imbibition, although the consevation law imposes a natural cutoff on their size and distribution [@Dougherty_98; @Dube_2005]. It is nevertheless interesting that a similar lack of temporal scaling is also seen in the phase field model of imbibition, thus surviving the introduction of a conservation law.
Continuum Description
---------------------
The DPD model (in its original version without “evaporation”) belongs to the same universality class as the quenched KPZ equation (see e.g. Chapter 10 in [@ref_gen] and references therein, as well as [@Csahok_93]) $$\frac{\partial h(x,t)}{\partial t} = \nu \nabla^2 h(x,t) -
\frac{\lambda}{2} (\nabla h(x,t))^2 + \eta(h(x,t),x).$$ A related equation with [*multiplicative*]{} noise has been introduced by Csahók et al. [@Csahok_93] in order to model the random porosity of the medium. Since these equations are local at the interface, they do not contain a conservation law for the fluid.
With these considerations in mind, a continuum equation was introduced by Zik et al. [@Zik_97] following Krug and Meakin [@Krug_91]. In Fourier space, it has the form $$\frac{dh_k}{dt} = -\frac{\kappa p_c}{\eta} |k| \frac{h_k}{h_0(t)}
+ \eta_k (t)
\label{cont-eq}$$ where $k$ is the wave-vector, $h_0 (t)$ is the mean height of the interface at time $t$ (cf., Eq. (\[t-half\])), and $\eta_k$ is the noise term, assumed to be annealed, with correlations $\langle
\eta_k(t) \eta_{k'}(t') \rangle \sim \delta_{k+k'} \delta(t \! - \!
t')$. This equation is essentially similar to the one derived by Krug and Meakin for the roughness of stable Laplacian fronts with noise and corresponds to the leading term of a Saffmann-Taylor analysis of the problem. Note that $|k|$ is nonlocal in space.
The other difficulty with respect to spontaneous imbibition, the quenched nature of the disorder, remains. As long as the interface sweeps fast enough through the medium, the disorder acts as annealed, time dependent noise [@Nattermann_94; @Leschhorn_97], although the question of the effective noise correlator is far from trivial [@Dube_2005] — Eq. \[cont-eq\], as it stands, gives the unphysical results of an interface that [*never*]{} saturates. In any cases, at late times under slow average interface propagation in a continuum model one is confronted with both the difficulties of a [*nonlocal*]{} interface equation with [*quenched*]{} disorder.
Experimental Efforts in Imbibition {#experimentalefforts}
==================================
Most of the experiments concerned with front propagation in random media seem to produce a self-affine interface, but the numerical value of the exponents often bears little resemblance to the standard universality classes and/or associated models. Imbibition of porous media would seem like an obvious first experimental choice, since the associated time and spatial scales are easily accessible in the laboratory. Many such imbibition experiments were performed, but it turns out that there is very little in common between the different experiments. This is partly due to the fact that different experiments had different goals, but it is also a reflection of the great complexity of the processes involved in imbibition.
Experiments on Pinned Interfaces
--------------------------------
The earliest statistical physics imbibition experiments were done by Buldyrev et al. [@Buldyrev_92; @Barabasi_92; @Buldyrev_ph] and Family et al.[@Family_92]. The first experiment was performed with a dye solution in a vertical capillary rise setup. The rising front moved from the reservoir and the roughness of the interface developed during the process. Eventually, the dye front stopped, due to gravity and/or evaporation (no dynamical measurements, either micro- or macroscopic, were done in this experiment), and the roughness of the pinned interface was measured. The main experimental conclusion of the paper was a roughness exponent with value $\chi = 0.63$, consistent with the DPD model (see above, Section \[previousmodels\]). However, the length scale over which the scaling behaviour was observed is extremely small. For a sheet of paper of total lateral extent $L = 40$ cm, $C(l) \sim l^{\chi}$ only for length scales $l$ smaller than $l_{max} \sim 1$ cm, after which it levels off to a constant or to a logarithmic function of $l$ [@note_1]. Since this scaling region is only a few times larger that the length scale of of the fibers themselves, it is not at all clear whether the exponent $\chi$ really is a universal value, or simply results from the microscopic fiber structure. As a comparison, the scaling region for the paper burning experiment [@jyvkl_97] was from $ 1.5$ cm to about $10$ cm, for a total length $30$ cm [@note_2]. A similar experiment was done in a three dimensional sponge-like material [@Buldyrev_ph]. Again, the stopped interface was observed, yielding a roughness exponent $\chi^{(2d)} \sim 0.5$, a result consistent with the higher-dimensional version of the DPD model. No other experimental details were however given.
In a further set of experiments, Amaral et al. [@Amaral_94] studied the role of evaporation in more detail. They again considered the stopped interface, but changed the height where pinning occured through the evaporation rate (presumably by modifying the humidity during the experiment, although this is not specified). Their main result is that the width of the pinned interface is related to the pinning height $H_p$ (and thus to the evaporation strength) through a new exponent $\gamma$, namely $w_{sat} \sim H_p^\gamma$. The experiments gave a value $\gamma = 0.49$, which – as reported in Section \[previousmodels\] – can also be obtained from a modified version of the DPD model [@Amaral_94]. (We will come back to this in Section \[Analysis\]). On the other hand, Kumar and Juma [@Kumar_96] also performed an experiment in presence of various evaporation rates and claimed that the roughness exponent was not universal but stronly dependent on the evaporation rate, i.e. $\chi
= \chi (\epsilon)$, although no systematic experimental investigation of this breakdown of universality was done.
Experiments on Moving Interfaces
--------------------------------
The experiment of Family et al. was performed in an horizontal capillary setup with water only, and the position of the air–water interface was recorded both temporally and spatially [@Family_92]. The main experimental results were an average interface progression $\bar{h} \sim t^{\delta}$, with $\delta = 0.7$, and a self–affine interface described by a Family-Vicsek scaling relation, and characterized by the exponents $\beta = 0.38$ and $\chi
= 0.76$. The first result $h(t) \sim t^{0.7}$ is certainly intriguing. However, it should be noticed that a reservoir was placed underneath the piece of paper, in order to prevent evaporation. It is of course possible that this caused condensation instead, thus increasing the velocity of the front. It is also in line with the results of non–Washburn water penetration in paper [@Salminen_88]. Also here the spatial scaling regime was rather small, for distances below $l_{max} \sim 2$ cm for a $40$ cm wide sheet of paper. It should also be noticed that these experiments were made with chinese paper, having a thickness of only a few fiber layers, which may also influence the results.
The temporal scaling of the interface was studied in details by Horváth and Stanley [@Horvath_95]. Moving a piece of paper such that the interface always remained at a fixed distance $H$ above a reservoir, they found a power law behaviour for the time correlation function $C_2(t) \sim t^{\beta}$ with $\beta = 0.56$. Another result is that the velocity $V$ at which the the piece of paper must be moved towards the reservoir in order to keep the interface at a constant $H$ varies as $V \sim H^{-\Omega}$, where $\Omega = 1.6$. Notice how this implies an interface propagating such that $H(t) \sim t^{2/5}$, since $V = dH/dt$. This is slower than $\delta = 1/2$ expected from Darcy’s equation, but consistent with our earlier discussion. Unfortunately, the scaling behaviour of the interface as a function of space was not discussed and/or measured in this otherwise very careful work.
It is certainly interesting that the value of $\beta$ is constant for all heights considered. The different heights did however affect the time at which the time correlation functions saturated, and the values at which it did so. Horváth and Stanley found for $C(t)$ a scaling form $$C(t) \sim V^{-\Theta_L} f (t V^{(\Theta_t+\Theta_L)/\beta} ),$$ where $f(y)$ is a scaling function such that $f(y) \sim y^{\beta}$ for $y \ll 1$ and $f(y) \sim const$ for $y \gg 1$. The values of the independent exponents were $\Theta_L=0.48$ and $\Theta_T=0.37$ [@note_3].
Another experiment was performed by Kwon et al. [@Kwon_96]. They deposited a paper towel on an inclined glass plate and followed the capillary rise of a dye solution, giving a mean interface dynamical propagation $\bar{h} (t) \sim t^{0.37}$. Again, two scaling regimes were present in the saturated width; on small length scales ($\leq 2$ cm), $\chi = 0.67$ while on larger length scales (up to $20$ cm) $\chi
\sim 0.2$. Within the simple Family-Vicsek picture they obtained $\beta = 0.24$ on the short lengthscale regime.
Finally, Zik et al. [@Zik_97] performed an horizontal capillary front experiment. They obtained rough interfaces only with highly anisotropic paper, for which they found $\chi = 0.4$. For isotropic paper, the roughness was at best logarithmic. It is remarkable, that the scaling for the anisotropic paper was observed through a large range of length scales, not only up to fiber length.
Analysis of the Experiments : Influence of Fluid Conservation {#Analysis}
-------------------------------------------------------------
As the previous paragraphs show, there is a large amount of experimental work on the subject of spontaneous imbibition and front roughening, but a coherent picture does not emerge naturally. The results are quite contradictory and difficult to analyse properly.
We have already mentioned a few experimental difficulties inherent to imbibition in paper, the most important being the complicated flow profile of a fluid inside the paper, due to fiber swelling and dissolution. Certainly all experiments suffer from these difficulties at least to some degree. Apart from the work of Family et al.[@Family_92], all experiments are consistent with an average front propagating roughly according to the Washburn–Rideal result, Eq.(\[washburn\]). They don’t agree perfectly, but definitely reflect the influence of the conservation of fluid on the propagation process.
One consistent feature to emerge from the experiments is an interface which has developed roughness only on very short length scales. A partial remedy to this problem is, as done by Amaral et al.[@Amaral_94], to introduce a length scale $l_p$ above which the interface saturates. It their experiment and model (see also Section \[previousmodels\]) it becomes visible in the width the interface takes at the height where it gets pinned by evaporation, and they establish the relation $w \sim H_p^\gamma \sim
l_p^\chi$.
The idea of such a length scale has of course a wider range of applicability than in the context of imbibition with evaporation. Indeed, the experiment of Horváth and Stanley [@Horvath_95], done with little influence of evaporation, and no discernable influence of gravity, showed the existence of a similar length scale, in this case, related to the velocity $V$ at which paper was being pulled down (or, equivalently, to the average height of the interface).
We have already shown that such a length scale $\xi_\times (H)$, connected to the existence of a conservation law existed. It can provide a natural connection between the work of Amaral et al. and Horváth and Stanley, provided that one is willing to revise the role of evaporation. If the sole role of evaporation is to stop the macroscopic progression of the interface, with little or no influence on the statistical fluctuations of the interface itself, then the width $w(t) \sim \xi_\times^{\chi} (H_p) \sim H_p^{\chi/2}$ and therefore $\gamma = \chi/2$. This length should also influence the setup examined by Horváth and Stanley [@Dube_2005]. Indeed, it provides an alternative explanation to the velocity (or alternativley height) dependence of the saturated value of the time correlation function, since $$C^2_2 (\tau \rightarrow \infty) \sim \int_{1/\xi_\times (H)}^{1/a}
\frac{dk}{k^{2\chi+1}}$$ which yields $C_2(\tau \rightarrow \infty) \sim H^{\chi/2} \sim
V^{-\chi/2}$. Note that as long as $\xi_\times < L$, the correlation function $C_2(t)$ is [*independent*]{} of the total length of the system. Since only one length of paper was considered experimentally, it is unfortunately impossible to check this hypothesis.
However, in order to demonstrate convincingly the existence of such a length scale (or of any other length scales) and the scaling below it, it is imperative to extend the region of scaling. We fail to see how a sub-centimeter description of the roughness might be considered universal. It is quite possible that the crossover $\xi_\times$ associated with paper really is in the sub–centimeter range, in which case, the alternative way to check for scaling is in the scaling of the width. Even in the freely rising case, the total width of the interface should show some early time power law behaviour, with some exponent $\beta$.
One should also note the fact the structure of paper does have short-scale power-law correlations [@Pro96]. These are manifest in the two-point-correlation function of the areal mass density of the paper, with the cut-off of the effective power-law correlations extending in practice to a few times the fiber length. It is by no means clear what role such correlations play in imbibition, since penetration of the liquid may take place both inside the sheet, and on the outer surface. Also, it is not clear how these correlations map into correlations in the hydraulic conductivity or local permeability. They however remind of the fact that scaling exponents established on lengthscales that are also relevant from the microstructural viewpoint should be taken with a grain of salt. Still, it is quite possible that the effective noise correlator at the interface will display power law behaviour, as in the case of forced flow through Hele-Shaw cells [@Horvath_91].
Summary and Perspective
=======================
In this paper, we have tried to extract a coherent picture of all previous studies about interface roughening in spontaneous imbibition experiments. Concerning theoretical understanding, the DPD picture plausibly reproduces the properties of the pinned interface. We strongly feel that not enough attention has been paid to the macroscopic dynamical features and their connection to the underlying microscopic structure. The flow of fluid in paper is influenced by many microscopic mechanisms, and a study of the average front velocity already indicates which ones may or may not be relevant. The main experimental characteristic of imbibition certainly is the slowing down of the front, a phenomenon not seen in other systems. Some very simple predictions concerning evaporation and gravity would already allow a solid comparison between the expected behaviour and the experimental results.
An important reason to consider the fluctuations of the propagating front is the presence of a lateral length scale $\xi_\times$, for which experimental evidence already exists [@Amaral_94; @Horvath_95], and which has a very strong influence on the roughening process.
Finally, we believe that a complete description of imbibition (including the dynamic interface) must include a conservation law for the whole fluid, not simply at the interface level. This is for two reasons, first the average interface ought to slow down continuously, and secondly the dynamics of the interface need to have long range interactions. Also, any model must deal the quenched nature of the noise in a suitable way. These requirements make it impossible to describe the interface roughening with a [*local*]{} equation of motion. To answer these criteria, we introduce in the next paper [@Dube_2005] a phase–field model of imbibition, which incorporates explicitely a conservation law for the liquid in a disordered medium, and thus produces the correct macroscopic physics.
We finish this overview by pointing out a few experimental directions. The use of “ordinary” paper for the medium creates some problems that are usually neglected. In this respect it comes to mind that other liquids than water or water/ink mixtures might well prove advantageous, in that they are less susceptible to evaporation and/or do not interact with the fibers themselves. Another option is to monitor the roughness of the front using an Hele-Shaw cell. This eliminates the problem of fiber-liquid interaction.
Acknowledgements
================
We wish to acknowledge interesting and stimulating discussions with J. Kertész, T. Ala-Nissila, K. R. Elder, S. Majaniemi, and J. Lohi. This work has been supported by the MATRA program of the Academy of Finland.
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[^1]: *Present address: Center for the Physics of Materials, McGill University, 3600 rue University, Montr’eal, Qu’ebec, Canada H3A 2T8*
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---
bibliography:
- 'catupdate.bib'
- 'Planck\_bib.bib'
title: '*Planck* 2013 results. XXIX. The *Planck* catalogue of Sunyaev–Zeldovich sources: *Addendum* '
---
Introduction
============
It is only recently that cluster samples selected via their Sunyaev–Zeldovich (SZ) signal have reached a significant sizes, e.g., the Early SZ (ESZ) catalogue from the Satellite[^1] [@planck2011-5.1a; @planck2013-p05a], and catalogues from the South Pole Telescope [ SPT, @rei13; @ble14] and the Atacama Cosmology Telescope [ACT, @mar11; @has13]. These are now considered as new reference samples for cluster studies and associated cosmological analyses.
The present note describes updates to the construction and properties of the catalogue of SZ sources PSZ1, [hereafter PXXIX2013, @planck2013-p05a], released in March 2013 as part of the first data delivery. The PSZ1 catalogue contains 1227 entries, including 683 so-called [*previously-known*]{} clusters. This category corresponds to the association of SZ source detections with known clusters from the literature. The association is set to the first identifier as defined in the hierarchy adopted by PXXIX2013, namely: (i) identification with MCXC clusters [@pif11]; (ii) identification with Abell and Zwicky objects; (iii) identification with clusters derived from SDSS-based catalogues (primarily from @wen12); (iv) identification with clusters from SZ catalogues [@has13; @rei13]; (v) searches in the NED and SIMBAD databases. Considerable added value, including consolidated redshift and mass estimates (Fig. \[fig:mz\]), has been obtained through compilation of this ancillary information.
Since its delivery March 2013, we have continued to update the PSZ1 catalogue by focusing on the confirmation of newly-discovered clusters in PSZ1. This process has first involved updating the redshifts of some previously-known clusters (Sect. \[asso\]). We have also made use of recent results from dedicated follow-up observations conducted by the [*Planck*]{} Collaboration with the RTT150 [@rtt150] and ENO telescopes (Planck Collaboration 2015, in prep.), which together have allowed us to observe and measure redshifts for $\sim 150$ PSZ1 sources (Sect. \[rtt\]). We have also used published results from PanSTARRS [@liu14] and from the latest SPT catalogue [@ble14], as described in Sects. \[pans\] and \[spt\]. For all clusters with measured redshifts, we have computed the estimated masses using the $Y_z$ mass proxy (@arn14 and PXXIX2013; Sec. \[masses\]). Finally, we have revisited the cluster candidate classification scheme, which in PXXIX2013 was organised into three classes ([*class-*]{}1, 2, 3) in order of decreasing reliability. As described in Sect. \[can\], we have now used the SZ spectral energy distribution (SED) to refine the quality assessment of the cluster candidates by adopting a new, novel quality flag derived from the Artificial Neural Network analysis developed by @agh14.
Redshift updates for *previously-known* clusters {#asso}
================================================
In the external validation process performed in PXXIX2013, a total of 683 PSZ1 sources were associated with clusters published in X-ray, optical, or SZ catalogues, or with clusters found in the NED or SIMBAD databases. We refer to these as [*previously-known clusters*]{}. Their redshifts, when available, were compiled from the literature and a consolidated value was provided with the PSZ1 catalogue. In the present update, we first re-examine the [ *previously-known*]{} clusters of the PSZ1 catalogue.
The dedicated follow-up of PSZ1 clusters with RTT150 described in @rtt150 provided updates to the redshifts of 19 [*previously-known*]{} clusters. The follow-up of PSZ1 clusters with ENO telescopes further updated the redshifts of five [*previously-known*]{} clusters.\
We have updated the redshifts of ten PSZ1 sources associated with SPT clusters provided in @ble14. Finally, we have queried the NED and SIMBAD databases, and searched in the cluster catalogues constructed from the SDSS data (namely @wen12 and @roz14a), for additional spectroscopic redshifts. When these were available, we report them in the updated version of the PSZ1 catalogue. The full process led us to change the redshifts of 34 [*previously-known*]{} PSZ1 clusters. We have also changed the published photometric redshift estimate of one ACT cluster (ACT-CL J0559-5249) to a spectroscopic redshift value.
In summary, 69 sources from the PSZ1 catalogue associated with [ *previously-known*]{} clusters now have updated redshifts. Most of these consist of updates from photometric to spectroscopic values; however, eight redshifts were measured for the first time for [*previously-known*]{} clusters.
[*Planck*]{}-discovered clusters
================================
The PSZ1 catalogue contained 366 cluster candidates, classified as [*class-*]{}1 to 3 in order of decreasing reliability, and 178 [*Planck*]{}-discovered clusters confirmed mostly with dedicated follow-up programmes undertaken by the [*Planck*]{} Collaboration. Since the delivery of the PSZ1 catalogue in March 2013, a number of additional confirmations, including results from the community, were performed and redshifts were updated from photometric estimates to spectroscopic values.
Combining the results from follow-up with the RTT150 @rtt150, ENO telescopes (Planck collaboration 2015, in prep.), @liu14, @roz14a, and @ble14, a total of 86 PSZ1 sources have been newly confirmed as [ *Planck-*]{}discovered clusters with measured redshifts.
From RTT150 results {#rtt}
-------------------
As part of the Collaboration optical follow-up programme, candidates were observed with the Russian Turkish Telescope [RTT150[^2], @rtt150] within the Russian quota of observational time, provided by Kazan Federal University and Space Research Institute (IKI, Moscow). Direct images and spectroscopic redshift measurements were obtained using TÜBİTAK Faint Object Spectrograph and Camera (TFOSC[^3]). For the highest-redshift clusters, complementary spectroscopic observations were performed with the BTA 6-m telescope of the SAO RAS using the SCORPIO focal reducer and spectrometer [@afa05].
These observations have confirmed and measured redshifts for a total of 24 new candidates. Eleven of these have spectroscopic redshifts. We have updated the PSZ1 catalogue by including these newly-measured redshifts.
From ENO telescopes
-------------------
Also as part of the Collaboration optical follow-up programme, candidates were observed at European Northern Observatory (ENO[^4]) telescopes, both in imaging (at IAC80, INT and WHT) and spectroscopy (at NOT, GTC, INT and TNG). The observations were obtained as part of proposals for the Spanish CAT time, and an [*International Time Programme (ITP)*]{}, accepted by the International Scientific Committee of the Roque de los Muchachos and Teide observatories. We summarise here the main results of these observing programmes. Further details will be presented in a companion article ( Collaboration 2015, in prep.).
These observations have confirmed and provided new redshifts for a total of 26 candidates, that are reported in the updated PSZ1 catalogue. These include the confirmation of 12 SZ sources as newly-discovered clusters: two [*class*]{}-1, high reliability candidates, five [*class*]{}-2, and five [*class*]{}-3 candidates.
From PanSTARRS {#pans}
--------------
Based on the Panoramic Survey Telescope & Rapid Response System (PanSTARRS, @kai02) data, @liu14 have searched for optical confirmation of the 237 SZ detections that overlap the PanSTARRS footprint.
We only report here the redshifts for unambiguously confirmed clusters. Of these, 15 objects were included in the RTT150 follow-up, for which the redshifts are published in @rtt150, and three objects were included in the ESO follow-up described above. In these cases, we report the Collaboration follow-up redshift values in the updated PSZ1 catalogue. An additional two [*Planck*]{} clusters confirmed by PanSTARRS have a counterpart in the @roz14a catalogue, with spectroscopic redshifts that we update in the PSZ1 catalogue.
A total of 40 [*Planck-discovered*]{} clusters are confirmed, for the first time, by @liu14 in the PanSTARRS survey. All of these have measured photometric redshifts that we have reported in the updated PSZ1 catalogue.
 
From SPT {#spt}
--------
A new catalogue of SZ clusters detected with the South Pole Telescope (SPT) cluster catalogue was published in @ble14. It provides an ensemble of spectroscopic and photometric redshifts. Four candidate [*class-*]{}1 and 2 clusters from the PSZ1 catalogue were confirmed and have photometric redshifts in @ble14. These are included in the updated PSZ1 catalogue.
From SDSS-RedMapper catalogue {#spt}
-----------------------------
Comparison with the SDSS-based catalogue from @roz14a provided confirmation and new redshift values for five [*Planck-discovered*]{} clusters. This includes confirmation of two [*Planck*]{} cluster candidates (one [*class-*]{}2 and one [*class-*]{}3 candidate). We use the spectroscopic redshift values available in the @roz14a in the updated PSZ1 catalogue.
![[Percentage of origin and type (photometric, spectroscopic) of the redshifts reported in PSZ1. To date associations with MCXC clusters provide 49.8% of the redshifts, all spectroscopic. Follow up observations by the collaboration (FUs) provide 24.6% of the redshifts, of which 64.73% are spectroscopic. Associations with clusters from SDSS-based catalogues result in 11.7% of all redshifts, of which 58.9% are spectroscopic. Redshifts from the NED and SIMBAD databases represent 5.9% of all redshifts, with 90.7% of them spectroscopic. The confirmation from PanSTARRS data provides 4.4% of the total number of redshifts, all of them photometric. Finally the association with SZ catalogues (SPT and ACT) represents 3.5% of all redshifts, of which 71.9% are spectroscopic. ]{}[]{data-label="fig:zdist"}](zredshifts.pdf){width="8.8cm"}
Mass estimate {#masses}
=============
The size–flux degeneracy discussed in, e.g., @planck2011-5.1a and PXXIX2013 can be broken when $z$ is known, using the relation between $\tv$ and $\YSZ$ see [@arn14]. The $\YSZ$ parameter, denoted $Y_z$, is derived from the intersection of the relation and the size–flux degeneracy curve. It is the SZ mass proxy $Y_z$ that is equivalent to the X-ray mass proxy $\YX$.
For all the clusters with measured redshifts, $Y_z$ was computed for our assumed cosmology, allowing us to derive an homogeneously-defined SZ mass proxy, denoted $M_{500}^{Y_z}$, based on X-ray calibration of the scaling relations (see discussion in PXXIX2013). We show in Fig. \[fig:z\_hist\] (right panel, in red) the distribution of masses obtained from the SZ-based mass proxy for all clusters with measured redshift. Note that since we use an X-ray calibration of the scaling relations, these masses are uncorrected for any bias due to the assumption of hydrostatic equilibrium in the X-ray mass analysis. The shaded black area shows the distribution of masses for clusters with redshifts higher than 0.5. They represent a total of 78 clusters.
Cluster candidates {#can}
==================
Since the delivery of the [*Planck*]{} catalogue and the confirmation in this *addendum* of 86 candidates as new clusters, the updated PSZ1 catalogue now contains 280 cluster candidates. In the original PSZ1, these latter were classified as [*class-*]{}1 to 3 in order of decreasing reliability; the reliability being defined empirically from the combination of internal [*Planck*]{} quality assessment and ancillary information (e.g., searches in RASS, WISE, SDSS data). The updated PSZ1 catalogue contains 24 high quality ([*class-*]{}1) SZ detections whereas lower reliability [*class*]{}-2 and 3 candidates represent 130 and 126 SZ sources, respectively.
With the updated PSZ1 catalogue, we now provide a new objective quality assessment of the SZ sources derived from an artificial neural-network analysis. The construction, training and validation of the network is based on the analysis of the Spectral Energy Distribution (SED) of the SZ signal in the [*Planck*]{} channels. The implementation is discussed in detail by @agh14. The neural network was trained with an ensemble of three samples: the confirmed clusters in the PSZ1 calatogue representing good/high-quality SZ signal; the [*Planck*]{} Catalogue of Compact Sources source, representing the IR and radio-source induced detections; and random positions on the sky as examples of noise-induced, very low reliability, detections.
In practice, we provide for each SZ source of the updated PSZ1 catalogue a neural-network quality flag, $Q_N$, defined as in @agh14. This flag separates the high quality SZ detections from the low quality sources such that $Q_N< 0.4$ identifies low-reliability SZ sources with a high degree of success. Figure \[fig:zqual\] summarises for each class of [*Planck*]{} cluster candidate the number of sources below and above the threshold velue of $Q_N=0.4$. The [*class-*]{}1 cluster candidates all have $Q_N>0.4$ except for one source for which $Q_N=0.39$. The fraction of ‘good’ $Q_N>0.4$ SZ detections in the [*class-*]{}2 category is about 80%, while the fraction of $Q_N>0.4$ candidates drops to about 30% for the [*class-*]{}3 cluster-candidates.
![[Number of [*Planck*]{} cluster-candidates below and above the neural-network quality flag threshold $Q_N=0.4$, denoting a high-quality SZ detection, for each reliability class.]{}[]{data-label="fig:zqual"}](zqual.pdf){width="8.8cm"}
Summary
=======
-3mm =
We have updated the catalogue of SZ-selected sources detected in the first 15.5 months of observations. The catalogue contains 1227 detections and was validated using external X-ray and optical/NIR data, alongside a multi-frequency follow-up programme for confirmation.
The updated PSZ1 catalogue now contains 947 confirmed clusters, including 264 brand-new clusters, of which 214 have been confirmed by the Collaboration’s follow-up programme. The remaining 280 cluster candidates have been divided into three classes according to their reliability, i.e., the quality of evidence that they are likely to be [*bona fide*]{} clusters. To date, high quality SZ detections in PSZ1 represent 24 sources, all of which are classified as high-quality by our neural-network quality assessment procedure. Lower reliability, [*class*]{}-2 and 3 candidates represent 130 and 126 SZ sources respectively (Table \[tab:summ\]). We find that $\sim 80$% of the [*class*]{}-2 candidates are classified as high-quality by our neural-network quality assessment procedure, whereas only 35% of the [*class*]{}-3 sources are considered as high-quality SZ detections. Based on this assessement, the purity of the updated PSZ1 catalogue is $\sim 94\%$.
A total of 913 clusters (i.e., 74.2% of all SZ detections) now have measured redshifts, of which 736 are spectroscopic values (i.e., 80.6% of all redshifts). The left-hand panel of Fig. \[fig:z\_hist\] shows the distribution in redshift of all clusters (red), and for the clusters with masses above $5 \times 10^{14} M_{\sun}$ (shaded black). The median redshift of the PSZ1 catalogue is about 0.23, and of order 35% of the clusters lie at redshifts above $z=0.3$.
The origins and types of redshifts are shown in Fig. \[fig:zdist\]. Association with MCXC clusters [@pif11] provides about $ 49.8\%$ of the redshifts, all of which are spectroscopic. Follow up observations undertaken by the Collaboration provide $ 24.6\%$ of the redshifts, about two thirds of them being spectroscopic. SDSS-based catalogues yield $ 11.7\%$ of the redshifts, of which more than half of which are spectroscopic. NED and SIMBAD database searches yield 5.9% of the redshifts, the vast majority of which are spectroscopic. PanSTARRS data provide $ 4.4\%$ of the redshifts, all of which are photometric. Finally, association with the SPT and ACT SZ catalogues represent $\sim 3.5\%$ of all redshifts, most of which are spectroscopic.
For the clusters with measured redshifts, we have provided a homogeneously-defined mass estimated from the Compton $Y$ parameter. The $M$–$z$ distribution of the clusters is shown by open red circles in Fig. \[fig:mz\], where it is compared with other large cluster surveys. Note that the masses are not homogenised and some clusters may appear several times due to differences in the mass estimation methods between surveys. We see that cluster distribution probes a unique region in the $M$–$z$ space occupied by massive, $M\ge5\times 10^{14}\,\msol$, high-redshift ($z\ge 0.5$) clusters. The detections almost double the number of massive clusters above redshift 0.5 with respect to other surveys.
The development of has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). The authors thank N. Schartel, ESA [*XMM-Newton*]{} project scientist, for granting the DDT used for confirmation of SZ candidates. The authors thank TUBITAK, IKI, KFU and AST for support in using RTT150; in particular we thank KFU and IKI for providing significant amount of their observing time at RTT150. We also thank BTA 6-m telescope TAC for support of optical follow-up project. The authors acknowledge the use of the INT and WHT telescopes operated on the island of La Palma by the Isaac Newton Group of Telescopes at the Spanish Observatorio del Roque de los Muchachos of the IAC; the NOT, operated on La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, at the Spanish Observatorio del Roque de los Muchachos; the TNG, operated on La Palma by the Fundacion Galileo Galilei of the INAF at the Spanish Observatorio del Roque de los Muchachos; the GTC telescope, operated on La Palma by the IAC at the Spanish Observatorio del Roque de los Muchachos; and the IAC80 telescope operated on the island of Tenerife by the IAC at the Spanish Observatorio del Teide. Part of this research has been carried out with telescope time awarded by the CCI International Time Programme. The authors thank the TAC of the MPG/ESO-2.2m telescope for support of optical follow-up with WFI under [*Max Planck*]{} time. Observations were also conducted with ESO NTT at the La Silla Paranal Observatory. This research has made use of SDSS-III data. Funding for SDSS-III <http://www.sdss3.org/> has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and DoE. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration.\
This research has made use of the following databases: the NED and IRSA databases, operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the NASA; SIMBAD, operated at CDS, Strasbourg, France; SZ cluster database <http://szcluster-db.ias.u-psud.fr> and SZ repository operated by IDOC operated by IAS under contract with CNES and CNRS.
Description of the updated PSZ1 catalogue
=========================================
The updated catalogue of SZ sources is available at PLA[^5] and SZ cluster database[^6].
The updated PSZ1 gathers in a single Table all the entries of the delivered catalogue based mainly on the data and of the external validation information, based on ancillary data (Appendices B and C of @planck2013-p05a respectively). It also contains additional entries. It is provided in a fits format, together with a readme file.\
The updated catalogue contains, when available, cluster external identifications[^7] and consolidated redshifts. We added two new entries namely the redshift type and the bibliographic reference. The three entries associated with the consolidated redshift reported in the catalogue are thus:
Type of redshift: a string providing the different cases.
[undef]{}: undefined
[estim]{}: estimated from galaxies magnitudes
[phot]{}: photometric redshift
[spec]{}: spectroscopic redshifts
Source for redshift: an integer value representing the origin of the redshifts.
[-1:]{} No redshift available
[1:]{} MCXC updated compilation
[2]{}: Databases NED and SIMBAD-CDS
[3:]{} SDSS cluster catalogue from @wen12
[4:]{} SDSS cluster catalogue from [@sza11]
[5:]{} SPT
[6:]{} ACT
[7:]{} Search in SDSS galaxy catalogue from Collab. from Fromenteau PhD 2010 and Fromenteau et al. (private comm.)
[8:]{} SDSS catalogue from @roz14a
[10:]{} Pan-STARRS1 Survey confirmation
[20:]{} XMM-Newton confirmation from Collab.
[50:]{} ENO confirmation from Collab.
[60:]{} WFI-imaging confirmation from Collab.
[65:]{} NTT-spectroscopic confirmation from Planck Collab.
[500:]{} RTT confirmation from Collab.
[600:]{} NOT confirmation from Collab.
[650:]{} GEMINI-spectroscopic confirmation from Collab.
Bibliographical references for the redshift
We also added a new entry describing further the quality of the SZ detection. This is the flag $Q_N$ derived from the artificial neural network SED-based quality assesssment described in [@agh14].
[^1]: (<http://www.esa.int/Planck>) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark.
[^2]: <http://hea.iki.rssi.ru/rtt150/en/index.php>.
[^3]: <http://hea.iki.rssi.ru/rtt150/en/>\
[index.php?page=tfosc](index.php?page=tfosc).
[^4]: ENO: <http://www.iac.es/eno.php?lang=en>.
[^5]: <http://www.sciops.esa.int/index.php?page=>\
[Planck\_Legacy\_Archive&project=planck](Planck_Legacy_Archive&project=planck)
[^6]: <http://szcluster-db.ias.u-psud.fr>
[^7]: The external identification corresponds to the first identifier as defined in the external validation hierarchy adopted in @planck2013-p05a.
|
---
abstract: 'The physical Hamiltonian of a gravity-matter system depends on the choice of time, with the vacuum naturally identified as its ground state. We study the expanding universe with scalar field in the volume time gauge. We show that the vacuum energy density computed from the resulting Hamiltonian is a non-linear function of the cosmological constant and time. This result provides a new perspective on the relation between time, the cosmological constant, and vacuum energy.'
author:
- Viqar Husain
- Babar Qureshi
bibliography:
- 'E-Lambda.bib'
title: |
Ground state of the Universe and the cosmological constant.\
A nonperturbative analysis
---
One of the outstanding problems in fundamental physics is that of the cosmological constant (CC). The problem arises in the context of quantum field theory (QFT) on a fixed background spacetime, which is usually taken to be flat [@Weinberg:1988cp; @Carroll:2000fy; @Rugh:2000ji; @Burgess:2013ara], or otherwise has a high degree of symmetry. The symmetry includes a global notion of time specified as a timelike Killing vector field. The dynamics of the gravitational field is included only in so far as it is viewed as a spin two field on the specified background; back reaction of quantum fields on spacetime is typically excluded.
QFT on a fixed background spacetime may be viewed as the leading order term coming from the semi-classical approximation defined by the equation G\_[ab]{} + g\_[ab]{} = 8G | \_[ab]{}(, g) | , \[semicl\] where $\Lambda$ is the (bare) cosmological constant. As written, this hybrid classical-quantum equation is ambiguous. To make it more precise we require (i) a quantization of the matter field $\phi$ on a general background $g_{ab}$, (ii) a suitably regularized self-adjoint operator $\hat{T}_{ab}$, and lastly (iii) computation of the expectation value of $\hat{T}_{ab}$ in some choice of matter vacuum state $|\psi\rangle$. This would give the tensor \_[ab]{}\^(g) | \_[ab]{}(, g) | , as the effective stress-energy tensor associated to the state $|\psi\rangle$, and hence a precise meaning for the r.h.s. of eqn. (\[semicl\]). One can then proceed to solve this equation for the “semi-classical" metric $g_{ab}$.
Although there is a large literature [@Parker:2009uva] on computations of the r.h.s for a given spacetime, the calculation of a semiclassical metric has not been carried to satisfactory completion, even for spacetimes with isometries. In fact the equation itself has been questioned [@Page:1981aj]. Nevertheless, an attempt to produce a self-consistent solution by expanding the metric and state as g\_[ab]{} &=& \_[ab]{} + h\^[(1)]{}\_[ab]{} + \^2 h\^[(2)]{} +\
|&=& |0+ |\^[(1)]{} + \^2 |\^[(2)]{} + ($\epsilon = m/m_P$) leads to $0th.$ order to \_[ab]{} = 8G 0| T\_[ab]{} | 0. This equation forms the basis of the connection between vacuum energy density $\rho_{vac}$ and $\Lambda$, specifically the broadly accepted linear relationship \_[vac]{} = . It leads to the cosmological constant problem via the elementary evaluation \_[vac]{}= = \_0\^[k\_p]{} (k) = k\_p\^4, where $k_p$ is a Planck scale cutoff. This huge quantity is often compared to the observed WMAP value = 1.27 0.07 10\^[-56]{} \^[-2]{} \[wmapL\] $( \sim 3.2 \times 10^{-122}\ l_P^{-2})$ as a significant failure of theory.
A more sophisticated argument presents this issue as a problem coming from running scales in the theory. Assuming a fixed background that defines energy $k$, the regulated vacuum energy density computed from $\langle 0| \hat{T}_{ab}|0\rangle$ is expected (on dimensional grounds) to be of the form \_[vac]{} = M\^4 f(k; g\_1,g\_2,) = . where $f$ is a function of energy scale $k$, matter coupling constants $g_1,g_2\cdots$, and some natural mass scale $M(k)$. The first equality comes from field theory, and the second from semiclassical general relativity. (This expression assumes the usual linear dependence of energy density on $\Lambda$, and is observer 4-velocity $v^a$ dependent: $\displaystyle \rho = v^a v^b\langle 0| \hat{T}_{ab}|0\rangle$, unless there is a preferred timelike vector field specified by a spacetime isometry.) In this setting there are two ways to state the CC problem: (i) it arises from the first equality due to the factor $M^4$ which gives a very large energy density even well below the Planck energy, for example for proton mass or $\Lambda_{QCD}$, or (ii) it arises from the second equality as a fine tuning problem; at low energies (1 meter to a few astronomical units) where $G$ and $\Lambda$ are observed to be constant, the corresponding dimensionless parameters flow canonically as $\lambda(k) = \Lambda /k^2$ and $g(k) = G k^2$. Thus the low energy renormalization group trajectory must be a hyperbola $\lambda(k) g(k)=$ constant, which reflects a fine tuning of the initial conditions for the flow [@Reuter:2004nx].
The field theory problem may be due to the fact that the function $f$ is usually computed in perturbation theory. A counterpoint is provided by a recent non-perturbative calculation in the Gross-Neveu model, which suggests that, non-perturbatively, $f$ is a non-analytic function of the coupling constant that suppresses $\rho_{vac}$ at low energy [@Holland:2013xya].
We question the basis of formulating the CC problem to first order in the semiclassical setting, and argue that in a non-perturbative quantum approach in which gravitational degrees of freedom are treated as a part of the dynamics, either the problem does not arise, or that its manifestation is substantially different from that coming from the usual arguments.
We take the view that to meaningfully talk about a vacuum, we need a physical Hamiltonian for the full gravity-matter system. This in turn requires a global notion of time in the context of a generally covariant theory. Hence [*there is a connection between non-perturbative vacuum energy, the cosmological constant, and a global time variable.* ]{} However as there is no “solution to the problem of time" in quantum gravity, one might impose a plausible time gauge, or use some other suitably defined “relational time." We will use geometry degrees of freedom to fix time gauge and derive the corresponding physical Hamiltonian. The spectrum of the corresponding operator then gives a formula for the vacuum energy density.
The suggestion that quantum gravity might play a role in its resolution is not new; see eg. [@Witten:2000zk] in the context of string theory, [@Husain:2009cf] in the Hamiltonian context which is developed further here, and a semiclassical approach using Regge calculus [@Mikovic:2014opa].
With this summary and context, we begin with the 3+1 Arnowitt-Deser-Misner (ADM) Hamiltonian for Einstein gravity and minimally coupled to a massive scalar field S = d\^3x dt ( \^[ab]{}\_[ab]{} + P\_ - NH -N\^a C\_a ), where $(q_{ab},\pi^{ab})$ and $(\phi,P_\phi)$ are the ADM gravitational and scalar field phase space variables, $N, N^a$ are the lapses and shift variables, and H &=& ( \^[ab]{}\_[ab]{} - \^2 ) + ( - R) + H\_\[Hcons\]\
C\_a &=& D\_b\^b\_[ a]{} + P\_\_a,\
H\_&=& ( + q\^[ab]{} \_a\_b+ m\^2 \^2). are respectively the Hamiltonian and diffeomorphism constraints, and the scalar field Hamiltonian density. (We work in geometric units where $G=\hbar=c=1$, and reintroduce these constants in the final result.)
From this starting point, our goal is to calculate the vacuum energy density of the scalar field $\rho_{vac}(\Lambda,m)$ derived from the physical Hamiltonian associated to the volume time gauge in a cosmological setting. We do this first in the homogeneous (zero mode) setting to illustrate the argument, and subsequently generalize it to include all matter modes.
The flat homogeneous model is derived by the parametrization q\_[ab]{} = a\^2 e\_[ab]{}, \^[ab]{} = e\^[ab]{} where $e_{ab}=\text{diag}(1,1,1)$. Substituting this into the constraints and ADM action gives the reduced theory S = V\_0 dt (p\_a + P\_- N H ), \[red-act\] where H = - + a\^3+ ( + a\^3 m\^2 \^2).\[frwH\] The last equation is obtained from substituting the reduction ansatz into the Hamiltonain constraint (\[Hcons\]), and $V_0$ is an unphysical coordinate volume. The reduced action is invariant under the scale transformations (V\_0, a, p\_a, ,P\_) (\^3 V\_0, , , , ) \[scale\]
At this stage we fix “physical volume time" gauge [@Hassan:2014sja] by setting t = d\^3x = V\_0a\^3. \[tgauge\] We note that this is both scale invariant (\[scale\]) and second class with the Hamiltonian constraint, as required of an adequate gauge fixing. It is also a physically natural time in the context of an expanding cosmology. Although we do not require it here, the lapse function corresponding to this canonical time gauge is given by the requirement that the gauge be preserved under evolution. This gives 1= {V\_0 a\^3, NH } = - N = -. We note that this lapse is invariant under the transformation (\[scale\]), as it should be.
This gauge condition, together with the solution of the Hamiltonian constraint, eliminate the variables $(a,p_a)$, leaving a theory for the scalar field variables evolving with respect to this time. The gauge fixed canonical action is obtained by substituting (\[tgauge\]) and the solution of the Hamiltonian constraint p\_a\^2 = 24 \_[|V\_0a\^3 =t]{} into the action (\[red-act\]). We choose the root that gives positive energy density.
It is useful to write the gauge fixed action using the scale invariant variables $p_\phi := V_0P_\phi$ and $t$. This gives S\^[GF]{} = dt ( p\_- H\_P ), where H\_P= \[Hp\] The energy density derived from this Hamiltonian is = = , \[rho\] since $V_0a^3$ is the physical volume (which is also the chosen time gauge). We note that this physical quantity does not depend on $V_0$.
To find the eigenvalues of this density operator we recall that for any operator $\hat{A}$ with a positive spectrum $a_n$, the spectrum of the square root operator $\sqrt{\hat{A}}$, is $\sqrt{a_n}$. In our case the argument of the square root in (\[Hp\]) is a shifted harmonic oscillator with time dependent mass and frequency. Therefore we can solve the eigenvalue problem for the density operator $\hat{\rho}$ = \_n by treating $t$ as a parameter. This gives the exact spectrum \_n = , \[rhon\] where $n=0, 1, \cdots $, and we have reintroduced the Planck mass, with $\Lambda, m,$ and $t$ specified in Planck units.
Let us note that this energy density operator may also be used to set up the time dependent Schrodinger equation, specify an initial state, such as the $n=0$ state, and evolve it to the present time. In general, such an evolved state may be approximated by a finite linear combination of the instantaneous energy eigenbasis $|\psi_n\rangle $ of the density operator, |(t)= \_[n=0]{}\^N c\_n(t) |\_n(t). Now for our purpose, which is to obtain a relationship between energy density and cosmological constant, we would need to evaluate the expectation value of the density operator in this state (t) | (t) |(t)= \_[n=0]{}\^N c\_n(t) \_n(t). \[state\] However this is not necessary to make the central point of the paper, as we now show.
The expression gives a non-perturbative quantum energy density of the scalar field with respect to the volume time gauge (\[tgauge\]). The eigenvalue $\rho_n(t)$ in this formula has some interesting features: (i) it depends only on variables invariant under the scale transformations (\[scale\]), (ii) there is a square root arising from the fact that all terms in the Hamiltonian constraint are quadratic in momenta, (iii) the energy density is not linear in $\Lambda$, (iv) there is a time factor suppression which for large times gives \_[vac]{} m\_P\^2 , independent of $n$.
These features are not what are expected from the usual flat space arguments for matter vacuum energy density, where this density is linear in $\Lambda$ and time independent. The last formula may be viewed as a prediction for the (zero mode) quantum vacuum energy density of the scalar field in an FRW universe, since this factor comes out of the sum (\[state\]) for late times. (We note that at each $t$ the state lives in the instantaneous Hilbert space at that time, so the remaining sum adds to unity. )
A numerical estimate of $\rho_{vac}$ using known cosmological parameters may be computed using the measured WMAP value for $\Lambda$ in eqn. (\[wmapL\]) and the present age of the universe $t = 10^{61}t_P$, ($t_P$ = Planck time). This gives \_[vac]{} \~510\^[-129]{} \_[P]{} = 2.510\^[-32]{} /\^3, where $\rho_P =m_P/l_P^3$ is the Planck density. (We note that experiments such as WMAP measure cosmological model parameters such as $\Lambda$; implications for vacuum energy density are then derived from theoretical models. That is, there is no direct measurement of the energy density in a box of empty space.)
In summary to this point, we have seen that the time dependence in (\[rhon\]) has its origin in factors of $\int d^3x \sqrt{q}=V_0a^3 =t$; the overall factor $1/t$ comes from converting the Hamiltonian scalar density (of weight one) to a scalar, and the factor in the oscillator frequency comes from the $\sqrt{q}$ terms in the matter Hamiltonian.
The semiclassical calculation of energy density is via $\rho = \langle 0|T_{ab} -\Lambda g_{ab} |0\rangle v^av^b \equiv \rho_\phi + \rho_\Lambda$ for an observer with four velocity $v^a$. How is this to be compared with our result eqn. (\[rhon\])? It is clear that the latter is additive in the contributions from matter and $\Lambda$, whereas our result (\[rhon\]) is not. It shows that imposing a time gauge, solving the Hamiltonian constraint, and then diagonalizing the resulting physical Hamiltonian is an entirely different process from QFT on a fixed background, and yields substantially different results.
The setting we have discussed so far is obviously limited without a field theory extension to include all matter modes. This requires inclusion of inhomogeneities in the matter and metric degrees of freedom. We now turn to this. We will see that the main features of the energy density formula (\[rhon\]) – explicit time dependence and the square root – remain unaltered.
We follow a hamiltonian approach similar to that developed in [@Langlois:1994ec], where the scalar field and metric perturbations are expanded in Fourier modes, and the Hamiltonian constraint is treated to second order in the perturbations. The resulting theory describes the dynamics of the gravity phase space variables $(a,p_a)$, and the scalar field and metric perturbation Fourier mode pairs $(\phi_k, p_k)$ and $(\delta q_{ab}^k, \delta \pi^{ab}_k)$; the mode decomposition is defined using the global chart on homogeneous space slices: ([**x**]{},t) &=& \_[**k**]{} \_[**k**]{} (t) e\^[ i [**k**]{} ]{}\
P\_( [**x**]{}, t) &=& \_[**k**]{} P\_[**k**]{} (t) e\^[i[**k**]{}]{}. This gives H\_= V\_0\_[**k**]{} ( + \_[**k**]{}\^2 + \_[**k**]{}\^2 ) \[Hk\] after a suitable mode relabelling. The Fourier modes so defined satisfy the equal time Poisson bracket {\_[**k**]{}(t), P\_[**k’**]{}(t) }= \_[[**k**]{},[**k**]{}’ ]{} . With this decomposition we define a Hamiltonian system by the phase space variables $(a,p_a)$ and $(\phi_{\bf k}, P_{\bf k})$ and action S = V\_0 dt(p\_a + \_[**k**]{} \_[**k**]{} P\_[**k**]{} - N H ). \[kact\] where H - + a\^3+ |[H]{}\_=0, with $\bar{H}_\phi = H_\phi/V_0$ from (\[Hk\]). This Hamiltonian constraint generalizes (\[frwH\]) to include an infinite number of degrees of freedom.
This system is not exactly that obtained from metric and matter perturbations in [@Langlois:1994ec]; in particular it does not include the spatial diffeomorphism constraint, which would impose further conditions between the matter and gravity modes. Nevertheless it is a consistent model that has the main features of interest for our purpose, which is to investigate the vacuum energy density of a matter-gravity system with an infinite number of degrees of freedom. We note also that the action (\[kact\]) has the scaling invariance (\[scale\]) with $(\phi, P_\phi)$ replaced by their Fourier modes $(\phi_{\bf k}, P_{\bf k})$.
Proceeding as for the homogeneous case, let us fix the (scale invariant) time gauge (\[tgauge\]) and solve the Hamiltonian constraint to eliminate $(a, p_a)$. Using the scale invariant momentum $p_{\bf k } \equiv V_0 P_{\bf k}$, the gauge fixed action for the matter modes is S\^[GF]{} = dt \_[k]{} ( \_[**k**]{} p\_[**k**]{} - H\_P ), H\_P = . where $\bar{\bf k} = {\bf k} V_0^{\frac{1}{3}}$ is the (scale invariant) wave vector. Upon quantization the corresponding operator has the spectrum
E\_n =,\[En\] \_[**k**]{}(t) = . \[freq\]
To find the vacuum energy density of the matter modes we again set $n=0$, and consider the massless case $m=0$ for simplicity. The $\sum_{\bf k}$ is a sum over comoving modes, which is evaluated by converting the sum to an integral in the usual way with a $k-$space volume $d^3k$: \_[**k**]{} \_[**k**]{} \_0\^[|[k]{}\_p]{} |[k]{}. Restoring factors of Planck mass, this gives for vacuum energy density the result \_[vac]{} = = , \[rho0-matter\] where $\bar{t}=t/t_P$. The overall factor is the same as that for the homogeneous case. The peculiar time factor multiplying the second term in the square root comes from the mode frequency (\[freq\]), which in turn has its origin in the scalar field gradient term $\displaystyle \sqrt{q}q^{ab}\p_a\phi\p_b\phi \rightarrow a^3 |{\bf k}|^2/a^2$.
It is apparent that the general features of the homogeneous case, the square root and explicit time dependence are still present. We may again compute a numerical estimate for the vacuum energy by substituting on the r.h.s. of (\[rho0-matter\]) the present age of the universe $\bar{t} = 10^{61}$ and the $\Lambda$ value from (\[wmapL\]). This gives \_[vac]{} \~\_P 10\^[-103]{} =510\^[-7]{} [Kg]{}/[m]{}\^3. This formula makes clear that the present vacuum energy density with the global choice of volume time is far smaller than the huge value from standard arguments. It shows that there is no cosmological constant problem.
Let us summarize our main result. We find for the non-perturbative matter-gravity system in the cosmological context that the physical Hamiltonian (i) is not a linear function of $\Lambda$, (ii) is explicitly time dependent, and (iii) yields the explicit formula (\[rho0-matter\]) for vacuum energy density. A numerical evaluation of this density shows that the vacuum energy problem is absent due to the time suppression factor. Beyond these details, our general argument reveals that there is an intimate connection between time, vacuum energy and the cosmological constant, which is revealed by extracting the physical Hamiltonian for a matter-gravity system is a physically reasonable time gauge. (For negative cosmological constant our results do not apply above a critical time value. This means that the volume time gauge does not provide a useful foliation.)
In closing we provide several comments on our approach, pointing out what we think are generic features and what are limitations which merit further work.
\(i) In FRW cosmology the matter energy density is identified as the right hand side of the Friedmann equation. This is fine for classical theory, but a non-perturbative quantum theory requires a physical Hamiltonian for the full matter-gravity system before one can talk about the true vacuum.
\(ii) Our approach does not address the question of why the observed cosmological constant is so small. But it does address the problem of the relation between vacuum energy density and the cosmological constant; this we show is time dependent and non linear.
\(iii) The functional form of the physical Hamiltonian, and hence the vacuum energy density is dependent on the time gauge. The square root and time dependent physical Hamiltonian are a common feature of canonical time gauge fixing. This is because the Hamiltonian constraint is quadratic in momenta for usual matter fields (see [@Husain:2011tk] for an unusual exception). As a result one ends up solving at least a quadratic equation for the momentum conjugate to the chosen time variable.
\(iv) Our results are derived in only one time gauge in the setting of FRW cosmology with perturbations. Although this is observationally relevant, for more general metrics it is not possible to use volume time because it does not provide a complete time gauge fixing. The general problem is more challenging. It requires fixing a suitable local matter or geometry scalar as time, and deriving the corresponding Hamiltonian density. The latter may not be a simple function, and the spectrum problem correspondingly difficult.
\(v) Beyond the homogeneous case, our development uses the fixed volume time gauge from the background to define the physical Hamiltonian of matter perturbations. The spectrum of this Hamiltonian provides only the energy part of the semiclassical equation (\[semicl\]). The pressures can be computed, and would come from analyzing the spatial diffeomorphism constraint $D_b\pi^{ab} = j^a(\phi)$ to leading order beyond the homogeneous approximation (where this constraint is trivially satisfied). This would be among the necessary steps for developing a canonical semiclassical approximation using our approach as a starting point.
\(vi) We used a Planck scale cutoff in deriving the vacuum energy density (\[rho0-matter\]). Our justification of this is the same as that in the usual treatment because the scalar perturbations are effectively being treated on the FRW background. That is, it is not yet full quantum gravity. But the novel feature in the formula, unlike the flat space case, is the time factor suppression of this term in (\[rho0-matter\]), which leaves the $\Lambda$ factor as the dominant one at late time.
\(vii) What becomes of the “low energy" CC problem in a small patch of spacetime where there is a local timelike Killing vector field? This local Minkowski time is obviously fine for short timescale particle physics during which the universe does not expand much. But our approach and results suggest that it is not useful to pose questions such as “does the vacuum gravitate" in a local flat patch of a cosmological spacetime.
The work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. We thank Jack Gegenberg, Tim Koslowski, Sanjeev Seahra and Jon Ziprick for discussion and comments on the manuscript.
|
---
abstract: 'We compute the hyperbolic covolume of the automorphism group of each even unimodular Lorentzian lattice. The result is obtained as a consequence of a previous work with Belolipetsky, which uses Prasad’s volume to compute the volumes of the smallest hyperbolic arithmetic orbifolds.'
address: |
Max Planck Institute for Mathematics\
Vivatsgasse 7\
53111 Bonn\
Germany
author:
- Vincent Emery
bibliography:
- 'unimod-lattices.bib'
title: Even unimodular Lorentzian lattices and hyperbolic volume
---
Introduction
============
Let ${\mathbf H}^n$ be the hyperbolic $n$-space, of constant curvature $-1$. We denote by $\operatorname{Isom}({\mathbf H}^n)$ the group of isometries of ${\mathbf H}^n$. One way to construct a lattice in $\operatorname{Isom}({\mathbf H}^n)$ is to consider the automorphism group $\operatorname{O}(L)$ of a Lorentzian lattice $L \subset {\mathbb R}^{n,1}$. Of particular interest are the unimodular Lorentzian lattices. There exist two such types of lattices: the odd unimodular Lorentzian lattice ${\mathrm{I}}_{n,1}$ and the even unimodular Lorentzian lattice ${\mathrm{II}}_{n,1}$. Their study appears in connection with the study of Euclidean lattices, as shown in the book of Conway and Sloane [@Conway-Sloane]. While ${\mathrm{I}}_{n,1}$ exists for every dimension $n$, the even lattice ${\mathrm{II}}_{n,1}$ exists only when $n \equiv 1
\mod 8$.
In [@BelEme] (see also [@EmePhD]) the following theorem was proved.
\[thm:smallest-volumes\] For each odd dimension $n = 2r-1 \ge 5$, there is a unique orientable non-compact arithmetic hyperbolic $n$-orbifold $\Delta_n {\backslash}{\mathbf H}^n$ of the smallest volume (with $\Delta_n$ an arithmetic lattice of $\operatorname{Isom}({\mathbf H}^n)$). Its volume is given by: $$\begin{aligned}
\frac{1}{2^{r-2}} \; \zeta(r) \; \prod_{j=1}^{r-1} \frac{(2j -1)!}{(2
\pi)^{2j}} \zeta(2j) &\qquad \mbox{if } n \equiv 1 \mod 8 ;\label{vol-min-II}\\
\frac{(2^r -1) (2^{r-1}
-1)}{3 \cdot 2^{r-1}} \; \zeta(r) \; \prod_{j=1}^{r-1}
\frac{(2j -1)!}{(2 \pi)^{2j}} \zeta(2j) &\qquad \mbox{if } n
\equiv 5 \mod 8; \label{vol-min-5}
\\
\frac{3^{r-1/2}}{2^{r-1}} \; L_{{\mathbb Q}(\sqrt{-3})|{\mathbb Q}}\!(r) \; \prod_{j=1}^{r-1}
\frac{(2j -1)!}{(2 \pi)^{2j}} \zeta(2j)
&\qquad \mbox{if } n \equiv 3 \mod 4. \label{vol-min-7}
\end{aligned}$$
It is remarkable that the smallest volume has the simplest form exactly for the dimensions $n$ where the even unimodular Lorentzian lattice ${\mathrm{II}}_{n,1}$ exists. The main purpose of this article is to show that for these $n$ the arithmetic group $\Delta_n$ of Theorem \[thm:smallest-volumes\] is actually given by the group $\operatorname{SO}({\mathrm{II}}_{n,1})$ of special automorphisms of ${\mathrm{II}}_{n,1}$ (cf. Theorem \[thm:identification\]). In particular, this allows to deduce in Corollary \[cor:volume-aut-even-unimod\] the hyperbolic covolume of the automorphism group $\operatorname{O}({\mathrm{II}}_{n,1})$. This complements the work of Ratcliffe and Tschantz [@RatTsch97], where the covolume of $\operatorname{O}({\mathrm{I}}_{n,1})$ was determined for every $n$.
In §\[sec:volume-Coxeter\]–\[sec:disussion-cusps\] we discuss some interesting consequences of our main result. Finally, in §\[sec:odd-unimod-sqrt3\] we discuss the case of Formulas –. In particular, we state in Proposition \[prop:odd-unimod-index-3\] the exact relation between $\Delta_n$ and $\operatorname{O}({\mathrm{I}}_{n,1})$ when $n \equiv 5 \mod 8$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank Curtis McMullen for the interesting discussions that are at the origin of this article. I thank Jiu-Kang Yu for his help concerning Bruhat-Tits theory, Steve Tschantz for the numerical computation mentioned in §\[sec:volume-Coxeter\], Anna Felikson and Pavel Tumarkin for helpful discussions, and Ruth Kellerhals, John Ratcliffe and the referee for helpful comments. I am thankful to the MPIM in Bonn for the hospitality and the financial support.
Main result and its proof {#sec:identification}
=========================
For $n \equiv 1 \mod 8$, we consider the even unimodular lattice ${\mathrm{II}}_{n,1}$ embedded in the real quadratic space equipped with the standard rational quadratic form: $$\begin{aligned}
q(x) &= -x_0^2 + x_1^2 + \cdots + x_n^2.
\label{eq:quad-form-stand}\end{aligned}$$ The group of automorphisms of this quadratic space acts then isometrically on ${\mathbf H}^n$, via an identification of ${\mathbf H}^n$ with its projective model. The group $\operatorname{O}({\mathrm{II}}_{n,1})$ (resp. $\operatorname{SO}({\mathrm{II}}_{n,1})$) of automorphisms (resp. special automorphisms) preserving $q$ and the lattice ${\mathrm{II}}_{n,1}$ acts discontinuously on ${\mathbf H}^n$. More precisely, the group $\operatorname{PO}({\mathrm{II}}_{n,1}) =
\operatorname{O}({\mathrm{II}}_{n,1})/\{\pm I \}$ (resp. $\operatorname{PSO}({\mathrm{II}}_{n,1}) =
\operatorname{SO}({\mathrm{II}}_{n,1})/\{\pm I \}$), where $I$ is the identity matrix, can be seen as a discrete subgroup of $\operatorname{Isom}({\mathbf H}^n)$.
\[thm:identification\] For $n \equiv 1 \mod 8$, the group $\Delta_n$ is conjugate in $\operatorname{Isom}({\mathbf H}^n)$ to $\operatorname{PSO}({\mathrm{II}}_{n,1})$.
We denote by $V$ the quadratic space over ${\mathbb Q}$ equipped with quadratic form $\frac{1}{2}q$, where $q$ is given in . Let ${\mathrm G}$ be the algebraic group defined over ${\mathbb Q}$ with ${\mathrm G}({\mathbb Q}) = \operatorname{Spin}(V)$, the group of spinors of $V$. Let ${\overline{{\mathrm G}}}$ be the adjoint form of ${\mathrm G}$. Then ${\overline{{\mathrm G}}}({\mathbb R})$ is isomorphic to $\operatorname{Isom}({\mathbf H}^n)$. For each prime $p$ we consider the quadratic space $V_p = V \otimes_{\mathbb Q}{\mathbb Q}_p$, and the Bruhat-Tits building ${\mathcal B}_p$ associated with $\operatorname{Spin}(V_p)$ and $\operatorname{SO}(V_p)$. Note that ${\mathrm G}$ and ${\overline{{\mathrm G}}}$ are split over ${\mathbb Q}_p$, for every prime $p$ (cf. [@BelEme Prop. 3.9]).
Let $L$ be the lattice in $V$ that identifies to ${\mathrm{II}}_{n,1}$ via the embedding in the quadratic space $(V \otimes_{\mathbb Q}{\mathbb R}, \;q)$. For each prime $p$, we consider the lattice $L_p = L \otimes
{\mathbb Z}_p$, which is a maximal lattice in $V_p$ [@BeloGan05 §5]. Bruhat-Tits theory allows to identify the lattice $L_p$ as an hyperspecial point of the building ${\mathcal B}_p$ (cf. [@BeloGan05 §5]), whose stabilizer in $\operatorname{SO}(V_p)$ is $\operatorname{SO}(L_p)$.
Let us denote by $K_p$ the hyperspecial parahoric subgroup of $\operatorname{Spin}(V_p)$ that stabilizes $L_p \in {\mathcal B}_p$. The set of all these $K_p$ for $p$ prime is a coherent collection of parahoric subgroups, and this defines a principal arithmetic subgroup of ${\mathrm G}({\mathbb Q})$ (see [@BelEme §2.2] for details): $$\begin{aligned}
\Lambda &= {\mathrm G}({\mathbb Q}) \cap \prod_p K_p\;,
\label{eq:princal-arithm}
\end{aligned}$$ which by construction maps into $\operatorname{SO}(L) = \operatorname{SO}({\mathrm{II}}_{n,1})$. But $\Lambda$ corresponds exactly to the group $\Lambda_1$ in [@BelEme], whose image in ${\overline{{\mathrm G}}}({\mathbb R})$ gives the group $\Delta_n$. It was proved in [@BelEme] that $\Lambda_1$ is maximal, and that up to conjugacy its construction does not depend on the choice of a coherent collection of hyperspecial subgroups. It follows that $\operatorname{PSO}({\mathrm{II}}_{n,1})$ is conjugate to $\Delta_n$ in $\operatorname{Isom}({\mathbf H}^n)$.
From Theorem \[thm:smallest-volumes\] and \[thm:identification\] we obtain the covolume of the group $\operatorname{PO}({\mathrm{II}}_{n,1})$, which contains $\operatorname{PSO}({\mathrm{II}}_{n,1})$ as a subgroup of index two. In order to simplify even more the volume formula, we use the well-known expression of $\zeta(2j)$ in terms of the Bernoulli number $B_{2j}$.
The covolume of the action of $\operatorname{PO}({\mathrm{II}}_{n,1})$ on ${\mathbf H}^n$ equals $$\begin{aligned}
\zeta(r) \prod_{j=1}^{r-1} \frac{\vert B_{2j}\vert}{8j} \;,
\label{vol-aut-even-unimod}
\end{aligned}$$ where $B_k$ is the $k$-th Bernoulli number. \[cor:volume-aut-even-unimod\]
Volume of Coxeter polytopes {#sec:volume-Coxeter}
===========================
Corollary \[cor:volume-aut-even-unimod\] was already known in dimension $n=9$ (see §\[sec:odd-unimod-sqrt3\]), where $\operatorname{PO}({\mathrm{II}}_{9,1})$ is the Coxeter group generated by reflections through the faces of a simplex. The only other group $\operatorname{PO}({\mathrm{II}}_{n,1})$ that is reflective is $\operatorname{PO}({\mathrm{II}}_{17,1})$, as it follows form the work of Conway and Vinberg (cf. [@Conway-Sloane Ch. 27] and [@Vinb93 Part II Ch. 6 §2.1]). It contains as a subgroup of index two the following Coxeter group: $$\label{eq:Coxeter-17}
\begin{split}
\xymatrix@C=14pt@R=12pt{
& & *={\bullet} \ar@{-}[d] & & & & & & & & & & & & *={\bullet} \ar@{-}[d] & & \\
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]&
*={\bullet} \ar@{-}[r]& *={\bullet}
}
\end{split}$$
\[cor:volume-Coxeter-gp\] Let $P \subset {\mathbf H}^{17}$ be a Coxeter polytope corresponding to the diagram . Then $$\begin{aligned}
{\mathrm{vol}_{\mathbf H}}(P) &= \frac{691 \cdot 3617}{2^{38}
\cdot 3^{10} \cdot 5^4 \cdot 7^2 \cdot 11 \cdot 13 \cdot
17} \; \zeta(9)
\\
&\approx 2.072451981 \cdot 10^{-18}.
\end{aligned}$$
It is not clear how one could compute precisely the volume of such an high- and odd-dimensional hyperbolic polytope without an identification with the fundamental domain of an arithmetic group. Steve Tschantz was able to compute the following numerical approximation, which agrees with the result of Corollary \[cor:volume-Coxeter-gp\]. $$\begin{aligned}
{\mathrm{vol}_{\mathbf H}}(P) &= 2.069 \cdot 10^{-18} \; \pm \; 2.4 \cdot 10^{-20}.
\label{eq:Tschantz}\end{aligned}$$ The computation took about 60 hours, showing that for this kind of polytopes even numerical computation is not an easy task.
It is most likely that $P$ realizes the smallest volume among all hyperbolic Coxeter polytopes (non-compact or not), independently of the dimension. The results of [@EmePhD; @BelEme] (odd dimensions) and [@Belo04] (even dimensions) determine the smallest possible arithmetic orientable hyperbolic orbifolds. From them we see that a Coxeter polytope smaller than $P$ and being the fundamental domain of an arithmetic group must necessarily lie in ${\mathbf H}^{17}$, be commensurable to $P$, and have exactly half of the volume of $P$. We don’t know if such a Coxeter polytope could exist.
The small size of $P$ can also be explained by Schläfli differential formula for the volume of polytopes (see [@Vinb93 Part I Ch. 7 §2.2]). According to this formula, the volume of Coxeter polytopes tends to be smaller for polytopes having large dihedral angles. The small size of $P$ results then from the combination of two factors: the only dihedral angles in $P$ are $\pi/2$ and $\pi/3$; and relatively to its dimension, $P$ is determined by few hyperplanes (actually the smallest possible number in ${\mathbf H}^{17}$). These two conditions are a very rare occurrence in high dimensions.
Comparison with the mass formula {#sec:disussion-cusps}
================================
The lattice ${\mathrm{II}}_{25,1}$ plays an important role in connection with the study of even unimodular Euclidean lattices in dimension $24$ (see [@Conway-Sloane Theorem 5, Ch. 26]). For $n \equiv 1 \mod 8$, let ${\mathcal{L}}_{n-1}$ denotes the set (up to isomorphism) of $(n-1)$-dimensional even unimodular Euclidean lattices. This is a finite set, and an important invariant is its *mass*, defined as $$\begin{aligned}
{\mathrm{mass}}({\mathcal{L}}_{n-1}) &= \sum_{L \in {\mathcal{L}}_{n-1}} \frac{1}{\vert
\operatorname{O}(L) \vert}\;.
\label{eq:mass-definition}\end{aligned}$$ For $n=9, 17$ and $25$ each group $\operatorname{O}(L)$ (with $L \in
{\mathcal{L}}_{n-1})$ appears as a subgroup of $\operatorname{O}({\mathrm{II}}_{n,1})$ as the stabilizer of a point at infinity of ${\mathbf H}^n$. Therefore, the groups $\operatorname{O}(L)$ correspond to cusps of the hyperbolic orbifold defined by $\operatorname{O}({\mathrm{II}}_{n,1})$, and ${\mathrm{mass}}({\mathcal{L}}_{n-1})$ could be regarded as a measurement of the contribution from these cusps to the volume. It is then quite natural to consider the ratio “covolume of $\operatorname{O}({\mathrm{II}}_{n,1})$ divided by ${\mathrm{mass}}({\mathcal{L}}_{n-1})$”. From the mass formula [@Conway-Sloane Theorem 2, Ch.16] we obtain the rather simple formula: $$\begin{aligned}
\frac{\mbox{covolume of } \operatorname{O}({\mathrm{II}}_{n,1})}{{\mathrm{mass}}({\mathcal{L}}_{n-1})} &=
2^{-r}
\frac{\vert B_{2r-2} \vert}{\vert B_{r-1} \vert}
\zeta(r).
\label{eq:ratio-volume-mass}\end{aligned}$$ Note that this ratio goes quickly to $\infty$ when $r$ grows.
We refer to [@StoverEnds] for more precise results on the behaviour of cusps of arithmetic orbifolds with respect to the dimension.
The case of the other odd dimensions {#sec:odd-unimod-sqrt3}
====================================
The covolume of $\operatorname{PO}({\mathrm{I}}_{n,1})$ was computed by Ratcliffe and Tschantz in all dimensions $n > 1$ [@RatTsch97]. They obtain the result by evaluating a formula due to Siegel. Note that Prasad’s volume formula, the main ingredient to obtain Theorem \[thm:smallest-volumes\], may be considered as a far-reaching extension of this formula of Siegel. Using the fact that $\operatorname{PO}({\mathrm{II}}_{n,1})$ and $\operatorname{PO}({\mathrm{I}}_{n,1})$ are commensurable, Ratcliffe and Tschantz could also deduce the covolume of $\operatorname{PO}({\mathrm{II}}_{9,1})$ (cf. [@Johnson-al-99 p. 345]). By the work of Vinberg and Kaplinskaya, the group $\operatorname{PO}({\mathrm{I}}_{n,1})$ is known to be reflective for $n \le 19$, and combining this fact with the work of Ratcliffe and Tschantz one can obtain the volume of several Coxeter polytopes.
By its construction in [@BelEme], it is clear that for $n \equiv 5
\mod 8$ the arithmetic group $\Delta_n$ of Theorem \[thm:smallest-volumes\] is commensurable to $\operatorname{PSO}({\mathrm{I}}_{n,1})$. Moreover, we can see that the ratio of the covolumes of these two groups is equal to $3$. In fact, using [@BeloGan05 Prop. 5.9] and the same kind of argument as in the proof of Theorem \[thm:identification\], we get the following result. It agrees with known facts about simplices in dimension $5$ (cf. [@Johnson-al-99 §5]).
\[prop:odd-unimod-index-3\] For $n \equiv 5 \mod 8$, the group $\operatorname{PSO}({\mathrm{I}}_{n,1})$ is conjugate in $\operatorname{Isom}({\mathbf H}^n)$ to a subgroup of index $3$ in $\Delta_n$.
For $n \equiv 3 \mod 4$, the group $\Delta_n$ is not commensurable to $\operatorname{PSO}({\mathrm{I}}_{n,1})$. Instead, it is commensurable to the group $\operatorname{PO}(f,{\mathbb Z})$ given by the integral automorphisms of the following quadratic form: $$\begin{aligned}
f &= -3 x_0^2 + x_1^2 + \cdots + x_n^2.
\label{eq:quadr-form-3}\end{aligned}$$ McLeod showed that the group $\operatorname{PO}(f,{\mathbb Z})$ is reflective when $n
\le 13$ [@McLeod11]. Recently, elaborating on their earlier work on Siegel’s formula (cf. [@Johnson-al-99 pp. 344–345]), Ratcliffe and Tschantz determined the covolume of $\operatorname{PO}(f,{\mathbb Z})$ (thus obtaining the covolumes of McLeod’s polytopes) [@RatTsch12]. For $n \equiv 3 \mod 4$, the ratio between the covolumes of $\operatorname{PO}(f,{\mathbb Z})$ and $\Delta_n$ is then computed to be equal to $a(n)/4$, where $a(n)$ is some odd integer tending to $\infty$ when $n \to \infty$ (see [@RatTsch12 (35)]). An alternative way to obtain the covolume of McLeod’s polytopes would be to determine the relation between $\operatorname{PO}(f,{\mathbb Z})$ and $\Delta_n$ in terms of subgroup inclusions, using the same kind of arguments as for Theorem \[thm:identification\] and Proposition \[prop:odd-unimod-index-3\].
|
---
abstract: |
Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to $1$, e.g. $G= {\mathrm{SL}}_2 ( {\mathbb{C}}) \times {\mathrm{SL}}_2 ({\mathbb{C}})$ or ${\mathrm{SL}}_3 ({\mathbb{C}})$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture made in [@BV], lattices in $G$ should have ‘little’ — in the very weak sense of ’subexponential in the co-volume’ — torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the *square root* of the volume.
This is deduced from a general theorem that compares twisted and untwisted $L^2$-torsions in the general base-change situation. This also makes uses of a precise equivariant ‘Cheeger-Müller Theorem’ proved by the second author [@Lip1].
address:
- |
Institut de Mathématiques de Jussieu\
Unité Mixte de Recherche 7586 du CNRS\
Université Pierre et Marie Curie\
4, place Jussieu 75252 Paris Cedex 05, France\
-
- |
Mathematics Department\
Duke University, Box 90320\
Durham, NC 27708-0320, USA\
author:
- Nicolas Bergeron
- Michael Lipnowski
bibliography:
- 'bibli.bib'
title: Twisted limit formula for torsion and cyclic base change
---
Introduction
============
Asymptotic growth of cohomology
-------------------------------
Let $\Gamma$ be a uniform lattice in a semisimple Lie group $G.$ Let $\Gamma_n \subset \Gamma$ be a decreasing sequence of normal subgroups with trivial intersection. It is known that $$\lim_{n \rightarrow \infty} \frac{\dim H^j(\Gamma_n, \mathbb{C})}{[\Gamma : \Gamma_n]}$$ converges to $b^{(2)}_j(\Gamma),$ the $j$th $L^2$-betti number of $\Gamma.$ If $b^{(2)}_j \neq 0$ for some $j,$ it follows that cohomology is abundant. However, it is often true that $b^{(2)}_j(\Gamma) = 0$ for all $j;$ this is the case whenever $\delta(G) := \mathrm{rank}_{{\mathbb{C}}} G - \mathrm{rank}_{{\mathbb{C}}}K \neq 0.$ What is the true rate of growth of $b_j(\Gamma_n) = \dim H^j(\Gamma_n, {\mathbb{C}})$ when $\delta(G) \neq 0$? In particular, is $b_j(\Gamma_n)$ non-zero for sufficiently large $n$?
We address this question for ‘cyclic base-change.’ Before stating a general result, let’s give two typical examples of this situation.
[*Examples.*]{} 1. The real semisimple Lie group $G={\mathrm{SL}}_2 ({\mathbb{C}})$ satisfies $\delta = 1$. Let $\sigma : G \to G$ be the real involution given by complex conjugation.
2\. The real semisimple Lie group $G={\mathrm{SL}}_2 ({\mathbb{C}}) \times \ldots \times {\mathrm{SL}}_2 ({\mathbb{C}})$ ($n$ times) satisfies $\delta = n$. Let $\sigma : G \to G$ be the order $2n$ automorphism of $G$ given by $\sigma (g_1 , \ldots , g_n ) = (\bar{g}_n , g_1 , \ldots , g_{n-1})$.
Now let $\Gamma_n \subset \Gamma$ be a sequence of finite index, $\sigma$-stable subgroups of $G$. It follows from the general Proposition \[prop:intro\] below that $$\sum_j b_j (\Gamma_n ) \gg {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash {\mathrm{SL}}_2 ({\mathbb{R}}) ).$$ Note that when the $\Gamma_n$’s are congruence subgroups of an arithmetic lattice $\Gamma$, then ${\mathrm{vol}}(\Gamma_n^{\sigma} \backslash {\mathrm{SL}}_2 ({\mathbb{R}}) )$ grows like ${\mathrm{vol}}(\Gamma_n \backslash G )^{\frac{1}{\mathrm{order}(\sigma)}}$.
In this paper we shall more generally consider the case where $G$ is obtained form a real algebraic group by ‘base change.’ Let $\mathbf{G}$ be a connected semisimple quasi-split algebraic group defined over ${\mathbb{R}}$. Let $\mathbb{E}$ be an étale ${\mathbb{R}}$-algebra such that $\mathbb{E} / {\mathbb{R}}$ is a cyclic Galois extension with Galois group generated by $\sigma \in {\mathrm{Aut}}(\mathbb{E} / {\mathbb{R}}).$ Concretely, $\mathbb{E}$ is either $\mathbb{R}^s$ or $\mathbb{C}^s.$ In the first case $\sigma$ is of order $s$ and acts on $\mathbb{R}^s$ by cyclic permutation. In the second case $\sigma$ is of order $2s$ and acts on ${\mathbb{C}}^s$ by $(z_1 , \ldots , z_s) \mapsto (\bar{z}_s , z_1 , \ldots , z_{s-1})$. The automorphism $\sigma$ induces a corresponding automorphism of the group $G$ of real points of $\mathrm{Res}_{\mathbb{E}/ {\mathbb{R}}} \mathbf{G}$. We will furthermore assume that $H^1 (\sigma , G) = \{1 \}$; see §\[par:C\] for comments on this condition. The following proposition is ‘folklore’ (see e.g. Borel-Labesse-Schwermer [@BLS], Rohlfs-Speh [@RohlfsSpeh] and Delorme [@Delorme]).
\[prop:intro\] Let $\Gamma_n \subset \Gamma$ be a sequence of finite index, $\sigma$-stable subgroups of $G.$ Suppose that $\delta(G^{\sigma} ) = 0$. Then we have: $$\sum_j \dim b_j (\Gamma_n ) \gg {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma}).$$
We prove Proposition \[prop:intro\] for certain families $\{ \Gamma_n \}$ in §\[WLF\] but our real interest here is rather how the [*torsion cohomology*]{} grows.
Asymptotic growth of torsion cohomology
---------------------------------------
Let $\rho: {\mathbf{G}}\rightarrow {\mathrm{GL}}(V)$ be a homomorphism of algebraic groups over $\mathbb{R}$ and suppose that the $\Gamma_n$’s stabilize some fixed lattice $\mathcal{O} \subset V.$ The first named author and Venkatesh [@BV] prove that for ‘strongly acyclic’ [@BV $\S 4$] representations $\rho,$ there is a lower bound $$\sum_j \log | H^j(\Gamma_n, \mathcal{O})_{\mathrm{tors}}| \gg c(G,\rho) \cdot [\Gamma : \Gamma_n].$$ for some constant $c(G,\rho).$ In fact, they prove a limiting identity $$\label{torsioneulerchar}
\frac{\sum_j (-1)^j \log | H^j(\Gamma_n, \mathcal{O})_{\mathrm{tors}}|}{[\Gamma:\Gamma_n]} \rightarrow c(G,\rho)$$ and prove that $c(G,\rho)$ is non-zero exactly when $\delta(G) = 1.$ The numerator of the left side of should be thought of as a ‘torsion Euler characteristic.’ The purpose of this article is to prove an analogous theorem about ‘torsion Lefschetz numbers.’
Suppose as above that $G$ is obtained form a real algebraic group by ‘cyclic base change,’ i.e. that $G$ is the group of real points of $\mathrm{Res}_{\mathbb{E}/ {\mathbb{R}}} \mathbf{G}$ where $\mathbb{E} / {\mathbb{R}}$ is a cyclic Galois extension of ${\mathbb{R}}$ by an étale ${\mathbb{R}}$-algebra. Let $\sigma$ be the generator of the corresponding Galois extension and let $G$ be the group of real points of $\mathrm{Res}_{\mathbb{E} / {\mathbb{R}}} {\mathbf{G}}$. In this introduction, we will furthermore assume that the $\Gamma_n$’s are principal congruence of level $p_n$ for some infinite sequence of primes.
Recall that $\rho: {\mathbf{G}}\rightarrow {\mathrm{GL}}(V)$ is a homomorphism of algebraic groups over $\mathbb{R}$. Assume moreover that the $\Gamma_n$’s are $\sigma$-stable and that $V$ is a $\sigma$-equivariant strongly twisted acyclic representation (see §\[def:SA\]) of ${\mathbf{G}}.$ It corresponds to a representation $\tilde{\rho}$ of $\mathrm{Res}_{\mathbb{E} / {\mathbb{R}}} \mathbf{G}$. In particular, $\tilde{\rho} = \rho \otimes \overline{\rho}$ is the corresponding representation of $\mathrm{Res}_{\mathbb{C}/ {\mathbb{R}}} \mathbf{G}$.
\[T:14\] We have: $$\label{coarsetorsion}
\limsup \frac{\sum_j \log |H^j(\Gamma_n, \mathcal{O})|}{{\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} > 0$$ whenever $\delta(G^{\sigma}) = 1$.
Analogously to , we prove by proving a limiting identity for torsion Lefschetz numbers. For example, when $\sigma^2 = 1,$ conditional on an assumption about the growth of the Betti numbers $\dim_{\mathbb{F}_2} H^j(\Gamma_n^{\sigma}, \mathcal{O}_{\mathbb{F}_2})$ we prove that $$\label{torsionlef}
\frac{\sum (-1)^j (\log|H^j(\Gamma_n, \mathcal{O})^{+}_{{\mathrm{tors}}}| - \log |H^j(\Gamma_n, \mathcal{O})^{-}_{\mathrm{tors}}|) }{|H^1 (\sigma , \Gamma_n)| [\Gamma^{\sigma}: \Gamma_n^{\sigma}]} \rightarrow c(G,\rho,\sigma),$$ where the superscript $\pm$ denote the $\pm 1$ eigenspaces.
Assume that the maximal compact subgroup $K \subset G$ is $\sigma$-stable and let $X = G/K$ and $X^{\sigma} =G^{\sigma} / K^{\sigma}$. The proof of crucially uses the equivariant Cheeger-Müller theorem, proven by Bismut-Zhang [@BZ2]. This allows us to compute the left side of (up to a controlled integer multiple of $\log 2$) by studying the eigenspaces of the Laplace operators of the metrized local system $V \rightarrow \Gamma_n \backslash X$ together with their $\sigma$ action. More precisely, the left side is nearly equal to the equivariant analytic torsion $\log T^{\sigma}_{\Gamma_n \backslash X} (\rho).$ Using the simple twisted trace formula and results of Bouaziz [@Bouaziz], we prove a ‘limit multiplicity formula.’
Assume that the $\Gamma_n$’s are $\sigma$-stable and that $V$ is a $\sigma$-equivariant strongly twisted acyclic representation (see §\[def:SA\]) of ${\mathbf{G}}.$ Then we have: $$\label{limitmult}
\frac{\log T^{\sigma}_{\Gamma_n \backslash X}(\rho)}{|H^1 (\sigma , \Gamma_n)| {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} \rightarrow s 2^{r} t^{(2)}_{X^{\sigma}} (\rho),$$ where $\mathbb{E} = {\mathbb{R}}^s$ or ${\mathbb{C}}^s$, and $r = 0$ in the first case and $r=\mathrm{rank}_{{\mathbb{R}}} {\mathbf{G}}({\mathbb{C}}) - \mathrm{rank}_{{\mathbb{R}}} {\mathbf{G}}({\mathbb{R}})$ in the second case.
Here $t^{(2)}_{X^{\sigma}} (\rho)$ is the (usual) $L^2$-analytic torsion of the symmetric space $X^{\sigma}$ twisted by the finite dimensional representation $\rho$. It is explicitly computed in [@BV]. Note that it is non-zero if and only if $\delta (G^{\sigma}) =1$.
The authors hope that the limit multiplicity formula together with the twisted endoscopic comparison implicit in Section 7 will be of interest independent of torsion in cohomology. These computations complement work by Borel-Labesse-Schwermer [@BLS] and Rohlfs-Speh [@RohlfsSpeh].
[*The authors would like to thank Abederrazak Bouaziz, Laurent Clozel, Colette Moeglin and David Renard for helful conversations.*]{}
The simple twisted trace formula
================================
Let $\mathbf{G}$ be a connected semisimple quasi-split algebraic group defined over ${\mathbb{R}}$. Let $\mathbb{E}$ be an étale ${\mathbb{R}}$-algebra such that $\mathbb{E} / {\mathbb{R}}$ is a cyclic Galois extension with Galois group generated by $\sigma \in {\mathrm{Aut}}(\mathbb{E} / {\mathbb{R}}).$ The automorphism $\sigma$ induces a corresponding automorphism of the group $G$ of real points of $\mathrm{Res}_{\mathbb{E}/ {\mathbb{R}}} \mathbf{G}$. We furthermore choose a Cartan involution $\theta$ of $G$ that commutes with $\sigma$ and denote by $K$ the group of fixed points of $\theta$ in $G$. Here we follow Labesse-Waldspurger [@LabesseWaldspurger].
Twisted spaces
--------------
We associate to these data the [*twisted space*]{} $$\widetilde{G} = G \rtimes \sigma \subset G \rtimes \mathrm{Aut} (G).$$ The left action of $G$ on $G$, $$(g , x \rtimes \sigma ) \mapsto gx \rtimes \sigma,$$ turns $\widetilde{G}$ into a left principal homogeneous $G$-space equipped with a $G$-equivariant map $\mathrm{Ad} : \widetilde{G} \to \mathrm{Aut} (G)$ given by $$\mathrm{Ad} (x \rtimes \sigma) (g) = \mathrm{Ad} (x) (\sigma (g)).$$ We also have a right action of $G$ on $\widetilde{G}$ by $$\delta g = \mathrm{Ad} (\delta ) (g) \delta \quad (\delta \in \widetilde{G} , \ g \in G ).$$ This allows to define an action by conjugation of $G$ on $\widetilde{G}$ and yields a notion of $G$-conjugacy class in $\widetilde{G}$. Note that taking $\delta = 1 \rtimes \sigma$ we have: $$(1 \rtimes \sigma) g = \delta g = \sigma (g) \delta = \sigma (g) \rtimes \sigma.$$
We similarly define the twisted space $\widetilde{K} = K \rtimes \sigma$.
Twisted representations
-----------------------
A [*representation of $\widetilde{G}$*]{}, in a vector space $V$, is the data for every $\delta \in \widetilde{G}$ of a invertible linear map $$\widetilde{\pi} (\delta) \in \mathrm{GL} (V)$$ and of a representation of $G$ in $V$: $$\pi : G \to \mathrm{GL} (V)$$ such that for $x,y \in G$ and $\delta \in \widetilde{G}$, $$\widetilde{\pi} (x \delta y) = \pi (x) \widetilde{\pi} (\delta ) \pi (y).$$ In particular $$\widetilde{\pi} (\delta x) = \pi ( \mathrm{Ad} (\delta ) (x)) \widetilde{\pi} (\delta).$$ Therefore $\widetilde{\pi} (\delta )$ intertwines $\pi$ and $\pi \circ \mathrm{Ad} (\delta )$. Note that $\widetilde{\pi}$ determines $\pi$; we will say that $\pi$ is the restriction of $\widetilde{\pi}$ to $G$.
Conversely $\widetilde{\pi}$ is determined by the data of $\pi$ and of an operator $A$ which intertwines $\pi$ and $\pi \circ \sigma$: $$A \pi (x) =(\pi \circ \sigma ) (x) A$$ and whose $p$-th power is the identity, where $p$ is the order of $\sigma$. We reconstruct $\widetilde{\pi}$ by setting $$\widetilde{\pi} (x \rtimes \sigma ) = \pi (x) A \quad \mbox{for } x \in G.$$
Say that $\widetilde{\pi}$ is *essential* if $\pi$ is irreducible. If $\widetilde{\pi}$ is unitary and essential, Schur’s lemma implies that $\pi$ determines $A$ up to a $p$-th root of unity.
There is a natural notion of equivalence between representations of $\widetilde{G}$ — see e.g. [@LabesseWaldspurger §2.3]. This is the obvious one; beware however that even if $\widetilde{\pi}$ is essential the class of $\pi$ does not determine the class of $\widetilde{\pi}$ since the intertwiner $A$ is only determined up to a root of unity.
We have a corresponding notion of a $(\mathfrak{g} , \widetilde{K})$-module.
If $\widetilde{\pi}$ is unitary and $f \in C_c^{\infty} (\widetilde{G})$ we set $$\widetilde{\pi} (f) = \int_{\widetilde{G}} f( y) \widetilde{\pi} (y) dy := \int_{G} f (x\rtimes \sigma) \widetilde{\pi} (x \rtimes \sigma) dx.$$ It follows from [@LabesseWaldspurger Lemma 2.3.2] that $\widetilde{\pi} (f)$ is of trace class. Moreover: $\mathrm{trace} \ \widetilde{\pi} (f) =0$ unless $\widetilde{\pi}$ is essential. In the following, we denote by $\Pi (\widetilde{G})$ the set of irreducible unitary representations $\pi$ of $G$ (considered up to equivalence) that can be extended to some (twisted) representation $\widetilde{\pi}$ of $\widetilde{G}$. Note that the extension is not unique.
Twisted trace formula (in the cocompact case) {#TTF}
---------------------------------------------
Let $\Gamma$ be a cocompact lattice of $G$ that is $\sigma$-stable. Associated to $\Gamma$ is the (right) regular representation $\widetilde{R}_\Gamma$ of $\widetilde{G}$ on $L^2(\Gamma \backslash G )$ where the restriction $R_\Gamma$ of $\widetilde{R}_\Gamma$ is the usual regular representation in $L^2(\Gamma \backslash G)$ and $$(\widetilde{R}_\Gamma (\sigma )) (f) (\Gamma x) = f( \Gamma \sigma (x)).$$ Note that $$(\widetilde{R}_\Gamma (\sigma ) R_\Gamma (g )) (f) (\Gamma x) = f( \Gamma \sigma (x) g) = (R_\Gamma (\sigma (g)) \widetilde{R}_\Gamma (\sigma ))(f) (\Gamma x).$$
Given $\delta \in \widetilde{G}$ we denote by $G^\delta$ its centralizer in $G$ (for the (twisted) action by conjugation of $G$ on $\widetilde{G}$). Corresponding to $\Gamma$ is a (non-empty) discrete twisted subspace $\widetilde{\Gamma} \subset \widetilde{G}$. Given $\delta \in \widetilde{\Gamma}$ we denote by $\{ \delta \}$ its $\Gamma$-conjugacy class (where here again $\Gamma$ acts by (twisted) conjugation on $\widetilde{\Gamma}$).
Let $f \in C_c^{\infty} (\widetilde{G})$. The twisted trace formula is obtained by computing the trace of $\widetilde{R}_\Gamma (f)$ in two different ways. It takes the following form (the LHS is the spectral side and the RHS is the geometric side): $$\label{TTF}
\sum_{\pi \in \Pi (\widetilde{G})} m (\pi , \widetilde{\pi} , \Gamma) \ \mathrm{trace} \ \widetilde{\pi} (f) = \sum_{\{ \delta \}} {\mathrm{vol}}(\Gamma^{\delta} \backslash G^{\delta}) \int_{G^{\delta} \backslash G} f(x^{-1} \delta x) d \dot{x}.$$ Here $\widetilde{\pi}$ is some extension of $\pi$ to a twisted representation of $\widetilde{G}$ and $$\begin{aligned}
m (\pi , \widetilde{\pi} , \Gamma) &= \sum_{\widetilde{\pi} ' | G = \pi} \lambda (\widetilde{\pi} ' , \widetilde{\pi}) m (\widetilde{\pi} ') \\
&= \mathrm{trace}\left(\sigma | {\mathrm{Hom}}_G(\widetilde{\pi}, L^2(\Gamma \backslash G)) \right), \end{aligned}$$ where $m (\widetilde{\pi} ')$ is the multiplicity of $\widetilde{\pi} '$ in $\widetilde{R}_\Gamma$ and $\lambda (\widetilde{\pi} ' , \widetilde{\pi}) \in {\mathbb{C}}^{\times}$ is the scalar s.t. for all $\delta \in \widetilde{G}$, we have $\widetilde{\pi} ' (\delta ) = \lambda (\widetilde{\pi} ' , \widetilde{\pi}) \widetilde{\pi} (\delta)$.[^1] Note that $\lambda (\widetilde{\pi} ' , \widetilde{\pi})$ is in fact a $p$-th root of unity.
The definition of $ \mathrm{trace} \ \widetilde{\pi} (f)$ depends on a choice of a Haar measure $dx$ on $G$. On the geometric side the volumes ${\mathrm{vol}}(\Gamma^{\delta} \backslash G^{\delta})$ depend on choices of Haar measures on the groups $G^{\delta}$. We will make precise choices later on. For the moment we just note that the measure $d\dot{x}$ on the quotient $G^{\delta} \backslash G$ is normalized by: $$\int_{G} \phi (x) dx = \int_{G^{\delta} \backslash G} \int_{G^{\delta}} \phi (gx) dg d\dot{x}.$$
Galois cohomology groups $H^1 (\sigma , \Gamma)$ {#par:C}
------------------------------------------------
Let $Z^1 (\sigma , \Gamma) = \{ \delta \in \widetilde{\Gamma} \; : \; \delta^p = \sigma\}$; it is invariant by conjugation by $\Gamma$. We denote by $H^1 (\sigma , \Gamma)$ the quotient of $Z^1 (\sigma , \Gamma)$ by the equivalence relation defined by conjugation by elements of $\Gamma$. We have similar definitions when $\Gamma$ is replaced by $G$.
We will assume that $$\label{condition}
H^1 (\sigma , G ) = \{1 \}.$$ Note that if $\mathbb{E} = {\mathbb{R}}^p,$ condition is always satisfied. Indeed: in that case $G = {\mathbf{G}}({\mathbb{R}})^p$ and an element $$(g_1 , \ldots , g_p) \rtimes \sigma \in \widetilde{G}$$ belongs to $Z^1 (\sigma , G)$ if and only if $g_1 g_2 \cdots g_p = e$. But then there is an equality $$\sigma (g_1 , g_1 g_2 , \ldots , g_1 \cdots g_p )^{-1} (g_1 , g_1 g_2 , \ldots , g_1 \cdots g_p) = (g_1 , \ldots , g_p).$$ Equivalently, $(g_1 , \ldots , g_p) \rtimes \sigma$ is conjugated to $\sigma$ in $\widetilde{G}$ by some element in $G$.
We furthermore note that $H^1 ({\mathbb{C}}/ {\mathbb{R}}, \mathrm{SL}_n ({\mathbb{C}})) = H^1 ({\mathbb{C}}/ {\mathbb{R}}, \mathrm{Sp}_n ({\mathbb{C}})) = \{1\}$, see e.g. [@Serre Chap. X]. Therefore, condition holds if ${\mathbf{G}}$ is a product of factors $\mathrm{SL}_n$ or $\mathrm{Sp}_n$ or of factors whose group of real points is isomorphic to a complex Lie group viewed as a real Lie group.
Under assumption , the map $$H^1 (\sigma , \Gamma) \to H^1 (\sigma , G)$$ necessarily has trivial image. In other words: if $\delta$ represents a class in $H^1 (\sigma , \Gamma)$ then $\delta$ is conjugate to $\sigma$ by some element of $G$. In particular, we have $${\mathrm{vol}}(\Gamma^{\delta} \backslash G^{\delta}) = {\mathrm{vol}}(\Gamma^{\sigma} \backslash G^{\sigma}) \mbox{ and } \int_{G^{\delta} \backslash G}
f (x^{-1} \delta x) d\dot{x} = \int_{G^{\sigma} \backslash G}
f (x^{-1} \sigma x) d\dot{x}.$$ We may therefore write the geometric side of the twisted trace formula as: $$\label{TTF2}
|H^1 (\sigma , \Gamma) | {\mathrm{vol}}(\Gamma^{\sigma} \backslash G^{\sigma}) \int_{G^{\sigma} \backslash G}
f (x^{-1} \sigma x) d\dot{x} + \sum_{\substack{\{\delta \} \\ \delta \notin Z^1 (\sigma , \Gamma)}} {\mathrm{vol}}(\Gamma^{\delta} \backslash G^{\delta}) \int_{G^{\delta} \backslash G}
f (x^{-1} \delta x) d\dot{x}.$$
Finite dimensional representations of $\widetilde{G}$ {#def:SA}
-----------------------------------------------------
Note that the complexification of $G$ may be identified with the complex points of $\mathrm{Res}_{\mathbb{E} / {\mathbb{R}}} \mathbf{G}$, i.e. $\mathbf{G} ({\mathbb{C}})^p$. Every complex finite dimensional $\sigma$-stable irreducible representation $(\widetilde{\rho} , F)$ of $\widetilde{G}$ can therefore be realized in a space $F = F_0^{\otimes p}$ where $(\rho_0 , F_0)$ is an irreducible complex linear representation of ${\mathbf{G}}({\mathbb{C}}).$ The action of $G$ is defined by the tensor product action $\rho_0^{\otimes p}$ if $\mathbb{E} = {\mathbb{R}}^p$ and by $\otimes_{i=1}^{p/2} (\rho_0 \otimes \bar{\rho}_0)$, where $\bar{\rho}_0$ is obtained by composing the complex conjugation in ${\mathbf{G}}({\mathbb{C}})$ by $\rho_0$, if $\mathbb{E} = {\mathbb{C}}^{p/2}$. In both cases, we choose the action of $\sigma$ on $F = F_0^{\otimes p}$ to be the cyclic permutation $A: x_1 \otimes \ldots \otimes x_p \mapsto x_p \otimes x_1 \otimes \ldots \otimes x_{p-1}$. Note that $$\mathrm{trace} (\sigma \; | \; F ) = \dim F_0.$$
Let ${\mathfrak{g}}$ be the Lie algebra of ${\mathbf{G}}.$ Say that $(\widetilde{\rho} , F)$ is [*strongly twisted acyclic*]{} if there is a positive constant $\eta$ depending only on $F$ such that: for every irreducible unitary $({\mathfrak{g}}, \widetilde{K})$-module $V$ for which $$\mathrm{trace} (\sigma \; | \; C^j ({\mathfrak{g}}({\mathbb{C}}) , K , V\otimes F)) \neq 0$$ for some $j \leq \dim {\mathbf{G}}({\mathbb{C}}),$ the inequality $$\Lambda_F - \Lambda_V \geq \eta$$ is satisfied. Here $\Lambda_F$, resp. $\Lambda_V$, is the scalar by which the Casimir acts on $F$, resp. $V$.
Write $\nu$ for the highest weight of $F_0$. The following lemma can be proven analogously to [@BV Lemma 4.1].
Suppose that $\nu$ is not preserved by the Cartan involution $\theta$ then $\widetilde{\rho}$ is strongly twisted acyclic.
Lefschetz number and twisted analytic torsion
=============================================
Let $G$, $\sigma$ and $\Gamma$ be as in §\[TTF\] and let $(\tilde{\rho} , F)$ be a complex finite dimensional $\sigma$-stable irreducible representation of $\widetilde{G}$. We denote by ${\mathfrak{g}}$ be the Lie algebra of $G$.
Twisted $({\mathfrak{g}}({\mathbb{C}}) , K)$-cohomology and Lefschetz number {#lefschetznumber}
----------------------------------------------------------------------------
We can define an action of $\sigma$ on each cohomology group $H^i (\Gamma \backslash X , F)$ and thus define a [*Lefschetz number*]{} $$\mathrm{Lef} (\sigma , \Gamma , F) = \sum_i (-1)^i {\mathrm{trace}}\ ( \sigma \; | \; H^i (\Gamma \backslash X , F)).$$
If $V$ is a $({\mathfrak{g}}, \widetilde{K})$-module, we have a natural action of $\sigma$ on the space of $({\mathfrak{g}}, K)$-cochains $C^\bullet ({\mathfrak{g}}({\mathbb{C}}) , K , V)$ which induces an action on the quotient $H^\bullet ({\mathfrak{g}}({\mathbb{C}}) , K , V)$. We denote by $$\mathrm{trace} \ (\sigma \; | \; H^\bullet ({\mathfrak{g}}, K , V))$$ the trace of the corresponding operator. We then define the [*Lefschetz number*]{} of $V$ by $$\mathrm{Lef} (\sigma , V ) = \sum_i (-1)^i \mathrm{trace} (\sigma \; | \; H^i ({\mathfrak{g}}, K , V)).$$ If $F$ is a finite dimensional representation of $\widetilde{G}$ then $F \otimes V$ is still a $({\mathfrak{g}}, \widetilde{K})$-module; we denote by $\mathrm{Lef}(\sigma , F , V)$ its Lefschetz number.
Labesse [@Labesse Sect. 7] proves that there exists a compactly supported function $L_{\rho} \in C_c^{\infty} (\widetilde{G})$ such that for every essential admissible representation $(\widetilde{\pi} , V)$ of $\widetilde{G}$ one has $$\mathrm{Lef} (\sigma , F , V) = {\mathrm{trace}}\ \widetilde{\pi} (L_{\rho}).$$ The function $L_{\rho}$ is called the [*Lefschetz function*]{} for $\sigma$ and $(\tilde{\rho}, F)$.
We then have: $$\begin{split}
{\mathrm{trace}}\ \widetilde{R}_{\Gamma} (L_{\rho}) & = \sum_i (-1)^i {\mathrm{trace}}( \sigma \; | \; H^i (\Gamma \backslash X , F)) \\
& = \mathrm{Lef} (\sigma , \Gamma , F) .
\end{split}$$
Twisted heat kernels
--------------------
Let $$H_t^{\rho , i} \in \left[ C^{\infty} (G) \otimes \mathrm{End} (\wedge^i ({\mathfrak{g}}/ {\mathfrak{k}})^* \otimes F) \right]^{K \times K}$$ be the heat kernel for $L^2$-forms of degree $i$ with values in the bundle associated to $(\rho , F)$. Note that we have a natural action of $\sigma$ on $\wedge^i ({\mathfrak{g}}/ {\mathfrak{k}})^* \otimes F$; we denote by $A_{\sigma}$ the corresponding linear operator and let $$h_t^{\rho, i, \sigma} : x \rtimes \sigma \mapsto {\mathrm{trace}}( H_t^{\rho , i} (x) \circ A_{\sigma}) .$$
Eventually we shall apply the twisted trace formula to $h_t^{\rho, i, \sigma}$. The heat kernel $H_t^{\rho , i}$ is not compactly supported. However, it follows from [@BM Proposition 2.4] that it belongs to all Harish-Chandra Schwartz spaces $\mathcal{C}^q \otimes \mathrm{End} (\wedge^i ({\mathfrak{g}}/ {\mathfrak{k}})^* \otimes F)$, $q>0.$ This is enough to ensure absolute convergence of both sides of the twisted trace formula.
\[L1\] Let $\widetilde{\pi}$ be an essential admissible representation of $\widetilde{G}$ and let $V$ be its associated $({\mathfrak{g}}, \widetilde{K})$-module. We have: $${\mathrm{trace}}\ \widetilde{\pi} (h_t^{\rho , i, \sigma}) = e^{t(\Lambda_V - \Lambda_F)} {\mathrm{trace}}(\sigma | [\wedge^i ({\mathfrak{g}}/ k )^* \otimes F \otimes V]^K).$$
It follows from the $K\times K$ equivariance of $H_t^{\rho , i}$, and Kuga’s Lemma that relative to the splitting $$\wedge^i ({\mathfrak{g}}/ k )^* \otimes F \otimes V = [\wedge^i ({\mathfrak{g}}/ k )^* \otimes F \otimes V]^K \oplus \left( [\wedge^i ({\mathfrak{g}}/ k )^* \otimes F \otimes V]^K \right)^{\perp},$$ we have: $$\pi (H_t^{\rho , i}) = \left(
\begin{array}{cc}
e^{t(\Lambda_V - \Lambda_F)} \mathrm{Id} & 0 \\
0 & 0
\end{array} \right).$$ We furthermore note that this decomposition is $\sigma$-invariant since $K$ is $\sigma$-stable. We conclude that we have: $$\widetilde{\pi} (H_t^{\rho ,i}) := \int_{{\mathbf{G}}({\mathbb{C}})} (\pi (g)\circ A) \otimes (H_t^{\rho, i} (g) \circ A_{\sigma}) dg = \left(
\begin{array}{cc}
e^{t(\Lambda_V - \Lambda_F)} A_{\sigma} & 0 \\
0 & 0
\end{array} \right).$$ Here $\pi$ is the restriction of $\widetilde{\pi}$ to $G$ and $A$ is the intertwining operator between $\pi$ and $\pi \circ \sigma$ that determines $\widetilde{\pi}$.
Now let $\{\xi_n \}_{n \in {\mathbb N}}$ and $\{e_j \}_{j=1, \ldots , m}$ be orthonormal bases of $V$ and $\wedge^i ({\mathfrak{g}}/ k )^* \otimes F$, respectively. Then we have: $$\begin{split}
{\mathrm{trace}}\ \widetilde{\pi} (H_t^{\rho , i}) & = \sum_{n=1}^{\infty} \sum_{j=1}^m \langle \widetilde{\pi} (H_t^{\rho ,i}) (\xi_n \otimes e_j ) , (\xi_n \otimes e_j ) \rangle \\
& = \sum_{n=1}^{\infty} \sum_{j=1}^m \int_{G} \langle (\pi (g) \circ A) \xi_n , \xi_n \rangle \langle (H_t^{\rho ,i} (g) \circ A_{\sigma}) e_j , e_j \rangle dg \\
& = \sum_{n=1}^{\infty} \int_{G} \langle (\pi (g) \circ A) \xi_n , \xi_n \rangle h_t^{\rho , i , \sigma} (g \rtimes \sigma) dg \\
& = {\mathrm{trace}}\ \widetilde{\pi} (h_t^{\rho , i, \sigma}).
\end{split}$$ The lemma follows.
Denoting by $H_t^0 \in \left[ C^{\infty} (G) \otimes \mathrm{End} (\wedge^0 ({\mathfrak{g}}/ {\mathfrak{k}})^*) \right]^{K \times K}$ the heat kernel for $L^2$-functions on $X$, the following proposition follows from [@MuellerPfaff Proposition 5.3] and the definition of strong twisted acyclicity.
\[P:larget\] Assume that $(\widetilde{\rho} , F)$ is strongly twisted acyclic. Then there exist positive constants $\eta$ and $C$ such that for every $x \in G$, $t \in (0, +\infty)$ and $i \in \{0 , \ldots , \dim X \}$, one has: $$|h_t^{\rho , i , \sigma} (x \rtimes \sigma) | \leq C e^{-\eta t} H_t^0 (x).$$
We define the kernel $k_t^{\rho, \sigma}$ by $$k_t^{\rho , \sigma} (g) = \sum_i (-1)^i i h_t^{\rho , i , \sigma} (g);$$ it defines a function in $\mathcal{C}^q (\widetilde{G})$, for all $q>0$.
Twisted analytic torsion
------------------------
The [*twisted analytic torsion*]{} $T_{\Gamma \backslash X}^{\sigma} (\rho)$ is then defined by $$\label{TT}
\log T_{\Gamma \backslash X}^{\sigma} (\rho) = \frac12 \frac{d}{ds} {}_{|s= 0} \left( \frac{1}{\Gamma (s)} \int_0^{\infty} t^{s-1} \left[ {\mathrm{trace}}\ \widetilde{R}_{\Gamma} (k_t^{\rho ,\sigma}) -
\sum_i (-1)^i i {\mathrm{trace}}( \sigma \; | \; H^i (\Gamma \backslash X , F)) \right] dt \right).$$ Note that if $(\widetilde{\rho} , F)$ is strongly twisted acyclic each ${\mathrm{trace}}( \sigma \; | \; H^i (\Gamma \backslash X , F))$ is trivial. From now on we will assume that $(\widetilde{\rho} , F)$ is strongly twisted acyclic. In particular, we have: $$\log T_{\Gamma \backslash X}^{\sigma} (\rho) = \frac12 \frac{d}{ds} {}_{|s= 0} \left( \frac{1}{\Gamma (s)} \int_0^{\infty} t^{s-1} {\mathrm{trace}}\ \widetilde{R}_{\Gamma} (k_t^{\rho ,\sigma}) dt \right).$$
Twisted $({\mathfrak{g}}, K)$-torsion
-------------------------------------
If $V$ is a $({\mathfrak{g}}, \widetilde{K})$-module and $F$ is a finite dimensional representation of $\widetilde{G}$ then $F \otimes V$ is still a $({\mathfrak{g}}, \widetilde{K})$-module; we define the [*twisted $({\mathfrak{g}}, K)$-torsion*]{} of $F \otimes V$ by $$\begin{split}
\mathrm{Lef} {}' (\sigma , F , V) & = \sum_i (-1)^i i \ \mathrm{trace} (\sigma \; | \; C^i ({\mathfrak{g}}, K , F \otimes V)) \\
& = \sum_i (-1)^i i \ \mathrm{trace} (\sigma \; | \; [\wedge^i ({\mathfrak{g}}/ {\mathfrak{k}})^* \otimes F \otimes V]^K).
\end{split}$$
[*Remark.*]{} We should explain the notation $\mathrm{Lef}'$. Given a group $G$ and a $G$-vector space $V$, we denote by $\det[1-V]$ the virtual $G$-representation (that is to say, element of $K_0$ of the category of $G$-representations) defined by the alternating sum $\sum_i (-1)^i [\wedge^i V]$ of exterior powers. This is multiplicative in an evident sense: $$\label{add1}
\det[1 - V \oplus W] = \det[1-V] \otimes \det[1-W].$$ Now given $g \in G$, the derivative $\frac{d}{dt} {}_{|t=1} \det (t1 - g)$ is equal to the character of $g$ acting on $\sum_{i} (-1)^i i \wedge^i V$. We therefore define ${\det{}'}[1-V] = \sum_{i} (-1)^i i \wedge^i V$.
Considering the virtual $\widetilde{K}$-representation ${\det{}'}[1- ({\mathfrak{g}}/ {\mathfrak{k}})^*]$ we have: $$\label{Lef'}
\mathrm{Lef} {}' (\sigma , F , V) = \mathrm{trace} \left( \sigma \; | \; \left[ {\det{}'}[1 - ({\mathfrak{g}}/ {\mathfrak{k}})^*] \otimes F \otimes V\right]^K \right).$$ This explains our notation $\mathrm{Lef}'$ for the twisted $({\mathfrak{g}}({\mathbb{C}}) , K)$-torsion.
For future reference we note that we have: $$\label{add}
{\det{}'}[1 - V \oplus W] = {\det{}'}[1-V] \otimes \det[1-W] \oplus \det[1-V] \otimes {\det{}'}[1-W].$$
We also note that Labesse’s proof of the existence of $L_{\rho}$ can be immediately modified to get a function $L_{\rho}' \in C_c^{\infty} (\widetilde{G})$ such that for every essential admissible representation $(\widetilde{\pi} , V)$ of $\widetilde{G}$ one has $$\mathrm{Lef}' (\sigma , F , V) = {\mathrm{trace}}\ \widetilde{\pi} (L_{\rho}').$$
It follows from Lemma \[L1\] that the spectral side of the twisted trace formula evaluated in $k_t^{\rho , \sigma}$ is $${\mathrm{trace}}\ \widetilde{R}_{\Gamma} (k_t^{\rho ,\sigma}) =
\sum_{\pi \in \Pi (\widetilde{G})} m (\pi , \widetilde{\pi} , \Gamma ) \mathrm{Lef}' (\sigma , F , V_{\pi}) e^{t(\Lambda_{\pi} - \Lambda_F)},$$ where $V_{\pi}$ is the $({\mathfrak{g}}, \widetilde{K})$-module associated to the extension $\widetilde{\pi}$.
$L^2$-Lefschetz number, $L^2$-torsion and limit formulas
========================================================
Let $G$, $\sigma$ and $\Gamma$ be as in §\[TTF\] and let $(\tilde{\rho} , F)$ be as in the preceding sections. Let $f \in C_c^{\infty} (\widetilde{G})$.
Given $g \in G$ we define $r(g) = \mathrm{dist}(gK,eK)$ with respect to the Riemannian symmetric distance of $X=G/K$. We extend $r$ to $G \rtimes \langle \sigma \rangle$ by setting $r(g \rtimes \sigma) = r(g)$. Note that $r(gg') \leq r(g) + r(g')$.
Now given $x \in G$ we set $\ell (x) = \inf \{ r(gxg^{-1}) \; : \; g \in G \}$; it only depends on the conjugacy class of $x$ in $G$. Recall that the [*injectivity radius of $\Gamma$*]{}, $$r_{\Gamma} = \frac12 \inf \{ \ell (\gamma ) \; : \; \gamma \in \Gamma - \{ e \} \},$$ is strictly positive. Now if $\delta \in \widetilde{\Gamma}$ with $\delta \notin Z^1 (\sigma , \Gamma)$ then $\delta^p \in \Gamma - \{ 1\}$ and therefore $\ell (\delta^p) \geq 2r_{\Gamma}$. In particular for any $x \in G$ we have: $$\label{twistedinjrad}
2r_\Gamma \leq \ell (\delta^p) \leq r(x\delta^px^{-1}) = r( (x \delta x^{-1} ) \cdots (x \delta x^{-1})) \leq p \cdot r(x \delta x^{-1}).$$
\[Lcount\] There exist constants $c_1,c_2 >0$, depending only on $G$, such that for any $x \in G$, we have: $$N(x; R ) := |\{ \delta \in \widetilde{\Gamma} \; : \; \delta \notin Z^1 (\sigma , \Gamma) \mbox{ and } r(x \delta x^{-1}) \leq R \} | \leq c_1 p^d r_\Gamma^{-d} e^{c_2 R} ,$$ where $d$ is the dimension of $X$.
It follows from that it suffices to prove the lemma for $R \geq 2r_\Gamma/p$. Set $\varepsilon := r_\Gamma/p$. By definition of $r(x \delta x^{-1})$ we have $B(\gamma (\sigma (x)) , \varepsilon ) \subset B(x, R+\varepsilon)$, for all $\delta = \gamma \rtimes \sigma \in \widetilde{\Gamma}$ with $\delta \notin Z^1 (\sigma , \Gamma)$ and $r(x \delta x^{-1}) \leq R$. Now, since $\varepsilon < r_\Gamma$, the balls $B(\gamma (\sigma (x)) , \varepsilon )$, $\gamma \in \Gamma$, are all disjoint of the same volume. We conclude that $$N(x;R) \cdot \mathrm{vol} B (\sigma (x) , \varepsilon ) \leq \mathrm{vol} B( x , R + \varepsilon ) \leq \mathrm{vol} B (x, 2R).$$ We conclude using standard estimates on volumes of balls (see e.g. [@7samurai Lemma 7.21] for more details).
\[WLF\] Let $\{\Gamma_n \}$ be a sequence of finite index $\sigma$-stable subgroups of $\Gamma$. Assume that for any $\delta \in \widetilde{\Gamma}$ with $\delta \notin Z^1 (\sigma , \Gamma)$ we have: $$\lim_{n \to \infty} \frac{| \{ \gamma \in \Gamma_n \backslash \Gamma \; : \; \gamma \delta \gamma^{-1} \in \widetilde{\Gamma}_n \}|}{[\Gamma^\sigma : \Gamma_n^\sigma]} = 0.$$ Then $$\frac{{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (f)}{|H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} \to \int_{G^{\sigma} \backslash G} f(x^{-1} \sigma x) d\dot{x}.$$
It follows from that we have: $$\begin{split}
{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (f ) & = |H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma}) \int_{G^{\sigma} \backslash G}
f(x^{-1} \sigma x) d\dot{x} + \sum_{\substack{\{\delta \}_{\Gamma_n} \\ \delta \notin Z^1 (\sigma , \Gamma_n)}} {\mathrm{vol}}(\Gamma_n^{\delta} \backslash G^{\delta}) \int_{G^{\delta} \backslash G} f (x^{-1} \delta x) d\dot{x} \\
& = |H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma}) \int_{G^{\sigma} \backslash G}
f(x^{-1} \sigma x) d\dot{x} + \sum_{\substack{\{\delta \}_\Gamma \\ \delta \notin Z^1 (\sigma , \Gamma)}} c_{\Gamma_n} (\delta ) {\mathrm{vol}}(\Gamma^{\delta} \backslash G^{\delta}) \int_{G^{\delta} \backslash G} f (x^{-1} \delta x) d\dot{x}
\end{split}$$ where $$c_{\Gamma_n} (\delta) = | \{ \gamma \in \Gamma_n \backslash \Gamma \; : \; \gamma \delta \gamma^{-1} \in \widetilde{\Gamma}_n \}|.$$ Since $f$ is compactly supported, the last sum above is finite: choosing $R>0$ so that the support of $f$ is contained in $$B_R = \{ g \rtimes \sigma \in \widetilde{G} \; : \; r(g) \leq R\}$$ we may restrict the sum on the right side of the above equation to $\delta$ that are contained in $B_R$. It follows from Lemma \[Lcount\] that the corresponding sum if finite.
Let $\{ \Gamma_n \}$ be a normal chain with $\cap_n \Gamma_n = \{ 1 \}.$ Then $r_{\Gamma_n} \to \infty$ as $n \to \infty$. The following lemma implies that the hypotheses of Proposition \[WLF\] are satisfied for $\{ \Gamma_n \}$.
\[normalchain\] Let $\delta \notin Z^1 (\sigma , \Gamma)$. If $r_{\Gamma_n} \geq \ell (\delta^p)$, then we have: $$| \{ \gamma \in \Gamma_n \backslash \Gamma \; : \; \gamma \delta \gamma^{-1} \in \widetilde{\Gamma}_n \}|= 0.$$
Indeed $$c_{\Gamma_n} (\delta) \leq | \{ \gamma \in \Gamma_n \backslash \Gamma \; : \; \gamma \delta^p \gamma^{-1} \in \Gamma_n \} |$$ and if $\delta \notin Z^1 (\sigma , \Gamma)$ then $\delta^p \neq 1$. Finally, if $\gamma \delta^p \gamma^{-1} \in \Gamma_n$ is non-trivial then $\ell (\delta^p) \geq r_{\Gamma_n}$.
$L^2$-Lefschetz number
----------------------
Proposition \[WLF\] motivates the following definition of the [*$L^2$-Lefschetz number*]{} associated to the triple $(G, \sigma, \rho)$: $$\mathrm{Lef}^{(2)} (\sigma, X , F) = SO_{e \rtimes \sigma}(L_{\rho}) = \int_{G^{\sigma} \backslash G} L_\rho (x^{-1} \sigma x) d\dot{x}.$$
\[C:47\] Let $\{ \Gamma_n \}$ be a normal chain of finite index $\sigma$-stable subgroups of $\Gamma$ with $\cap_n \Gamma_n = \{ 1 \}$. Then we have: $$\frac{\mathrm{Lef}(\sigma , \Gamma_n , F)}{|H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} \to
\mathrm{Lef}^{(2)} (\sigma, X , F) .$$
Apply Proposition \[WLF\] (and Lemma \[normalchain\]) to the Lefschetz function $L_{\rho}$.
Twisted $L^2$-torsion
---------------------
Analogously we define the [*twisted $L^2$-torsion*]{} $T_{\Gamma \backslash X}^{(2) \sigma} (\rho) \in {\mathbb{R}}^+$ by $$\log T_{\Gamma \backslash X}^{(2) \sigma} (\rho) = |H^1 (\sigma , \Gamma) | {\mathrm{vol}}(\Gamma^{\sigma} \backslash G^{\sigma}) t_X^{(2) \sigma} (\rho)$$ where $t_X^{(2) \sigma} (\rho)$ — which depends only on the symmetric space $X$, the involution $\sigma,$ and the finite dimensional representation $\rho$ — is defined by $$\label{t}
t_X^{(2) \sigma} (\rho) = \frac12 \frac{d}{ds} {}_{|s=0} \left( \frac{1}{\Gamma (s)} \int_0^{\infty} \int_{G^{\sigma} \backslash G}
k_t^{\rho , \sigma} (x^{-1} \sigma x) d\dot{x} t^{s-1} dt \right) .$$
Note that $k_t^{\rho , \sigma}$ is not compactly supported and that we have to prove that the RHS of is indeed well defined. Recall however that $k_t^{\rho , \sigma}$ belongs to $\mathcal{C}^q (\widetilde{G})$. Lemma \[Lcount\] therefore implies that the series $$\sum_{\delta \in \widetilde{\Gamma}} k_t^{\rho , \sigma} (x^{-1} \delta x)$$ converges absolutely and locally uniformly. This implies that the integral of this series along a (compact) fundamental domain $D$ for the action of $\Gamma$ on $G$ is absolutely convergent. Restricting the sum to the $\delta$’s that belong to the (twisted) $\Gamma$-conjugacy class of $\sigma$ we conclude in particular that, for every positive $t$, the integral $$\int_{G^{\sigma} \backslash G} k_t^{\rho , \sigma} (x^{-1} \sigma x) d\dot{x} = \frac{1}{\mathrm{vol} (\Gamma^{\sigma} \backslash G^\sigma)} \int_{D} \sum_{\delta \in \{ \sigma \}} k_t^{\rho , \sigma} (x^{-1} \delta x) dx$$ is absolutely convergent. We postpone the proof of the fact that is indeed well defined until sections 6 and 7 where we will explicitly compute $t_X^{(2) \sigma}
(\rho)$. In the course of the computations we will also prove the following lemma.
\[Lkt\] There exist constants $C,c>0$ such that $$\left| \int_{G^{\sigma} \backslash G} k_t^{\rho , \sigma} (x^{-1} \sigma x) d\dot{x} \right| \leq Ce^{-ct}, \quad t \geq 1.$$
Granted this we conclude this section by the proof of the following ‘limit multiplicity theorem’.
\[twistedtorsionlimitmultiplicity\] Assume that $(\widetilde{\rho} , F)$ is strongly twisted acyclic. Let $\{\Gamma_n \}$ be a sequence of finite index $\sigma$-stable subgroups of $\Gamma$. Assume that there exists a constant $A$ s.t. for every $\delta \in \widetilde{\Gamma}$ with $\delta \notin Z^1 (\sigma , \Gamma)$ the sequence $$\left(\frac{| \{ \gamma \in \Gamma_n \backslash \Gamma \; : \; \gamma \delta \gamma^{-1} \in \widetilde{\Gamma}_n \}|}{[\Gamma^\sigma : \Gamma_n^\sigma]} \right)_{n \geq 0}$$ remains uniformly bounded by $A$ and converges to $0$ as $n$ tends to infinity.[^2] Then $$\frac{\log T_{\Gamma_n \backslash X}^{\sigma} (\rho)}{|H^1 (\sigma , \Gamma_n ) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} \to t_X^{(2) \sigma} (\rho) .$$
Since $k_t^{\rho , \sigma} \in \mathcal{C}^q (\widetilde{G})$, for all $q>0$, we still have: $$\begin{gathered}
{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (k_t^{\rho ,\sigma}) = |H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma}) \int_{G^{\sigma} \backslash G}
k_t^{\rho ,\sigma} (x^{-1} \sigma x) d\dot{x} \\
+ \sum_{\substack{\{\delta \}_\Gamma \\ \delta \notin Z^1 (\sigma , \Gamma)}} c_{\Gamma_n} (\delta ) {\mathrm{vol}}(\Gamma^{\delta} \backslash G^{\delta}) \int_{G^{\delta} \backslash G} k_t^{\rho ,\sigma} (x^{-1} \delta x) d\dot{x}.\end{gathered}$$ Note that at this point it is not clear that the sum on the right (absolutely) converges. This is however indeed the case: it first follows from that if $\delta \notin Z^1 (\sigma , \Gamma)$ and $x \in G$ we have $r(x \delta x^{-1}) \geq 2r_\Gamma/p$. Now recall from [@BV Lemma 3.8] or [@MuellerPfaff Proposition 3.1 and (3.14)] that we have, for $t \in (0, 1]$, $$\label{kt}
|k_t^{\rho , \sigma} (x^{-1} \delta x)| \leq C t^{-d} \exp \left( -c \frac{r(x \delta x^{-1})^2}{t} \right) \leq C e^{-c'/t} \exp \left( - c'' r(x \delta x^{-1})^2\right).$$ (Here $c'$ depends on $r_\Gamma$.) From this and Lemma \[Lcount\], it follows that the geometric side of the trace formula evaluated in $k_t^{\rho ,\sigma}$ indeed absolutely converges. Moreover, it follows from together with our uniform boundedness assumption that we have: $${\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (k_t^{\rho ,\sigma}) = |H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma}) \int_{G^{\sigma} \backslash G}
k_t^{\rho , \sigma} (x^{-1} \sigma x) d\dot{x} + O (e^{-c' /t})$$ for $0 < t \leq 1$. It follows that $$\int_0^1 t^{s-1} \left( \frac{{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (k_t^{\rho ,\sigma}) }{|H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} -
\int_{G^{\sigma} \backslash G}
k_t^{\rho ,\sigma} (x^{-1} \sigma x) d\dot{x} \right) dt$$ is holomorphic is $s$ in a half-plane containing $0$, so $$\begin{gathered}
\frac12 \frac{d}{ds} {}_{|s=0} \frac{1}{\Gamma (s)} \int_0^{+\infty} t^{s-1} \left( \frac{{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (k_t^{\rho ,\sigma}) }{|H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} -
\int_{G^{\sigma} \backslash G}
k_t^{\rho ,\sigma} (x^{-1} \sigma x) d\dot{x} \right) dt \\
= \int_0^{+\infty} \left( \frac{{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (k_t^{\rho ,\sigma}) }{|H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} -
\int_{G^{\sigma} \backslash G}
k_t^{\rho ,\sigma} (x^{-1} \sigma x) d\dot{x} \right) \frac{dt}{t}.\end{gathered}$$ Now it follows from Proposition \[P:larget\] that there exists some positive $\eta$ such that $|k_t^{\rho , \sigma} (x \rtimes \sigma )| \ll e^{- \eta t} H_t^0 (x)$. In particular $|k_t^{\rho , \sigma} (x \rtimes \sigma )| \ll e^{- \eta t} H_1^0 (x)$ if $t \geq 1$ and we have: $$\frac{|{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (k_t^{\rho ,\sigma})|}{|H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} \ll e^{-\eta t} \sum_{
\{\delta \}_\Gamma } \int_{G^{\delta} \backslash G} H_1^{0} (x^{-\sigma} \delta x) d\dot{x}.$$ The above sum is absolutely convergent and independent of $t$ and $n$, implying that $$\frac{|{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (k_t^{\rho ,\sigma})|}{|H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} \ll e^{-\eta t}$$ where the implicit constant does not depend on $n$. Using Lemma \[Lkt\], we conclude that both $$\int_1^{+\infty} \frac{{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (k_t^{\rho ,\sigma}) }{|H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})}\frac{dt}{t} \mbox{ and } \int_1^{+\infty} \int_{G^{\sigma} \backslash G}
k_t^{\rho ,\sigma} (x^{-1} \sigma x) d\dot{x} \frac{dt}{t}$$ are absolutely convergent uniformly in $n.$ We are therefore reduced to proving that $$\frac{{\mathrm{trace}}\ \widetilde{R}_{\Gamma_n} (k_t^{\rho ,\sigma}) }{|H^1 (\sigma , \Gamma_n) | {\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} -
\int_{G^{\sigma} \backslash G}
k_t^{\rho ,\sigma} (x^{-1} \sigma x) d\dot{x} \to 0$$ uniformly in $t$ when $t$ belongs to a compact interval of $[0, +\infty)$. But this follows from the proof of Proposition \[WLF\] and the fact that for every $\delta \in \Gamma$ the sequence $$\left(\frac{| \{ \gamma \in \Gamma_n \backslash \Gamma \; : \; \gamma \delta \gamma^{-1} \in \widetilde{\Gamma}_n \}|}{[\Gamma^\sigma : \Gamma_n^\sigma]} \right)_{n \geq 0}$$ remains uniformly bounded by $A$ and converges to $0$ as $n$ tends to infinity.
[*Remark.*]{} Though reminiscent of a natural condition in the non-twisted case, we do not know how to check the condition on $c_{\Gamma_n} (\delta )$ stated in Theorem \[twistedtorsionlimitmultiplicity\] for any non-arithmetic $\Gamma.$ However, we show in the next section that it holds for many sequences of congruence subgroups of arithmetic groups.
Bounding the growth of $c_{\Gamma_n}(\delta)$ for congruence subgroups of arithmetic groups
===========================================================================================
We begin with a lemma to be used heavily in the proof of Proposition \[boundinggrowth\], which bounds the growth of $c_{\Gamma_p}(\delta).$
\[innerforms\] Let $\mathbf{P}$ be any connected, affine algebraic group over a finite field $k.$ If $\mathbf{P}'$ over $k$ is any inner form of $\mathbf{P},$ then $|\mathbf{P}(k)| = |\mathbf{P}'(k)|.$
We first reduce to the smooth case. Denote by $\mathbf{P}_{\rm red}$ the underlying reduced scheme of $\mathbf{P}$. We have an isomorphism $(\mathbf{P}_{\rm red} \times_k \mathbf{P}_{\rm red})_{\rm red}
\stackrel{\sim}{\to} (\mathbf{P} \times_k \mathbf{P})_{\rm red}$, see [@EGA Chap. I, Cor. 5.1.8]. Since the field $k$ is perfect, we moreover have $(\mathbf{P}_{\rm red} \times_k \mathbf{P}_{\rm red})_{\rm red}
\stackrel{\sim}{\to} \mathbf{P}_{\rm red} \times_k \mathbf{P}_{\rm red}$ so that the group law $\mathbf{P} \times \mathbf{P} \to \mathbf{P}$ induces a group law on $\mathbf{P}_{\rm red}.$ Hence, $\mathbf{P}_{\rm red}$ is a closed subgroup $k$-scheme of $\mathbf{P}.$ Since every reduced finite type scheme over a perfect field is smooth over a dense open subscheme, the standard homogeneity argument implies the group $\mathbf{P}_{\rm red}$ is smooth. But $\mathbf{P}(k) = \mathbf{P}_{\mathrm{red}}(k), \mathbf{P}'_{\mathrm{red}}(k) = \mathbf{P}'(k),$ and $\mathbf{P}_{\mathrm{red}}$ is an inner form of $\mathbf{P}'_{\mathrm{red}}.$ We may therefore assume that $\mathbf{P}$ and $\mathbf{P}'$ are smooth.
Let $\mathbf{P}$ and $\mathbf{P}'$ have respective unipotent radicals $\mathbf{U}, \mathbf{U}'$ and respective reductive quotients $\mathbf{R}, \mathbf{R}'.$ Because $H^1(k,\mathbf{U}) = H^1(k,\mathbf{U}') = 0,$ there are exact sequences of finite groups $$\begin{aligned}
1 \rightarrow \mathbf{U}(k) &\rightarrow \mathbf{P}(k) \rightarrow \mathbf{R}(k) \rightarrow 1 \\
1 \rightarrow \mathbf{U}'(k) &\rightarrow \mathbf{P}'(k) \rightarrow \mathbf{R}'(k) \rightarrow 1.\end{aligned}$$ Because $\mathbf{U}$ and $\mathbf{U}'$ are forms, their dimensions are equal, say to $d.$ Furthermore, all unipotent groups over the perfect field $k$ are split unipotent. Therefore, applying the vanishing of $H^1$ to filtrations of $\mathbf{U},\mathbf{U}'$ by subgroups whose successive quotients are $\mathbb{G}_a,$ we find that $$|\mathbf{U}(k)| = |k|^d = |\mathbf{U}'(k)|.$$ Moreover, because $\mathbf{P}, \mathbf{P}'$ are inner forms, so are $\mathbf{R}, \mathbf{R}'.$ But $H^1(k, \mathrm{Inn}(\mathbf{R})) = 0$ since $\mathrm{Inn}(\mathbf{R})$ is connected. Therefore, $\mathbf{R} \cong \mathbf{R}'$ over $k.$ In particular, $$|\mathbf{R}(k)| = |\mathbf{R}'(k)|.$$ The result follows.
Let ${\mathbf{G}}/ \mathbb{Z}[1/N]$ be a semisimple group. For ease of exposition, we also assume that ${\mathbf{G}}$ is simply connected. Let $E/\mathbb{Q}$ be a cyclic Galois extension with ${\mathrm{Gal}}(E/\mathbb{Q}) = \langle \sigma \rangle.$ Let $\mathbf{H} \subset {\mathbf{G}}$ be a connected algebraic subgroup smooth over $\mathbb{Z}[1/N].$ We fix integral structures so we may speak of ${\mathbf{G}}(\mathbb{Z}), \Gamma = {\mathbf{G}}(O_E)$ and $\mathbf{H}(\mathbb{Z}), \mathbf{H}(O_E).$
Let $p$ be a rational prime. Let $$\Gamma_p = \{ \gamma \in {\mathbf{G}}(O_E): \gamma \in \mathbf{H}(O_E / pO_E) \text{ mod } pO_E \}.$$ Every $\Gamma_p \subset {\mathbf{G}}(E_{\mathbb{R}})$ is a Galois-stable lattice.
\[boundinggrowth\] Suppose $\mathbf{H}_{\overline{{\mathbb{Q}}}}$ does not contain any normal subgroup of ${\mathbf{G}}_{\overline{{\mathbb{Q}}}}.$ Then the hypotheses of Proposition \[WLF\] and Theorem \[twistedtorsionlimitmultiplicity\] hold for the sequence $\Gamma_p.$ Namely,
- There is a uniform upper bound $$\frac{c_{\Gamma_p}(\delta)}{[\Gamma^{\sigma}:\Gamma_p^{\sigma}]} = \frac{|\{ \gamma \in \Gamma_p \backslash \Gamma: \gamma \delta \gamma^{-1}\in \widetilde{\Gamma}_p \}|}{[\Gamma^{\sigma}:\Gamma_p^{\sigma}]} \leq C$$ for some constant $C$ depending only on ${\mathbf{G}}$ and $\mathbf{H}.$
- For every $\delta,$ $$\lim_{p \rightarrow \infty} \frac{c_{\Gamma_p}(\delta)}{[\Gamma^{\sigma}:\Gamma_p^{\sigma}]} = 0.$$
We abuse notation and write $\delta = \delta \rtimes \sigma$, so that now $\delta$ belongs to $\Gamma$. We then identify $$c_{\Gamma_p}(\delta) = | \mathrm{Fix}(\delta \sigma | \mathbf{H}(O_E/p) \backslash {\mathbf{G}}(O_E/p)) | = {\mathrm{trace}}\left( \delta \sigma | \mathrm{Ind}_{\mathbf{H}(O_E/p)}^{{\mathbf{G}}(O_E/p)} 1 \right).$$ In the case where $O_E/p = \mathbb{F}_p \times \mathbb{F}_p,$ $$c_{\Gamma_p}(\delta) = {\mathrm{trace}}\left( \mathrm{Norm}(\delta) | \mathrm{Ind}_{\mathbf{H}(\mathbb{F}_p)}^{{\mathbf{G}}(\mathbb{F}_p)} 1 \right);$$ this follows, for example, from the identity $${\mathrm{trace}}\left( A_1 \otimes A_2 \circ \text{cyclic permutation} | V^{\otimes 2} \right) = {\mathrm{trace}}\left( A_1 \circ A_2 | V \right)$$ valid for arbitrarily endomorphisms $A_i$ of a finite dimensional vector space $V.$ So in this case, $$c_{\Gamma_p}(\delta) = | \mathrm{Fix}\left( \mathrm{Norm}(\delta) | \mathbf{H}(\mathbb{F}_p) \backslash {\mathbf{G}}(\mathbb{F}_p) \right)| \leq [{\mathbf{G}}(\mathbb{F}_p): \mathbf{H}(\mathbb{F}_p)] = [\Gamma^{\sigma}:\Gamma_p^{\sigma}].$$
We turn to the more interesting case where $O_E/p = k'$ is a finite field extension of $k = \mathbb{F}_p.$ The quantity $c_{\Gamma_p}(\delta)$ can be expressed explicitly as a sum over twisted conjugacy classes: $$\label{traceofinduced}
{\mathrm{trace}}\left( \delta \sigma | \mathrm{Ind}_{\mathbf{H}(O_E/p)}^{{\mathbf{G}}(O_E/p)} 1 \right) = \frac{|\mathbf{Z}_{\delta \rtimes \sigma}(k)|}{|\mathbf{H}(k')|} \times \left| \mathbf{H}(k') \cap \{ {\mathbf{G}}(k')\text{-twisted conjugacy class of $\delta$}\} \right|.$$ In , $\mathbf{Z}_{{\mathbf{G}},\delta \rtimes \sigma}$ denotes the twisted centralizer in $R_{k'/k}{\mathbf{G}}$ of $\delta.$ The set $\mathbf{H}(k') \cap \{ {\mathbf{G}}(k')\text{-twisted conjugacy class of $\delta$} \}$ is invariant under twisted conjugation and so decomposes as a finite disjoint union of $\mathbf{H}(k')$-twisted conjugacy classes. Let $\{ y \}_{y \in I}$ be a full set of representatives for these conjugacy classes.
All of the $\mathbf{H}(k)$-conjugacy classes $[\mathrm{Norm}(y)], y \in I,$ are $\mathbf{G}(k)$-conjugate to $\mathrm{Norm}(\delta).$ Furthermore, the fiber over the conjugacy class $[\mathrm{Norm}(y)]$ under the norm map is in bijection with $H^1(\sigma, \mathbf{Z}_{\mathbf{H}, y \rtimes \sigma}(k))$ [@Lan $\S 4$ Lemma 4.2]. By the vanishing of $H^1$ of connected groups over finite fields, $$H^1(\sigma,\mathbf{Z}_{y \rtimes \sigma}(k)) \cong H^1(\sigma,\pi_0(\mathbf{Z}_{\mathbf{H},y \rtimes \sigma})(k)).$$
Observe that $$\begin{aligned}
\label{firstinequality}
\frac{|\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma}(k)|}{|\mathbf{H}(k')|} \times \left| \mathbf{H}(k') \cap \{ {\mathbf{G}}(k')\text{-twisted conjugacy class of $\delta$}\} \right| &= |\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma}(k)| \times \sum_{y \in I} \frac{| \{ \mathbf{H}(k')\text{-twisted conjugacy class of } y\} | }{|\mathbf{H}(k')|} \nonumber \\
&= |\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma}(k)| \times \sum_{y \in I} \frac{1}{| \mathbf{Z}_{\mathbf{H}, y \rtimes \sigma}(k) |}.\end{aligned}$$
The groups $\mathbf{Z}_{\mathbf{H}, y \rtimes \sigma}$ and $\mathbf{Z}_{\mathbf{H},\mathrm{Norm}(y)}$ are inner forms [@AC $\S 1$ Lemma 1.1]. By Lemma \[innerforms\], $$\label{secondinequality}
\frac{1}{|\mathbf{Z}_{\mathbf{H},y \rtimes \sigma}(k)|} \leq |\pi_0(\mathbf{Z}_{\mathbf{H},\mathrm{Norm}(y)})(k)| \times \frac{1}{|\mathbf{Z}_{\mathbf{H},\mathrm{Norm}(y)}(k)|}.$$ Let $M_{\mathbf{H}}$ be the maximum of $|\pi_0(\mathbf{Z}_{\mathbf{H},\mathrm{Norm}(y)})(k)|$ and $C_{\mathbf{H},\sigma}$ the maximum of $|H^1(\sigma,\pi_0(\mathbf{Z}_{\mathbf{H}, y \rtimes \sigma})(k))|$ for all $y \in I.$ Combining and gives
$$\begin{aligned}
\label{secondinequality}
& \frac{|\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma}(k)|}{|\mathbf{H}(k')|} \times \left| \mathbf{H}(k') \cap \{ {\mathbf{G}}(k')\text{-twisted conjugacy class of $\delta$}\} \right| \nonumber \\
&\leq M_{\mathbf{H}} \times |\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma}(k)| \times \sum_{y \in I} \frac{1}{|\mathbf{Z}_{\mathrm{Norm}(y)}(k)|} \nonumber \\
&= M_{\mathbf{H}} \times |\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma}(k)| \times \sum_{y \in I} \frac{|\{ \mathbf{H}(k) \text{-conjugacy class of } \mathrm{Norm}(y) \}|}{|\mathbf{H}(k)|} \nonumber \\
&\leq M_{\mathbf{H}} \times \frac{|\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma}(k)|}{|\mathbf{Z}_{\mathbf{G},\mathrm{Norm}(\delta)}(k)|} \times C_{\mathbf{H},\sigma} \times \left( \frac{|\mathbf{Z}_{\mathbf{G},\mathrm{Norm}(\delta)}(k)|}{|\mathbf{H}(k)|} \times \left| \mathbf{H}(k) \cap \{ {\mathbf{G}}(k)\text{-conjugacy class of $\delta$}\} \right| \right) \nonumber \\
&= M_{\mathbf{H}} \times \frac{|\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma}(k)|}{|\mathbf{Z}_{\mathbf{G},\mathrm{Norm}(\delta)}(k)|} \times C_{\mathbf{H},\sigma} \times {\mathrm{trace}}\left( \mathrm{Norm}(\delta) | \mathrm{Ind}_{\mathbf{H}(k)}^{\mathbf{G}(k)} 1 \right).\end{aligned}$$
Because $\mathbf{Z}_{{\mathbf{G}}, \delta \rtimes \sigma}$ and $\mathbf{Z}_{{\mathbf{G}}, \mathrm{Norm}(\delta)}$ are inner forms [@AC $\S 1$ Lemma 1.1], Lemma \[innerforms\] gives $$\label{thirdinequality}
\frac{|\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma}(k)|}{|\mathbf{Z}_{\mathbf{G},\mathrm{Norm}(\delta)}(k)|} \leq |\pi_0(\mathbf{Z}_{\mathbf{G},\delta \rtimes \sigma})(k)| \leq M_{{\mathbf{G}},\sigma},$$ where $M_{\mathbf{G},\sigma}$ is the maximum of $\pi_0(Z_{{\mathbf{G}}, \delta \rtimes \sigma})$ over all $\delta \in {\mathbf{G}}(k').$ Combining , , and gives $$\begin{aligned}
c_{\Gamma_p}(\delta) = {\mathrm{trace}}\left( \delta \sigma | \mathrm{Ind}_{\mathbf{H}(O_E/p)}^{{\mathbf{G}}(O_E/p)} 1 \right) &\leq M_{\mathbf{H}} \times M_{{\mathbf{G}},\sigma} \times C_{\mathbf{H},\sigma} \times {\mathrm{trace}}\left( \mathrm{Norm}(\delta) | \mathrm{Ind}_{\mathbf{H}(k)}^{\mathbf{G}(k)} 1 \right) \\
&= M_{\mathbf{H}} \times M_{{\mathbf{G}},\sigma} \times C_{\mathbf{H},\sigma} \times |\mathrm{Fix}\left(\mathrm{Norm}(\delta) | \mathbf{H}(k) \backslash {\mathbf{G}}(k) \right) |.\end{aligned}$$
In Lemma \[centralizercomponents\], we show that the number of geometric components of the centralizer of every element of $\mathbf{H}(\overline{k})$ or $\mathbf{G}(\overline{k})$ is uniformly bounded over all $p$; this proves the same for twisted centralizers too, since the twisted centralizer of $\delta$ is an (inner) form of the centralizer of $\mathrm{Norm}(\delta).$ Thus, $M_{\mathbf{H}}, M_{{\mathbf{G}},\sigma},$ and $C_{\mathbf{H},\sigma}$ are bounded by some constant $M = M({\mathbf{G}}, \mathbf{H})$ depending only on ${\mathbf{G}}$ and $\mathbf{H}.$ We are reduced to bounding the right side of $$\frac{c_{\Gamma_p}}{[\Gamma^{\sigma}:\Gamma_p^{\sigma}]} \leq M \times \frac{ |\mathrm{Fix}\left(\mathrm{Norm}(\delta) | \mathbf{H}(k) \backslash {\mathbf{G}}(k) \right) |}{|\mathbf{H}(k) \backslash {\mathbf{G}}(k) |}.$$
- Evidently, $$M \times \frac{ |\mathrm{Fix}\left(\mathrm{Norm}(\delta) | \mathbf{H}(k) \backslash {\mathbf{G}}(k) \right) |}{|\mathbf{H}(k) \backslash {\mathbf{G}}(k) |} \leq M,$$ uniformly for all $\delta$ and all $p.$
- To ease notation, let $\gamma = \mathrm{Norm}(\delta).$ By Lang’s theorem, $$\frac{ |\mathrm{Fix}\left(\gamma | \mathbf{H}(k) \backslash {\mathbf{G}}(k) \right) |}{|\mathbf{H}(k) \backslash {\mathbf{G}}(k) |} = \frac{ |\mathrm{Fix}\left(\gamma | (\mathbf{H} \backslash {\mathbf{G}}) \right)(k) |}{|(\mathbf{H} \backslash {\mathbf{G}})(k) |}.$$ If $\gamma$ acts trivially on $\mathbf{H} \backslash {\mathbf{G}},$ then $\gamma \in \bigcap_{g \in {\mathbf{G}}(\overline{k})} g \mathbf{H}_{\overline{{\mathbb{Q}}}}g^{-1},$ a normal subgroup of ${\mathbf{G}}_{\overline{{\mathbb{Q}}}}$ contained in $\mathbf{H}_{\overline{{\mathbb{Q}}}},$ implying that $\gamma = 1$ by hypothesis. Therefore, $\mathrm{Fix}(\gamma | \mathbf{H} \backslash {\mathbf{G}})$ is a proper subvariety of $\mathbf{H} \backslash {\mathbf{G}}$ for every $\gamma \neq 1.$ Since $\mathbf{H} \backslash {\mathbf{G}}$ is irreducible, $\mathrm{Fix}(\gamma | \mathbf{H} \backslash {\mathbf{G}})$ must have strictly positive codimension in $\mathbf{H} \backslash {\mathbf{G}}.$ It follows that $$\frac{c_{\Gamma_p}(\delta)}{[\Gamma^{\sigma}:\Gamma_p^{\sigma}]} \leq M \times \frac{|\mathrm{Fix}(\mathrm{Norm}(\delta)| \mathbf{H} \backslash {\mathbf{G}})(\mathbb{F}_p)|}{(\mathbf{H} \backslash {\mathbf{G}})(\mathbb{F}_p)} \xrightarrow{p \rightarrow \infty} 0.$$
\[centralizercomponents\] Let ${\mathbf{G}}$ be an affine algebraic group over an arbitrary field $F.$ The number of components of $\mathbf{Z}_{{\mathbf{G}},x},$ where $x$ ranges over all elements of ${\mathbf{G}}(F),$ is uniformly bounded.
${\mathbf{G}}$ is a closed subvariety of ${\mathrm{SL}}_n \subset \mathrm{End}_n$ for some fixed $n.$ Suppose ${\mathbf{G}}$ is the simultaneous vanishing locus of polynomials $f_1,\ldots,f_m$ on the vector space $\mathrm{End}_n.$ Let $f_i$ have degree $d_i,$ the maximum degree of all the monomials in its support. Note that $$\mathbf{Z}_{{\mathbf{G}},x} = \mathbf{Z}_{\mathrm{End}_n,x} \cap V(f_1,\ldots,f_m).$$ Clearly, $\mathbf{Z}_{{\mathbf{G}},x}$ is Zariski open and dense in its projective completion $$\widetilde{\mathbf{Z}_{{\mathbf{G}},x}} = \mathbb{P}(\mathbf{Z}_{\mathrm{End}_n,x}) \cap V(\widetilde{f}_1,\ldots,\widetilde{f}_n)$$ obtained by adding a hyperplane at infinity to $\mathrm{End}_n.$ Therefore, the number of connected components of $\widetilde{\mathbf{Z}_{{\mathbf{G}},x}}$ equals the number of connected components of $\mathbf{Z}_{{\mathbf{G}},x}.$ Thus, $$\begin{aligned}
|\pi_0(\mathbf{Z}_{{\mathbf{G}},x})| &= | \pi_0(\widetilde{\mathbf{Z}_{{\mathbf{G}},x}}) | \\
&\leq \text{degree of } \mathbb{P}(\mathbf{Z}_{\mathrm{End}_n,x}) \cap V(\widetilde{f}_1) \cap \cdots \cap V(\widetilde{f}_n) \\
&\leq 1 \cdot d_1 \cdots d_m, \end{aligned}$$ where the final inequality follows by Bézout’s theorem. This upper bound is independent of $x.$
[*Remark.*]{} The proof of Proposition \[boundinggrowth\] is an adaptation of Shintani’s arguments [@Shintani] proving the existence of a “base change transfer" $$\text{representations of } {\mathrm{GL}}_n(k) \leadsto \text{Galois-fixed representations of } {\mathrm{GL}}_n(k').$$ On the one hand, he sidesteps all component and endoscopy issues by working with ${\mathbf{G}}= {\mathrm{GL}}_n, \mathbf{H} =$ parabolic subgroup. On the other hand, he proves an exact trace identity between matching principal series representations, an analogue of the fundamental lemma for ${\mathrm{GL}}_n(k).$ We plan to pursue this analogue of the fundamental lemma for more general finite groups of Lie-type in a subsequent paper.
In the next sections we compute the $L^2$-Lefschetz numbers and twisted $L^2$-torsion and in particular prove Lemma \[Lkt\]. We distinguish two cases: we first deal with the case where $\mathbb{E} = \mathbb{R}^p$ (the product case) and then deal with case where $\mathbb{E} = \mathbb{C}$. The general case easily reduces to these two cases.
Computations on a product
=========================
Here we suppose that $\mathbb{E} = \mathbb{R}^p$. Then $G$ is the $p$-fold product of ${\mathbf{G}}(\mathbb{R})$ and $\sigma$ cyclically permutes the factors of $G$. We will abusively denote by $G^{\sigma}$ the group ${\mathbf{G}}(\mathbb{R})$. Let $(\rho_0 , F_0)$ be an irreducible complex linear representation of ${\mathbf{G}}(\mathbb{C})$. We denote by $(\widetilde{\rho} , F)$ the corresponding complex finite dimensional $\sigma$-stable irreducible representation of $\widetilde{G}$. Recall that $F = F_0^{\otimes p}$, that $G$ acts by the tensor product representation $\rho_0^{\otimes p}$ and that $\sigma$ acts by the cyclic permutation $A: x_1 \otimes \ldots \otimes x_p \mapsto x_p \otimes x_1 \otimes \ldots \otimes x_{p-1}$. We finally let $X$ and $X^{\sigma}$ be the symmetric spaces corresponding to $G$ and $G^{\sigma}$ respectively, so $X = (X^{\sigma})^p$.
Heat kernels of a product
-------------------------
The heat kernels $H_t^{\rho, j}$ decompose as $$\label{productheatkernel}
H_t^{\rho , j}(g_1,\cdots,g_p) = \sum_{a_1 + \ldots + a_p = j} H_t^{\rho_0 , a_1}(g_1) \otimes \cdots \otimes H_t^{\rho_0 , a_p}(g_p).$$ Now the twisted orbital integral of $H_t^{\rho , j}$ associated to the class of the identity element is given by $$\left(\int_{G^{\sigma} \backslash G} H_t^{\rho , j} (g^{- \sigma} g) dg \right) \circ A_{\sigma}.$$ But because $H_t^{\rho , j} (g^{- \sigma} g)$ preserves all of the summands in the decomposition of and $\sigma$ maps the $(a_1, \ldots ,a_p)$-summand to the $(a_p,a_1,\ldots ,a_{p-1})$-summand, only those summands for which $j = pa$ and $a_1 = \ldots = a_p = a$ can contribute to the trace of the above twisted orbital integral. Furthermore, by a computation identical to that done for scalar-valued functions in [@Lan §8], we see that $$\left( \int_{G^{\sigma} \backslash G} H_t^{\rho_0 ,a}(g_p^{-1}g_1) \otimes \ldots \otimes H_t^{\rho_0 , a}(g_{p-1}^{-1}g_p) dg \right) \circ A_\sigma = (-1)^{a^2(p-1)} H_t^{\rho_0 ,a} * \ldots * H_t^{\rho_0 ,a} (e).$$ This implies that $$\begin{aligned}
\int_{G^{\sigma} \backslash G} h_t^{\rho , pa}(x^{-1} \sigma x) d\dot{x} &=& {\mathrm{trace}}\left[ \left( \int_{G^{\sigma} \backslash G} H_t^{\rho_0 , a}(g_p^{-1}g_1) \otimes \ldots \otimes H_t^{\rho_0 , a}(g_{p-1}^{-1}g_p) d\dot{g} \right) \circ A_\sigma \right] \\
&=& (-1)^{a^2(p-1)} {\mathrm{trace}}\left( H_t^{\rho_0 ,a} * \ldots * H_t^{\rho_0 ,a} (e) \right) \\
&=& (-1)^{a^2(p-1)} {\mathrm{trace}}\left( H_{pt}^{\rho_0 ,a} (e) \right) \\
&=& (-1)^{a^2(p-1)} h_{pt}^{\rho_0 ,a} (e). \end{aligned}$$ Here $H_{pt}^{\rho_0 ,a}$ is an [*untwisted*]{} heat kernel on $X^{\sigma}$. Lemma \[Lkt\] therefore follows from standard estimates (see e.g. [@BV]). Moreover, computations of the $L^2$-Lefschetz number and of the twisted $L^2$-torsion immediately follow from the above explicit computation.
\[T:62\] We have: $$\mathrm{Lef}^{(2)} (\sigma , X , F) = \left\{ \begin{array}{ll}
(-1)^{\frac12 \dim X^{\sigma}} (\dim F_0 ) \frac{\chi (X_u^{\sigma})}{{\mathrm{vol}}(X_u^{\sigma})} & \mbox{ if } \delta (G^{\sigma}) = 0, \\
0 & \mbox{ if not}.
\end{array} \right.$$ Here $X_u^{\sigma}$ is the compact dual of $X^{\sigma}$ whose metric is normalized such that multiplication by $i$ becomes an isometry $T_{eK^{\sigma}} (X^{\sigma}) \cong {\mathfrak{p}}\to i {\mathfrak{p}}\cong T_{eK^{\sigma}} (X_u^{\sigma})$.
First note[^3] that $$\begin{aligned}
\mathrm{Lef}^{(2)} (\sigma , X , F) & = & \lim_{t \to +\infty} \int_{G^{\sigma} \backslash G} k_{t}^{\rho}(x^{-1} \sigma x) d\dot{x} \\
& = & \lim_{t \to +\infty} \sum_a (-1)^{pa} \int_{G^{\sigma} \backslash G} h_{t}^{\rho, pa}(x^{-1} \sigma x) d\dot{x} \\
& = & \lim_{t \to +\infty} \sum_a (-1)^{a} h_{pt}^{\rho_0 ,a} (e).\end{aligned}$$ The computation then reduces to the untwisted case for which we refer to [@Olbricht].
The computation of the twisted $L^2$-torsion similarly reduces to the untwisted case:
\[twistedtorsionproduct\] We have: $$t_{X}^{(2)\sigma}(\rho) = p \cdot t_{X^{\sigma}}^{(2)}(\rho).$$
Computations in the case $\mathbb{E} = \mathbb{C}$
==================================================
Throughout this section, $\mathbb{E} = \mathbb{C}$. Then $G= {\mathbf{G}}({\mathbb{C}})$ is the group of complex points, $\sigma : G \to G$ is the real involution given by the complex conjugation and $G^{\sigma} = {\mathbf{G}}({\mathbb{R}})$. Recall that we fix a choice of Cartan involution $\theta$ of $G$ that commutes with $\sigma$.
Irreducible $\sigma$-stable tempered representations of $G$
-----------------------------------------------------------
Choose $\theta$-stable representatives ${\mathfrak{h}}_1^0 , \ldots , {\mathfrak{h}}_s^0$ of the ${\mathbf{G}}({\mathbb{R}})$-conjugacy classes of Cartan subalgebras in the Lie algebra ${\mathfrak{g}}^0$ of ${\mathbf{G}}({\mathbb{R}})$. For each $j \in \{1 , \ldots , s\}$ we write ${\mathfrak{h}}_j^0 = {\mathfrak{t}}_j \oplus {\mathfrak{a}}_j$ for the decomposition of ${\mathfrak{h}}_j^0$ w.r.t. $\theta$, i.e. ${\mathfrak{a}}_j$ is the split part of ${\mathfrak{h}}_j^0$ and ${\mathfrak{t}}_j$ is the compact part of ${\mathfrak{h}}_j^0$. We denote by ${\mathfrak{h}}_j$ the complexification of ${\mathfrak{h}}_j^0$; note that ${\mathfrak{a}}_j \oplus i {\mathfrak{t}}_j$ and ${\mathfrak{t}}_j \oplus i {\mathfrak{a}}_j$ are resp. the split and compact part of ${\mathfrak{h}}_j$.
We now fix some $j$. To ease notations we will omit the $j$ index. Choose a Borel subgroup $B$ of $G={\mathbf{G}}({\mathbb{C}})$ containing the torus $H$ which corresponds to ${\mathfrak{h}}_j$. Let $A$ and $T$ be resp. the split and compact tori corresponding to ${\mathfrak{a}}\oplus i {\mathfrak{t}}$ and ${\mathfrak{t}}\oplus i {\mathfrak{a}}$. Write $\mu$ for the differential of a character of $T$ and $\lambda$ for the differential of a character of $A$. Note that $\mu$ is $\sigma$-stable if and only if $\mu$ is zero on $i {\mathfrak{a}}$.
Associated to $(\mu , \lambda)$ is a representation $$\pi_{\mu, \lambda} = \mathrm{ind}_{B}^{G} (\mu \otimes \lambda \otimes 1).$$
\[Delorme\] Every irreducible $\sigma$-stable tempered representation of $G$ is equivalent to some $\pi_{\mu , \lambda}$ as above (for some $j$) where $\mu$ is zero on $i {\mathfrak{a}}_j$ and $\lambda$ is zero on ${\mathfrak{t}}_j$ and has pure imaginary image.
Note that if $\lambda$ is zero on ${\mathfrak{t}}_j$ we may think of $\lambda$ as a real linear form ${\mathfrak{a}}\to {\mathbb{C}}$.
We denote by $I_{\mu , \lambda}$ the underlying $({\mathfrak{g}}, K)$-module. It is $\sigma$-stable and Delorme [@Delorme §5.3] define a particular extension to a $({\mathfrak{g}}, \widetilde{K})$-module, but we won’t follow his convention here (see Convention I below).
Computations of the Lefschetz numbers
-------------------------------------
If ${\mathbf{G}}({\mathbb{R}})$ has no discrete series Delorme [@Delorme Proposition 7] proves that for any admissible $({\mathfrak{g}}, \widetilde{K})$-module and any finite dimensional representation $(\widetilde{\rho} , F)$ of $\widetilde{G}$, we have: $$\mathrm{Lef}(\sigma , F , V) = 0.$$ Even if ${\mathbf{G}}({\mathbb{R}})$ has discrete series Delorme’s proof — see also [@RohlfsSpeh Lemma 4.2.3] — shows that $$\mathrm{Lef} (\sigma , F , I_{\mu , \lambda}) = 0$$ unless ${\mathfrak{h}}^0 = {\mathfrak{t}}$ is a compact Cartan subalgebra (so that $i {\mathfrak{t}}$ is the split part of ${\mathfrak{h}}$). In the latter case $\lambda=0$ (recall that $I_{\mu , \lambda}$ is assumed to be $\sigma$-stable); we will simply denote by $I_\mu$ the $({\mathfrak{g}}, \widetilde{K})$-module $I_{\mu, 0}$. The following proposition — due to Delorme [@Delorme Th. 2][^4] — computes the Lefschetz numbers in the remaining cases.
\[P:delorme\] We have: $$\mathrm{Lef} (\sigma , F , I_\mu ) = \left\{ \begin{array}{ll}
\pm 2^{\dim {\mathfrak{t}}} & \mbox{ if } w \mu = 2 (\nu + \rho)_{| {\mathfrak{t}}} \quad (w \in W) \\
0 & \mbox{ otherwise}.
\end{array} \right.$$ Here $W$ is the Weyl group of $({\mathfrak{g}}, {\mathfrak{h}})$ and the sign depends on the chosen extension of $I_{\mu}$ to a $({\mathfrak{g}}, \widetilde{K})$-module.
### Convention I {#convention-i .unnumbered}
In the following we will always assume that the extension of a $\sigma$-discrete $I_{\mu}$ to a $({\mathfrak{g}}, \widetilde{K})$-module is s.t. that the sign in Proposition \[P:delorme\] is positive. (See [@RohlfsSpeh §4.2.5] for more details.)
Computations of the twisted $({\mathfrak{g}}, K)$-torsion
---------------------------------------------------------
Consider an arbitrary irreducible $\sigma$-stable tempered representation of $G$ associated to some $j$ and some $(\mu, \lambda)$ as in Proposition \[Delorme\]. Let $\mathbf{P}$ be the parabolic subgroup of ${\mathbf{G}}$ whose Levi subgroup $\mathbf{M} = {}^0 \mathbf{M} \mathbf{A}_P$ is the centralizer in ${\mathbf{G}}$ of ${\mathfrak{a}}$. We have $B \subset \mathbf{P} ({\mathbb{C}})$ and we may write $\pi_{\mu , \lambda}$ as the induced representation $$\pi_{\mu , \lambda} = \mathrm{ind}_{\mathbf{P} ({\mathbb{C}})}^{{\mathbf{G}}({\mathbb{C}})} ( \pi_{\mu}^{{}^0 \mathbf{M} ({\mathbb{C}})} \otimes \lambda),$$ where $$\pi_{\mu,0}^{{}^0 \mathbf{M} ({\mathbb{C}})} = \mathrm{ind}_{B \cap {}^0 \mathbf{M} ({\mathbb{C}})}^{{}^0 \mathbf{M} ({\mathbb{C}})} (\mu_{|{\mathfrak{t}}} \otimes 0)$$ is a tempered ($\sigma$-discrete) representation of ${}^0 \mathbf{M} ({\mathbb{C}})$ and we think of $\lambda$ — seen as real linear form ${\mathfrak{a}}\to {\mathbb{C}}$ — as (the differential of) a character of $\mathbf{A}_P ({\mathbb{C}})$.
### Convention II {#convention-ii .unnumbered}
In the following we fix the extension of $I_{\mu , \lambda}$ to a $({\mathfrak{g}}, \widetilde{K})$-module to be the one associated to the interwining operator $A_G = \mathrm{ind} _{\mathbf{P} ({\mathbb{C}})}^{{\mathbf{G}}({\mathbb{C}})} (A_M \otimes 1)$ where $A_M$ is chosen according to Convention I.
Let $K_M = K \cap {}^0 \mathbf{M} ({\mathbb{C}})$. Since $\sigma$ stabilizes ${}^0 \mathbf{M} ({\mathbb{C}})$, $\mu$, [*etc*]{}$\ldots$ it follows from Frobenius reciprocity and that we have: $$\label{Lef'2}
\mathrm{Lef} {}' (\sigma , F , I_{\mu , \lambda}) =\mathrm{trace} \left( \sigma \; | \; \left[ {\det{}'}[1 - ({\mathfrak{g}}/ {\mathfrak{k}})^*] \otimes F \otimes \pi_{\mu,0}^{{}^0 \mathbf{M} ({\mathbb{C}})} \right]^{K_M} \right).$$ Write $${\mathfrak{g}}/ {\mathfrak{k}}= {}^0\mathfrak{m} / {\mathfrak{k}}_M \oplus {\mathfrak{a}}\oplus \mathfrak{n}.$$
It follows from that — as a $\widetilde{K}_M$-module — we have: $${\det{}'}[1-({\mathfrak{g}}/ {\mathfrak{k}})^* ] = \det[1-({}^0\mathfrak{m} / {\mathfrak{k}}_M)^*] \otimes {\det{}'}[1- {\mathfrak{a}}^*\oplus \mathfrak{n}^*] \oplus {\det{}'}[1-({}^0\mathfrak{m}/ {\mathfrak{k}}_M)^*] \otimes \det [1- {\mathfrak{a}}^*\oplus \mathfrak{n}^*],$$ with $$\label{tracezeropart}
\det [1- {\mathfrak{a}}^*\oplus \mathfrak{n}^*]= \det [1- {\mathfrak{a}}^*] \otimes \det [1- \mathfrak{n}^*]$$ and $${\det{}'}[1- {\mathfrak{a}}^*\oplus \mathfrak{n}^*] = \det[1- \mathfrak{n}^*] \otimes {\det{}'}[1- {\mathfrak{a}}^*] \oplus {\det{}'}[1- \mathfrak{n}^*] \otimes \det [1- {\mathfrak{a}}^*].$$ Now two simple lemmas:
\[L:triv\] (1) We have $\det [1- {\mathfrak{a}}^*] = 0$ as a virtual $\widetilde{K}_M$-module unless ${\mathfrak{a}}^{\sigma} = 0$.
\(2) We have ${\det{}'}[1- {\mathfrak{a}}^*] = 0$ as a virtual $\widetilde{K}_M$-module unless $\dim {\mathfrak{a}}^{\sigma} \leq 1$.
For any $\delta \in \widetilde{K}_M,$ $${\mathrm{trace}}(\delta | \det[1-{\mathfrak{a}}^*]) = \det(1-\delta | {\mathfrak{a}}^*), {\mathrm{trace}}( \delta | {\det{}'}[1-{\mathfrak{a}}^*]) = \frac{d}{dt}|_{t = 1} \det(t \cdot 1 - \delta | {\mathfrak{a}}^*)$$ cf. §\[lefschetznumber\]. Write $\delta = \epsilon k,$ where $\epsilon \in \{ 1, \sigma \}$ and $k \in K_M.$
For any $X \in {\mathfrak{a}}^{*},$ $$\delta X = \epsilon k X = \epsilon X$$ since $K_M$ centralizes ${\mathfrak{a}}^*.$ Thus, $$\dim \{ \text{+1-eigenspace of } \delta \} \geq \dim \; ({\mathfrak{a}}^*)^{\sigma}.$$ In particular,
- if $\dim {\mathfrak{a}}^{\sigma} > 0,$ then $$\det(1 - \delta | {\mathfrak{a}}^*) = 0.$$
- if $\dim {\mathfrak{a}}^{\sigma} > 1,$ then $\det(t \cdot 1 - \delta | {\mathfrak{a}}^*)$ vanishes to order at least 2 at $t = 1,$ whence $$\frac{d}{dt}|_{t = 1}\det(t \cdot 1 - \delta | {\mathfrak{a}}^*) = 0.$$
\[virtuallytrivial\] Let $V$ be a finite dimensional $\widetilde{K}_M$-module and $\tau$ any admissible $\widetilde{K}_M$-module, i.e. a $\widetilde{K}_M$-module all of whose $K_M$-isotypic subspaces are finite dimensional. Suppose $V$ is virtually trivial. Then $[V \otimes \tau]^{K_M}$ is finite dimensional and $${\mathrm{trace}}(\sigma | [V \otimes \tau]^{K_M}) = 0.$$
Finite dimensionality is immediate since $\tau$ is admissible. Let $\zeta$ be a finite dimensional subrepresentation of $\tau$ such that $[V \otimes \tau]^{K_M} = [V \otimes \zeta]^{K_M}.$
Since $K_M$ is compact, taking $K_M$-invariants is an exact functor from the category of finite dimensional $K_M$-modules to the category of finite dimensional $\sigma$-modules. Virtually trivial $\widetilde{K}_M$-modules therefore map to virtually trivial $\sigma$-modules. Thus, $[V \otimes \zeta]^{K_M}$ is virtually trivial. In particular, $${\mathrm{trace}}(\sigma | [V \otimes \tau]^{K_M}) = {\mathrm{trace}}(\sigma | [V \otimes \zeta]^{K_M}) = 0.$$
In particular we conclude that is zero unless $\dim {\mathfrak{a}}\leq 1$. In the following we compute in the two remaining cases.
Computation of when $\dim {\mathfrak{a}}= 1$
--------------------------------------------
Assume that $\dim {\mathfrak{a}}=1$. It then follows from Lemmas \[L:triv\] and \[virtuallytrivial\] that
$${\mathrm{trace}}(\sigma \; | \; {\det{}'}[1-({\mathfrak{g}}/ {\mathfrak{k}})^* ]) = {\mathrm{trace}}(\sigma \; | \; \det[1-({}^0\mathfrak{m} / {\mathfrak{k}}_M)^*] \otimes {\det{}'}[1- {\mathfrak{a}}^*\oplus \mathfrak{n}^*]).$$ We can therefore compute $$\label{Lef'3}
\begin{split}
\mathrm{Lef} {}' (\sigma , F , I_{\mu , \lambda}) & = {\mathrm{trace}}\left( \sigma \; | \; \left[ \det[1-({}^0\mathfrak{m}/ {\mathfrak{k}}_M)^*] \otimes {\det{}'}[1- {\mathfrak{a}}^*\oplus \mathfrak{n}^*] \otimes F \otimes \pi_{\mu,0}^{{}^0 \mathbf{M} ({\mathbb{C}})} \right]^{K_M} \right) \\
& = \mathrm{Lef} (\sigma , {\det{}'}[1- {\mathfrak{a}}^* \oplus \mathfrak{n}^* ] \otimes F , I_{\mu}^{{}^0 \mathbf{M} ({\mathbb{C}})}).
\end{split}$$
For each $w \in W$ we let $\nu_w$ be the restriction of $w(\rho + \nu)$ to ${\mathfrak{t}}$. Let $[W_{K_M} \backslash W]$ be the set of $w \in W_U$ such that $\nu_w$ is dominant as a weight on ${\mathfrak{t}}$ (with respect to the roots of ${\mathfrak{t}}$ on ${\mathfrak{k}}_M$), i.e.: $$[W_{K_M} \backslash W] = \{ w \in W \; : \; \langle \nu_w, \beta \rangle \geq 0 \mbox { for all } \beta \in \Delta^{+}(\mathfrak{t}, \mathfrak{k}_M) \}.$$ This is therefore a set of coset representatives for $W_{K_M}$ in $W$.
\[P68\] Assume $\dim {\mathfrak{a}}= 1$. Then we have: $$\mathrm{Lef} {}' (\sigma , F , I_{\mu , \lambda}) = \left\{ \begin{array}{ll}
\mathrm{sgn} (w) 2^{\dim {\mathfrak{t}}} & \mbox{ if } \mu = 2\nu_w \mbox{ for some } w \in [W_{K_M} \backslash W] \\
0 & \mbox{ otherwise}.
\end{array} \right.$$
We shall apply Proposition \[P:delorme\] to the twisted space associated to ${}^0 \mathbf{M} ({\mathbb{C}})$ to compute . To do so directly, we would need to decompose the virtual representation $$\label{*}
{\det{}'}[1- {\mathfrak{a}}^* \oplus \mathfrak{n}^*] \otimes F$$ into irreducibles. This can be done by hand by reducing to the cases where ${\mathbf{G}}$ is simple of type ${\mathrm{SO}}(p,p)$ with $p$ odd, or of type ${\mathrm{SL}}(3)$. Instead we note that Proposition \[P:delorme\] implies that only the essential $\sigma$-stable subrepresentations of contribute to the final expression in . We may therefore reduce to considering the virtual representation $$({\det{}'}[1- {\mathfrak{a}}_0^* \oplus {\mathfrak{n}}_0^*] \otimes ({\det{}'}[1-{\mathfrak{a}}_0^* \oplus {\mathfrak{n}}_0^*])^{\sigma}) \otimes (F_0 \otimes F_0^{\sigma}) = ({\det{}'}[1- {\mathfrak{a}}_0^* \oplus {\mathfrak{n}}_0^*] \otimes F_0) \otimes ({\det{}'}[1- {\mathfrak{a}}_0^* \oplus {\mathfrak{n}}_0^*] \otimes F_0)^{\sigma}.$$ Here we have realized the $\widetilde{K}_M$-representations ${\mathfrak{a}}$ and ${\mathfrak{n}}$ as representations in ${\mathfrak{a}}_0 \otimes {\mathfrak{a}}_0^{\sigma}$ and ${\mathfrak{n}}_0 \otimes {\mathfrak{n}}_0^{\sigma}$, resp.
Now it follows from Proposition \[P:delorme\] that $\mathrm{Lef} (\sigma , F , I_{\mu , \lambda}) =0$ unless $\mu = 2\mu_0$ where $\mu_0-\rho$ is the highest weight of a finite dimensional representation of ${}^0 \mathbf{M} ({\mathbb{R}})$. Next we use that if $\theta_0$ denotes the discrete series of ${}^0 \mathbf{M} ({\mathbb{R}})$ with infinitesimal character $\mu_0$ and $H_0$ a finite dimensional representation of ${}^0 \mathbf{M} ({\mathbb{R}})$ of highest weight $\nu$ then the (untwisted) Euler-Poincaré characteristic $$\begin{split}
\chi (H_0 , \theta_0) & := \dim [\theta_0 \otimes \det[ 1- ({}^0\mathfrak{m} ({\mathbb{R}})/ {\mathfrak{k}}_{M, {\mathbb{R}}})^* ] \otimes H_0]^{K_{M}^{\sigma}} \\
& = \left\{ \begin{array}{ll}
(-1)^{\frac12 \dim ({}^0\mathfrak{m} ({\mathbb{R}})/ {\mathfrak{k}}_{M, {\mathbb{R}}})} & \mbox{ if } w \mu_0 = \nu + \rho \quad (w \in W_M) \\
0 & \mbox{ otherwise}.
\end{array} \right.
\end{split}$$ We can extend $\chi (\cdot , \theta_0)$ to any virtual representation. Applying the above to $H_0 = {\det{}'}[1-{\mathfrak{a}}_0^* \oplus {\mathfrak{n}}_0^*] \otimes F_0$, it follows from Proposition \[P:delorme\] and that we have: $$\begin{split}
\mathrm{Lef} {}' (\sigma , F , I_{\mu , \lambda}) & = 2^{\dim {\mathfrak{t}}} \times \# \{ \text{irreducible } \sigma \text{-stable } {}^0 \mathbf{M}({\mathbb{C}}) \text{ subrepresentations of } {\det{}'}[1-\mathfrak{a}^* \oplus \mathfrak{n}^*] \otimes F \\
& \quad \text{ with same infinitesimal character as } I_{2\mu_0,0} \} \\
&= 2^{\dim {\mathfrak{t}}} \times \# \{ \text{irreducible } \sigma \text{-stable } {}^0 \mathbf{M}({\mathbb{C}}) \text{ subrepresentations of } \\
& \quad ({\det{}'}[1- \mathfrak{a}_0^* \oplus \mathfrak{n}_0^*] \otimes F_0) \otimes (\mathrm{det}'[1- \mathfrak{a}_0^* \oplus \mathfrak{n}_0^*] \otimes F_0)^{\sigma} \text{ with same infinitesimal character as } I_{2\mu_0,0} \} \\
&= 2^{\dim {\mathfrak{t}}} \times \# \{ \text{irreducible } {}^0 \mathbf{M}({\mathbb{R}}) \text{ subrepresentations of } {\det{}'}[1- \mathfrak{a}_0^* \oplus \mathfrak{n}_0^* ] \otimes F_0 \\
& \quad \text{ with infinitesimal character equal to that of } \theta_0 \} \\
&= 2^{\dim {\mathfrak{t}}} (-1)^{\frac12 \dim ({}^0\mathfrak{m} ({\mathbb{R}})/ {\mathfrak{k}}_{M, {\mathbb{R}}})} \times \chi( {\det{}'}[1- \mathfrak{a}_0^* \oplus \mathfrak{n}_0^* ] \otimes F_0, \theta_0) \\
& = 2^{\dim {\mathfrak{t}}} (-1)^{\frac12 \dim ({}^0\mathfrak{m} ({\mathbb{R}})/ {\mathfrak{k}}_{M, {\mathbb{R}}})} \dim [\theta_0 \otimes \det[ 1- ({}^0\mathfrak{m} ({\mathbb{R}})/ {\mathfrak{k}}_{M, {\mathbb{R}}}) ] \otimes {\det{}'}[1-{\mathfrak{a}}_{0}^* \oplus {\mathfrak{n}}_{0}^*] \otimes F_0]^{K_{M}^{\sigma}} \\
& = 2^{\dim {\mathfrak{t}}} (-1)^{\frac12 \dim ({}^0\mathfrak{m} ({\mathbb{R}})/ {\mathfrak{k}}_{M, {\mathbb{R}}})+1} \dim [I_{\mu_0} \otimes {\det{}'}[1 - ({\mathfrak{g}}({\mathbb{R}})/ {\mathfrak{k}}_{{\mathbb{R}}})^*] \otimes F_0]^{K^{\sigma}} \\
& = 2^{\dim {\mathfrak{t}}} (-1)^{\frac12 \dim ({}^0\mathfrak{m} ({\mathbb{R}})/ {\mathfrak{k}}_{M, {\mathbb{R}}})} \times \mathrm{det}'(F_0, I_{\mu_0}).
\end{split}$$ We are therefore reduced to the untwisted case. And the proposition finally follows from the computations made in [@BV §5.6].
[*Remark.*]{} 1. The proof above and the transfer of infinitesimal characters under base change shows that $$(\text{twisted heat kernel for } F)(2t) \xrightarrow{\text{transfer}} 2^{\dim {\mathfrak{t}}} (-1)^{\frac12 \dim ({}^0\mathfrak{m} ({\mathbb{R}})/ {\mathfrak{k}}_{M, {\mathbb{R}}})} \times ( \text{heat kernel for } F_0)(t)$$ for base change ${\mathbb{C}}/ {\mathbb{R}}$.
2\. Note that this base change calculation includes, as a special case, that of a product ${\mathbf{G}}= {\mathbf{G}}' \times {\mathbf{G}}'$. But we’ve worked out separately that $$\mathrm{Lef}'(\sigma,F_0 \otimes F_0, \pi \otimes \pi) = 2 \mathrm{det}'(F_0, \pi).$$ There is no contradiction here: if either side is non zero then $\dim {\mathfrak{a}}= 1,$ but in that special case $\dim {\mathfrak{a}}= \dim {\mathfrak{t}}$ since the real group is in fact a complex group.
Computation of when $\dim {\mathfrak{a}}= 0$
--------------------------------------------
We now assume that $\dim {\mathfrak{a}}=0$ and follow an observation made by Mueller and Pfaff [@MuellerPfaff]. In that case $\mathbf{M} = {\mathbf{G}}$, $K_M = K$ and $\pi_{\mu , \lambda} = \pi_{\mu , 0}$ is $\sigma$-discrete. Now $\dim {\mathfrak{g}}({\mathbb{C}}) / {\mathfrak{k}}$ equals $2d,$ where $d$ is the dimension of the symmetric space associated to ${\mathbf{G}}({\mathbb{R}})$. Note that — as $\widetilde{K}$-modules — we have $$\wedge^i ({\mathfrak{g}}/ {\mathfrak{k}})^* \cong \wedge^{2d-i} ({\mathfrak{g}}/ {\mathfrak{k}})^*, \quad i=0, \ldots , 2d.$$ It follows that as $\widetilde{K}$-representations we have: $${\det{}'}[1 - ({\mathfrak{g}}/ {\mathfrak{k}})^*] = d \det [1-({\mathfrak{g}}/ {\mathfrak{k}})^*].$$ This implies that $$\mathrm{Lef} {}' (\sigma , F , I_{\mu , 0}) = d \ \mathrm{Lef} (\sigma , F , I_{\mu , 0}).$$ Proposition \[P:delorme\] therefore implies:
\[P610\] Let $\pi_{\mu}$ be a $\sigma$-discrete representation of $G$. Then we have: $$\mathrm{Lef} {}' (\sigma , F , I_{\mu}) = \left\{ \begin{array}{ll}
2^{\dim {\mathfrak{t}}} \dim ({\mathfrak{g}}^0 / {\mathfrak{k}}^0 ) & \mbox{ if } w \mu = 2 (\nu + \rho)_{| {\mathfrak{t}}} \quad (w \in W) \\
0 & \mbox{ otherwise}.
\end{array} \right. .$$
Proof of Lemma \[Lkt\]
----------------------
If $\phi$ is any smooth compactly supported function on $G$, Bouaziz [@Bouaziz] shows that $$\label{eq:bouaziz}
\int_{G^{\sigma} \backslash G}
\phi (x^{-1} \sigma x) d\dot{x} = \phi^G (e)$$ where $\phi^G \in C_c^{\infty} ({\mathbf{G}}({\mathbb{R}}))$ is the transfer of $\phi$. Now we can use the Plancherel theorem of Herb and Wolf [@HerbWolf] (as in [@RohlfsSpeh Proposition 4.2.14]) and get $$\phi^G (e) = \sum_{\pi \ {\rm discrete}} d(\pi) {\mathrm{trace}}\ \pi (\phi^G) + \int_{\rm tempered} {\mathrm{trace}}\ \pi (\phi^G) d\pi.$$ We can group the terms into stable terms since all terms in a $L$-packet have the same Plancherel measure [@HC]. We write $\pi_\varphi$ for the sum of the representations in an $L$-packet $\varphi$ and $d\pi_\varphi = d\pi$ for the corresponding measure. We then obtain $$\phi^G (e) = \sum_{\substack{{\rm elliptic} \\ L-{\rm packets} \ \varphi}} d(\varphi) {\mathrm{trace}}\ \pi_\varphi (\phi^G) + \int_{\substack{{\rm non \ elliptic} \\ {\rm bounded} \ L-{\rm packets} \ \varphi}} {\mathrm{trace}}\ \pi_\varphi (\phi^G) d\pi_\varphi.$$ Now we use transfer again. Indeed, Clozel [@Clozel] shows that if $\varphi$ is a tempered $L$-packet and $\widetilde{\pi}_\varphi$ the sum of the twisted representations of $\widetilde{G}$ associated to $\varphi$ by base-change, we have: $${\mathrm{trace}}\ \pi_{\varphi} (\phi^G) = {\mathrm{trace}}\ \widetilde{\pi}_\varphi (\phi).$$ We conclude: $$\label{eq:bouaziz2}
\int_{G^{\sigma} \backslash G}
\phi (x^{-1} \sigma x) d\dot{x} = \sum_{\substack{{\rm elliptic} \\ L-{\rm packets} \ \varphi}} d(\varphi) {\mathrm{trace}}\ \widetilde{\pi}_\varphi (\phi) + \int_{\substack{{\rm non \ elliptic} \\ {\rm bounded} \ L-{\rm packets} \ \varphi}} {\mathrm{trace}}\ \widetilde{\pi}_\varphi (\phi) d\pi_\varphi.$$ We want to apply this to the function $\phi = k_t^{\rho , \sigma}$. Since it is not compactly supported we have to explain why still holds for functions $\phi$ in the Harish-Chandra Schwartz space. We first note we have already checked (in §4.8) that the distribution $$\phi \mapsto \int_{G^{\sigma} \backslash G}
\phi (x^{-1} \sigma x) d\dot{x}$$ extends continuously to the Harish-Chandra Schwartz space, i.e. it defines a *tempered* distribution. Now for $\phi$ compactly supported Bouaziz [@Bouaziz Théorème 4.3] proves that we have (recall that we suppose that $H^1 (\sigma , G) = \{1 \}$): $$\label{eq:inversion}
\int_{G^{\sigma} \backslash G}
\phi (x^{-1} \sigma x) d\dot{x} = \int_{(\mathfrak{g}_a^* / G)^\sigma} \left( \frac12 \sum_{\tau \in X(f)} Q_\sigma (f , \tau) {\mathrm{trace}}\ \Pi_{f , \tau} (\phi ) \right) dm (G \cdot f).$$ We refer to [@Bouaziz] for all undefined notations and simply note that
- the $\Pi_{f , \tau}$ are tempered (twisted) representation, and
- if $\phi$ belongs to Harish-Chandra Schwartz space, the (twisted) characters $\Theta_{f,\tau} (\phi) = {\mathrm{trace}}\ \Pi_{f , \tau} (\phi) $ define rapidly decreasing functions of $f$.
Bouaziz does not explicitly compute the function $Q_\sigma (f, \tau)$ but proves however that it grows at most polynomially in $f$. It therefore follows that the distribution defined by the right hand side of also extends continuously to the Harish-Chandra Schwartz space.[^5] We conclude that still holds when $\phi$ belongs to Harish-Chandra Schwartz space.
We may now group the characters $\Theta_{f, \tau}$ into finite packets to get (all) stable tempered characters, as in [@Bouaziz §7.3] and denoted $\widetilde{\Theta}_{\lambda}$ there. Then the right hand side of becomes $$\sum_{\mathfrak{a} \in \mathrm{Car}(\mathfrak{g}^0)/G^0 } 2^{-\frac12 (\dim G^\sigma + \mathrm{rank} G^{\sigma})} |W (G , \mathfrak{a})|^{-1} \int_{\mathfrak{a}^*} p_{\sigma, \sigma} (\lambda) \Pi_{\mathfrak{g}^0}^\sigma (\lambda) \widetilde{\Theta}_\lambda (\phi) d \eta_{\mathfrak{a}} (\lambda).$$ Here again we refer to [@Bouaziz] for notations. Then [@Bouaziz Eq. (3), (4), (5) and (6) p. 287] imply that the right hand side of is equal to $$\sum_{\mathfrak{a} \in \mathrm{Car}(\mathfrak{g}^0)/G^0 } |W (G , \mathfrak{a})|^{-1} \int_{\mathfrak{a}^*} p_{1, 1} (\lambda) \Pi_{\mathfrak{g}^0 / \mathfrak{a}} (\lambda) \widetilde{\Theta}_{2\lambda} (\phi) d \eta_{\mathfrak{a}} (\lambda).$$ Here the stable character $\widetilde{\Theta}_{2\lambda}$ is the transfer of the character of a tempered packet, see [@Bouaziz 7.3(a)]) that is parametrized by $\lambda$ there. Finally in this parametrization the measure against which we integrate $\widetilde{\Theta}_{2\lambda} (\phi)$ can be identified with the Plancherel measure (see the very beginning of the proof of [@Bouaziz Théorème 7.4]) and we conclude that extends to Harish-Chandra Schwartz space. It applies in particular to $\phi= k_t^{\rho , \sigma}$, and using Lemma \[L1\] and Propositions \[P68\] and \[P610\], we conclude that $$\label{maineq}
\int_{G^{\sigma} \backslash G}
k_t^{\rho , \sigma} (x^{-1} \sigma x) d\dot{x} = 2^{\dim \mathfrak{t}} \int_{{\rm tempered}} e^{-t (\Lambda_\rho - \Lambda_\pi)} {\det{}'}(F_0 , \pi) d\pi.$$ Lemma \[Lkt\] therefore follows from the untwisted case for which we refer to [@BV].
We furthermore deduce from (and the computation in the untwisted case) the following theorem.
\[torsioncomparison\] We have : $$t_X^{(2) \sigma} (\rho) = 2^{\dim {\mathfrak{t}}} t_{X^{\sigma}}^{(2)} (\rho).$$
Similarly we have:
\[lefschetzcomparison\] We have: $$\mathrm{Lef}_X^{(2)\sigma}(\widetilde{\rho}) = 2^{\dim \mathfrak{t}} \cdot \chi_{X^{\sigma}}^{(2)}(\rho).$$
[*Remark.*]{} It follows from Theorems \[T:62\] and \[lefschetzcomparison\] that if $\delta (G^{\sigma})=0$ then $\mathrm{Lef}_X^{(2)\sigma}(\widetilde{\rho})$ is non zero. Proposition \[prop:intro\] of the Introduction therefore follows from the limit formula proved in Corollary \[C:47\].
General base change
===================
Twisted torsion of product automorphisms
----------------------------------------
Let ${\mathbf{G}}/{\mathbb{R}}$ be semisimple, $\sigma$ be an automorphism of ${\mathbf{G}},$ and $\rho$ a $\sigma$-equivariant representation of ${\mathbf{G}}.$ Let ${\mathbf{G}}' /{\mathbb{R}}$ be semisimple, $\sigma'$ be an automorphism of ${\mathbf{G}}',$ and $\rho'$ a $\sigma'$-equivariant representation of ${\mathbf{G}}'.$
\[productaut\] There is an equality $$t_{X_{G \times G'}}^{(2) \sigma \times \sigma'}(\rho \boxtimes \rho') = t_{X_G}^{(2) \sigma}(\rho) \cdot \mathrm{Lef}_{X_{G'}}^{(2) \sigma'}(\rho') + \mathrm{Lef}_{X_G}^{(2) \sigma}(\rho) \cdot t_{X_{G'}}^{(2) \sigma'}(\rho')$$
Let $M,M'$ be compact Riemannian manifolds together with equivariant metrized local systems $L \rightarrow M$ and $L' \rightarrow M'.$ Lück [@luck Proposition 1.32] proves that
$$t^{\sigma \times \sigma'}(M \times M';L \boxtimes L') = t^{\sigma}(M,L) \cdot \mathrm{Lef}(\sigma',M',L') + \mathrm{Lef}(\sigma,M,L) \cdot t^{\sigma'}(M',L').$$
Furthermore, Theorem \[twistedtorsionlimitmultiplicity\] shows that
$$t^{(2) \sigma \times \sigma}_{X_{G \times G'}}(\rho \boxtimes \rho') = \lim_{n \rightarrow \infty} \frac{\log \tau^{\sigma \times \sigma'}(\Upsilon_n \backslash X_{G \times G'} )}{{\mathrm{vol}}(\Upsilon_n \backslash G \times G')}$$
for any sequence of subgroups $\Upsilon_n$ satisfying the hypotheses therein. The sequence $\Upsilon_n = \Gamma_n \times \Gamma_n',$ where $\Gamma_n$ (resp. $\Gamma'_n$) is a chain of $\sigma$-stable (resp. $\sigma'$-stable) normal subgroups of $G$ (resp. $G'$) with trivial intersection, satisfies the hypotheses of Proposition \[WLF\] and Theorem \[twistedtorsionlimitmultiplicity\]. Therefore,
$$\begin{aligned}
t^{(2) \sigma \times \sigma'}_{X_{G \times G'}}(\rho \boxtimes \rho') &=& \lim_{n \rightarrow \infty} \frac{t^{\sigma}(\Gamma_n \backslash X_G, \rho)}{{\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} \cdot \frac{\mathrm{Lef}(\sigma',\Gamma'_n \backslash X_{G'} ,\rho')}{{\mathrm{vol}}(\Gamma^{'\sigma'}_n \backslash G^{'\sigma'})} + \frac{\mathrm{Lef}(\sigma,\Gamma_n \backslash X_G,\rho)}{{\mathrm{vol}}(\Gamma_n^{\sigma} \backslash G^{\sigma})} \cdot \frac{t^{\sigma'}(\Gamma_n \backslash X_{G'},\rho')}{{\mathrm{vol}}(\Gamma_n^{'\sigma'} \backslash G^{'\sigma'})} \\
&=& t^{(2) \sigma}_{X_G}(\rho) \cdot \mathrm{Lef}_{X_{G'}}^{(2) \sigma'}(\rho') + \mathrm{Lef}_{X_G}^{(2) \sigma}(\rho) \cdot t^{(2) \sigma'}_{X_{G'}}(\rho').\end{aligned}$$
Now let $\mathbb{E}$ be an étale ${\mathbb{R}}$-algebra; concretely, $\mathbb{E} = \mathbb{R}^r \times \mathbb{C}^s.$ Fix $\sigma \in {\mathrm{Aut}}(\mathbb{E} / {\mathbb{R}})$. The automorphism $\sigma$ permutes the factors of $\mathbb{E}$ and so induces a decomposition of the factors of $\mathbb{E}$ into its set of orbits $\mathcal{O}: \mathbb{E} = \prod_{o \in \mathcal{O}} \mathbb{E}_o.$ Each orbit is either
- a product of real places acted on by cyclic permutation,
- a product of complex places acted on by cyclic permutation, or
- a single complex place acted on by complex conjugation.
Let ${\mathbf{G}}$ be a semisimple group over ${\mathbb{R}}.$ Let $\rho$ be a representation of ${\mathbf{G}}/ {\mathbb{R}}$ and $\widetilde{\rho}_o$ the corresponding representation of $\mathrm{Res}_{\mathbb{E}_o / {\mathbb{R}}}.$ In particular, $\widetilde{\rho} = \rho \otimes \bar{\rho}$ is the corresponding representation of $\mathrm{Res}_{{\mathbb{C}}/ {\mathbb{R}}}{\mathbf{G}}.$ The automorphism $\sigma$ induces a corresponding automorphism of the group $\mathrm{Res}_{\mathbb{E} / {\mathbb{R}}} {\mathbf{G}}.$ There is a decomposition $$\mathrm{Res}_{\mathbb{E} / {\mathbb{R}}} {\mathbf{G}}= \prod_{o \in \mathcal{O}} \mathrm{Res}_{\mathbb{E}_o / {\mathbb{R}}} {\mathbf{G}}$$ with respect to which $\sigma$ acts as a product automorphism.
- Theorem \[twistedtorsionproduct\] shows that $$t^{(2) \sigma}_{X_{{\mathbf{G}}(\mathbb{E}_o)}}(\widetilde{\rho}_o) =|o| \cdot t^{(2)}_{X_{{\mathbf{G}}({\mathbb{R}})}}(\rho)$$ and $$\mathrm{Lef}^{(2) \sigma}_{X_{{\mathbf{G}}(\mathbb{E}_o)}}(\widetilde{\rho}_o) = \chi^{(2)}_{X_{{\mathbf{G}}({\mathbb{R}})}}(\rho)$$ for the orbits of type (a).
- Theorem \[twistedtorsionproduct\] shows that $$t^{(2) \sigma}_{X_{{\mathbf{G}}(\mathbb{E}_o)}}(\widetilde{\rho}_o) =|o| \cdot t^{(2)}_{X_{{\mathbf{G}}({\mathbb{C}})}}(\widetilde{\rho})$$ and $$\mathrm{Lef}^{(2)\sigma}_{X_{{\mathbf{G}}(\mathbb{E}_o)}}(\widetilde{\rho}_o) = \chi^{(2)}_{X_{{\mathbf{G}}({\mathbb{C}})}}(\widetilde{\rho})$$ for the orbits of type (b).
- Theorem \[torsioncomparison\] proves that $$t^{(2) \sigma}_{X_{{\mathbf{G}}(\mathbb{E}_o)}}(\widetilde{\rho}_o) = 2^{\dim \mathfrak{t}} t_{X_{{\mathbf{G}}({\mathbb{R}})}}^{(2)}(\rho)$$ and $$\mathrm{Lef}^{(2)\sigma}_{X_{{\mathbf{G}}(\mathbb{E}_o)}}(\widetilde{\rho}_o) = \chi^{(2)}_{X_{{\mathbf{G}}({\mathbb{R}})}}(\rho)$$ for the orbits of type (c).
The aggregate of these three examples, together with Theorem \[productaut\], allows us to compute the twisted $L^2$-torsion for arbitrary base change.
We have $$\tau^{(2)\sigma}_{X_{{\mathbf{G}}(\mathbb{E})}}(\widetilde{\rho}_{\mathbb{E}}) \neq 0$$ if and only if $\delta( {\mathbf{G}}(\mathbb{E})^{\sigma}) = 1.$
Suppose that there are $n$ orbits. Using Lemma \[productaut\], we expand $\tau^{(2)\sigma}_{X_{{\mathbf{G}}(\mathbb{E})}}(\widetilde{\rho}_{\mathbb{E}})$ as a sum of $n$ terms. Each summand is a product of $Lef^{(2) \sigma}_{X_{{\mathbf{G}}(\mathbb{E}_o)}}(\widetilde{\rho}_o)$ for $n - 1$ of the orbits $o$ and of $t^{(2)\sigma}_{X_{{\mathbf{G}}(\mathbb{E}_o)}}(\widetilde{\rho}_o)$ for the remaining orbit $o.$ Thus, exactly $n-1$ of the $\mathrm{Lef}^{(2) \sigma}$’s must be non-zero and the remaining $t^{(2) \sigma}$ must be non-zero; say $t^{(2) \sigma}_{X_{{\mathbf{G}}(\mathbb{E}_{o^{*}})}}(\rho_{o^{*}}) \neq 0.$ But by the preceding computations relating $t^{(2) \sigma}$ and $\mathrm{Lef}^{(2) \sigma}$ to their untwisted analogues, this is possible if and only if $\delta( {\mathbf{G}}(\mathbb{E}_{o^{*}})^{\sigma}) = 1$ and $\delta( {\mathbf{G}}(\mathbb{E}_o)^{\sigma}) = 0$ for all $o \neq o^{*}.$ This is equivalent to $\delta({\mathbf{G}}(\mathbb{E})^{\sigma}) = 1.$
Application to torsion in cohomology
====================================
Generalities on Reidemeister torsion and the Cheeger-Müller theorem
-------------------------------------------------------------------
Let $A^{\bullet}$ be a finite chain complex of $F$ vector spaces for a field $F.$ Suppose the chain groups $A^i$ and the cohomology groups $H^i(A^{\bullet})$ are equipped with volume forms, i.e. with top non-zero element $\omega_i \in \mathrm{det}(A^i)^{*}$ and $\mu_i \in (\mathrm{det}(H^i(A^{\bullet}) )^{*}.$
\[detofcomplex\] For a finite complex $C^{\bullet}$ situated in degrees $\geq 0,$ we define $\mathrm{det}(C^{\bullet}) := \mathrm{det}(C_0) \otimes \mathrm{det}(C_1)^{*} \otimes \mathrm{det}(C_2) \otimes ...$
There is a natural isomorphism [@KnudsenMumford $\S 1$]
$$\label{detofcohomology}
\mathrm{det}(A^{\bullet}) \otimes \mathrm{det}(H(A^{\bullet})) \cong F.$$
We let $s_{A^{\bullet}}$ denote the preimage of $1$ under the above isomorphism
The Reidemeister torsion $RT(A^{\bullet}, \omega_{\bullet}, \mu_{\bullet})$ is defined to be $$\omega \otimes \mu^{-1} (s_{A^{\bullet}}).$$
It is readily checked that if $\mu'_i = c_i \mu_i$ for some non-zero constants $c_i,$ then
$$\label{rtscaling}
RT(A^{\bullet}, \omega, \mu') = \frac{c_0 c_2 \cdots }{c_1 c_3 \cdots} \cdot RT(A^{\bullet}, \omega, \mu).$$
\[combinatorialvolume\]
Suppose $N$ is any finite free abelian group. We can define a volume $\omega$ form on $A_{{\mathbb{C}}}$ by the formula $$\omega(e_1 \wedge ... \wedge e_n) = 1$$ for any basis $e_1,...,e_n.$ This is well-defined up to sign. Endow each $A^{\bullet}_{\mathbb{C}}$ and $H^i(A^{\bullet}_{\mathbb{C}}) = H^i(A^{\bullet})_{\mathbb{C}}$ with the above volume forms, $\omega_{\mathbb{Z}}$ and $\mu_{\mathbb{Z}}.$ Then
$$RT(A^{\bullet}, \omega_{\mathbb{Z} }, \mu_{\mathbb{Z}} ) = \frac{|H^1(A^{\bullet})_{\mathrm{tors}}| \cdot |H^3(A^{\bullet})_{\mathrm{tors}}| \cdots}{|H^0(A^{\bullet})_{\mathrm{tors}}| \cdot |H^2(A^{\bullet})_{\mathrm{tors}}| \cdots}.$$
Fix a triangulation $K$ of a Riemannian manifold $(M,g)$ together with a metrized local system of free abelian group $L \rightarrow M.$ We can endow the cochain groups $C^i(M,L; K)_{\mathbb{C}}$ with the combinatorial volume form described in described in Example \[combinatorialvolume\]. Identifying $H^i(M,L_{{\mathbb{C}}})$ with the vector space of harmonic $L_{{\mathbb{C}}}$-valued $i$-forms on $M$ defines a volume form $\mu_g$ on $H^i(M, L_{\mathbb{C}}) = H^i(C^{\bullet}(M,L;K)).$ We define $$RT(M,L) := RT(C^{\bullet}(M,L;K), \omega_{\mathbb{Z}}, \mu_g),$$ which clearly does not depend on the triangulation $K.$ By , we see that $$RT(M,L) = \frac{|H^1(A^{\bullet})_{\mathrm{tors}}| \cdot |H^3(A^{\bullet})_{\mathrm{tors}}| \cdots}{|H^0(A^{\bullet})_{\mathrm{tors}}| \cdot |H^2(A^{\bullet})_{\mathrm{tors}}| \cdots} \times \frac{R^0(M,L) R^2(M,L) \cdots}{R^1(M,L) R^3(M,L) \cdots},$$ where $R^i(M,L) := {\mathrm{vol}}(H^i(M,L)_{\mathrm{free}}),$ where this volume is defined by identifying $H^i(M, L_{\mathbb{C}})$ with harmonic $L$-valued $i$-forms on $M.$
\[cmtheorem\] Let $L \rightarrow M$ be a unimodular local system over a compact Riemannian manifold. There is an equality $$t(M,L) = RT(M,L).$$
Equivariant Reidemeister torsion
--------------------------------
Let ${\mathcal{L}}\rightarrow {\mathcal{M}}$ be a unimodular, metrized local system of free abelian groups, equivariant with respect to an automorphism of ${\mathcal{M}}$ of finite order $p.$
Let $P(x) = x^{p -1} + x^{p-2} + ... + 1.$ For a polynomial $h \in \mathbb{Z}[x]$ and a $\mathbb{Z}[\sigma]$-module $A,$ we define $A[h(\sigma)]: = \{ a \in A: h(\sigma) \cdot a = 0 \}.$
Let $MS({\mathcal{M}},{\mathcal{L}})$ denote the Morse-Smale complex [@BZ1 $\S 1.6$] of a $\sigma$-invariant gradient vector field which we suppress from the notation. We may endow the chain groups of $MS({\mathcal{M}},{\mathcal{L}}_{\mathbb{C}})[\sigma - 1]$ and $MS({\mathcal{M}},{\mathcal{L}}_{\mathbb{C}})[P(\sigma)]$ with volume forms induced from the metric on ${\mathcal{L}}$ and the combinatorial volume form induced by the unstable cells of the gradient vector field [@Lip1 $\S 1.4$].
\[deftwistedrt\] The equivariant Reidemeister torsion of the equivariant local system ${\mathcal{L}}\rightarrow {\mathcal{M}}$ of ${\mathbb{C}}$-vector spaces is defined by $$\log RT_{\sigma}({\mathcal{M}},{\mathcal{L}}) := \log RT(MS({\mathcal{M}},{\mathcal{L}})[\sigma - 1]) - \frac{1}{p-1} \log RT(MS({\mathcal{M}},{\mathcal{L}})[P(\sigma)]).$$
[*Remark.*]{} Illman [@Illman] proves that any two smooth, $\Gamma$-equivariant triangulations of a smooth manifold $M$ acted on by a finite group $\Gamma$ admit a common equivariant refinement. Therefore, Definition \[deftwistedrt\] is independent of of the choice of (transversal) gradient vector field.
\[concretert\] Let ${\mathcal{L}}\rightarrow {\mathcal{M}}$ be a metrized, unimodular local system of free abelian groups acted on isometrically by $\langle \sigma \rangle \cong \mathbb{Z} / p\mathbb{Z}.$ Suppose that the fixed point set ${\mathcal{M}}^{\sigma}$ has Euler characteristic 0. Then $$\begin{aligned}
\log RT_{\sigma}({\mathcal{M}},{\mathcal{L}}) &=& - \sum_i {}^{*} \left(\log \left|H^i({\mathcal{M}}, {\mathcal{L}})[p^{-1}]^{\sigma - 1} \right| - \frac{1}{p-1} \log \left|H^i({\mathcal{M}}, {\mathcal{L}})[p^{-1}]^{P(\sigma)} \right| \right) \\
&+& \sum {}^{*} \log R^i({\mathcal{M}}, {\mathcal{L}}) \\
&+& O\left( \log|H^{*}(\mathcal{M}, {\mathcal{L}})[p^{\infty}]| +\log |H^{*}(\mathcal{M}, {\mathcal{L}}_{\mathbb{F}_p})| + \log |H^{*}({\mathcal{M}}^{\sigma}, {\mathcal{L}}_{\mathbb{F}_p})| \right)\end{aligned}$$
[@Lip1 Corollary 3.8].
The equivariant Cheeger-Müller theorem on locally symmetric spaces
------------------------------------------------------------------
Let ${\mathbf{G}}/ \mathbb{Q}$ be a semisimple group and $\sigma$ be an automorphism of ${\mathbf{G}}$ of prime degree $p.$ Let $G = {\mathbf{G}}({\mathbb{R}})$ and $X_G = G / K$ for $K$ a $\sigma$-stable maximal compact subgroup of $G.$ Let $\widetilde{\rho}: {\mathbf{G}}\rightarrow {\mathrm{GL}}(V)$ be a homomorphism of algebraic groups over $\mathbb{Q}.$ Let $\Gamma \subset {\mathbf{G}}(\mathbb{Q})$ be a $\sigma$-stable cocompact lattice and let $\mathcal{O} \subset V$ be $\Gamma$-stable $\mathbb{Z}$-lattice. Let ${\mathcal{M}}= \Gamma \backslash G / K$ and ${\mathcal{L}}_{\rho} \rightarrow {\mathcal{M}}$ the $\sigma$-equivariant local system associated to $\rho.$
\[twistedatequalstwistedrt\] Let ${\mathcal{L}}\rightarrow {\mathcal{M}}$ be an equivariant, metrized local system of free abelian groups over a locally symmetric space ${\mathcal{M}}$ acted on equivariantly and isometrically by $\langle \sigma \rangle \cong \mathbb{Z} / p\mathbb{Z}.$ Suppose further that the restriction to the fixed point set ${\mathcal{L}}|_{{\mathcal{M}}^{\sigma}} = L^{\otimes p}$ for $L \rightarrow {\mathcal{M}}^{\sigma}$ self-dual and that ${\mathcal{M}}^{\sigma}$ is odd dimensional. It follows that $$\log \tau_{\sigma}({\mathcal{M}}, {\mathcal{L}}) = \log RT_{\sigma}({\mathcal{M}},{\mathcal{L}}).$$
The essense of this theorem is the Bismut-Zhang formula [@BZ2 Theorem 0.2]. In [@Lip1 Corollary 5.5], the aforementioned formula is massaged to prove that the error term is zero under the hypotheses of the theorem.
Growth of torsion in cohomology {#growthoftorsion}
-------------------------------
\[refinedcohomologygrowth\] Let $\Gamma_n \subset \Gamma$ be a sequence of $\sigma$-stable subgroups satisfying the hypotheses of Proposition \[WLF\]. Let ${\mathcal{M}}_n = \Gamma_n \backslash X.$ Suppose further that the restriction to the fixed point set ${\mathcal{L}}|_{{\mathcal{M}}_n^{\sigma}} = L^{\otimes p}$ for $L \rightarrow {\mathcal{M}}^{\sigma}$ self-dual and that ${\mathcal{M}}_n^{\sigma}$ is odd dimensional. Suppose that $$\log |H^{*}({\mathcal{M}}_n, {\mathcal{L}})[p^{\infty}]|, \log |H^i({\mathcal{M}}_n^{\sigma}, {\mathcal{L}}_{\mathbb{F}_p})| = o({\mathrm{vol}}({\mathcal{M}}_n^{\sigma})).$$ and that ${\mathcal{L}}\rightarrow {\mathcal{M}}_n$ is rationally acyclic. Then $$\frac{- \sum_i {}^{*} \left(\log \left|H^i({\mathcal{M}}, {\mathcal{L}})[p^{-1}]^{\sigma - 1} \right| - \frac{1}{p-1} \log \left|H^i({\mathcal{M}}, {\mathcal{L}})[p^{-1}]^{P(\sigma)} \right| \right)}{{\mathrm{vol}}({\mathcal{M}}_n^{\sigma})} \xrightarrow{n \rightarrow \infty} t_{X_G}^{(2) \sigma}(\widetilde{\rho}).$$
This follows immediately by combining Theorem \[concretert\] with Theorem \[twistedatequalstwistedrt\]. The hypotheses imply that the $p$-power torsion error can be ignored.
\[unrefinedcohomologygrowth\] Enforce the same notation and hypotheses as in Corollary \[refinedcohomologygrowth\], with *no a priori cohomology growth assumptions*. Furthermore, assume that $t^{(2) \sigma}_{X_G}(\widetilde{\rho}) \neq 0.$ Then $$\limsup \frac{\log H^{*}({\mathcal{M}}_n, {\mathcal{L}})_{\mathrm{tors}}}{{\mathrm{vol}}({\mathcal{M}}_n^{\sigma})} > 0.$$
Suppose that not both of the growth hypotheses of Corollary \[refinedcohomologygrowth\] hold.
- If $\limsup \frac{\log |H^{*}({\mathcal{M}}_n, {\mathcal{L}})[p^{\infty}]|}{{\mathrm{vol}}({\mathcal{M}}_n^{\sigma})} > 0,$ we are done.
- If the mod $p$ cohomology of $({\mathcal{M}}_n^{\sigma}, {\mathcal{L}}_{\mathbb{F}_p})$ is large, i.e. if $$\frac{\log |H^{*}({\mathcal{M}}_n^\sigma, {\mathcal{L}}_{\mathbb{F}_p}) |}{ {\mathrm{vol}}({\mathcal{M}}_n^{\sigma}) } \nrightarrow 0,$$ then the conclusion follows by Smith theory [@bredon $\S$ III]. Indeed [@bredon $\S$ III.4.1], $$\label{smith}
\dim_{\mathbb{F}_p} H^{*}({\mathcal{M}}_{{\mathcal{U}}_N}^{\sigma}, {\mathcal{L}}_{\mathbb{F}_p}) \leq \dim_{\mathbb{F}_p} H^{*}({\mathcal{M}}_{{\mathcal{U}}_N}, {\mathcal{L}}_{\mathbb{F}_p}).
\footnote{Recall the Smith sequence \cite[$\S$ III.3.1]{bredon}, from which \eqref{smith} follows:
\begin{align*}
0 \rightarrow C_{\bullet}({\mathcal{M}}^{\sigma}, {\mathcal{L}}; K) \oplus &(1-\sigma)^i C_{\bullet}({\mathcal{M}}, {\mathcal{L}}; K) \rightarrow \\
& C_{\bullet}({\mathcal{M}}, {\mathcal{L}}; K) \xrightarrow{(1 - \sigma)^{p-i}} (1 - \sigma)^{p-i} C_{\bullet}({\mathcal{M}}, {\mathcal{L}}; K) \rightarrow 0,
\end{align*}
where $K$ is a sufficiently fine $\sigma$-stable triangulation of $M.$ The first map is the sum of the obvious inclusions.
Bredon's argument proving exactness of the above sequence when ${\mathcal{L}}$ is the trivial local system carries over to any situation where $C_{\bullet}({\mathcal{M}}^{\sigma}, {\mathcal{L}})$ is a direct sum of copies of trivial $\mathbb{F}_p[\sigma]$ modules and a free $\mathbb{F}_p[\sigma]$-module of finite rank. This holds in our context because ${\mathcal{L}}|_{{\mathcal{M}}^{\sigma}}$ is isomorphic to ${\mathcal{L}}^{\otimes p}.$ The ``standard basis of a tensor product" (relative to a fixed basis of the original vector space) realizes the required direct sum decomposition.}$$
Since $({\mathcal{M}}_n,{\mathcal{L}})$ has no rational cohomology by assumption, the desired conclusion follows by the universal coefficient theorem.
Otherwise, both a priori cohomology growth hypothesis from Corollary \[refinedcohomologygrowth\] are satisfied. We thus apply Corollary \[refinedcohomologygrowth\], whose conclusion is more refined.
[*Remark.*]{} Corollaries \[refinedcohomologygrowth\] and \[unrefinedcohomologygrowth\] were one major source of inspiration for this paper. We sought to understand when $t^{(2)}_{X_G}(\widetilde{\rho}) \neq 0$ in order to detect torsion cohomology growth. Note that it follows from Theorems \[twistedtorsionproduct\] and \[torsioncomparison\] (and the computations in the non-twisted case done in [@BV]) that $t^{(2) \sigma}_{X_G}(\widetilde{\rho}) \neq 0$ whenever $\delta (G^{\sigma}) =1$. Theorem \[T:14\] of the Introduction therefore follows from Corollary \[unrefinedcohomologygrowth\].
[^1]: Note that $m (\pi , \widetilde{\pi} , \Gamma) \ \mathrm{trace} \ \widetilde{\pi} (f)$ does not depend on the chosen particular extension $\widetilde{\pi}$ but only on $\pi$.
[^2]: Note that, in the untwisted case, the condition $$\frac{| \{ \gamma \in \Gamma_n \backslash \Gamma \; : \; \gamma \delta \gamma^{-1} \in \Gamma_n \}|}{[\Gamma : \Gamma_n]} \to 0$$ is equivalent to the BS-convergence of the [*compact*]{} quotients $\Gamma_n \backslash X$ towards the symmetric space $X$, see [@7samurai].
[^3]: Beware that $\rho$ is not assumed to be strongly acyclic here !
[^4]: Note that Delorme considers $\sigma$-invariants rather than traces, this introduces a factor $1/2$.
[^5]: In fact, we need that (twisted) tempered characters are rapidly decreasing in the parameters “Schwartz-uniformly” in $\phi$. But this holds because of known properties of discrete series characters combined with the fact that the constant term operator $\phi \mapsto \phi^{(P)}$ are all Schwartz-continuous.
|
---
abstract: 'It is shown that Extensive Air Shower (EAS) longitudinal development has a critical point where an equilibrium between the main hadronic component and the secondary electromagnetic one exhibits a brake. This results in a change of slope in quasi-power law function $N_{e}(E_{o})$. The latter leads to a knee in the EAS size spectrum at primary energy of about 100 TeV/nucleon. Many “strange” experimental results can be successfully explained in the frames of current approach.'
---
**EAS Longitudinal Development and the Knee[^1]**
*Yuri V. Stenkin*
\
[*60th October anniv. prospect, 7a, Moscow 117312, Russia*]{}\
Introduction
============
In 1958 there was claimed $\cite{KK}$ the existence of the “knee” in primary cosmic ray spectrum and its possible explanation by an existence of Galactic and Extragalactic cosmic rays. It should be noted that in the cited paper there was no any doubt that visible break in EAS size spectrum could be connected with any other reason but with primary spectrum steepening. At those times the recalculation from EAS size to primary energy was very simple. People merely used a constant coefficient for recalculation from EAS size at maximum to primary energy. Other ground level experiments later confirmed the “knee” existence in EAS size spectrum while direct measurements of primary cosmic ray nuclei spectra at satellites and balloons made up to energy $\sim$1 PeV do not confirm deviation from a pure power law at energies above 10 TeV. All experimental data confirming the “knee” existence are originated from *indirect measurements* using the EAS technique. Some physicists tried to explain the visible knee by a dramatic change in parameters of particle interactions $\cite{Nik,kaz,pet}$. Absolutely new approach to this problem has been proposed in 2003 and details of the approach can be found elsewhere $\cite{st1}$. It has been also shown $\cite{st2,st3}$ that a lot of experiments contradict the hypothesis of the “knee” in primary spectrum and its astrophysical origin.
The EAS method
==============
An advantage of the EAS method is a possibility to work up to the highest energy. But, the indirect measurements have to be recalculated to primary spectrum. This is a very complicated and model dependent problem. If the primary spectrum follows a power law function of a type: I$\ensuremath{\sim}E_{0}^{\ensuremath{-\gamma}}$ and a secondary component N$_{x}$ also follows a power law: N$_{x}\ensuremath{\sim} E_{0}^{\ensuremath{\alpha}}$, then I$\sim$N$_{x}^{\ensuremath{-\beta}}$, where $\beta=\gamma/\alpha$. If a break in a power law of experimental data distribution exists , then a change in any of the two indices ($\gamma$ or $\alpha$) may be responsible for this.
Suppose the primary spectrum index $\gamma$ changes at a point E$_{0}$=E$_{knee}$ from $\gamma$ to $\gamma$+$\Delta\gamma$. Then, one could expect a predictable break in the index $\beta$ for each component: $\Delta\beta$=$\Delta \gamma
/ \alpha$. Typical values for $\alpha$ are the following: $\alpha$$_{e}$$\approx$1.1-1.25 for electron component and $\alpha$$_{h}$$\approx$0.8 - 0.9 for hadronic and muonic components. If $\Delta\gamma$=0.5, then expected values are: $\Delta\beta$$_{e}$$\approx$0.44 for electrons and $\Delta\beta$$_{h}$$\approx$0.6 for hadrons and muons. But this contradicts observations (see $\cite{st2}$ and references there) where the knee in muonic and in hadronic components is equal to only $\Delta\beta$$_{h}$$\approx$0.1-0.2.
The problem of primary spectrum recovering from observable EAS parameters is additionally complicated due to uncertainties in primaries mass composition. It is very difficult to define primary particle mass using traditional EAS method. Only hybrid arrays such as Tibet AS or Chacaltaya array could more or less adequate solve this problem using emulsion chambers for primary mass separation. This problem is connected very tightly with the “knee” problem. If one accepts [*a priori*]{} a hypothesis of the charge- or mass-dependent knee in primary spectrum then he accepts also [*a priori*]{} the change of mass composition. If experimental data are processed under this hypothesis then both the knee and chemical composition change will be “shown” by these data after such processing.
The knee in PeV region: what is responsible for it?
===================================================
The question put above could be split in two parts: 1. Is the knee origin astrophysical or methodical (or any)? 2. Does it caused by proton or iron primaries? I’ll try here to answer the second part question. As it has been shown in our previous works $\cite{st1}$ there should be observed a break in EAS size spectrum at primary energy of $\sim$100 TeV/nucleon at sea level. The origin of the “knee” is a break of equilibrium (see $\cite{zat}$) between hadronic and electromagnetic components at a point where the number of cascading hadrons becomes close to 1. This point is critical one for EAS development because the number of particle is discreet value and less than 1 is only 0. Therefore, below this point the cascade development follows pure electromagnetic scenario and all EAS parameters change dramatically. Due to spread of primary masses from A=1 to A=56 there should be observed 2 “knees”: “proton knee” at 100 TeV and “iron knee” at $\sim$5 PeV. That means the visible knee in PeV region is connected with iron primary. The knee positions in shower size $N_{e}$ are equal to $\sim 10^{5}$ and $\sim 10^{6.3}$ consequently. These values are more or less constant in the current approach and depend weakly on the altitude. But, the corresponding primary energies are sure different. Recent results of Tibet AS $\cite{tib}$ on proton and helium spectra are not understandable in the frames of commonly used astrophysical knee hypothesis. But, it becomes absolutely clear in the frames of current approach. Actually, this hybrid experiment uses different sub-arrays for different purposes: emulsion chambers are used for core location and primary mass selection, while the EAS size is measured by the traditional EAS array. As we noted above, the “proton knee” position for Tibet altitude should be also close to 100 TeV. Therefore, the EAS size spectrum to the right of this point (their threshold is equal to 200 TeV) should be steep. To demonstrate this we performed Monte Carlo simulations with CORSIKA codes.
Results of simulations
======================
The latest version of CORSIKA program$\cite{hec}$ were used for calculations(v.6.501). Standard HDPM as well as VENUS and DPMJet
{height="8.cm"} {height="8.cm"}
models were used for high-energy hadron interactions and no significant difference were seen. Simulations were performed for proton and iron primary nuclei with the pure power law energy spectrum having the slope $\gamma= -2.7$ for altitudes from 100 m to 4.3 Km a. s. l. The number of all electromagnetic particles ($e^{+}, e^{-}$ and $\gamma$) summarized inside radius 1000 m were assigned to $N_{e}$. Note that such a definition is close but not equal to the $N_{e}$ usually obtained by experimenters from the NKG-function. As one can see from Fig.1, the distributions at altitude 4.3 Km have clear visible kinks at $N_{e}\sim
3\times10^{5}$ for protons and $N_{e}\sim 3\times10^{6}$ for iron. It is seen that the break of slopes coincides with the appearance of coreful showers and disappearance of coreless EAS’, the curves for those are also shown. This graph shows that EAS size spectrum slope becomes steeper at $N_{e}\sim 3.5\times10^{5}$ (corresponding $E_{0}\sim 100 TeV$). Above 200 TeV the slope is steep enough to explain the Tibet AS data: if one takes $\alpha=1.2$ then primary spectrum slope $\gamma=1.69\times1.2=2.03$. This is very close to the value obtained by Tibet AS. In other words, the spectrum slope measured by the Tibet AS experiment being recalculated to primary index taking current approach into account, is very close to $\gamma=
-2.7$ . Therefore, this result is in agreement with direct spectrum measurements.
A visible kink for iron primaries coincides with the “knee” position as one can see in fig. 1 lower panel. In this point our conclusion is the same as in $\cite{tib}$. Similar shape of the distributions can be obtained for any other altitudes, but at different primary energies. The effect depends also on the radius of integration (on the array dimensions): the smaller radius, the bigger is effect. Corresponding curves for muons and hadrons have no visible “knees” $\cite{st1}$.
{height="8.5cm"}
Due to different behavior of different components as a function of primary energy, the correlation plot between different EAS components also exhibits the “knee” as one can see in Fig.2. We plot these distributions divided by $N_{e}^{0.75}$ to emphasize the slope change and we made the calculations for another altitude to show that the effect exists at any altitude. And again the “knees” are visible. Only its positions are little bit different for another altitude. Similar curves obtained experimentally are usually interpreted as an evidence of the fact that primary mass composition becomes heavier while here, we obtained it for [**constant mass composition**]{} and for [**pure power law spectra**]{}. This is an example how experimental data could be erroneously understood and interpreted if one supposes a priori the knee existence.
Summary
=======
$\bullet$ EAS size spectrum has “a knee” at any altitude even for pure power law spectrum of primary cosmic ray.\
$\bullet$ The index of all-particle primary cosmic ray energy spectrum does not likely change significantly in a range of 0.1$\div$10 PeV.\
$\bullet$ The “knee” observed experimentally in electromagnetic EAS component is caused by EAS structure change at energy $\sim 0.1 PeV / nucleon$ where the number of cascading hadrons becomes equal to zero. Below this energy, EAS’s at sea level are mostly *coreless* while above this threshold EAS’s are mostly *coreful*.\
$\bullet$ Primary particle mass composition “change” measured by the EAS method using $N_{\mu}/N_{e}$ ratio is probably methodical one, while the composition of primaries at the top of atmosphere could be constant.\
$\bullet$ The steep spectra of protons and $\alpha$-particles measured by Tibet AS Group confirm our hypothesis that EAS size spectra must be steep above the threshold of $\sim$100 TeV / nucleon, while the primary spectrum does not change a slope.\
$\bullet$ We coincide with Tibet AS in the conclusion that the “knee” in PeV-region is connected with iron nuclei.
Acknowledgements {#acknowledgements .unnumbered}
================
I’d like to thank once again the Developers of CORSIKA program for very useful instrument for EAS study.
The work was supported in part by RFBR grants Nos 05-02-17395 and 07-02-00964, by the Scientific School Support grant NSh-4580.2006.2 and by the RAS Basic Research Program “Neutrino Physics”.
[30]{}
Khristiansen, G.B. and Kulikov, G.V., Sov. J. JETP, 35 (1958) Nikolsky, S.I. Nucl. Phys. B (Proc. Suppl.) 39A, (1995), 105. Kazanas, D. and Nikolaidis, A.Proc. of ICRC2001, Hamburg (2001), 1760. Petrukhin, A.A. Proc of ICRC2001, Hamburg, (2001), 1768. Stenkin, Yu.V. Mod. Phys. Lett. A, (2003), 18, 1225; Proc. of 28 ICRC, Tsukuba, p. 267; Yadernaya Fizika, (2007), in press Stenkin, Yu.V. Nucl. Phys. B (Proc. Suppl.), (2006), 151, 65. Stenkin, Yu.V.. Proc. of 29th ICRC, Pune, (2005), v. 6, p. 261. Zatsepin, G.T. Dokl. Akad. Nauk SSSR, v.67, No 6, (1949),993. Amenomori, M. et al. Proc. of 29th ICRC, Pune, (2005), v. 6, 177. Heck, D. et al. FZKA report 6019, Forshungzentrum Karsruhe (1998).
[^1]: Talk given at 30th ICRC (2007), Merida, Mexico
|
---
abstract: 'We develop a theory for state-based noninterference in a setting where different security policies—we call them local policies—apply in different parts of a given system. Our theory comprises appropriate security definitions, characterizations of these definitions, for instance in terms of unwindings, algorithms for analyzing the security of systems with local policies, and corresponding complexity results.'
author:
- Sebastian Eggert
- Henning Schnoor
- Thomas Wilke
title: Noninterference with Local Policies
---
Introduction
============
Research in formal security aims to provide rigorous definitions for different notions of security as well as methods to analyse a given system with regard to the security goals. Restricting the information that may be available to a user of the system (often called an agent) is an important topic in security. Noninterference [@GogMes; @goguen84] is a notion that formalizes this. Noninterference uses a security policy that specifies, for each pair of agents, whether information is allowed to flow from one agent to the other. To capture different aspects of information flow, a wide range of definitions of noninterference has been proposed, see, e.g., [@DBLP:conf/csfw/YoungB94; @DBLP:conf/csfw/Millen90; @DBLP:conf/esorics/Oheimb04; @DBLP:conf/sp/WittboldJ90].
In this paper, we study systems where in different parts different policies apply. This is motivated by the fact that different security requirements may be desired in different situations, for instance, a user may want to forbid interference between his web browser and an instant messenger program while visiting banking sites but when reading a news page, the user may find interaction between these programs useful.
As an illustrating example, consider the system depicted in Fig. \[fig:admin changes policy\], where three agents are involved: an administrator $A$ and two users $H$ and $L$. The rounded boxes represent system states, the arrows represent transitions. The labels of the states indicate what agent $L$ observes in the respective state; the labels of the arrows denote the action, either action $a$ performed by $A$ or action $h$ performed by $H$, inducing the respective transition. Every action can be performed in every state; if it does not change the state (i.e., if it induces a loop), the corresponding transition is omitted in the picture.
The lower part of the system constitutes a secure subsystem with respect to the bottom policy: when agent $H$ performs the action $h$ in the initial state, the observation of agent $L$ changes from $0$ to $1$, but this is allowed according to the policy, as agent $H$ may interfere with agent $L$—there is an edge from $H$ to $L$.
Similarly, the upper part of the system constitutes a secure subsystem with respect to the top policy: interference between $H$ and $L$ is not allowed—no edge from $H$ to $L$—and, in fact, there is no such interference, because $L$’s observation does not change when $h$ performs an action.
[r]{}[6.2cm]{} \[fig:admin changes policy\]
However, the entire system is clearly insecure: agent $A$ must not interfere with anyone—there is no edge starting from $A$ in either policy—but when $L$ observes “$1$” in the lower right state, $L$ can conclude that $A$ did *not* perform the $a$ action depicted.
Note that interference between $H$ and $L$ is allowed, unless $A$ performs action $a$. But $L$ must not get to know whether $a$ was performed. To achieve this, interference between $H$ and $L$ must never be allowed. Otherwise, as we have just argued, $L$ can—by observing $H$’s actions—conclude that in the current part of the system, interference between $H$ and $L$ is still legal and thus $A$ did not perform $a$. In other words, in the policy of the lower part, the edge connecting $H$ and $L$ can never be “used” for an actual information flow. We call such edges *useless*.—Useless edges are a key issue arising in systems with local policies.
#### Our results.
We develop a theory of noninterference with local policies which takes the aforementioned issues into account. Our contributions are as follows:
1. We provide new and natural definitions for noninterference with local policies, both for the transitive [@GogMes; @goguen84] (agent $L$ may only be influenced by agent $H$ if there is an edge from $H$ to $L$ in the policy) and for the intransitive setting [@HY87] (interference between $H$ and $L$ via “intermediate steps” is also allowed).
2. We show that policies can always be rewritten into a normal form which does not contain any “useless” edges (see above).
3. We provide characterizations of our definitions based on unwindings, which demonstrate the robustness of our definitions and from which we derive efficient verification algorithms.
4. We provide results on the complexity of verifying noninterference. In the transitive setting, noninterference can be verified in nondeterministic logarithmic space ([[$\mathrm{NL}$]{}]{}). In the intransitive setting, the problem is [[$\mathrm{NP}$]{}]{}-complete, but fixed-parameter tractable with respect to the number of agents.
Our results show significant differences between the transitive and the intransitive setting. In the transitive setting, one can, without loss of generality, always assume a policy is what we call uniform, which means that each agent may “know” (in a precise epistemic sense) the set of agents that currently may interfere with him. Assuming uniformity greatly simplifies the study of noninterference with local policies in the transitive setting. Moreover, transitive noninterference with local policies can be characterized by a simple unwinding, which yields very efficient algorithms.
In the intransitive setting, the situation is more complicated. Policies cannot be assumed to be uniform, verification is [[$\mathrm{NP}$]{}]{}-complete, and, consequently, we only give an unwinding condition that requires computing exponentially many relations. However, for *uniform* policies, the situation is very similar to the transitive setting: we obtain simple unwindings and efficient algorithms.
As a consequence of our results for uniform policies, we obtain an unwinding characterization of IP-security [@HY87] (which uses a single policy for the entire system). Prior to our results, only an unwinding characterization that was *sound*, but not *complete* for IP-security was known [@rushby92]. Our new unwinding characterization immediately implies that IP-security can be verified in nondeterministic logarithmic space, which improves the polynomial-time result obtained in [@emsw11].
*Related Work*. Our intransitive security definitions generalize IP-security [@HY87] mentioned above. The issues raised against IP-security in [@meyden2007] are orthogonal to the issues arising from local policies. We therefore study local policies in the framework of IP-security, which is technically simpler than, e.g., TA-security as defined in [@meyden2007].
Several extensions of intransitive noninterference have been discussed, for instance, in [@RoscoeG99; @DBLP:journals/jcs/MyersSZ06]. In [@Leslie-DYNAMICNONINTERFERENCE-SSE-2006], a definition of intransitive noninterference with local policies is given, however, the definition in [@Leslie-DYNAMICNONINTERFERENCE-SSE-2006] does not take into account the aforementioned effects, and that work does not provide complete unwinding characterizations nor complexity results.
State-based Systems with Local Policies {#sect:preliminaries}
=======================================
We work with the standard state-observed system model, that is, a system is a deterministic finite-state automaton where each action belongs to a dedicated agent and each agent has an observation in each state. More formally, a *system* is a tuple $M=(S,s_0,{\ensuremath{A}},\mathtt{step},\mathtt{obs},\mathtt{dom})$, where $S$ is a finite set of *states*, $s_0\in S$ is the *initial state*, ${A}$ is a finite set of *actions*, $\mathtt{step}\colon S\times{\ensuremath{A}}\rightarrow S$ is a *transition function*, $\mathtt{obs}\colon S\times D\rightarrow O$ is an *observation function*, where $O$ is an arbitrary set of observations, and $\mathtt{dom}\colon{\ensuremath{A}}\rightarrow D$ associates with each action an agent, where $D$ is an arbitrary finite set of agents (or security domains).
For a state $s$ and an agent $u$, we write ${{\tt obs}}_u(s)$ instead of ${{\tt obs}}(s,u)$. For a sequence $\alpha\in{\ensuremath{A}}^*$ of actions and a state $s\in S$, we denote by $s\cdot\alpha$ the state obtained when performing $\alpha$ starting in $s$, i.e., $s\cdot\epsilon=s$ and $s\cdot\alpha a=\mathtt{step}(s\cdot\alpha,a)$.
A *local policy* is a reflexive relation ${{{\rightarrowtail}_{}}}\subseteq D\times D$. To keep our notation simple, we do not define subsystems nor policies for subsystems explicitly. Instead, we assign a local policy to every state and denote the policy in state $s$ by ${{\rightarrowtail}_{s}}$. We call the collection of all local policies ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ the *policy* of the system. If $(u,v) \in {{{\rightarrowtail}_{s}}}$ for some $u, v \in {D}$, $s \in {S}$, we say $u {{\rightarrowtail}_{s}} v$ is an *edge* in ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$. A system has a *global policy* if all local policies ${{\rightarrowtail}_{s}}$ are the same in all states, i.e., if $u{{\rightarrowtail}_{s}}v$ does not depend on $s$. In this case, we denote the single policy by ${\rightarrowtail}$ and only write $u {\rightarrowtail}v$. We define the set ${\ensuremath{u^{\leftarrowtail}_{s}}}$ as the set of agents that *may interfere* with $u$ in $s$, i.e., the set ${\ensuremath\left\{v\ \vert\ v{{\rightarrowtail}_{s}}u\right\}}$.
In the following, we fix an arbitrary system $M$ and a policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$.
In our examples, we often identify a state with an action sequence leading to it from the initial state $s_0$, that is, we write $\alpha$ for $s_0 \cdot \alpha$, which is well-defined, because we consider deterministic systems. For example, in the system from Fig. \[fig:admin changes policy\], we denote the initial state by $\epsilon$ and the upper right state by $ah$. In each state, we write the local policy in that state as a graph. In the system from Fig. \[fig:admin changes policy\], we have $H{{\rightarrowtail}_{\epsilon}}L$, but $H\not{{\rightarrowtail}_{a}}L$. In general, we only specify the agents’ observations as far as relevant for the example, which usually is only the observation of the agent $L$. We adapt the notation from Fig. \[fig:admin changes policy\] to our definition of local policies, which assigns a local policy to every state: we depict the graph of the local policy inside the rounded box for the state, see Fig. \[fig:dpsecure\_system\].
The Transitive Setting {#sect:transitive}
======================
In this section, we define noninterference for systems with local policies in the transitive setting, give several characterizations, introduce the notion of useless edge, and discuss it. The basic idea of our security definition is that an occurrence of an action which, according to a local policy, should not be observable by an agent $u$ must not have any influence on $u$’s future observations.
\[def:local-dpurge-secure\] The system $M$ is [t-secure]{}iff for all $u \in {D}$, $s \in {S}$, $a \in {A}$ and $\alpha \in {A}^*$ the following implication holds: $$\text{If } {{\tt dom}}(a) \not {{\rightarrowtail}_{s}} u, \text{ then }
{{\tt obs}}_u(s \cdot \alpha) = {{\tt obs}}_u(s \cdot a \alpha) \enspace.$$
[r]{}[0pt]{}
Fig. \[fig:dpsecure\_system\] shows a [t-secure]{}system. In contrast, the system in Fig. \[fig:admin changes policy\] is not [t-secure]{}, since $A \not {{\rightarrowtail}_{\epsilon}} L$, but ${{\tt obs}}_L(ah) \neq {{\tt obs}}_L(h)$.
Characterizations of [t-Security]{}
-----------------------------------
In Theorem \[thm:dp\_characterizations\], we give two characterizations of [t-security]{}, underlining that our definition is quite robust. The first characterization is based on an operator which removes all actions that must not be observed. It is essentially the definition from Goguen and Meseguer [@GogMes; @goguen84] of the ${{\tt purge}}$ operator generalized to systems with local policies.
\[sec:dynamic-transitive-purge\] For all $u \in {D}$ and $s \in {S}$ let $ {{\tt purge}}(\epsilon, u,
s) = \epsilon$ and for all $a \in {A}$ and $\alpha \in
{A}^*$ let $$\begin{aligned}
{{\tt purge}}(a \alpha, u, s) & =
\begin{cases}
a {\ }{{\tt purge}}(\alpha, u, s \cdot a) &\text{if } {{\tt dom}}(a) {{\rightarrowtail}_{s}} u
\\
{{\tt purge}}(\alpha, u, s) & \text{otherwise} \enspace.
\end{cases}
\end{aligned}$$
The other characterization is in terms of unwindings, which we define for local policies in the following, generalizing the definition of Haigh and Young [@HY87].
A *transitive unwinding* for $M$ with a policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ is a family of equivalence relations $({\sim}_u)_{u\in{D}}$ such that for every agent $u \in {D}$, all states $s, t \in {S}$ and all $a \in {A}$, the following holds:
- If ${{\tt dom}}(a) \not{{\rightarrowtail}_{s}} u$, then $s {\sim}_u s\cdot a$. [[[(LR$_{\textnormal{t}}$)]{.nodecor}]{}—local respect]{.nodecor}
- If $s {\sim}_u t$, then $s\cdot a {\sim}_u t \cdot a$. [[[(SC$_{\textnormal{t}}$)]{.nodecor}]{}—step consistency]{.nodecor}
- If $s {\sim}_u t$, then ${{\tt obs}}_u(s) = {{\tt obs}}_u(t)$. [[[(OC$_{\textnormal{t}}$)]{.nodecor}]{}—output consistency]{.nodecor}
Our characterizations of [t-security]{}are spelled out in the following theorem.
\[thm:dp\_characterizations\] The following are equivalent:
1. The system $M$ is [t-secure]{}. \[thm:dp\_characterization\_def\]
2. For all $u \in {D}$, $s \in {S}$, and $\alpha, \beta \in {A}^*$ with ${{\tt purge}}(\alpha, u, s) = {{\tt purge}}(\beta, u, s)$, we have ${{\tt obs}}_u(s \cdot \alpha) = {{\tt obs}}_u(s \cdot \beta)$. \[thm:dp\_characterization\_purge\]
3. There exists a transitive unwinding for $M$ with the policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$. \[thm:dp\_characterization\_unwind\]
Unwinding relations yield efficient verification procedure. For verifying t-security, it is sufficient to compute for every $u \in {D}$ the smallest equivalence relation satisfying [[(LR$_{\textnormal{t}}$)]{.nodecor}]{}and [[(SC$_{\textnormal{t}}$)]{.nodecor}]{}and check that the function ${{\tt obs}}_u$ is constant on every equivalence class. This can be done with nearly the same algorithm as is used for global policies, described in [@emsw11]. The above theorem directly implies that [t-security]{}can be verified in nondeterministic logarithmic space.
Useless Edges
-------------
An “allowed” interference $v{{\rightarrowtail}_{s}}u$ may contradict a “forbidden” interference $v\not{{\rightarrowtail}_{s'}}u$ in a state $s'$ that should be indistinguishable to $s$ for $u$. In this case, the edge $v{{\rightarrowtail}_{s}}u$ is useless. What this means is that an edge $v{{\rightarrowtail}_{s}}u$ in the policy may be deceiving and should not be interpreted as “it is allowed that $v$ interferes with $u$”, rather, it should be interpreted as “it is not explicitly forbidden that $v$ interferes with $u$”. To formalize this, we introduce the following notion:
States $s$, $s'$ are *[t-similar]{}* for an agent $u \in {D}$, denoted $s {\approx}_u s'$, if there exist $t \in {S}$, $a \in {A}$, and $\alpha \in {A}^*$ such that ${{\tt dom}}(a) \not{{\rightarrowtail}_{t}} u$, $s = t \cdot a \alpha$, and $s' = t\cdot \alpha$.
Observe that [t-similarity]{}is identical with the smallest equivalence relation satisfying [[(LR$_{\textnormal{t}}$)]{.nodecor}]{}and [[(SC$_{\textnormal{t}}$)]{.nodecor}]{}. Also observe that the system $M$ is [t-secure]{}if and only if for every agent $u$, if $s{\approx}_u s'$, then ${{\tt obs}}_u(s)={{\tt obs}}_u(s')$.
The notion of [t-similarity]{}allows us to formalize the notion of a useless edge:
An edge $v{{\rightarrowtail}_{s}}u$ is *useless* if there is a state $s'$ with $s{\approx}_u s'$ and $v\not{{\rightarrowtail}_{s'}}u$.
For example, consider again the system in Fig. \[fig:admin changes policy\]. Here, the local policy in the initial state allows information flow from $H$ to $L$. However, if $L$ is allowed to observe $H$’s action in the initial state, then $L$ would know that the system is in the initial state, and would also know that $A$ has not performed an action. This is an information flow from $A$ to $L$, which is prohibited by the policy.
Useless edges can be removed without any harm:
\[theorem:uniformpolicies\] Let ${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$ be defined by $$\begin{aligned}
{{{\rightarrowtail}_{s}}'} = {{{\rightarrowtail}_{s}}} \setminus \{v {{\rightarrowtail}_{s}} u \mid v
{{\rightarrowtail}_{s}} u \text{ is useless}\} \qquad \text{for all $s \in S$.}
\end{aligned}$$ Then $M$ is [t-secure]{}[w.r.t.]{}${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ iff $M$ is [t-secure]{}[w.r.t.]{}${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$.
The policy ${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$ in Theorem \[theorem:uniformpolicies\] has no useless edges, hence every edge in one of its local policies represents an *allowed* information flow—no edge contradicts an edge in another local policy. Another interpretation is that any information flow that is *forbidden* is *directly* forbidden via the absence of the corresponding edge. In that sense, the policy is closed under logical deduction.
We call a policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ *uniform* if ${\ensuremath{u^{\leftarrowtail}_{s}}}={\ensuremath{u^{\leftarrowtail}_{s'}}}$ holds for all states $s$ and $s'$ with $s{\approx}_u s'$. In other words, in states that $u$ should not be able to distinguish, the exact same set of agents may interfere with $u$. Hence $u$ may “know” the set of agents that currently may interfere with him. Note that a policy is uniform if and only if it does not contain useless edges. (This is not true in the intransitive setting, hence the seemingly complicated definition of uniformity.) Uniform policies have several interesting properties, for example, with a uniform policy the function ${{\tt purge}}$ behaves very similarly to the setting with a global policy: it suffices to verify action sequences that start in the initial state of the system and ${{\tt purge}}$ satisfies a natural associativity condition on a uniform policy.
The Intransitive Setting {#sect:intransitive case}
========================
In this section, we consider the intransitive setting, where, whenever an agent performs an action, this event may transmit information about the actions the agent has performed himself as well as information about actions by other agents that was previously transmitted to him. The definition follows a similar pattern as that of [t-security]{}: if performing an action sequence $a\alpha$ starting in a state $s$ should not transmit the action $a$ (possibly via several intermediate steps) to the agent $u$, then $u$ should be unable to deduce from his observations whether $a$ was performed. To formalize this, we use Leslie’s extension [@Leslie-DYNAMICNONINTERFERENCE-SSE-2006] of Rushby’s definition [@rushby92] of ${\texttt{sources}}$.
For an agent $u$ let ${\ensuremath{{\tt src}(\epsilon,u,s)}} ={\ensuremath\left\{u\right\}}$ and for $a \in {A}$, $\alpha \in {A}^*$, if ${{\tt dom}}(a) {{\rightarrowtail}_{s}} v$ for some $v \in {\ensuremath{{\tt src}(\alpha,u,s\cdot
a)}}$, then let ${\ensuremath{{\tt src}(a\alpha,u,s)}} = {\ensuremath{{\tt src}(\alpha,u,s\cdot a)}} \cup {\ensuremath\left\{dom(a)\right\}}$, and else let ${\ensuremath{{\tt src}(a\alpha,u,s)}} = {\ensuremath{{\tt src}(\alpha,u,s\cdot a)}}$.
The set ${\ensuremath{{\tt src}(a\alpha,u,s)}}$ contains the agents that “may know” whether the action $a$ has been performed in state $s$ after the run $a\alpha$ is performed: initially, this is only the set of agents $v$ with ${{\tt dom}}(a){{\rightarrowtail}_{s}}v$. The knowledge may be spread by every action performed by an agent “in the know:” if an action $b$ is performed in a later state $t$, and ${{\tt dom}}(b)$ already may know that the action $a$ was performed, then all agents $v$ with ${{\tt dom}}(b){{\rightarrowtail}_{t}}v$ may obtain this information when $b$ is performed. Following the discussion above, we obtain a natural definition of security:
The system $M$ is [i-secure]{}iff for all $s \in {S}$, $a\in{A}$, and $\alpha\in{A}^*$, the following implication holds. $$\begin{aligned}
\text{If ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$, then ${{\tt obs}}_u(s\cdot
a\alpha)={{\tt obs}}_u(s\cdot\alpha)$.}\end{aligned}$$
The definition formalizes the above: if, on the path $a\alpha$, the action $a$ is not transmitted to $u$, then $u$’s observation must not depend on whether $a$ was performed; the runs $a\alpha$ and $\alpha$ must be indistinguishable for $u$.
Consider the example in Fig. \[fig:admin changes policy\]. The system remains insecure in the intransitive setting: as $A$ must not interfere with any agent in any state, we have ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(ah,L,\epsilon)}}$, where again, according to our convention, $\epsilon$ denotes the initial state. So, the system is insecure, since ${{\tt obs}}_L(ah)\neq{{\tt obs}}_L(h)$.
Characterizations and Complexity of [i-Security]{} {#sect:intransitive unwinding exponential}
--------------------------------------------------
\[sect:dipsecty characterizations\]
We now establish two characterizations of intransitive noninterference with local policies and study the complexity of verifying [i-security]{}. Our characterizations are analogous to the ones obtained for the transitive setting in Theorem \[thm:dp\_characterizations\]. The first one is based on a purge function, the second one uses an unwinding condition. This demonstrates the robustness of our definition and strengthens our belief that [i-security]{}is indeed a natural notion.
We first extend Rushby’s definition of ${{\tt ipurge}}$ to systems with local policies.
For all $u \in {D}$ and all $s \in {S}$, let ${\ensuremath{{\tt ipurge}(\epsilon,u,s)}}=\epsilon$ and, for all $a \in {A}$ and $\alpha \in {A}^*$, let $$\begin{aligned}
{\ensuremath{{\tt ipurge}(a\alpha,u,s)}} & =
\begin{cases}
a {\ }{\ensuremath{{\tt ipurge}(\alpha,u,s\cdot a)}} & {\ensuremath{\mathrm{\text{ if }}}}{{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\alpha,u,s)}}, \\
{\ensuremath{{\tt ipurge}(\alpha,u,s)}} & {\ensuremath{\mathrm{\text{ otherwise}}}}.
\end{cases}\end{aligned}$$
The crucial point is that in the case where $a$ must remain hidden from agent $u$, we define ${\ensuremath{{\tt ipurge}(a\alpha,u,s)}}$ as ${\ensuremath{{\tt ipurge}(\alpha,u,s)}}$ instead of the possibly more intuitive choice ${\ensuremath{{\tt ipurge}(\alpha,u,s\cdot a)}}$, on which the security definition in [@Leslie-DYNAMICNONINTERFERENCE-SSE-2006] is based.
We briefly explain the reasoning behind this choice. To this end, let ${\ensuremath{{\tt ipurge}}}'$ denote the alternative definition of [${\tt ipurge}$]{}outlined above. Consider the sequence $ah$, performed from the initial state in the system in Fig. \[fig:admin changes policy\]. Clearly, the action $a$ is purged from the trace, thus the result of ${\ensuremath{{\tt ipurge}}}'$ is the same as applying ${\ensuremath{{\tt ipurge}}}'$ to the sequence $h$ starting in the upper left state. However, in this state, the action $h$ is invisible for $L$, hence ${\ensuremath{{\tt ipurge}}}'$ removes it, and thus purging $ah$ results in the empty sequence. On the other hand, if we consider the sequence $h$ also starting in the initial state, then $h$ is not removed by ${\ensuremath{{\tt ipurge}}}'$, since $H$ may interfere with $L$. Hence $ah$ and $h$ do not lead to the same purged trace—a security definition based on ${\ensuremath{{\tt ipurge}}}'$ does not require $ah$ and $h$ to lead to states with the same observation. Therefore, the system is considered secure in the ${\ensuremath{{\tt ipurge}}}'$-based security definition from [@Leslie-DYNAMICNONINTERFERENCE-SSE-2006]. However, a natural definition must require $ah$ and $h$ to lead to the same observation for agent $L$, as the action $a$ must always be hidden from $L$.
We next define unwindings for [i-security]{}and then give a characterization of [i-security]{}based on them.
An *intransitive unwinding* for the system $M$ with a policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ is a family of relations $({\precsim_{{D}'}})_{{D}'\subseteq {D}}$ such that ${\precsim_{{D}'}}\subseteq {S}\times {S}$ and for all ${D}'\subseteq {D}$, all $s, t \in {S}$ and all $a \in {A}$, the following hold:
- $s{\precsim_{{\ensuremath\left\{u\in {D}\ \vert\ {{\tt dom}}(a)\not{{\rightarrowtail}_{s}} u\right\}}}} s\cdot a$. [[(LR$_{\textnormal{i}}$)]{.nodecor}]{}
- If $s{\precsim_{{D}''}} t$, then $s\cdot
b{\precsim_{{D}''}} t\cdot b$, where ${D}''= {D}'$ if ${{\tt dom}}(b)\in{D}'$,\
and else ${D}''= {D}'\cap{\ensuremath\left\{u\ \vert\ {{\tt dom}}(b)\not{{\rightarrowtail}_{s}} u\right\}}$. [[(SC$_{\textnormal{i}}$)]{.nodecor}]{}
- If $s{\precsim_{{D}'}} t$ and $u\in{D}'$, then ${{\tt obs}}_u(s)={{\tt obs}}_u(t)$, [[(OC$_{\textnormal{i}}$)]{.nodecor}]{}
Intuitively, $s{\precsim_{{D}'}} t$ expresses that there is a common reason for all agents in ${D}'$ to have the same observations in $s$ as in $t$, i.e., if there is a state $\tilde s$, an action $a$ and a sequence $\alpha$ such that $s= \tilde s\cdot a\alpha$, $t=\tilde s
\cdot\alpha$, and ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,\tilde s)}}$ for *all* agents $u\in{D}'$.
\[theorem:intransitive unwinding and ipurge characterization\] The following are equivalent:
1. The system $M$ is [i-secure]{}.
2. For all agents $u$, all states $s$, and all action sequences $\alpha$ and $\beta$ with\
${\ensuremath{{\tt ipurge}(\alpha,u,s)}}={\ensuremath{{\tt ipurge}(\beta,u,s)}}$, we have ${{\tt obs}}_u(s\cdot\alpha)={{\tt obs}}_u(s\cdot\beta)$.
3. There exists an intransitive unwinding for $M$ and ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$.
In contrast to the transitive setting, the unwinding characterization of [i-security]{}does not lead to a polynomial-time algorithm to verify security of a system, because the number of relations needed to consider is exponential in the number of agents in the system. Unless ${{\ensuremath{\mathrm{P}}}}={{\ensuremath{\mathrm{NP}}}}$, we cannot do significantly better, because the verification problem is [[$\mathrm{NP}$]{}]{}-complete; our unwinding characterization, however, yields an FPT-algorithm.
\[theorem:intransitive case np complete\] Deciding whether a given system is [i-secure]{}with respect to a policy is [[$\mathrm{NP}$]{}]{}-complete and fixed-parameter tractable with the number of agents as parameter.
Intransitively Useless Edges
----------------------------
[r]{}[0pt]{}
In our discussion of [t-security]{}we observed that local policies may contain edges that can never be used. This issue also occurs in the intransitive setting, but the situation is more involved. In the transitive setting, it is sufficient to “remove any incoming edge for $u$ that $u$ must not know about” (see Theorem \[theorem:uniformpolicies\]). In the intransitive setting it is not: when the system in Fig. \[fig:redundant edge\] is in state $h_1$, then agent $L$ must not know that the edge $D{{\rightarrowtail}_{}} L$ is present, since states $\epsilon$ and $h_1$ should be indistinguishable for $L$, but clearly, the edge cannot be removed without affecting security. However, useless edges still exist in the intransitive setting, even in the system from Figure \[fig:redundant edge\], as we will show below.
To formally define useless edges, we adapt [t-similarity]{}to the intransitive setting in the natural way.
For an agent $u$, let $\approx^i_u$ be the smallest equivalence relation on the states of $M$ such that for all $s \in {S}$, $a
\in {A}$, $\alpha\in {A}^*$, if ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$, then $s\cdot a\alpha\approx^i_us\cdot\alpha$. We call states $s$ and $s'$ with $s\approx^i_u s'$ *[i-similar]{}for $u$*.
Using this, we can now define intransitively useless edges:
Let $e$ be an edge in a local policy of [$({{\rightarrowtail}_{s}})_{s\in S}$]{}and let $({\hat \rightarrowtail}_s)_{s\in S}$ be the policy obtained from [$({{\rightarrowtail}_{s}})_{s\in S}$]{}by removing $e$. Let $\approx^i_u$ and ${\hat \approx}^i_u$ be the respective i-similarity relations. Then $e$ is *intransitively useless* if $s\approx^i_u s'$ if and only if $s {\hat \approx}^i_u s'$ for all states $s$ and $s'$ and all agents $u$.
An edge is intransitively useless if removing it does not forbid any information flow that was previously allowed. In particular, such an edge itself cannot be used directly. Whether an edge is useless does not depend on the observation function of the system, but only on the policy and the transition function, whereas a definition of security compares observations in different states.
If the policy does not contain any intransitively useless edges, then there is no edge in any of its local policies that is contradicted by other aspects of the policy. In other words, the set of information flows *forbidden* by such a policy is closed under logical deduction—every edge that can be shown to represent a forbidden information flow is absent in the policy.
Fig. \[fig:redundant edge\] shows a secure system with an intransitively useless edge. The system is secure (agent $L$ knows whether in the initial state, $h_1$ or $h_2$ was performed, as soon as this information is transmitted by agent $D$). The edge $H{{\rightarrowtail}_{h_1}}L$ is intransitively useless, as explained in what follows.
The edge allows $L$ to distinguish between the states $h_1, h_1h_1, h_1h_2$. However, one can verify that $h_2h_1\approx^i_L h_1$, $h_2h_1h_1\approx^i_L h_2h_1$, $h_2h_1h_1\approx^i_L h_1h_1$, $h_2h_1h_2\approx^i_L h_2h_1$, and $h_2h_1h_2\approx^i_L h_1h_2$ all hold. Symmetry and transitivity of $\approx^i_L$ imply that all the three states $h_1,h_1h_1,h_1h_2$ are $\approx^i_L$-equivalent. Hence the edge $H{{\rightarrowtail}_{h_1}}L$ is indeed intransitively useless (and the system would be insecure if $h_1$, $h_1h_1$, and $h_1h_2$ would not have the same observations).
Intransitively useless edges can be removed without affecting security:
\[theorem:redundant edges\] Let ${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$ be obtained from ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ by removing a set of edges which are intransitively useless. Then $M$ is [i-secure]{}with respect to ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ if and only if $M$ is [i-secure]{}with respect to ${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$.
This theorem implies that for every policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$, a policy ${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$ without intransitively useless edges that is equivalent to ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ can be obtained from ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ by removing all intransitively useless edges.
Sound Unwindings and Uniform Intransitive Policies
--------------------------------------------------
The exponential size unwinding of [i-security]{}given in Section \[sect:intransitive unwinding exponential\] does not yield a polynomial-time algorithm for security verification. Since the problem is [[$\mathrm{NP}$]{}]{}-complete, such an algorithm—and hence an unwinding that is both small and easy to compute—does not exist, unless ${{\ensuremath{\mathrm{P}}}}={{\ensuremath{\mathrm{NP}}}}$. In this section, we define unwinding conditions that lead to a polynomial-size unwinding and are *sound* for [i-security]{}, and are *sound and complete* for [i-secure]{}in the case of uniform policies. Uniform policies are (as in the transitive case) policies in which every agent “may know” the set of agents who may currently interfere with him, that is, if an agent $u$ must not distinguish two states by the security definition, then the set of agents that may interfere with $u$ must be identical in these two states. Formally, we define this property as follows.
A policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ is *intransitively uniform*, if for all agents $u$ and states $s$, $s'$ with $s\approx^i_u s'$, we have that ${\ensuremath{u^{\leftarrowtail}_{s}}}={\ensuremath{u^{\leftarrowtail}_{s'}}}$.
Note that this definition is very similar to the uniformity condition for the transitive setting, but while in the transitive setting, uniform policies and policies without useless edges coincide, this is not true for intransitive noninterference (in fact, neither implication holds).
Uniformity, on an abstract level, is a natural requirement and often met in concrete systems, since an agent usually knows the sources of information available to him. In the uniform setting, many of the subtle issues with local policies do not occur anymore; as an example, [i-security]{} and the security definition from [@Leslie-DYNAMICNONINTERFERENCE-SSE-2006] coincide for uniform policies. Uniformity also has nice algorithmic properties, as both, checking whether a system has a uniform policy and checking whether a system with a uniform policy satisfies [i-security]{}, can be performed in polynomial time. This follows from the characterizations of i-security in terms of the unwindings we define next.
A *uniform intransitive unwinding* for $M$ with a policy [$({{\rightarrowtail}_{s}})_{s\in S}$]{} is a family of equivalence relations $\sim^{\tilde s,v}_u$ for each choice of states $\tilde s$ and agents $v$ and $u$, such that for all $s, t \in {S}$, and all $a \in {A}$, the following holds:
- If $s\sim^{\tilde s,v}_ut$, then ${{\tt obs}}_u(s)={{\tt obs}}_u(t)$. [[(OC$_{{\textnormal{i}}}^{{\textnormal{u}}}$)]{.nodecor}]{}
- If $s\sim^{\tilde s,v}_ut$, then ${\ensuremath{u^{\leftarrowtail}_{s}}}={\ensuremath{u^{\leftarrowtail}_{t}}}$. [[(PC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}
- If $s\sim^{\tilde s,v}_ut$ and $a\in
A$ with $v\not{{\rightarrowtail}_{\tilde s}}{{\tt dom}}(a)$, then $s\cdot
a\sim^{\tilde s,v}_ut\cdot a$. [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}
- If ${{\tt dom}}(a)\not{{\rightarrowtail}_{\tilde s}}u$, then $\tilde s\sim^{\tilde s,{{\tt dom}}(a)}_u \tilde s\cdot a$. [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}
In the following theorem intransitive uniformity and [i-security]{}(for uniform policies) are characterized by almost exactly the same unwinding. The only difference is that for uniformity we require policy consistency [[(PC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, since we are concerned with having the same *local policies* in certain states, while for security, we require [[(OC$_{{\textnormal{i}}}^{{\textnormal{u}}}$)]{.nodecor}]{}, since we are interested in *observations*.
\[theorem:polynomial unwinding characterization of intransitive uniformity and security\]
1. The policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ is intransitively uniform if and only if there is a uniform intransitive unwinding for $M$ and ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ that satisfies [[(PC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, and [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}.
2. If ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ is intransitively uniform, then $M$ is [i-secure]{}if and only if there is a uniform intransitive unwinding that satisfies [[(OC$_{{\textnormal{i}}}^{{\textnormal{u}}}$)]{.nodecor}]{}, [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}and [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}.
In particular, if an unwinding satisfying all four conditions exists, then a system is secure. Due to Theorem \[theorem:intransitive case np complete\], we cannot hope that the above unwindings completely characterize [i-security]{}, and indeed the system in Fig. \[fig:redundant edge\] is [i-secure]{}but not intransitively uniform. However, for uniform policies, Theorem \[theorem:polynomial unwinding characterization of intransitive uniformity and security\] immediately yields efficient algorithms to verify the respective conditions via a standard dynamic programming approach:
1. Verifying whether a policy is intransitively uniform can be performed in nondeterministic logarithmic space.
2. For systems with intransitively uniform policies, verifying whether a system is [i-secure]{}can be performed in nondeterministic logarithmic space.
The above shows that the complexity of intransitive noninterference with local policies comes from the *combination* of local policies that do not allow agents to “see” their allowed sources of information with an intransitive security definition. In the transitive setting, this interplay does not arise, since there a system always can allow agents to “see” their incoming edges (see Theorem \[theorem:uniformpolicies\]).
Unwinding for [IP-Security]{}
-----------------------------
In the setting with a global policy, [i-security]{}is equivalent to IP-security as defined in [@HY87]. For IP-security, Rushby gave unwinding conditions that are sufficient, but not necessary. This left open the question whether there is an unwinding condition that *exactly* characterizes IP-security, which we can now answer positively as follows. Clearly, a policy that assigns the same local policy to every state is intransitively uniform. Hence our results immediately yield a characterization of IP-security with the above unwinding conditions, and from these, an algorithm verifying IP-security in nondeterministic logarithmic space can be obtained in the straight-forward manner.
1. A system is IP-secure if and only if it has an intransitive unwinding satisfying [[(OC$_{{\textnormal{i}}}^{{\textnormal{u}}}$)]{.nodecor}]{}, [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, and [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}.
2. IP-security can be verified in nondeterministic logarithmic space.
Conclusion
==========
We have shown that noninterference with local policies is considerably different from noninterference with a global policy: an allowed interference in one state may contradict a forbidden interference in another state. Our new definitions address this issue. Our purge- and unwinding-based characterizations show that our definitions are natural, and directly lead to our complexity results.
We have studied generalizations of Rusby’s IP-security [@rushby92]. An interesting question is to study van der Meyden’s TA-security [@meyden2007] in a setting with local policies. Preliminary results indicate that such a generalization needs to use a very different approach from the one used in this paper.
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Additional Results
==================
In this Section we present and prove additional results which were informally mentioned in the main paper.
Initial-State Verification Suffices for Uniform Policies
--------------------------------------------------------
One noteworthy difference to the case of a system with a global policy is that it is necessary to evaluate the ${{\tt purge}}$-function in every state, and not only in the initial state: The system in Figure \[fig:dpurge\_in\_all\_states\] is secure with respect to the [[purge]{}]{}-based characterization of [t-security]{}, if we only consider traces starting in the initial state, but can easily be seen to not be [t-secure]{}.
(q0) [ [ (h1) [$H_1$]{}; (h2) \[right of=h1\] [$H_2$]{}; (l) \[below of=h1\] [$L$]{}; ]{}\
${{\tt obs}}_L \colon 0$\
]{}; (h2) edge node (l);
(q1) \[right=of q0\] [ [ (h1) [$H_1$]{}; (h2) \[right of=h1\] [$H_2$]{}; (l) \[below of=h1\] [$L$]{}; ]{}\
${{\tt obs}}_L \colon 0$\
]{}; (q2) \[right=of q1\] [ [ (h1) [$H_1$]{}; (h2) \[right of=h1\] [$H_2$]{}; (l) \[below of=h1\] [$L$]{}; ]{}\
${{\tt obs}}_L \colon 1$\
]{}; (q3) \[below=of q1\] [ [ (h1) [$H_1$]{}; (h2) \[right of=h1\] [$H_2$]{}; (l) \[below of=h1\] [$L$]{}; ]{}\
${{\tt obs}}_L \colon 1$\
]{};
(q0) edge node [$h_1$]{} (q1) edge node [$h_2$]{} (q3) (q1) edge node [$h_2$]{} (q2) ;
However, in the case of a uniform policy, it suffices to consider traces starting in the initial state, as we now show.
\[theorem:dpurge\_uniform\_security\] Let $M$ be a system with a uniform policy. Then $M$ is [t-secure]{} iff for all $u \in {D}$ and all $\alpha \in {A}^*$: ${{\tt obs}}_u(s_0 \cdot \alpha) = {{\tt obs}}_u(s_0 \cdot {{\tt purge}}(\alpha, u, s_0))$.
Assume that $M$ is a secure system. Then from $s_0 \cdot \alpha {\sim}_u s_0 \cdot {{\tt purge}}(\alpha, u,
s_0)$ follows from the output consistency that ${{\tt obs}}_u(s_0 \cdot \alpha) = {{\tt obs}}_u(s_0 \cdot {{\tt purge}}(\alpha, u,
s_0))$.
For the other direction of the proof, we consider $\alpha, \beta
\in{A}^*$ with ${{\tt purge}}(\alpha, u, s)$ $= {{\tt purge}}(\beta, u, s)$. Then it exists $\gamma \in {A}^*$ with $s = s_0 \cdot \gamma$. It follows that $s_0 \cdot \gamma {\sim}_u {{\tt purge}}(\gamma, u,
s_0)$. This gives $$\begin{aligned}
{{\tt obs}}_u(s \cdot \alpha)
& = {{\tt obs}}_u(s_0 \cdot \gamma \alpha) \\
& = {{\tt obs}}_u(s_0 \cdot {{\tt purge}}(\gamma \alpha, u, s_0)) \\
& = {{\tt obs}}_u(s_0 \cdot {{\tt purge}}(\gamma, u, s_0) {{\tt purge}}(\alpha, u,
s_0 \cdot {{\tt purge}}(\gamma, u, s_0))) \\
& = {{\tt obs}}_u(s_0 \cdot {{\tt purge}}(\gamma,u, s_0) {{\tt purge}}(\alpha, u, s_0
\cdot \gamma)) \\
& = {{\tt obs}}_u(s_0 \cdot {{\tt purge}}(\gamma,u, s_0) {{\tt purge}}(\beta, u, s_0
\cdot \gamma)) \\
& = {{\tt obs}}_u(s_0 \cdot \beta)
\enspace.
\end{aligned}$$
Some Properties of the [[purge]{}]{} Function
----------------------------------------------
Here we show that our [[purge]{}]{} function in the transitive setting behaves very naturally in the case of a uniform policy.
\[lemma:dpurge\_properties\] Let $M$ be a system with a policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$. For every $u \in {D}$, $s, t \in {S}$ and $\alpha, \beta \in {A}^*$, we have
1. ${{\tt purge}}({{\tt purge}}(\alpha, u, s), u, s) = {{\tt purge}}(\alpha, u,
s)$,
2. ${{\tt purge}}(\alpha \beta, u, s) = {{\tt purge}}(\alpha, u, s)
{{\tt purge}}(\beta, u, s \cdot {{\tt purge}}(\alpha, u, s))$,
3. if ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ is uniform and if ${\sim}_u$ is an equivalence relation on ${S}$ that satisfies [[(LR$_{\textnormal{t}}$)]{.nodecor}]{}and [[(SC$_{\textnormal{t}}$)]{.nodecor}]{}and if $s {\sim}_u t$, then $s \cdot \alpha {\sim}_u t \cdot {{\tt purge}}(\alpha, u, t)$ and ${{\tt purge}}(\alpha, u, s) = {{\tt purge}}(\alpha, u, t)$.
<!-- -->
1. We show this by an induction on the length of $\alpha$. Since the base case is obvious, we proceed with the inductive step. We consider $a \alpha$ with $a \in {A}$ and $\alpha \in
{A}^*$ and assume that the claim holds for $\alpha$. In the following two cases, we get
1. If ${{\tt dom}}(a) {{\rightarrowtail}_{s}} u$, we have $$\begin{aligned}
{{\tt purge}}({{\tt purge}}(a \alpha, u, s), u, s)
& = {{\tt purge}}(a {{\tt purge}}(\alpha, u, s\cdot a), u s)\\
& = a {{\tt purge}}({{\tt purge}}(\alpha, u, s\cdot a), u, s\cdot a) \\
& \stackrel{\text{I.H.}}{=} a {{\tt purge}}(\alpha, u, s\cdot a)\\
& = {{\tt purge}}(a \alpha, u, s)
\enspace.
\end{aligned}$$
2. If ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}} u$, we have $$\begin{aligned}
{{\tt purge}}({{\tt purge}}(a \alpha, u, s), u, s)
& = {{\tt purge}}({{\tt purge}}(\alpha, u, s), u, s) \\
& \stackrel{\text{I.H.}}{=} {{\tt purge}}(\alpha, u, s)
\enspace.
\end{aligned}$$
2. We show this claim by an induction on the length of $\alpha$ and consider again $a \alpha$. We get the following two cases
1. If ${{\tt dom}}(a) {{\rightarrowtail}_{s}} u$, we have $$\begin{aligned}
{{\tt purge}}(a \alpha \beta, u, s)
& = a {{\tt purge}}(\alpha \beta, u, s \cdot a) \\
& \stackrel{\text{I.H.}}{=} a {{\tt purge}}(\alpha, u, s\cdot a)
{{\tt purge}}(\beta, u, s \cdot a {{\tt purge}}(\alpha, u, s\cdot a)) \\
& = {{\tt purge}}(a \alpha, u, s) {{\tt purge}}(\beta, u, s \cdot
{{\tt purge}}(a \alpha, u, s))
\enspace.
\end{aligned}$$
2. If ${{\tt dom}}(a) \not{{\rightarrowtail}_{s}} u$, we have $$\begin{aligned}
{{\tt purge}}(a \alpha \beta, u, s)
& = {{\tt purge}}(\alpha \beta, u, s) \\
& \stackrel{\text{I.H.}}{=} {{\tt purge}}(\alpha, u,
s){{\tt purge}}(\beta, u, s \cdot {{\tt purge}}(\alpha, u, s)) \\
& = {{\tt purge}}(a \alpha, u, s) {{\tt purge}}(\beta, u, s \cdot {{\tt purge}}(a
\alpha, u, s))
\enspace.
\end{aligned}$$
3. This can be shown by an induction on the length of $\alpha$.
Equivalence of Intransitive Security Definitions for Uniform Policies
---------------------------------------------------------------------
We now show that in case of an intransitively uniform policy, a system is secure with respect to the definition of [@Leslie-DYNAMICNONINTERFERENCE-SSE-2006] if and only if it is [i-secure]{}.
We first show the following Lemma, which intuitively says that if the first action of $a\alpha$ is not transmitted to $u$ on the path $a\alpha$, then the same actions on the remaining path $\alpha$ are transmitted to $u$ when evaluating $\alpha$ from the state $s$ or from the state $s\cdot a$ in the case of a uniform policy. This is the key reason why, for uniform policies, the difference between Leslie’s function ${\ensuremath{{\tt ipurge'}}}$ and our ${\ensuremath{{\tt ipurge}}}$ is irrelevant.
\[lemma:same dsources on paths in uniform policies\] Let $M$ be a system with an intransitively uniform policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$. Let ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$, where $\alpha=\beta b\beta'$. Then $${{\tt dom}}(b)\in{\ensuremath{{\tt src}(b\beta',u,s\cdot\beta)}}{\ensuremath{\mathrm{\text{ iff
}}}}{{\tt dom}}(b)\in{\ensuremath{{\tt src}(b\beta',u,s\cdot a \beta)}}.$$
Assume this is not the case, and let $b\beta'$ be a minimal counter-example. First assume that ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(b\beta',u,s\cdot a\beta)}}$ and ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\beta',u,s\cdot \beta)}}$. Then there is some ${{\tt dom}}(c)\in{\ensuremath{{\tt src}(\beta',u,s\cdot a\beta b)}}$ with ${{\tt dom}}(b){{\rightarrowtail}_{s\cdot a\beta}}{{\tt dom}}(c)$, and due to minimality of $b\beta'$ it follows that ${{\tt dom}}(c)\in{\ensuremath{{\tt src}(\beta',u,s\cdot \beta b)}}$. Since ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\beta',u,s\cdot\beta)}}$, it thus follows that ${{\tt dom}}(b)\not{{\rightarrowtail}_{s\cdot\beta}}{{\tt dom}}(c)$. This is a contradiction to the intransitive uniformity of ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$, since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta,{{\tt dom}}(c),s)}}$, and hence $s\cdot a\beta\approx^i_{{{\tt dom}}(c)}s\cdot\beta$.
The second case is essentially identical: Assume that ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(b\beta',u,s\cdot\beta)}}$ and ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\beta',u,s\cdot a\beta)}}$. Then there is some ${{\tt dom}}(c)\in{\ensuremath{{\tt src}(\beta',u,s\cdot\beta b)}}$ with ${{\tt dom}}(b){{\rightarrowtail}_{s\cdot\beta}}{{\tt dom}}(c)$. Due to the minimality of $b\beta'$, it follows that ${{\tt dom}}(c)\in{\ensuremath{{\tt src}(\beta',u,s\cdot a\beta b)}}$, hence ${{\tt dom}}(b)\not{{\rightarrowtail}_{s\cdot a\beta}}{{\tt dom}}(c)$. Since $s\cdot a\beta\approx^i_{{{\tt dom}}(c)}s\cdot\beta$ due to the above, we have a contradiction to the uniformity of ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$.
From the above Lemma, we can now easily show that for uniform policies, [i-security]{} and security in the sense of [@Leslie-DYNAMICNONINTERFERENCE-SSE-2006] coincide:
Let $M$ be a system with an intransitively uniform policy ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$. Then $M$ is [i-secure]{} if and only if $M$ is secure with respect to the definition in [@Leslie-DYNAMICNONINTERFERENCE-SSE-2006].
Due to Theorem \[theorem:intransitive unwinding and ipurge characterization\], it suffices to show that in the case of a uniform policy, the functions ${\ensuremath{{\tt ipurge}}}$ and ${\ensuremath{{\tt ipurge'}}}$ coincide. Assume indirectly that this is not the case, and let $\alpha$ be a minimal sequence such that there exists a state $s$ and an agent $u$ with ${\ensuremath{{\tt ipurge}(\alpha,u,s)}}\neq{\ensuremath{{\tt ipurge'}(\alpha,u,s)}}$. Clearly $\alpha\neq\epsilon$, hence assume that $\alpha=a\alpha'$.
First assume that ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\alpha',u,s)}}$. In this case, we have (by definition and minimality of $\alpha$), that
$$\begin{array}{llllllllllllll}
{\ensuremath{{\tt ipurge}(a\alpha',u,s)}} & = & a {\ }{\ensuremath{{\tt ipurge}(\alpha',u,s\cdot a)}} \\
& = & {\ensuremath{{\tt ipurge'}(\alpha',u,s\cdot a)}} & = &{\ensuremath{{\tt ipurge}(a\alpha',u,s)}} \enspace,
\end{array}$$
which is a contradiction to the choice of $\alpha$.
Hence assume that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha',u,s)}}$. By definition, it follows that ${\ensuremath{{\tt ipurge}(a\alpha',u,s)}}={\ensuremath{{\tt ipurge}(\alpha',u,s)}}$ and ${\ensuremath{{\tt ipurge'}(a\alpha',u,s)}}={\ensuremath{{\tt ipurge'}(\alpha',u,s\cdot a)}}={\ensuremath{{\tt ipurge}(\alpha',u,s\cdot a)}}$ (the final equaility is due to the minimality of $\alpha$).
It hence suffices to show that ${\ensuremath{{\tt ipurge}(\alpha',u,s)}}={\ensuremath{{\tt ipurge}(\alpha',u,s\cdot a)}}$. This easily follows by induction on Lemma \[lemma:same dsources on paths in uniform policies\]: The same actions of $\alpha'$ are transmitted to $u$ when evaluating $\alpha'$ starting in the state $s$ and in $s\cdot a$.
Proofs
======
In this section we give proofs for the results claimed in the paper.
Proof of Theorem \[thm:dp\_characterizations\]
----------------------------------------------
First, we will show that \[thm:dp\_characterization\_def\]. implies \[thm:dp\_characterization\_unwind\].. Let $M$ be a [t-secure]{}system. Let $u \in {D}$. Define for every $s, t \in {S}$: $$s {\sim}_u t \text{ iff for all } \alpha \in {A}^* :
{{\tt obs}}_u(s \cdot \alpha) = {{\tt obs}}_u(t \cdot \alpha) \enspace.$$ The condition [[(OC$_{\textnormal{t}}$)]{.nodecor}]{}is satisfied if $\alpha = \epsilon$. For the condition [[(SC$_{\textnormal{t}}$)]{.nodecor}]{}, we consider $s, t \in {S}$ with $s {\sim}_u t$ and let $a \in {A}$. Then for all $\alpha \in {A}^*$, we have $s \cdot \alpha
{\sim}_u t \cdot\alpha$ and also $s\cdot a \alpha {\sim}_u t \cdot
a \alpha$. Therefore, $s \cdot a {\sim}_u t\cdot a$. For the condition [[(LR$_{\textnormal{t}}$)]{.nodecor}]{}, we consider $a \in {A}$ and $s \in {S}$ with ${{\tt dom}}(a) \not{{\rightarrowtail}_{s}} u$. Since $s$ is a reachable state, it exists $\alpha \in {A}^*$ with $s = s_0 \cdot \alpha$. The definition of [[t-security]{}]{}states, that for every $\beta \in {A}^*$ the equality of ${{\tt obs}}_u(s\cdot a \beta)$ and ${{\tt obs}}_u(s \cdot \beta)$ holds. Therefore, $s {\sim}_u s\cdot a$.
We assume that \[thm:dp\_characterization\_unwind\]. holds and will proof \[thm:dp\_characterization\_purge\].. Let $u\in {D}$ and assume that there exists a transitive unwinding ${\sim}_u$ that satisfies [[(LR$_{\textnormal{t}}$)]{.nodecor}]{}, [[(SC$_{\textnormal{t}}$)]{.nodecor}]{}and [[(OC$_{\textnormal{t}}$)]{.nodecor}]{}. We will show by an induction on the combined length of $\alpha$ and $\beta$, that for every state $s \in {S}$: ${{\tt purge}}(\alpha, u, s) = {{\tt purge}}(\beta, u, s)$ implies $s \cdot
\alpha {\sim}_u s \cdot \beta$. The base case with $\alpha = \beta = \epsilon$ is clear. For the inductive step consider $\alpha$ and $\beta$ with ${{\tt purge}}(\alpha, u, s) = {{\tt purge}}(\beta, u, s)$ for some state $s$. We have to consider two cases:
1. $\alpha = a \alpha'$ for some $a \in {A}$, $\alpha' \in
{A}^*$ and ${{\tt dom}}(a) \not{{\rightarrowtail}_{s}} u$. Then we have ${{\tt purge}}(a \alpha', u ,s) = {{\tt purge}}(\alpha', u, s)$. From the property [[(LR$_{\textnormal{t}}$)]{.nodecor}]{}follows that $s {\sim}_u
s \cdot a$ and from [[(LR$_{\textnormal{t}}$)]{.nodecor}]{}follows $s \cdot \alpha' {\sim}_u s
\cdot a \alpha'$. Applying the induction hypothesis gives $s\cdot \alpha' {\sim}_u
s \cdot \beta$ which can be combined to $s \cdot \alpha {\sim}_u
s \cdot \beta$.
2. $\alpha = a \alpha'$ and $\beta = b \beta'$ with ${{\tt dom}}(a) {{\rightarrowtail}_{s}} u$ and ${{\tt dom}}(b) {{\rightarrowtail}_{s}} u$. From $$\begin{aligned}
a {\ }{{\tt purge}}(\alpha', u, s\cdot a)
& = {{\tt purge}}(a \alpha', u, s) \\
& = {{\tt purge}}(\alpha, u, s) \\
& = {{\tt purge}}(\beta, u, s) \\
& = b {\ }{{\tt purge}}(\beta', u, s \cdot b)
\end{aligned}$$ follows that $a = b$ and ${{\tt purge}}(\alpha', u, s\cdot a) =
{{\tt purge}}(\beta', u, s\cdot a)$. Applying the induction hypothesis gives $s \cdot a \alpha' {\sim}_u s \cdot b \beta'$.
In both cases follows from [[(OC$_{\textnormal{t}}$)]{.nodecor}]{}that ${{\tt obs}}_u(s \cdot \alpha) =
{{\tt obs}}_u(s \cdot \beta)$.
For proofing the implication from \[thm:dp\_characterization\_purge\] to \[thm:dp\_characterization\_def\], we assume, that $M$ does not satisfy [[t-security]{}]{}. Therefore, there exists an agent $u \in {D}$ and states $s, s'\in
{S}$ with $s \approx_u s'$ and ${{\tt obs}}_u(s) \neq {{\tt obs}}_u(s')$. By the definition of [[t-security]{}]{}, there exists $t \in {S}$, $a
\in {A}$ and $\alpha \in {A}^*$ with ${{\tt dom}}(a)
\not{{\rightarrowtail}_{t}} u$, $s = t \cdot a \alpha$ and $s' = t \cdot
\alpha$. By applying of ${{\tt purge}}$, we have ${{\tt purge}}(a \alpha, u, t) =
{{\tt purge}}(\alpha, u, t)$ and from ${{\tt obs}}_u(t \cdot a \alpha) \neq
{{\tt obs}}_u(t \cdot \alpha)$, follows that \[thm:dp\_characterization\_purge\] does not hold.
For proofing the missing implication, we assume that \[thm:dp\_characterization\_def\]. does not hold. Therefore, it exists $u \in {D}$, $s \in {S}$, $a \in {A}$ and $\alpha \in {A}^*$ with ${{\tt dom}}(a) \not{{\rightarrowtail}_{s}} u$ and ${{\tt obs}}_u(s \cdot a \alpha) \neq {{\tt obs}}_u(s \cdot \alpha)$. Therefore, $s\cdot a \alpha \approx_u s \cdot \alpha$ and \[thm:dp\_characterization\_def\] does not hold.
Proof of Theorem \[theorem:uniformpolicies\]
--------------------------------------------
Let $M$ be a [t-secure]{}system with respect to the policy [$({{\rightarrowtail}_{s}})_{s\in S}$]{}. Then there exists a transitive unwinding $({\sim}_u)_{u \in
{D}}$ for $M$. Note, that for every $u \in {D}$, the smallest eqivalence relation ${\sim}_u$ that satisfies [[(LR$_{\textnormal{t}}$)]{.nodecor}]{}and [[(SC$_{\textnormal{t}}$)]{.nodecor}]{}is equal to the smallest equivalence relation on ${S}$ that includes $\approx_u$. Let ${\sim}_u'$ be the a smallest equivalence relation that satisfies [[(SC$_{\textnormal{t}}$)]{.nodecor}]{}and [[(LR$_{\textnormal{t}}$)]{.nodecor}]{}with respect to the policy [$({{\rightarrowtail}_{s}}')_{s\in S}$]{}. We will show that ${{\sim}_u'} \subseteq {{\sim}_u}$. Let $s, t \in {S}$ with $s {\sim}_u' t$ and $t = s \cdot a$ form some $a \in {A}$ with ${{\tt dom}}(a) \not{{\rightarrowtail}_{s}}' u$. Therefore, there exists $s' \in {S}$ with $s' {\sim}_u s$ and ${{\tt dom}}(a) \not{{\rightarrowtail}_{s'}} u$. From $s' {\sim}_u s' \cdot a$ and $s' \cdot a {\sim}_u s \cdot a$ follows $s {\sim}_u t$.
The other direction of the proof follows directly from the fact, that the policy [$({{\rightarrowtail}_{s}}')_{s\in S}$]{}is at least as restrictive as the policy [$({{\rightarrowtail}_{s}})_{s\in S}$]{}.
Proof of Theorem \[theorem:intransitive unwinding and ipurge characterization\]
-------------------------------------------------------------------------------
We first consider the [${\tt ipurge}$]{}-characterization and then the intransitive unwinding characterization.
1. We first show that [i-security]{} implies the [${\tt ipurge}$]{}-characterization. Hence indirectly assume that the system is [i-secure]{}, and indirectly assume that the [${\tt ipurge}$]{}-condition is not satisfied. Then there exists a state $s$, an agent $u$, and sequences $\alpha$ and $\beta$ with ${\ensuremath{{\tt ipurge}(\alpha,u,s)}}={\ensuremath{{\tt ipurge}(\beta,u,s)}}$, and ${{\tt obs}}_u(s\cdot\alpha)\neq{{\tt obs}}_u(s\cdot\beta)$. We choose $\alpha$ and $\beta$ such that ${\left| \alpha \right|}+{\left| \beta \right|}$ is minimal among all such examples. Clearly, if *both* $\alpha$ and $\beta$ start with an action that is transmitted to $u$, then this action must be the same: If $\alpha=a\alpha'$ with ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\alpha',u,s)}}$ and $\beta=b\beta'$ with ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(b\beta',u,s)}}$, then ${\ensuremath{{\tt ipurge}(\alpha,u,s)}}$ starts with $a$, and ${\ensuremath{{\tt ipurge}(\beta,u,s)}}$ starts with $b$. It thus follows that $a=b$, and hence we could use the state $s'=s\cdot a$ and the sequences $\alpha'$ and $\beta'$ as a counter-example, which contradicts the minimality of $\alpha$ and $\beta$. Hence we can, without loss of generality, assume that $\alpha=a\alpha'$ for some $a$ with ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha',u,s)}}$. It thus follows that ${\ensuremath{{\tt ipurge}(\alpha',u,s)}}={\ensuremath{{\tt ipurge}(\alpha,u,s)}}={\ensuremath{{\tt ipurge}(\beta,u,s)}}$. Since the system is secure, we also have ${{\tt obs}}_u(s\cdot\alpha')={{\tt obs}}_u(s\cdot a\alpha')={{\tt obs}}_u(s\cdot\alpha)\neq{{\tt obs}}_u(s\cdot\beta)$, and hence we again obtain a contradiction to the minimality of $\alpha$ and $\beta$ (with choosing $\alpha'$ instead of $\alpha$).
We now show the converse, i.e., that the [${\tt ipurge}$]{}-characterization implies [i-security]{}. Hence assume that the system satisfies the [${\tt ipurge}$]{}-condition. To show interference security, let ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$ for some agent $u$ and state $s$, we show that ${{\tt obs}}_u(s\cdot a\alpha)={{\tt obs}}_u(s\cdot\alpha)$. Note that since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$, it follows that ${\ensuremath{{\tt ipurge}(a\alpha,u,s)}}={\ensuremath{{\tt ipurge}(\alpha,u,s)}}$. Hence from the prerequisites of the theorem it follows that ${{\tt obs}}_u(s\cdot a\alpha)={{\tt obs}}_u(s\cdot\alpha)$ as required.
2. We prove that the intransitive unwinding characterization is also equivalent to [i-security]{}. First assume that there is an intransitive unwinding $({\precsim_{{D}'}})_{{D}'\subseteq {D}}$ for $M$ with respect to ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$. We show that the system is [i-secure]{}. For this it suffices to show that if ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$, then $s\cdot a\alpha{\precsim_{{D}'}} s\cdot\alpha$ for some set ${D}'$ with $u\in {D}'$. For each prefix $\alpha'$ of $\alpha$, let ${D}_{\alpha'}$ be defined as $${D}_{\alpha'}={\ensuremath\left\{v\in D\ \vert\ {{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha',v,s)}}\right\}} \enspace.$$ Clearly, if $\alpha'$ is a prefix of $\alpha''$, then ${D}_{\alpha''}\subseteq {D}_{\alpha'}$. Since $u\in {D}_\alpha$, it suffices to show that $s\cdot a\alpha'{\precsim_{{D}_{\alpha'}}} s\cdot\alpha'$ for all prefixes $\alpha'$ of $\alpha$. We show the claim by induction. For $\alpha'=\epsilon$, the claim follows from [[(LR$_{\textnormal{i}}$)]{.nodecor}]{}, since ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}}u$. Hence assume that $\alpha'=\beta b$ for some sequence $\beta$ and action $b$. By induction, we have that $s\cdot a\beta{\precsim_{{D}_\beta}}s\cdot\beta$, where ${D}_\beta$ contains all agents $v$ with ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta,v,s)}}$. Now let $u\in {D}_{\alpha'}$, it then also follows that $u\in {D}_{\beta}$. Let ${D}'$ be defined as in the condition [[(SC$_{\textnormal{i}}$)]{.nodecor}]{}. Since the condition implies $s\cdot a\beta b{\precsim_{{D}'}}s\cdot\beta b$, it suffices to show that $u\in {D}'$. Clearly this is the case if ${{\tt dom}}(b)\in {D}_\beta$, i.e., if ${D}_\beta={D}'$. Hence assume this is not the case, by definition of ${D}_\beta$ it then follows that ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\beta,{{\tt dom}}(b),s)}}$. Since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b,u,s)}}$, this implies that ${{\tt dom}}(b)\not{{\rightarrowtail}_{s\cdot a\beta}} u$, hence $u\in {D}'$ follows in this case as well.
For the other direction, assume that the system is [i-secure]{}. We define $s{\precsim_{{D}'}} t$ if there is a state $\tilde s$, an action $a$ and a sequence $\alpha$, such that $s=\tilde s \cdot a\alpha$, $t=\tilde s\cdot\alpha$, and for all $u\in {D}'$, we have ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,\tilde s)}}$. We claim that this defines an intransitive unwinding for $M$ with respect to ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$. Since the system is [i-secure]{}, the condition [[(OC$_{\textnormal{i}}$)]{.nodecor}]{}is obviously satisfied. The condition [[(LR$_{\textnormal{i}}$)]{.nodecor}]{}follows from the fact that if ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}}u$, then ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a,u,s)}}$. It remains to show [[(SC$_{\textnormal{i}}$)]{.nodecor}]{}. Hence let $s {\precsim_{{D}'}} t$, and let $\tilde s$, $a$ and $\alpha$ be chosen with the above properties. Let $b$ be an action, and let ${D}''$ be the set resulting from applying [[(SC$_{\textnormal{i}}$)]{.nodecor}]{}. It remains to show that for each $u\in {D}''$, we have ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha b,u,\tilde s)}}$. First assume that ${{\tt dom}}(b)\in {D}'$, it then follows from the definition of ${\precsim_{{D}'}}$ that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,{{\tt dom}}(b),\tilde s)}}$, and hence ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha b,u,\tilde s)}}$. On the other hand, if ${{\tt dom}}(b)\notin {D}'$, then from $u\in {D}''$, we know that ${{\tt dom}}(b)\not{{\rightarrowtail}_{\tilde s\cdot a\alpha}}u$, and hence from ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,\tilde s)}}$ (since $u\in {D}'$) and ${\ensuremath{{\tt src}(a\alpha b,u,\tilde s)}}={\ensuremath{{\tt src}(a\alpha,u,\tilde s)}}$, it follows that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha b,u,\tilde s)}}$ as required.
Proof of Theorem \[theorem:intransitive case np complete\] {#sect:complexity}
----------------------------------------------------------
Checking whether a system is not [i-secure]{}can be done in [[$\mathrm{NP}$]{}]{}.
The algorithm simply guesses the corresponding values of $a$, $u$, $s$, and $\alpha$, and verifies that these satisfy ${{\tt obs}}_u(s\cdot a\alpha)\neq{{\tt obs}}_u(s\cdot\alpha)$ and ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$ in the straight-forward way. To show that this gives an [[$\mathrm{NP}$]{}]{}-algorithm, it suffices to show that the length of $\alpha$ can be bounded polynomially in the size of the system. We show that if the system is insecure, then $\alpha$ can be chosen with ${\left| \alpha \right|}\leq{\left| {S}\right|}^2$.
To show this, let $\alpha$ be a path of minimal length satisfying the above. Let $F_s$ and $F_{s\cdot a}$ be the finite state machines obtained when starting the system in the states $s$ and $s\cdot a$, respectively, and let $F=F_s\times F_{s\cdot a}$, with initial state $(s,s\cdot a)$. Clearly, in $F$, we have $(s,s\cdot a)\cdot \alpha=(s\cdot\alpha, s\cdot a\alpha)$. If ${\left| \alpha \right|}\ge {\left| {S}\right|}^2$, then $\alpha$ visits a state from $F$ twice, i.e., $\alpha$ contains a nontrivial loop. Such a loop can be removed from $\alpha$ without changing the states that are reached. Clearly, removing a loop does not add information flow, hence the thus-obtained $\alpha'$ also satisfies the prerequisites for $\alpha$, which is a contradiction to $\alpha$’s minimality.
\[theorem:np hardness\] For every security definition that is at least as strict as information-flow-security and at least as permissive as interference-security, the problem to determine whether a given system is insecure is [[$\mathrm{NP}$]{}]{}-hard under ${\ensuremath{\leq_{m}^{\log}}}$-reductions.
[r]{}[6.25cm]{}
;
(incoming) at (0,0) [ $\cdot$ ]{};
\(h) at (3,3) [ $h$ ]{}; (uneq0) at (3,2) [ $u_{\neq0}$ ]{} ; (h) edge (uneq0); (u-0-1) \[round-boxed, fit = (h) (uneq0)\] ;
(incoming) edge node\[above\] [ $u_{=0}$]{} (u-0-1);
\(h) at (3,0.5) [ $h$ ]{}; (uneq1) at (3,-0.5) [ $u_{\neq1}$ ]{} ; (h) edge (uneq1); (u-1-1) \[round-boxed, fit = (h) (uneq1)\] ;
(incoming) edge node\[above\] [ $u_{=1}$]{} (u-1-1);
\(h) at (3,-2) [ $h$ ]{}; (uneq2) at (3,-3) [ $u_{\neq2}$ ]{} ; (h) edge (uneq2); (u-2-1) \[round-boxed, fit = (h) (uneq2)\] ;
(incoming) edge node\[above\] [ $u_{=2}$]{} (u-2-1);
(outgoing) at (6,0) [ $\cdot$ ]{};
(u-0-1) edge node\[above\] [ $h$ ]{} (outgoing); (u-1-1) edge node\[above\] [ $h$ ]{} (outgoing); (u-2-1) edge node\[above\] [ $h$ ]{} (outgoing);
We reduce from the 3-colorability problem for graphs. Let a graph $G$ with vertices $u_1,\dots,u_n$ and edges $(v^1_1,v^2_1),\dots,(v^m_1,v^m_2)$ be given. We construct a system $M^G$ as follows:
- for each vertex $u$, there is an agent $u$ with actions $u_{=0}$, $u_{=1}$, and $u_{=2}$, and there are agents $u_{\neq 0}$, $u_{\neq 1}$, $u_{\neq 2}$, each having exactly one action, which for simplicity we denote with the agent’s name. Additionally, there is an agent $h$ with a single action $h$, and an agent $L$ with a single action $L$.
- for each vertex $u$, we construct a subsystem ${\ensuremath{C(u)}}$ (see Figure \[fig:colorsys\]), that models the choice of coloring of $u$ in the graph. In ${\ensuremath{C(u)}}$ and all following systems, all transitions that are not explicitly indicated in the graphical representation loop in the corresponding state.
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- for each edge $(u,v)$, we construct a subsystem ${\ensuremath{E(u,v)}}$ (see Figure \[fig:diffcols\]), which enforces that the colors of $u$ and $v$ must be different. The edges labelled with a transition of the form $u_{\neq i,j}$ represent two consecutive edges, the first one with the transition $u_{\neq i}$, and the second one labelled with the transition $u_{\neq j}$, where the policy is repeated between the two transitions.
- the system $M^G$ is now designed as shown in Figure \[fig:complete system MG\]. We denote the left-most state with $s_0$. The unlabelled arrows between the different ${\ensuremath{C(u)}}$ and ${\ensuremath{E(u,v)}}$-nodes express that the final node of one is the starting node of the other. The subsystems ${\ensuremath{C'(u)}}$ and ${\ensuremath{E'(u,v)}}$ are defined in the same way as ${\ensuremath{C(u)}}$ and ${\ensuremath{E(u,v)}}$, except that here, in all states we have policies that allow interference between any two agents. With [$last$]{}, we denote the final state of ${\ensuremath{E(v^m_1,v^m_2)}}$, and with [$last'$]{}, the final state of ${\ensuremath{E'(v^m_1,v^m_2)}}$. We define the observation functions as follows: ${{\tt obs}}_L({\ensuremath{last'}\xspace})=1$, and for all other combinations of agent $u$ and state $s$, ${{\tt obs}}_u(s)=0$.
The main property of $M^G$ is that it is possible to find a path $h\alpha$ from $s_0$ to ${\ensuremath{last}\xspace}$ that does *not* transmit $h$ to $L$ if and only if $G$ is $3$-colorable:
A path $h\alpha$ is *hiding*, if ${{\tt dom}}(h)\notin{\ensuremath{{\tt src}(h\alpha,L,s_0)}}$, and $s_0\cdot h\alpha={\ensuremath{last}\xspace}$.
Intuitively, the subsystem ${\ensuremath{C(u)}}$ forces the agent $u$ to “choose” a color $i\in{\ensuremath\left\{0,1,2\right\}}$, by performing the action $u_{=i}$. For each edge $(u,v)$ or $(v,u)$ in which $u$ is involved, the agent $u$ later repeats the same transition in the subsystem ${\ensuremath{E(u,v)}}$ (or ${\ensuremath{E(v,u)}}$). These systems ensure that no two agents that are connected with an edge can choose the same color—if they do, then a dead-end is reached. To ensure that agents are consistent in their choice of colors (i.e., choose the same color in later ${\ensuremath{E(u,v)}}$-systems as in the ${\ensuremath{C(u)}}$ system, and consequently chooses the same color for each ${\ensuremath{E(u,v)}}$-system), we use the following construction: When agent $u$ chooses color $i$ in ${\ensuremath{C(u)}}$, the agent $u_{\neq i}$ “receives” interference from $h$. If the agent $u$ later claims to have a color different from $i$, then the only available path is one that allows an interference between $u_{\neq i}$ and $L$, which transmits the information about $h$ to $L$.
\[lemma:hiding path iff colorable\] There is a hiding path if and only if $M^G$ is $3$-colarable.
First assume that $G$ is $3$-colorable, hence let $c\colon {\ensuremath\left\{u_1,\dots,u_n\right\}}\rightarrow{\ensuremath\left\{0,1,2\right\}}$ be a coloring function such that for all edges $(u,v)\in E$, we have that $c(u)\neq c(v)$. We construct the path $a\alpha$ as the unique path from $s_0\cdot a$ to ${\ensuremath{last}\xspace}$ that starts with $L$, does not use loops in any state, and where each agent $u$ chooses the action $u_{=c(u)}$ whenever the current state has more than one non-looping actions. Since $c$ is a $3$-coloring, this path does not hit a dead-end in any of the ${\ensuremath{E(u,s)}}$-systems, and in particular, reaches the state ${\ensuremath{last}\xspace}$. Due to the construction of the path, whenever a transaction $u_{\neq i}$ is performed, the action $u_{=i}$ has never been performed on the path, and thus $u_{\neq i}$ has not received $h$. Hence none of the agents interfering with $L$ has received the action $h$, and thus ${{\tt dom}}(h)\notin{\ensuremath{{\tt src}(a\alpha,L,s_0)}}$, i.e., $a\alpha$ is hiding.
For the other direction, assume that there is a hiding path $a\alpha$. Without loss of generality, we can assume that $a\alpha$ does not use any actions that loop in the current state. Since $a\alpha$ is hiding, we know that $s_0\cdot a\alpha={\ensuremath{last}\xspace}$, in particular, every subsystem ${\ensuremath{C(u)}}$ and ${\ensuremath{E(u,v)}}$ is passed when following $a\alpha$ from $s_0$. We can thus define a coloring $c\colon {\ensuremath\left\{u_1,\dots,u_n\right\}}\rightarrow{\ensuremath\left\{0,1,2\right\}}$ by $c(u)=i$, where $i$ is the unique value such that at the start of ${\ensuremath{C(u)}}$, the action $u_{=i}$ is performed by $u$. We claim that this is a $3$-coloring of $G$.
For this, first observe that on $a\alpha$, no action $u_{=j}$ is performed for $j\neq c(u)$: Due to the above, no looping action is performed. Now observe that after the performance of $u_{=c(u)}$ in ${\ensuremath{C(u)}}$, the agent $u_{\neq c(u)}$ has received the $h$-event. Now after a later performance of the action $u_{=j}$, every path that proceeds to ${\ensuremath{last}\xspace}$ uses a transition $u_{\neq c(u)}$ in a state where $u_{\neq c(u)}{{\rightarrowtail}_{}} L$, which is a contradiction to the assumption that $h\alpha$ is hiding.
We now show that for each edge $(u,v)$ of $G$, we have that $c(u)\neq c(v)$. Since $a\alpha$ is hiding, $a\alpha$ passes through the subsystem ${\ensuremath{E(u,v)}}$. Due to the above, in this subsystems the actions $u_{=c(u)}$ and $v_{=c(v)}$ are performed at the relevant states. If $c(u)$ and $c(v)$ were equal, this would reach a dead-end state, which is a contradiction, as $a\alpha$ is hiding, and hence $s_0\cdot a\alpha={\ensuremath{last}\xspace}$.
Since $M^G$ can clearly be constructed from $G$ in logarithmic space, the following lemma now proves Theorem \[theorem:np hardness\]:
- If $G$ is $3$-colorable, then $M^G$ is not [i-secure]{}.
- If $G$ is not $3$-colorable, then $M^G$ is [i-secure]{}.
First assume that $G$ is $3$-colorable. By Lemma \[lemma:hiding path iff colorable\], there is a hiding path $h\alpha$. In particular, $s_0\cdot h\alpha={\ensuremath{last}\xspace}$. Since the action $h$ loops in the state $s_0\cdot h$, we can without loss of generality assume that $\alpha$ does not start with $h$, and hence $s_0\cdot\alpha={\ensuremath{last'}\xspace}$. Since $h\alpha$ is hiding, we know that ${{\tt dom}}(h)\notin{\ensuremath{{\tt src}(h\alpha,L,s_0)}}$. Since in $s_0$, there is no outgoing edge from $h$, we also know that ${{\tt dom}}(h)^{s_0}_{\downarrow}\cap{\ensuremath{{\tt src}(\alpha,L,s_0)}}=\emptyset$. Since ${{\tt obs}}_L({\ensuremath{last}\xspace})\neq{{\tt obs}}_L({\ensuremath{last'}\xspace})$, it follows that the $M^G$ is not [i-secure]{}.
Now assume that $G$ is not $3$-colorable, and indirectly assume that $M^G$ is not [i-secure]{}. Since $L$ is the only agent whose observation function is not constant, this implies that there is a state $s$, an action $a$, and a sequence $\alpha$ such that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,L,s)}}$ and ${{\tt obs}}_L(s\cdot a\alpha)\neq{{\tt obs}}_L(s\cdot\alpha)$. Since ${\ensuremath{last'}\xspace}$ is the only state with an observation different from $0$, we know that ${\ensuremath{last'}\xspace}\in{\ensuremath\left\{s\cdot a\alpha,s\cdot\alpha\right\}}$. In particular, $s$ is an ancestor of ${\ensuremath{last'}\xspace}$ in $M^G$. Since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,L,s)}}$, we know that in particular, ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}}L$. Since the only ancestor state of ${\ensuremath{last'}\xspace}$ in which the local policy is not the complete relation is $s_0$, we know that $s=s_0$. Since in $s_0$, all agents except for $h$ may interfere with $L$, we also know that $a=h$. Since $s_0\cdot h\alpha\neq{\ensuremath{last'}\xspace}$ for any $\alpha$, we know that $s_0\cdot\alpha={\ensuremath{last'}\xspace}$. From the design of $M^G$, it follows that $s_0\cdot h\alpha={\ensuremath{last}\xspace}$. Since $h\notin{\ensuremath{{\tt src}(h\alpha,L,s_0)}}$, it follows that $h\alpha$ is hiding, and thus Lemma \[lemma:hiding path iff colorable\], implies that $G$ is $3$-colorable as required.
We now prove the FPT result, from which the case for a logarithmic number of agents immediately follows:
It clearly suffices to provide an FPT algorithm. Such an algorithm can be obtained by the standard dynamic programming approach, by first creating a table with an entry for every choice $s$, $t$ and ${D}'$, that indicates whether $s{\precsim_{{D}'}}t$ has already been established. The size of the table is $2^{{\left| {D}\right|}}\cdot{\left| {S}\right|}^2$. Now initialize the table with ${\left| {S}\right|}\cdot{\left| {A}\right|}$ operations (using the [[(LR$_{\textnormal{i}}$)]{.nodecor}]{}property), and use the [[(SC$_{\textnormal{i}}$)]{.nodecor}]{}condition to add entries to the table until no changes are performed anymore. Then the condition [[(OC$_{\textnormal{i}}$)]{.nodecor}]{}can be verified by checking, for each agent $u$, and each set ${D}'$ for which $u\in {D}'$, whether for all $s{\precsim_{{D}'}} t$, we have ${{\tt obs}}_u(s)={{\tt obs}}_u(t)$. For each choice of $u$ and ${D}'$, this requires ${\left| {S}\right|}^2$ accesses to the table. Since the access to the table can be implemented in time $2^{{\left| {D}\right|}}\cdot{\ensuremath{\mathrm{\text{poly}}}}{{\left| M \right|}}$, this completes the proof.
Proof of Theorem \[theorem:redundant edges\]
--------------------------------------------
Clearly, if $M$ is not [i-secure]{}with respect to ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$, then $M$ is also not [i-secure]{}with respect to ${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$. Using induction, we can assume that ${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$ arose from ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$ by removing a single intransitively useless edge $e$. Assume that $M$ is not [i-secure]{}with respect to ${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$. Hence there are $a\in {A}$, $\alpha in {A}^*$, $s \in {S}$, $u \in {D}$ such that ${{\tt dom}}(a) \notin {\ensuremath{{\tt src}(a \alpha,u,s)}}$ (with respect to ${\ensuremath{({{\rightarrowtail}_{s}}')_{s\in S}}\xspace}$) and ${{\tt obs}}_u(s \cdot a \alpha) \neq {{\tt obs}}_u(s\cdot \alpha)$. Since $M$ is [i-secure]{}, we know that ${{\tt dom}}(a) \in {\ensuremath{{\tt src}(a \alpha,u,s)}}$ (with respect to ${\ensuremath{({{\rightarrowtail}_{s}})_{s\in S}}\xspace}$). In particular, we know that $s\cdot a\alpha \not{{\rightarrowtail}_{u}} s\cdot \alpha$. It follows thtat $e$ is not intransitively useless, a contradiction.
Proof of Theorem \[theorem:polynomial unwinding characterization of intransitive uniformity and security\]
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The proof of this theorem highlights an interesting difference between intransitive noninterference with a global policy (IP-security) and with local policies: It can easily be shown (see [@emsw11]) that if a system is not IP-secure, then there exist a “witness” for the insecurity consisting of a state $s$, an agent $u$, an action $a$, and a sequence $\alpha$ such that
1. ${{\tt dom}}(a)\notin{{\tt src}}(a\alpha,u)$ and ${{\tt obs}}_u(s\cdot a\alpha)\neq{{\tt obs}}_u(s\cdot\alpha)$ (i.e., these values demonstrate insecurity of the system), and
2. $\alpha$ contains no $b$ with ${{\tt dom}}(a){{\rightarrowtail}_{}}{{\tt dom}}(b)$.
Intuitively, this means that to verify insecurity, it suffices to consider sequences in which the “secret” action $a$ is not transmitted even one step. This feature is crucial for the polynomial-time algorithm in [@emsw11] to verify IP-security. In a setting with local policies, the situation is different, the above-mentioned property does not hold. This is in fact the key reason why no “small” unwinding for [i-security]{}exists, and why the verification problem is [[$\mathrm{NP}$]{}]{}-hard. However, in systems with a uniform policy, we again can prove an analogous property, even though the proof is more complicated than for the setting with a global policy:
\[lemma:minimal path different observations\] Let $M$ be a system with a policy that is intransitively uniform. Then $M$ is [i-secure]{}if and only if there are $a$, $u$, $s$, and $\alpha$ with ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$, ${{\tt obs}}_u(s\cdot\alpha)\neq{{\tt obs}}_u(s\cdot a\alpha)$, and no $b$ with ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$ appears in $\alpha$.
Clearly if such $a$, $u$, $s$, and $\alpha$ exist, then the system is not [i-secure]{}. For the converse, let $\alpha$ be of minimal length such that there exist $u$, $s$, and $a$ with ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$ and ${{\tt obs}}_u(s\cdot a\alpha)\neq{{\tt obs}}_u(a\cdot\alpha)$. Indirectly, assume that $\alpha=\beta b\beta'$ for some $b$ with ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$. We consider three cases.
- *Assume ${{\tt obs}}_u(s\cdot a\beta b\beta')\neq {{\tt obs}}_u(s\cdot a\beta\beta')$.* Note that ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\beta',u,s\cdot a\beta)}}$. Hence choosing $s'=s\cdot a\beta$, $a'=b$, and $\alpha'=\beta'$ is a contradiction to the minimality of $\alpha$.
- *Assume ${{\tt obs}}_u(s\cdot\beta b\beta')\neq {{\tt obs}}_u(s\cdot\beta\beta')$.* To show that this again is a contradiction to the minimality of $\alpha$ (starting in the state $s\cdot\beta$), it suffices to show that ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\beta',u,s\cdot\beta)}}$. Hence, indirectly assume that ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(b\beta',u,s\cdot\beta)}}$, and let $\gamma$ be a minimal prefix of $b\beta'$ such that there is some agent $v$ with
- ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(\gamma,v,s\cdot\beta)}}$,
- ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\gamma,v,s)}}$.
Since choosing $v=u$ and $\gamma=\beta'$ satisfies these conditions, such a minimal $\gamma$ exists. Again, consider the point where $v$ “learns” that $a$ was performed, i.e., let $\gamma=\pi c\pi'$ with
- ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(\pi,{{\tt dom}}(c),s\cdot\beta)}}$, and
- ${{\tt dom}}(c){{\rightarrowtail}_{s\cdot\beta\pi}}v$.
Since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\cdot\beta\gamma,v,s)}}$, and $\pi$ is a prefix of $\gamma$, the prerequisites to the lemma imply that $v^\uparrow_{s\cdot a\beta\pi}=v^\uparrow_{s\cdot\beta\pi}$, in particular, ${{\tt dom}}(c){{\rightarrowtail}_{s\cdot a\beta\pi}}v$. Since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\gamma,v,s)}}$, this implies $${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\pi,{{\tt dom}}(c),s)}},$$ hence we have a contradiction to the minimality of $\gamma$.
- *Assume ${{\tt obs}}_u(s\cdot a\beta b\beta')={{\tt obs}}_u(s\cdot a\beta\beta')$ and ${{\tt obs}}_u(s\cdot\beta b\beta')={{\tt obs}}_u(s\cdot\beta\beta')$.* Since ${{\tt obs}}_u(s\cdot a\beta b\beta')\neq {{\tt obs}}_u(s\cdot\beta b\beta')$, this implies ${{\tt obs}}_u(s\cdot a\beta\beta')\neq{{\tt obs}}_u(s\cdot\beta\beta')$. To obtain a contradiction to the minimality of $\alpha$, it suffices to show that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\beta',u,s)}}$. Hence, indirectly assume that ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\beta\beta',u,s)}}$, and let $\gamma$ be a minimal prefix of $\beta'$ such that there is an agent $v$ with
- ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b\gamma,v,s)}}$, and
- ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\beta\gamma,v,s)}}$.
Since choosing $v=u$ and $\gamma=\beta'$ satisfies these conditions, such a minimal $\gamma$ exists. Now consider the step where $v$ “learns” $a$, which clearly happens inside $\gamma$ (as ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b\gamma,v,s)}}$). Hence $\gamma=\pi c\pi'$ with
- ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\beta\pi,{{\tt dom}}(c),s)}}$, and
- ${{\tt dom}}(c){{\rightarrowtail}_{s\cdot a\beta\pi}}v$.
Since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b\gamma,v,s)}}$, we have ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\gamma,v,s\cdot a\beta)}}$. Since $\pi$ is a prefix of $\gamma$, this implies ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\pi,v,s\cdot a\beta)}}$. The conditions of the lemma this imply that $v^\uparrow_{s\cdot a\beta b\pi}=v^\uparrow_{s\cdot a\beta\pi}$. In particular, this implies ${{\tt dom}}(c){{\rightarrowtail}_{s\cdot a\beta b\pi}}v$. Since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b\gamma,v,s)}}$, this implies ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b\pi,{{\tt dom}}(c),s)}}$, which is a contradiction to the minimality of $\gamma$.
We now show a similar fact which allows us to easily verify whether a policy is intransitively uniform: To verify uniformity, it again suffices to consider action sequences in which the “secret” action is not even transmitted a single step. This is shown in the following Lemma:
\[lemma:minimal path different policies\] If a policy for a system is not intransitively uniform, there is an agent $u$, an action $a$, a sequence $\alpha$, and a state $s$ such that
1. ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$,
2. ${\ensuremath{u^{\leftarrowtail}_{s\cdot a\alpha}}}\neq {\ensuremath{u^{\leftarrowtail}_{s\cdot\alpha}}}$,
and contains no $b$ with ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$.
Choose $u$, $a$, $s$, and $\alpha$ such that ${\left| \alpha \right|}$ is minimal, and indirectly assume that $\alpha=\beta b\beta'$ for some sequences $\beta$ and $\beta'$, where ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$. Note that this implies
$${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\beta',u,s\cdot a\beta)}},$$ which we will use throughout the proof. We consider three cases:
- *Assume that ${\ensuremath{u^{\leftarrowtail}_{s\cdot a\beta b\beta'}}}\neq {\ensuremath{u^{\leftarrowtail}_{s\cdot a\beta\beta'}}}$.* We choose $s'=s\cdot a\beta$, $a'=b$, and $\alpha'=\beta'$. This is a contradiction to the minimality of $\alpha$, since ${\left| \alpha' \right|}<{\left| \beta' \right|}$.
- *Assume that ${\ensuremath{u^{\leftarrowtail}_{s\cdot\beta b\beta'}}}\neq {\ensuremath{u^{\leftarrowtail}_{s\cdot\beta\beta'}}}$.* We choose $s'=s\cdot\beta$, $a'=b$, and $\alpha=\beta'$ and obtain a contradiction in the same way as in the above case. For this, it suffices to prove that ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\beta',u,s\cdot\beta)}}$. Hence assume indirectly that ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(b\beta',u,s\cdot\beta)}}$. Let $\gamma$ be a minimal prefix of $b\beta'$ such that there is an agent $v$ with
- ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(\gamma,v,s\cdot\beta)}}$,
- ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\gamma,v,s)}}$.
Since $\gamma=b\beta'$ and $v=u$ satisfies these conditions, such a minimal choice of $\gamma$ and $v$ exists. Now consider the position where $v$ “learns” $b$, i.e., let $\gamma=\pi c\pi'$ such that the action $c$ transmits the $b$-action to $v$, i.e., we have that
- ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(\pi,{{\tt dom}}(c),s\cdot\beta)}}$,
- ${{\tt dom}}(c){{\rightarrowtail}_{s\cdot\beta\pi}}v$.
Note that $\pi$ is a proper prefix of $\gamma$. Since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\gamma,v,s)}}$, it follows that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\pi,v,s)}}$. Hence we know by the minimality of $\alpha$ that $v^\uparrow_{s\cdot\beta\pi}=v^\uparrow_{s\cdot a\beta\pi}$, In particular, ${{\tt dom}}(c){{\rightarrowtail}_{s\cdot a\beta\pi}}v$. We now have the following:
- Due to the above, we know that ${{\tt dom}}(b)\in{\ensuremath{{\tt src}(\pi,{{\tt dom}}(c),s\cdot\beta)}}$,
- since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\gamma,v,s)}}$, we know that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\pi,{{\tt dom}}(c),s)}}$.
Since $\pi$ is a proper prefix of $\gamma$, this is a contradiction to the minimality of $\gamma$.
- *Assume that ${\ensuremath{u^{\leftarrowtail}_{s\cdot a\beta b\beta'}}}={\ensuremath{u^{\leftarrowtail}_{s\cdot a\beta\beta'}}}$ and ${\ensuremath{u^{\leftarrowtail}_{s\cdot\beta b\beta'}}}={\ensuremath{u^{\leftarrowtail}_{s\cdot\beta\beta'}}}$.* Since ${\ensuremath{u^{\leftarrowtail}_{s\cdot a\beta b\beta'}}}\neq {\ensuremath{u^{\leftarrowtail}_{s\cdot\beta b\beta'}}}$, it then follows that ${\ensuremath{u^{\leftarrowtail}_{s\cdot a\beta\beta'}}}\neq {\ensuremath{u^{\leftarrowtail}_{s\cdot\beta\beta'}}}$. It suffices to show that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta\beta',u,s)}}$, we then have a contradiction to the minimality of $\alpha$. Hence indirectly assume that ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\beta\beta',u,s)}}$. Let $\gamma$ be a minimal prefix of $\beta'$ such that there is some $v$ such that
- ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b\gamma,v,s)}}$,
- ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\beta\gamma,v,s)}}$.
Since $\gamma=\beta'$ and $v=u$ satisfy these conditions, such a minimal choice exists. Similarly as before, look at the action where $a$ is forwared to $v$, i.e., let $\gamma=\pi c\pi'$ such that
- ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\beta\pi,{{\tt dom}}(c),s)}}$,
- ${{\tt dom}}(c){{\rightarrowtail}_{s\cdot a\beta\pi}}v$.
Since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b\gamma,v,s)}}$ and ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$, it follows that ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\gamma,v,s\cdot a\beta)}}$. Since $\pi$ is a prefix of $\gamma$, this implies ${{\tt dom}}(b)\notin{\ensuremath{{\tt src}(b\pi,v,s\cdot a\beta)}}$. The minimality of $\alpha$ implies that $v^\uparrow_{s\cdot a\beta b\pi}=v^\uparrow_{s\cdot a\beta\pi}$, in particular, ${{\tt dom}}(c) {{\rightarrowtail}_{s\cdot a\beta b\pi}} v$. Since ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b\gamma,v,s)}}$, we obtain
- ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\beta b\pi,{{\tt dom}}(c),s)}}$,
- from the above, we know that ${{\tt dom}}(a)\in{\ensuremath{{\tt src}(a\beta\pi,{{\tt dom}}(c),s)}}$.
This contradicts the minimality of $\gamma$, since $\pi$ is a proper prefix of $\gamma$.
Using these lemmas, we can now prove Theorem \[theorem:polynomial unwinding characterization of intransitive uniformity and security\]:
1. First assume that there is a uniform intransitive unwinding satisfying [[(PC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, and [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, and indirectly assume that the policy is not intransitively uniform. Due to Lemma \[lemma:minimal path different policies\], there exist $a,u,s$, and $\alpha$ such that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$, ${\ensuremath{u^{\leftarrowtail}_{s\cdot a\alpha}}}\neq {\ensuremath{u^{\leftarrowtail}_{s\cdot\alpha}}}$, and $\alpha$ does not contain any $b$ with ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$. Let $v={{\tt dom}}(a)$. Let $\sim^{s,v}_u$ be an equivalence relation satisfying [[(PC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, and [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}. It suffices to show that $s\cdot a\alpha\sim^{s,v}_us\cdot\alpha$ to obtain a contradiction to [[(PC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}.
Clearly, ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}}u$, hence [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{} implies $s\sim^{s,{{\tt dom}}(a)}_us\cdot a$, i.e., $s^{s,v}_us\cdot a$. Note that for all $a'$ appearing in $\alpha$, we have that ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}}{{\tt dom}}(a')$. Hence applying [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{} for each $a'$, we obtain $s\cdot a\alpha\sim^{s,v}_us\cdot\alpha$ as required.
For the converse, assume that for all ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$, we have that ${\ensuremath{u^{\leftarrowtail}_{s\cdot a\alpha}}}={\ensuremath{u^{\leftarrowtail}_{s\cdot\alpha}}}$, and let $s_0$ be a state, and let $v$ and $u$ be agents. We define
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$s\sim^{s_0,v}_ut$ iff for all sequences $\alpha$ that contain no $b$ with $v{{\rightarrowtail}_{s_0}}{{\tt dom}}(b)$, we have ${\ensuremath{u^{\leftarrowtail}_{s\cdot\alpha}}}={\ensuremath{u^{\leftarrowtail}_{t\cdot\alpha}}}$.
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Clearly, $\sim^{s_0,v}_u$ is an equivalence relation and satisfies [[(PC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{} (choose $\alpha=\epsilon$). For showing [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, let $s\sim^{s_0,v}_ut$, and let $v\not{{\rightarrowtail}_{s_0}}{{\tt dom}}(a)$. To show the required condition $s\cdot a\sim^{s_0,v}_ut\cdot a$, let $\alpha$ be a sequence containing no $b$ with $v{{\rightarrowtail}_{s_0}}b$. Since $v\not{{\rightarrowtail}_{s_0}}{{\tt dom}}(a)$, the sequence $a\alpha$ satisfies the same condition, and hence from $s\sim^{s_0,v}_ut$, it follows that ${\ensuremath{u^{\leftarrowtail}_{s\cdot a\alpha}}}=v^\uparrow_{s\cdot a\alpha}$ as required.
Finally, consider [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}. Let ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}}u$. To show that $s\sim^{s,{{\tt dom}}(a)}_us\cdot a$, let $\alpha$ be such that no $b$ with ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$ appears in $\alpha$, we need to show that ${\ensuremath{u^{\leftarrowtail}_{s\cdot\alpha}}}={\ensuremath{u^{\leftarrowtail}_{s\cdot a\alpha}}}$. This follows from the prerequites, since clearly, ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$.
2. 1. Assume that the system is [i-secure]{}. Let $s_0$ be a state, and let $v$ and $u$ be agents. We define:
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$s\sim^{s_0,v}_ut$ iff for all sequences $\alpha$ that contain no $b$ with $v{{\rightarrowtail}_{s_0}}{{\tt dom}}(b)$, we have ${{\tt obs}}_u(s\cdot\alpha)={{\tt obs}}_u(t\cdot\alpha)$.
-------------------- ----- --------------------------------------------------------------------------------------------------------------------------------------------------------------------
Clearly, $\sim^{s_0,v}_u$ is an equivalence relation and satisfies [[(OC$_{{\textnormal{i}}}^{{\textnormal{u}}}$)]{.nodecor}]{} (choose $\alpha=\epsilon$). For showing [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, let $s\sim^{s_0,v}_ut$, and let $a\in A$ with $v\not{{\rightarrowtail}_{s_0}}{{\tt dom}}(a)$. We need to show that for all $\alpha$ containing no $b$ with $v{{\rightarrowtail}_{s_0}}{{\tt dom}}(b)$, we have ${{\tt obs}}_u(s\cdot a\alpha)={{\tt obs}}_u(t\cdot a\alpha)$. This trivially follows from $s\sim^{s_0,v}_ut$, since $\alpha'=a\alpha$ also does not contain a $b$ with $v{{\rightarrowtail}_{s_0}}{{\tt dom}}(b)$.
Finally, consider [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}. Let ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}}u$. We need to show that $s\sim^{s,{{\tt dom}}(a)}_us\cdot a$. Hence let $\alpha$ be a sequence containing no $b$ with ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$. We need to show that ${{\tt obs}}_u(s\cdot\alpha)={{\tt obs}}_u(s\cdot a\alpha)$. Since the system is [i-secure]{}, it suffices to show that ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$. This follows trivially since ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}}u$, and $\alpha$ does not contain any $b$ with ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$.
2. Assume that the system is not [i-secure]{}. Due to Lemma \[lemma:minimal path different observations\], there is a state $s$, an agent $u$, an action $a$ and a sequence $\alpha$ with ${{\tt dom}}(a)\notin{\ensuremath{{\tt src}(a\alpha,u,s)}}$, ${{\tt obs}}_u(s\cdot a\alpha)\neq{{\tt obs}}_u(s\cdot\alpha)$, and $\alpha$ does not contain any $b$ with ${{\tt dom}}(a){{\rightarrowtail}_{s}}{{\tt dom}}(b)$. Let $v={{\tt dom}}(a)$, and let $\sim^{s,v}_u$ be an equivalence relation on $S$ that satisfies [[(OC$_{{\textnormal{i}}}^{{\textnormal{u}}}$)]{.nodecor}]{}, [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}, and [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}. It suffices to show that $s\alpha\sim^{s,v}_us\cdot a\alpha$. Clearly we have that $v\not{{\rightarrowtail}_{s}}u$. Therefore, (recall that $v={{\tt dom}}(a)$), [[(LR$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}implies $s\sim^{s,v}_us\cdot a$. Note that for all $b\in\alpha$, we have that ${{\tt dom}}(a)\not{{\rightarrowtail}_{s}}{{\tt dom}}(b)$. Hence applying [[(SC$_{\textnormal{i}}^{\textnormal{u}}$)]{.nodecor}]{}repeatedly, we obtain $s\cdot a\alpha\sim^{s,v}_us\cdot\alpha$, which completes the proof.
|
---
abstract: 'We study implications of exact conformal invariance of scalar quantum field theories at the critical point in non-integer dimensions for the evolution kernels of the light-ray operators in physical (integer) dimensions. We demonstrate that all constraints due the conformal symmetry are encoded in the form of the generators of the collinear $sl(2)$ subgroup. Two of them, $S_-$ and $S_0$, can be fixed at all loops in terms of the evolution kernel, while the generator of special conformal transformations, $S_+$, receives nontrivial corrections which can be calculated order by order in perturbation theory. Provided that the generator $S_+$ is known at the $\ell-1$ loop order, one can fix the evolution kernel in physical dimension to the $\ell$-loop accuracy up to terms that are invariant with respect to the tree-level generators. The invariant parts can easily be restored from the anomalous dimensions. The method is illustrated on two examples: The $O(n)$-symmetric $\varphi^4$ theory in $d=4$ to the three-loop accuracy, and the $su(n)$ matrix $\varphi^3$ theory in $d=6$ to the two-loop accuracy. We expect that the same technique can be used in gauge theories e.g. in QCD.'
author:
- 'V. M. Braun'
- 'A. N. Manashov'
date: 'Received: date / Accepted: date'
title: Evolution equations beyond one loop from conformal symmetry
---
Introduction {#intro}
============
It is well known that conformal symmetry of the QCD Lagrangian imposes strong constraints on the leading-order (LO) correlation functions and operator renormalization, see Ref. [@Braun:2003rp] for a review. A schematic structure of the perturbation theory for a generic quantity $\mathcal{Q}$ beyond the LO is usually conjectured to be $$\begin{aligned}
\mathcal{Q} = \mathcal{Q}^{\rm con} + \frac{\beta(g)}{g}\Delta \mathcal{Q}
\label{conformal}\end{aligned}$$ where $\mathcal{Q}$ is the result in the formal conformal limit, obtained by setting the $\beta$-function to zero by hand. It is expected to have full symmetries of a conformally invariant theory. The extra term involving the $\beta$-function can be calculated separately and e.g. the leading contribution to $\Delta \mathcal{Q}$ can be evaluated very easily via quark bubble insertions.
A prominent example is provided by the generalized Crewther relation [@Crewther:1972kn; @Broadhurst:1993ru; @Crewther:1997ux] between the Bjorken sum rule in deep inelastic scattering and the total cross section of $e^+e^-$ annihilation: The expected structure has been confirmed by direct calculations to the $\mathcal{O}(\alpha_s^4)$ accuracy [@Baikov:2010je; @Baikov:2012zn].
Another important application concerns the evolution equation for meson distribution amplitudes and generalized parton distributions. As shown by D. Müller, off-diagonal terms of the anomalous dimension matrix of leading twist operators to the $\ell$-loop accuracy are determined by the special conformal transformation anomaly at one order less [@Mueller:1991gd]. This approach was later used to calculate the complete two-loop mixing matrix for twist-two operators in QCD [@Mueller:1993hg; @Mueller:1997ak; @Belitsky:1997rh], and derive the two-loop evolution kernels in momentum space for the generalized parton distributions [@Belitsky:1998vj; @Belitsky:1999hf; @Belitsky:1998gc].
In this paper we present an alternative technique to study implications of conformal invariance and illustrate it on two examples: Calculation of the anomalous dimension matrix in the $O(n)$-symmetric $\varphi^4$ theory in $d=4$ to the three-loop accuracy, and the $su(n)$ matrix $\varphi^3$ theory in $d=6$ to the two-loop accuracy. The motivation for our study, apart from various phenomenological applications, is the following.
First of all, we think that the calculations can be considerably simplified by going over to evolution equations for non-local light-ray operators in position space. In particular the intricate procedure for the restoration of the evolution kernels from local operators can be avoided.
Second, we want to make the separation of perturbation theory in “conformal part” and “corrections proportional to the $\beta$-function” as in Eq. (\[conformal\]) to be more transparent. Our starting point is the observation that QCD and toy-model scalar theories that we consider for illustration possess a nontrivial fixed point in non-integer dimension, $d=4-2\epsilon$ ($d=6-2\epsilon$ for $\varphi^3$) [@Banks:1981nn; @Hasenfratz:1992jv]. Conformal symmetry is an *exact* symmetry of the interacting theory for the fine-tuned (critical) value of coupling. As a consequence, the renormalization group equations are exactly conformally invariant: the evolution kernels commute with the generators of the conformal group. The generators are, however, modified by quantum corrections as compared to their canonical expressions, $S_\alpha=S_\alpha^{(0)}+\Delta S_\alpha$, and the corrections can be calculated order by order in the perturbative expansion. From the pure technical point of view, this calculation replaces evaluation of the conformal anomaly in the theory with broken symmetry in integer dimensions via the Conformal Word identities (CWI) in the approach of D. M[ü]{}ller. We show that the non-invariant part of the evolution equations with respect to canonical transformations in $\ell-$th order of perturbation theory is uniquely fixed by the generators in the order $\ell-1$, and the invariant part is determined (and can be easily restored) from the spectrum of anomalous dimensions.
Last but not least, in $\text{MS}$-like schemes the evolution kernels (anomalous dimensions) do not depend on the space-time dimension by construction. Thus all expressions derived in the $d$-dimensional (conformal) theory remain exactly the same for the theory in integer dimensions; considering the theory at the critical point one does not lose any information.
As already mentioned, the present paper is exploratory. We work out the necessary formalism for the simplest, scalar field theories. We expect, however, that the same technique can be used in gauge theories and in particular in QCD. The corresponding generalization and applications will be considered elsewhere.
General formalism {#sec:1}
=================
This section is introductory and contains mostly some general remarks.
Scalar field theories
---------------------
We will consider the conventional $O(n)$-symmetric $\varphi^4$ theory in $d=4-2\epsilon$ dimensions $$\begin{aligned}
\label{Sphi4}
S^{(4)}(\varphi)=\int d^dx\left[\frac12(\partial\varphi)^2+\frac{gM^{2\epsilon}}{24}(\varphi^2)^2\right]\,,\end{aligned}$$ where $\varphi^2=\sum_{a=1}^{n}(\varphi^{a})^2$, and the (somewhat exotic) $su(n)$-matrix $\varphi^{3}$ theory $$\begin{aligned}
\label{Sphi3}
S^{(3)}(\varphi)=&\int d^dx\left[\operatorname{tr}(\partial\varphi)^2+\frac{2}{3}g M^\epsilon \operatorname{tr}\varphi^3\right]\,\end{aligned}$$ in $d=6-2\epsilon$ dimensions. Here $\varphi=\sum_{a}\varphi^a t^a$ and the matrices $t^a$ are the $su(n)$ generators in the fundamental representation normalized as $\operatorname{tr}t^{a}t^{b}=1/2$. One can rewrite (\[Sphi3\]) as follows $$\begin{aligned}
S^{(3)}(\varphi)
=&\int d^dx\left[\frac12(\partial\varphi^a)^2+\frac{gM^\epsilon}6 d^{abc}\varphi^a\varphi^b\varphi^c\right]\,,\end{aligned}$$ where $d^{abc}=2\operatorname{tr}t^a\{t^b t^c\}$. Both theories are multiplicatively renormalizable $$\begin{aligned}
\label{}
S_R(\varphi)=\int d^d x\left[\frac12Z_1(\partial\varphi)^2+Z_3M^{k\epsilon} g \,V(\varphi)\right]\,,\end{aligned}$$ where $k=1,2$ for $\varphi^3$ and $\varphi^4$ interaction, respectively.
The renormalization constants $Z_1$ and $Z_3$ for the $\varphi^4$ theory can be found in literature, see e.g. [@AN]: $$\begin{aligned}
Z^{(4)}_1=&1-\frac{u^2}{24\epsilon}\frac{n+2}{3}-\frac{u^3(2-\epsilon)}{48\epsilon^2}\frac{n+2}{3}\frac{n+8}{9}
+\mathcal{O}(u^4),
\notag\\
Z^{(4)}_3
=&1+\frac{u}{\epsilon}\frac{n+8}{6}+\left(\frac{u}{\epsilon}\frac{n+8}{6}\right)^2\!\!-\!
\frac{u^2}{\epsilon}\frac{5n+22}{18}\!+\!\mathcal{O}(u^3).\end{aligned}$$ For the $\varphi^3$ theory we find $$\begin{aligned}
Z^{(3)}_1=&1-\frac{n^2\!-\!4}{2 n}\biggl[\frac{u}{6\epsilon}-\frac{u^2}{36}
\biggl(\frac1{\epsilon^2}\frac{n^2\!-\!16}{ n}
-\frac1\epsilon\frac{n^2\!-\!100}{12 n}\biggr)\biggr]
\notag\\ &
+\mathcal{O}(u^3)\,,
\notag\\
Z^{(3)}_3=&1-\frac{u}{4\epsilon}\frac{n^2-12}{n}+\frac{u^2}{16}
\biggl(\frac1{\epsilon^2}\frac{n^2-4}{n}\frac{n^2-12}{n}
\notag\\
&-\frac1\epsilon
\frac{n^4-100n^2+960}{6n^2}\biggr)+\mathcal{O}(u^3)\,,\end{aligned}$$ where $$\begin{aligned}
u =\frac{g}{(4\pi)^2} \ [\varphi^4\,\text{theory}],
\qquad
u = \frac{g^2}{(4\pi)^3} \ [\varphi^3\,\text{theory}].\end{aligned}$$ The beta-function and the anomalous dimension of the basic field are defined as follows $$\begin{aligned}
\beta(u)=\frac{du}{d \ln M}\,,\qquad
\gamma_\varphi=\frac12 \frac{d\ln Z_1}{d\ln M }\,.\end{aligned}$$ One obtains $$\begin{aligned}
\label{rgf-4}
\beta(u)=&-2\epsilon u+\frac{u^2(n+8)}3-\frac{u^3(3n+14)}{3}+\mathcal{O}(u^4)\,,
\notag\\
\gamma_\varphi=&\frac{u^2(n+2)}{36}\left[1-\frac{u(n+8)}{12}+\mathcal{O}(u^2)\right]\,\end{aligned}$$ and $$\begin{aligned}
\beta(u)=&-2\epsilon u-u^2\frac{n^2-20}{2n}
\notag\\&-u^3\frac{5n^4-496n^2+5360}{72 n^2}+\mathcal{O}(u^4)\,,
\notag\\
\gamma_\varphi=&u\frac{n^2-4}{12 n}\left(1+u\frac{n^2-100}{36 n}\right)+\mathcal{O}(u^3)\,
\label{rgf-3}\end{aligned}$$ for the $\varphi^4$ and $\varphi^3$ theories, respectively.
The critical coupling is defined by the condition $\beta(u_*) =0$. Solving this equation for $u_*$, one obtains the well-known expansion for the critical coupling $u_*$ in powers of $4-d$ in the $\varphi^4$ theory $$\begin{aligned}
\label{ust4}
u^{(\varphi^4)}_*=&\frac{6\epsilon}{n+8}+\left(\frac{6\epsilon}{n+8}\right)^2\frac{3n+14}{n+8}+\mathcal{O}(\epsilon^3)\,,\end{aligned}$$ whereas for the $\varphi^3$ theory we obtain $$\begin{aligned}
\label{ust3}
u^{(\varphi^3)}_*\!=\frac{4n\epsilon}{20-n^2}+\frac{4n\epsilon^2}{9}\frac{(5n^4-496n^2+5360)}{(20-n^2)^3}+O(\epsilon^3).\end{aligned}$$ In the latter case a nontrivial critical point for $d<6$ only exists for $n=3$ and $n=4$. (For $n=2$ the theory is free as the $d_{abc}$-symbols vanish identically). Staying within perturbation theory on can, however, consider $n$ as a continuous parameter. In this sense all further results hold for arbitrary $n$.
Renormalization ensures finiteness of the correlation functions of the basic field that are encoded in the partition function $$\begin{aligned}
Z(A)=\mathcal{N}^{-1}\int D\varphi \, e^{-S_R(\varphi)+\int d^d x A(x)\varphi(x)}\,.
\label{partition1}\end{aligned}$$ Here $A$ is an external source and, as usual, the normalization is chosen in such a way that $Z(0)=1$. Correlation functions with an insertion of a composite operator, $\mathcal{O}_k$, possess additional divergences that are removed by the operator renormalization, $$\begin{aligned}
{}[\mathcal{O}_k]=\sum_j Z_{kj} \mathcal{O}_j,\end{aligned}$$ where the sum goes over all operators with the same quantum numbers that get mixed. Here and below we use square brackets to denote renormalized composite operators (in a minimal subtraction scheme).
Light-ray operators {#sec:nonlocal}
-------------------
Light-ray operators (see e.g. [@Balitsky:1987bk]) will always be understood here as generating functions for the leading-twist local operators: $$\begin{aligned}
\mathcal{O}^{ab}(x;z_1,z_2) &\equiv& \varphi^a(x+z_1n)\varphi^b(x+z_2n)
\nonumber\\&\equiv& \sum_{mk} z_1^m z_2^k \mathcal{O}_{mk}^{ab}(x)\,,
\label{LRO}\end{aligned}$$ where $$\begin{aligned}
\mathcal{O}_{mk}^{ab}(x)=\frac{1}{m!k!}(n\partial)^m\varphi^a(x)(n\partial)^k\varphi^b(x)\,.\end{aligned}$$ Here $n^\mu$ is an auxiliary light-like vector, $n^2=0$, that ensures symmetrization and subtraction of traces of local operators.
A renormalized light-ray operator is the generating function for renormalized local operators. It can be written in the form $$\begin{gathered}
{}[\mathcal{O}^{ab}(x;z_1,z_2)] =\sum_{mk} z_1^m z_2^k [\mathcal{O}_{mk}^{ab}(x)]=
\\
=Z^{ab}_{a'b'}\,\mathcal{O}^{a'b'}(x;z_1,z_2)\,,\end{gathered}$$ where $Z$ (the renormalization constant) is an integral operator acting on the coordinates $z_1,z_2$ that has an expansion in inverse powers of $\epsilon$ $$\begin{aligned}
Z=1+\sum_{k=1}^\infty \epsilon^{-k} Z_k(u)\,.
\label{Zk}\end{aligned}$$ It is a matrix in isotopic space. The renormalized light-ray operator $[\mathcal{O}(x;z_1,z_2)]$ satisfies the renormalization-group (RG) equation $$\begin{aligned}
\label{RGO}
\Big(M{\partial_M}+\beta(u)\partial_u +\mathbb{H}\Big)[\mathcal{O}(x;z_1,z_2)]=0\,,\end{aligned}$$ where we suppressed the isotopic indices. The coupling $u$ for the theories in question is defined in Eq.(\[rgf-3\]) and the evolution kernel (Hamiltonian) $\mathbb{H}$ is given by $$\begin{aligned}
\mathbb{H}=-\left({M}\frac{d}{dM} \mathbb{Z}\right) \mathbb{Z}^{-1}=2u\partial_u Z_1(u)+2\gamma_\varphi\,,
\label{H}\end{aligned}$$ where $\mathbb{Z}=Z Z_1^{-1}$. In perturbation theory $\mathbb{H}$ can be written as a series $$\begin{aligned}
\label{Hexp}
\mathbb{H}=u\mathbb{H}^{(1)}+u^2\mathbb{H}^{(2)}+\ldots\,.\end{aligned}$$ The kernels $\mathbb{H}^{(k)}$ in minimal subtraction schemes do not depend $\epsilon$. As a consequence these kernels are exactly the same for the theories in $d$ dimensions that we consider at the intermediate step and physical theories in integer dimensions that are our final goal. We stress that Eq. (\[RGO\]) is completely equivalent to the RG equation for the local twist-two operators, $$\begin{aligned}
\Big([M{\partial_M}+\beta(u)\partial_u]\delta_{mk}^{m'k'}\delta^{ab}_{a'b'} +
(\gamma_{mk}^{m'k'})^{ab}_{a'b'}\Big)[\mathcal{O}_{m'k'}^{a'b'}]=0\,,\end{aligned}$$ where $\gamma$ is the usual anomalous dimension matrix.
Conformal symmetry
------------------
The usual Poincare symmetry of the theory is enhanced at the critical point $u=u_*$, $\beta(u_*)=0$ by the dilatation (scale invariance) and space-time inversion. For our purposes it is sufficient to consider the transformations that act nontrivially on the twist-two (symmetric and traceless) operators. These transformations form the so-called collinear $sl(2)$ subgroup of the full conformal group that leaves the light-ray $x^\mu=z n^\mu $ invariant, see Ref. [@Braun:2003rp] for a review.
Collinear conformal transformations are generated by translations along the light-ray direction $n^\mu$, special conformal transformations in the alternative light-like direction $\bar n$, $\bar n^2=0$, $(n\bar n)=1$, and the combination of the dilation and rotation in the $(n,\bar n)$ plane $$\begin{aligned}
\mathbf{L}_-=-i\mathbf{P}_n,\! &&\!\mathbf{L}_+=\frac12i\mathbf{K}_{\bar n},\! &&\!
\mathbf{L_0}=\frac{i}2\left(\mathbf{D}-\mathbf{M}_{n\bar n}\right).
\label{gene}\end{aligned}$$ Here and below we use a shorthand notation $\mathbf{P}_n = n^\mu \mathbf{P}_\mu$ etc. The generators defined in this way satisfy standard $sl(2)$ commutation relations $$\begin{aligned}
\label{}
{}[\mathbf{L}_{\pm},\mathbf{L}_0]=\pm \mathbf{L}_{\pm}\,, &&
{}[\mathbf{L}_{+},\mathbf{L}_-]=-2\mathbf{L}_0\,.\end{aligned}$$ Local composite operators can be classified according to irreducible representations of the $sl(2)$ algebra. A traceless and symmetric (renormalized) operator $$[\mathcal{O}_N](x)=n_{\mu_1}\ldots n_{\mu_N} [\mathcal{O}^{\mu_1\ldots\mu_N}_N](x),$$ (for a while we suppress isotopic indices) is called conformal if it transforms covariantly under the special conformal transformation [@Derkachov:1992he]: $$\begin{gathered}
i\big[\mathbf{K}^\mu,[\mathcal{O}_N](x)\big]=
\biggl[2 x^\mu (x
\partial)-x^2\partial^\mu+2\Delta^\ast_N x^\mu
\\
+2x^\nu \left(n^\mu \frac{\partial}{\partial n^\nu}- n_\nu \frac{\partial}{\partial n_\mu}
\right)\biggr][\mathcal{O}_N](x)\,.\end{gathered}$$ Here $\Delta^\ast_N$ is the scaling dimension of the operator (at the critical point): $$\begin{aligned}
i\big[\mathbf{D},[\mathcal{O}_N](x)\big]=\big(x\partial_x+\Delta^\ast_N\big)[\mathcal{O}_N](x)\,.\end{aligned}$$ As a consequence of having definite scaling dimension, the conformal operator $\mathcal{O}_{N}$ satisfies the RG equation $$\begin{aligned}
\label{OM}
\Big(M{\partial_M}+\gamma^*_N\Big)[\mathcal{O}_{N}]=0\,,\end{aligned}$$ where $\gamma_N^*$ is the anomalous dimension at the critical point, $\gamma_N^*=\gamma_N(u_*)$. The scaling dimension is given by the sum of the canonical and anomalous dimensions, $\Delta^\ast_N=\Delta_N+\gamma_N^*$. For the operators under consideration $\Delta_N=2\Delta+N$ where $\Delta=d/2-1$ is the canonical dimension of the basic field $\varphi(x)$.
Each conformal operator $[\mathcal{O}_N]$ generates an irreducible representation of the $sl(2)$ algebra (conformal tower), consisting of operators obtained by adding total derivatives: $$\begin{aligned}
\mathcal{O}_{Nk}=(n\partial)^k [\mathcal{O}_N(0)],\,\qquad k=0,1,\ldots\end{aligned}$$ such that $$\begin{aligned}
\label{3L}
\delta_-\mathcal{O}_{Nk}=~&\big[\mathbf{L}_-,\mathcal{O}_{Nk}\big]=~-\mathcal{O}_{Nk+1}\,,
\notag\\
\delta_{0\phantom{i}}\mathcal{O}_{Nk}=~&\big[\mathbf{L}_0,\,\mathcal{O}_{Nk}\big]=~(j_N+k)\mathcal{O}_{Nk}\,,
\notag\\
\delta_+\mathcal{O}_{Nk}=~&\big[\mathbf{L}_+,\mathcal{O}_{Nk}\big]=~k(2j_N+k-1)\mathcal{O}_{Nk-1}\,,\end{aligned}$$ with the operator $[\mathcal{O}_N]$ itself being the highest weight vector, $\big[\mathbf{L}_+,[\mathcal{O}_{N}]\big]=0$. Here $j_N$ is the so-called conformal spin of the operator — the half-sum of its scaling dimension and spin $$\begin{aligned}
j_N=\frac12(\Delta^\ast_N+N) = \Delta + N + \frac12 \gamma_N^*.\end{aligned}$$ All operators $\mathcal{O}_{Nk}$ in a conformal tower have, obviously, the same anomalous dimension $\gamma_N^\ast$.
Going over from the description in terms of conformal towers of local operators to the light-ray operators essentially corresponds to going over to a different realization of conformal symmetry. Due to Poincare invariance one can put, without loss of generality, $x=0$ in a definition on light-ray operator (\[LRO\]). Hereafter we consider $$[\mathcal{O}(z_1,z_2)]\equiv [\mathcal{O}(x=0;z_1,z_2)].$$ The light-ray operator $[\mathcal{O}(z_1,z_2)]$ can be expanded in terms of local operators $\mathcal{O}_{Nk}$ $$\begin{aligned}
\label{nlo2}
[\mathcal{O}(z_1,z_2)]=\sum_{Nk}\Psi_{Nk}(z_1,z_2)\,[\mathcal{O}_{Nk}]\,,\end{aligned}$$ where $\Psi_{Nk}(z_1,z_2)$ are homogeneous polynomials of degree $N+k$ $$\begin{aligned}
\left(z_1\partial_{z_1} + z_2\partial_{z_2} -N-k\right)\Psi_{Nk}(z_1,z_2) =0\,.\end{aligned}$$ The action of the generators $\mathbf{L}_{\pm,0}$ on the quantum fields in the light-ray operator can be traded for the differential operators $S_{\pm,0}$ acting on the field coordinates, i.e. on the coefficient functions $\Psi_{Nk}(z_1,z_2)$: $$\begin{aligned}
\delta_{\pm,0} [\mathcal{O}(z_1,z_2)] = S_{\pm,0} [\mathcal{O}(z_1,z_2)]\,.\end{aligned}$$ The generators $S_{\pm,0}$ obey the usual $sl(2)$ commutation relations $$\begin{aligned}
\label{sl2-comm}
{}[S_0,S_{\pm}]=\pm S_{\pm}\,, &&
{}[S_{+},S_-]= 2S_0\,\end{aligned}$$ and their action on the coefficient functions in the expansion (\[nlo2\]) takes the form $$\begin{aligned}
S_- \Psi_{Nk}(z_1,z_2) =& - \Psi_{Nk-1}(z_1,z_2)\,,
\notag\\
S_0\, \Psi_{Nk}(z_1,z_2) =& (j_N+k) \Psi_{Nk}(z_1,z_2)\,,
\notag\\
S_+ \Psi_{Nk}(z_1,z_2) =& (k+1)(2j_N+k)\Psi_{Nk+1}(z_1,z_2)\,.
\label{steps}\end{aligned}$$ With the exception of $S_-$, the form of the generators in the interacting theory (at the critical point) differs, however, from the canonical expressions (see e.g. [@Braun:2003rp]) $$\begin{aligned}
\label{S_0}
S_-^{(0)}=&-\partial_{z_1}-\partial_{z_2}\,,
\notag\\
S_0^{(0)}=&~z_1\partial_{z_1}+z_2\partial_{z_2}+2j\,,
\notag\\
S_+^{(0)}=&~z_1^2\partial_{z_1}+z_2^2\partial_{z_2}+2j(z_1+z_2)\,.\end{aligned}$$ Here $j$ is the conformal spin of the field $\varphi$, $j=1/2$ ($\varphi^4$-theory) and $j=1$ ($\varphi^3$-theory). We obtain (see below) $$\begin{aligned}
S_- =& S_-^{(0)}\,,
\notag\\
S_0\, =& S_0^{(0)} + \Delta S_0^{(0)}\,,
\notag\\
S_+ =& S_+^{(0)} + \Delta S_+^{(0)}\,,
\label{DeltaS}\end{aligned}$$ where $$\begin{aligned}
\label{DeltaS0+}
\Delta S_0=& -\epsilon+\frac12 \mathbb{H}(u_*)\,.
\notag\\
\Delta S_+=&(z_1+z_2)\Big(-\epsilon+\frac12 u_*\, \mathbb{H}^{(1)}\Big)+\mathcal{O}(\epsilon^2)\,.\end{aligned}$$ Note that $\Delta S_0$ is given in closed form in terms of the evolution kernel $\mathbb{H}$, whereas $\Delta S_+$ can only be calculated as a series expansion in $\epsilon$ and/or $u_\ast =\mathcal{O}(\epsilon)$. To the $\mathcal{O}(\epsilon)$ accuracy the result (given above) turns out to be the same in both theories that we consider in this work. For the case of the $\varphi^4$ theory we have also calculated the next, $\mathcal{O}(\epsilon^2)$, correction.
If $\Delta S_+$ is known (to a given order), one can use the last equation in (\[steps\]) to construct the whole set of the coefficient functions $\Psi_{Nk}$ to the same accuracy starting from the lowest one, $\Psi_{Nk=0}=(z_1-z_2)^N$. This in turn is sufficient in order to be able to obtain explicit expressions for the multiplicatively renormalizable operators $\mathcal{O}_{Nk}$ in terms of the operators $\mathcal{O}_{nm}$, cf. Eq. (\[LRO\]), by comparing the coefficients of $z_1^n z_2^m$. Thus the operator $S_+$ effectively encodes all information on the form of the eigenoperators of the evolution equation at the critical point.
Conformal symmetry implies that the full evolution kernel for light-ray operators for the critical coupling $$\begin{aligned}
\label{Hcrit}
\mathbb{H}(u_\ast) =u_\ast\,\mathbb{H}^{(1)}+u_\ast^2\,\mathbb{H}^{(2)}+\ldots\,\end{aligned}$$ commutes with the exact $sl(2)$ generators $$\begin{aligned}
{}[\mathbb{H}(u_\ast),S_{\pm,0}] = 0\,.
\label{com22}\end{aligned}$$ Hence the evolution kernels at each order in perturbation theory, $\mathbb{H}^{(k)}$, can be split in the $sl(2)$-invariant and non-invariant parts with respect to the canonical transformations (\[S\_0\]) $$\begin{aligned}
\mathbb{H}^{(k)} = \mathbb{H}^{(k)}_{\rm inv}+\Delta \mathbb{H}^{(k)}\,,
\label{inv-noninv}\end{aligned}$$ such that $$\begin{aligned}
{}[\mathbb{H}^{(k)}_{\rm inv},S^{(0)}_{\pm,0}] = 0\,.\end{aligned}$$ We will show that the non-invariant part of the $k$-loop kernels $\Delta \mathbb{H}^{(k)}$ is uniquely determined by the $(k-1)$-loop result for $S_+$, after which the invariant part $\mathbb{H}^{(k)}_{\rm inv}$ can easily be restored from the anomalous dimensions.
Deformed $sl(2)$ generators
===========================
Thanks to Poincare invariance the generator $S_-$ does not receive any corrections in the interacting theory, i.e. $S_-=S_-^{(0)}$.
Indeed, since $\mathcal{O}(x;z_1,z_2)$ actually depends on the two field coordinates $x+z_1n$ and $x+z_2n$, a translation along the light ray $x^\mu \to x^\mu + a n^\mu$ can be compensated by redefinition of the $z$-coordinates $z_{1,2} \to z_{1,2} -a$. This means that action of the quantum operator $\mathbf{L}_-=-i\mathbf{P}_n$ on the quantum fields in $\mathcal{O}(z_1,z_2)$ can be traded for the shift in the field coordinates $$\begin{aligned}
\delta_- [\mathcal{O}(z_1,z_2)] &\equiv& \big[\mathbf{L}_-, [\mathcal{O}(z_1,z_2)]\big]
\nonumber\\ &=& -(\partial_{z_1}+\partial_{z_2})[\mathcal{O}(z_1,z_2)]\,.\end{aligned}$$ Since, on the other hand $$\begin{aligned}
\delta_- [\mathcal{O}(z_1,z_2)] &=& - \sum_{Nk}\Psi_{Nk}(z_1,z_2)\,[\mathcal{O}_{N,k+1}]\,,
\label{dminus2}\end{aligned}$$ we conclude that $S_- = - (\partial_{z_1}+\partial_{z_2})$ acts as a step-down operator in the space of coefficient functions, cf. the first Eq. (\[steps\]). Note that the expansion on the r.h.s. in (\[dminus2\]) starts with $\mathcal{O}_{Nk=1}$ which means that the coefficient function of the conformal operator $\mathcal{O}_{N} \equiv \mathcal{O}_{Nk=0}$ is annihilated by $S_-$. Hence $$\begin{aligned}
\Psi_{N}(z_1,z_2) \equiv \Psi_{N0}(z_1,z_2) = c_N (z_1-z_2)^N \equiv c_N z_{12}^N \,,\end{aligned}$$ where the coefficients $c_N$ depend on the normalization convention for the conformal operators.
Next, let us consider $S_0$. Using Eq. (\[3L\]) we obtain $$\begin{aligned}
\delta_0 [\mathcal{O}(z_1,z_2)] &\equiv& \big[\mathbf{L}_0, [\mathcal{O}(z_1,z_2)]\big]
\nonumber\\ &=&
\sum_{Nk}\Psi_{Nk}(z_1,z_2)\,(j_N+k)[\mathcal{O}_{N,k}]\,,\end{aligned}$$ where one can rewrite $$\begin{aligned}
j_N+k=N+k+\Delta+\frac12\gamma_N^*\,.\end{aligned}$$ It follows from Eqs. (\[RGO\]) and (\[OM\]) that the functions $\Psi_{Nk}(z_1,z_2)$ are the eigenfunctions of the evolution kernel $\mathbb{H}$ for the critical value of coupling $$\begin{aligned}
\label{EigenPsi}
[\mathbb{H}(u_*)\Psi_{Nk}](z_1,z_2)=\gamma_N^*\,\Psi_{Nk}(z_1,z_2)\,.\end{aligned}$$ Thus one obtains $$\begin{aligned}
\delta_0 [\mathcal{O}(z_1,z_2)] & = & S_0 [\mathcal{O}(z_1,z_2)]\end{aligned}$$ with $$\begin{aligned}
\label{S0}
S_0 &=& z_1\partial_{z_1}+z_2\partial_{z_2} +\Delta+\frac12\mathbb{H}(u_*) \,,\end{aligned}$$ which is the result quoted in Eq. (\[DeltaS0+\]).
Unfortunately, the deformation of the generator $S_+$ in interacting theory cannot be found using similar general arguments and requires an explicit calculation. It can be done using the special conformal Ward identity. To this end we consider the partition function with the insertion of the renormalized light-ray operator $[\mathcal{O}(z_1,z_2)]$, cf. Eq. (\[partition1\]): $$\begin{aligned}
\label{Gc}
Z_{\mathcal{O}}(z_1,z_2;A)&=&{\left\langle\left\langle{[\mathcal{O}(z_1,z_2)]}\right\rangle\right\rangle}=
\nonumber\\&=&
\mathcal{N}^{-1}\!\int D\varphi \,[\mathcal{O}(z_1,z_2)]e^{-S_R(\varphi)+A\varphi}\,.\end{aligned}$$ Let us make a change of variables in the functional integral (\[Gc\]): $$\begin{aligned}
\label{ftf}
\varphi(x)\to
\varphi(x)+\delta_c\varphi(x)=\varphi(x)+\omega K_{\bar n}(\Delta)\varphi(x)\,.\end{aligned}$$ Here $\omega$ is a small parameter and the operator $K_{\bar n} = \bar n^\mu K_\mu$ is the generator of special conformal transformations $$\begin{aligned}
K_\mu(\Delta)=2x_\mu(x\partial)-x^2\partial_\mu+2\Delta x_\mu\,.
\label{Kmu}\end{aligned}$$ Note that we have chosen the parameter $\Delta$ entering the definition of $K_\mu$ equal to the canonical dimension of the field $\varphi$, $\Delta=d/2-1$. With this choice the kinetic term in the action is invariant under special conformal transformations.
Further, note that $\delta_c[\mathcal{O}(z_1,z_2)]=Z\delta_c\mathcal{O}(z_1,z_2)$. Variation of the bare light-ray operator can easily be calculated using the definition in Eq. (\[Kmu\]): $$\begin{aligned}
\delta_c\mathcal{O}(z_1,z_2)=2\omega(n\bar n) \bar S_+ \mathcal{O}(z_1,z_2)\,,\end{aligned}$$ where $$\begin{aligned}
\bar S_+=z_1^2\partial_{z_1}+z_{2}^2\partial_{z_2}+\Delta(z_1+z_2) = S^{(0)}_+ -\epsilon(z_1+z_2)\,.\end{aligned}$$ Thus $$\begin{aligned}
\delta_c[\mathcal{O}(z_1,z_2)]=2\omega(n\bar n) Z\bar S_+ Z^{-1}[\mathcal{O}(z_1,z_2)]\,.\end{aligned}$$ Since the partition function (\[Gc\]) does not change under the change of variables (\[ftf\]), one obtains an identity $$\begin{aligned}
\label{CWI0}
0&=& \Big(\int\! d^dy\, A(y) K^{y}_{\bar n}(\Delta)\frac{\delta}{\delta A(y)}
+2(n \bar n)Z\bar S_+ Z^{-1}\Big)
\nonumber\\&&{}
\times Z_{\mathcal{O}}(z_1,z_2;A)
-{\left\langle\left\langle{\delta_c S_R(\varphi)[\mathcal{O}(z_1,z_2)]}\right\rangle\right\rangle}\,,\end{aligned}$$ where the superscript in $K^{y}_{\bar n}$ indicates the variable the operator acts on and $\delta_c S_R(\varphi)$ is the variation of the action under the special conformal transformation (\[ftf\]) $$\begin{aligned}
\label{deltaSE}
\delta_c S_R(\varphi) =&-\frac13 \epsilon
{Z_3 M^\epsilon g}
\int\! d^d x\, x_{\bar n}d^{abc}\varphi^a(x)\varphi^b(x)\varphi^c(x)\,,
\notag\\
\delta_c S_R(\varphi) =&-\frac16 \epsilon
{Z_3M^{2\epsilon} g}
\int\! d^d x\, x_{\bar n}\varphi^4(x)\end{aligned}$$ for the $\varphi^3$- and $\varphi^4$-theories, respectively. It should be stressed that the CWI (\[CWI0\]) holds for arbitrary value of the coupling constant. Note also that since it is derived by a variation of the finite (renormalized) partition function, all singular $1/\epsilon^k$-terms in Eq. (\[CWI0\]) have to cancel each other.
Conformal symmetry of the theory at the critical point implies that Eq. (\[CWI0\]) can be rewritten in the form $$\begin{aligned}
\label{CWIC}
0 &=& \Big(\int d^dy\, A(y) K^{y}_{\bar n}(\Delta^\ast)\frac{\delta}{\delta A(y)}+2(n \bar n) S_+ \Big)
\nonumber\\&&{}\times
Z_{\mathcal{O}}(z_1,z_2;A)\,,\end{aligned}$$ where $\Delta^*$ is the critical scaling dimension of the field $\varphi$, $\Delta^* = \Delta +\gamma^\ast_\varphi$. Evaluating the entries in Eq. (\[CWI0\]) in perturbation theory and bringing the result to the form (\[CWIC\]) one obtains the generator $S_+$ as a series in $\epsilon$ or, equivalently, $u_\ast(\epsilon)$.
![One-loop diagrams for the 1PI Green function $\Gamma_2(\underline{z},\underline{p})$ in the $\varphi^4$-theory (left) and $\varphi^3$-theory (right). The boxes denote the insertion of the light-ray operator $[\mathcal{O}(z_1,z_2)]$.[]{data-label="fig:1"}](Oneloop){width="32.00000%"}
It turns out to be more convenient to analyze the corresponding identities for one-particle irreducible (1PI) Green functions, $\Gamma_{\mathcal{O}}(z_1,z_2,\varphi)$. The CWIs (\[CWI0\]) and (\[CWIC\]) are replaced in this case by $$\begin{aligned}
\label{CWI0a}
0&=& \Big(\int\! d^dy\, \varphi(y) K^{y}_{\bar n}(\widetilde\Delta)\frac{\delta}{\delta \varphi(y)}
+2(n \bar n)Z\bar S_+ Z^{-1}\Big)
\nonumber\\&&{}
\times \Gamma_{\mathcal{O}}(z_1,z_2;\varphi)
-{\left\langle\left\langle{\delta_c S_R(\varphi)[\mathcal{O}(z_1,z_2)]}\right\rangle\right\rangle}_{1PI}\,,\end{aligned}$$ $$\begin{aligned}
\label{CWICa}
0 &=& \Big(\int d^dy\, \varphi(y) K^{y}_{\bar n}(\widetilde\Delta^\ast)\frac{\delta}{\delta \varphi(y)}+2(n \bar n) S_+ \Big)
\nonumber\\&&{}\times
\Gamma_{\mathcal{O}}(z_1,z_2;\varphi)\,,\end{aligned}$$ respectively, where the $\widetilde \Delta$ is the shadow dimension $$\widetilde
\Delta=d-\Delta.$$
As the first step, let us rewrite $$\begin{aligned}
\label{Su}
S_+(u)\equiv Z(u)\bar S_+ Z^{-1}(u)=\mathbb{Z}(u)\bar S_+ \mathbb{Z}^{-1}(u)\,,\end{aligned}$$ where $\mathbb{Z}=Z Z_1^{-1}$ in terms of the evolution kernel $\mathbb{H}$ (\[H\]). To this end, note that $S_+(u)$ obeys the following differential equation: $$\begin{aligned}
M\frac{d}{dM}S_+(u)=\beta(u)\frac{d}{du}S_+(u)=-[\mathbb{H}(u),\,S_+(u)]\,,\end{aligned}$$ which follows readily from Eq. (\[Su\]). Taking into account that in the free theory $S_+(u=0)=\bar S_+$ one obtains $$\begin{gathered}
\label{Sexp}
S_+(u)=\bar S_+-\int_0^{u}\frac{dv}{\beta(v)}[\mathbb{H}(v),\bar S_+]\\
+ \int_0^{u}\frac{dv}{\beta(v)}\int_0^v
\frac{dw}{\beta(w)}[\mathbb{H}(v),[\mathbb{H}(w),\bar S_+]]+ \ldots\end{gathered}$$ Substituting in this equation the evolution kernel $\mathbb{H}(u)$ and the beta-function $\beta(u)$ by their perturbative expansion $$\begin{aligned}
\mathbb{H}(u)=&~u\,\mathbb{H}^{(1)}+u^2\,\mathbb{H}^{(2)}+\ldots
\notag\\
\beta(u)=&u\big(-2\epsilon + u\beta_0+u^2\beta_1+\ldots\big)\end{aligned}$$ we get $$\begin{aligned}
\label{Su+}
S_+(u)&=&\bar S_++\frac{u}{2\epsilon}[\mathbb{H}^{(1)},\bar S_+]\left(1+\frac{\beta_0
u}{4\epsilon}\right) +\frac{u^2}{4\epsilon}[\mathbb{H}^{(2)},\bar S_+]
\nonumber\\&&{}
+\frac{u^2}{8\epsilon^2}[\mathbb{H}^{(1)}[\mathbb{H}^{(1)},\bar S_+]]+\mathcal{O}(u^3)\end{aligned}$$ Taking into account that the leading-order evolution kernel $\mathbb{H}^{(1)}$ commutes with the canonical generator $S_+^{(0)}$ (\[S\_0\]) and writing $\bar S_+ = S_+^{(0)}-\epsilon(z_1+z_2)$ one can further simplify Eq. (\[Su+\]) as follows: $$\begin{aligned}
\label{S+exp}
S_+(u)=& \bar S_+-\frac{u}{2}[\mathbb{H}^{(1)},(z_1+z_2)]
-\frac{u^2}{4}[\mathbb{H}^{(2)},(z_1+z_2)]
\notag\\
&
-\frac{u^2}{8\epsilon}
\biggl\{\beta_0[\mathbb{H}^{(1)},(z_1+z_2)]+2[S_+^{(0)},\mathbb{H}^{(2)}]
\notag\\
&
+[\mathbb{H}^{(1)}[\mathbb{H}^{(1)},(z_1+z_2)]]
\biggr\}+\mathcal{O}(u^3)\,.\end{aligned}$$ Note that this expression contains both regular and singular parts in $1/\epsilon$, $S_+(u)=S^{(\text{reg})}_+(u)+S^{(\text{sing})}_+(u)$. It is easy to see that the regular part comes solely from the first two terms in the expansion (\[Sexp\]) so that to all orders in the coupling $$\begin{aligned}
S^{(\text{reg})}_+(u)=S_+^{(0)}-\epsilon(z_1\!+\!z_2)-\frac12\int_{0}^u\frac{dv}v[\mathbb{H}(v),z_1\!+\!z_2]\,.\end{aligned}$$ The singular part, $S^{(\text{sing})}_+(u)$, receives contributions from all terms in the expansion (\[Sexp\]). Since all $\epsilon$-singular terms in the CWI (\[CWI0a\]) must cancel, the sum of the last two terms in this identity, $2(n\bar n) S_+(u)\Gamma_\mathcal{O}(z_1,z_2,\varphi)$ and ${\left\langle\left\langle{\delta_c S_R(\varphi)[\mathcal{O}(z_1,z_2)]}\right\rangle\right\rangle}_{1PI}$, has to be finite. This implies that the counterterm to the operator $\delta_c S_R(\varphi)[\mathcal{O}(z_1,z_2)]$ must have the following form: $$\begin{aligned}
2(n\bar n) S^{(\text{sing})}_+(u)[\mathcal{O}(z_1,z_2)]\,.\end{aligned}$$ As seen from Eq. (\[S+exp\]) $S^{(\text{sing})}_+(u)\sim \mathcal{O}(u^2)$ and thus the correlation function ${\left\langle\left\langle{\delta_c S_R(\varphi)[\mathcal{O}(z_1,z_2)]}\right\rangle\right\rangle}_{1PI}$ must be finite to the leading order in $u$.
Let us analyze this contribution in detail. To this end it is sufficient to consider the two-point $1PI$ function $$\begin{gathered}
\label{DeltaSC}
{\left\langle{\delta_cS_R(\varphi)\,[\mathcal{O}(z_1,z_2)]\varphi(x_1)\varphi(x_2)}\right\rangle}_{1PI}=
\\=\int d^dp_1 d^d p_2 \, e^{-ip_1x_1-ip_2 x_2}
\delta\Gamma_{2}(z_1,z_2,p_1,p_2)\,.\end{gathered}$$ Let also $$\begin{gathered}
\Gamma_{2}(z_1,z_2,x_1,x_2) = {\left\langle{[\mathcal{O}(z_1,z_2)]\varphi(x_1)\varphi(x_2)}\right\rangle}_{1PI}
\\=\int d^dp_1 d^d p_2 e^{-ip_1x_1-ip_2 x_2}
\Gamma_{2}(z_1,z_2,p_1,p_2)\,.
\label{Gamma2}\end{gathered}$$ To save space below we use a shorthand notation for the arguments $$\underline{z} = \{z_1,z_2\}\,\qquad \underline{p} = \{p_1,p_2\}\,,$$ etc.
As follows from the explicit expressions in Eq. (\[deltaSE\]), variation of the action can be written in both theories as $$\begin{aligned}
\label{Veff}
\delta_c S_R(\varphi)=-2k\epsilon Z_3 M^{k\epsilon} g\int d^dx\, (\bar n x) V(\varphi),\end{aligned}$$ where $k=1,2$ for $\varphi^3-$ and $\varphi^4$-theories, respectively, and $V(\varphi)$ is the corresponding potential.
The one-loop Feynman diagrams for $\delta
\Gamma_2(\underline{z},\underline{p})$ are obtained from the diagrams shown in Fig. \[fig:1\] by the replacement of one of the interaction vertices by an effective vertex derived from (\[Veff\]). The factor $(\bar n x)$ in the effective vertex can be represented as the derivative with respect to the incoming momentum [@Mueller:1991gd] $$\int d^dx\, e^{i(px)}(\bar n x)V(\varphi)=-i(\bar n \partial_p)\int d^dx\,
e^{i(px)}V(\varphi)\,.$$ Thus to the one-loop accuracy $$\begin{aligned}
\delta\Gamma^{(1)}_{2}(\underline{z},\underline{p})=
-2i\epsilon\Big(\bar n,\frac{\partial}{\partial p_1}+\frac{\partial}{\partial p_2}\Big)
\Gamma^{(1,{\rm bare})}_{2}(\underline{z},\underline{p})\,,\end{aligned}$$ where $(*,*)$ is the usual Minkowski scalar product and $\Gamma^{(1,{\rm bare})}_{2}$ is the unrenormalized Green function, $\Gamma^{(1)}_{2}=\Gamma^{(1,{\rm bare})}_{2}-\,\text{counterterm}$. Note that this relation holds for the both theories considered here. Taking into account that $\Gamma^{(1,{\rm bare})}_{2}\sim M^{2\epsilon}$ and that the tree-level function $\Gamma^{(0)}_2(\underline{z},\underline{p})$ (and, hence, the counterterm) does not depend on $M$, we can rewrite this expression as $$\begin{aligned}
\lefteqn{\delta\Gamma^{(1)}_{2}(\underline{z},\underline{p})=}
\nonumber\\&=&
-i (\bar n,\partial_{p_1}+\partial_{p_2})M\partial_M\Gamma^{(1)}_2(\underline{z},\underline{p})
\nonumber\\&=&
i(\bar n,\partial_{p_1}+\partial_{p_2})\big[\beta(u)\partial_u + \mathbb{H}-2\gamma_\varphi\big]
\Gamma^{(0)}_2(\underline{z},\underline{p}).\end{aligned}$$ Since $\Gamma^{(0)}_2(\underline{z},\underline{p})\sim e^{i(n,p_1 z_1+p_2 z_2)}$ the derivatives in the momenta are trivial so that $$\begin{aligned}
\lefteqn{\delta\Gamma^{(1\ell)}_{2}(\underline{z},\underline{p})=}
\nonumber\\&=&-(n\bar n)\big[\beta(u)\partial_u+ \mathbb{H}-2\gamma_\varphi\big]
(z_1+z_2) \Gamma^{(0)}_2(\underline{z},\underline{p})\,.\end{aligned}$$ The tree-level 1PI function in coordinate space is given by the product of delta-functions, $$\Gamma^{(0)}_2(\underline{z},\underline{x})\sim
\delta(x_1-z_1n)\delta(x_2-z_2 n),$$ hence $$\begin{aligned}
\label{DScorr}
\lefteqn{{\left\langle{\delta_cS_R(\varphi)\,[\mathcal{O}(z_1,z_2)]\varphi(x_1)\varphi(x_2)}\right\rangle}_{1PI}=}
\nonumber\\&=&
\Big\{\gamma_\varphi(x_1 +x_2,\bar n)
-(n\bar n)\big[\beta(u)\partial_u+\mathbb{H}\,(z_1+z_2)\big]\Big\}
\nonumber\\&&{}\times\Gamma^{(0)}_2(\underline{z},\underline{x})\,,\end{aligned}$$ where to our accuracy $\mathbb{H} = u \mathbb{H}^{(1)}$.
The CWI (\[CWI0a\]) for the two-point 1PI Green function takes the form $$\begin{aligned}
\label{CWI2}
0&=& \Big(K^{x_1}_{\bar n}(\widetilde \Delta)+K^{x_2}_{\bar n}(\widetilde \Delta)+2(n\bar n) Z\bar S_+
Z^{-1}\Big) \Gamma_2(\underline{z},\underline{x})
\nonumber\\&&{}
-{\left\langle{\delta_cS_R(\varphi)\,[\mathcal{O}(z_1,z_2)]\varphi(x_1)\varphi(x_2)}\right\rangle}_{1PI}\,,\end{aligned}$$ where (\[S+exp\]) $$\begin{aligned}
Z\bar S_+Z^{-1} = S_+^{(0)} - \epsilon(z_1+z_2) -\frac12 u[\mathbb{H}^{(1)},z_1+z_2]\,.\end{aligned}$$ Substituting the expression (\[DScorr\]) into Eq. (\[CWI2\]) we can replace, to the required accuracy, $\Gamma^{(0)}_2(\underline{z},\underline{x}) \to \Gamma_2(\underline{z},\underline{x})$ in the last term. One sees then that the resulting contribution $\sim \gamma_\varphi$ to the CWI (\[CWI2\]) can be absorbed by modifying the parameter $\widetilde \Delta$ in the conformal generators $K_{\bar n}(\widetilde \Delta)\to K_{\bar n}(\widetilde \Delta_\varphi)$ where $$\widetilde \Delta_\varphi = \widetilde \Delta-\gamma_\varphi$$ is the (one-loop) shadow scaling dimension of the scalar field. In this way we obtain $$\begin{aligned}
\label{CWI3}
0 &=& \Big\{K^{x_1}_{\bar n}(\widetilde \Delta_\varphi)+K^{x_2}_{\bar n}(\widetilde \Delta_\varphi)
+2(n\bar n)[\beta(u)\partial_u+ S_+] \Big\}
\nonumber\\&&{}\times\Gamma_2(\underline{z},\underline{x})+\mathcal{O}(u^2)\,,\end{aligned}$$ where $$\begin{aligned}
\label{S+oneloop}
S_+=z_1^2\partial_{z_1}+z_2^2\partial_{z_2} + (z_1+z_2)\Big(\Delta+\frac12u \mathbb{H}^{(1)}\Big),\end{aligned}$$ At the critical point, $u\to u_*$, $\beta(u_*)=0$, the contribution of the beta-function to Eq. (\[CWI3\]) vanishes and we end up with the result for the deformation $\Delta S_+$ that has been quoted in Eq. (\[DeltaS0+\]).
The one-loop expression in (\[S+oneloop\]) is the same for both scalar theories that we consider in this paper. It is tempting to assume that this result can be generalized to all orders as $$\begin{aligned}
\label{S+conjecture}
S_+=z_1^2\partial_{z_1}+z_2^2\partial_{z_2} + (z_1+z_2)\Big(\Delta+\frac12\mathbb{H}(u_*)\Big)\,.\end{aligned}$$ Indeed, the expression in Eq. (\[S+conjecture\]) obeys the necessary commutation relation $[S_+,S_-]=2S_0$ and its action on the eigenfunctions $\Psi_{Nk}(z_1,z_2)$ takes the expected form $$\begin{aligned}
\label{Conj84}
S_+\Psi_{Nk}=\left(z_1^2\partial_{z_1}+z_2^2\partial_{z_2}+(z_1+z_2)\Delta_N\right)\Psi_{Nk}\,.\end{aligned}$$ It can be shown that Eq. (\[Conj84\]) results in the form of a conformal operator proposed in Ref. [@Brodsky:1984xk].
It turns out, however, that this (plausible) conjecture is wrong. We have calculated the operator $S_+$ in the $\varphi^4$ theory at the order $\epsilon^2$ with the result $$\begin{aligned}
\label{S+NL}
S_+=&z_1^2\partial_{z_1}+z_2^2\partial_{z_2}+(z_1+z_2)\Big(\Delta+\frac12\mathbb{H}(u_*)\Big)
\notag\\
&+\frac14 u_*^2\, [\mathbb{H}^{(2)},z_1+z_2]+\mathcal{O}(\epsilon^3)\,,\end{aligned}$$ where $\mathbb{H}^{(2)}$ is the two-loop evolution kernel, Eq. (\[Hexp\]). Details of this calculation can be found in \[app:B\].
Constraints for the evolution kernels
=====================================
It is easy to see that translations along the $n$ direction, $(z_1,z_2)\mapsto (z_1+a,z_2+a)$, and scale transformations $(z_1,z_2)\mapsto (\lambda z_1,\lambda z_2)$ commute with the evolution operator $\mathbb{H}$. The first property is an obvious consequence of Poincare invariance, and the second one is equivalent to the statement that only the operators of the same canonical dimension mix under renormalization. A generic integral operator satisfying these two restrictions can be represented in the form $$\begin{aligned}
\label{int-H-form}
[\mathbb{H} f](z_1,z_2)=\int d\alpha d\beta\, h(\alpha,\beta)
f(z_{12}^\alpha,z_{21}^\beta)\,,\end{aligned}$$ where $$\begin{aligned}
z_{12}^\alpha\equiv z_1\bar\alpha+z_2\alpha && \bar\alpha=1-\alpha\,,\end{aligned}$$ and $h(\alpha,\beta)$ is a certain weight function. Note that the powers $f(z_1,z_2) = (z_{1}-z_{2})^N$ are eigenfunctions of the evolution kernel $\mathbb{H}$, and the corresponding eigenvalues $$\begin{aligned}
\label{hmoments}
\gamma_N=\int d\alpha d\beta\, h(\alpha,\beta)(1-\alpha-\beta)^N\,.\end{aligned}$$ are nothing else as the anomalous dimensions, $\gamma_N$.
In general the function $h(\alpha,\beta)$ is a function of two variables. However, if $\mathbb{H}$ is an invariant operator with respect to the canonical conformal transformations (\[S\_0\]), $[\mathbb{H}_{\rm inv },S_\alpha^{(0)}]=0$, then it can be shown that the function $h(\alpha,\beta)$ takes the form [@Braun:1999te] $$\begin{aligned}
\label{hinv}
h_{\rm inv }(\alpha,\beta)=(\bar\alpha\bar\beta)^{2j-2}\, \bar h\left(\frac{\alpha\beta}{\bar\alpha\bar\beta}\right)\,\end{aligned}$$ and is effectively a function of one variable. This function can easily be reconstructed from its moments (\[hmoments\]), alias from the anomalous dimensions.
In the interacting theory $[\mathbb{H},S_\alpha^{(0)}]\slashed{=}0$ beyond the leading order. As it was shown in the previous section one can define, however, three generators $S_\alpha = S_{\alpha}^{(0)}+ \Delta S_{\alpha} $ (\[DeltaS\]), (\[DeltaS0+\]) which satisfy the canonical $sl(2)$ commutation relations (\[sl2-comm\]) (for the theory at the critical coupling in non-integer dimensions). The commutation relations impose certain self-consistency relations on the corrections $\Delta S_{\alpha}$. Since the evolution kernel $\mathbb{H}(u_\ast)$ commutes with two of the generators, $S_-$ and $S_0$, and since, as it is easy to see [^1], $$\begin{aligned}
{}[S_0^{(0)},\Delta S_+]=\Delta S_+\,,\end{aligned}$$ there are two such relations only: $$\begin{aligned}
\label{restr}
{}[S_+^{(0)},\Delta S_0]=&~[\Delta S_0, \Delta S_+]\,,
\notag\\
{}[\Delta S_+, S_-]=&~2\Delta S_0\,.\end{aligned}$$ Taking into account that $\Delta S_0 = -\epsilon + (1/2)\mathbb{H}(u_*)$, see Eq. (\[DeltaS0+\]), the first relation in (\[restr\]) can be rewritten as $[S_+^{(0)},\mathbb{H}(u_*)]=[\mathbb{H}(u_*), \Delta S_+]$. It implies that the exact evolution kernel $\mathbb{H}(u_*)$ commutes with the full generator $S_+$. The second relation provides a constraint on the possible deformation of $S_+$.
Writing $\Delta S_+$ as a power series in the critical coupling $u_\ast$ $$\begin{aligned}
\Delta S_+= \sum_{k=1}^\infty u_*^k
\Delta S_+^{(k)}\,\end{aligned}$$ and expanding $[S_+,\mathbb{H}(u_\ast)]=0$ in powers of $u_*$ one obtains $$\begin{aligned}
\label{HE}
[S_+^{(0)},\mathbb{H}^{(1)}]=&~0\,,
\notag\\
[S_+^{(0)},\mathbb{H}^{(2)}]=&~[\mathbb{H}^{(1)},\Delta S_+^{(1)}]\,,
\notag\\
[S_+^{(0)},\mathbb{H}^{(3)}]=&~[\mathbb{H}^{(1)},\Delta S_+^{(2)}]+[\mathbb{H}^{(2)},\Delta S_+^{(1)}]\,,\end{aligned}$$ etc. Note that the commutator of the canonical generator $S_+^{(0)}$ with the evolution kernel at order $\ell$ is given in terms of the evolution kernels $\mathbb{H}^{(k)}$ and the corrections to the generators $\Delta S_+^{(k)}$ at one order less, $k \le \ell-1$. The commutation relations Eq. (\[HE\]) can be viewed as, essentially, inhomogeneous first-order differential equations on the evolution kernels. Their solution determines $\mathbb{H}^{(k)}$ up to an $sl(2)$-invariant term, $[\mathbb{H}_{inv}^{(k)},S_\alpha^{(0)}]=0$, which can be restored from the spectrum of the anomalous dimensions.
As discussed above, an evolution kernel $\mathbb{H}$ can be represented in the form of an integral operator (\[int-H-form\]). The corresponding weight function $h(\alpha,\beta)$ can be split in the $sl(2)$-invariant and non-invariant parts with respect to the canonical transformations (\[S\_0\]) $$\begin{aligned}
h^{(k)}(\alpha,\beta) = h^{(k)}_{\rm inv}(\alpha,\beta)+\Delta h^{(k)}(\alpha,\beta)\,,
\label{hsplit}\end{aligned}$$ cf. Eq. (\[inv-noninv\]). We will show that the non-invariant part can be determined from the commutation relations and requires a $(k-1)$-loop calculation. In turn, the invariant function $h^{(k)}_{{\rm inv}}(\alpha,\beta)$ takes the form (\[hinv\]) and can easily be restored from the moments. Let $\Delta\gamma^{(k)}_{N}$ be the eigenvalue of $\Delta\mathbb{H}^{(k)}$ on the function $(z_1-z_2)^N$, i.e. $$\begin{aligned}
\Delta\gamma^{(k)}_{N} = \int d\alpha d\beta\, \Delta h^{(k)}(\alpha,\beta) (1-\alpha-\beta)^N\,.\end{aligned}$$ Since the invariant function $h^{(k)}_{{\rm inv}}(\alpha,\beta)$ is effectively a function of one variable (\[hinv\]) it can be recovered inverting the equation for the moments $$\begin{aligned}
\int d\alpha d\beta\, h^{(k)}_{{\rm inv}}(\alpha,\beta) (1-\alpha-\beta)^N = \gamma^{(k)}_{N} - \Delta\gamma^{(k)}_{N}\end{aligned}$$ Determination of the anomalous dimensions $\gamma_N^{(k)}$ still requires evaluation of $k$-loop integrals. It is much simpler, nevertheless, than calculation of the full kernel $h^{(k)}(\alpha,\beta)$ alias the full anomalous dimension matrix at the same order. In what follows we demonstrate the utility of this procedure on two examples.
Three-loop evolution equations in the $\varphi^4$ theory
--------------------------------------------------------
Twist-two operators in the $O(n)$-symmetric $\varphi^4$ theory can be divided in three classes that transform differently under rotations in the isotopic space: scalar (sc), symmetric and traceless (st) and antisymmetric (as) $$\begin{aligned}
\mathcal{O}_{(sc)}(z_1,z_2) &=& \varphi^a(z_1n)\varphi^a(z_2n)\,,
\notag\\
\mathcal{O}_{(st)}^{ab}(z_1,z_2) &=& [\varphi^a(z_1n)\varphi^b(z_2n) + (a\leftrightarrow b)] -\,\text{trace}\,,
\notag\\
\mathcal{O}_{(as)}^{ab}(z_1,z_2) &=& \varphi^a(z_1n)\varphi^b(z_2n) - (a\leftrightarrow b)\,,\end{aligned}$$ respectively. Anomalous dimensions for all these operators were calculated in Ref. [@Kehrein:1995ia] at two loops, and for the scalar operators in Ref. [@Derkachov:1997pf] at four loops. The anomalous dimensions of the symmetric traceless and the antisymmetric operators can easily be derived from the expressions presented in Ref. [@Derkachov:1997pf] by taking into account appropriate isotopic factors. We collect below the anomalous dimensions $\gamma_k$ ($k$ is the number of derivatives) to three-loop accuracy that is relevant for this study.
For the scalar operators one obtains [@Derkachov:1997pf] $$\begin{aligned}
\gamma^{(sc)}_{k = 0}(u)=&(n+2)\biggl\{\frac{u}3-\frac{5u^2}{18}+\frac{u^3(5n+37)}{36}+\ldots\biggr\},
\notag\\
\gamma^{(sc)}_{k \ge 1}(u)=&(n+2)\biggl\{\frac{u^2}{18}\frac{(k-2)(k+3)}{k(k+1)}-\frac{2u^3(n+8)}{27 k(k+1)}
\notag\\ &
\times\left[
{S_1(k)}
+\frac{k^4+2k^3-39k^2-16k+12}{16k(k+1)}\right]\biggr\}
\notag\\ & +\ldots,\end{aligned}$$ where $S_1(k)=\sum_{m=1}^k 1/m=\psi(k+1)-\psi(1)$. Note that the anomalous dimension $\gamma_{k=2}$ vanishes: The corresponding operator is nothing but the energy momentum tensor of the scalar field and it is conserved in quantum theory.
For the symmetric traceless operators we get $$\begin{aligned}
\label{}
\gamma_{k=0}^{(st)}(u)=&\,\frac{2u}{3}-\frac{u^2(n+10)}{18}
-\frac{u^3(5n^2-84n-444)}{216}
\nonumber\\ &+\ldots\,,
\nonumber\\
\gamma_{k\ge 1}^{(st)}(u)=&\frac{u^2}{9}\biggl[\frac{n+2}2-\frac{n+6}{k(k+1)}\biggr]-\!\frac{u^3}{54}\biggl[
\frac{(n+2)(n+8)}4
\notag\\
&+\frac{8(n+4)}{k(k+1)}\left(2S_1(k)-\frac{4k^2+2k-1}{k(k+1)}\right)
\notag\\
&-\frac{(2k^2-1)(n^2+6n+16)}{k^2(k+1)^2}\biggr]+
\ldots\end{aligned}$$ and, finally, for the antisymmetric operators $$\begin{aligned}
\gamma_{k\ge1}^{(as)}(u)=&(n+2)\biggl\{\frac{u^2}{9}\left[\frac12-\frac{1}{k(k+1)}\right]-\frac{u^3}{54}\biggl[
\frac{(n+8)}4
\notag\\
&+\frac{4}{k(k+1)}\left(2S_1(k) -\frac{4k^2+2k-1}{k(k+1)}\right)
\notag\\
&-\frac{(2k^2-1)(n+4)}{k^2(k+1)^2}\biggr]+
\ldots \biggr\}.\end{aligned}$$ Note that the antisymmetric operator without derivatives $(k=0)$ does not exist. These results are in agreement with the $1/n$-expansion of the anomalous dimensions [@Derkachov:1997ch].
We now proceed with the calculation of the evolution kernels. As the first step one has to reconstruct the leading-order evolution kernel $\mathbb{H}^{(1)}$. This operator commutes, cf. the first relation in (\[HE\]), with canonical $sl(2)$ generators $S_\alpha^{(0)}$, $\alpha = 0,\pm$ and its spectrum is $$\begin{aligned}
\label{c-coef}
\gamma^{(1)}_k =&c_n\,\delta_{k0}\,, &&
c_n=\left\{\frac{n+2}3,\,\frac23,\,0\right\},\end{aligned}$$ for the scalar, symmetric traceless, and antisymmetric operators, respectively. The general form of the invariant operator is given by Eq. (\[hinv\]) where, for the case at hand, $j=1/2$. It is easy to convince oneself that the spectrum $\gamma_k\sim \delta_{k0}$ corresponds to the choice $\bar h(\tau)\sim\delta(1-\tau)$. We define a $sl(2)$ invariant operator [@Derkachov:1995zr; @Braun:2009vc] $$\begin{aligned}
\label{Hd}
{}[\mathcal{H}^{(d)}f](z_1,z_2)=&\int_0^1 \frac{d\alpha}{\bar\alpha}\int_0^{\bar\alpha}\frac{d\beta}{\bar\beta}
\delta\left(1-\frac{\alpha\beta}{\bar\alpha\bar\beta}\right)
f(z_{12}^\alpha,z_{21}^\beta)
\notag\\
=&\int_0^1 d\alpha\,f(z_{12}^\alpha,z_{12}^\alpha)\,.\end{aligned}$$ Obviously $$\mathcal{H}^{(d)}z_{12}^k=\delta_{k0} z_{12}^k\,,\qquad z_{12} \equiv z_1-z_2\,,$$ so that the one-loop evolution kernel can be written as $$\begin{aligned}
\label{H1(4)}
\mathbb{H}^{(1)}=c_n\, \mathcal{H}^{(d)}\,,\end{aligned}$$ where the coefficient $c_n$ depends on the symmetry of the operators, cf. Eq. (\[c-coef\]).
Note that the operator $\mathcal{H}^{(d)}$ is a $sl(2)$-invariant projector. Indeed, one can easily check that $$(\mathcal{H}^{(d)})^2=\mathcal{H}^{(d)}\,,\qquad\mathcal{H}^{(d)}(1-\mathcal{H}^{(d)})=0.$$ For later use we define the operator $$\begin{aligned}
\mathrm{\Pi}_0 = 1-\mathcal{H}^{(d)}
\label{Pi0}\end{aligned}$$ which is also a projector. As the next step, we calculate the non-invariant part of the two-loop kernel $\mathbb{H}^{(2)}$. Re-expanding $\epsilon =(4-d)/2$ in terms of the critical coupling $$\begin{aligned}
\epsilon=\frac{n+8}{6} u_*-\frac{3n+14}{6}u_*^2+\mathcal{O}(u_*^3)\,,\end{aligned}$$ cf. Eq. (\[ust4\]), one obtains the one-loop deformation of the generator of special conformal transformations: $$\begin{aligned}
\Delta S_+^{(1)}=(z_1+z_2)\biggl[-\frac16 {(n+8)}+\frac12\mathbb{H}^{(1)}\biggr].\end{aligned}$$ Using this expression, the second commutator relation in Eq. (\[HE\]) takes the form $$\begin{aligned}
\label{HE2}
{}[S_+^{(0)},\mathbb{H}^{(2)}]&=&[\mathbb{H}^{(1)},{z_1+z_2}]\left\{-\frac16 {(n+8)}+\frac12\mathbb{H}^{(1)}\right\}
\notag\\
&=&\frac{c_n}{2}[\mathcal{H}^{(d)},z_1+z_2]\left\{c_n-\frac13 {(n+8)}\right\},\end{aligned}$$ where in the second line we have taken into account that $[\mathcal{H}^{(d)},z_1+z_2]\mathcal{H}^{(d)}=[\mathcal{H}^{(d)},z_1+z_2]$. The remaining commutator on the r.h.s. of (\[HE2\]) can be written as $$\begin{aligned}
{}[\mathcal{H}^{(d)},z_1+z_2]&=&z_{12}\widetilde{\mathcal{H}}^{(d)}\,,
\nonumber\\
{}[\widetilde{\mathcal{H}}^{(d)} f](z_1,z_2)&=&\int_0^1 d\alpha\,(\bar\alpha-\alpha)f(z_{12}^\alpha,z_{12}^\alpha)\,,
\label{Htilde}\end{aligned}$$ which is easy to verify using the explicit expression in Eq. (\[Hd\]).
Eq. (\[HE2\]) can be viewed as an equation on $\mathbb{H}^{(2)}$. We look for the solution as the sum (\[inv-noninv\]) $$\begin{aligned}
\mathbb{H}^{(2)}= \mathbb{H}^{(2)}_{inv}+ \Delta\mathbb{H}^{(2)}
\label{H2split}\end{aligned}$$ such that $\mathbb{H}^{(2)}_{inv}$ is a solution of the homogeneous equation $[\mathbb{H}^{(k)}_{\rm inv},S^{(0)}_{\pm,0}] = 0$.
It is easy to check that for an operator that has the structure $$\begin{aligned}
\label{VV}
[\mathbb{H}f](z_1,z_2)=\int_0^1 d\alpha\,\upsilon(\alpha)\,f(z_{12}^\alpha,z_{12}^\alpha)\,\end{aligned}$$ the commutator with $S_{+}^{(0)}$ equals to $$\begin{aligned}
\label{VVV}
[[S_{+}^{(0)},\mathbb{H}]f](z_1,z_2)=
z_{12}\!\int_0^1 d\alpha\,\alpha\bar\alpha\, \partial_\alpha\upsilon(\alpha)
f(z_{12}^\alpha,z_{12}^\alpha)\,.\end{aligned}$$ It follows from Eqs. (\[HE2\]) and (\[Htilde\]) that $\Delta\mathbb{H}^{(2)}$ corresponds to the weight function $$\begin{aligned}
\upsilon(\alpha)=\tilde c_n\, (\ln\alpha+\ln\bar\alpha)\,,\end{aligned}$$ where $$\begin{aligned}
\label{}
\tilde c_n =\frac{c_n}{6}\left[3c_n- {(n+8)}\right]=\Big\{-\frac{n\!+\!2}3,-\frac{n\!+\!6}9,0
\Big\}\end{aligned}$$ for the scalar, symmetric traceless and antisymmetric operators, respectively. Thus $$\begin{aligned}
\label{deltaH2}
\Delta\mathbb{H}^{(2)} = \tilde c_n\, \mathcal{V}^{(d,1)}\,,\end{aligned}$$ where $$\begin{aligned}
\label{Hd1}
{}[\mathcal{V}^{(d,1)}f](z_1,z_2)=&\int_0^1d\alpha\, \ln\alpha\bar\alpha f(z_{12}^\alpha,z_{12}^\alpha)\,.\end{aligned}$$ A straightforward calculation yields $$\begin{aligned}
{}[\Delta\mathbb{H}^{(2)}z_{12}^N](z_1,z_2)= -4 \tilde c_n\,\delta_{N0} \, z^N_{12} \equiv \Delta\gamma^{(2)}_N \, z^N_{12}\,.\end{aligned}$$ As the last step, the invariant kernel $\mathbb{H}^{(2)}_{inv}$ (\[H2split\]) can be reconstructed from the known two-loop anomalous dimensions $\gamma^{(2)}_N$ by inverting the equation for the moments $$\begin{aligned}
{}[\mathbb{H}^{(2)}_{inv}z_{12}^N](z_1,z_2) = (\gamma^{(2)}_N - \Delta\gamma^{(2)}_N) \, z^N_{12}\,.\end{aligned}$$ One gets after some algebra[^2] $$\begin{aligned}
\label{invH2}
\mathbb{H}^{sc,(2)}=&-\frac{n+2}3\left[3\mathcal{H}^{(d)}+\mathcal{H}^{(+)}\mathrm{\Pi}_0+\mathcal{V}^{(d,1)}-\frac16\right],
\notag\\
\mathbb{H}^{st,(2)}=&-\frac{n+6}9\left[3\mathcal{H}^{(d)}+\mathcal{H}^{(+)}\mathrm{\Pi}_0+\mathcal{V}^{(d,1)}\right]+\frac{n+2}{18}\,,
\notag\\
\mathbb{H}^{as,(2)}=&-\frac{n+2}9\left[\mathcal{H}^{(+)}-\frac12\right],\end{aligned}$$ where we have introduced a new invariant kernel $\mathcal{H}^{(+)}$ that corresponds to the choice $\bar h(\tau)=(1-\tau)^{-1}$ in Eq. (\[hinv\]): $$\begin{aligned}
{}[\mathcal{H}^{(+)}f](z_1,z_2)=&\int_0^1d\alpha\int_0^{\bar\alpha}\frac{d\beta}{1-\alpha-\beta}
{f(z_{12}^\alpha,z_{21}^\beta)}\,.\end{aligned}$$ Note that $\mathcal{H}^{(+)}$ is only well defined on the space of functions that vanish at $z_1\to z_2$. It can be checked that the projector $\mathrm{\Pi}_0$ eliminates a constant term at $z_1\to z_2$ from any function, so that the product $\mathcal{H}^{(+)}\mathrm{\Pi}_0$ is always well defined. We have removed the projector $\mathcal{H}^{(+)}\mathrm{\Pi}_0 \to \mathcal{H}^{(+)}$ in the last expression in Eq. (\[invH2\]) (for antisymmetric operators) since the relevant functions are in this case antisymmetric under the permutations $z_1\leftrightarrow z_2$.
Proceeding in the same way one can derive the three-loop evolution kernels. The second-order correction to the generator $S_+^{(0)}$ takes the form $$\begin{aligned}
\Delta S_+^{(2)}=(z_1+z_2)\frac{3n+14}{6}+\frac14\big\{z_1+z_2,\mathbb{H}^{(2)}\big\}\,,\end{aligned}$$ where $\{\ast,\ast\}$ stands for the anticommutator. Taking into account that $$\begin{aligned}
\mathbb{H}^{(1)}[\mathbb{H}^{(2)},z_1+z_2]= c_n \, \mathcal{H}^{(d)} [\mathbb{H}^{(2)},z_1+z_2] = 0\end{aligned}$$ one obtains an equation for $\mathbb{H}^{(3)}$: $$\begin{aligned}
\label{SS3}
[S_+^{(0)},\mathbb{H}^{(3)}]&=&\frac16[({3n+14})\mathbb{H}^{(1)}-
({n+8})\mathbb{H}^{(2)}\!,z_1\!+\!z_2]
\nonumber\\
&+&\frac14[\mathbb{H}^{(2)}\!,z_1\!+\!z_2]\mathbb{H}^{(1)}
+\frac12[\mathbb{H}^{(1)}\!,z_1\!+\!z_2]\mathbb{H}^{(2)}\!.
\nonumber\\\end{aligned}$$ Let us consider the antisymmetric operators at first. This case is simpler because $\mathbb{H}^{as(1)}=0$ so that one gets $$\begin{aligned}
\label{S3asym}
[S_+^{(0)},\mathbb{H}^{as(3)}]=\varkappa_n\, [\mathcal{H}^{+},z_1+z_2] = \varkappa_n z_{12}\widetilde{\mathcal{H}}^{+}\,,\end{aligned}$$ where $$\varkappa_n={(n+2)(n+8)}/{54}$$ and $$\begin{aligned}
{}[\widetilde{\mathcal{H}}^{(+)}f](z_1,z_2)=
\int_0^1d\alpha\int_0^{\bar\alpha}d\beta \frac{\beta-\alpha}{1-\alpha-\beta}f(z_{12}^\alpha,z_{21}^\beta)\,.
\label{tildeHplus}\end{aligned}$$ The commutator of $S_+^{(0)}$ with an integral operator that has a generic structure (\[int-H-form\]) $$\begin{aligned}
[\mathbb{H} f](z_1,z_2)= \int_0^1d\alpha \int_0^{\bar\alpha}d\beta\, h(\alpha,\beta)
f(z_{12}^\alpha,z_{21}^\beta)\,
\label{hform}\end{aligned}$$ can be written as $$\begin{aligned}
[[S_+^{(0)},\mathbb{H}]f](z_1z_2) &=&
z_{12}\int_0^1\!d\alpha\!\int_0^{\bar\alpha }\!\!d\beta f(z_{12}^\alpha,z_{21}^\beta)
\nonumber\\&& {}\hspace*{-0.8cm}
\times\big[\alpha\bar\alpha\partial_\alpha-\beta\bar\beta\partial_\beta+\beta-\alpha\big]h(\alpha,\beta)\,.\end{aligned}$$ Looking for the solution of Eq. (\[S3asym\]) in this form one obtains $$\begin{aligned}
h(\alpha,\beta)=\varkappa_n \frac{\ln(1\!-\!\alpha\!-\!\beta)}{1\!-\!\alpha\!-\!\beta}
+ \frac{1}{\bar\alpha\bar\beta} \bar h\left(\frac{\alpha\beta}{\bar\alpha\bar\beta}\right),
$$ where the function $\bar h$ is arbitrary (a solution of the homogeneous equation). It corresponds to the invariant kernel. Thus $$\begin{aligned}
\Delta\mathbb{H}^{as(3)} = \varkappa_n \mathcal{V}^{(+,1)}\end{aligned}$$ with $$\begin{aligned}
\mathcal{V}^{(+,1)}=
\int_0^1d\alpha\int_0^{\bar\alpha }d\beta\, \frac{\ln(1\!-\!\alpha\!-\!\beta)}{1\!-\!\alpha\!-\!\beta} f(z_{12}^\alpha,z_{21}^\beta)\,.
\label{Vplus1}\end{aligned}$$ As above, the $sl(2)$-invariant contribution can be restored from the known anomalous dimensions. We obtain after some algebra $$\mathbb{H}^{as(3)}=\varkappa_n\biggl[
\mathcal{V}^{(+,1)}
+\frac{4}{n+8}\mathcal{H}^{(1)}+2\frac{n+12}{n+8}\mathcal{H}^{(+)} -\frac14\biggr],
\label{Has3}$$ where $$\begin{aligned}
\mathcal{H}^{(1)}f =
\int_0^1d\alpha\int_0^{\bar\alpha }d\beta\,
\frac{1}{1\!-\!\alpha\!-\!\beta}\ln\left(\frac{\alpha\beta}{\bar\alpha\bar\beta}\right) f(z_{12}^\alpha,z_{21}^\beta)\,.\end{aligned}$$ The kernels in Eq. (\[Has3\]) have the following eigenvalues on $z_{12}^k$: $$\begin{aligned}
\mathcal{H}^{(+)} z_{12}^k&=& \frac{1}{k(k+1)}\,z_{12}^k\,,
\nonumber\\
\mathcal{H}^{(1)}z_{12}^k &=& -\frac{2S_1(k)}{k(k+1)}\,z_{12}^k\,,
\nonumber\\
\mathcal{V}^{(+,1)}z_{12}^k &=& -\frac{2k+1}{k^2(k+1)^2}\,z_{12}^k\,.\end{aligned}$$
[c|cccc]{} & $r_1$ & $r_2$ & $r_3$\
& $ \frac{(n+2)(11n+100)}{54}$& $\frac{(n+2)(n+14)}{36}$ &$\frac{(n+2)(n+8)}{18}$\
\
[st]{} & $\frac{3n^2+56n+200}{54}$& $\frac{(n+6)(n+7)}{54}$ & $\frac{(n+6)(n+8)}{54}$\
The calculation of the three-loop evolution kernel for the scalar and symmetric traceless operators goes along the same lines so that we will only sketch the main steps. For the operators of the type $$\begin{aligned}
{}[\mathcal{H}(w) f](z_1,z_2)=\int_0^1 d\alpha \,w(\alpha)\, f(z_{12}^\alpha,z_{12}^\alpha)\end{aligned}$$ the following identity holds: $$\begin{aligned}
\label{HwHw}
\mathcal{H}(w_1)\mathcal{H}(w_2)=\left(\int_0^1d\alpha\, w_2(\alpha)\right)\mathcal{H}(w_1)\,\end{aligned}$$ which appears to be quite useful.
The commutator involving $\mathbb{H}^{(2)}$ on the r.h.s. of (\[SS3\]) can be written as $$\begin{aligned}
{}[\mathbb{H}^{(2)},z_1\!+\!z_2]=z_{12}\xi_n\left[3\widetilde{\mathcal{H}}^{(d)}\!+
\widetilde{\mathcal{H}}^{(+)}\mathrm{\Pi}_0+\!\widetilde{\mathcal{H}}^{(d,1)}\right]\end{aligned}$$ where $$\begin{aligned}
\xi_n=\big\{-({n+2})/{3}, -({n+6})/{9}\big\}\end{aligned}$$ for the scalar and symmetric traceless operators, respectively. The kernels $\widetilde{\mathcal{H}}^{(d)}$ and $\widetilde{\mathcal{H}}^{(+)}$ are defined above in Eqs. (\[Htilde\]), (\[tildeHplus\]) and $$\begin{aligned}
[\widetilde{\mathcal{H}}^{(d,1)}f](z_1,z_2)=\int_0^1d\alpha(\bar\alpha\ln\alpha-\alpha\ln\bar\alpha)
f(z_{12}^\alpha,z_{12}^\alpha).\end{aligned}$$ A straightforward calculation yields: $$\begin{aligned}
\label{Hrk}
[S_+^{(0)}\!,\mathbb{H}^{(3)}]=\!z_{12}\!\left[r_1\widetilde{\mathcal{H}}^{(d)}\!+r_2\widetilde{\mathcal{H}}^{(d,1)}+
r_3\widetilde{\mathcal{H}}^{(+)}\mathrm{\Pi}_0\right]\!.\end{aligned}$$
[c|cccc]{} & $p_1$ & $p_2$ & $p_3$\
& $ \frac{(n+2)(107n+862)}{216}$& $\frac{5(n+2)(n+8)}{27}$ &$\frac{(n+2)(n+8)}{27}$\
\
[st]{} & $\frac{6n^2+219n+862}{108}$& $\frac{n^2+22n+80}{27}$ & $\frac{4(n+4)}{27}$\
The expressions for the coefficients $r_1, r_2, r_3 $ are collected in Table \[tab:rk-coef\]. In this way we obtain for the non-invariant part of the kernel $$\begin{aligned}
\Delta\mathbb{H}^{(3)}=r_1{\mathcal{V}}^{(d,1)}+r_2{\mathcal{V}}^{(d,2)}+
r_3{\mathcal{V}}^{(+,1)}\mathrm{\Pi}_0\,.
\label{DeltaH3}\end{aligned}$$ The operators ${\mathcal{V}}^{(d,1)}$ and ${\mathcal{V}}^{(+,1)}$ are defined in Eqs. (\[Hd1\]) and (\[Vplus1\]), respectively. The new contribution $\mathcal{V}^{(d,2)}$ comes in play as solution to the equation $[S_+^{(0)},\mathcal{H}^{(d,2)}]=z_{12}\widetilde{\mathcal{H}}^{(d,1)}$ and has the form $$\begin{aligned}
[\mathcal{V}^{(d,2)}f](z_1,z_2)=\frac12\int_0^1\! d\alpha\,(\ln^2\alpha+\ln^2\bar\alpha)f(z_{12}^\alpha,z_{12}^\alpha).\end{aligned}$$ The remaining invariant part of the kernel $\mathbb{H}^{(3)}$ can be restored from the anomalous dimensions. The result can be written as follows $$\begin{aligned}
\label{Hinv3}
\mathbb{H}^{(3)}_{inv}=p_1\mathcal{H}^{(d)}+p_2\mathcal{H}^{(+)}\mathrm{\Pi}_0+p_3\mathcal{H}^{(1)}+
2\gamma_{\varphi}^{(3)}\,,\end{aligned}$$ where $\gamma_{\varphi}^{(3)}=-{(n+2)(n+8)}/{432}$ is the corresponding coefficient in the anomalous dimension of the scalar field (\[rgf-4\]) and the coefficients $p_k$ are given in Table \[tab:pk-coef\]. The total kernel $\mathbb{H}^{(3)}$ is given by the sum of the expressions in Eqs. (\[DeltaH3\]) and (\[Hinv3\]), $\mathbb{H}^{(3)}=\mathbb{H}^{(3)}_{inv}+\Delta\mathbb{H}^{(3)}$.
To summarize, making use of the [*exact*]{} conformal invariance of the scalar theory at the critical coupling in $d=4-2\epsilon$ dimensions we have been able to restore the complete three-loop evolution kernels (alias the full anomalous dimension matrix) at arbitrary coupling $u$ using three-loop anomalous dimensions as input. The required calculation is mainly algebraic. The only place where Feynman diagrams appear is the calculation of the deformation of the $S_+$ generator. This calculation is, however, considerably simpler as compared to a direct evaluation of the three-loop evolution kernels.
Two-loop evolution equations in the $\varphi^3$ theory
------------------------------------------------------
We use this example to discuss a somewhat different technique that is based on the representation of $sl(2)$ invariant kernels in terms of the Casimir operators [@Bukhvostov:1985rn]. To start with, we need to classify the existing twist-2 operators $\mathcal{O}^{ab}(z_1,z_2)$ according to the irreducible representations of the isotopic $su(n)$ group. We define $$\begin{aligned}
\label{Pj}
\mathcal{O}^{ab}_{j}(z_1,z_2)=(P_j)_{a'b'}^{ab}\mathcal{O}^{a'b'}(z_1,z_2),\end{aligned}$$ where $P_j$, $j=1,\ldots,7$, are projectors onto the seven irreducible representations in the tensor product of two adjoint representations. Explicit expressions are given in \[app:C\]. Operators corresponding to different representations do not mix under renormalization and can be considered separately.
The one-loop evolution kernel (anomalous dimensions) for all operators except $\mathcal{O}_{j=3}$ is determined by the second diagram in Fig. \[fig:1\]. For the case of $\mathcal{O}_{j=3}$ there is an additional contribution corresponding to the transition $\varphi^{a}\varphi^{b}\to d^{abc}\partial^2\varphi^c$. This extra term vanishes for other operators thanks to the isotopic projector. Although it does not present any particular complication for our analysis, for simplicity we do not consider $\mathcal{O}_{j=3}$ in what follows.
The one-loop evolution kernel corresponding to the diagram in Fig. \[fig:1\] takes the form $$\begin{aligned}
\mathbb{H}^{(1)}=-2\lambda_j\,\mathcal{H}+2\gamma^{(1)}_\varphi\,,
\label{3H11}\end{aligned}$$ where $\lambda_j$ are numbers that depend on the representation and the rank of the group. They are collected in Eq. (\[lambdas\]) in \[app:C\]. The operator $\mathcal{H}$ is defined as $$\begin{aligned}
[\mathcal{H}f](z_1,z_2)=\int_0^1d\alpha\int_0^{\bar\alpha}d \beta f(z_{12}^\alpha,z_{21}^\beta)\,.\end{aligned}$$ $\mathcal{H}$ commutes with the canonical $sl(2)$ generators $S^{(0)}_\alpha$ and has the following eigenvalues $$\begin{aligned}
\mathcal{H} z_{12}^k = E_k z_{12}^k\,, && E_k=\frac1{(k+1)(k+2)}\,,
\label{Ek}\end{aligned}$$ so that the one-loop anomalous dimensions of the twist-two operators are equal to $$\begin{aligned}
\gamma^{(1)}_{j,k} = -2\lambda_j\,E_k+2\gamma^{(1)}_\varphi\,, \qquad j \slashed{=}3\,,\end{aligned}$$ where $k$ is the number of derivatives.
Comparing (\[Ek\]) with the spectrum of the quadratic Casimir operator, $$\begin{aligned}
&\mathbb{C}^{(0)}_2~=~S_+^{(0)}S_-^{(0)}+S_0^{(0)}(S_0^{(0)}-1)=-\partial_1\partial_2 z_{12}^2,
\notag\\
&\mathbb{C}^{(0)}_2z_{12}^k~=~(k+1)(k+2)z_{12}^k\,,\end{aligned}$$ we conclude that the operator $\mathcal{H}$ is nothing else as the inverse of $\mathbb{C}^{(0)}_2$, $$\begin{aligned}
\mathcal{H}=(\mathbb{C}^{(0)}_2)^{-1}.
\label{3H12}\end{aligned}$$
The [complete]{} evolution kernel at the critical point, $\mathbb{H}(u_*)$, commutes with the deformed generators $S_\alpha$ and hence is a function of the complete (deformed) Casimir operator, $$\begin{aligned}
\mathbb{H}_j(u_*)=h_j(\mathbb{C}_2)\,,
&&\mathbb{C}_2=S_+S_-+S_0(S_0-1)\,,
$$ where the subscript $j$ enumerates the isotopic structures (\[Pj\]).
The function $h_j(x)$ has a perturbative expansion $$\begin{aligned}
h_j(x)=&~u_*\, h_j^{(1)}(x)+u_*^2 \,h_j^{(2)}(x)+\ldots\,\end{aligned}$$ and the leading contribution $h_j^{(1)}(x)$ is uniquely fixed by one-loop result (\[3H11\]), (\[3H12\]), alias by the one-loop anomalous dimensions: $$\begin{aligned}
h^{(1)}_j(x)= - 2 \lambda_j \frac{1}{x} + 2 \gamma_\varphi^{(1)}.\end{aligned}$$ Expanding the Casimir operator $\mathbb{C}_2$ $$\begin{aligned}
\mathbb{C}_2=&~\mathbb{C}_2^{(0)}+u_* \,\mathbb{C}_2^{(1)}+\ldots\end{aligned}$$ one gets the following expression for the evolution kernel $\mathbb{H}(u_*)$ to the $\mathcal{O}(u_\ast^3)$ accuracy: $$\begin{aligned}
\label{Hhh}
\mathbb{H}(u_*)&=&u_* h^{(1)}\left(\mathbb{C}_2^{(0)}\!+\!u_* \,\mathbb{C}_2^{(1)}\right)+
u_*^2\, h^{(2)}\,\Big(\mathbb{C}_2^{(0)}\Big).\end{aligned}$$ Similar to the case of the $\varphi^4$ theory considered in the previous section, our strategy here is to look for the the two-loop kernels $\mathbb{H}_j^{(2)}$ in the form $$\begin{aligned}
\mathbb{H}_j^{(2)} = \mathbb{H}_{j,inv}^{(2)} + \Delta\mathbb{H}_j^{(2)}\,,\end{aligned}$$ where the two terms correspond to the $sl(2)$-invariant and non-invariant contributions, respectively. Calculation of the non-invariant kernel $\Delta\mathbb{H}_j^{(2)}$ is the main task, after which the invariant kernel can easily be reconstructed from the spectrum of two-loop anomalous dimensions.
The last term in Eq. (\[Hhh\]) is obviously invariant under canonical $sl(2)$ transformations so that one does not need to know $h^{(2)}(x)$; $\Delta\mathbb{H}_j^{(2)}$ arises exclusively from the first term. Using Eq. (\[3H12\]) yields $$\begin{aligned}
\label{Cexp}
\left(\mathbb{C}_2^{(0)}+u_* \,\mathbb{C}_2^{(1)}\right)^{-1}=\mathcal{H}-u_* \mathcal{H}\mathbb{C}_2^{(1)}\mathcal{H}
+\mathcal{O}(u_*^2)\,.\end{aligned}$$ Next, making use of explicit expressions for the deformed generators, Eqs. (\[S\_0\]), (\[DeltaS0+\]), and writing $$\begin{aligned}
\epsilon-\gamma_\varphi=u_*\kappa+\mathcal{O}(u_*^2),
&& \kappa=(16-n^2)/{3n}\,,\end{aligned}$$ we obtain a correction to the Casimir operator $$\begin{aligned}
\label{C21}
\mathbb{C}_2^{(1)}=-\Big(\partial_1 z_{12}+\partial_2z_{21}+1\Big)\left(\kappa+\lambda_j \mathcal{H}\right).\end{aligned}$$ Since we are interested here in the $sl(2)$-breaking contributions to Eq. (\[Cexp\]) only, any $sl(2)$-invariant terms in $\mathbb{C}_2^{(1)}$ can be dropped. It is convenient to rewrite Eq. (\[C21\]) as follows: $$\begin{aligned}
\mathbb{C}_2^{(1)}=-
\big(S_{12}+S_{21}\big)\left(\kappa+\lambda_j \mathcal{H}\right)
+\ldots,\end{aligned}$$ where the ellipses stand for the $sl(2)$-invariant contributions and $$\begin{aligned}
S_{12}={z_{12}^{-1}}\partial_1 z^2_{12}\,, && S_{21}={z_{21}^{-1}}\partial_2z^2_{21}\,\end{aligned}$$ are intertwining operators[^3] $$\begin{aligned}
S_{12} T^{j=1}\otimes T^{j=1}=T^{j=3/2}\otimes
T^{j=1/2}S_{12}\,,\notag\\
S_{21} T^{j=1}\otimes T^{j=1}=T^{j=1/2}\otimes
T^{j=3/2}S_{21}\,.\end{aligned}$$ Thus we have to evaluate the following expression: $$\begin{aligned}
\mathcal{H}\mathbb{C}_2^{(1)}\mathcal{H}=
-\mathcal{H}\big(S_{12}+S_{21}\big)\left(\kappa+\lambda_j \mathcal{H}\right)\mathcal{H}\,.\end{aligned}$$ To this end, the following technique proves to be very efficient.
As the first step, consider the operators $$\begin{aligned}
\mathbb{W}_1=S_{12}\mathcal{H}\,, && \mathbb{W}_2 = S_{12}\mathcal{H}^2.\end{aligned}$$ They are, both, $sl(2)$-invariant operators that act on $T^{j=1}\otimes T^{j=1}\to T^{j=3/2}\otimes T^{j=1/2}$. The general form of such an operator is given by the following expression (see e.g. [@Braun:2009vc]): $$\begin{aligned}
[\mathbb{W}f](z_1,z_2)=\int d\alpha d\beta\, \frac{\beta}{\bar\beta}\,
w\left(\frac{\alpha\beta}{\bar\alpha\bar\beta}\right) f(z_{12}^\alpha, z_{21}^\beta)\end{aligned}$$ and the kernel $w(\tau)$ is completely determined by the spectrum $$\begin{aligned}
\mathbb{W}z_{12}^k = w_k z_{12}^k.\end{aligned}$$ By a direct calculation one finds for the operators in question $$\begin{aligned}
w_{1,k} = \frac{1}{k+1}\,, && w_{2,k} =\frac{1}{(k+2)(k+1)^2}\,.\end{aligned}$$ It is easy to check that the corresponding kernels are$$\begin{aligned}
w_{1,k} \mapsto w_1(\tau)=\delta(\tau)\,, && w_{2,k} \mapsto w_2(\tau)=1\,,\end{aligned}$$ so that we obtain $$\begin{aligned}
{}[\mathbb{W}_1f](z_1,z_2)=&\int_0^1 d\beta \ f(z_{1}, z_{21}^\beta)\,,
\notag\\
{}[\mathbb{W}_2f](z_1,z_2)=&\int_0^1 d\alpha \int_0^{\bar\alpha}d\beta\, \frac{\beta}{\bar\beta}\, f(z_{12}^\alpha, z_{21}^\beta).\end{aligned}$$ Next, since $S_{21}=P_{12}S_{12}P_{21}$ where $P_{12}$ is the permutation operator $z_1\leftrightarrow z_2$, and $[\mathcal{H},P_{12}]=0$, we can write $$\begin{aligned}
\mathcal{H}\mathbb{C}_2^{(1)}\mathcal{H}=-\kappa\, \mathbb{U}_1-\lambda_j\,\mathbb{U}_2\,,\end{aligned}$$ where $$\begin{aligned}
\mathbb{U}_k=\mathcal{H}\,\mathbb{W}_k+P_{12}\,\mathcal{H}\,\mathbb{W}_k\, P_{12}\,.\end{aligned}$$ One obtains after some algebra $$\begin{aligned}
{}[\mathbb{U}_k f](z_1,z_2)=\int_{0}^1\!d\alpha\int_0^{\bar\beta}\!\!d\beta\, u_k(\alpha,\beta)f(z_{12}^\alpha, z_{21}^\beta)\,,\end{aligned}$$ where[^4] $$\begin{aligned}
\label{u12}
u_1(\alpha,\beta)=&-\ln(1\!-\!\alpha\!-\!\beta)+\ldots\,,
\\
u_2(\alpha,\beta)=&\frac12\Big[\ln^2(1\!-\!\alpha\!-\!\beta)-\ln^2\bar\alpha-\ln^2\bar\beta\Big]+\ldots.
\notag\end{aligned}$$ The ellipses stand for contributions that are functions of the invariant (conformal) ratio $ r =\alpha\beta/(\bar\alpha\bar\beta)$. They give rise to $sl(2)$-invariant contributions to $\mathbb{C}_2^{(1)}$ and can be dropped in the present context.
Collecting everything we obtain for the $sl(2)$-breaking part of the evolution kernel $$\begin{aligned}
\lefteqn{ [\Delta\mathbb{H}^{(2)}f](z_1,z_2)=}
\nonumber\\&=&
\lambda_j\int_0^1d\alpha\!\int_{0}^{\bar\alpha}\!\!d\beta \Big[2\kappa \ln(1\!-\!\alpha\!-\!\beta)
- \lambda_j \ln^2(1\!-\!\alpha\!-\!\beta)
\nonumber\\&& \hspace*{2cm}
+ \lambda_j \ln^2\bar\alpha + \lambda_j \ln^2\bar\beta\Big]f(z_{12}^\alpha,z_{21}^\beta).\end{aligned}$$ We have checked that this expression coincides with the result of the direct calculation of the relevant Feynman diagrams.
The invariant part of the kernel (see \[app:two-loop\]) has the form $$\begin{aligned}
{}[\mathbb{H}_{inv}^{(2)}f](z_1,z_2)=\!\int_0^1\!d\alpha\int_{0}^{\bar\alpha}\!\!d\beta\, w(\alpha,\beta)
f(z_{12}^\alpha, z_{21}^\beta),\end{aligned}$$ where $$\begin{aligned}
w(\alpha,\beta)=&
4\left[\frac12\nu_j-\lambda_j^2-\lambda_j\frac{n^2-4}{24n}\right]\ln\left(1-\frac{\alpha\beta}{\bar\alpha\bar\beta}\right)
\notag\\
&+\frac{\lambda_j}{3n}\!\left[92-5n^2-(n^2-16)\ln\left(\frac{\alpha\beta}{\bar\alpha\bar\beta}\right)\right].\end{aligned}$$ Explicit expressions for the isotopic $su(n)$ factors $\lambda_j$ and $\nu_j$ are given in Eqs. (\[lambdas\]) and (\[nus\]), respectively. It can be checked that the anomalous dimensions $\gamma_{j=1,k=2}$ and $\gamma_{j=2,k=1}$ vanish as they should, since the corresponding operators are the energy momentum tensor and isotopic current, respectively.
Summary
=======
We have studied implications of exact conformal invariance of scalar quantum field theories at the critical point in non-integer dimensions for the evolution kernels of the light-ray operators. The possibility to make this connection is based on the observation that in $\text{MS}$-like schemes the evolution kernels (anomalous dimensions) do not depend on the space-time dimension. Thus all expressions derived in the $d$-dimensional (conformal) theory remain exactly the same for the theory in integer dimensions. We demonstrate that all conformal symmetry constraints for the twist-two light-ray operators are encoded in the form of the generators of the so-called collinear $sl(2)$ subgroup. Two of them, $S_-$ and $S_0$, can be fixed at all loops in terms of the evolution kernel, while the generator of special conformal transformations, $S_+$, receives nontrivial corrections which can only be calculated order by order in perturbation theory. Provided that the generator $S_+$ is known at the $\ell-1$ loop order, one can determine the evolution kernel to the $\ell$-loop accuracy up to terms that are invariant with respect to the tree-level generators. The invariant parts can eventually be restored from the anomalous dimensions. This procedure is advantageous as compared to a direct calculation because the calculation of the anomalous dimensions is, as a rule, considerably simpler than of the full evolution kernel in general (non-forward) kinematics.
The method suggested in this work is similar to the approach of D. Müller who was the first to use conformal constraints to determine the form of the renormalized operators to the next-to-leading order (NLO) [@Mueller:1991gd]. Our technique seems, however, to be better suited for dealing with nonlocal light-ray operators in coordinate representation. We demonstrated its efficiency by restoring the evolution kernels for twist-two operators in two toy models: $O(n)$ symmetric $\varphi^4$ theory to the three-loop accuracy and in the matrix $\varphi^3$ model to two loops.
We have calculated the two-loop correction to the operator of special conformal transformations, $S_+$, in the $\varphi^4$ theory and observed that it form deviates from the “natural” ansatz (\[Conj84\]). Thus the form of a conformal operator suggested in Ref. [@Brodsky:1984xk] does not hold beyond the NLO even in scalar theories.
We expect that the same technique can be applied to gauge theories and in particular to QCD. The QCD beta function vanishes for large number of flavors for the critical value of the coupling $\alpha_s$ in the $d=4-2\epsilon$ dimensions. As a consequence, correlation functions of gauge-invariant operators are scale-invariant at the critical point. It is believed that QCD correlation functions at the critical point have to be invariant under conformal transformations as well, although, to our knowledge, this statement has not been rigorously proven (or disproved) so far.
A.M. is grateful to Dieter Müller and Sergey Derkachov for helpful discussions. This work was supported by the DFG, grant BR2021/5-2.
The generator of special conformal transformation in the $\varphi^4$-theory to the two-loop accuracy {#app:B}
====================================================================================================
{width="85.00000%"}
In this Appendix we calculate the deformation $\Delta S_+$ of the generator of special conformal transformations in the $\varphi^4$ theory to the $\mathcal{O}(\epsilon^2)$ accuracy. To this end we need to evaluate the CWI (\[CWI2\]) in the two-loop approximation.
The starting observation is that to the required accuracy the operator $S_+(u)$ (\[S+exp\]) turns out to be finite in the $\varphi^4$ theory: $$\begin{aligned}
\label{Su1}
S_+(u)=& S_+^{(0)}-\epsilon(z_1+z_2)-\frac{u}{2}[\mathbb{H}^{(1)},(z_1+z_2)]
\notag\\
&
-\frac{u^2}{4}[\mathbb{H}^{(2)},(z_1+z_2)]+\mathcal{O}(u^3)\,.\end{aligned}$$ Indeed, making use of the identities in Eqs. (\[HE2\]), (\[HwHw\]) one can simplify the divergent terms in last two lines in Eq. (\[S+exp\]) to a single term $\sim \mathbb{H}^{(1)}[\mathbb{H}^{(1)},z_1+z_2]$ which happens to be zero in the $\varphi^4$ theory. This implies, in particular, that the two-point 1PI Green function $\delta\Gamma_2(\underline{z},\underline{p})$ (\[DeltaSC\]) is finite to the two-loop accuracy as well.
A diagrammatic representation for $\delta\Gamma_2(\underline{z},\underline{p})$ is shown in Fig. \[fig:2\]. Taking into account that the one- and two-loop diagrams enter the expansion with the factors $M^{2\epsilon}$ and $M^{4\epsilon}$, respectively, one gets $$\begin{aligned}
\delta\Gamma_2(\underline{z},\underline{p})&=&-i(\bar
n,\partial_{p_1}+\partial_{p_2})M\partial_M\Gamma_2(\underline{z},\underline{p})
\nonumber\\&&{}
+\Delta\Gamma_2(\underline{z},\underline{p})+\mathcal{O}(u^3)\,,
\label{twoterms}\end{aligned}$$ where $\Gamma_2(\underline{z},\underline{p})$ is the usual 1PI Green function (\[Gamma2\]) and the extra term $\Delta\Gamma_2(\underline{z},\underline{p})$ stands for the sum of two diagrams in the last square brackets in the second line in Fig. \[fig:2\].
Using the RG-equation we can rewrite $$\begin{aligned}
M\partial_M\Gamma_2(\underline{z},\underline{p})=-\Big(\beta(u)\partial_u+\mathbb{H}-2\gamma_\varphi\Big)\Gamma_2(\underline{z},\underline{p})\,.\end{aligned}$$ The Green function $\Gamma_2$ on the r.h.s. of this equation can be taken in the one-loop approximation. Explicit calculation gives $$\begin{aligned}
\Gamma^{(1)}_2(\underline{z},\underline{p})=\Big(1- u \zeta_n\,\mathbb{K}(p_1,p_2)\Big)\Gamma^{(0)}_2(\underline{z},\underline{p})\end{aligned}$$ where, cf. Eq.(\[c-coef\]), $$\begin{aligned}
\zeta_n = \{n+2,\,2,\, 0\} = 3 c_n\end{aligned}$$ for the scalar, symmetric traceless and antisymmetric operators, respectively, and $\Gamma^{(0)}_2(\underline{z},\underline{p})$ is the tree-level 1PI Green function. Finally, $\mathbb{K}(p_1,p_2)$ is an integral operator that can be written as follows: $$\begin{aligned}
[\mathbb{K}(p_1,p_2) f](z_1,z_2)=
\int_0^1 \!d\alpha\, K(p_1+p_2,\alpha) f(z_{12}^\alpha,z_{12}^\alpha)\,,\end{aligned}$$ with $$\begin{aligned}
K(p,\alpha)=\frac{1}{6\epsilon}
\Big[\Gamma(1+\epsilon)\left(\frac{\alpha\bar\alpha p^2}{4\pi M^2}\right)^{-\epsilon}-1\Big]\,.\end{aligned}$$ Omitting the terms in the $\beta-$function (which vanish at the critical point) one obtains for the contribution in the first line in Eq. (\[twoterms\]): $$\begin{aligned}
\label{BB}
\lefteqn{
-i(\bar n,\partial_{p_1}+\partial_{p_2})M\partial_M\Gamma_2(\underline{z},\underline{p}) =
}
\nonumber\\&=&
\Big(\mathbb{H}-2\gamma_\varphi\Big)i(\bar
n,\partial_{p_1}+\partial_{p_2})\Gamma^{(1)}_2(\underline{z},\underline{p})
\nonumber\\&=&
\Big(\mathbb{H}-2\gamma_\varphi\Big)\biggl\{-u\zeta_n \Big[i(\bar
n,\partial_{p_1}+\partial_{p_2})\mathbb{K}(p_1,p_2)\Big]
\nonumber\\&&{}
+ \Big[1-u\zeta_n \mathbb{K}(p_1,p_2)\Big]i(\bar
n,\partial_{p_1}\!+\!\partial_{p_2})\!\biggr\}\Gamma^{(0)}_2(\underline{z},\underline{p})\,.\end{aligned}$$ Neglecting in this expression terms of higher order than $\mathcal{O}(u^2)$ (note that in the $\varphi^4$ theory $\gamma_\varphi\sim O(u^2)$) and taking into account that $$\begin{aligned}
i(\bar n,\partial_{p_1}+\partial_{p_2})\Gamma^{(0)}_2(\underline{z},\underline{p})~=~
-(n\bar n)(z_1+z_2)\Gamma^{(0)}_2(\underline{z},\underline{p})\,
\notag\end{aligned}$$ and $$\begin{aligned}
\mathbb{H}^{(1)}\mathbb{K}(p_1,p_2)(z_1+z_2)~=~\mathbb{H}^{(1)}(z_1+z_2)\mathbb{K}(p_1,p_2)
\notag\end{aligned}$$ one obtains $$\begin{aligned}
\label{BBB}
\lefteqn{
-i(\bar n,\partial_{p_1}+\partial_{p_2})M\partial_M\Gamma_2(\underline{z},\underline{p}) =
}
\nonumber\\&=&
(n\bar n) \Big[2(z_1+z_2)\gamma_\varphi-\mathbb{H}\,(z_1+z_2)\Big] \Gamma^{(0+1)}_2(\underline{z},\underline{p})
\notag\\&&{}
-u^2\zeta_n \mathbb{H}^{(1)}\Big[i(\bar n,\partial_{p_1}\!+\!\partial_{p_2})\mathbb{K}(p_1,p_2)\Big]
\Gamma^{(0)}_2(\underline{z},\underline{p})\,.\end{aligned}$$ Using explicit expression for the one-loop kernel $\mathbb{H}^{(1)}$ (\[H1(4)\]) the contribution in the last line in Eq. (\[BBB\]) can be simplified to $$\begin{aligned}
\label{}
\Delta' \Gamma_2(\underline{z},\underline{p})=
\frac23 u^2\zeta_n \gamma(\epsilon)\frac{i(\bar n, p_1+p_2)}{(p_1\!+\!p_2)^{2(1+\epsilon)}}\,
\mathbb{H}^{(1)}\Gamma^{(0)}_2(\underline{z},\underline{p})\,,\end{aligned}$$ where $\gamma(\epsilon)=\Gamma(1+\epsilon)\Gamma^2(1-\epsilon)/\Gamma(2-2\epsilon)$. It can be checked that this term is canceled by the remaining contribution $\Delta \Gamma_2(\underline{z},\underline{p})$ in Eq. (\[twoterms\]) up to terms $\mathcal{O}(u^2\epsilon)$: $$\begin{aligned}
\Delta' \Gamma_2(\underline{z},\underline{p})+\Delta \Gamma_2(\underline{z},\underline{p})= \mathcal{O}(u^2\epsilon).\end{aligned}$$ Thus one obtains for $\delta \Gamma_2$ at the critical point with the two-loop accuracy $$\begin{aligned}
\delta\Gamma_2(\underline{z},\underline{p})&=&
(n\bar n) \Big[2(z_1\!+\!z_2)\gamma^*_\varphi
-\mathbb{H}(u_*)(z_1\!+\!z_2)\Big]\Gamma_2(\underline{z},\underline{p})
\nonumber\\&&{}+\mathcal{O}(\epsilon^3)\,.\end{aligned}$$ Collecting all terms and going over to the coordinate space representation [^5] one gets for the CWI (\[CWI2\]) $$\begin{aligned}
\Big(K^{x_1}_-(\widetilde \Delta_\varphi)+K^{x_2}_-(\widetilde \Delta_\varphi)
+2(n\bar n) S_+ \Big) \Gamma_2(\underline{z},\underline{x})=\mathcal{O}(\epsilon^3)\,,\end{aligned}$$ where the $S_+$ generator takes the form (\[S+NL\]).
{width="75.00000%"}
$su(n)$ projectors {#app:C}
==================
A decomposition of the tensor product of two adjoint representations of the $su(n)$ group contains seven irreducible representations. The projectors onto the scalar and two adjoint representations have the form $$\begin{aligned}
(P_{1})^{ab}_{a'b'}=&~\frac{1}{n^2-1}\delta^{ab}_{a'b'}\,,
&&
(P_2)^{ab}_{a'b'}=\frac1{n}f^{abc}\, f^{a'b'c}\,,
\notag\\
(P_3)^{ab}_{a'b'}=&~\frac{n}{n^2-4}d^{abc}\,d^{a'b'c}\,.\end{aligned}$$ In addition, we define the projectors $P_4, P_5$ onto symmetric tensors and the projectors $P_6, P_7$ onto antisymmetric tensors of rank two: $$\begin{aligned}
\label{}
P_4=S_+ \Pi_S\,, &&P_5=S_- \Pi_S\,, \notag\\
P_6=A_+ \Pi_A\,, &&P_7=A_- \Pi_A\,,\end{aligned}$$ where $$\begin{aligned}
\Pi_S&=&\frac12\left(1\!+\!\mathbb{P}\right)\!-\!P_1\!-\!P_3\,,
\quad
\Pi_A~=~\frac12\left(1-\mathbb{P}\right)-P_2\,,
\nonumber\\
S_\pm&=&\frac1{2n}\Big({n\pm 2}\pm n\,\mathbb{R}\Big)\,,
\quad\,
A_{\pm}~=~\frac12\Big(\II\pm i \mathbb{K}\Big),\end{aligned}$$ $\mathbb{P}$ is the permutation operator, $\mathbb{P}^{ab}_{a'b'}=\delta^{a}_{b'}\delta^{b}_{a'}$, and $$\begin{aligned}
\mathbb{R}^{ab}_{a'b'}=d^{aa'c}d^{bb'c}\,, &&
\mathbb{K}^{ab}_{a'b'}=d^{aca'}f^{bcb'}\,.\end{aligned}$$ The invariant operator $\mathbb{R}$ which arises in the calculation of the one-loop diagram in Fig. \[fig:1\] has the following eigenvalues on the invariant subspaces, $\mathbb{R}P_j=\lambda_j\, P_j$, with $$\begin{aligned}
\label{lambdas}
&\lambda_1=\frac{n^2-4}{n}\,, &&\lambda_2=\frac{n^2-4}{2n}\,, &&\lambda_3=\frac{n^2-12}{2n}
\notag\\
&\lambda_4=1-\frac{2}n\,,&&\lambda_5=-1-\frac{2}n\,, &&\lambda_{6,7}=-\frac2{n}\,.\end{aligned}$$ For completeness we give the dimensions of the corresponding subspaces: $$\begin{aligned}
&\text{dim}V_0=~1\\
&\text{dim}V_{2,3}=~n^2-1,\\
&\text{dim} V_{4(5)}=~n^2(n\pm3)(n\mp 1)/4,\\
&\text{dim} V_{6,7}=~(n^2-1)(n^2-4)/4\,.\end{aligned}$$ One can check that $\sum_j \lambda_j\text{dim}V_j=0$ that follows from $\operatorname{tr}\mathbb{R}=0$.
Two-loop evolution kernel in the $\varphi^3$ theory {#app:two-loop}
===================================================
In this Appendix we collect contributions of individual two-loop diagrams to the renormalization of light-ray operators $\mathcal{O}_j(z_1,z_2)$ in the $\varphi^3$ theory. The relevant diagrams are shown in Fig. \[fig:3\] where in all cases we imply that the symmetric diagrams are added. Let $\Gamma_j^{(a)}$ be a divergent part of the diagram after subtraction of divergent subgraphs. The results can be presented in the form $$\begin{aligned}
\Gamma_j^{(a)}=u^2\int_0^1d \alpha\int_0^{\bar\alpha} d\beta \chi_j^{(a)}(\alpha,\beta)
\mathcal{O}_j(z_{12}^\alpha,z_{21}^\beta)\,.\end{aligned}$$ We obtain the following expressions:
$\bullet~$ Self-energy insertions, Fig. \[fig:3\]a,b: $$\begin{aligned}
\chi_j^{(SE)}(\alpha,\beta)&=&\lambda_j\frac{n^2-4}{24n}\biggl\{\frac3{\epsilon^2}-\frac1\epsilon
\Big[2\ln(1-\alpha-\beta)
\notag\\&&{}
+8-\ln\left(\frac{\bar\alpha\bar\beta}{\alpha\beta}-1\right)\Big]\biggr\}.\end{aligned}$$ $\bullet~$ Vertex correction, Fig. \[fig:3\]c: $$\begin{aligned}
\chi_j^{(V)}(\alpha,\beta)&=&\lambda_j\frac{n^2-12}{8n}\biggl\{-\frac2{\epsilon^2}+\frac1\epsilon
\Big[2\ln(1-\alpha-\beta)
\notag\\&&{}
+6+\ln\left(\frac{\alpha\beta}{\bar\alpha\bar\beta}\right)\Big]\biggr\}.\end{aligned}$$ $\bullet~$ Ladder diagram, Fig. \[fig:3\]d: $$\begin{aligned}
\chi_j^{(L)}(\alpha,\beta)&=&\frac12\lambda^2_j\biggl\{\left(\frac1{\epsilon^2}+\frac2{\epsilon}\right)
\ln\left(1-\frac{\alpha\beta}{\bar\alpha\bar\beta}\right)
\notag\\&+&
\frac1{2\epsilon}\Big[\ln^2(1\!-\!\alpha\!-\!\beta)-\ln^2\bar\alpha-\ln^2\bar\beta\Big]\biggr\}.\end{aligned}$$ $\bullet~$ Crossed diagram, Fig. \[fig:3\]e: $$\chi_j^{(C)}(\alpha,\beta)=-\nu_j\frac1{2\epsilon}\ln\left(1-\frac{\alpha\beta}{\bar\alpha\bar\beta}\right).$$ Here $$\begin{aligned}
\label{nus}
&\nu_2=\nu_6=\nu_7=0\,,
\notag\\
&\nu_1= {(n^2-4)(n^2-12)}/{(2n^2)}\,,
\notag\\
&\nu_3=- {4(n^2-10)}/{n^2}\,,
\notag\\
&\nu_{4(5)}= \pm{(n\mp2)(n^2\pm4n-8)}/{(2n^2)}\,.
$$
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[^1]: Indeed, $S_0^{(0)}$ counts the canonical dimension on an object
[^2]: Here and below we use a generic notation $\mathcal{H}^{(a)}$ for the $sl(2)$-invariant and $\mathcal{V}^{(a)}$ for the $sl(2)$-breaking kernels
[^3]: These relations follow readily from the intertwining relations for the generators $$z_{12}\,\left(S^{(j)}_{1,\alpha}+S^{(j)}_{2,\alpha}\right)=\left(S^{(j-1/2)}_{1,\alpha}+S^{(j-1/2)}_{2,\alpha}\right)\,z_{12}$$ and $\partial_z S^{j=0}=S^{j=1}\partial_z$. Both are easy to check.
[^4]: The full expression for $u_2$ involves the dilogarithm function and can be brought to the the form in Eq. (\[u12\]) using the pentagon identity for $\operatorname{Li}_2$.
[^5]: Since $\gamma_\varphi^*\sim \epsilon^2$ it is sufficient to use the tree level Green function $\Gamma^{(0)}_2(\underline{z},\underline{x})\sim \delta(x_1-z_1n)\delta(x_2-z_2 n)$ in the term $\sim \gamma_\varphi^*\Gamma_2(\underline{z},\underline{x})$. Hence one can replace in this contribution $\gamma_\varphi^*(z_1n+z_2n,\bar n)\to \gamma_{\varphi}^*(x_1+x_2,\bar n)$ and absorb it in the redefinition of $K_{\bar n}$ generators.
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------------------------------------------------------------------------
<span style="font-variant:small-caps;">A Markov Process Approach to the asymptotic Theory of abstract Cauchy Problems driven by Poisson Processes</span>
by
<span style="font-variant:small-caps;">Alexander Nerlich[^1][^2][^3][^4]</span>
------------------------------------------------------------------------
[ABSTRACT]{}
In this paper, we employ Markov process theory to prove asymptotic results for a class of stochastic processes which arise as solutions of a stochastic evolution inclusion and are given by the representation formula $$\begin{aligned}
\mathbb{X}_{x}(t)=\sum \limits_{m=0}\limits^{\infty}T((t-\alpha_{m})_{+})(\text{\scalebox{0.62}{$\mathbb{X}$}}_{x,m})1\hspace{-0,9ex}1_{[\alpha_{m},\alpha_{m+1})}(t),\end{aligned}$$ where $(T(t))_{t \geq 0}$ is a (nonlinear) time-continuous, contractive semigroup acting on a separable Banach space $(V,||\cdot||_{V})$, $(\alpha_{m})_{m \in \mathbb{N}}$ is the sequence of arrival times of a homogeneous Poisson process, $x$ is a $V$-valued random variable and $(\text{\scalebox{0.62}{$\mathbb{X}$}}_{x,m})_{m \in \mathbb{N}}$ is a recursively defined sequence of $V$-valued random variables, fulfilling $\text{\scalebox{0.62}{$\mathbb{X}$}}_{x,0}=x$.\
It will be demonstrated that $\mathbb{X}_{x}$ is, under some distributional assumptions on the involved random variables, a time-continuous Markov process and that it obeys, under polynomial decay conditions on $T$, the strong law of large numbers (SLLN) and, if the decay rate is sufficiently fast, also the central limit theorem (CLT). Finally, we consider two examples: A nonlinear ordinary differential equation and the (weighted) $p$-Laplacian evolution equation for $p \in (2,\infty)$.\
**Mathematical Subject Classification (2010).** 60J25, 47J35, 60H15, 35B40, 60F05, 60F15\
**Keywords.** Markov Processes, Nonlinear evolution equation, Stochastic evolution inclusion, Pure jump noise, Asymptotic results, Strong law of large numbers, Central limit theorem, Weighted p-Laplacian evolution equation
Introduction
============
The theory of Markov processes is a beautiful tool to gain asymptotic results for stochastic processes. Particularly in the area of SPDEs, this theory enables one to get profound insights into the long time behavior of the SPDE’s solution. In this paper, we apply Markov process theory to a class of stochastic processes which arise as solutions of abstract Cauchy problems driven by Poisson processes.\
Let us embark on the endeavor ahead of us by rigorously describing the processes considered here: To this end, let $(V,||\cdot||_{V})$ be a separable Banach space and let $(T(t))_{t\geq 0}$ be a time-continuous, contractive semigroup on $V$. Moreover, let $(\Omega,{\mathcal{F}},{\mathbb{P}})$ be a complete probability space, let $(\eta_{m})_{m \in \mathbb{N}}$ and $(\beta_{m})_{m \in \mathbb{N}}$ be i.i.d. sequences that are independent of each other; where the former consists of $V$-valued random variables and the latter of $(0,\infty)$-valued, exponentially distributed random variables. Moreover, we coin the term “independent initial” as a $V$-valued random variable $x$ which is independent of $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}})$ and introduce, for any independent initial $x$, the recursively defined sequence $({\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m})_{m\in \mathbb{N}_{0}}$, by ${\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,0}:=x$ and ${\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m}:=T(\beta_{m}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m-1}+\eta_{m}$ for all $m \in \mathbb{N}$. Now, introduce ${\mathbb{X}}_{x}:[0,\infty)\times \Omega \rightarrow V$ by ${\mathbb{X}}_{x}(t):= \sum \limits_{m=0}\limits^{\infty}T((t-\alpha_{m})_{+})({\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m}){1\hspace{-0,9ex}1}_{[\alpha_{m},\alpha_{m+1})}(t)$ for all $t \geq 0$, where $\alpha_{0}:=0$ and $\alpha_{m}:=\sum \limits_{k=1}\limits^{m}\beta_{k}$. Finally, set $N(t):=\sum \limits_{m=0}\limits^{\infty}m{1\hspace{-0,9ex}1}\{ \alpha_{m}\leq t<\alpha_{m+1} \}$, then $(N(t))_{t \geq 0}$ is a Poisson process and we have ${\mathbb{X}}_{x}(t)=T(t-\alpha_{N(t)}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t)}$, for all $t \geq 0$, almost surely.\
Processes like $({\mathbb{X}}_{x}(t))_{t \geq 0}$ arise as solutions of abstract Cauchy problems driven by Poisson processes, see [@ich2] for more details; particularly [@ich2 Remark 3.14]. Moreover, there is also an intuitive interpretation of the phenomena that are modeled by $({\mathbb{X}}_{x}(t))_{t \geq 0}$: Assume $(T(t))_{t \geq 0}$ describes the time-change of some physical process, which is a reasonable assumption, since nonlinear semigroups arise naturally as solutions of evolution equations. Now assume that this physical process is at each of the succeeding times $\alpha_{m}$ exposed to the shock $\eta_{m}$. Then, the process describing the shocked system is $({\mathbb{X}}_{x}(t))_{t\geq 0}$. More concretely: The (solution of the) $p$-Laplacian equation we consider as an example, can be used to model the evolution of a hill that consists mostly out of sand, see [@birnirtheory]. In this case $T(t)u$ describes the hill’s surface at time $t$, $u$ is the hill’s initial shape and the $\eta_{k}$’s then could be rain showers, or storms, etc.\
Now, let us describe this paper’s highlights as well as the techniques employed to prove them: Firstly, $({\mathbb{X}}_{x}(t))_{t \geq 0}$ is (w.r.t. the completion of its natural filtration and any independent initial $x$) a time-continuous Markov process. For proving this, it is crucial that $(\beta_{m})_{m \in \mathbb{N}}$ is not any i.i.d. sequence, but one consisting of exponentially distributed random variables. Moreover, due to the contractivity and time-continuity of $(T(t))_{t \geq 0}$, the transition semigroup of $({\mathbb{X}}_{x}(t))_{t \geq 0}$ has the e-property and the Feller property.\
For proving these results, we only need the assumptions that have been stated in this introduction so far. But, obtaining more sophisticated results requires the following decay assumption on $(T(t))_{t \geq 0}$: There is a w.r.t. $(T(t))_{t\geq 0}$ invariant, separable and dense sub-Banach space $(W,||\cdot||_{W})\subseteq V$, with continuous injection, such that there are constants $\kappa,\rho \in (0,\infty)$ such that $$\begin{aligned}
\label{intro_eqbound}
||T(t)w_{1}-T(t)w_{2}||_{W}\leq \left(\kappa t+||w_{1}-w_{2}||_{W}^{-\frac{1}{\rho}}\right)^{-\rho},~\forall t \in [0,\infty)\end{aligned}$$ and $w_{1},w_{2}\in W$. Moreover, we have to assume that $||\eta_{k}||_{V}\in L^{2}(\Omega)$ and that $T(t)0=0$ for all $t \in [0,\infty)$. The latter is due to the nonlinearity indeed not necessarily true, but it is “usually” easily verified whether it holds.\
As we shall see, (\[intro\_eqbound\]) enables us to derive upper bounds for $||{\mathbb{X}}_{x}(t)||_{V}$ and $||{\mathbb{X}}_{x}(t)-{\mathbb{X}}_{x}(t)||_{V}$. These bounds, together with the e-property allow us to conclude by the aid of the results in [@Szarek], that the transition function of $({\mathbb{X}}_{x}(t))_{t \geq 0}$ possesses a unique invariant probability measure $\bar{\mu}:{\mathfrak{B}}(V)\rightarrow [0,1]$. From there, we infer that $$\begin{aligned}
\tag{SLLN}
\label{intro_slln}
\lim \limits_{t \rightarrow \infty } \frac{1}{t} \int \limits_{0} \limits^{t} \psi ({\mathbb{X}}_{x}(\tau)) d\tau =\overline{(\psi)}:= \int \limits_{V}\psi(v)\bar{\mu}(dv),\end{aligned}$$ with probability one, for any Lipschitz continuous $\psi:V \rightarrow {\mathbb{R}}$ and any independent initial $x$. Once this is achieved we will employ the results in [@Holzmann] to prove that: If, in addition the constant $\rho$ appearing in (\[intro\_eqbound\]) fulfills $\rho > \frac{1}{2}$, then there is a $\sigma^{2}(\psi) \in [0,\infty)$ such that $$\begin{aligned}
\tag{CLT}
\label{intro_clt}
\lim \limits_{t \rightarrow \infty }\frac{1}{\sqrt{t}}\left( \int \limits_{0}\limits^{t}\psi({\mathbb{X}}_{x}(\tau))d\tau-t\overline{(\psi)}\right)=Y\sim N(0,\sigma^{2}(\psi)) ,\end{aligned}$$ in distribution, for any Lipschitz continuous $\psi:V \rightarrow {\mathbb{R}}$ and any independent initial $x$.\
Finally, we will illustrate the applicability of these results with two examples: In the first one $(T(t))_{t \geq 0}$ is the semigroup of solutions of a first order, nonlinear ODE. Consequently, in this case $(T(t))_{t \geq0}$ is a semigroup on ${\mathbb{R}}$. By aid of this example, we will demonstrate that (\[intro\_clt\]) can fail if (\[intro\_eqbound\]) only holds for a $\rho\in (0,\frac{1}{2}]$; particularly, even if $\rho=\frac{1}{2}$.\
In our second (more sophisticated) example, $(T(t))_{t\geq 0}$ acts on an infinite dimensional Banach space and is the semigroup of strong solutions of the weighted $p$-Laplacian evolution equation with Neumann boundary conditions for large $p$, i.e. $p \in (2,\infty)$. We will prove that in this case, (\[intro\_slln\]) holds for any $p \in (2,\infty)$ and (\[intro\_clt\]) holds if $p \in (2,4)$.\
Results analogous to (\[intro\_slln\]) and (\[intro\_clt\]) are proven in [@ich2]. But there it is assumed that the involved semigroup fulfills a finite extinction assumption and not a polynomial decay assumption. Polynomial decay and finite extinction are probably the most common asymptotic behaviors exhibited by nonlinear semigroups. Even though the i.i.d.-splitting method employed in [@ich2] and the Markov process technique used in this paper have essentially nothing in common, one can consider these two papers as natural complements of each other. Particularly, the example considered in [@ich2] is the $p$-Laplacian evolution equation for ”small” $p$.\
The results proven in the current paper mainly rely on [@Szarek] and [@Holzmann]. Of course there are many other general criteria dealing with ergodicity as well as the SLLN and the CLT for Markov processes. Particularly interesting criteria can be found in the book [@Kulik].\
Finally, let us briefly outline this paper’s structure: Section \[sec\_notaprel\] clarifies this paper’s notation, states some basic results with a focus on nonlinear semigroups and concludes with some elementary properties of $({\mathbb{X}}_{x}(t))_{t\geq 0}$ - for technical conveniences the results in Section \[sec\_notaprel\] are formulated without any distributional assumptions on the involved random variables. We then proceed in Section \[sec\_mp\] by proving that $({\mathbb{X}}_{x}(t))_{t\geq 0}$ is a time-homogeneous Markov process and demonstrate that it possesses, among others, the Feller and the e-property. Section \[sec\_sllnclt\] is this section’s centerpiece, it is proven there that the transition function of $({\mathbb{X}}_{x}(t))_{t\geq 0}$ possesses a unique invariant probability measure (Proposition \[prop\_uniqueinvpropmeas\]) that it fulfills the SLLN (Theorem \[theorem\_slln\]) as well as the CLT (Theorem \[theorem\_clt\]). Finally, in Section \[sec\_examples\] we start with a general differential inequality result useful to prove (\[intro\_eqbound\]), then consider the nonlinear ODE example (Remark \[remarkex1\]) and devote the remainder of this section to the $p$-Laplacian example.
Notation and preliminary Results {#sec_notaprel}
================================
This section starts with some remarks regarding nonlinear semigroups, proceeds with some general words on the functional analytic and probability theoretic notations used throughout this paper and concludes with some results regarding the stochastic processes considered in this paper.\
Throughout this section $(V,||\cdot||_{V})$ denotes a separable (real) Banach space and ${\mathfrak{B}}(V)$ its Borel $\sigma$-Algebra.\
A family of mappings $(T(t))_{t \geq 0}$, where $T(t):V\rightarrow V$ is called a semigroup on $V$, if $T(0)v=v$ and $T(t+h)v=T(t)T(h)v$ for all $t,h \in [0,\infty)$ and $v \in V$. A semigroup $(T(t))_{t \geq 0}$ on $V$ is called
1. time-continuous, if $[0,\infty) \ni t \mapsto T(t)v$ is a continuous map for all $v \in V$,
2. contractive, if $||T(t)v_{1}-T(t)v_{2}||_{V}\leq ||v_{1}-v_{2}||_{V}$ for all $t \in [0,\infty)$ and $v_{1},~v_{2}\in V$,
3. linear, if the mapping $T(t):V \rightarrow V$ is linear for all $t \in [0,\infty)$.
Particularly in the linear theory, one frequently uses the phrase “strongly continuous” instead of “time continuous” semigroup and implicitly assumes that a (linear) strongly continuous semigroup consists of linear and continuous operators. Of course, using our notation a linear, time-continuous semigroup does not necessarily consist of continuous operators. Therefore, to avoid confusion, we chose to use “time continuous” instead of “strongly continuous.”
\[remark\_measuc0sg\] Let $(T(t))_{t \geq 0}$ be a time-continuous and contractive semigroup on $V$. Then it is easily verified that $T$ is also jointly continuous, i.e. $[0,\infty) \times V \ni (t,v)\mapsto T(t)v$ is a continuous map. Consequently, this map is a fortiori $\mathfrak{B}([0,\infty)\times V)$-$\mathfrak{B}(V)$-measurable. Moreover, by separability we have $\mathfrak{B}([0,\infty)\times V)= \mathfrak{B}([0,\infty))\otimes \mathfrak{B}(V)$, see [@Billingsley page 244]; which gives that this map is $\mathfrak{B}([0,\infty))\otimes \mathfrak{B}( V)$-$\mathfrak{B}(V)$-measurable.
The following remark gives the connection between nonlinear semigroups and nonlinear evolution equations. It is not needed for the Sections \[sec\_mp\] and \[sec\_sllnclt\], but for the $p$-Laplacian example considered in Section \[sec\_examples\]. It is stated now, as it reveals the significance of nonlinear semigroups and therefore motivates why we consider them.
\[remark\_msee\] A mapping $\mathcal{A}:V\rightarrow 2^{V}$ is called multi-valued operator and $D(\mathcal{A}):=\{v\in V: \mathcal{A}v\neq \emptyset\}$ is called its domain. ${\mathcal{A}}$ is single-valued if $\mathcal{A}v$ contains precisely one element for all $v\in D(\mathcal{A})$. Moreover, instead of $\mathcal{A}:V\rightarrow 2^{V}$ we may write $\mathcal{A}:D(\mathcal{A})\rightarrow 2^{V}$. In addition, by identifying ${\mathcal{A}}$ with its graph $G({\mathcal{A}}):=\{(v,\hat{v}):~v \in D({\mathcal{A}}),~\hat{v}\in {\mathcal{A}}v\}$ we may write $(v,\hat{v})\in {\mathcal{A}}$ instead of $v \in D({\mathcal{A}})$ an $\hat{v}\in {\mathcal{A}}v$. Furthermore, ${\mathcal{A}}:D({\mathcal{A}})\rightarrow 2^{V}$ is called accretive, if $||v_{1}-v_{2}||_{V}\leq ||v_{1}-v_{2}+\alpha(\hat{v}_{1}-\hat{v}_{2})||_{V}$ for all $\alpha>0$, $(v_{1},\hat{v}_{1}),(v_{2},\hat{v}_{2}) \in {\mathcal{A}}$; m-accretive, if it is accretive and $Range(Id+\alpha{\mathcal{A}})=V$, for all $\alpha>0$; and densely defined if $\overline{D({\mathcal{A}})}=V$.\
Moreover, we have the following celebrated result connection nonlinear semigroups and evolution equations: Let $\mathcal{A}:V\rightarrow 2^{V}$ be densely defined and m-accretive. Then, the initial value problem $$\begin{aligned}
\label{remark_eqms}
0 \in u^{\prime}(t)+\mathcal{A}u(t),~\text{for a.e. }t\in (0,\infty),~u(0)=v,
\end{aligned}$$ has precisely one mild solution, see [@BenilanBook Definition 1.3] and [@BenilanBook Prop. 3.7]. Moreover, the family of mappings $(T_{{\mathcal{A}}}(t))_{t \geq 0}$ such that $T_{{\mathcal{A}}}(\cdot)v$ is, for each $v \in V$, the mild solution of (\[remark\_eqms\]) forms a time-continuous, contractive semigroup on $V$, see [@BenilanBook Theorems 1.10 and 3.10], and will be called “the semigroup associated to ${\mathcal{A}}$”.
The reader is referred to [@BenilanBook] for a comprehensive introduction to nonlinear semigroups. Moreover, the book [@acmbook] deals with existence, uniqueness, asymptotic and qualitative results for numerous evolution equations and this book’s appendix contains a more concise introduction to this topic.\
Even though we also consider linear semigroups, namely the semigroup associated to a Markov process, no profound knowledge of linear semigroups is required to understand this paper.\
Given a measure space $(K,\Sigma,\nu)$, we denote by $L^{q}(K,\Sigma,\nu)$, where $q \in [1,\infty]$, the usual Lebesgue spaces of ($\nu$-equivalence classes of) real-valued, $\Sigma$-${\mathfrak{B}}(\mathbb{R})$-measurable functions $f:K \rightarrow \mathbb{R}$, such that: $|f|^{q}$ is Lebesgue integrable, if $q \neq \infty$; $\nu$-essentially bounded, if $q= \infty$.\
Moreover, we introduce the spaces $\text{BM}(V)$, $C_{b}(V)$, $\text{Lip}_{b}(V)$ and $\text{Lip}(V)$ as the spaces of all functions $\psi:V\rightarrow \mathbb{R}$ which are bounded and measurable, continuous and bounded, Lipschitz continuous and bounded, and Lipschitz continuous, respectively. Moreover, for any Lipschitz continuous function $\psi$, we denote its Lipschitz constant by $L_{\psi}$.\
Throughout everything which follows $(\Omega,{\mathcal{F}},{\mathbb{P}})$ denotes a complete probability space. Moreover, we introduce the short cut notation $L^{q}(\Omega,{\mathcal{F}},{\mathbb{P}}):=L^{q}(\Omega)$ for all $q \in [1,\infty)$. In addition, $\mathcal{M}(\Omega;V)$ denotes the space of $V$-valued random variables, i.e. all ${\mathcal{F}}$-${\mathfrak{B}}(V)$-measurable mappings $Y:\Omega \rightarrow V$.\
If $Y_{i}$ is a $V_{i}$-valued random variable for each $i \in I$, where $I$ is an arbitrary index set and the $V_{i}$’s are separable Banach spaces, then $\sigma(Y_{j};j \in I)\subseteq {\mathcal{F}}$ is the smallest $\sigma$-Algebra, such that each $Y_{i}$ is $\sigma(Y_{j};j \in I)-{\mathfrak{B}}(V_{i})$-measurable. In addition, $\sigma_{0}(Y_{j};j \in I)$ denotes its completion, i.e. $$\begin{aligned}
\sigma_{0}(Y_{j};j \in I):=\{A \in {\mathcal{F}}:~ \exists B \in \sigma(Y_{j};j \in I)\text{, such that } {\mathbb{P}}(A \Delta B)=0 \},\end{aligned}$$ where $\Delta$ denotes the symmetric difference. It is easily verified that the right-hand-side of the previous equation is indeed a $\sigma$-Algebra and the smallest one containing all ${\mathbb{P}}$-null-sets as well as all elements of $\sigma(Y_{j};j \in I)$. Moreover, it is well known that an $Y \in \mathcal{M}(\Omega;V)$ is independent of a $\sigma$-algebra, if and only if it is independent of the $\sigma$-algebra’s completion.\
Finally, for any $Y \in \mathcal{M}(\Omega;V)$, we denote by ${\mathbb{P}}_{Y}$ its law, i.e. ${\mathbb{P}}_{Y}(B):={\mathbb{P}}(Y\in B)$, for all $B \in {\mathfrak{B}}(V)$.\
Even though we mostly consider real-valued functionals of vector-valued processes, some results on random variables (and stochastic processes) taking values in Banach spaces are needed in the sequel. For a concise introduction, see [@SIBS Chapter 2].\
Now, let us spend some words on the stochastic process which is the central object of this paper:
\[def\_proc\] Let $(\beta_{m})_{m \in \mathbb{N}}$, where $\beta_{m}:\Omega \rightarrow (0,\infty)$, be a sequence of real-valued random variables. Moreover, let $(\eta_{m})_{m \in \mathbb{N}}\subseteq\mathcal{M}(\Omega;V)$, introduce $\alpha_{m}:=\sum \limits_{k=1}\limits^{m}\beta_{k}$ for all $m \in \mathbb{N}$ and set $\alpha_{0}:=0$. Finally, let $x \in \mathcal{M}(\Omega;V)$ and let $(T(t))_{t \geq 0}$ be a time-continuous, contractive semigroup on $V$. Then the sequence $({\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m})_{m\in \mathbb{N}_{0}}$ defined by ${\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,0}:=x$ and $$\begin{aligned}
{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m}:=T(\alpha_{m}-\alpha_{m-1}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m-1}+\eta_{m}=T(\beta_{m}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m-1}+\eta_{m},~\forall m \in \mathbb{N},\end{aligned}$$ is called the sequence generated by $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}},x,T)$ in $V$. Moreover, the stochastic process ${\mathbb{X}}_{x}:[0,\infty)\times \Omega \rightarrow V$ defined by $$\begin{aligned}
{\mathbb{X}}_{x}(t):= \sum \limits_{m=0}\limits^{\infty}T((t-\alpha_{m})_{+})({\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m}){1\hspace{-0,9ex}1}_{[\alpha_{m},\alpha_{m+1})}(t),~\forall t\geq 0,\end{aligned}$$ is called the process generated by $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}},x,T)$ in $V$.
\[remark\_Xmeascad\] Let $({\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m})_{m\in \mathbb{N}_{0}}$ and ${\mathbb{X}}_{x}:[0,\infty)\times \Omega \rightarrow V$ be the sequence and the process generated by some $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}},x,T)$ in $V$. Then it follows easily from Remark \[remark\_measuc0sg\] that each ${\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m}$ and each ${\mathbb{X}}_{x}(t)$ is ${\mathcal{F}}$-${\mathfrak{B}}(V)$-measurable.
Let us conclude this section with the following lemma, which reveals that the stochastic process generated by some $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}},x,T)$ in $V$, depends continuously on $(\eta_{m})_{m \in \mathbb{N}}$ and $x$.
\[lemma\_xcont\] Let $(\beta_{m})_{m \in \mathbb{N}}$, where $\beta_{m}:\Omega \rightarrow (0,\infty)$, be a sequence of real-valued random variables. Moreover, let $(\eta_{m})_{m \in \mathbb{N}},~(\hat{\eta}_{m})_{m \in \mathbb{N}}\subseteq\mathcal{M}(\Omega;V)$ and $x,~\hat{x} \in \mathcal{M}(\Omega;V)$. In addition, introduce $\alpha_{m}:=\sum \limits_{k=1}\limits^{m}\beta_{k}$ for all $m \in \mathbb{N}$, set $\alpha_{0}:=0$ and define $N(t):= \sum \limits_{m=0}\limits^{\infty}m{1\hspace{-0,9ex}1}\{ \alpha_{m}\leq t<\alpha_{m+1} \}$ for all $t \in [0,\infty)$. Finally, let ${\mathbb{X}}_{x}$ and $\hat{{\mathbb{X}}}_{\hat{x}}$ be the processes generated by $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}},x,T)$ in $V$ and $((\beta_{m})_{m \in \mathbb{N}},(\hat{\eta}_{m})_{m \in \mathbb{N}},\hat{x},T)$ in $V$, respectively; where $(T(t))_{t \geq 0}$ is a time-continuous, contractive semigroup on $V$. Then the assertion $$\begin{aligned}
\label{lemma_twoprocessboundeq}
||{\mathbb{X}}_{x}(t)-\hat{{\mathbb{X}}}_{\hat{x}}(t)||_{V}\leq ||x-\hat{x}||_{V}+\sum \limits_{k=1}\limits^{N(t)}||\eta_{k}-\hat{\eta}_{k}||_{V},~\forall t \in [0,\infty)\end{aligned}$$ holds on $\Omega$.
Let $t \in [0,\infty)$ be given and set $M_{t}:=\{\omega \in \Omega: t< \sup \limits_{m \in \mathbb{N}} \alpha_{m}(\omega) \}$. Note that ${\mathbb{X}}_{x}(t)=\hat{{\mathbb{X}}}_{\hat{x}}(t)=0$ on $\Omega \setminus M_{t}$, thus (\[lemma\_twoprocessboundeq\]) holds on $\Omega \setminus M_{t}$. Moreover, on $M_{t}$ we have by contractivity of $(T(t))_{t \geq 0}$ that $$\begin{aligned}
||{\mathbb{X}}_{x}(t)-\hat{{\mathbb{X}}}_{\hat{x}}(t)||_{V} = ||T(t-\alpha_{N(t)}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t)}-T(t-\alpha_{N(t)})\hat{{\text{\scalebox{0.62}{$\mathbb{X}$}}}}_{\hat{x},N(t)}||_{V}\leq ||{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t)}-\hat{{\text{\scalebox{0.62}{$\mathbb{X}$}}}}_{\hat{x},N(t)}||_{V},\end{aligned}$$ where $({\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m})_{m\in \mathbb{N}_{0}}$ and $(\hat{{\text{\scalebox{0.62}{$\mathbb{X}$}}}}_{\hat{x},m})_{m\in \mathbb{N}_{0}}$ denote the sequences generated by $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}},x,T)$ in $V$ and $((\beta_{m})_{m \in \mathbb{N}},(\hat{\eta}_{m})_{m \in \mathbb{N}},\hat{x},T)$ in $V$, respectively. Consequently, it suffices to prove that $$\begin{aligned}
||{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m}-\hat{{\text{\scalebox{0.62}{$\mathbb{X}$}}}}_{\hat{x},m}||_{V} \leq ||x-\hat{x}||_{V}+\sum \limits_{k=1}\limits^{m}||\eta_{k}-\hat{\eta}_{k}||_{V},~\forall m \in \mathbb{N}_{0}.\end{aligned}$$ For $m=0$ this is clear and for $m \in \mathbb{N}_{0}$ we get $||{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m+1}-\hat{{\text{\scalebox{0.62}{$\mathbb{X}$}}}}_{,\hat{x},m+1}||_{V} \leq ||{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m}-\hat{{\text{\scalebox{0.62}{$\mathbb{X}$}}}}_{\hat{x},m}||_{V}+||\eta_{m+1}-\hat{\eta}_{m+1}||_{V}$, which yields the claim by induction.
The Markov Property {#sec_mp}
===================
Throughout this section, $(V,||\cdot||_{V})$ is a separable Banach space and $(\eta_{m})_{m \in \mathbb{N}} \subseteq \mathcal{M}(\Omega;V)$ denotes an i.i.d. sequence. In addition $(\beta_{m})_{m \in \mathbb{N}}$, where $\beta_{m}:\Omega \rightarrow (0,\infty)$, is an i.i.d. sequence of exponentially distributed random variables, with parameter $\theta \in (0,\infty)$. Furthermore, we assume that $(\eta_{m})_{m \in \mathbb{N}}$ and $(\beta_{m})_{m \in \mathbb{N}}$ are independent of each other.\
Now, set $\alpha_{m}:= \sum \limits_{k=1}\limits^{m}\beta_{k}$ for all $m \in \mathbb{N}$, introduce $\alpha_{0}:=0$ and $N: [0,\infty)\times \Omega \rightarrow \mathbb{N}_{0}$ by $$\begin{aligned}
\label{eq_poissonp}
N(t):= \sum \limits_{m=0}\limits^{\infty}m{1\hspace{-0,9ex}1}\{ \alpha_{m}\leq t<\alpha_{m+1} \},~\forall t \in [0,\infty).\end{aligned}$$ Moreover, let $(T(t))_{t \geq 0}$ be a time-continuous, contractive semigroup on $V$.\
An $x \in \mathcal{M}(\Omega;V)$ is called an independent initial, if $x$ is jointly independent of $(\eta_{m})_{m \in \mathbb{N}}$ and $(\beta_{m})_{m \in \mathbb{N}}$. Moreover, for any independent initial $x \in \mathcal{M}(\Omega;V)$ we denote by $({\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m})_{m\in \mathbb{N}_{0}}$ the sequence and by ${\mathbb{X}}_{x}:[0,\infty)\times \Omega \rightarrow V$ the process generated by $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}},x,T)$ in $V$. Moreover, we may identify a $v \in V$ with the random variable which is constantly $v$ and note that any $v \in V$ is obviously an independent initial. In addition, we introduce the filtration $({\mathcal{F}}^{x}_{t})_{t \geq 0}$, by ${\mathcal{F}}^{x}_{t}:= \sigma_{0}({\mathbb{X}}_{x}(\tau);~\tau \in [0,t])$, for all $t \in [0,\infty)$ and any independent initial $x \in \mathcal{M}(\Omega;V)$. Finally, let $P:[0,\infty)\times V \times {\mathfrak{B}}(V)\rightarrow [0,1]$ be defined by $$\begin{aligned}
\label{eq_pdef}
P(t,v,B):={\mathbb{P}}({\mathbb{X}}_{v}(t)\in B)={\mathbb{P}}_{X_{v}(t)}(B),\end{aligned}$$ for all $v \in V$, $t \in [0,\infty)$ and $B \in {\mathfrak{B}}(V)$.\
The purpose of this section is to show that ${\mathbb{X}}_{x}$ is for any independent initial $x \in \mathcal{M}(\Omega;V)$ a time homogeneous Markov process with transition function $P$ and initial distribution ${\mathbb{P}}_{x}$. In addition, we will establish some basic properties of these quantities.\
Before embarking on theses tasks, let us clarify the following: Throughout this entire section, we do not assume that $(T(t))_{t \geq 0}$ exhibits the introductory mentioned (or any other) decay behavior. Consequently, one can also apply this section’s results under possibly different decay assumptions on the involved semigroup.
\[remark\_xrepresentation\] Let $x \in \mathcal{M}(\Omega;V)$ be an independent initial. Appealing to the strong law of large numbers yields $\lim \limits_{m \rightarrow \infty}\alpha_{m}=\infty$ almost surely. Consequently, on a set $\tilde{\Omega}\in {\mathcal{F}}$ of full ${\mathbb{P}}$-measure, we can find for each $\omega \in \tilde{\Omega}$ and $t \in [0,\infty)$ precisely one $m \in \mathbb{N}$, s.t. $t \in [\alpha_{m}(\omega),\alpha_{m+1}(\omega))$. Thus, we get $$\begin{aligned}
{\mathbb{P}}\left({\mathbb{X}}_{x}(t)=T(t-\alpha_{N(t)}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t)},~\forall t \geq 0\right)=1.\end{aligned}$$ In addition, it is well known that $(N(t))_{t\geq 0}$ is a homogeneous Poisson process with intensity $\theta$.
\[lemma\_meas\] The mapping defined by $[0,\infty)\times V \times \Omega \ni (t,v,\omega)\mapsto {\mathbb{X}}_{v}(t,\omega)$ is ${\mathfrak{B}}([0,\infty))\otimes {\mathfrak{B}}(V)\otimes {\mathcal{F}}$-${\mathfrak{B}}(V)$-measurable. Consequently, if $\psi\in \text{BM}(V)$, then $[0,\infty)\times V \ni (t,v) \mapsto \mathbb{E}\psi({\mathbb{X}}_{v}(t))$ is ${\mathfrak{B}}([0,\infty))\otimes {\mathfrak{B}}(V)$-${\mathfrak{B}}({\mathbb{R}})$-measurable
For the moment, let $v \in V$ and $\omega \in \Omega$ be fixed. If, for a given $t \in [0,\infty)$, we have $t \geq \sup \limits_{m \in \mathbb{N}}\alpha_{m}(\omega)$, then the same holds for $t+h$, for any $h\geq 0$. Thus, we get ${\mathbb{X}}_{v}(t+h,\omega)={\mathbb{X}}_{v}(t,\omega)=0$. Moreover, if $t < \sup \limits_{m \in \mathbb{N}}\alpha_{m}(\omega)$, we have $N(t+h,\omega)=N(t,\omega)$ for all $h\geq 0$ sufficiently small. Consequently, the time-continuity of $T$ yields that $[0,\infty)\ni t \mapsto {\mathbb{X}}_{v}(t,\omega)$ is right-continuous for all $v \in V$ and $\omega \in \Omega$. Consequently, as each ${\mathbb{X}}_{v}(t)$ is ${\mathcal{F}}$-${\mathfrak{B}}(V)$-measurable we get that ${\mathbb{X}}_{v}$ is ${\mathfrak{B}}([0,\infty))\otimes {\mathcal{F}}$-${\mathfrak{B}}(V)$-measurable, see [@SIBS Prop. 2.2.3.ii]. In addition, appealing to Lemma \[lemma\_xcont\] yields that $V \ni v\mapsto {\mathbb{X}}_{v}(t,\omega)$ is continuous for all $t \in [0,\infty)$ and $\omega \in \Omega$. Consequently, $[0,\infty)\times V \times \Omega \ni (t,v,\omega)\mapsto {\mathbb{X}}_{v}(t,\omega)$ is $ {\mathfrak{B}}([0,\infty))\otimes {\mathfrak{B}}(V)\otimes {\mathcal{F}}$-${\mathfrak{B}}(V)$-measurable, by [@IDA Lemma 4.51].\
Finally, let $\psi \in \text{BM}(V)$, then the boundedness of $\psi$ yields that the expectation at hand exists; and the already proven measurability result (together with [@SIBS Prop. 2.1.4]) enables us to conclude the remaining claim.
\[theorem\_mp\] Let $x \in \mathcal{M}(\Omega;V)$ be an independent initial. Then $({\mathbb{X}}_{x}(t))_{t \geq 0}$ is a Markov process with respect to $({\mathcal{F}}_{t}^{x})_{t \geq 0}$, i.e. $$\begin{aligned}
\label{theorem_mpeq1}
{\mathbb{P}}({\mathbb{X}}_{x}(t+h)\in B|{\mathcal{F}}_{t}^{x})={\mathbb{P}}({\mathbb{X}}_{x}(t+h)\in B|{\mathbb{X}}_{x}(t)),\end{aligned}$$ almost surely, for all $t,h \in [0,\infty)$ and $B \in {\mathfrak{B}}(V)$. In addition, $P$ is a time homogeneous transition function, that is
i) \[theorem\_mpenumi1\] ${\mathfrak{B}}(V)\ni B \mapsto P(t,v,B)$ is a probability measure on $(V,{\mathfrak{B}}(V))$, for all $t \in [0,\infty)$ and $v \in V$,
ii) \[theorem\_mpenumi2\] ${\mathbb{P}}(0,v,B)={1\hspace{-0,9ex}1}_{B}(v)$ for all $v \in V$ and $B \in {\mathfrak{B}}(V)$,
iii) \[theorem\_mpenumi3\] $P(\cdot,\cdot,B)$ is ${\mathfrak{B}}([0,\infty))\otimes {\mathfrak{B}}(V)$-${\mathfrak{B}}([0,1])$-measurable for any $B \in {\mathfrak{B}}(V)$ and
iv) \[theorem\_mpenumi4\] $P$ fulfills has the Chapman-Kolmogorov property, i.e. $P(t+h,v,B)=\int \limits_{V} P(h,\hat{v},B)dP(t,v,d\hat{v})$ for all $t,h \in [0,\infty)$, $v \in V$ and $B \in {\mathfrak{B}}(V)$.
Moreover, $({\mathbb{X}}_{x}(t))_{t \geq 0}$ is time homogeneous (with initial distribution ${\mathbb{P}}_{x}$) and transition function $P$, i.e. $$\begin{aligned}
\label{theorem_mpeq2}
{\mathbb{P}}({\mathbb{X}}_{x}(t+h)\in B|{\mathcal{F}}_{t}^{x})=P(h,{\mathbb{X}}_{x}(t),B),\end{aligned}$$ almost surely, for all $t,h \in [0,\infty)$ and $B \in {\mathfrak{B}}(V)$.
The assertions \[theorem\_mpenumi1\]) and \[theorem\_mpenumi2\]) are trivial. Moreover, the third follows from Lemma \[lemma\_meas\].\
Proving the remaining assertions is more involved and will occupy us for some time. Let us start with some preparatory observations. To this end, let $t,h \in [0,\infty)$, $v \in V$ and $B \in {\mathfrak{B}}(V)$ be given; and introduce $F_{m}:V\times [0,\infty)^{m}\times V^{m}\rightarrow V$, for all $m \in \mathbb{N}$, by $F_{1}(y,b,n):=T(b)y+n$ and $F_{m}(y,b_{1},..,b_{m},n_{1},..,n_{m}):=T(b_{m})F_{m-1}(y,b_{1},..,b_{m-1},n_{1},..,n_{m-1})+n_{m}$ for all $y,n,n_{1},..,n_{m}\in V$, $b_{1},..,b_{m}\in [0,\infty)$ and $m \in \mathbb{N}\setminus \{1\}$.\
Appealing to Remark \[remark\_measuc0sg\] yields that $F_{1}$ is continuous and it then follows inductively that each $F_{m}$ has this property and is therefore ${\mathfrak{B}}(V)\otimes {\mathfrak{B}}([0,\infty)^{m})\otimes {\mathfrak{B}}(V^{m})$-${\mathfrak{B}}(V)$-measurable.\
Now, for the sake of space let $\hat{\eta}_{\tau,m}:=(\eta_{N(\tau)+1},..,\eta_{N(\tau)+m})$, for all $m \in \mathbb{N}$, $\tau \in [0,\infty)$ and$\hat{\beta}_{\tau,m}:=(\alpha_{N(\tau)+1}-\tau,\beta_{N(\tau)+2},..,\beta_{N(\tau)+m})$ if $m \geq 2$ and $\hat{\beta}_{\tau,1}:=\alpha_{N(\tau)+1}-\tau$ for all $\tau \in [0,\infty)$ and let us prove inductively that $$\begin{aligned}
\label{theorem_mpprroof1}
{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(\tau)+m}=F_{m}({\mathbb{X}}_{x}(\tau),\hat{\beta}_{\tau,m},\hat{\eta}_{\tau,m}),~\forall \tau \in [0,\infty),\end{aligned}$$ almost surely for all $m \in \mathbb{N}$.\
If $m=1$, we get by the semigroup property and Remark \[remark\_xrepresentation\] that $$\begin{aligned}
{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(\tau)+1}=T(\alpha_{N(\tau)+1}-\tau){\mathbb{X}}_{x}(\tau)+\eta_{N(\tau)+1}=F_{1}({\mathbb{X}}_{x}(\tau),\alpha_{N(\tau)+1}-\tau,\eta_{N(\tau)+1})\end{aligned}$$ almost surely. Moreover, if (\[theorem\_mpprroof1\]) holds for an $m \in \mathbb{N}$ we get $$\begin{aligned}
{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(\tau)+m+1}
& = & ~ T(\beta_{N(\tau)+m+1}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(\tau)+m}+\eta_{N(\tau)+m+1} \\
& = & ~ T(\beta_{N(\tau)+m+1})F_{m}({\mathbb{X}}_{x}(\tau),\hat{\beta}_{\tau,m},\hat{\eta}_{\tau,m})+\eta_{N(\tau)+m+1} \\
& = & ~ F_{m+1}({\mathbb{X}}_{x}(\tau),\hat{\beta}_{\tau,m},\beta_{N(\tau)+m+1},\hat{\eta}_{\tau,m},\eta_{N(\tau)+m+1}),\end{aligned}$$ which yields (\[theorem\_mpprroof1\]). Consequently, on $\{N(\tau+h)=N(\tau)\}$, we have $$\begin{aligned}
\label{theorem_mpprroof5}
{\mathbb{X}}_{x}(\tau+h)=T(\tau+h-\alpha_{N(\tau)}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(\tau)}=T(h){\mathbb{X}}_{x}(\tau),~\forall \tau,h\in [0,\infty).\end{aligned}$$ up-to a ${\mathbb{P}}$-null-set and on $\{N(\tau+h)=N(\tau)+m\}$, where $m \in \mathbb{N}$, we have $$\begin{aligned}
\label{theorem_mpprroof6}
{\mathbb{X}}_{x}(\tau+h)=T(\tau+h-\alpha_{N(\tau)+m})F_{m}({\mathbb{X}}_{x}(\tau),\hat{\beta}_{\tau,m},\hat{\eta}_{\tau,m}),~\forall \tau,h\in [0,\infty),\end{aligned}$$ up-to a ${\mathbb{P}}$-null-set. These two results will turn out to be useful to prove (\[theorem\_mpeq1\]) and (\[theorem\_mpeq2\]). But before we can do so some distribution results have to be established, namely
I) For all $m \in \mathbb{N}$, we have that $(\alpha_{N(t)+1}-t,..,\alpha_{N(t)+m}-t,\eta_{N(t)+1},..,\eta_{N(t)+m},N(t+h)-N(t))$ is in distribution equal to $(\alpha_{1},..,\alpha_{m},\eta_{1},..,\eta_{m},N(h))$.
II) For all $m \in \mathbb{N}$, we have that $(\alpha_{N(t)+1},\beta_{N(t)+2},..,\beta_{N(t)+m},\eta_{N(t)+1},..,\eta_{N(t)+m},N(t+h)-N(t))$ is independent of ${\mathcal{F}}^{x}_{t}$.
Proof of I). Let $z_{1},..,z_{m}\in [0,\infty)$, $B_{1},..,B_{m}\in {\mathfrak{B}}(V)$ and $C \subseteq \mathbb{N}_{0}$. Then, as $(\eta_{m})_{m \in \mathbb{N}}$ is i.i.d and independent of $(\alpha_{m})_{m\in \mathbb{N}}$ we get $$\begin{aligned}
& & ~
{\mathbb{P}}(\alpha_{N(t)+k}-t\leq z_{k},~\eta_{N(t)+k} \in B_{k},~k=1,..,m,~ N(t+h)-N(t)\in C) \\
& = & ~ {\mathbb{P}}(\eta_{k} \in B_{k},~k=1,..,m)\sum \limits_{j=0}\limits^{\infty}{\mathbb{P}}(\alpha_{j+k}-t\leq z_{k},~k=1,..,m,~ N(t+h)-N(t)\in C,~N(t)=j)\\
& = & ~ {\mathbb{P}}(\eta_{k} \in B_{k},~k=1,..,m){\mathbb{P}}(N(z_{k}+t)-N(t)\geq k,~k=1,..,m,~ N(t+h)-N(t)\in C)\end{aligned}$$ where the last equality follows from $$\begin{aligned}
\label{theorem_mpprroof4}
\{\alpha_{k}\leq \tau\}=\{N(\tau)\geq k\},~\forall \tau \in [0,\infty),~k \in \mathbb{N}_{0},\end{aligned}$$ up to a ${\mathbb{P}}$-null-set. Since $(N(t))_{t \geq 0}$ is a homogeneous Poisson process, it is now easily verified that the distribution of $(N(z_{1}+t)-N(t),..,N(z_{m}+t)-N(t),N(t+h)-N(t))$ is independent of $t$. Using this and (\[theorem\_mpprroof4\]) yields $$\begin{aligned}
{\mathbb{P}}(N(z_{k}+t)-N(t)\geq k,~k=1,..,m,~ N(t+h)-N(t)\in C) = {\mathbb{P}}(\alpha_{k}\leq z_{k},~k=1,..,m,~ N(h)\in C).\end{aligned}$$ Combining the preceding two calculations, while having in mind the independence of $(\eta_{m})_{m\in \mathbb{N}}$ and $(\alpha_{m})_{m\in \mathbb{N}}$ gives i).\
Proof of II). Since $\beta_{N(t)+k}=\alpha_{N(t)+k}-\alpha_{N(t)+k-1}$ for all $k \in \mathbb{N}\setminus\{1\}$, it suffices to prove that $(\alpha_{N(t)+1},..,\alpha_{N(t)+m},\eta_{N(t)+1},..,\eta_{N(t)+m},N(t+h)-N(t))$ is independent of ${\mathcal{F}}_{t}^{x}$. The latter is obviously true if $(\alpha_{N(t)+1}-t,..,\alpha_{N(t)+m}-t,\eta_{N(t)+1},..,\eta_{N(t)+m},N(t+h)-N(t))$ is independent of ${\mathcal{F}}_{t}^{x}$.\
Now introduce $\Sigma_{\tau}:=\sigma(A \cap B:~ A \in \Sigma^{N}_{\tau},~ B \in \sigma_{0}(\eta_{k},~k \in \mathbb{N}_{0}))$, for all $\tau \in [0,\infty)$, where $(\Sigma^{N}_{\tau})_{\tau \geq 0}$ denotes the completion of the natural filtration of $(N(\tau))_{\tau \geq 0}$ and $\eta_{0}:=x$ and let us prove that $$\begin{aligned}
\label{theorem_mpprroof2}
{\mathcal{F}}^{x}_{t} \subseteq \Sigma_{t} \text{ and } \eta_{N(t)+j} \text{ is } \Sigma_{t}-{\mathfrak{B}}(V)-\text{measurable for all }j \in \mathbb{N}.\end{aligned}$$ The second assertion is clearly true, since $$\begin{aligned}
\{\eta_{N(t)+j}\in B\}= \bigcup\limits_{k=0}\limits^{\infty}\{\eta_{k+j}\in B,~N(t)=k\} \in \Sigma_{t},~\forall B \in {\mathfrak{B}}(V).\end{aligned}$$ Now, note that $\Sigma_{t}$ contains by construction every ${\mathbb{P}}$-null-set. Consequently, the first assertion follows if ${\mathbb{X}}_{x}(s)$ is $\Sigma_{t}$-${\mathfrak{B}}(V)$-measurable, for each $s \in [0,t]$, which will be verified now: So let $s \in [0,t]$, then appealing to (\[theorem\_mpprroof5\]) and (\[theorem\_mpprroof6\]) (with $\tau=0$ and $h=s$ there) yields, for a given $B \in {\mathfrak{B}}(V)$, that $$\begin{aligned}
\{X_{x}(s)\in B\}=\{T(s)x \in B,N(s)=0\}\cup\left( \bigcup\limits_{n=1}\limits^{\infty}\{T(s-\alpha_{n})F_{n}(x,\beta_{1},..,\beta_{n},\eta_{1},..,\eta_{n})\in B,~N(s)=n\}\right),\end{aligned}$$ up to a ${\mathbb{P}}$-null-set. It is plain that the first set in the preceding equation is an element of $\Sigma_{t}$. Moreover, for each $k \in \{1,..,n\}$ and $z \in [0,\infty)$ we have $\{\alpha_{k}\leq z,~N(s)=n\}=\{N(z)\geq k, N(s)=n\}$. If $z \leq s$, this set is clearly in $\Sigma^{N}_{s}$ and if $z>s$, we have $N(z)\geq N(s)$, which gives $\{N(z)\geq k, N(s)=n\}=\{N(s)=n\}\in \Sigma^{N}_{s}$; thus in any case $\{\alpha_{k}\leq z,~N(s)=n\}\in \Sigma_{t}$. Consequently, as $\beta_{k}=\alpha_{k}-\alpha_{k-1}$, we obtain that $\beta_{k}{1\hspace{-0,9ex}1}_{\{N(s)=n\}}$ is $\Sigma_{t}$-${\mathfrak{B}}([0,\infty))$-measurable, for all $k=1,..,n$ and $n \in \mathbb{N}$.\
Hence, we get $$\begin{aligned}
& & ~
\{T(s-\alpha_{n})F_{n}(x,\beta_{1},..,\beta_{n},\eta_{1},..,\eta_{n})\in B,~N(s)=n\}\\
& = & ~ \{T(s-(\beta_{1}+..+\beta_{n}){1\hspace{-0,9ex}1}_{\{N(s)=n\}})F_{n}(x,\beta_{1}{1\hspace{-0,9ex}1}_{\{N(s)=n\}},..,\beta_{n}{1\hspace{-0,9ex}1}_{\{N(s)=n\}},\eta_{1},..,\eta_{n})\in B,~N(s)=n\},\end{aligned}$$ is in $\Sigma_{t}$, for all $n \in \mathbb{N}$ by the measurability of $F_{n}$ and $T$, which concludes the proof of (\[theorem\_mpprroof2\]).\
Now let $n \in \mathbb{N}$ $z_{1},..,z_{m}\in [0,\infty)$, $B_{1},..B_{m},D_{1},..,D_{n}\in {\mathfrak{B}}(V)$, $C \subseteq \mathbb{N}_{0}$ and $s_{1},..,s_{n}\in [0,t]$. As $(N(\tau))_{\tau \geq 0}$ is a Poisson process, it is clear that $(N(z_{1}+t)-N(t),..,N(z_{m}+t)-N(t),N(t+h)-N(t))$ is independent of $\Sigma_{t}^{N}$. Consequently, as $\sigma_{0}(\eta_{k},~k \in \mathbb{N}_{0})$ is independent of all $\Sigma^{N}_{\tau}$, for $\tau \in [0,\infty)$, we get that $(N(z_{1}+t)-N(t),..,N(z_{m}+t)-N(t),N(t+h)-N(t))$ is independent of $\Sigma_{t}$ and that this random vector’s distribution does not depend on $t$. Hence, employing (\[theorem\_mpprroof4\]) and (\[theorem\_mpprroof2\]) yields $$\begin{aligned}
& & ~
{\mathbb{P}}(\alpha_{N(t)+k}-t\leq z_{k},\eta_{N(t)+k}\in B_{k},k=1,..,m,~N(t+h)-N(t) \in C,~{\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n)\\
& = & ~ {\mathbb{P}}(N(z_{k}+t)-N(t)\geq k,\eta_{N(t)+k}\in B_{k},k=1,..,m,~N(t+h)-N(t) \in C,{\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n)\\
& = & ~ {\mathbb{P}}(N(z_{k})\geq k,k=1,..,m,~N(h) \in C){\mathbb{P}}(\eta_{N(t)+k}\in B_{k},k=1,..,m,~{\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n)\\
& = & ~ {\mathbb{P}}(\alpha_{k}\leq z_{k},~k=1,..,m,N(h) \in C){\mathbb{P}}(\eta_{N(t)+k}\in B_{k},k=1,..,m,~{\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n).\end{aligned}$$ Moreover, we have $$\begin{aligned}
& & ~
{\mathbb{P}}(\eta_{N(t)+k}\in B_{k},k=1,..,m,{\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n)\\
& = & ~ \sum \limits_{i=0}\limits^{\infty}\sum \limits_{i_{1},.,i_{n}=0}\limits^{i}{\mathbb{P}}(\eta_{i+k}\in B_{k},k=1,..,m,~ N(t)=i,N(s_{j})=i_{j},T(s_{j}-\alpha_{i_{j}}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,i_{j}}\in D_{j},j=1,..,n)\end{aligned}$$ Now it is easily verified that ${\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,i}$ is $\sigma(x,\beta_{1},..,\beta_{i},\eta_{1},..,\eta_{i})$ for any $i \in \mathbb{N}_{0}$. (For $i=0$ this is trivial, for $i \in \mathbb{N}$ this follows from (\[theorem\_mpprroof1\]) by putting $\tau=0$ there.)\
Consequently, since $i_{1},..,i_{n} \in \{0,..,i\}$ in the sum of the preceding calculation, we get by the imposed independence assumptions $$\begin{aligned}
{\mathbb{P}}(\eta_{N(t)+k}\in B_{k},k=1,..,m,{\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n)={\mathbb{P}}(\eta_{k}\in B_{k},k=1,..,m){\mathbb{P}}( {\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n).\end{aligned}$$ Finally, putting it all together while having in mind (I) yields $$\begin{aligned}
& & ~
{\mathbb{P}}(\alpha_{N(t)+k}-t\leq z_{k},\eta_{N(t)+k}\in B_{k},k=1,..,m,~N(t+h)-N(t) \in C,~{\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n)\\
& = & ~ {\mathbb{P}}(\alpha_{k}\leq z_{k},\eta_{k}\in B_{k}~k=1,..,m,N(h) \in C,){\mathbb{P}}( {\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n)\\
& = & ~ {\mathbb{P}}(\alpha_{N(t)+k}-t\leq z_{k},\eta_{N(t)+k}\in B_{k}~k=1,..,m,N(t+h)-N(t) \in C,){\mathbb{P}}( {\mathbb{X}}_{x}(s_{j})\in D_{j},j=1,..,n),\end{aligned}$$ which proves (II).\
Now (\[theorem\_mpeq1\]) and (\[theorem\_mpeq2\]) will be deduced from I)-II) as well as (\[theorem\_mpprroof5\]) and (\[theorem\_mpprroof6\]).\
Firstly, I) enables us to conclude that $(t+h-\alpha_{N(t)+m},\hat{\beta}_{t,m},\hat{\eta}_{t,m},N(t+h)-N(t))$ is in distribution equal to $(h-\alpha_{m},\hat{\beta}_{0,m},\hat{\eta}_{0,m},N(h))$, since $\beta_{N(t)+m}=(\alpha_{N(t)+m}-t)-(\alpha_{N(t)+m-1}-t)$, for all $m \in \mathbb{N}$.\
Now, thanks to II) and I) we can apply well known properties of conditional probabilities (cf. [@SOC Prop. 1.43]) in the following two calculations; where in the first line of the first calculation (\[theorem\_mpprroof5\]) and in the first line of the second one (\[theorem\_mpprroof6\]) is used. $$\begin{aligned}
\mathbb{E}({1\hspace{-0,9ex}1}_{B}({\mathbb{X}}_{x}(t+h)){1\hspace{-0,9ex}1}_{\{N(t+h)=N(t)\}}|{\mathcal{F}}_{t}^{x})(\omega)
& = & ~ \mathbb{E}({1\hspace{-0,9ex}1}_{B}(T(h){\mathbb{X}}_{x}(t)){1\hspace{-0,9ex}1}_{\{N(t+h)=N(t)\}}|{\mathcal{F}}_{t}^{x})(\omega)\\
& = & ~ \int \limits_{\Omega} {1\hspace{-0,9ex}1}_{B}(T(h){\mathbb{X}}_{x}(t,\omega)){1\hspace{-0,9ex}1}_{\{N(t+h)=N(t)\}}(\tilde{\omega}) {\mathbb{P}}(d\tilde{\omega})\\
& = & ~ \int \limits_{\Omega} {1\hspace{-0,9ex}1}_{B}(T(h){\mathbb{X}}_{x}(t,\omega)){1\hspace{-0,9ex}1}_{\{N(h)=0\}}(\tilde{\omega}) {\mathbb{P}}(d\tilde{\omega})\\\end{aligned}$$ and $$\begin{aligned}
& & ~
\mathbb{E}({1\hspace{-0,9ex}1}_{B}({\mathbb{X}}_{x}(t+h)){1\hspace{-0,9ex}1}_{\{N(t+h)=N(t)+m\}}|{\mathcal{F}}_{t}^{x})(\omega)\\
& = & ~\mathbb{E}({1\hspace{-0,9ex}1}_{B}(T(t+h-\alpha_{N(t)+m})F_{m}({\mathbb{X}}_{x}(t),\hat{\beta}_{t,m},\hat{\eta}_{t,m})){1\hspace{-0,9ex}1}_{\{N(t+h)=N(t)+m\}}|{\mathcal{F}}_{t}^{x})(\omega)\\
& = & ~ \int \limits_{\Omega} {1\hspace{-0,9ex}1}_{B}(T(t+h-\alpha_{N(t)+m}(\tilde{\omega}))F_{m}({\mathbb{X}}_{x}(t,\omega),\hat{\beta}_{t,m}(\tilde{\omega}),\hat{\eta}_{t,m}(\tilde{\omega}))){1\hspace{-0,9ex}1}_{\{N(t+h)=N(t)+m\}}(\tilde{\omega}) {\mathbb{P}}(d\tilde{\omega})\\
& = & ~ \int \limits_{\Omega} {1\hspace{-0,9ex}1}_{B}(T(h-\alpha_{m}(\tilde{\omega}))F_{m}({\mathbb{X}}_{x}(t,\omega),\hat{\beta}_{0,m}(\tilde{\omega}),\hat{\eta}_{0,m}(\tilde{\omega}))){1\hspace{-0,9ex}1}_{\{N(h)=m\}}(\tilde{\omega}) {\mathbb{P}}(d\tilde{\omega}),\end{aligned}$$ for all $m \in \mathbb{N}$ and ${\mathbb{P}}$-a.e. $\omega \in \Omega$.\
Moreover, for any $v \in V$, it is easily verified by induction that ${\text{\scalebox{0.62}{$\mathbb{X}$}}}_{v,m}=F_{m}(v,\hat{\beta}_{0,m},\hat{\eta}_{0,m})$ for all $m \in \mathbb{N}$ a.s. Consequently, we get $$\begin{aligned}
P(h,v,B)
& = & ~\sum \limits_{m=0}\limits^{\infty}\mathbb{E}\left({1\hspace{-0,9ex}1}_{B}(T(h-\alpha_{m}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{v,m}){1\hspace{-0,9ex}1}_{\{N(h)=m\}}\right)\\
& = & ~\mathbb{E}\left({1\hspace{-0,9ex}1}_{B}(T(h)v){1\hspace{-0,9ex}1}_{\{N(h)=0\}}\right)+\sum \limits_{m=1}\limits^{\infty}\mathbb{E}\left({1\hspace{-0,9ex}1}_{B}(T(h-\alpha_{m})F_{m}(v,\hat{\beta}_{0,m},\hat{\eta}_{0,m})){1\hspace{-0,9ex}1}_{\{N(h)=m\}}\right),\end{aligned}$$ for all $v \in V$. Hence, combining the preceding three calculations yields $$\begin{aligned}
{\mathbb{P}}(X_{x}(t+h)\in B|{\mathcal{F}}_{t}^{x})(\omega) = \sum \limits_{m=0}\limits^{\infty}\mathbb{E}({1\hspace{-0,9ex}1}_{B}({\mathbb{X}}_{x}(t+h)){1\hspace{-0,9ex}1}_{\{N(t+h)=N(t)+m\}}|{\mathcal{F}}_{t}^{x})(\omega)=P(h,{\mathbb{X}}_{x}(t,\omega),B)\end{aligned}$$ for ${\mathbb{P}}$-a.e. $\omega \in \Omega$, which proves (\[theorem\_mpeq2\]). Consequently, invoking \[theorem\_mpenumi3\]) gives that the random variable ${\mathbb{P}}(X_{x}(t+h)\in B|{\mathcal{F}}_{t}^{x})$ is (after a possible modification on a ${\mathbb{P}}$-null-set) $\sigma({\mathbb{X}}_{x}(t))$-${\mathfrak{B}}([0,1])$-measurable, which yields ${\mathbb{P}}({\mathbb{X}}_{x}(t+h)\in B|{\mathcal{F}}^{x}_{t}) =\mathbb{E} ({\mathbb{P}}({\mathbb{X}}_{x}(t+h)\in B|{\mathcal{F}}^{x}_{t})|{\mathbb{X}}_{x}(t))$ a.s., which implies (\[theorem\_mpeq1\]) by the tower property of conditional expectations.\
Finally, as $x$ was arbitrary, (\[theorem\_mpeq2\]) holds for all independent initials, which is well known to imply \[theorem\_mpenumi4\]) - for the sake of completeness: Appealing to (\[theorem\_mpeq2\]) yields $$\begin{aligned}
P(t+h,v,B)=\mathbb{E}({\mathbb{P}}({\mathbb{X}}_{v}(t+h)\in B|{\mathcal{F}}^{v}_{t}))= \mathbb{E}P(h,{\mathbb{X}}_{v}(t),B)=\int \limits_{V}, P(h,\hat{v},B)P(t,v,d\hat{v}),\end{aligned}$$ for all $v \in V$, where the equality of the third and the fourth expression follow from the change of measure formula for expectations, which also holds for vector-valued random variables, see [@Billingsley p. 25].
In the sequel $(Q(t))_{t \geq 0}$, where $Q(t):\text{BM}(V)\rightarrow \text{BM}(V)$, denotes the family of mappings, defined by $$\begin{aligned}
\label{rema_qdefeq}
(Q(t)\psi)(v):= \mathbb{E}\psi({\mathbb{X}}_{v}(t))= \int \limits_{V} \psi(\hat{v})P(t,v,d\hat{v}),\end{aligned}$$ for all $\psi\in \text{BM}(V)$, $v \in V$ and $t \in [0,\infty)$.
Now, this section concludes by deriving some basic properties of our Markov process. Particularly, the e-property established in the following lemma, opens the door to useful results which enable one to conclude that a (transition function of a) Markov process on a polish state space possesses a unique invariant probability measure, see [@Szarek] for more details.
\[lemma\_basicprop\] The family of mappings $(Q(t))_{t \geq 0}$ has the Feller and the e-property, that is
i) \[lemma\_basicpropmpenumi1\] Feller Property: $Q(t)\psi \in C_{b}(V)$ for all $\psi \in C_{b}(V)$.
ii) \[lemma\_basicpropmpenumi2\] e-property: For all $\psi \in \text{Lip}_{b}(V)$, $v \in V$ and $\varepsilon>0$, there is a $\delta>0$, such that for all $\hat{v}\in V$, with $||\hat{v}-v||_{V}<\delta$, we have $|(Q(t)\psi)(v)-(Q(t)\psi)(\hat{v})|<\varepsilon$ for all $t \geq 0$.
Moreover, the following assertions hold for any independent initial $x \in \mathcal{M}(\Omega;V)$.
iii) \[lemma\_basicpropmpenumi3\] $({\mathbb{X}}_{x}(t))_{t\geq 0}$ has almost surely càdlàg paths and is continuous in probability.
iv) \[lemma\_basicpropmpenumi4\] The mapping $[0,\infty) \ni t \mapsto \mathbb{E}\psi({\mathbb{X}}_{x}(t))$ is continuous, whenever $\psi \in C_{b}(V)$; in particular, $(Q(\cdot)\psi)(v)$ is continuous for all $\psi \in C_{b}(V)$ and $v \in V$.
v) \[lemma\_basicpropmpenumi5\] The filtration $({\mathcal{F}}^{x}_{t})_{t\geq 0}$ fulfills the usual conditions, i.e. it is complete and right right-continuous.
vi) \[lemma\_basicpropmpenumi6\] The stochastic process $({\mathbb{X}}_{x}(t))_{t\geq 0}$ is $({\mathcal{F}}^{x}_{t})_{t\geq 0}$-progressive.
The required boundedness in \[lemma\_basicpropmpenumi1\]) is plain and the desired continuity follows from Lemma \[lemma\_xcont\] and dominated convergence.\
Proof of \[lemma\_basicpropmpenumi2\]). Let $\psi \in \text{Lip}_{b}(V)$ and assume that it is not constantly zero, since the claim is trivial in this case. Moreover, let $v \in V$ and $\varepsilon>0$ be given and introduce $\delta:= \frac{\varepsilon}{2L_{\psi}}$. Then employing the services of Lemma \[lemma\_xcont\] once more yields $$\begin{aligned}
|(Q(t)\psi)(v)-(Q(t)\psi)(\hat{v})| \leq L_{\psi}\mathbb{E}||{\mathbb{X}}_{v}(t)-{\mathbb{X}}_{\hat{v}}(t)||_{V}\leq L_{\psi}||v-\hat{v}||_{V} < \varepsilon,~\forall t \geq 0\end{aligned}$$ for all $\hat{v}\in V$, with $||v-\hat{v}||_{V}<\delta$, which proves \[lemma\_basicpropmpenumi2\]).\
Proof of \[lemma\_basicpropmpenumi3\]). It follows analogously to the beginning of the proof of Lemma \[lemma\_meas\] that $$\begin{aligned}
\label{lemma_basicpropeq1}
[0,\infty) \ni t \mapsto {\mathbb{X}}_{x}(t,\omega) \text{ is right continuous for all }\omega \in \Omega.\end{aligned}$$ Moreover, it is easily verified that the left limits exists on the set $\tilde{\Omega}:=\{\omega \in \Omega:~\lim \limits_{m \rightarrow \infty }\alpha_{m}(\omega)=\infty\}$, which is by the strong law of large numbers a set of full ${\mathbb{P}}$-measure. Consequently, it remains to prove the continuity in probability, which is in light of (\[lemma\_basicpropeq1\]) true, if ${\mathbb{X}}_{x}$ is left-continuous in probability. So let $t_{0}\in (0,\infty)$, $t \in [0,t_{0}]$ and $\varepsilon>0$ and note that $$\begin{aligned}
{\mathbb{P}}(||{\mathbb{X}}_{x}(t)-{\mathbb{X}}_{x}(t_{0})||_{V}>\varepsilon)\leq {\mathbb{P}}(||{\mathbb{X}}_{x}(t)-{\mathbb{X}}_{x}(t_{0})||_{V}>\varepsilon,~N(t)=N(t_{0}))+ P(|N(t)-N(t_{0})|\geq 1).\end{aligned}$$ Moreover, the contractivity of $(T(t))_{t\geq 0}$ yields $$\begin{aligned}
|| {\mathbb{X}}_{x}(t)-{\mathbb{X}}_{x}(t_{0})||_{V}
& = & ~ ||T(t-\alpha_{N(t_{0})}){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t_{0})}-T(t-\alpha_{N(t_{0})})T(t_{0}-t){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t_{0})}||_{V}\\
& \leq & ~ ||{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t_{0})}-T(t_{0}-t){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t_{0})}||_{V}\\\end{aligned}$$ on $\{N(t)=N(t_{0})\}$, up-to a ${\mathbb{P}}$-null-set. Conclusively, as Poisson processes are well-known to be stochastically continuous and as $(T(t))_{t\geq 0}$ is time-continuous, we get $$\begin{aligned}
\lim \limits_{t \nearrow t_{0}}{\mathbb{P}}(||{\mathbb{X}}_{x}(t)-{\mathbb{X}}_{x}(t_{0})||_{V}>\varepsilon)\leq \lim \limits_{t \nearrow t_{0}} {\mathbb{P}}(||{\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t_{0})}-T(t_{0}-t){\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,N(t_{0})}||_{V}>\varepsilon,~N(t)=N(t_{0})) = 0,\end{aligned}$$ which proves \[lemma\_basicpropmpenumi3\]).\
Proof of \[lemma\_basicpropmpenumi4\]). Let $(t_{m})_{m \in \mathbb{N}}$ be converging to a given $t \in [0,\infty)$. Then, we get by \[lemma\_basicpropmpenumi3\]) (and by passing to a subsequence if necessary) that $\lim \limits_{m \rightarrow \infty}||{\mathbb{X}}_{x}(t_{m})-{\mathbb{X}}_{x}(t)||_{V}=0$ almost surely. Consequently, $\lim \limits_{m \rightarrow \infty} \psi({\mathbb{X}}_{x}(t_{m}))=\psi({\mathbb{X}}_{x}(t))$ a.s. and by dominated convergence also in $L^{1}(\Omega)$, which gives \[lemma\_basicpropmpenumi4\]).\
Finally, the desired completeness in \[lemma\_basicpropmpenumi5\]) holds by construction, the right-continuity follows from [@Doob Theorem, p. 556], which is indeed applicable due to \[lemma\_basicpropmpenumi1\]) and (\[lemma\_basicpropeq1\]); and \[lemma\_basicpropmpenumi6\]) follows from \[lemma\_basicpropmpenumi5\]) and (\[lemma\_basicpropeq1\]) by [@SIBS Prop. 2.2.3].
The SLLN and the CLT {#sec_sllnclt}
====================
Let the notations of the previous section prevail, that means: $(V,||\cdot||_{V})$ is a separable Banach space and $((\eta_{m})_{m \in \mathbb{N}},(\beta_{m})_{m \in \mathbb{N}},T)$ is a fixed triplet, where $(\eta_{m})_{m \in \mathbb{N}}\subseteq \mathcal{M}(\Omega;V)$ is an i.i.d. sequence, $(\beta_{m})_{m \in \mathbb{N}}$ is an i.i.d. sequence which is independent of $(\eta_{m})_{m \in \mathbb{N}}$ and each $\beta_{m}$ is exponentially distributed with parameter $\theta \in (0,\infty)$, and $(T(t))_{t \geq 0}$ is a time-continuous contractive semigroup on $V$. Moreover, $(N(t))_{t \geq 0}$ is the Poisson process arising from $(\beta_{m})_{m \in \mathbb{N}}$ and $(\alpha_{m})_{m \in \mathbb{N}_{0}}$ is the process’ sequence of arrival times.\
Again we refer to an $x \in \mathcal{M}(\Omega;V)$ which is independent of $((\eta_{m})_{m \in \mathbb{N}},(\beta_{m})_{m \in \mathbb{N}})$ as an independent initial, and denote by ${\mathbb{X}}_{x}$ and $({\text{\scalebox{0.62}{$\mathbb{X}$}}}_{x,m})_{m \in \mathbb{N}_{0}}$ the sequence and the process generated by $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}},x,T)$ in $V$.\
Finally, set $P(t,v,B):={\mathbb{P}}({\mathbb{X}}_{v}(t)\in B)$ and $(Q(t)\psi)(v):= \mathbb{E}\psi({\mathbb{X}}_{v}(t))$, for every $v \in V$, $t \in [0,\infty)$, $B \in {\mathfrak{B}}(V)$ and $\psi \in \text{BM}(V)$.\
In addition, we assume throughout this entire section that $$\begin{aligned}
\label{eq_etal2int}
||\eta_{k}||_{V} \in L^{2}(\Omega),~\forall k \in \mathbb{N},\end{aligned}$$ where $L^{q}(\Omega):=L^{q}(\Omega,{\mathcal{F}},{\mathbb{P}})$ for every $q \in [1,\infty)$. Moreover, we impose the following assumption regarding $(T(t))_{t \geq 0}$.
\[assumption\] There is a separable Banach space $(W,||\cdot||_{W})$, with $ W \subseteq V$, such that the following assertions hold.
i) \[assumption\_enumi1\] The injection $W \hookrightarrow V$ is continuous and $W$ is dense in $(V,||\cdot||_{V})$.
ii) \[assumption\_enumi2\] $T$ is invariant with respect to $W$, that is $T(t)w \in W$ for all $w \in W$ and $t \in [0,\infty)$.
iii) \[assumption\_enumi3\] There are constants $\kappa,\rho \in (0,\infty)$ such that $||T(t)w_{1}-T(t)w_{2}||_{W}\leq \left(\kappa t+||w_{1}-w_{2}||_{W}^{-\frac{1}{\rho}}\right)^{-\rho}$ for all $w_{1},w_{2}\in W$ and $t \in [0,\infty)$.
iv) \[assumption\_enumi4\] $T(t)0=0$ for all $t \in [0,\infty)$.
Throughout this entire section, Assumption \[assumption\] is assumed to hold; particularly, $(W,||\cdot||_{W})$ and $\kappa,\rho \in (0,\infty)$ are such that \[assumption\].\[assumption\_enumi1\])-\[assumption\_enumi3\]) are fulfilled. In addition, $C> 0$ denotes the operator norm of the injection $W \hookrightarrow V$; hereby, we exclude the trivial case $C=0$, since $C=0$ implies $W=\{0\}$ and by density $V=\{0\}$.\
In Assumption \[assumption\].\[assumption\_enumi3\]) we did not assume $w_{1}\neq w_{2}$. If $w_{1}=w_{2}$, we set $\big(\kappa t+||w_{1}-w_{2}||_{W}^{-\frac{1}{\rho}}\big)^{-\rho}:=0$, which is reasonable, since: For any $x \in [0,\infty)$ the mapping $(0,\infty) \ni y \mapsto \left(x+y^{-\frac{1}{\rho}}\right)^{-\rho}$ can be extended continuously by zero in $y=0$.\
Moreover, Assumption\[assumption\].\[assumption\_enumi1\]) yields that $W \in {\mathfrak{B}}(V)$ and that if $Y \in \mathcal{M}(\Omega;V)$, with ${\mathbb{P}}(Y\in W)=1$ then the real-valued mapping $||Y||_{W}$ is up-to a ${\mathbb{P}}$-null-set well-defined and ${\mathfrak{B}}(V)$-${\mathfrak{B}}({\mathbb{R}})$-measurable, see [@ich3 Remark 2.7].\
The following estimates will play a fundamental role in this entire section, it is needed in the proofs of all of our main results, which are: Proposition \[prop\_uniqueinvpropmeas\], Theorem \[theorem\_slln\], Theorem \[theorem\_clt\] and Corollary \[corollary\_clt\]. The remaining results of this section simply serve to keep the exposition more structured, but are probably not of independent interest.\
As mentioned introductory, proving the CLT requires the additional assumption $\rho>\frac{1}{2}$. It will be stated explicitly whenever this additional assumption is needed.
\[lemma\_fundamentalestimate\] Let $x \in \mathcal{M}(\Omega;V)$ be an independent initial. Then the inequality $$\begin{aligned}
\label{lemma_fundamentalestimateeq1}
||{\mathbb{X}}_{x}(t)||_{V} \leq C \kappa^{-\rho}(t-\alpha_{N(t)})^{-\rho},~\forall t>0,\end{aligned}$$ takes place with probability one. In addition, if $y \in \mathcal{M}(\Omega;V)$ is another independent initial we have $$\begin{aligned}
\label{lemma_fundamentalestimateeq2}
||{\mathbb{X}}_{x}(t)-{\mathbb{X}}_{y}(t)||_{V} \leq C \kappa^{-\rho}t^{-\rho},~\forall t>0,\end{aligned}$$ almost surely.
Let us start by proving (\[lemma\_fundamentalestimateeq1\]). To this end, let $(\tilde{\eta}_{m})_{m \in \mathbb{N}}\subseteq \mathcal{M}(\Omega;V)$ and $\tilde{x}\in \mathcal{M}(\Omega;V)$, assume $\tilde{\eta}_{m},\tilde{x}\in W$ almost surely and introduce $(\tilde{X}(t))_{t\geq 0}$ and $(\tilde{x}_{m})_{m \in \mathbb{N}_{0}}$ as the process and the sequence generated by $((\beta_{m})_{m \in \mathbb{N}},(\tilde{\eta}_{m})_{m \in \mathbb{N}},\tilde{x},T)$ in $V$, respectively.\
Then, note that $\tilde{x}_{m} \in W$ for all $m \in \mathbb{N}_{0}$ almost surely, since: If $m=0$ this is trivial and if it holds for an $m \in \mathbb{N}_{0}$, we have $\tilde{x}_{m+1}=T(\beta_{m+1})\tilde{x}_{m}+\tilde{\eta}_{m+1}$ and both summands are elements of $W$, for the first this follows from Assumption \[assumption\].\[assumption\_enumi2\]) and the induction hypothesis, and for the second this holds by construction.\
Consequently, appealing to Assumption \[assumption\].\[assumption\_enumi2\]) again yields $\tilde{X}(t) \in W$ for all $t\geq 0$, almost surely, since clearly $\tilde{X}(t)=T(t-\alpha_{N(t)})\tilde{x}_{N(t)}$ for all $t \geq 0$ with probability one.\
Hence, employing Assumption \[assumption\].\[assumption\_enumi3\]) and \[assumption\_enumi4\]) yields $$\begin{aligned}
\label{lemma_fundamentalestimateproof1}
||\tilde{X}(t)||_{V}\leq C||T(t-\alpha_{N(t)})\tilde{x}_{N(t)}||_{W} \leq C \left(\kappa (t-\alpha_{N(t)})+||\tilde{x}_{N(t)}||_{W}^{-\frac{1}{\rho}}\right)^{-\rho} \leq C\kappa^{-\rho}(t-\alpha_{N(t)})^{-\rho},\end{aligned}$$ for all $t>0$ almost surely.\
Now let us infer (\[lemma\_fundamentalestimateeq1\]) from (\[lemma\_fundamentalestimateproof1\]). To this end, let $(\Gamma_{n})_{n \in \mathbb{N}}$, where $\Gamma_{n}:V\rightarrow V$, be a sequence of ${\mathfrak{B}}(V)$-${\mathfrak{B}}(V)$-measurable mappings, such that $$\begin{aligned}
\label{lemma_fundamentalestimateproof2}
\Gamma_{n}(V)\subseteq W,~\forall n \in \mathbb{N} \text{ and } \lim \limits_{n \rightarrow \infty} \Gamma_{n}(v)=v,~\forall v \in V.\end{aligned}$$ Since $W$ is dense in $(V,||\cdot||_{V})$, such a sequence exists, see [@ich3 Lemma 3.12]. Now, for every $n \in \mathbb{N}$, let $(X^{n}(t))_{t \geq 0}$ be the process generated by $((\beta_{m})_{m \in \mathbb{N}},\left(\Gamma_{n}(\eta_{m})\right)_{m \in \mathbb{N}},\Gamma_{n}(x),T)$ in $V$. Then, as $\Gamma_{n}(V)\subseteq W$, (\[lemma\_fundamentalestimateproof1\]) yields $||X^{n}(t)||_{V}\leq C\kappa^{-\rho}(t-\alpha_{N(t)})^{-\rho}$ for all $t>0$ and $n \in \mathbb{N}$ almost surely. (If one sets $(\tilde{\eta}_{m})_{m \in \mathbb{N}}=\left(\Gamma_{n}(\eta_{m})\right)_{m \in \mathbb{N}}$ and $\tilde{x}=\Gamma_{n}(x)$ for a given $n \in \mathbb{N}$, then $\tilde{X}=X^{n}$). Moreover, appealing to Lemma \[lemma\_xcont\], while having in mind (\[lemma\_fundamentalestimateproof2\]), yields $$\begin{aligned}
||{\mathbb{X}}_{x}(t)||_{V} \leq \lim \limits_{n \rightarrow \infty} ||x-\Gamma_{n}(x)||_{V}+\sum \limits_{m=1}\limits^{N(t)}||\eta_{m}-\Gamma_{n}(\eta_{m})||_{V}+C\kappa^{-\rho}(t-\alpha_{N(t)})^{-\rho} =C\kappa^{-\rho}(t-\alpha_{N(t)})^{-\rho},\end{aligned}$$ for all $t>0$, with probability one. Consequently, (\[lemma\_fundamentalestimateeq1\]) is proven and it remains to verify (\[lemma\_fundamentalestimateeq2\]).\
In addition, to the existing notations, let $\tilde{y}\in \mathcal{M}(\Omega;V)$, assume $\tilde{y}\in W$ almost surely and introduce $(\tilde{Y}(t))_{t\geq 0}$ and $(\tilde{y}_{m})_{m \in \mathbb{N}_{0}}$ as the process and the sequence generated by $((\beta_{m})_{m \in \mathbb{N}},(\tilde{\eta}_{m})_{m \in \mathbb{N}},\tilde{y},T)$ in $V$, respectively.\
Of course, we then also have $\tilde{Y}(t),\tilde{y}_{m}\in W$ for all $t \geq 0$ and $m \in \mathbb{N}_{0}$, with probability one. Now let us verify inductively that $$\begin{aligned}
\label{lemma_fundamentalestimateeq3}
||\tilde{x}_{m}-\tilde{y}_{m}||_{W}\leq \left(\kappa \alpha_{m}+||\tilde{x}-\tilde{y}||_{W}^{-\frac{1}{\rho}}\right)^{-\rho},~\text{a.s. }\forall m \in \mathbb{N}.\end{aligned}$$ If $m=0$, (\[lemma\_fundamentalestimateeq3\]) is even an equality. And if it holds for an $m \in \mathbb{N}_{0}$ we get by applying Assumption \[assumption\].\[assumption\_enumi3\]) and then the induction hypothesis that $$\begin{aligned}
||\tilde{x}_{m+1}-\tilde{y}_{m+1}||_{W} \leq \left(\kappa \beta_{m+1}+||\tilde{x}_{m}-\tilde{y}_{m}||_{W}^{-\frac{1}{\rho}}\right)^{-\rho} \leq \left(\kappa \alpha_{m+1}+||\tilde{x}-\tilde{y}||_{W}^{-\frac{1}{\rho}}\right)^{-\rho},\end{aligned}$$ with probability one, which proves (\[lemma\_fundamentalestimateeq3\]). Using this, while employing the services of \[assumption\].\[assumption\_enumi3\]) once more gives $$\begin{aligned}
||\tilde{X}(t)-\tilde{Y}(t)||_{V}
&\leq& ~ C||T(t-\alpha_{N(t)})\tilde{x}_{N(t)}-T(t-\alpha_{N(t)})\tilde{y}_{N(t)}||_{W} \\
&\leq& ~ C\left(\kappa (t-\alpha_{N(t)})+||\tilde{x}_{N(t)}-\tilde{y}_{N(t)}||_{W}^{-\frac{1}{\rho}}\right)^{-\rho} \\
&\leq& ~ C\left(\kappa (t-\alpha_{N(t)})+\kappa \alpha_{N(t)}+||\tilde{x}-\tilde{y}||_{W}^{-\frac{1}{\rho}}\right)^{-\rho} \\
&\leq& ~ C\left(\kappa t\right)^{-\rho},\end{aligned}$$ for all $t>0$ with probability one. Now, for every $n \in \mathbb{N}$, let $(Y^{n}(t))_{t \geq 0}$ be the process generated by $((\beta_{m})_{m \in \mathbb{N}},\left(\Gamma_{n}(\eta_{m})\right)_{m \in \mathbb{N}},\Gamma_{n}(y),T)$ in $V$. Then, as $\Gamma_{n}(V)\subseteq W$, the preceding calculation yields $||X^{n}(t)-Y^{n}(t)||_{V}\leq C\left(\kappa t\right)^{-\rho}$ for all $t>0$ and $n \in \mathbb{N}$ almost surely. Finally, Lemma \[lemma\_xcont\] enables us to conclude that $$\begin{aligned}
||{\mathbb{X}}_{x}(t)-{\mathbb{X}}_{y}(t)||_{V}\leq C\left(\kappa t\right)^{-\rho}+||x-\Gamma_{n}(x)||_{V}+||y-\Gamma_{n}(y)||_{V}+2\sum \limits_{m=1}\limits^{N(t)}||\eta_{m}-\Gamma_{n}(\eta_{m})||_{V},\end{aligned}$$ for all $t>0$ and $n \in \mathbb{N}$ with probability one, which yields the claim by recalling (\[lemma\_fundamentalestimateproof2\]) and letting $n$ to infinity.
\[prop\_uniqueinvpropmeas\] The transition function $P$ possesses a unique invariant probability measure, i.e. there is one, and only one, probability measure $\mu: {\mathfrak{B}}(V)\rightarrow [0,1]$, such that $$\begin{aligned}
\label{prop_uniqueinvpropmeaseq}
\int \limits_{V} P(t,v,B) \mu (dv)=\mu(B),~\forall t \geq 0,~B \in {\mathfrak{B}}(V).\end{aligned}$$
Appealing to Theorem \[theorem\_mp\] as well as Lemma \[lemma\_basicprop\].\[lemma\_basicpropmpenumi1\])-\[lemma\_basicpropmpenumi3\]) yields, by virtue of [@Szarek Theorem 1], the existence of a unique invariant probability measure, if we can prove that $$\begin{aligned}
\liminf \limits_{t \rightarrow \infty} \frac{1}{t} \int \limits_{0}\limits^{t} P(||{\mathbb{X}}_{v}(t)||_{V}< \varepsilon)d\tau >0,~\forall \varepsilon>0,~v \in V.\end{aligned}$$ So fix $\varepsilon>0$ as well as $v \in V$ and recall the well-known fact that ${\mathbb{P}}(\tau-\alpha_{N(\tau)}>q)=\exp(-\theta q){1\hspace{-0,9ex}1}_{[0,\tau)}(q)$ for all $\tau,q\in [0,\infty)$. Now, introduce $q:=\kappa^{-1}\varepsilon^{-\frac{1}{\rho}}C^{\frac{1}{\rho}}$.\
Then we get by Lemma \[lemma\_fundamentalestimate\] that $$\begin{aligned}
\liminf \limits_{t \rightarrow \infty} \frac{1}{t} \int \limits_{0}\limits^{t} P(||{\mathbb{X}}_{v}(\tau)||_{V}< \varepsilon)d\tau
& \geq & ~ \liminf \limits_{t \rightarrow \infty} \frac{1}{t} \int \limits_{0}\limits^{t} P(C \kappa^{-\rho}(\tau-\alpha_{N(\tau)})^{-\rho}< \varepsilon)d\tau \\
& = & ~ \liminf \limits_{t \rightarrow \infty} \frac{1}{t} \int \limits_{0}\limits^{t} P(\tau-\alpha_{N(\tau)}> q)d\tau\\
& = & ~ \exp(-\theta q),\end{aligned}$$ which is obviously strictly positive.
In the remainder of this section, $\bar{\mu}:{\mathfrak{B}}(V)\rightarrow [0,1]$, denotes the uniquely determined probability measure fulfilling (\[prop\_uniqueinvpropmeaseq\]). Moreover, we call an $\bar{x} \in \mathcal{M}(\Omega;V)$ which is an independent initial with ${\mathbb{P}}(\bar{x} \in B)=\bar{\mu}(B)$, for all $B \in {\mathfrak{B}}(V)$, an independent, stationary initial.\
As $\bar{\mu}$ is unique, it is ergodic, see [@Prato Theorem 3.2.6] for a proof and [@Prato Theorem 3.2.4] for a couple of useful equivalent definitions of ergodicity, commonly used in the literature.\
Furthermore, if $\bar{x} \in \mathcal{M}(\Omega;V)$ is an independent, stationary initial, then the Markov process $({\mathbb{X}}_{\bar{x}}(t))_{t \geq 0}$ is strictly stationary, see [@Kallenberg Lemma 8.11]. Moreover, $({\mathbb{X}}_{\bar{x}}(t))_{t \geq 0}$ is also ergodic (in the sense that the shift invariant $\sigma$-algebra is ${\mathbb{P}}$-trivial), which one easily deduces from [@ergodicequi Prop. 2.2] by appealing to [@Prato Theorem 3.2.4.ii)].\
Finally, $L^{2}(\bar{\mu}):=L^{2}(V,{\mathfrak{B}}(V),\bar{\mu})$ and for any $\psi \in L^{2}(\bar{\mu})$ we set $\overline{(\psi)}:= \int \limits_{V}\psi(v)\bar{\mu}(dv)$ and introduce $L^{2}_{0}(\bar{\mu}):=\{\psi \in L^{2}(\bar{\mu}):\overline{(\psi)}=0\}$.
\[lemma\_boundL2\] Let $x \in \mathcal{M}(\Omega;V)$ be an independent initial. Then $||{\mathbb{X}}_{x}(t)||_{V} \in L^{2}(\Omega)$ for all $t \in (0,\infty)$. In particular, the following assertions hold.
i) \[lemma\_boundL2\_enumi1\] $\psi({\mathbb{X}}_{\bar{x}}(t)) \in L^{2}(\Omega)$, for all $t \in [0,\infty)$, $\psi \in Lip(V)$ and independent stationary initials $\bar{x}\in \mathcal{M}(\Omega;V)$.
ii) \[lemma\_boundL2\_enumi2\] $Lip(V) \subseteq L^{2}(\bar{\mu})$.
Let $t>0$ and $x \in \mathcal{M}(\Omega;V)$ be an independent initial. Then we get by employing the services of Lemma \[lemma\_xcont\] and Lemma \[lemma\_fundamentalestimate\] that $$\begin{aligned}
||{\mathbb{X}}_{x}(t)||_{V} \leq ||{\mathbb{X}}_{x}(t)-{\mathbb{X}}_{0}(t)||_{V}+||{\mathbb{X}}_{0}(t)||_{V} \leq C \kappa^{-\rho}t^{-\rho}+\sum \limits_{m=1}\limits^{N(t)}||\eta_{m}||_{V}\end{aligned}$$ almost surely. Consequently, $||{\mathbb{X}}_{x}(t)||_{V} \in L^{2}(\Omega)$ holds, if $\sum \limits_{m=1}\limits^{N(t)}||\eta_{m}||_{V} \in L^{2}(\Omega)$. But the latter is true by the Blackwell-Girshick equation, which is applicable since $(||\eta_{k}||)_{k \in \mathbb{N}}\subseteq L^{2}(\Omega)$ is i.i.d. and independent of $(N(t))_{t \geq 0}$, which is (as it is a Poisson process) in particular square integrable.\
Now, note that, due to stationary, \[lemma\_boundL2\].\[lemma\_boundL2\_enumi1\]) holds for one $t \in [0,\infty)$ if and only if, it holds for every $t \in [0,\infty)$. So assume $t>0$, then we get $|\psi({\mathbb{X}}_{\bar{x}}(t))|\leq L_{\psi}||{\mathbb{X}}_{\bar{x}}(t)||_{V}+|\psi(0)|$, which is already known to be square integrable. Finally, \[lemma\_boundL2\].\[lemma\_boundL2\_enumi2\]) follows from \[lemma\_boundL2\].\[lemma\_boundL2\_enumi1\]), since $||\psi||_{L^{2}(\bar{\mu})}^{2}=\mathbb{E}\left(\psi(\bar{x})^{2}\right)$.
\[theorem\_slln\] Let $\psi \in Lip(V)$ and $x \in \mathcal{M}(\Omega;V)$ be an independent initial. Then the convergence $$\begin{aligned}
\label{theorem_sllneq}
\lim \limits_{t \rightarrow \infty } \frac{1}{t} \int \limits_{0} \limits^{t} \psi ({\mathbb{X}}_{x}(\tau)) d\tau =\overline{(\psi)},\end{aligned}$$ takes place with probability one.
Firstly, note that the left hand side integral exists, since Lemma \[lemma\_meas\] and Lemma \[lemma\_basicprop\] yield that $[0,t]\ni \tau \mapsto \psi({\mathbb{X}}_{x}(\tau,\omega))$ is ${\mathfrak{B}}([0,t])$-${\mathfrak{B}}({\mathbb{R}})$-measurable and for ${\mathbb{P}}$-a.e. $\omega \in \Omega$ bounded, respectively.\
Now let $\bar{x} \in \mathcal{M}(\Omega;V)$ be an independent stationary initial. Then appealing to [@Prato Theorem 3.3.1] yields $$\begin{aligned}
\label{theorem_sllnproofeq1}
\lim \limits_{t \rightarrow \infty } \frac{1}{t} \int \limits_{0} \limits^{t} \psi ({\mathbb{X}}_{\bar{x}}(\tau)) d\tau = \overline{(\psi)},\end{aligned}$$ almost surely, for all $\psi \in Lip(V)$. (This theorem is indeed applicable, since $\bar{\mu}$ is ergodic, $({\mathbb{X}}_{\bar{x}}(t))_{t \geq 0}$ is stationary, stochastically continuous and since $Lip(V)\subseteq L^{2}(\bar{\mu})$.)\
Conclusively, recalling Lemma \[lemma\_fundamentalestimate\] gives $$\begin{aligned}
\lim \limits_{t \rightarrow \infty } \left|\frac{1}{t} \int \limits_{0} \limits^{t} \psi ({\mathbb{X}}_{\bar{x}}(\tau))- \psi ({\mathbb{X}}_{x}(\tau)) d\tau\right| \leq L_{\psi}C\kappa^{-\rho}\lim \limits_{t \rightarrow \infty } \frac{1}{t} \int \limits_{1} \limits^{t} \tau^{-\rho} d\tau=0,\end{aligned}$$ almost surely, which yields combined with (\[theorem\_sllnproofeq1\]) the claim.
The task ahead of us that remains is proving the CLT, which will be achieved by the results in [@Holzmann]. Applying the results in [@Holzmann] requires to extend the family of mappings $(Q(t))_{t \geq 0}$ to a linear, time-continuous, contractive semigroup on $L^{2}(\bar{\mu})$. To aid the reader who is not too familiar with Markov processes, let us outline why this is possible.
\[remark\_extension\] Let $t \in [0,\infty)$ be given. Then for any $\hat{V} \in {\mathfrak{B}}(V)$, with $\bar{\mu}(\hat{V})=1$, we get by the invariance of $\bar{\mu}$ that there is a set $\tilde{V} \in {\mathfrak{B}}(V)$, with $\mu(\tilde{V})=1$, such that ${\mathbb{P}}({\mathbb{X}}_{v}(t)\in \hat{V})=1,~\forall v \in \tilde{V}$.\
Moreover, if $\psi = {1\hspace{-0,9ex}1}_{B}$, where $B \in {\mathfrak{B}}(V)$, then the invariance of $\bar{\mu}$ gives $$\begin{aligned}
\label{remark_extensioneq1}
\int \limits_{V} \mathbb{E} \psi({\mathbb{X}}_{v}(t))\bar{\mu}(dv)= \int \limits_{V} \psi(v)\bar{\mu}(dv).\end{aligned}$$ Moreover, by linearity in $\psi$, (\[remark\_extensioneq1\]) also holds for all step functions. Now let $\psi \in L^{2}(\bar{\mu})$ be arbitrary, then there are step functions $(\psi_{m})_{m \in \mathbb{N}}$ with $\lim \limits_{m \rightarrow \infty}\psi_{m}=\psi$ in $L^{2}(\bar{\mu})$ and $\bar{\mu}$-a.e. Hence, for $\mu$-a.e. $v \in V$ we get $\lim \limits_{m \rightarrow \infty} \psi_{m}({\mathbb{X}}_{v}(t))= \psi({\mathbb{X}}_{v}(t))$ a.s. Consequently, applying Fatou’s Lemma (twice) and (\[remark\_extensioneq1\]) yields $$\begin{aligned}
\int \limits_{V} \mathbb{E}\left( \psi({\mathbb{X}}_{v}(t))^{2}\right)\bar{\mu}(dv) \leq \liminf \limits_{m \rightarrow \infty} \int \limits_{V} \mathbb{E}\left( \psi_{m}({\mathbb{X}}_{v}(t))^{2}\right)\bar{\mu}(dv) = \liminf \limits_{m \rightarrow \infty} \int \limits_{V} \psi_{m}(v)^{2}\bar{\mu}(dv) =||\psi||_{L^{2}(\bar{\mu})}^{2}<\infty.\end{aligned}$$ Hence, for $\bar{\mu}$-almost every $v \in V$, $\mathbb{E}\psi({\mathbb{X}}_{v}(t))$ exists and we infer from Jensen’s inequality that $$\begin{aligned}
\label{remark_extensioneq2}
\int \limits_{V} \big(\mathbb{E} \psi({\mathbb{X}}_{v}(t))\big)^{2}\bar{\mu}(dv) \leq ||\psi||_{L^{2}(\bar{\mu})}^{2},~\forall \psi \in L^{2}(\bar{\mu}).\end{aligned}$$ Consequently, we can extend the domain of each $Q(t)$ to $L^{2}(\bar{\mu})$, i.e. from now on $Q(t):L^{2}(\bar{\mu})\rightarrow L^{2}(\bar{\mu})$, with $(Q(t)\psi)(v):= \mathbb{E} \psi({\mathbb{X}}_{v}(t))$, for all $t \in [0,\infty)$, $v \in V$ and $\psi \in L^{2}(\bar{\mu})$.\
Using this and Theorem \[theorem\_mp\].\[theorem\_mpenumi4\]) yields that $(Q(t))_{t \geq 0}$ is a linear, contractive semigroup on $L^{2}(\bar{\mu})$, see [@Yosida Theorem 1, p. 381] for a detailed proof.
It seems to be mathematical common knowledge that this semigroup is (due to stochastic continuity and contractivity) time-continuous. But, the present author was unable to find any proof of this assertion, therefore let’s do that:
\[lemma\_extension\] The family of mappings $(Q(t))_{t \geq 0}$ is a linear, time-continuous contractive semigroup on $L^{2}(\bar{\mu})$.
In light of Remark \[remark\_extension\], it remains to prove the time continuity. So let $(h_{m})_{m \in \mathbb{N}}$ be a null-sequence, let $t \in [0,\infty)$ and assume w.l.o.g. that $t+h_{m}\geq 0$ for all $m \in \mathbb{N}$. Now let $\psi \in L^{2}(\bar{\mu})$, choose $\varepsilon>0$ and $\varphi \in C_{b}(V)$ such that $||\psi-\varphi||_{L^{2}(\bar{\mu})}<\frac{\varepsilon}{2}$. Then, by stochastic continuity of $({\mathbb{X}}_{v}(t))_{t\geq 0}$, and passing to a subsequence if necessary, we have $\lim \limits_{m \rightarrow \infty}\varphi({\mathbb{X}}_{v}(t+h_{m}))=\varphi({\mathbb{X}}_{v}(t))$ almost surely. Thus, the boundedness of $\varphi$ yields (by dominated convergence) that $\lim \limits_{m \rightarrow \infty}(Q(t+h_{m})\varphi)(v)=(Q(t)\varphi)(v)$ for all $v \in V$. Consequently, employing Lebesgue’s theorem once more gives $\lim \limits_{m \rightarrow \infty}Q(t+h_{m})\varphi=Q(t)\varphi$ in $L^{2}(\bar{\mu})$. Using this, as well as the contractivity of $Q$ gives $$\begin{aligned}
\lim \limits_{m \rightarrow \infty} ||Q(t+h_{m})\psi-Q(t)\psi||_{L^{2}(\bar{\mu})}\leq 2||\psi-\varphi||_{L^{2}(\bar{\mu})}+\lim \limits_{m \rightarrow \infty} ||Q(t+h_{m})\varphi-Q(t)\varphi||_{L^{2}(\bar{\mu})}\leq \varepsilon,\end{aligned}$$ which yields the desired time continuity, as $\varepsilon>0$ was arbitrary.
\[lemma\_cltprepbpund\] Let $\psi \in Lip(V)$ and set $\psi_{c}:=\psi-\overline{(\psi)}$. Then $\psi_{c} \in L^{2}_{0}(\bar{\mu})$ and $$\begin{aligned}
||Q(t)\psi_{c}||_{L^{2}(\bar{\mu})}\leq L_{\psi}C \kappa^{-\rho}t^{-\rho},\end{aligned}$$ for all $t>0$.
Clearly, $\psi_{c} \in Lip(V)$, thus $\psi_{c} \in L^{2}(\bar{\mu})$ by Lemma \[lemma\_boundL2\].\[lemma\_boundL2\_enumi2\]). Moreover, $\psi_{c}$ is obviously centered. In addition, by stationary we get $\overline{(\psi)}= \mathbb{E}\psi({\mathbb{X}}_{\bar{x}}(t))$, where $\bar{x}\in \mathcal{M}(\Omega;V)$ is an independent, stationary initial. Using this and invoking Lemma \[lemma\_fundamentalestimate\] yields $$\begin{aligned}
||Q(t)\psi_{c}||_{L^{2}(\bar{\mu})}^{2} = \int \limits_{V} \big(\mathbb{E}[\psi({\mathbb{X}}_{v}(t))-\psi({\mathbb{X}}_{\bar{x}}(t))]\big)^{2}\bar{\mu}(dv) \leq (L_{\psi}C\kappa^{-\rho}t^{-\rho})^{2}\end{aligned}$$ and the claim follows.
\[theorem\_clt\] Assume $\rho > \frac{1}{2}$, let $\psi \in Lip(V)$ and $x \in \mathcal{M}(\Omega;V)$ be an independent initial. Then there is a $\sigma^{2}(\psi) \in [0,\infty)$ such that $$\begin{aligned}
\label{theorem_clteq1}
\lim \limits_{t \rightarrow \infty }\frac{1}{\sqrt{t}}\left( \int \limits_{0}\limits^{t}\psi({\mathbb{X}}_{x}(\tau))d\tau-t\overline{(\psi)}\right)=Y\sim N(0,\sigma^{2}(\psi)) ,\end{aligned}$$ in distribution. Moreover, we have $$\begin{aligned}
\label{theorem_clteq2}
\sigma^{2}(\psi) := \lim \limits_{t \rightarrow \infty}\frac{1}{t}\mathbb{E}\left( \int \limits_{0}\limits^{t}\psi_{c}({\mathbb{X}}_{\bar{x}}(\tau))d\tau\right)^{2}=\lim \limits_{t \rightarrow \infty}\frac{1}{t}{\text{Var}}\left( \int \limits_{0}\limits^{t}\psi({\mathbb{X}}_{\bar{x}}(\tau))d\tau\right),\end{aligned}$$ where $\bar{x}\in \mathcal{M}(\Omega;V)$ is an arbitrary stationary, independent initial and $\psi_{c}:=\psi-\overline{(\psi)}$.
Appealing to Lemma \[lemma\_cltprepbpund\] gives $\psi_{c} \in L^{2}_{0}(\bar{\mu})$ as well as $$\begin{aligned}
\int \limits_{1}\limits^{\infty} \frac{1}{\sqrt{t}}||Q(t)\psi_{c}||_{L^{2}(\bar{\mu})}dt \leq L_{\psi}C \kappa^{-\rho} \int \limits_{1}\limits^{\infty} t^{-\rho-\frac{1}{2}}dt,\end{aligned}$$ which is finite, since $\rho>\frac{1}{2}$. Consequently, as we already know that $({\mathbb{X}}_{\bar{x}}(t))_{t \geq 0}$ is a stationary, ergodic, $({\mathcal{F}}_{t}^{\bar{x}})_{t \geq 0}$-progressive Markov process with time-continuous, contractive semigroup $(Q(t))_{t \geq 0}$, we get by [@Holzmann Corollary 3.2 and Theorem 3.1] that $$\begin{aligned}
\label{theorem_cltproofeq1}
\lim \limits_{t \rightarrow \infty }\frac{1}{\sqrt{t}} \int \limits_{0}\limits^{t}\psi_{c}({\mathbb{X}}_{\bar{x}}(\tau))d\tau=Y\sim N(0,\sigma^{2}(\psi)),\end{aligned}$$ in distribution and that $\sigma^{2}(\psi)$ is indeed given by the first equality in (\[theorem\_clteq2\]). Moreover, the second equality in (\[theorem\_clteq2\]) is trivial, since $\psi_{c}({\mathbb{X}}_{\bar{x}}(\tau))=\psi({\mathbb{X}}_{\bar{x}}(\tau))-\mathbb{E}(\psi({\mathbb{X}}_{\bar{x}}(\tau)))$ by stationarity.\
Now, note that clearly $$\begin{aligned}
\frac{1}{\sqrt{t}}\left( \int \limits_{0}\limits^{t}\psi({\mathbb{X}}_{x}(\tau))d\tau-t\overline{(\psi)}\right) = \frac{1}{\sqrt{t}}\int \limits_{0}\limits^{t}\psi({\mathbb{X}}_{x}(\tau))-\psi({\mathbb{X}}_{\bar{x}}(\tau))d\tau+\frac{1}{\sqrt{t}} \int \limits_{0}\limits^{t}\psi_{c}({\mathbb{X}}_{\bar{x}}(\tau))d\tau,~\forall t>0\end{aligned}$$ which yields, in light of (\[theorem\_cltproofeq1\]), that (\[theorem\_clteq1\]) holds, if the first summand in the previous express converges almost surely to zero. But recalling that $\rho>\frac{1}{2}$ and invoking Lemma \[lemma\_fundamentalestimate\] yields $$\begin{aligned}
\lim \limits_{t \rightarrow \infty } \left|\frac{1}{\sqrt{t}} \int \limits_{0} \limits^{t} \psi ({\mathbb{X}}_{x}(\tau))- \psi ({\mathbb{X}}_{\bar{x}}(\tau)) d\tau\right| \leq L_{\psi}C\kappa^{-\rho}\lim \limits_{t \rightarrow \infty } \frac{1}{\sqrt{t}} \int \limits_{1} \limits^{t} \tau^{-\rho} d\tau=0,\end{aligned}$$ with probability one.
Now this section concludes by summarizing Theorem \[theorem\_slln\] and Theorem \[theorem\_clt\] for the probably most prominent Lipschitz continuous map from $V$ to ${\mathbb{R}}$, namely $||\cdot||_{V}$.
\[corollary\_clt\] Let $x \in \mathcal{M}(\Omega;V)$ be an independent initial and $\bar{x}\in \mathcal{M}(\Omega;V)$ a stationary independent initial. Then the following assertions hold.
i) $\lim \limits_{t \rightarrow \infty } \frac{1}{t} \int \limits_{0} \limits^{t} ||{\mathbb{X}}_{x}(\tau)||_{V} d\tau =\nu$ with probability one, where $\nu:=\int \limits_{V} ||v||_{V} \bar{\mu}(dv)=\mathbb{E}||\bar{x}||_{V}$.
ii) \[corollary\_clt\_enumi2\] If $\rho>\frac{1}{2}$, then $\lim \limits_{t \rightarrow \infty }\frac{1}{\sqrt{t}}\left( \int \limits_{0}\limits^{t}||{\mathbb{X}}_{x}(\tau)||_{V}d\tau-t\nu\right)=Y\sim N(0,\sigma^{2})$ in distribution, where $\sigma^{2}\in [0,\infty)$, with $\sigma^{2} =\lim \limits_{t \rightarrow \infty}\frac{1}{t}{\text{Var}}\left( \int \limits_{0}\limits^{t}||{\mathbb{X}}_{\bar{x}}(\tau)||_{V}d\tau\right)$.
Examples and a useful Criteria {#sec_examples}
==============================
The first result of this section is the introductory mentioned differential-inequality-result, which is probably not only in our examples useful to verify Assumption \[assumption\].\[assumption\_enumi3\]). Even though this result seems to be in common use, we were unable to find it anywhere in the literature, stated precisely as we need it and with a rigorous proof. Therefore, the simple proof will be given. Once this is achieved we proceed with our ODE example and devote the remainder of this section to the $p$-Laplacian example.
\[lemma\_diffinequality\] Let $f:[0,\infty)\rightarrow [0,\infty)$ be locally Lipschitz continuous on $[0,\infty)$. Moreover, assume that there are constants $\kappa,\tilde{\rho} \in (0,\infty)$ such that $$\begin{aligned}
\label{lemma_diffinequalityeq1}
f^{\prime}(t)\leq - \kappa \tilde{\rho} f(t)^{1+\frac{1}{\tilde{\rho}}},\end{aligned}$$ for a.e. $t \in (0,\infty)$. Then we have $$\begin{aligned}
\label{lemma_diffinequalityeq2}
f(t)\leq \left(\kappa t+f(0)^{-\frac{1}{\tilde{\rho}}}\right)^{-\tilde{\rho}},\end{aligned}$$ for all $t \in [0,\infty)$.
Firstly, as $f$ is a real-valued, locally Lipschitz continuous function it is indeed differentiable almost everywhere. Moreover, (\[lemma\_diffinequalityeq1\]) yields that $f$ is monotonically decreasing.\
Now set $I:=\inf\{t\geq0:~ f(t)=0\}$. If $I=0$, then $f(t)=0$ for all $t>0$ and by continuity for all $t \geq 0$. Consequently, in this case (\[lemma\_diffinequalityeq2\]) trivially holds. So assume $I>0$ and let $\tilde{I} \in [0,I)$ be arbitrary but fixed and introduce $F:[0,\tilde{I}]\rightarrow [0,\infty)$ with $F(t):=f(t)^{-\frac{1}{\tilde{\rho}}}$. As $f(t)\geq f(\tilde{I})>0$ for all $t \in [0,\tilde{I}]$, $F$ is, as is the composition of Lipschitz continuous functions, itself Lipschitz continuous. Consequently, we get $$\begin{aligned}
F(t)-F(0)= \int \limits_{0}\limits^{t} F^{\prime}(\tau)d\tau =-\frac{1}{\tilde{\rho}} \int \limits_{0}\limits^{t} f(\tau)^{-\frac{1}{\tilde{\rho}}-1}f^{\prime}(\tau)d\tau\geq \kappa t,~\forall t \in [0,\tilde{I}].\end{aligned}$$ Thus (\[lemma\_diffinequalityeq2\]) holds on $[0,\tilde{I}]$ and as $\tilde{I}$ was arbitrary, it holds on $t \in [0,I)$. Finally, if $I=\infty$ the proof is complete and if $I<\infty$, the infimum is (by continuity) a minimum and by monotonicity $f=0$ on $[I,\infty)$, in which case (\[lemma\_diffinequalityeq2\]) is trivial.
\[remarkex1\] Let $\rho \in (0,\infty)$ and introduce the family of mappings $(T(t))_{t \geq 0}$, where $T(t):{\mathbb{R}}\rightarrow{\mathbb{R}}$, by $T(t)v:=sgn(v)\left(t+|v|^{-\frac{1}{\rho}}\right)^{-\rho}$ for all $v \in \mathbb{R}$. Then, obviously $T(0)v=sgn(v)|v|=v$ and a direct calculation shows that $T(\cdot)v$ fulfills the ODE $$\begin{aligned}
y^{\prime}(t)=-\rho y(t)|y(t)|^{\frac{1}{\rho}},~\forall t \in (0,\infty) \text{ and } y(0)=v.\end{aligned}$$ Moreover, we have $$\begin{aligned}
T(t)(T(h)v)=sgn(T(h)v)\left(t+|T(h)v|^{-\frac{1}{\rho}}\right)^{-\rho}=sgn(v)\left(t+h+|v|^{-\frac{1}{\rho}}\right)^{-\rho}=T(t+h)v,\end{aligned}$$ for all $t,h \in [0,\infty)$ and $v \in {\mathbb{R}}$. Thus, as $t \mapsto T(t)v$ is trivially continuous, $(T(t))_{t \geq 0}$ is a time-continuous semigroup.\
Now set $\kappa:=2^{-\frac{1}{\rho}}$ and let us verify Assumption \[assumption\].\[assumption\_enumi3\]), with $V=W={\mathbb{R}}$. Doing this requires to prove
i) \[remarkex1\_enumi1\] $T(t)u_{1}+T(t)u_{2}\leq \left(\kappa t+(u_{1}+u_{2})^{-\frac{1}{\rho}}\right)^{-\rho}$, for all $t \in [0,\infty)$, $u_{1},u_{2} \geq 0$ and
ii) \[remarkex1\_enumi2\] $T(t)u_{1}-T(t)u_{2}\leq \left(\kappa t+(u_{1}-u_{2})^{-\frac{1}{\rho}}\right)^{-\rho}$, for all $t \in [0,\infty)$, $u_{1},u_{2} \geq 0$ with $u_{1}\geq u_{2}$.
Proof of \[remarkex1\_enumi1\]). Firstly, the convexity of $[0,\infty) \ni x \mapsto x^{1+\frac{1}{\rho}}$ yields $x^{1+\frac{1}{\rho}}+y^{1+\frac{1}{\rho}}\geq 2^{-\frac{1}{\rho}}(x+y)^{1+\frac{1}{\rho}}$ for all $x,y \in [0,\infty)$. Now set $f(t):=T(t)u_{1}+T(t)u_{2}$, for all $t \in [0,\infty)$. Then we get $$\begin{aligned}
f^{\prime}(t)= -\rho \left((T(t)u_{1})^{1+\frac{1}{\rho}}+(T(t)u_{2})^{1+\frac{1}{\rho}}\right)\leq -\rho \kappa\left(T(t)u_{1}+T(t)u_{2}\right)^{1+\frac{1}{\rho}}=-\rho \kappa f(t)^{1+\frac{1}{\rho}},~\forall t>0.\end{aligned}$$ Consequently, as $f$ is (particularly locally) Lipschitz continuous, \[remarkex1\_enumi1\]) follows from Lemma \[lemma\_diffinequality\].\
Proof of \[remarkex1\_enumi2\]). Firstly, it is easily verified that $x^{1+\frac{1}{\rho}}-y^{1+\frac{1}{\rho}}\geq (x-y)^{1+\frac{1}{\rho}}\geq \kappa (x-y)^{1+\frac{1}{\rho}}$ for all $x \geq y \geq 0$. Moreover, note that $T(t)u_{1}\geq T(t)u_{2}$, since $u_{1}\geq u_{2}\geq 0$. Now, set $f(t):=T(t)u_{1}-T(t)u_{2}$, then we get $$\begin{aligned}
f^{\prime}(t)=-\rho \left((T(t)u_{1})^{1+\frac{1}{\rho}}-(T(t)u_{2})^{1+\frac{1}{\rho}}\right)\leq -\rho \kappa \left(T(t)u_{1}-T(t)u_{2}\right)^{1+\frac{1}{\rho}}=-\rho \kappa f(t)^{1+\frac{1}{\rho}},~\forall t>0.\end{aligned}$$ Consequently, employing Lemma \[lemma\_diffinequality\] once more yields \[remarkex1\_enumi2\]).\
Now, one easily infers from \[remarkex1\_enumi1\]), \[remarkex1\_enumi2\]) and $T(t)(-v)=-T(t)v$, for all $v \in {\mathbb{R}}$ that $$\begin{aligned}
|T(t)u-T(t)v|\leq \left(\kappa t+|u-v|^{-\frac{1}{\rho}}\right)^{-\rho},~\forall t \in [0,\infty),~u,v \in {\mathbb{R}}.\end{aligned}$$ In particular, $(T(t))_{t \geq 0}$ is contractive and by construction we have $T(t)0=0$. Consequently, $(T(t))_{t \geq 0}$ is a time-continuous, contractive semigroup fulfilling Assumption \[assumption\] with $V=W={\mathbb{R}}$. Now, let $(\eta_{m})_{m \in \mathbb{N}}\subseteq L^{2}(\Omega)$ be an i.i.d. sequence and let $(\beta_{m})_{m \in \mathbb{N}}$ be an i.i.d. sequence of $Exp(\theta)$-distributed random variables, where $\theta \in (0,\infty)$. In addition, assume that both sequences are independent of each other and let, for any independent initial $x \in \mathcal{M}(\Omega;{\mathbb{R}})$, ${\mathbb{X}}_{x}:[0,\infty)\times \Omega \rightarrow {\mathbb{R}}$ denote the process generated by $((\beta_{m})_{m \in \mathbb{N}},(\eta_{m})_{m \in \mathbb{N}},x,T)$ in ${\mathbb{R}}$. Then, as the identity is Lipschitz continuous, it follows from Theorem \[theorem\_slln\] and Theorem \[theorem\_clt\] that
iii) \[remarkex1\_enumi3\] $\lim \limits_{t \rightarrow \infty } \frac{1}{t} \int \limits_{0} \limits^{t} {\mathbb{X}}_{x}(\tau) d\tau =\mathbb{E}\bar{x}$ a.s., for any independent initial $x \in \mathcal{M}(\Omega;{\mathbb{R}})$ where $\bar{x}\in \mathcal{M}(\Omega;{\mathbb{R}})$ is a stationary, independent initial, and
iv) \[remarkex1\_enumi4\] if in addition $\rho>\frac{1}{2}$, then we have $\lim \limits_{t \rightarrow \infty }\frac{1}{\sqrt{t}}\left( \int \limits_{0}\limits^{t}{\mathbb{X}}_{x}(\tau)d\tau-t\mathbb{E}\bar{x}\right)=Y\sim N(0,\sigma^{2})$ in distribution, for any independent initial $x \in \mathcal{M}(\Omega;{\mathbb{R}})$, where $\sigma^{2} = \lim \limits_{t \rightarrow \infty}\frac{1}{t}{\text{Var}}\left( \int \limits_{0}\limits^{t}{\mathbb{X}}_{\bar{x}}(\tau)d\tau\right)$.
Now let us demonstrate that the assumption $\rho>\frac{1}{2}$ in \[remarkex1\_enumi4\]) cannot be dropped. To this end, assume $\eta_{k}=0$ for all $k \in \mathbb{N}$, then ${\mathbb{X}}_{x}(t)=T(t)x$ for any independent initial $x \in \mathcal{M}(\Omega;{\mathbb{R}})$. Since $T(t)0=0$ for all $t \geq 0$, $\bar{x}=0$ is the (in this case even almost surely unique) stationary, independent initial. Consequently, we have $\mathbb{E}\bar{x}={\text{Var}}\left( \int \limits_{0}\limits^{t}{\mathbb{X}}_{\bar{x}}(\tau)d\tau\right)=0$ and \[remarkex1\_enumi4\]), with $x=1$ and without additional assuming $\rho>\frac{1}{2}$, would imply $$\begin{aligned}
\label{remarkex1_eq1}
\lim \limits_{t \rightarrow \infty }\frac{1}{\sqrt{t}} \int \limits_{0}\limits^{t}\left(\tau+1\right)^{-\rho}d\tau=0,~\forall \rho>0,\end{aligned}$$ which is now, due to the lack of randomness, simply convergence in ${\mathbb{R}}$. But obviously, (\[remarkex1\_eq1\]) is true if and only if $\rho>\frac{1}{2}$.
Even though the semigroup considered in the previous example only acted on ${\mathbb{R}}$ and not an infinite dimensional Banach space, it is worth mentioning that neither \[remarkex1\].\[remarkex1\_enumi3\]) nor \[remarkex1\].\[remarkex1\_enumi4\]) are trivial.\
Now let us turn to the weighted $p$-Laplacian example, in which case the semigroup acts on an infinite dimensional Banach space.\
Throughout the remainder of this section, let $n \in \mathbb{N}\setminus \{1\}$ and $\emptyset \neq S \subseteq \mathbb{R}^{n}$ be a non-empty, open, connected and bounded sets of class $C^{1}$. Moreover, let $p \in (2,\infty)$ and set $L^{q}(S,\mathbb{R}^{m}):=L^{q}(S,{\mathfrak{B}}(S),\lambda;\mathbb{R}^{m})$, for any $q \in [1,\infty]$ and $m \in \mathbb{N}$, where $\lambda$ denotes the Lebesgue measure. This is further abbreviated by $L^{q}(S)$, if $m=1$. In addition, introduce $L^{q}_{0}(S):=\{f\in L^{q}(S):~\overline{(f)}=0\}$, where $\overline{(f)}:=\frac{1}{\lambda(S)}\int \limits_{S}fd\lambda$. Clearly, $(L^{q}_{0}(S),||\cdot||_{L^{q}(S)})$ is a separable Banach space.\
Now, let $\gamma: S \rightarrow (0,\infty)$ be such that $\gamma \in L^{\infty}( S )$, $\gamma^{\frac{2}{2-p}} \in L^{1}( S)$ and assume that there is an $A_{p}$-Muckenhoupt weight (see, [@ich1 page 4]) $\gamma_{0}:\mathbb{R}^{n}\rightarrow \mathbb{R}$ such that $\gamma_{0}|_{ S }=\gamma$ a.e. on $S$. Moreover, we introduce the weighted Sobolev space $W_{\gamma}^{1,p}( S )$ as $$\begin{aligned}
W_{\gamma }^{1,p}( S ):=\{f \in L^{p}( S ): f \text{ is weakly diff. and } ~\gamma^{\frac{1}{p}}\nabla f \in L^{p}( S;\mathbb{R}^{n})\}. \end{aligned}$$ In addition, whenever $q \in [1,\infty)$, $W^{1,q}(S)$ denotes the usual Sobolev space and $C_{S,q}$ is the Poincaré constant of $S$ in $L^{q}_{0}(S)$, i.e. the smallest constant such that $||\varphi||_{L^{q}(S)}\leq C_{S,q}||\nabla \varphi ||_{L^{q}(S)}$ for all $\varphi \in W^{1,q}(S)\cap L^{q}_{0}(S)$.\
Throughout the sequel, $|\cdot|_{n}$ is the Euclidean norm on $\mathbb{R}^{n}$ and for any $x,y\in{\mathbb{R}}^{n}$, $x\cdot y$ denotes the canonical inner product of these vectors.\
In the sequel, we frequently apply results from [@mazon] and [@ich1]. Applying them requires the assumption $\gamma^{\frac{1}{1-p}}\in L^{1}(S)$, which is indeed easily inferred from $\gamma^{\frac{2}{2-p}}\in L^{1}(S)$.\
The following weighted $p$-Laplace operator is the central object of the remainder of this paper:
\[definifition\_plaplaceop\] Let $A: D(A)\rightarrow 2^{L^{1}(S)}$ be defined by: $(f,\hat{f}) \in A$ if and only if the following assertions hold.
1. $f \in W^{1,p}_{\gamma}( S ) \cap L^{\infty}( S )$.
2. $\hat{f} \in L^{1}( S )$.
3. $\int \limits_{ S} \gamma|\nabla f|_{n}^{p-2}\nabla f\cdot\nabla \varphi d \lambda = \int \limits_{ S } \hat{f} \varphi d \lambda$ for all $\varphi\in W^{1,p}_{\gamma }( S )\cap L^{\infty}( S )$.
Moreover, ${\mathcal{A}}:D({\mathcal{A}})\rightarrow 2^{L^{1}(S)}$ denotes the closure of $A$, i.e. $(f,\hat{f})\in {\mathcal{A}}$ if there is a sequence $((f_{m},\hat{f}_{m}))_{m \in \mathbb{N}}\subseteq A$ such that $\lim \limits_{m \rightarrow \infty} (f_{m},\hat{f}_{m})=(f,\hat{f})$, in $L^{1}(S)\times L^{1}(S)$
One verifies that $A$ is single-valued, see [@ich1 Lemma 3.1]. In addition, if one chooses $\gamma=1$ on $S$, then $A$ is simply the $p$-Laplacian operator with Neumann boundary conditions.\
Moreover, it is possible to determine the closure explicitly, see [@mazon Proposition 3.6]. But the explicit description is fairly technical and not needed for our purposes, therefore it will be omitted. What is important to our purposes is that ${\mathcal{A}}$ is densely defined and m-accretive, see [@mazon Section 3][^5]. Consequently, recalling Remark \[remark\_msee\], we can introduce $(T_{{\mathcal{A}}}(t))_{t \geq 0}$ as the semigroup associated to ${\mathcal{A}}$, which is, according to the same remark, a time-continuous, contractive semigroup on $L^{1}(S)$. In fact, $(T_{{\mathcal{A}}}(t))_{t \geq 0}$ even forms a family of strong solutions, not just of mild ones, see [@mazon Section 3].\
The following three results enable us to apply the result of Sections \[sec\_mp\] and \[sec\_sllnclt\] to the current setting, which is achieved in Theorem \[theoremplaplace\]. In particular, in Proposition \[prop\_plaplacebound\] it is demonstrated how to use Lemma \[lemma\_diffinequality\] in the current situation.
\[lemma\_invsssg\] For each $q \in [1,\infty)$, the space $L^{q}_{0}(S)$ is invariant w.r.t. $T_{{\mathcal{A}}}(t)$. Moreover, the restriction of $T_{{\mathcal{A}}}$ to $L^{q}_{0}(S)$ is a time-continuous, contractive semigroup on $(L^{q}_{0}(S),||\cdot||_{L^{q}(S)})$ which fulfills $T_{{\mathcal{A}}}(t)0=0$ for all $t \in [0,\infty)$.
Let $u\in L^{q}_{0}(S)$, then [@ich1 Lemma 3.4] yields $\overline{(T_{{\mathcal{A}}}(t)u)}=0$ and by [@ich1 Lemma 3.3.2] we get $T_{{\mathcal{A}}}(t)u \in L^{q}(S)$ for all $t \geq 0$. Consequently, $T_{{\mathcal{A}}}(t)u \in L^{q}_{0}(S)$ for all $t \geq 0$. In addition, $(T_{{\mathcal{A}}}(t))_{t \geq 0}$ inherits the semigroup property of $(T_{{\mathcal{A}}}(t))_{t\geq 0}$ and appealing to [@ich1 Lemma 3.3.1] yields that $(T_{{\mathcal{A}}}(t))_{t \geq 0}$ is contractive. Moreover, $T_{{\mathcal{A}}}(t)0=0$ is inferred easily from $0 \in D(A)$, $A0=0$.\
It remains to prove the time-continuity. So let $(h_{m})_{m \in \mathbb{N}}$ be a null-sequence, $t \geq 0$ and $\varepsilon>0$ be given, and assume w.l.o.g. that $t+h_{m}\geq 0$ for all $m \in \mathbb{N}$. Moreover, choose $v \in L^{\infty}_{0}(S)$ such that $||u-v||_{L^{q}(S)}<\frac{\varepsilon}{2}$. Then we get by the time continuity of $T_{{\mathcal{A}}}$, and by passing to a subsequence if necessary, that $\lim \limits_{m \rightarrow \infty} T_{{\mathcal{A}}}(t+h_{m})v=T_{{\mathcal{A}}}(t)v$ almost everywhere. In addition, invoking [@ich1 Lemma 3.3.3] gives $||T_{q}(t+h_{m})v||_{L^{\infty}(S)}\leq ||v||_{L^{\infty}(S)}$ for all $m \in \mathbb{N}$ and employing dominated convergence gives $\lim \limits_{m \rightarrow \infty} T_{{\mathcal{A}}}(t+h_{m})v=T_{{\mathcal{A}}}(t)v$ w.r.t. $||\cdot||_{L^{q}(S)}$. Conclusively, we get by contractivity that $$\begin{aligned}
\lim \limits_{m \rightarrow \infty} ||T_{{\mathcal{A}}}(t+h_{m})u-T_{{\mathcal{A}}}(t)u||_{L^{q}(S)}\leq 2||u-v||_{L^{q}(S)}+\lim \limits_{m \rightarrow \infty} ||T_{{\mathcal{A}}}(t+h_{m})v-T_{{\mathcal{A}}}(t)v||_{L^{q}(S)} \leq \varepsilon,\end{aligned}$$ which yields the desired time continuity.
\[lemma\_lipdifae\] Let $u,v \in L^{2}_{0}(S)\cap D(A)$ and $f:[0,\infty)\rightarrow [0,\infty)$, with $f(t):=||T_{{\mathcal{A}}}(t)u-T_{{\mathcal{A}}}(t)v||^{2}_{L^{2}(S)}$ for all $t \geq 0$. Then $f$ is locally Lipschitz continuous. Consequently, it is differentiable almost everywhere and we have $$\begin{aligned}
\label{prop_plaplaceboundproofeq1}
f^{\prime}(t)=-2 \int \limits_{S}\gamma \left(|\nabla T_{{\mathcal{A}}}(t)u|_{n}^{p-2}\nabla T_{{\mathcal{A}}}(t)u-|\nabla T_{{\mathcal{A}}}(t)v|_{n}^{p-2}\nabla T_{{\mathcal{A}}}(t)v\right)\cdot(\nabla T_{{\mathcal{A}}}(t)u-\nabla T_{{\mathcal{A}}}(t)v)d\lambda,\end{aligned}$$ for a.e. $t \in (0,\infty)$.
Firstly, let us verify the desired local Lipschitz continuity. To this end, fix $c>0$ and note that $[0,c]\ni t \mapsto T_{{\mathcal{A}}}(t)u$ and $[0,c]\ni t \mapsto T_{{\mathcal{A}}}(t)v$ are by [@BenilanBook Lemma 7.8], w.r.t. $||\cdot||_{L^{1}(S)}$, Lipschitz continuous. So let $C_{u},C_{v}\geq 0$ denote their Lipschitz constants. Then, we get for any $t_{1},t_{2} \in [0,c]$ that $$\begin{aligned}
|f(t_{1})-f(t_{2})|\leq |t_{1}-t_{2}|(C_{u}+C_{v})(2||u||_{L^{\infty}(S)}+2||v||_{L^{\infty}(S)}),\end{aligned}$$ where we used [@ich1 Lemma 3.3.3], which reads $||T_{{\mathcal{A}}}(t)w||_{L^{\infty}(S)}\leq ||w||_{L^{\infty}(S)}$ for all $t \geq 0$, $w \in L^{\infty}(S)$.\
Consequently, $f$ is locally Lipschitz continuous and as it is real-valued, it is also differentiable almost everywhere.\
Proof of (\[prop\_plaplaceboundproofeq1\]). Firstly, for all $w \in D(A)$ we have $T_{{\mathcal{A}}}(t)w \in D(A)$ and $-T^{\prime}_{{\mathcal{A}}}(t)w=A T_{{\mathcal{A}}}(t)w$ for a.e. $t \in (0,\infty)$, see [@ich1 Lemma 3.3.4]. Thus, as $D(A)\subseteq W^{1,p}_{\gamma}(S)$, the integral occurring on the right-hand-side of (\[prop\_plaplaceboundproofeq1\]) exists for a.e. $t \in (0,\infty)$ and for almost every $t \in (0,\infty)$ we have $T_{{\mathcal{A}}}(t)u, T_{{\mathcal{A}}}(t)v \in D(A)$, $-T^{\prime}_{{\mathcal{A}}}(t)u=A T_{{\mathcal{A}}}(t)u$ and $-T^{\prime}_{{\mathcal{A}}}(t)v=A T_{{\mathcal{A}}}(t)v$. In light of this, it is intuitively clear that $$\begin{aligned}
\label{prop_plaplaceboundproofeq3}
f^{\prime}(t)= -2 \int \limits_{S} (T_{{\mathcal{A}}}(t)u-T_{{\mathcal{A}}}(t)v)(AT_{{\mathcal{A}}}(t)u-AT_{{\mathcal{A}}}(t)v)d\lambda,\end{aligned}$$ for a.e. $t \in (0,\infty)$ and making rigorous that one is allowed to perform the needed exchange of the integral and the differential works by the aid of dominated convergence and identical to the proof of [@ich1 Lemma 5.3].\
Finally, (\[prop\_plaplaceboundproofeq3\]) implies (\[prop\_plaplaceboundproofeq1\]) by using $(T_{{\mathcal{A}}}(t)u-T_{{\mathcal{A}}}(t)v)$ as a test function in the definition of $A$.
\[prop\_plaplacebound\] Let $u,v \in L^{2}_{0}(S)$. Then we have $$\begin{aligned}
\label{prop_plaplaceboundeq}
||T_{{\mathcal{A}}}(t)u-T_{{\mathcal{A}}}(t)v||_{L^{2}(S)}\leq \left(\kappa t+||u-v||_{L^{2}(S)}^{2-p}\right)^{\frac{1}{2-p}},~\forall t\geq 0,\end{aligned}$$ where $\kappa:=(p-2)2^{2-p}\left(\int \limits_{S}\gamma^{\frac{2}{2-p}}d\lambda\right)^{\frac{2-p}{2}}C_{S,2}^{-p}$ and $C_{S,2}$ is the Poincaré constant of $S$ in $L^{2}_{0}(S)$.
For now assume in addition $u,v \in L^{2}_{0}(S)\cap D(A)$, set $f(t):=||T_{{\mathcal{A}}}(t)u-T_{{\mathcal{A}}}(t)v||^{2}_{L^{2}(S)}$ for all $t \geq 0$ and let us derive an upper bound on $f^{\prime}$, which exists a.e. on $(0,\infty)$ due to Lemma \[lemma\_lipdifae\].\
Firstly, we have $W^{1,p}_{\gamma}(S)\subseteq W^{1,2}(S)$, since appealing to Hölder’s inequality gives $$\begin{aligned}
\int \limits_{S} |\nabla \varphi|_{n}^{2}d\lambda = \int \limits_{S} \gamma^{-\frac{2}{p}}\gamma^{\frac{2}{p}} |\nabla \varphi|_{n}^{2}d\lambda \leq \left(\int \limits_{S} \gamma^{\frac{2}{2-p}}d\lambda\right)^{\frac{p-2}{p}}\left(\int \limits_{S} \gamma |\nabla \varphi|_{n}^{p}d\lambda\right)^{\frac{2}{p}}<\infty,~\forall \varphi \in W^{1,p}_{\gamma}(S).\end{aligned}$$ Consequently, employing Poincaré’s inequality yields $$\begin{aligned}
\label{prop_plaplaceboundproofeq4}
\int \limits_{S}\gamma |\nabla \varphi|_{n}^{p}d\lambda \geq C_{S,2}^{-p}\left(\int \limits_{S} \varphi^{2}d\lambda\right)^{\frac{p}{2}}\left(\int \limits_{S} \gamma^{\frac{2}{2-p}}d\lambda\right)^{\frac{2-p}{2}},~\forall \varphi \in W^{1,p}_{\gamma}(S)\cap L^{2}_{0}(S).\end{aligned}$$ Moreover, it is well known that $(|x|_{n}^{p-2}x-|y|_{n}^{p-2}y)\cdot(x-y)\geq 2^{2-p}|x-y|_{n}^{p}$ for all $x,y \in \mathbb{R}^{n}$, see [@plaplaceinequality Lemma 3.6]. By [@ich1 Lemma 3.3.4], we get $T_{{\mathcal{A}}}(t)u,T_{{\mathcal{A}}}(t)v \in W^{1,p}_{\gamma}(S)$ for a.e. $t \in (0,\infty)$ and by the aid of Lemma \[lemma\_invsssg\] we then obtain $T_{{\mathcal{A}}}(t)u-T_{{\mathcal{A}}}(t)v \in W^{1,p}_{\gamma}(S)\cap L^{2}_{0}(S)$ for a.e. $t \in (0,\infty)$. These observations enable us to conclude from (\[prop\_plaplaceboundproofeq4\]) and Lemma \[lemma\_lipdifae\] that $$\begin{aligned}
f^{\prime}(t)\leq - 2^{3-p} \int \limits_{S}\gamma |\nabla T_{{\mathcal{A}}}(t)u-\nabla T_{{\mathcal{A}}}(t)v|_{n}^{p}d\lambda \leq - 2^{3-p}C_{S,2}^{-p}\left(\int \limits_{S} \gamma^{\frac{2}{2-p}}d\lambda\right)^{\frac{2-p}{2}}f(t)^{\frac{p}{2}},\end{aligned}$$ for a.e. $t \in (0,\infty)$. Thus, by setting $\tilde{\rho}:= \frac{2}{p-2}$, we get $f^{\prime}(t)\leq -\kappa \tilde{\rho}f(t)^{1+\frac{1}{\tilde{\rho}}}$ for a.e. $t \in (0,\infty)$. Hence, invoking Lemma \[lemma\_diffinequality\] yields $f(t)\leq (\kappa t+f(0)^{-\frac{1}{\tilde{\rho}}})^{-\tilde{\rho}}$; thus by taking the square root and noting that $u,v\in L^{2}_{0}(S)\cap D(A)$ were arbitrary, we get $$\begin{aligned}
\label{prop_plaplaceboundproofeq5}
||T_{{\mathcal{A}}}(t)u-T_{{\mathcal{A}}}(t)v||_{L^{2}(S)}\leq \left(\kappa t+||u-v||_{L^{2}(S)}^{2-p}\right)^{\frac{1}{2-p}},~\forall u,v \in L^{2}_{0}(S)\cap D(A),\end{aligned}$$ for all $t \in [0,\infty)$. It remains to generalize the preceding inequality to all $u,v \in L^{2}_{0}(S)$. So fix $t \in [0,\infty)$, let $u,v \in L^{2}_{0}(S)$ and introduce $(u_{m})_{m \in \mathbb{N}},(v_{m})_{m \in \mathbb{N}}\subseteq D(A)$ such that $\lim \limits_{m \rightarrow \infty} u_{m}=u$ and $\lim \limits_{m \rightarrow \infty} v_{m}=v$ in $L^{2}(S)$, such sequences exist by [@ich1 Lemma 5.6]. Now, one instantly verifies that $u_{m}-\overline{(u_{m})} \in D(A)$, with $A(u_{m}-\overline{(u_{m})})=A u_{m}$. Consequently, $u_{m}-\overline{(u_{m})} \in D(A)\cap L^{2}_{0}(S)$ for all $m \in \mathbb{N}$ and $\lim \limits_{m \rightarrow \infty} u_{m}-\overline{(u_{m})}=u$, in $L^{2}(S)$ since $\overline{(u)}=0$. Conclusively, as the analogous statements hold for $v,v_{m}$, (\[prop\_plaplaceboundeq\]) follows from (\[prop\_plaplaceboundproofeq5\]) and Lemma \[lemma\_invsssg\].
\[theoremplaplace\] Let $q \in [1,2]$ and let $(\eta_{k})_{k \in \mathbb{N}}\subseteq \mathcal{M}(\Omega;L_{0}^{q}(S))$ be an i.i.d. sequence. Moreover, let $(\beta_{m})_{m \in \mathbb{N}}$ be another i.i.d. sequence which is independent of $(\eta_{k})_{k \in \mathbb{N}}$ and assume that $\beta_{m}\sim Exp(\theta)$ for all $m \in \mathbb{N}$, where $\theta \in (0,\infty)$. In addition, assume $||\eta_{k}||_{L^{q}(S)} \in L^{2}(\Omega)$ for all $k \in \mathbb{N}$. Moreover, let $x \in \mathcal{M}(\Omega;L_{0}^{q}(S))$ be an independent initial, i.e. independent of $((\eta_{k})_{k \in \mathbb{N}},(\beta_{k})_{k \in \mathbb{N}})$ and let ${\mathbb{X}}_{x}:[0,\infty)\times \Omega \rightarrow L_{0}^{q}(S)$ be the process generated by $((\beta_{k})_{k \in \mathbb{N}},(\eta_{k})_{k \in \mathbb{N}},x,T_{{\mathcal{A}}})$ in $L_{0}^{q}(S)$.\
Then $({\mathbb{X}}_{x}(t))_{t \geq 0}$ is a time-homogeneous Markov process (w.r.t. the completion of its natural filtration) which possesses a unique invariant probability measure $\bar{\mu}:{\mathfrak{B}}(L^{q}_{0}(S))\rightarrow [0,1]$. In addition, for any $\psi \in Lip(L_{0}^{q}(S))$, the convergence $$\begin{aligned}
\label{theoremplaplace_eq1}
\lim \limits_{t \rightarrow \infty} \frac{1}{t}\int \limits_{0}\limits^{t}\psi({\mathbb{X}}_{x}(\tau))d\tau=\int \limits_{L^{q}_{0}(S)} \psi(v)\bar{\mu}(dv):= \overline{(\psi)},\end{aligned}$$ takes place with probability one, and if additionally $p \in (2,4)$, then there is a $\sigma^{2}(\psi)\in [0,\infty)$ such that $$\begin{aligned}
\label{theoremplaplace_eq2}
\lim \limits_{t \rightarrow \infty }\frac{1}{\sqrt{t}}\left( \int \limits_{0}\limits^{t}\psi({\mathbb{X}}_{x}(\tau))d\tau-t\overline{(\psi)}\right)=Y\sim N(0,\sigma^{2}(\psi)),\end{aligned}$$ in distribution.
By Lemma \[lemma\_invsssg\] $(T_{{\mathcal{A}}}(t))_{t \geq 0}$ is a time continuous, contractive semigroup on $L^{q}_{0}(S)$. Consequently, by choosing $V=L^{q}_{0}(S)$ in Section \[sec\_mp\] it follows from Theorem \[theorem\_mp\] that ${\mathbb{X}}_{x}$ is a time-continuous Markov process.\
Moreover, it follows from Lemma \[lemma\_invsssg\] and Proposition \[prop\_plaplacebound\] that $(T_{{\mathcal{A}}}(t))_{t \geq 0}$ fulfills Assumption \[assumption\], where we choose $V=L^{q}_{0}(S)$, $W=L^{2}_{0}(S)$, $\rho:=\frac{1}{p-2}$ and $\kappa$ as in Proposition \[prop\_plaplacebound\]. Consequently, appealing to Proposition \[prop\_uniqueinvpropmeas\] yields the existence of a unique invariant probability measure and Theorem \[theorem\_slln\] implies (\[theoremplaplace\_eq1\]). Finally, (\[theoremplaplace\_eq2\]) follows from Theorem \[theorem\_clt\], since $p \in (2,4)$ implies $\rho>\frac{1}{2}$.
<span style="font-variant:small-caps;">Acknowledgment</span>
The present author is grateful to Prof. Dr. Alexei Kulik for fruitful conversations during a research stay of the present author at Technische Universität Berlin.
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[^1]: Affiliation: Ulm University
[^2]: Affiliation’s address: 89081 Ulm, Helmholtzstr. 18, Germany
[^3]: Author’s E-Mail: [email protected]
[^4]: Author’s ORCID: 0000-0001-7823-0648
[^5]: This is also stated in [@ich1 Theorem 2.3], which summarizes the highlights of [@mazon Section 3].
|
---
abstract: 'We investigate a simple theory where Baryon number (B) and Lepton number (L) are local gauge symmetries. In this theory B and L are on the same footing and the anomalies are cancelled by adding a single new fermionic generation. There is an interesting realization of the seesaw mechanism for neutrino masses. Furthermore there is a natural suppression of flavour violation in the quark and leptonic sectors since the gauge symmetries and particle content forbid tree level flavor changing neutral currents involving the quarks or charged leptons. Also one finds that the stability of a dark matter candidate is an automatic consequence of the gauge symmetry. Some constraints and signals at the Large Hadron Collider are briefly discussed.'
author:
- 'Pavel Fileviez Pérez$^{1}$'
- 'Mark B. Wise$^{2}$'
title: Baryon and Lepton Number as Local Gauge Symmetries
---
I. Introduction
===============
In the Standard Model (SM) of particle physics baryon number (B) and lepton number (L) are accidental global symmetries. In addition the individual lepton numbers, $U(1)_{L_i}$ where $L_i=L_e$, $L_{\mu}$, $L_{\tau}$, are also automatic global symmetries of the renormalizable couplings. Since neutrinos oscillate the individual lepton numbers $U(1)_{L_i}$, for a given $i$, cannot be exact symmetries at the electroweak scale and furthermore, the neutrinos are massive. At the non-renormalizable level in the SM one can find operators that violate baryon number and lepton number. For example, $QQQL/\Lambda_{B}^2$ and $LLHH/\Lambda_L$, where $\Lambda_B$ and $\Lambda_L$ are the scales where B and L are broken, respectively [@Weinberg]. These operators give rise to new phenomena (that are not permitted by the renormalizable couplings) such as proton decay [@proton] and neutrinoless double beta decay [@Vogel].
Baryon number must be broken in order to explain the origin of the matter-antimatter asymmetry in the Universe. Also having Majorana neutrino masses is very appealing since one can explain the smallness of neutrino masses through the seesaw mechanism [@TypeI]. Hence lepton number is also expected to be broken. In the future it may be possible to observe the violation of baryon number and lepton number (For future experimental proposals and current bounds, see Ref. [@experiments].).
In this paper we examine the possibility that B and L are spontaneously broken gauge symmetries where the scale for B and L breaking is low (i.e., around a [TeV]{}). Then one does not need a “desert region" between the weak and GUT scales to adequately suppress the contribution of dimension six baryon number violating operators to proton decay [@proton]. (Note that gauging B-L does not address this issue since the dimension six operators mentioned above are B-L invariant.) In this paper we propose a new model where B and L are gauged and are spontaneously broken at a low-scale.
We construct and discuss a simple extension of the standard model based on the gauge symmetry, $SU(3) \bigotimes SU(2) \bigotimes U(1)_Y \bigotimes U(1)_B \bigotimes U(1)_L$. The anomalies are cancelled by adding a single new fermionic generation of opposite chirality, where the new leptons have ${\rm L}=3$ and the additional quarks have ${ \rm B}=1$. In this model we find that there is a natural suppression of flavor violation at tree level in the quark and charged leptonic sectors. The realization of the seesaw mechanism for the generation of neutrino masses is investigated and we show that the spontaneous breaking of the gauge symmetry does not generate dangerous baryon number violating operators. The model has a dark matter candidate and its stability is an automatic consequence of the gauge symmetry. The most generic signals at the Large Hadron Collider and the relevant constraints are briefly discussed.
II. Gauging Baryon and Lepton Numbers
=====================================
In this section we show how to cancel the anomalies in a simple extension of the standard model theory based on the gauge group $G_{SM} \bigotimes U(1)_B \bigotimes U(1)_L$. In this model the SM fields transform as: $Q_L^T = \left( u, \ d \right)_L \sim (3,2,1/6,1/3,0), ~
l_L^T = \left( \nu, \ e \right)_L \sim (1,2,-1/2,0,1),~
u_R \sim (3,1,2/3,1/3,0),~ d_R \sim (3,1,-1/3,1/3,0),$ and $e_R \sim (1,1,-1,0,1)$ under the gauge group. We add three generations of right handed neutrinos, $\nu_R\sim (1,1,0,0,1)$, to the standard model particles above to study the generation of neutrino masses. In order to find an anomaly free theory one needs to add additional new fermions to cancel the following anomalies (Of course one has to keep in mind that the anomalies in the SM gauge group should be satisfied as well.):
- Baryonic Anomalies:
- ${\cal A}_1 \left( SU(3)^2 \bigotimes U(1)_B \right)$: In this case all the quarks contribute and one finds ${\cal A}_1^{SM} =0$.
- ${\cal A}_2 \left( SU(2)^2 \bigotimes U(1)_B \right)$: Since in the SM there is only one quark doublet, $Q_L$, for each family one cannot cancel this anomaly, ${\cal A}_2^{SM} =\frac{3}{2}$. Therefore, here one needs extra states in a non-trivial representation of $SU(2)$.
- ${\cal A}_3 \left( U(1)_Y^2 \bigotimes U(1)_B \right)$: In the Abelian sector one has the contributions of all quarks and one finds $
{\cal A}_3^{SM} = - \frac{3}{2}.
$
- ${\cal A}_4 \left(U(1)_Y \bigotimes U(1)_B^2\right)$: In this case, since all SM quarks have the same baryon number, this anomaly is equivalent to the $U(1)_Y$ anomaly condition in the SM, i.e. ${\cal A}_4^{SM}=0$.
- ${\cal A}_5 \left( U(1)_B \right)$: The baryon-gravity anomaly is also cancelled in the SM, ${\cal A}_5^{SM}=0$.
- ${\cal A}_6 \left(U(1)_B^3\right)$: In this case one finds that this anomaly is zero in the SM, ${\cal A}_6^{SM}=0$.
- Leptonic Anomalies:
- ${\cal A}_7 \left( SU(2)^2 \bigotimes U(1)_L\right)$: It is easy to show that ${\cal A}_7^{SM}=3/2$.
- ${\cal A}_8 \left( U(1)_Y^2 \bigotimes U(1)_L \right)$: As in the previous case, in the SM this anomaly is not zero, ${\cal A}_8^{SM}=-3/2$.
- ${\cal A}_{9} \left(U(1)_Y \bigotimes U(1)_L^2\right)$: In the SM this anomaly is cancelled.
- ${\cal A}_{10} \left( U(1)_L\right)$: The lepton-gravity anomaly is cancelled since we add three families of right-handed neutrinos, $\nu_R \sim (1,1,0,0,1)$.
- ${\cal A}_{11} \left(U(1)_L^3\right)$: As in the previous case this anomaly is cancelled once we add three families of right-handed neutrinos.
We have to cancel all the anomalies discussed above (and not induce standard model gauge anomalies as well) in order to find an anomaly free theory with B and L gauged. For a previous study of these anomalies see Ref. [@Foot].
There are two simple ways to cancel the anomalies. They are:
- Case 1) All baryonic anomalies are cancelled adding new quarks, $Q_L^{'T} = ( u^{'}, d^{'})_L \sim (3,2,1/6,-1,0)$, $u_R^{'} \sim (3,1,2/3,-1,0) $, and $d_R^{'} \sim (3,1,-1/3,-1,0)$, which transform as the SM quarks but with baryon number, $B=-1$. At the same time the leptonic anomalies are cancelled if one adds new leptons $l_L^{'T} = ( \nu^{'}, e^{'})_L \sim (1,2,-1/2,0,-3)$, $e_R^{'} \sim (1,1,-1,0,-3)$ and $\nu_R^{'}\sim (1,1,0,0,-3)$. All anomalies in the SM gauge group are cancelled since we have introduced one new full family. It differs from the usual standard model families since the new quarks have baryon number minus one and the new leptons have lepton number minus three[[^1]]{}.
- Case 2) The baryonic anomalies are cancelled adding new quarks, $Q_R^{'T} = ( u^{'}, d^{'})_R \sim (3,2,1/6,1,0)$, $u_L^{'} \sim (3,1,2/3,1,0) $, and $d_L^{'} \sim (3,1,-1/3,1,0)$, which transform as the SM quarks of opposite chirality but with baryon number, $B=1$. At the same time the leptonic anomalies are cancelled if one adds new leptons $l_R^{'T} = ( \nu^{'}, e^{'})_R \sim (1,2,-1/2,0,3)$, $e_L^{'} \sim (1,1,-1,0,3)$ and $\nu_L^{'}\sim (1,1,0,0,3)$. All anomalies in the SM gauge group are cancelled since we have introduced one new full family but with opposite chirality. It also differs from the usual standard model families since the new quarks have baryon number one and the new leptons have lepton number three.
These are the two simplest fermionic contents that give an anomaly free theory with gauge group $SU(3) \bigotimes SU(2) \bigotimes U(1)_Y \bigotimes U(1)_B \bigotimes U(1)_L$. For the study of the phenomenological properties of models with an extra generation see Ref. [@Sher].
III. Mass Generation and Flavour Violation
===========================================
A. Quark Sector
---------------
We begin by considering the case 1), new generation where the chirality is the same as in the standard model generations. Masses for the new quarks present in the model are generated from their couplings to the SM Higgs: $$\begin{aligned}
-\Delta {\cal L}_{q' {\rm mass}}^{(1)}&=& h_U^{'} \ \overline{Q^{'}_L} \ \tilde{H} \ u_R^{'} \ \nonumber \\
&+&\ h_D^{'} \ \overline{Q^{'}_L} \ {H} \ d_R^{'} \ + \ \rm{h.c.},\end{aligned}$$ where $H \sim (1,2,1/2,0,0)$ is the SM Higgs, and $\tilde{H} = i \sigma_2 H^*$.
To avoid having a stable colored particle we couple the SM fermions to the new fourth generation quarks. However, at the same time it is important to avoid tree level flavor changing neutral currents in the quark sector. We achieve this goal by adding a new scalar field, $\phi \sim (1,2,1/2,4/3,0)$ which does not get a vev. The interactions of this scalar are, $
Y_1 \ \overline{Q^{'}_L} \ \tilde{\phi} \ u_R \ + \ \rm{h.c.},
$ which permit the new fourth generation quarks to decay to a scalar and a standard model quark. One might think that this particle is the dark matter. Unfortunately, it is not consistent with experiment [@Drees] to have the $\phi^0$ be the dark matter because the mass degeneracy of the imaginary and real parts [@Cirelli].
Next we consider case 2), where the new fermions have the opposite chirality to the standard model. We shall see that in this case there is a dark matter candidate. Now, mass for the new quarks is generated through the terms, $$\begin{aligned}
-\Delta {\cal L}_{q' {\rm mass}}^{(2)}&=&Y_U^{'} \ \overline{Q^{'}_R} \ \tilde{H} \ u_L^{'} \ \nonumber \\
&+&\ Y_D^{'} \ \overline{Q^{'}_R} \ {H} \ d_L^{'} \ + \ \rm{h.c.}.\end{aligned}$$ Decays of the new quarks are induced by adding a new scalar field $X$ with gauge quantum numbers, $X \sim(1,1,0,-2/3,0)$ and the following terms occur in the Lagrange density: $$\begin{aligned}
-\Delta{\cal L}_{{DM}}^{(2)}&=& \lambda_Q \ X \ \overline{Q_L }\ Q_R^{'} \ + \ \lambda_U \ X \ \overline{u_R} \ u_L^{'} \nonumber \\
& + & \lambda_D \ X \ \overline{d_R }\ d_L^{'} \ + \ \rm{h.c.}.
\label{DM}\end{aligned}$$ The field $X$ does not get a vev and so there is no mass mixing between the new exotic generation quarks and the standard model ones. When $X$ is the lightest new particle with baryon number, it is stable. This occurs because the model has a global U(1) symmetry where the $Q'_R$, $u'_L$, $d'_L$ and $X$ get multiplied by a phase. This $U(1)$ symmetry is an automatic consequence of the gauge symmetry and the particle content. Notice that the new fermions have $V+A$ interactions with the W-bosons. The $X$ particle is a dark matter candidate and its properties will be investigated in a future publication.
The field $X$ has flavor changing couplings that cause transitions between quarks with baryon number 1 and the usual quarks with baryon number 1/3. However, since there is no mass mixing between these two types of quarks integrating out the $X$ does not generate any tree level flavor changing neutral currents for the ordinary quarks. Those first occur at the one loop level.
B. Leptonic Sector
------------------
The interactions that generate masses for the new charged leptons in case 1) are: $$\begin{aligned}
-\Delta{\cal L}_{l}^{(1)}&=& Y_E^{'} \ \overline{l^{'}_L} \ {H} \ e_R^{'} \ + \ \rm{h.c.}\end{aligned}$$ while for the neutrinos they are $$\begin{aligned}
-\Delta{\cal L}_{\nu}^{(1)}&=& Y_\nu \ l H \nu^C \ + \ Y_\nu^{'} \ l^{'} H N \ + \
\nonumber
\\
& + & \ \frac{\lambda_a}{2} \ \nu^C \ S_L \ \nu^C \ + \ {\lambda_b} \ \nu^C \ S_L^\dagger \ N \ + \ \rm{h.c.},\end{aligned}$$ where $S_L \sim (1,1,0,0,2)$ is the Higgs that breaks $U(1)_L$, generating masses for the right-handed neutrinos and the quark-phobic $Z^{'}_L$. We introduce the notation $\nu^C = (\nu_R)^C$ and $N= (\nu_R^{\prime})^C $. After symmetry breaking the mass matrix for neutrinos in the left handed basis, $(\nu, \nu^{'}, N, \nu^C)$, is given by the eight by eight matrix $${\cal M}_{N} =
\begin{pmatrix}
%11
0
&
%12
0
&
%13
0
&
%14
M_D
\\
%21
0
&
%22
0
&
%23
M_D^{'}
&
%24
0
\\
%31
0
&
%32
(M_D^{'})^T
&
%33
0
&
%34
M_b
\\
%41
M^T_D
&
%42
0
&
%43
M_b^T
&
%44
M_a
\end{pmatrix}.
\label{neutralino}$$ Here, $M_D=Y_\nu v_H/\sqrt{2}$ and $M_a=\lambda_a v_L/\sqrt{2}$ are $3\times3$ matrices, $M_b=\lambda_b v_L^*/ \sqrt{2}$ is a $1\times 3$ matrix, $M_D^{'}=Y_\nu^{'} v_H/\sqrt{2}$ is a number and $\langle S_L\rangle= v_L/{\sqrt{2}}$. Lets assume that the three right-handed neutrinos $\nu^C$ are the heaviest. Then, integrating them out generates the following mass matrix for the three light-neutrinos : $${\cal M}_\nu = M_D \ M_a^{-1} \ M_D^T.$$ In addition, a Majorana mass $M'$ for the fourth generation right handed neutrino $N$, $$M^{'}=M_b M_a^{-1} M_b^T,$$ is generated. Furthermore, suppose that $M^{'} << M_D^{'}$, then the new fourth generation neutrinos $\nu^{'}$ and $N$ are quasi-Dirac with a mass equal to $M_D^{'}$. Of course we need this mass to be greater than $M_Z/2$ to be consistent with the measured $Z$-boson width. In this model we have a consistent mechanism for neutrino masses which is a particular combination of Type I seesaws [@TypeI].
The interactions that generate masses for the new leptons in case 2) are: $$\begin{aligned}
-\Delta{\cal L}_{l}^{(2)}&=& Y_E^{''} \ \overline{l^{'}_R} \ {H} \ e_L^{'} \ + \ \rm{h.c.}
\\
-\Delta{\cal L}_{\nu}^{(2)}&=& Y_\nu \ l H \nu^C \ + \ Y_\nu^{''} \ \bar{l}_R^{'} \tilde{H} \nu_L^{'} \ + \
\nonumber
\\
& + & \ \frac{\lambda_a}{2} \ \nu^C \ S_L \ \nu^C \ + \ {\lambda_c} \ \nu^C \ S_L^\dagger \ \nu_L^{'}
\nonumber
\\
& + & \lambda_l \ \bar{l}_R^{'} \ l_L S_L \ + \ \lambda_e \ \bar{e}_R \ e_L^{'} \ S_L^{\dagger}\ + \ \rm{h.c.},\end{aligned}$$ Notice that in this case $S_L$ does not get a vev in order to avoid tree level lepton flavour violation. Then, the neutrinos can be Dirac fermions and one has to introduce a new scalar field to break $U(1)_L$. Let us say $S_L^{'} \sim (1,1,0,0,n_L)$, where $n_L \neq \pm 2,\pm 6$. Notice that if in this case we do not introduce $S_L$, the heavy extra Dirac neutrino is stable and it is difficult to satisfy the experimental bounds from dark matter direct detection in combination with the collider bounds on a heavy stable Dirac neutrino.
In order to complete the discussion of symmetry breaking we introduce a new Higgs, $S_B$, with non-zero baryon number (but no other gauge quantum numbers) which gets the vev, $v_B$, breaking $U(1)_B$ and giving mass to the leptophobic $Z^{'}_B$. In summary, the Higgs sector in case 2) is composed of the SM Higgs, $S_L$, $S_L^{'}$, $S_B$ and $X$. This is the minimal Higgs sector needed to have a realistic theory where B and L are both gauged, and have a DM candidate.
IV. Flavour Violation and Signals at the LHC
============================================
- Flavour Violation:
Even though there are no quark flavor changing neutral currents at tree level, they do occur at the one loop level. The scalar $X$ is used to couple the fourth generation quarks to the ordinary ones and we find that at one loop there are box diagrams that give contributions (after integrating out the heavy particles) to the effective Lagrangian for $K-{\bar K}$ mixing of the form, $\lambda^4 {{\bar d_{L,R}}\gamma^{\mu}s_{L,R} {\bar d_{L,R}}\gamma_{\mu}s_{L,R} /(16 \pi^2 M^2}) +{\rm h.c.}$, where the product of the four elements of the Yukawa matrix $\lambda_i$ (and CKM angles) that enter into the coefficient are denoted by $\lambda^4$ and $M$ denotes a mass scale set by the masses of the new fields in the loop. For $M$ of order $100 ~{\rm GeV}$ this is negligible provided $\lambda <10^{-2}$.
Charged lepton neutral currents are induced at one loop level. For example, there is a one loop contribution to the amplitude for $\mu \rightarrow e \gamma$. It involves the usual factor of the muon mass and one loop suppression factor. In addition it requires two factors of the mixing between the essentially massless ordinary neutrinos and the new fourth generation neutrino. In the limit we discussed above where the fourth generation neutrino in quasi-Dirac this mixing is small. A detailed study of the flavour violation issue in this context is beyond the scope of this letter.
- $Z_L$ and $Z_B$:
In this model one can observe lepton number violating processes at the LHC through the channels with same-sign dileptons: $ pp \ \to \ Z_L^* \ \to \ \nu_R \ \nu_R \ \to \ W^{\pm} \ W^{\pm} \ e^{\mp}_i \ e^{\mp}_j$. However, for this Drell-Yan production one needs the mixing between $Z$ and $Z_L$, or $Z_L$ and $Z_B$. One can also have pair production of $Z_L$ through its couplings to the physical Higgses. For a study of these channels see Ref. [@Tao]. In the case of the lepto-phobic $Z_B$ its coupling to the third generation of quarks can be used to observe it at the LHC. In particular, the channel: $pp \ \to \ Z_B^* \ \to \ t \ \bar{t}$. For previous studies of a lepto-phobic $Z^{'}$ see Ref. [@ZB]. For a review on the properties of new $U(1)$ gauge bosons see for example Ref. [@Langacker].
- Decays of the new quarks:
As we have discussed above to avoid a stable colored particle we introduced the new interactions in Eq.(\[DM\]). Now we discuss signals at the LHC coming from these interactions. Here we will focus on the case where the new quarks can decay into $X$ and the top quark. Then, one can have the interesting channel: $pp \ \to \ \bar{t}^{'} \ t^{'} \ \to \ X \ X \ \bar{t} \ t$. Since $X$ is stable, the final state has missing energy and a $t \bar{t}$ pair (After posting the first version of this article the paper [@Feng] appeared.).
- Baryon number violating processes:
Since baryon number is spontaneously broken it is important to consider possible baryon number violating operators that might arise after symmetry breaking. Using the minimal Higgs sector discussed above one can see that the operator, $QQQL/\Lambda_B^2$, is never generated since we do not have a Higgs which carries B and L quantum numbers. Also, in general we expect the $\Delta B=2$ operators only when $S_B$ has $B=2$, but in general we do not induce this operator as well. Hence the model we have introduced does not give rise to dangerous baryon number operators. At the same time, in order to avoid the vev for $X$, one should forbid the terms with an odd power of $X$ in the scalar potential. Then, one must impose the condition $B(S_B) \neq \pm 2/3, \pm 1/3, \pm 2/9, \pm 2$.
- Other aspects:
It is well known, that in any model with an extra fermionic generation one finds that the gluon fusion cross section for the SM Higgs is larger by a factor 9. See for example Ref. [@Tilman]. However, the new results from CDF and D0 [@Higgs], do not rule out our model when the Higgs mass is $114 \ \rm{GeV} \ < \ M_H \ < 120$ GeV, or when $M_H > 200$ GeV. Notice that for a large mixing between $H$ and the singlets $S_L$ and $S_B$ one can relax those constraints. For the study of other aspects, such as electroweak precision constraints, see for example Ref. [@Tilman].
V. Summary
==========
We have constructed and investigated a simple theory where Baryon (B) number and Lepton (L) number are local gauge symmetries. In this theory, B and L are treated on the same footing, anomalies are cancelled by adding a single new fermionic generation, there is a simple realization of the seesaw mechanism for neutrino masses and there is a natural suppression for flavour violation at tree level in the quark and leptonic sectors. It is important to emphasize that in this theory the B and L violation scales can be as low as TeV and one does not have dangerous processes, such as proton decay. Also one finds that the stability of a dark matter candidate is an automatic consequence of the gauge symmetry. Constraints and signals at the Large Hadron Collider have been briefly discussed. The results of our paper can be applied to different theories for physics beyond the Standard Model, such as the Minimal Supersymmetric Standard Model, where new interactions can give rise to fast proton decay.
Acknowledgment {#acknowledgment .unnumbered}
--------------
P. F. P. would like to thank T. Han and S. Spinner for useful discussions and pointing out Ref. [@Feng]. The work of P. F. P. was supported in part by the U.S. Department of Energy contract No. DE-FG02-08ER41531 and in part by the Wisconsin Alumni Research Foundation. The work of M.B.W. was supported in part by the U.S. Department of Energy under contract No. DE-FG02-92ER40701.
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address:
- '$^{\dag}$Deceased, May 2015. $^{\ddag}$Deceased, March 2015. $^{\sharp}$Deceased, May 2012.'
- '$^{1}$LIGO, California Institute of Technology, Pasadena, CA 91125, USA '
- '$^{2}$Louisiana State University, Baton Rouge, LA 70803, USA '
- '$^{3}$Università di Salerno, Fisciano, I-84084 Salerno, Italy '
- '$^{4}$INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy '
- '$^{5}$University of Florida, Gainesville, FL 32611, USA '
- '$^{6}$LIGO Livingston Observatory, Livingston, LA 70754, USA '
- '$^{7}$Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Université Savoie Mont Blanc, CNRS/IN2P3, F-74941 Annecy-le-Vieux, France '
- '$^{8}$Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany '
- '$^{9}$Nikhef, Science Park, 1098 XG Amsterdam, Netherlands '
- '$^{10}$LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA '
- '$^{11}$Instituto Nacional de Pesquisas Espaciais, 12227-010 São José dos Campos, São Paulo, Brazil '
- '$^{12}$INFN, Gran Sasso Science Institute, I-67100 L’Aquila, Italy '
- '$^{13}$INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy '
- '$^{14}$Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India '
- '$^{15}$International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560012, India '
- '$^{16}$University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA '
- '$^{17}$Leibniz Universität Hannover, D-30167 Hannover, Germany '
- '$^{18}$Università di Pisa, I-56127 Pisa, Italy '
- '$^{19}$INFN, Sezione di Pisa, I-56127 Pisa, Italy '
- '$^{20}$Australian National University, Canberra, Australian Capital Territory 0200, Australia '
- '$^{21}$The University of Mississippi, University, MS 38677, USA '
- '$^{22}$California State University Fullerton, Fullerton, CA 92831, USA '
- '$^{23}$LAL, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, 91400 Orsay, France '
- '$^{24}$Chennai Mathematical Institute, Chennai 603103, India '
- '$^{25}$Università di Roma Tor Vergata, I-00133 Roma, Italy '
- '$^{26}$University of Southampton, Southampton SO17 1BJ, United Kingdom '
- '$^{27}$Universität Hamburg, D-22761 Hamburg, Germany '
- '$^{28}$INFN, Sezione di Roma, I-00185 Roma, Italy '
- '$^{29}$Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-14476 Potsdam-Golm, Germany '
- '$^{30}$APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cité, F-75205 Paris Cedex 13, France '
- '$^{31}$Montana State University, Bozeman, MT 59717, USA '
- '$^{32}$Università di Perugia, I-06123 Perugia, Italy '
- '$^{33}$INFN, Sezione di Perugia, I-06123 Perugia, Italy '
- '$^{34}$European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy '
- '$^{35}$Syracuse University, Syracuse, NY 13244, USA '
- '$^{36}$SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom '
- '$^{37}$LIGO Hanford Observatory, Richland, WA 99352, USA '
- '$^{38}$Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Miklós út 29-33, Hungary '
- '$^{39}$Columbia University, New York, NY 10027, USA '
- '$^{40}$Stanford University, Stanford, CA 94305, USA '
- '$^{41}$Università di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy '
- '$^{42}$INFN, Sezione di Padova, I-35131 Padova, Italy '
- '$^{43}$CAMK-PAN, 00-716 Warsaw, Poland '
- '$^{44}$University of Birmingham, Birmingham B15 2TT, United Kingdom '
- '$^{45}$Università degli Studi di Genova, I-16146 Genova, Italy '
- '$^{46}$INFN, Sezione di Genova, I-16146 Genova, Italy '
- '$^{47}$RRCAT, Indore MP 452013, India '
- '$^{48}$Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia '
- '$^{49}$SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom '
- '$^{50}$University of Western Australia, Crawley, Western Australia 6009, Australia '
- '$^{51}$Department of Astrophysics/IMAPP, Radboud University Nijmegen, 6500 GL Nijmegen, Netherlands '
- '$^{52}$Artemis, Université Côte d’Azur, CNRS, Observatoire Côte d’Azur, CS 34229, Nice cedex 4, France '
- '$^{53}$MTA Eötvös University, “Lendulet” Astrophysics Research Group, Budapest 1117, Hungary '
- '$^{54}$Institut de Physique de Rennes, CNRS, Université de Rennes 1, F-35042 Rennes, France '
- '$^{55}$Washington State University, Pullman, WA 99164, USA '
- '$^{56}$Università degli Studi di Urbino “Carlo Bo,” I-61029 Urbino, Italy '
- '$^{57}$INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy '
- '$^{58}$University of Oregon, Eugene, OR 97403, USA '
- '$^{59}$Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research University, Collège de France, F-75005 Paris, France '
- '$^{60}$Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland '
- '$^{61}$VU University Amsterdam, 1081 HV Amsterdam, Netherlands '
- '$^{62}$University of Maryland, College Park, MD 20742, USA '
- '$^{63}$Center for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA '
- '$^{64}$Institut Lumière Matière, Université de Lyon, Université Claude Bernard Lyon 1, UMR CNRS 5306, 69622 Villeurbanne, France '
- '$^{65}$Laboratoire des Matériaux Avancés (LMA), IN2P3/CNRS, Université de Lyon, F-69622 Villeurbanne, Lyon, France '
- '$^{66}$Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain '
- '$^{67}$Università di Napoli “Federico II,” Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy '
- '$^{68}$NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA '
- '$^{69}$Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada '
- '$^{70}$Tsinghua University, Beijing 100084, China '
- '$^{71}$Texas Tech University, Lubbock, TX 79409, USA '
- '$^{72}$The Pennsylvania State University, University Park, PA 16802, USA '
- '$^{73}$National Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China '
- '$^{74}$Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia '
- '$^{75}$University of Chicago, Chicago, IL 60637, USA '
- '$^{76}$Caltech CaRT, Pasadena, CA 91125, USA '
- '$^{77}$Korea Institute of Science and Technology Information, Daejeon 305-806, Korea '
- '$^{78}$Carleton College, Northfield, MN 55057, USA '
- '$^{79}$Università di Roma “La Sapienza,” I-00185 Roma, Italy '
- '$^{80}$University of Brussels, Brussels 1050, Belgium '
- '$^{81}$Sonoma State University, Rohnert Park, CA 94928, USA '
- '$^{82}$Northwestern University, Evanston, IL 60208, USA '
- '$^{83}$University of Minnesota, Minneapolis, MN 55455, USA '
- '$^{84}$The University of Melbourne, Parkville, Victoria 3010, Australia '
- '$^{85}$The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA '
- '$^{86}$The University of Sheffield, Sheffield S10 2TN, United Kingdom '
- '$^{87}$University of Sannio at Benevento, I-82100 Benevento, Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy '
- '$^{88}$Montclair State University, Montclair, NJ 07043, USA '
- '$^{89}$Università di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy '
- '$^{90}$INFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy '
- '$^{91}$Cardiff University, Cardiff CF24 3AA, United Kingdom '
- '$^{92}$National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan '
- '$^{93}$School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom '
- '$^{94}$Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424, India '
- '$^{95}$Institute for Plasma Research, Bhat, Gandhinagar 382428, India '
- '$^{96}$University of Szeged, Dóm tér 9, Szeged 6720, Hungary '
- '$^{97}$Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA '
- '$^{98}$University of Michigan, Ann Arbor, MI 48109, USA '
- '$^{99}$Tata Institute of Fundamental Research, Mumbai 400005, India '
- '$^{100}$American University, Washington, D.C. 20016, USA '
- '$^{101}$University of Massachusetts-Amherst, Amherst, MA 01003, USA '
- '$^{102}$University of Adelaide, Adelaide, South Australia 5005, Australia '
- '$^{103}$West Virginia University, Morgantown, WV 26506, USA '
- '$^{104}$University of Bia[ł]{}ystok, 15-424 Bia[ł]{}ystok, Poland '
- '$^{105}$SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom '
- '$^{106}$IISER-TVM, CET Campus, Trivandrum Kerala 695016, India '
- '$^{107}$Institute of Applied Physics, Nizhny Novgorod, 603950, Russia '
- '$^{108}$Pusan National University, Busan 609-735, Korea '
- '$^{109}$Hanyang University, Seoul 133-791, Korea '
- '$^{110}$NCBJ, 05-400 Świerk-Otwock, Poland '
- '$^{111}$IM-PAN, 00-956 Warsaw, Poland '
- '$^{112}$Rochester Institute of Technology, Rochester, NY 14623, USA '
- '$^{113}$Monash University, Victoria 3800, Australia '
- '$^{114}$Seoul National University, Seoul 151-742, Korea '
- '$^{115}$University of Alabama in Huntsville, Huntsville, AL 35899, USA '
- '$^{116}$ESPCI, CNRS, F-75005 Paris, France '
- '$^{117}$Università di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy '
- '$^{118}$Southern University and A&M College, Baton Rouge, LA 70813, USA '
- '$^{119}$College of William and Mary, Williamsburg, VA 23187, USA '
- '$^{120}$Instituto de Física Teórica, University Estadual Paulista/ICTP South American Institute for Fundamental Research, São Paulo SP 01140-070, Brazil '
- '$^{121}$University of Cambridge, Cambridge CB2 1TN, United Kingdom '
- '$^{122}$IISER-Kolkata, Mohanpur, West Bengal 741252, India '
- '$^{123}$Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11 0QX, United Kingdom '
- '$^{124}$Whitman College, 345 Boyer Avenue, Walla Walla, WA 99362 USA '
- '$^{125}$National Institute for Mathematical Sciences, Daejeon 305-390, Korea '
- '$^{126}$Hobart and William Smith Colleges, Geneva, NY 14456, USA '
- '$^{127}$Janusz Gil Institute of Astronomy, University of Zielona Góra, 65-265 Zielona Góra, Poland '
- '$^{128}$Andrews University, Berrien Springs, MI 49104, USA '
- '$^{129}$Università di Siena, I-53100 Siena, Italy '
- '$^{130}$Trinity University, San Antonio, TX 78212, USA '
- '$^{131}$University of Washington, Seattle, WA 98195, USA '
- '$^{132}$Kenyon College, Gambier, OH 43022, USA '
- '$^{133}$Abilene Christian University, Abilene, TX 79699, USA '
- '$^{134}$Louisiana Tech University, Ruston, LA 71272, USA '
author:
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B P Abbott$^{1}$, R Abbott$^{1}$, T D Abbott$^{2}$, M R Abernathy$^{1}$, F Acernese$^{3,4}$, K Ackley$^{5}$, M Adamo$^{4,21}$, C Adams$^{6}$, T Adams$^{7}$, P Addesso$^{3}$, R X Adhikari$^{1}$, V B Adya$^{8}$, C Affeldt$^{8}$, M Agathos$^{9}$, K Agatsuma$^{9}$, N Aggarwal$^{10}$, O D Aguiar$^{11}$, L Aiello$^{12,13}$, A Ain$^{14}$, P Ajith$^{15}$, B Allen$^{8,16,17}$, A Allocca$^{18,19}$, P A Altin$^{20}$, S B Anderson$^{1}$, W G Anderson$^{16}$, K Arai$^{1}$, M C Araya$^{1}$, C C Arceneaux$^{21}$, J S Areeda$^{22}$, N Arnaud$^{23}$, K G Arun$^{24}$, S Ascenzi$^{25,13}$, G Ashton$^{26}$, M Ast$^{27}$, S M Aston$^{6}$, P Astone$^{28}$, P Aufmuth$^{8}$, C Aulbert$^{8}$, S Babak$^{29}$, P Bacon$^{30}$, M K M Bader$^{9}$, P T Baker$^{31}$, F Baldaccini$^{32,33}$, G Ballardin$^{34}$, S W Ballmer$^{35}$, J C Barayoga$^{1}$, S E Barclay$^{36}$, B C Barish$^{1}$, D Barker$^{37}$, F Barone$^{3,4}$, B Barr$^{36}$, L Barsotti$^{10}$, M Barsuglia$^{30}$, D Barta$^{38}$, J Bartlett$^{37}$, I Bartos$^{39}$, R Bassiri$^{40}$, A Basti$^{18,19}$, J C Batch$^{37}$, C Baune$^{8}$, V Bavigadda$^{34}$, M Bazzan$^{41,42}$, B Behnke$^{29}$, M Bejger$^{43}$, A S Bell$^{36}$, C J Bell$^{36}$, B K Berger$^{1}$, J Bergman$^{37}$, G Bergmann$^{8}$, C P L Berry$^{44}$, D Bersanetti$^{45,46}$, A Bertolini$^{9}$, J Betzwieser$^{6}$, S Bhagwat$^{35}$, R Bhandare$^{47}$, I A Bilenko$^{48}$, G Billingsley$^{1}$, J Birch$^{6}$, R Birney$^{49}$, S Biscans$^{10}$, A Bisht$^{8,17}$, M Bitossi$^{34}$, C Biwer$^{35}$, M A Bizouard$^{23}$, J K Blackburn$^{1}$, L Blackburn$^{10}$, C D Blair$^{50}$, D G Blair$^{50}$, R M Blair$^{37}$, S Bloemen$^{51}$, O Bock$^{8}$, T P Bodiya$^{10}$, M Boer$^{52}$, G Bogaert$^{52}$, C Bogan$^{8}$, A Bohe$^{29}$, P Bojtos$^{53}$, C Bond$^{44}$, F Bondu$^{54}$, R Bonnand$^{7}$, B A Boom$^{9}$, R Bork$^{1}$, V Boschi$^{18,19}$, S Bose$^{55,14}$, Y Bouffanais$^{30}$, A Bozzi$^{34}$, C Bradaschia$^{19}$, P R Brady$^{16}$, V B Braginsky$^{48}$, M Branchesi$^{56,57}$, J E Brau$^{58}$, T Briant$^{59}$, A Brillet$^{52}$, M Brinkmann$^{8}$, V Brisson$^{23}$, P Brockill$^{16}$, A F Brooks$^{1}$, D A Brown$^{35}$, D D Brown$^{44}$, N M Brown$^{10}$, C C Buchanan$^{2}$, A Buikema$^{10}$, T Bulik$^{60}$, H J Bulten$^{61,9}$, A Buonanno$^{29,62}$, D Buskulic$^{7}$, C Buy$^{30}$, R L Byer$^{40}$, L Cadonati$^{63}$, G Cagnoli$^{64,65}$, C Cahillane$^{1}$, J Calderón Bustillo$^{66,63}$, T Callister$^{1}$, E Calloni$^{67,4}$, J B Camp$^{68}$, K C Cannon$^{69}$, J Cao$^{70}$, C D Capano$^{8}$, E Capocasa$^{30}$, F Carbognani$^{34}$, S Caride$^{71}$, J Casanueva Diaz$^{23}$, C Casentini$^{25,13}$, S Caudill$^{16}$, M Cavaglià$^{21}$, F Cavalier$^{23}$, R Cavalieri$^{34}$, G Cella$^{19}$, C B Cepeda$^{1}$, L Cerboni Baiardi$^{56,57}$, G Cerretani$^{18,19}$, E Cesarini$^{25,13}$, R Chakraborty$^{1}$, T Chalermsongsak$^{1}$, S J Chamberlin$^{72}$, M Chan$^{36}$, S Chao$^{73}$, P Charlton$^{74}$, E Chassande-Mottin$^{30}$, S Chatterji$^{10}$, H Y Chen$^{75}$, Y Chen$^{76}$, C Cheng$^{73}$, A Chincarini$^{46}$, A Chiummo$^{34}$, H S Cho$^{77}$, M Cho$^{62}$, J H Chow$^{20}$, N Christensen$^{78}$, Q Chu$^{50}$, S Chua$^{59}$, S Chung$^{50}$, G Ciani$^{5}$, F Clara$^{37}$, J A Clark$^{63}$, F Cleva$^{52}$, E Coccia$^{25,12,13}$, P-F Cohadon$^{59}$, A Colla$^{79,28}$, C G Collette$^{80}$, L Cominsky$^{81}$, M Constancio Jr.$^{11}$, A Conte$^{79,28}$, L Conti$^{42}$, D Cook$^{37}$, T R Corbitt$^{2}$, N Cornish$^{31}$, A Corsi$^{71}$, S Cortese$^{34}$, C A Costa$^{11}$, M W Coughlin$^{78}$, S B Coughlin$^{82}$, J-P Coulon$^{52}$, S T Countryman$^{39}$, P Couvares$^{1}$, E E Cowan$^{63}$, D M Coward$^{50}$, M J Cowart$^{6}$, D C Coyne$^{1}$, R Coyne$^{71}$, K Craig$^{36}$, J D E Creighton$^{16}$, J Cripe$^{2}$, S G Crowder$^{83}$, A Cumming$^{36}$, L Cunningham$^{36}$, E Cuoco$^{34}$, T Dal Canton$^{8}$, S L Danilishin$^{36}$, S D’Antonio$^{13}$, K Danzmann$^{17,8}$, N S Darman$^{84}$, V Dattilo$^{34}$, I Dave$^{47}$, H P Daveloza$^{85}$, M Davier$^{23}$, G S Davies$^{36}$, E J Daw$^{86}$, R Day$^{34}$, D DeBra$^{40}$, G Debreczeni$^{38}$, J Degallaix$^{65}$, M De Laurentis$^{67,4}$, S Deléglise$^{59}$, W Del Pozzo$^{44}$, T Denker$^{8,17}$, T Dent$^{8}$, H Dereli$^{52}$, V Dergachev$^{1}$, R T DeRosa$^{6}$, R De Rosa$^{67,4}$, R DeSalvo$^{87}$, S Dhurandhar$^{14}$, M C Díaz$^{85}$, L Di Fiore$^{4}$, M Di Giovanni$^{79,28}$, A Di Lieto$^{18,19}$, S Di Pace$^{79,28}$, I Di Palma$^{29,8}$, A Di Virgilio$^{19}$, G Dojcinoski$^{88}$, V Dolique$^{65}$, F Donovan$^{10}$, K L Dooley$^{21}$, S Doravari$^{6,8}$, R Douglas$^{36}$, T P Downes$^{16}$, M Drago$^{8,89,90}$, R W P Drever$^{1}$, J C Driggers$^{37}$, Z Du$^{70}$, M Ducrot$^{7}$, S E Dwyer$^{37}$, T B Edo$^{86}$, M C Edwards$^{78}$, A Effler$^{6}$, H-B Eggenstein$^{8}$, P Ehrens$^{1}$, J Eichholz$^{5}$, S S Eikenberry$^{5}$, W Engels$^{76}$, R C Essick$^{10}$, T Etzel$^{1}$, M Evans$^{10}$, T M Evans$^{6}$, R Everett$^{72}$, M Factourovich$^{39}$, V Fafone$^{25,13,12}$, H Fair$^{35}$, S Fairhurst$^{91}$, X Fan$^{70}$, Q Fang$^{50}$, S Farinon$^{46}$, B Farr$^{75}$, W M Farr$^{44}$, M Favata$^{88}$, M Fays$^{91}$, H Fehrmann$^{8}$, M M Fejer$^{40}$, I Ferrante$^{18,19}$, E C Ferreira$^{11}$, F Ferrini$^{34}$, F Fidecaro$^{18,19}$, I Fiori$^{34}$, D Fiorucci$^{30}$, R P Fisher$^{35}$, R Flaminio$^{65,92}$, M Fletcher$^{36}$, J-D Fournier$^{52}$, S Franco$^{23}$, S Frasca$^{79,28}$, F Frasconi$^{19}$, Z Frei$^{53}$, A Freise$^{44}$, R Frey$^{58}$, V Frey$^{23}$, T T Fricke$^{8}$, P Fritschel$^{10}$, V V Frolov$^{6}$, P Fulda$^{5}$, M Fyffe$^{6}$, H A G Gabbard$^{21}$, J R Gair$^{93}$, L Gammaitoni$^{32,33}$, S G Gaonkar$^{14}$, F Garufi$^{67,4}$, A Gatto$^{30}$, G Gaur$^{94,95}$, N Gehrels$^{68}$, G Gemme$^{46}$, B Gendre$^{52}$, E Genin$^{34}$, A Gennai$^{19}$, J George$^{47}$, L Gergely$^{96}$, V Germain$^{7}$, Archisman Ghosh$^{15}$, S Ghosh$^{51,9}$, J A Giaime$^{2,6}$, K D Giardina$^{6}$, A Giazotto$^{19}$, K Gill$^{97}$, A Glaefke$^{36}$, E Goetz$^{98}$, R Goetz$^{5}$, L Gondan$^{53}$, G González$^{2}$, J M Gonzalez Castro$^{18,19}$, A Gopakumar$^{99}$, N A Gordon$^{36}$, M L Gorodetsky$^{48}$, S E Gossan$^{1}$, M Gosselin$^{34}$, R Gouaty$^{7}$, C Graef$^{36}$, P B Graff$^{62}$, M Granata$^{65}$, A Grant$^{36}$, S Gras$^{10}$, C Gray$^{37}$, G Greco$^{56,57}$, A C Green$^{44}$, P Groot$^{51}$, H Grote$^{8}$, S Grunewald$^{29}$, G M Guidi$^{56,57}$, X Guo$^{70}$, A Gupta$^{14}$, M K Gupta$^{95}$, K E Gushwa$^{1}$, E K Gustafson$^{1}$, R Gustafson$^{98}$, J J Hacker$^{22}$, B R Hall$^{55}$, E D Hall$^{1}$, G Hammond$^{36}$, M Haney$^{99}$, M M Hanke$^{8}$, J Hanks$^{37}$, C Hanna$^{72}$, M D Hannam$^{91}$, J Hanson$^{6}$, T Hardwick$^{2}$, J Harms$^{56,57}$, G M Harry$^{100}$, I W Harry$^{29}$, M J Hart$^{36}$, M T Hartman$^{5}$, C-J Haster$^{44}$, K Haughian$^{36}$, A Heidmann$^{59}$, M C Heintze$^{5,6}$, H Heitmann$^{52}$, P Hello$^{23}$, G Hemming$^{34}$, M Hendry$^{36}$, I S Heng$^{36}$, J Hennig$^{36}$, A W Heptonstall$^{1}$, M Heurs$^{8,17}$, S Hild$^{36}$, D Hoak$^{101}$, K A Hodge$^{1}$, D Hofman$^{65}$, S E Hollitt$^{102}$, K Holt$^{6}$, D E Holz$^{75}$, P Hopkins$^{91}$, D J Hosken$^{102}$, J Hough$^{36}$, E A Houston$^{36}$, E J Howell$^{50}$, Y M Hu$^{36}$, S Huang$^{73}$, E A Huerta$^{103,82}$, D Huet$^{23}$, B Hughey$^{97}$, S Husa$^{66}$, S H Huttner$^{36}$, T Huynh-Dinh$^{6}$, A Idrisy$^{72}$, N Indik$^{8}$, D R Ingram$^{37}$, R Inta$^{71}$, H N Isa$^{36}$, J-M Isac$^{59}$, M Isi$^{1}$, G Islas$^{22}$, T Isogai$^{10}$, B R Iyer$^{15}$, K Izumi$^{37}$, T Jacqmin$^{59}$, H Jang$^{77}$, K Jani$^{63}$, P Jaranowski$^{104}$, S Jawahar$^{105}$, F Jiménez-Forteza$^{66}$, W W Johnson$^{2}$, D I Jones$^{26}$, R Jones$^{36}$, R J G Jonker$^{9}$, L Ju$^{50}$, Haris K$^{106}$, C V Kalaghatgi$^{24,91}$, V Kalogera$^{82}$, S Kandhasamy$^{21}$, G Kang$^{77}$, J B Kanner$^{1}$, S Karki$^{58}$, M Kasprzack$^{2,23,34}$, E Katsavounidis$^{10}$, W Katzman$^{6}$, S Kaufer$^{17}$, T Kaur$^{50}$, K Kawabe$^{37}$, F Kawazoe$^{8,17}$, F Kéfélian$^{52}$, M S Kehl$^{69}$, D Keitel$^{8,66}$, D B Kelley$^{35}$, W Kells$^{1}$, R Kennedy$^{86}$, J S Key$^{85}$, A Khalaidovski$^{8}$, F Y Khalili$^{48}$, I Khan$^{12}$, S Khan$^{91}$, Z Khan$^{95}$, E A Khazanov$^{107}$, N Kijbunchoo$^{37}$, C Kim$^{77}$, J Kim$^{108}$, K Kim$^{109}$, Nam-Gyu Kim$^{77}$, Namjun Kim$^{40}$, Y-M Kim$^{108}$, E J King$^{102}$, P J King$^{37}$, D L Kinzel$^{6}$, J S Kissel$^{37}$, L Kleybolte$^{27}$, S Klimenko$^{5}$, S M Koehlenbeck$^{8}$, K Kokeyama$^{2}$, S Koley$^{9}$, V Kondrashov$^{1}$, A Kontos$^{10}$, M Korobko$^{27}$, W Z Korth$^{1}$, I Kowalska$^{60}$, D B Kozak$^{1}$, V Kringel$^{8}$, B Krishnan$^{8}$, A Królak$^{110,111}$, C Krueger$^{17}$, G Kuehn$^{8}$, P Kumar$^{69}$, L Kuo$^{73}$, A Kutynia$^{110}$, B D Lackey$^{35}$, M Landry$^{37}$, J Lange$^{112}$, B Lantz$^{40}$, P D Lasky$^{113}$, A Lazzarini$^{1}$, C Lazzaro$^{63,42}$, P Leaci$^{29,79,28}$, S Leavey$^{36}$, E O Lebigot$^{30,70}$, C H Lee$^{108}$, H K Lee$^{109}$, H M Lee$^{114}$, K Lee$^{36}$, A Lenon$^{35}$, M Leonardi$^{89,90}$, J R Leong$^{8}$, N Leroy$^{23}$, N Letendre$^{7}$, Y Levin$^{113}$, B M Levine$^{37}$, T G F Li$^{1}$, A Libson$^{10}$, T B Littenberg$^{115}$, N A Lockerbie$^{105}$, J Logue$^{36}$, A L Lombardi$^{101}$, J E Lord$^{35}$, M Lorenzini$^{12,13}$, V Loriette$^{116}$, M Lormand$^{6}$, G Losurdo$^{57}$, J D Lough$^{8,17}$, H Lück$^{17,8}$, A P Lundgren$^{8}$, J Luo$^{78}$, R Lynch$^{10}$, Y Ma$^{50}$, T MacDonald$^{40}$, B Machenschalk$^{8}$, M MacInnis$^{10}$, D M Macleod$^{2}$, F Magaña-Sandoval$^{35}$, R M Magee$^{55}$, M Mageswaran$^{1}$, E Majorana$^{28}$, I Maksimovic$^{116}$, V Malvezzi$^{25,13}$, N Man$^{52}$, I Mandel$^{44}$, V Mandic$^{83}$, V Mangano$^{36}$, G L Mansell$^{20}$, M Manske$^{16}$, M Mantovani$^{34}$, F Marchesoni$^{117,33}$, F Marion$^{7}$, S Márka$^{39}$, Z Márka$^{39}$, A S Markosyan$^{40}$, E Maros$^{1}$, F Martelli$^{56,57}$, L Martellini$^{52}$, I W Martin$^{36}$, R M Martin$^{5}$, D V Martynov$^{1}$, J N Marx$^{1}$, K Mason$^{10}$, A Masserot$^{7}$, T J Massinger$^{35}$, M Masso-Reid$^{36}$, F Matichard$^{10}$, L Matone$^{39}$, N Mavalvala$^{10}$, N Mazumder$^{55}$, G Mazzolo$^{8}$, R McCarthy$^{37}$, D E McClelland$^{20}$, S McCormick$^{6}$, S C McGuire$^{118}$, G McIntyre$^{1}$, J McIver$^{1}$, D J McManus$^{20}$, S T McWilliams$^{103}$, D Meacher$^{72}$, G D Meadors$^{29,8}$, J Meidam$^{9}$, A Melatos$^{84}$, G Mendell$^{37}$, D Mendoza-Gandara$^{8}$, R A Mercer$^{16}$, E Merilh$^{37}$, M Merzougui$^{52}$, S Meshkov$^{1}$, C Messenger$^{36}$, C Messick$^{72}$, P M Meyers$^{83}$, F Mezzani$^{28,79}$, H Miao$^{44}$, C Michel$^{65}$, H Middleton$^{44}$, E E Mikhailov$^{119}$, L Milano$^{67,4}$, J Miller$^{10}$, M Millhouse$^{31}$, Y Minenkov$^{13}$, J Ming$^{29,8}$, S Mirshekari$^{120}$, C Mishra$^{15}$, S Mitra$^{14}$, V P Mitrofanov$^{48}$, G Mitselmakher$^{5}$, R Mittleman$^{10}$, A Moggi$^{19}$, M Mohan$^{34}$, S R P Mohapatra$^{10}$, M Montani$^{56,57}$, B C Moore$^{88}$, C J Moore$^{121}$, D Moraru$^{37}$, G Moreno$^{37}$, S R Morriss$^{85}$, K Mossavi$^{8}$, B Mours$^{7}$, C M Mow-Lowry$^{44}$, C L Mueller$^{5}$, G Mueller$^{5}$, A W Muir$^{91}$, Arunava Mukherjee$^{15}$, D Mukherjee$^{16}$, S Mukherjee$^{85}$, N Mukund$^{14}$, A Mullavey$^{6}$, J Munch$^{102}$, D J Murphy$^{39}$, P G Murray$^{36}$, A Mytidis$^{5}$, I Nardecchia$^{25,13}$, L Naticchioni$^{79,28}$, R K Nayak$^{122}$, V Necula$^{5}$, K Nedkova$^{101}$, G Nelemans$^{51,9}$, M Neri$^{45,46}$, A Neunzert$^{98}$, G Newton$^{36}$, T T Nguyen$^{20}$, A B Nielsen$^{8}$, S Nissanke$^{51,9}$, A Nitz$^{8}$, F Nocera$^{34}$, D Nolting$^{6}$, M E Normandin$^{85}$, L K Nuttall$^{35}$, J Oberling$^{37}$, E Ochsner$^{16}$, J O’Dell$^{123}$, E Oelker$^{10}$, G H Ogin$^{124}$, J J Oh$^{125}$, S H Oh$^{125}$, F Ohme$^{91}$, M Oliver$^{66}$, P Oppermann$^{8}$, Richard J Oram$^{6}$, B O’Reilly$^{6}$, R O’Shaughnessy$^{112}$, D J Ottaway$^{102}$, R S Ottens$^{5}$, H Overmier$^{6}$, B J Owen$^{71}$, A Pai$^{106}$, S A Pai$^{47}$, J R Palamos$^{58}$, O Palashov$^{107}$, C Palomba$^{28}$, A Pal-Singh$^{27}$, H Pan$^{73}$, C Pankow$^{82}$, F Pannarale$^{91}$, B C Pant$^{47}$, F Paoletti$^{34,19}$, A Paoli$^{34}$, M A Papa$^{29,16,8}$, H R Paris$^{40}$, W Parker$^{6}$, D Pascucci$^{36}$, A Pasqualetti$^{34}$, R Passaquieti$^{18,19}$, D Passuello$^{19}$, B Patricelli$^{18,19}$, Z Patrick$^{40}$, B L Pearlstone$^{36}$, M Pedraza$^{1}$, R Pedurand$^{65}$, L Pekowsky$^{35}$, A Pele$^{6}$, S Penn$^{126}$, A Perreca$^{1}$, M Phelps$^{36}$, O Piccinni$^{79,28}$, M Pichot$^{52}$, F Piergiovanni$^{56,57}$, V Pierro$^{87}$, G Pillant$^{34}$, L Pinard$^{65}$, I M Pinto$^{87}$, M Pitkin$^{36}$, R Poggiani$^{18,19}$, P Popolizio$^{34}$, A Post$^{8}$, J Powell$^{36}$, J Prasad$^{14}$, V Predoi$^{91}$, S S Premachandra$^{113}$, T Prestegard$^{83}$, L R Price$^{1}$, M Prijatelj$^{34}$, M Principe$^{87}$, S Privitera$^{29}$, G A Prodi$^{89,90}$, L Prokhorov$^{48}$, O 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Smith$^{1}$, E J Son$^{125}$, B Sorazu$^{36}$, F Sorrentino$^{46}$, T Souradeep$^{14}$, A K Srivastava$^{95}$, A Staley$^{39}$, M Steinke$^{8}$, J Steinlechner$^{36}$, S Steinlechner$^{36}$, D Steinmeyer$^{8,17}$, B C Stephens$^{16}$, R Stone$^{85}$, K A Strain$^{36}$, N Straniero$^{65}$, G Stratta$^{56,57}$, N A Strauss$^{78}$, S Strigin$^{48}$, R Sturani$^{120}$, A L Stuver$^{6}$, T Z Summerscales$^{128}$, L Sun$^{84}$, P J Sutton$^{91}$, B L Swinkels$^{34}$, M J Szczepańczyk$^{97}$, M Tacca$^{30}$, D Talukder$^{58}$, D B Tanner$^{5}$, M Tápai$^{96}$, S P Tarabrin$^{8}$, A Taracchini$^{29}$, R Taylor$^{1}$, T Theeg$^{8}$, M P Thirugnanasambandam$^{1}$, E G Thomas$^{44}$, M Thomas$^{6}$, P Thomas$^{37}$, K A Thorne$^{6}$, K S Thorne$^{76}$, E Thrane$^{113}$, S Tiwari$^{12}$, V Tiwari$^{91}$, K V Tokmakov$^{105}$, C Tomlinson$^{86}$, M Tonelli$^{18,19}$, C V Torres$^{\ddag}$$^{85}$, C I Torrie$^{1}$, D Töyrä$^{44}$, F Travasso$^{32,33}$, G Traylor$^{6}$, D Trifirò$^{21}$, M C Tringali$^{89,90}$, L Trozzo$^{129,19}$, M Tse$^{10}$, M Turconi$^{52}$, D Tuyenbayev$^{85}$, D Ugolini$^{130}$, C S Unnikrishnan$^{99}$, A L Urban$^{16}$, S A Usman$^{35}$, H Vahlbruch$^{17}$, G Vajente$^{1}$, G Valdes$^{85}$, N van Bakel$^{9}$, M van Beuzekom$^{9}$, J F J van den Brand$^{61,9}$, C Van Den Broeck$^{9}$, D C Vander-Hyde$^{35,22}$, L van der Schaaf$^{9}$, J V van Heijningen$^{9}$, A A van Veggel$^{36}$, M Vardaro$^{41,42}$, S Vass$^{1}$, M Vasúth$^{38}$, R Vaulin$^{10}$, A Vecchio$^{44}$, G Vedovato$^{42}$, J Veitch$^{44}$, P J Veitch$^{102}$, K Venkateswara$^{131}$, D Verkindt$^{7}$, F Vetrano$^{56,57}$, A Viceré$^{56,57}$, S Vinciguerra$^{44}$, D J Vine$^{49}$, J-Y Vinet$^{52}$, S Vitale$^{10}$, T Vo$^{35}$, H Vocca$^{32,33}$, C Vorvick$^{37}$, D Voss$^{5}$, W D Vousden$^{44}$, S P Vyatchanin$^{48}$, A R Wade$^{20}$, L E Wade$^{132}$, M Wade$^{132}$, M Walker$^{2}$, L Wallace$^{1}$, S Walsh$^{16,8,29}$, G Wang$^{12}$, H Wang$^{44}$, M Wang$^{44}$, X Wang$^{70}$, Y Wang$^{50}$, R L Ward$^{20}$, J Warner$^{37}$, M Was$^{7}$, B Weaver$^{37}$, L-W Wei$^{52}$, M Weinert$^{8}$, A J Weinstein$^{1}$, R Weiss$^{10}$, T Welborn$^{6}$, L Wen$^{50}$, P We[ß]{}els$^{8}$, T Westphal$^{8}$, K Wette$^{8}$, J T Whelan$^{112,8}$, S Whitcomb$^{1}$, D J White$^{86}$, B F Whiting$^{5}$, R D Williams$^{1}$, A R Williamson$^{91}$, J L Willis$^{133}$, B Willke$^{17,8}$, M H Wimmer$^{8,17}$, W Winkler$^{8}$, C C Wipf$^{1}$, H Wittel$^{8,17}$, G Woan$^{36}$, J Worden$^{37}$, J L Wright$^{36}$, G Wu$^{6}$, J Yablon$^{82}$, W Yam$^{10}$, H Yamamoto$^{1}$, C C Yancey$^{62}$, M J Yap$^{20}$, H Yu$^{10}$, M Yvert$^{7}$, A Zadrożny$^{110}$, L Zangrando$^{42}$, M Zanolin$^{97}$, J-P Zendri$^{42}$, M Zevin$^{82}$, F Zhang$^{10}$, L Zhang$^{1}$, M Zhang$^{119}$, Y Zhang$^{112}$, C Zhao$^{50}$, M Zhou$^{82}$, Z Zhou$^{82}$, X J Zhu$^{50}$, N Zotov$^{\sharp}$$^{134}$ M E Zucker$^{1,10}$, S E Zuraw$^{101}$, and J Zweizig$^{1}$\
[(LIGO Scientific Collaboration and Virgo Collaboration)]{}
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abstract: 'We present a study on the mechanical configuration and the electronic properties of semiconducting carbon nanotubes supported by partially depassivated silicon substrates, as inferred from topographic and spectroscopic data acquired with a room-temperature ultrahigh vacuum scanning tunneling microscope and density-functional theory calculations. A mechanical distortion and doping for semiconducting carbon nanotubes on Si(100)-(2$\times$1):H with hydrogen-depassivated stripes up to 100 Å wide are ascertained from both experiment and theory. The results presented here point towards novel and local functionalities of nanotube-semiconductor interfaces.'
author:
- 'Salvador Barraza-Lopez$^1$'
- 'Peter M. Albrecht$^2$'
- 'Joseph W. Lyding$^2$'
title: 'Carbon nanotubes on partially depassivated *n*-doped Si(100)-(2$\times$1):H substrates'
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The modifications of the intrinsic electronic and thermal properties of single-walled carbon nanotubes (SWNTs) due to their interaction with the semiconducting surface by which they are supported have been the focal point of a sizeable number of experimental[@a1; @a2; @a4; @a6; @a7; @a8; @aplrecent; @peternanotechnology; @a18] and theoretical[@a8; @a9; @a13; @a17; @GalliPRL] studies in recent years due to the technological interest in hybrid SWNT-semiconductor devices. Dry contact transfer (DCT)[@a2] allows for the *in situ* deposition of SWNTs from solid sources onto technologically relevant surfaces, such as Si(100)[@a2; @a6; @a8; @peternanotechnology; @a18] and the (110) surfaces of GaAs and InAs[@a4], forming an atomically pristine interface. Comprehensive studies of semiconducting SWNTs (s-SWNTs) on Si(100) and Si(100)-(2$\times$1):H at the density-functional theory (DFT) level have been reported[@a17; @a8]. The adsorption properties of s-SWNTs and metallic SWNTs (m-SWNTs) of similar diameter on Si(100) are remarkably different[@a17]. This holds for $\sim$10 Å-diameter SWNTs with varying chiralities. Unlike m-SWNTs on Si(100), no covalent bonds are formed at the interface between Si(100) and s-SWNTs. (In recent studies of a (10,0) semiconducting SWNT on Si(100), covalent bonds between the s-SWNT and Si surface atoms were artificially induced[@GalliPRL] .) Si-C covalent bonding for SWNTs grown on Si was asserted only for a subset of the population based upon micro-Raman imaging[@aplrecent], suggesting that both the electronic character of the SWNT and the local termination of the Si surface markedly influence the degree of nanotube-substrate interaction. When SWNTs are placed on the inert monohydride Si(100)-(2$\times$1):H surface[@a2], the respective electronic characteristics of the nanotube and the surface are preserved and charge transfer is largely suppressed[@a8; @a13]. In contrast, a shift of the Fermi level away from the nanotube midgap position has been experimentally observed for SWNTs supported by metallic surfaces (Au(111)[@DekkerLieber] and Ag(100)[@Kawai]), unpassivated III-V compound semiconductors (GaAs(110) and InAs(110)[@a4]) and an ultrathin insulating film (NaCl(100)/Ag(100)[@Kawai]). Faithful to experimental conditions, our computational studies included dopants within the Si(100) slab. Ultrahigh vacuum scanning tunneling microscope (UHV-STM) based nanolithography on hydrogen-passivated Si(100) enables the definition of patterns of reactive depassivated Si[@Lyding1; @Lyding2] with possible consequences for the adhesion and electronic properties of the adsorbed SWNTs[@a6; @peternanotechnology; @a18]. In this Letter, we report on the properties of isolated s-SWNTs interfaced with nanoscale regions of selectively depassivated Si as determined from room-temperature UHV-STM measurements and DFT calculations.
{width=".8\textwidth"}
![\[fig:Fig2\] (color online) Top: Ball-and-stick model of the (8,4) SWNT on a partially depassivated (10 Å wide) $n$-doped Si(100)-(2$\times$1):H substrate. The red line indicates the direction of the Si dimmer rows, and it forms a 45$^{o}$ angle with respect to the SWNT axis. Only the lower half of the SWNT is shown for clarity. Side: The side view of the system, showing in red the location of the phosporous ($n-$type) dopant. Horizontal dashed lines serve to indicate distortion in the SWNT. Maximum distortion in SWNT occurs at $x_1$ and $x_2$. Inset: The (8,4) SWNT was contracted by 3.8% (blue vertical line) in order to be placed on top of the Si substrate. This implied a 30% reduction of its gap and a 0.75% increase in diameter to compensate longitudinal compression.](Fig2){width=".45\textwidth"}
Fig. \[fig:fig1\] summarizes the experimental observations; reproducible results were obtained for several unique s-SWNTs on partially depassivated Si(100)-(2$\times$1):H. Degenerately $n$-type doped Si(100) substrates (As, $10^{19}$ cm$^{-3}$) were employed, and subjected to UHV H-passivation[@a2]. Isolated HiPco SWNTs[@FN1] were deposited by DCT. The STM was operated at room temperature in constant-current mode with the bias voltage ($V$) applied to the substrate and the electrochemically etched W tip grounded through a current ($I$) preamplifier. Partial surface depassivation, as seen in the filled-states topograph of Fig. 1(a), was achieved with the methods described in Refs. and . Fig. \[fig:fig1\](b) depicts the relative STM height along the top of the SWNT, as indicated by the blue line in Fig. 1(a). When the SWNT is on the H-passivated substrate[@a2] the height fluctuations are of the order of 0.2 Å, and they are directly related to the underlying honeycomb lattice of the SWNT. The dip in the apparent SWNT height stressed by the horizontal red line, beyond the 0.2 Å fluctuations, correlates with the location where the SWNT traverses the depassivated stripe (brighter region) in Fig. \[fig:fig1\](a). Similar trends were reported before[@peternanotechnology]. Given that the substrate is degenerately $n$-doped, in the absence of a mechanical deformation, one would anticipate the negative charging of the Si surface states within the depassivated region[@Dujardin] and a protrusion, rather than a dip, in the height profile: to maintain constant current, the tip should retract due to the higher density of states of the Si dangling bonds. Hence the data in Fig. \[fig:fig1\](b) provides evidence for a slight conformal deformation (of the order of 0.5 Å) of the SWNT along the depassivated region. In a subsequent STM scan, the absolute current vs. bias was recorded (\[$-2,+2$\] $V$, $\Delta V$ = 20 mV) along the blue line in Fig. \[fig:fig1\](c). In Fig. \[fig:fig1\](c), we notice the apparent widening of the s-SWNT as the STM tip moves away from the depassivated region (the region within the two dashed vertical lines). This was previously observed[@a18] and is consistent with the fact that the SWNT on Si(100)-(2$\times$1):H only weakly interacts with the substrate[@a8; @a13]. The d$I$/d$V$ characteristics for the SWNT shown in Fig. 1(d) are consistent with those of a s-SWNT, as determined in Ref. . In Fig. 1(e), the absolute current vs bias, as a function of position along the SWNT, is shown[@FN2]. The white traces on Fig. 1(e) highlight the onset of the gap (upper and lower curves, at 1pA), as well as the midgap bias, equidistant from the conduction (upper) and valence (lower) band edges. The average onset biases are $-0.74\pm0.10$ V and $+0.64\pm0.08$ V, when an average is made from 0 to 150 Å; $-0.80\pm0.08$ and $0.60\pm0.06$ V, for an average made from 150 to 250 Å in Fig. 1(e) (standard deviations are also indicated). The average values indicate a $\sim$0.06 V lowering of the band edges (and hence of the midgap) in the fully depassivated section. The presence of standard deviations of this order in our measurements at room temperature (the oscillations presented here are also seen at lower temperatures, see Ref. [@Korea]) calls for an independent confirmation of this effect from DFT calculations.
Parameter Undoped $n$-doped
------------------- ---------------- ----------------
$\phi z_{max}$ 8.59 (3.0%) 8.60 (3.1%)
$\phi y_{max}$ 8.63 (3.5%) 8.47 (1.6%)
$\phi z_{min}$ 8.11 ($-$2.8%) 8.13 ($-$2.5%)
$\phi y_{min}$ 8.09 ($-$3.0%) 8.22 ($-$1.4%)
$\Delta z_{min}$ 2.69 2.75
$\Delta z'_{max}$ 1.96 1.94
: \[tab:table2\] Parameters of the structural deformation on the SWNT (Å).
 Band structures and PDOS of systems shown in Fig. 2. Flat bands around the Fermi level arise from localized states in the uppermost, unpassivated Si atoms. The highlighted PDOS of C atoms, shift downwards in energy in the $n$-doped plot, along with the bands in the substrate; compare to the C PDOS when no dopants are present. (b) The doping effect observed in Fig. 1(e) is independently confirmed from our calculations: Both C band edges go down by 0.11 eV once the depassivated stripe is present in the substrate. (c) The band structure of a 2-unit-cell long (8,4) SWNT in the absence of compression, and the corresponding band structure of a SWNT with the parametric distortion as in Eq. (1). An ever slight reduction of the semiconducting gap is seen, as well as the breaking of the degeneracy in the points highlighted by horizontal arrows, also present in (a) when doping is present. (d) Schematic view of the parametric distortion (the distortion seen is larger than that in Eq. (1) for clarity).](Fig3){width=".45\textwidth"}
Calculations are performed for a (8,4) SWNT (diameter: 8.30 Å and length $L_0$=11.29 Å). The supercell has six Si monolayers. The lowermost layer is passivated with H in the dihydride configuration[@Northrup]. The uppermost Si layer is also H passivated, but in the monohydride configuration. The area spanned by the supercell is 43.21$\times$21.61 Å$^2$. The supercell employed contains 1024 atoms when both Si surfaces are passivated with hydrogen, and 1012 atoms when a stripe of depassivated Si is formed. The SIESTA code[@a19] is employed in the local density approximation (LDA) for exchange-correlation as parametrized by Perdew and Zunger[@PerdewZunger] from the Ceperley-Alder data[@a20]. Double-$\zeta$ plus polarization numerical atomic orbitals were used to expand the electronic wavefunctions, and a mesh cutoff of 220 Ry was employed to compute the overlap integrals. Si dimer rows form an angle of 45$^o$ with respect to the $x$ direction, as seen in Fig. 2. A section 10 Å wide (yellow rectangle in Fig. 2) is rendered chemically reactive by the removal of H atoms. Afterwards, the SWNT is placed on this substrate, crossing the depassivated section. A single phosphorous atom in the slab provides an $n$-type doping density of 10$^{20}$cm$^{-3}$. Due to the lack of covalent bonding we find between s-SWNTs and Si(100), a better functional (i.e, that found in Ref. [@vdW]) should in principle be used. We choose LDA as it well describes (by cancellation of errors) the spacing between the s-SWNT and Si(100) better than GGA functionals[@Nicksuggestion]. No corrections for basis set superposition error were added either. More details of the calculations can be found in Ref. . Commensurability of the system required a non-negligible longitudinal contraction of the SWNT by 3.8%. As seen in the inset in Fig. 2, this entails a reduction of 30% in the semiconducting gap, and an increase in diameter by 0.75% to minimize the additional forces caused by the longitudinal contraction[@FN3]. The entire system shown in Fig. 2 was relaxed employing only the $\Gamma$ point until individual forces did not exceed 0.04 eV/Å. Because of periodic boundary conditions, the hydrogen stripe appears along the SWNT axis multiple times. This is to be contrasted with experiment, where a single stripe is fabricated. The SWNT displays a periodic distortion: it appears oblate with its maximum distortion occurring at $x_1$ and $x_2$ (Fig. 2). At $x_1$, the major axis is parallel to the $y-$direction; while at $x_2$ it is the minor axis that is parallel to the $y-$direction. The reason for the distortion is that the SWNT bends towards surface depassivated Si atoms; to relieve the most stress, this vertical elongation is accompanied by a horizontal elongation at the edges of the unit cell. The Si atoms in the depassivated section also protrude towards the SWNT. Specific values are in Table I. Although smaller than experimental values in magnitude, the DFT results are consistent with a mechanical distortion of the SWNT caused by the substrate (in simulations the depassivated stripe is 10 Å wide; in experiment it is about 100 Å wide). The resulting band structure, computed with a 4$\times$4$\times$1 Monkhorst-Pack $k-$point mesh[@MP], with and without dopants is given in Fig. 3(a). Flat bands are due to dangling bonds in Si atoms which pin the Fermi level. The projected density of states (PDOS) of C atoms is highlighted in Fig. 3(a). For an undoped substrate, electrons from the periphery of the SWNT escape to the substrate, and as a result the SWNT becomes $p$-doped as the carbon HOMO level moves upward towards the Fermi energy (see also Ref. ). Upon $n$-doping of the Si substrate, the location of the carbon band edges in Fig. 3(a) moves downwards with respect to the system’s Fermi level, as the Coulomb repulsion caused by excess electrons in the substrate suppress to some extent electron transfer from the SWNT.
In order to provide theoretical support to the lowering of the band edges when the substrate is locally depassivated, the PDOS for the SWNT on a $n-$doped substrate with full H-coverage on its upper surface was obtained. (In this case, no further relaxation to the fully passivated H substrate was performed upon placement of the dopant atom.) The location of the SWNT conduction and valence band edges are shown in Fig. 3(b). It can be seen that the band edges shift down by 0.11 eV when the H-depassivated strip is present. The larger value than the one found in experiment may be due to the fact that in calculations the depassivated strip repeats infinitely along the nanotube’s length. Discrepancies may also be due to the approximations in the calculations. (The change in the nanotube band gap of about 10 meV lays within our precision in computing the PDOS). A similar calculation was performed for the case when the substrate was undoped. In that case the band edges remained in their original positions even when the depassivated strip was present: The band edges in this latter case were at $-4.69$ ($-4.69$) eV and $-5.32$ ($-5.33$) eV on the full H-passivated (partially depassivated) systems.
The mechanical distortion in the SWNT and its relation to the electronic band structure can be understood by introducing a parametric distortion along the $y$ and $z$ directions to an uncompressed, isolated SWNT: $$\begin{aligned}
y=y_0[1+0.04\cos(\pi x_0/L_0)],\nonumber\\
z=z_0[1+0.07\sin(\pi x_0/L_0)].\end{aligned}$$ ($x_0,y_0,z_0$) are the coordinates of an undistorted, uncompressed SWNT. Eq. (1) implies a periodicity in the $x$ direction over two SWNT unit cells, aimed to reduce the local distortion for C atoms, while keeping a relatively small supercell. The dissimilar amplitude of the modulation along the $y$ and $z$ directions is responsible for the lifting of the degeneracy at $k-$points, highlighted by horizontal arrows in Fig. 3(a), n-doping and Fig. 3(c). This parametric distortion results in a modest reduction of the semiconducting gap, also consistent with results from full-scale calculations (Fig. 3(a)). Fig. 3(d) schematically depicts the shape of the SWNT after a distortion as that shown in Eq. (1) is applied. The distortion also results in a shift of the nanotube’s conduction and valence band edges away from the $\Gamma-$point, as is the case in Fig. 3(a).
In conclusion, it has been shown from STM data and DFT calculations that partial depassivation of degenerately $n-$doped Si(100)-(2$\times$1):H produces a mechanical distortion within an adsorbed s-SWNT that slightly modifies the magnitude of the semiconducting gap, and produces minor modifications to the band structure of the nanotube. More importantly, the partial depassivation allows for a slight local doping of the SWNT adsorbed on this substrate. We expect that an increase in the area of the clean stripe will result in a further population of electronic states at the Si(100)-SWNT interface, until the behavior described in Ref. is recovered. We thank M. Kuroda, N. A. Romero and K. Ritter for discussions. Calculations were performed on the Turing cluster and the Intel-64 Abe cluster at U of I. Support by the NCSA (grants TG-PHY090002 and TG-PHY090034) is acknowledged.
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abstract: 'Giving customers queue length information about a service system has the potential to influence the decision of a customer to join a queue. Thus, it is imperative for managers of queueing systems to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze a two-dimensional deterministic fluid model that incorporates customer choice behavior based on **delayed** queue length information. All of the previous literature assumes that all queues have identical parameters and the underlying dynamical system is symmetric. However, in this paper, we relax this symmetry assumption by allowing the arrival rates, service rates, and the choice model parameters to be different for each queue. Our methodology exploits the method of multiple scales and asymptotic analysis to understand how to break the symmetry. We find that the asymmetry can have a large impact on the underlying dynamics of the queueing system.'
author:
- |
Philip Doldo\
Center for Applied Mathematics\
Cornell University\
657 Rhodes Hall, Ithaca, NY 14853\
[email protected]\
- |
Jamol Pender\
School of Operations Research and Information Engineering\
Center for Applied Mathematics\
Cornell University\
228 Rhodes Hall, Ithaca, NY 14853\
[email protected]\
- |
Richard Rand\
Department of Mathematics\
Sibley School of Mechanical and Aerospace Engineering\
Cornell University\
417 Upson Hall, Ithaca, NY 14853\
[email protected]\
bibliography:
- 'Breaking\_Symmetry\_2.bib'
title: Breaking the Symmetry in Queues with Delayed Information
---
Introduction {#sec_intro}
============
In many service systems, customers are given information about the system, which can influence their decision to join the queue. In many of these systems, either the queue length or the waiting time might be given to customers to estimate the time that they might lose waiting to receive service. Consequently, it is important for service system managers to understand how the information that they provide to customers can affect the underlying queueing system’s dynamics. The most common way that service systems interact with their customers are through delay announcements.
Delay announcements provide customers with information about the estimated time that a customer will wait for service. This is usually important when customers have multiple decisions about what service they might want to receive. For example, if one takes a trip to Disney World, one has the option to take many different rides as seen by @nirenberg2018impact. Thus, the information provided by the company will influence park goers to join the line for specific rides. Since this issue is quite important, the impact of delay announcements on the performance of queueing systems has been studied quite extensively in the applied probability literature. See for example work by @armony2004customer [@guo2007analysis; @hassin2007information; @armony2009impact; @guo2009impacts; @jouini2009queueing; @jouini2011call; @allon2011impact; @allon2011we; @whitt1999improving]. Unfortunately, most of the literature assumes that the information that the customer receives is in real time and is 100% accurate.
However, there are two important scenarios where real-time information is unreasonable to assume. First, in reality, most of the information communicated to customers is through some electronic device. Generally these devices need time to process the wait time information and send it to the customer. In both processing and sending the information, it is possible that the information experiences some time lag. The second scenario is where customers must commit to a queue before physically joining the queue. In this setting, the information itself is not delayed, but the customer must experience a travel delay. During this delay, the system state will most certainly change by the time the customer arrives to the queue. From the perspective of Disney World, customers who use the My Disney app, first see the wait times of the rides, choose a ride, and then walk to that ride. The time the customer spends walking to the ride represents the time delay.
Unlike the previous literature, this paper takes on the challenge of trying to understand the impact of giving delayed information to customers and how this impacts the dynamics of the queueing system. However, we are not the first to consider delayed information in the context of queueing systems. The impact of delayed information has been studied in previous work by @pender2017queues [@pender2018analysis; @novitzky2019nonlinear; @penderstochastic]. However, one crucial assumption in all of the previous work is that all queues have the same arrival rates, service rates, and the same choice model function. Thus, the previous models have a symmetry that is easy to exploit for mathematical and analysis purposes. In this paper, we attempt to explore the same delayed information queueing model, but instead where the queues have different arrival rates, service rates and different choice functions, thereby *breaking the symmetry*. This is a significant challenge as symmetry was the key ingredient in previous analyses of our model.
To this end, we consider an asymmetric two-dimensional system of delay differential equations in which the customer choice model is informed by delayed queue length information. We apply techniques from asymptotic analysis such as the method of multiple scales to analyze how the stability of the system depends on how delayed the information in the choice model is provided to customers. In doing so, we derive a first-order approximation of the asymmetric equilibrium, the critical delay for a Hopf bifurcation to occur, and the amplitude of oscillations when a limit cycle is born. To find these quantities, we employ Lindstedt’s method to derive an analytic expression for an approximation of the amplitude of limit cycles when the delay is close to the critical delay. Our analysis provides additional insight into how the asymmetry of our model can impact the performance of the underlying queueing system.
Main Contributions of Paper
---------------------------
The contributions of this work can be summarized as follows.
- We analyze an asymmetric two-dimensional fluid model that uses delayed queue length information to inform customers about the queue length at each queue. We show how the asymmetry affects the queueing model’s equilibrium.
- Using the method of multiple scales, we derive an approximation for the critical delay, which determines the location of a Hopf bifurcation, in terms of the queueing model parameters.
- We derive an asymptotic closed form approximation for the amplitudes of the limit cycles that arise when the delay is larger than the critical delay.
Organization of Paper
---------------------
The remainder of this paper is organized as follows. In Section \[sec\_modeling\], we review the symmetric two dimensional queueing model and then introduce the asymmetric model that we will analyze. Section \[sec\_equilibrium\] derives an expression for the new approximate equilibrium point of the asymmetric system. We demonstrate through several numerical examples to show the equilibrium changes as the queueing model’s parameters are varied. Section \[sec\_delta\_mod\] uses the method of multiple scales and asymptotic analysis to find the critical delay at which the stability of the delay differential equations changes. We also demonstrate numerically that our approximate critical delays are quite accurate and determine the location of Hopf bifurcations. In Section \[sec\_amplitude\], we use Lindstedt’s method to approximate the amplitude of limit cycles when the delay is near the critical delay. We show through numerical examples that our amplitude approximations are accurate near the critical delay, even with the model asymmetry. Finally in Section \[sec\_conclusion\], we conclude with directions for future research related to this work.
Asymmetric Queueing Model {#sec_modeling}
=========================
In this section, we describe the asymmetric queueing model that we will analyze in this paper. In previous literature on queueing systems with delayed information, such as @novitzky2019nonlinear and @pender2018analysis, a common assumption is that the queueing system is symmetric. This symmetry was assumed for convenience in the analysis since the analysis of an N-dimensional DDE system can be reduced to a one dimensional DDE. In this paper, our focus is on understanding the impact of asymmetry on the dynamics of the queuing system. In fact, it can be shown that the asymmetric model does not yield an explicit closed form formula for the equilibrium and the equilibrium can only be written as the solution to a fixed point equation. This is true even in the two dimensional case. Thus, the asymmetric model presents significant mathematical challenges that the symmetric model does not.
The symmetric model used in previous literature consists of two infinite-server queues. The two queues are coupled through the arrival rate function, which is equal to the product of the arrival rate $\lambda > 0$ and the probabilistic choice model for joining each queue. The choice model that determines the probabilities of joining each queue is based on a Multinomial Logit Model (MNL) @ben1999discrete, @train2009discrete that makes the decision off of delayed queue length information, as shown in the following system @novitzky2019limiting. Customers are served immediately at each queue at rate $\mu > 0$ and therefore the total departure rate at each queue is queue length times the service rate. The infinite server queue is widely used as a canonical model that represents the best one can hope for, see for example @iglehart1965limiting [@fralix2009infinite; @daw2018queues; @daw2019distributions; @daw2019new]. This is because the infinite server queue is a lower bound for multi-server queues without abandonment. From a dynamical system perspective, it was shown in @novitzky2019limiting that it is unnecessary to study fluid models with a finite number of servers as the finite server model can be reduced to an infinite server dynamical system model, with modified different parameters. The two delay differential equations in the symmetric case are given by the following equations
$$\begin{aligned}
{\raisebox{-0.4pt}{$\stackrel{\bullet}{q}$}}_1(t) &= \lambda \cdot \frac{\exp(- \theta q_1(t-\Delta) + \alpha )}{\exp(-\theta q_1(t-\Delta) + \alpha ) + \exp(-\theta q_2(t-\Delta) + \alpha)} - \mu q_1(t) \label{symmetric equation 1}\\
{\raisebox{-0.4pt}{$\stackrel{\bullet}{q}$}}_2(t) &= \lambda \cdot \frac{\exp(-\theta q_2(t-\Delta) + \alpha)}{\exp(-\theta q_1(t-\Delta) + \alpha ) + \exp(-\theta q_2(t-\Delta) + \alpha)} - \mu q_2(t) \label{symmetric equation 2}\end{aligned}$$
where we assume that $q_1(t)$ and $q_2(t)$, which represent the queue lengths as functions of time, start with different initial continuous functions on the interval $[-\Delta, 0].$ One should note that if the two queue lengths in the symmetric model start with identical initial functions, then they will remain the same for all time. Now we will describe the symmetric model’s parameters. The parameter $\lambda$ represents the arrival rate, which is the rate at which customers arrive to each queue. The parameter $\mu$ is the service rate at which servers will serve each customer in the system. The parameter $\theta$ is the customer sensitivity to the queue length. When the parameter $\theta$ is large, then customers are highly sensitive to the queue length. In fact, when we let $\theta \to \infty$, the MNL model converges to the indicator function for the smallest queue. In addition, when we let $\theta \to 0$, the MNL model converges to $\frac{1}{N}$ and the system becomes a system of N independent and uncoupled infinite server queues. The parameter $\alpha$ is the customer preference parameter. Whichever queue has the largest preference parameter $\alpha$, then customers are more likely to go to that queue regardless of the queue length. The parameter $\alpha$ may initially seem pointless as it cancels in the the symmetric system, however, we include it for clarity because it will not cancel in the asymmetric system. With those four model parameters, we can break the symmetry by perturbing the parameters associated with the first queue, yielding the following asymmetric queueing system
$$\begin{aligned}
{\raisebox{-0.4pt}{$\stackrel{\bullet}{q}$}}_1(t) &= (\lambda + \epsilon \hat{\lambda}) \cdot \frac{\exp(- (\theta + \epsilon \hat{\theta}) q_1(t-\Delta) + (\alpha + \epsilon \hat{\alpha}))}{\exp(-(\theta + \epsilon \hat{\theta}) q_1(t-\Delta) + (\alpha + \epsilon \hat{\alpha})) + \exp(-\theta q_2(t-\Delta) + \alpha)} \nonumber \\ &- (\mu + \epsilon \hat{\mu}) q_1(t) \label{perturbed equation 1} \\
{\raisebox{-0.4pt}{$\stackrel{\bullet}{q}$}}_2(t) &= \lambda \cdot \frac{\exp(-\theta q_2(t-\Delta) + \alpha)}{\exp(-(\theta + \epsilon \hat{\theta}) q_1(t-\Delta) + (\alpha + \epsilon \hat{\alpha})) + \exp(-\theta q_2(t-\Delta) + \alpha)} - \mu q_2(t) \label{perturbed equation 2}\end{aligned}$$
where $\epsilon$ is assumed to be a small parameter. This is the asymmetric system that we will be concerned with throughout this paper.
Before we move to the analysis of the asymmetric model, we believe that it is important to observe that the asymmetric model can be viewed as a symmetric model where the parameters are uncertain or random. Thus, our asymmetric model can be used to provide confidence intervals around the symmetric model when the model parameters are unknown. This is also useful from a statistical perspective when the model parameters are obtained through some inference analysis and they are not exactly symmetric. In the age of of uncertainty quantification, the asymmetry analysis provides information about the DDEs with random parameters.
Asymptotic Analysis of the Equilibrium {#sec_equilibrium}
=======================================
In this paper, our goal is to analyze the stability of the queueing system as a function of the model parameters and the delayed information and to approximate the amplitude of the limit cycles near the bifurcation point. In order to understand the stability of the queueing model, we must calculate the equilibrium or an approximation equilibrium for our queueing model. For the symmetric model, @novitzky2019nonlinear shows that the symmetric model given in Equations \[symmetric equation 1\]-\[symmetric equation 2\] has a unique equilibrium point at $q_1 = q_2 = \frac{\lambda}{2 \mu}$. In this section, we explore the effects that the asymmetry has on this equilibrium point. In doing so, we obtain a first-order (in $\epsilon$) approximation of the equilibrium of our perturbed system which is described in Theorem \[equilibrium\_theorem\].
\[equilibrium\_theorem\]
The system of Equations \[perturbed equation 1\]-\[perturbed equation 2\] has an approximate (up to order $\epsilon^2$) equilibrium point at $$\left(q_1^*, q_2^* \right)= \left(\frac{\lambda}{2 \mu} + a\epsilon + O(\epsilon^2), \frac{\lambda}{2 \mu} + b \epsilon + O(\epsilon^2)\right)$$ where $$a = \frac{\lambda \theta + 4 \mu}{4 \mu (\lambda \theta + 2 \mu)} \hat{\lambda} + \frac{- \lambda(\lambda \theta + 4 \mu)}{4 \mu^2 (\lambda \theta + 2 \mu)} \hat{\mu} + \frac{- \lambda^2}{4 \mu (\lambda \theta + 2 \mu)} \hat{\theta} + \frac{\lambda}{2(\lambda \theta + 2 \mu)} \hat{\alpha}$$ and $$b = \frac{\lambda \theta}{4 \mu (\lambda \theta + 2 \mu)} \hat{\lambda} + \frac{- \lambda^2 \theta}{4 \mu^2 (\lambda \theta + 2 \mu)} \hat{\mu} + \frac{\lambda^2}{4 \mu (\lambda \theta + 2 \mu)} \hat{\theta} + \frac{- \lambda}{2 (\lambda \theta + 2 \mu)} \hat{\alpha}.$$
\[equilibrium theorem\]
If we substitute in the constants $q_1^*$ and $q_2^*$ for $q_1$ and $q_2$ in the system \[symmetric equation 1\]-\[symmetric equation 2\], respectively, we have that $q_1(t) = q_1(t - \Delta) = \frac{\lambda}{2 \mu} + a \epsilon + O(\epsilon^2)$ and $q_2(t) = q_2(t - \Delta) = \frac{\lambda}{2 \mu} + b \epsilon + O(\epsilon^2)$ which gives us
$$\begin{aligned}
0 &= (\lambda + \epsilon \hat{\lambda}) \left[ 1 + \exp\left(\epsilon\left(\theta (a-b) + \frac{\lambda}{2 \mu} \hat{\theta} - \hat{\alpha}\right) + O(\epsilon^2)\right) \right]^{-1} - (\mu + \epsilon \hat{\mu}) \left( \frac{\lambda}{2 \mu} + a \epsilon + O(\epsilon^2) \right)\\
&= (\lambda + \epsilon \hat{\lambda}) \left( \frac{1}{2} - \frac{1}{4} \left( \theta (a - b) + \frac{\lambda}{2 \mu} \hat{\theta} - \hat{\alpha} \right)\epsilon + O(\epsilon^2) \right) - (\mu + \epsilon \hat{\mu}) \left( \frac{\lambda}{2 \mu} + a \epsilon + O(\epsilon^2) \right)\\\end{aligned}$$
and
$$\begin{aligned}
0 &= \lambda \left[ 1 + \exp\left( \epsilon \left( \theta(b-a) - \frac{\lambda}{2 \mu} \hat{\theta} + \hat{\alpha} \right) + O(\epsilon^2) \right) \right]^{-1} - \mu \left( \frac{\lambda}{2 \mu} + b \epsilon + O(\epsilon^2) \right)\\
&= \lambda \left( \frac{1}{2} - \frac{1}{4} \left( \theta (b-a) - \frac{\lambda}{2 \mu} \hat{\theta} + \hat{\alpha} \right) \epsilon + O(\epsilon^2) \right)- \mu \left( \frac{\lambda}{2 \mu} + b \epsilon + O(\epsilon^2) \right).\end{aligned}$$
Matching $O(\epsilon)$ terms, we get a system of two equations with two unknowns $a$ and $b$.
$$\begin{aligned}
0 &= 2 \mu \lambda \theta (b - a) - 8 \mu^2 a - \lambda^2 \hat{\theta} + 2 \mu \lambda \hat{\alpha} + 4 \mu \hat{\lambda} - 4 \lambda \hat{\mu}\\
0 &= 2 \mu \lambda \theta (a-b) - 8 \mu^2 b + \lambda^2 \hat{\theta} - 2 \mu \lambda \hat{\alpha}\end{aligned}$$
Solving this two dimensional system of equations gives us the desired values for $a$ and $b$.
Our expression for the new approximate equilibrium makes it very easy to understand what happens to the equilibrium when only a single parameter is perturbed. We observe that the $\hat{\lambda}$ terms in $a$ and $b$ are both positive so that a positive perturbation in the arrival rate $\lambda$ will cause the equilibrium for each queue to increase, with the first queue’s equilibrium increasing more as the $\hat{\lambda}$ term in $a$ is larger than the corresponding term in $b$. One should note this asymmetry in the arrival rate change. In fact, this is because the increase in the arrival rate is direct to first queue, but is indirect for the second queue. By similar reasoning, we see that positively perturbing the service rate $\mu$ will cause the equilibrium corresponding to each queue to decrease, with the first queue’s equilibrium decreasing more than that of the second queue. Thus, we observe the effects of increasing either $\hat{\lambda}$ or $\hat{\mu}$ are not symmetric with respect to each queue, but they have the same sign in this model. However, if we only perturb either $\theta$ or $\alpha$, we observe that one of the queue length’s equilibrium will increase while the other will decrease. Despite the opposite signs of direction, the magnitude of the change is identical and symmetric.
Numerical Verification of Equilibrium
-------------------------------------
In this section, we analyze the validity of Theorem \[equilibrium theorem\] by plotting several numerical examples. Below we show plots of queue length versus time to illustrate the shift in the equilibrium due to the perturbations of the model parameters.
In Figure \[Fig1\], we consider the symmetric system for two values of $\Delta$, each of which shows us a qualitatively different behavior of the system as the queue lengths decay in one case and grow in the other. This trend will be discussed more in the following section. In this case, the equilibrium is at $q_1 = q_2 = \frac{\lambda}{2 \mu}$.
In Figure \[Fig2\], we consider the perturbed system where the only perturbed parameter is the queueing system’s arrival rate $\lambda$. Since we increased the arrival rate for the first queue, it makes sense that the equilibrium for $q_1$ increases. However, we observe that the equilibrium for $q_2$ also increases, to a slightly less extent, and this is due to the fact that the probability of joining the second queue depends on the delayed length of the first queue in a way so that if the delayed length of the first queue increases (which of course happens because we increased the arrival rate into the first queue), then the probability of joining the second queue increases.
In Figure \[Fig3\], the only perturbed parameter is the service rate $\mu$. Since we increased the service rate for the first queue, we see that the equilibrium for it decreases. We also see that the equilibrium for the second queue decreases because the probability of joining the second queue decreases when the delayed length of the first queue decreases.
In Figure \[Fig4\], the only perturbed parameter is $\theta$. Perturbing this parameter positively causes the probability of joining the first queue to decrease and the probability of joining the second queue to increase which gives us some intuition for why we see the equilibrium for the first queue decrease and the equilibrium for the second queue increase.
In Figure \[Fig5\], the only perturbed parameter is $\alpha$. Perturbing this parameter positively causes the probability of joining the first queue to increase and the probability of joining the second queue to decrease, which causes the equilibrium for the first queue to increase and the equilibrium for the second queue to decrease.
We see that the effects of perturbing $\alpha$ are the opposite of the effects of perturbing $\theta$. Similarly, perturbing $\lambda$ seems to qualitatively affect the equilibrium in a way opposite to how perturbing $\mu$ does.
In Figure \[Fig6\], all four of the aforementioned parameters are perturbed and in this case the resulting behavior depends on how big the permutations are for each parameter.
We summarized the various values used in these figures along with the resulting equilibrium values and approximations and errors in Table \[Table1\]. A natural concern is how the error of the equilibrium approximation varies as a parameter is perturbed by varying amounts. We explore this by varying $\hat{\lambda}$ while keeping other parameters fixed. The results can be seen in Table \[Table2\] and they are plotted on the left side of Figure \[Fig\_amp\_error\]. We do the same for $\hat{\mu}$ in Table \[Table3\] and we plotted those results in the right side of Figure \[Fig\_amp\_error\]. In both cases, we see that the error increases as the parameter is perturbed more. This is expected as we expect our approximation to get its best results when the perturbations are small.
[ | l | l | l | l | l | l | l | l | l | l | l | l | l | l | l | l |]{}\
\[-2.5ex\] $\lambda$ & $\hat{\lambda}$ & $\mu$ & $\hat{\mu}$ & $\theta$ & $\hat{\theta}$ & $\alpha$ & $\hat{\alpha}$ & $\epsilon$ & $\hat{q}_1$ & $q_1$ & $\hat{q}_1$ error & $\hat{q}_2$ & $q_2$ & $\hat{q}_2$ error\
10 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0.1 & $5.0292$ & $5.0290$ & $2 \cdot 10^{-4}$ & 5.0208 & 5.0207 & $1 \cdot 10^{-4}$\
10 & 0 & 1 & 0.1 & 1 & 0 & 0 & 0 & 0.1 & $4.9708$ & $ 4.9710$ & $2 \cdot 10^{-4}$ & 4.9792 & 4.9793 & $1 \cdot 10^{-4}$\
10 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0.1 & $ 4.7917$ & $4.8000$ & 0.0083 & 5.2084 & 5.2000 & 0.0084\
10 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0.1 & $5.0417$ & $ 5.0417 $ & 0.0000 & 4.9583 & 4.9583 & 0.0000\
10 & 1 & 1 & 0.1 & 1 & 1 & 0 & 1 & 0.1 & $ 4.8333$& $ 4.8400$ & 0.0067 & 5.1667 & 5.1600 & 0.0067\
[| l | l | l | l | l | l | l |]{}\
\[-2.5ex\] $\hat{\lambda}$ & $\hat{q}_1$ & $q_1$ & $\hat{q}_1$ error & $\hat{q}_2$ & $q_2$ & $\hat{q}_2$ error\
1 & $ 5.0292$ & $5.0290$ & $2 \cdot 10^{-4}$ & 5.0208 & 5.0207 & $1 \cdot 10^{-4}$\
1.5 & $ 5.0438$ & $5.0435$& $3 \cdot 10^{-4}$ & 5.0313 & 5.0311 & $2 \cdot 10^{-4}$\
2 & $ 5.0583 $ & $ 5.0578 $ & $5 \cdot 10^{-4}$ & 5.0417 & 5.0413 & $4 \cdot 10^{-4}$\
2.5 & $ 5.0729 $ & $ 5.0722 $ & $7 \cdot 10^{-4}$ & 5.0521 & 5.0515 & $6 \cdot 10^{-4}$\
3 & $ 5.0875$ & $ 5.0864 $ & 0.0011 & 5.0625 & 5.0617 & $8 \cdot 10^{-4}$\
3.5 & $ 5.1021 $ & $ 5.1006 $ & 0.0015 & 5.0729 & 5.0719 & 0.0010\
4 & $ 5.1167 $ & $ 5.1147 $ & 0.0020 & 5.0833 & 5.0820 & 0.0013\
4.5 & $ 5.1313$ & $5.1288 $ & 0.0025 & 5.0938 & 5.0920 & 0.0018\
5 & $ 5.1458$ & $ 5.1429\ $ & 0.0029 & 5.1042 & 5.1020 & 0.0022\
[| l | l | l | l | l | l | l |]{}\
\[-2.5ex\] $\hat{\mu}$ & $\hat{q}_1$ & $q_1$ & $\hat{q}_1$ error & $\hat{q}_2$ & $q_2$ & $\hat{q}_2$ error\
0.1 & $4.9708 $ & $ 4.9710 $ & $2 \cdot 10^{-4}$ & 4.9792 & 4.9793 & $1 \cdot 10^{-4}$\
0.15 & $ 4.9562 $ & $ 4.9566 $& $4 \cdot 10^{-4}$ & 4.9688 & 4.9690 & $2 \cdot 10^{-4}$\
0.2 & $ 4.9417 $ & $ 4.9423 $ & $6 \cdot 10^{-4}$ & 4.9583 & 4.9588 & $5 \cdot 10^{-4}$\
0.25 & $ 4.9271 $ & $ 4.9281 $ & 0.001 & 4.9479 & 4.9487 & $8 \cdot 10^{-4}$\
0.3 & $ 4.9125 $ & $ 4.9140 $ & 0.0015 & 4.9375 & 4.9386 & 0.0011\
0.35 & $ 4.8979 $ & $ 4.9000 $ & 0.0021 & 4.9271 & 4.9285 & 0.0014\
0.4 & $ 4.8833 $ & $ 4.8860 $ & 0.0027 & 4.9167 & 4.9186 & 0.0019\
0.45 & $ 4.8688 $ & $ 4.8721 $ & 0.0033 & 4.9063 & 4.9087 & 0.0024\
0.5 & $4.8542 $ & $ 4.8583 $ & 0.0041 & 4.8958 & 4.8988 & 0.0030\
![Plots of the error of the equilibrium approximation against $\hat{\lambda}$ and $\hat{\mu}$ from Tables \[Table2\] (Left) and \[Table3\] (Right)[]{data-label="Fig_amp_error"}](./Code/Paper_Figures/equilibrium_error_lambda_hat2.pdf "fig:") ![Plots of the error of the equilibrium approximation against $\hat{\lambda}$ and $\hat{\mu}$ from Tables \[Table2\] (Left) and \[Table3\] (Right)[]{data-label="Fig_amp_error"}](./Code/Paper_Figures/equilibrium_error_mu_hat3.pdf "fig:")
![$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 0$, $\lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig1"}](./Code/Paper_Figures/Equilibrium_sym_Fig_1.pdf "fig:") ![$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 0$, $\lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig1"}](./Code/Paper_Figures/Equilibrium_sym_Fig_2.pdf "fig:")
![$\hat{\lambda} = 1, \hat{\mu} = \hat{\theta} = \hat{\alpha} = 0$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig2"}](./Code/Paper_Figures/Equilibrium_lambda_Fig_1.pdf "fig:") ![$\hat{\lambda} = 1, \hat{\mu} = \hat{\theta} = \hat{\alpha} = 0$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig2"}](./Code/Paper_Figures/Equilibrium_lambda_Fig_2.pdf "fig:")
![$\hat{\mu} = 1, \hat{\lambda}= \hat{\theta} = \hat{\alpha} = 0$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig3"}](./Code/Paper_Figures/Equilibrium_mu_Fig_1.pdf "fig:") ![$\hat{\mu} = 1, \hat{\lambda}= \hat{\theta} = \hat{\alpha} = 0$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig3"}](./Code/Paper_Figures/Equilibrium_mu_Fig_2.pdf "fig:")
![$\hat{\theta}=1, \hat{\lambda} = \hat{\mu} = \hat{\alpha} = 0$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig4"}](./Code/Paper_Figures/Equilibrium_theta_Fig_1.pdf "fig:") ![$\hat{\theta}=1, \hat{\lambda} = \hat{\mu} = \hat{\alpha} = 0$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig4"}](./Code/Paper_Figures/Equilibrium_theta_Fig_2.pdf "fig:")
![$\hat{\alpha} = 1, \hat{\lambda} = \hat{\mu} = \hat{\theta} = 0$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig5"}](./Code/Paper_Figures/Equilibrium_alpha_Fig_1.pdf "fig:") ![$\hat{\alpha} = 1, \hat{\lambda} = \hat{\mu} = \hat{\theta} = 0$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig5"}](./Code/Paper_Figures/Equilibrium_alpha_Fig_2.pdf "fig:")
![$\hat{\mu} = 0.1, \hat{\lambda} = \hat{\theta} = \hat{\alpha} = 1$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig6"}](./Code/Paper_Figures/Equilibrium_all_Fig_1.pdf "fig:") ![$\hat{\mu} = 0.1, \hat{\lambda} = \hat{\theta} = \hat{\alpha} = 1$, $\epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$ and $q_2 = 5.01$, Left: $\Delta = .25$, Right: $\Delta = .4$[]{data-label="Fig6"}](./Code/Paper_Figures/Equilibrium_all_Fig_2.pdf "fig:")
Hopf Bifurcation of Asymmetric Model {#sec_delta_mod}
====================================
Now that we have derived an approximate equilibrium for our asymmetric queueing model, we can now analyze the stability of this approximate equilibrium. In the symmetric model, @novitzky2019nonlinear shows that if $\lambda \theta > 2 \mu$, then the symmetric system given in Equations \[symmetric equation 1\]-\[symmetric equation 2\] will exhibit a Hopf bifurcation for values of $\Delta > \Delta_{\text{cr}}$ where $$\Delta_{\text{cr}} = \frac{ \arccos\left( \frac{-2 \mu}{\lambda \theta} \right)}{\omega_{\text{cr}}} \hspace{5mm} \text{ and } \hspace{5mm} \omega_{\text{cr}} = \frac{1}{2} \sqrt{\lambda^2 \theta^2 - 4 \mu^2}.$$ Our goal in this section is to derive an analogous critical delay expression for the asymmetric model, which we will denote as $\Delta_{\text{mod}}$. We will show that the new critical delay, $\Delta_{\text{mod}}$, marks a change in stability for the queueing model and we verify this result using numerical integration of DDEs. This means that we show from numerical integration that a limit cycle is born at this modified critical value of $\Delta_{mod}$. Our analysis for deriving the approximate critical delay makes use of the method of multiple scales. We show this result below in Theorem \[hopf\_eqn\]
\[hopf\_eqn\] If $ \lambda \theta> 2 \mu$, then, for sufficiently small $\epsilon$, the stability of the queueing system given in Equations \[perturbed equation 1\]-\[perturbed equation 2\] changes when $\Delta = \Delta_{\text{mod}}$ where
$$\begin{aligned}
\Delta_{\text{mod}} &= \Delta_{\text{cr}} - \epsilon \left( \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \lambda \omega_{\text{cr}}^2} \hat{\lambda} - \frac{1 + \mu \Delta_{\text{cr}}}{2 \omega_{\text{cr}}^2} \hat{\mu} + \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \theta \omega_{\text{cr}}^2} \hat{\theta} \right) + O(\epsilon^2).\end{aligned}$$
\[delta mod theorem\]
We begin by linearizing the system of Equations \[perturbed equation 1\]-\[perturbed equation 2\] about the approximate equilibrium point $$(q_1^*, q_2^*) = \left(\frac{\lambda}{2 \mu} + a\epsilon + O(\epsilon^2), \frac{\lambda}{2 \mu} + b \epsilon + O(\epsilon^2)\right)$$ where $a$ and $b$ are as defined in Theorem 2.1. In doing so, we introduce the functions $\tilde{u}_1(t)$ and $\tilde{u}_2(t)$ so that $$q_1(t) = q_1^* + \tilde{u}_1(t), \hspace{5mm} q_2(t) = q_2^* + \tilde{u}_2(t)$$ and we approximate $\overset{\bullet}{\tilde{u}}_1$ and $\overset{\bullet}{\tilde{u}}_2$ by a linear Taylor expansion about the equilibrium point $\tilde{u}_1(t) = \tilde{u}_2(t) = \tilde{u}_1(t - \Delta) = \tilde{u}_2(t - \Delta) = 0$ and we denote the linear approximations by $\overset{\bullet}{u}_1(t)$ and $\overset{\bullet}{u}_2(t)$, respectively, and we Taylor expand coefficients with nonlinear dependence on $\epsilon$ about $\epsilon = 0$ and neglect terms that are $O(\epsilon^2)$ yielding the following first-order (in $\epsilon$) approximation of the linear system.
$$\begin{aligned}
\overset{\bullet}{u}_1(t) &= \frac{(\lambda + \hat{\lambda} \epsilon) \theta}{4}\left[ u_2(t - \Delta) - u_1(t - \Delta) \right] - \frac{\lambda \hat{\theta}}{4} \epsilon u_1(t - \Delta) - (\mu + \hat{\mu} \epsilon) u_1(t) \label{u1 equation 45}\\
\overset{\bullet}{u}_2(t) &= \frac{\lambda \theta}{4} [u_1(t- \Delta) - u_2(t - \Delta)] + \frac{\lambda \hat{\theta}}{4} \epsilon u_1(t - \Delta) - \mu u_2(t) \label{u2 equation 46}\end{aligned}$$
We then proceed by making the change of variables $$v_1(t) = u_1(t) + u_2(t), \hspace{5mm} v_2(t) = u_1(t) - u_2(t)$$ to get the following system
$$\begin{aligned}
\overset{\bullet}{v}_1(t) + \left( \mu + \frac{\hat{\mu}}{2} \epsilon \right) v_1(t) &= - \frac{\theta \hat{\lambda}}{4}\epsilon v_2(t - \Delta) - \frac{\hat{\mu}}{2} \epsilon v_2(t) \label{v1 equation 1}\\
\overset{\bullet}{v}_2(t) + \left( \frac{\lambda \theta}{2} + \frac{\theta \hat{\lambda}}{4} \epsilon + \frac{\lambda \hat{\theta}}{4} \epsilon \right) v_2(t - \Delta) + \left( \mu + \frac{\hat{\mu}}{2} \epsilon \right) v_2(t) &= - \frac{\lambda \hat{\theta}}{4} \epsilon v_1(t - \Delta) - \frac{\hat{\mu}}{2} \epsilon v_1(t). \label{v2 equation 1}\end{aligned}$$
Before proceeding, we introduce new variables $$\xi = t, \hspace{5mm} \eta = \epsilon t$$ to represent a regular time and a slow time, respectively, so we have $$v_i(t) = v_i(\xi, \eta) \hspace{5mm} \text{ and } \hspace{5mm} v_i(t - \Delta) = v_i(\xi - \Delta, \eta - \epsilon \Delta)$$ and the derivatives become $$\overset{\bullet}{v}_i(t) = \frac{d v_i}{dt} = \frac{\partial u_i}{\partial \xi} \frac{d \xi}{d t} + \frac{\partial u_i}{\partial \eta} \frac{d \eta}{d t} = \frac{\partial v_i}{\partial \xi} + \epsilon \frac{\partial v_i}{\partial \eta}, \hspace{5mm} i = 1,2.$$ In addition to this change of variables, we expand our functions and detune our delay from the critical delay for the symmetric system as follows. $$v_1(t) = v_{1,0}(t) + \epsilon v_{1,1}(t) + O(\epsilon^2)$$ $$v_2(t) = v_{2,0}(t) + \epsilon v_{2,1}(t) + O(\epsilon^2)$$ $$\Delta = \Delta_{\text{cr}} + \epsilon \Delta_1 + O(\epsilon^2)$$ Taylor expanding our delayed terms yields $$v_i(t-\Delta) = v_i(\xi - \Delta, \eta - \epsilon \Delta) = \bar{v}_i - \epsilon \left( \Delta_1 \frac{\partial \bar{v}_i}{\partial \xi} + \Delta_{\text{cr}} \frac{\partial \bar{v}_i}{\partial \eta} \right) + O(\epsilon^2)$$ where $\bar{v}_i := v_i(\xi - \Delta_{\text{cr}}, \eta)$ for $ i = 1, 2.$ Applying these expansions to equations \[v1 equation 1\] and \[v2 equation 1\] and then collecting $O(1)$ terms and $O(\epsilon)$ terms yields the following four equations.
$$\begin{aligned}
\frac{\partial v_{1,0}}{\partial \xi} + \mu v_{1,0} &= 0 \label{v10 equation 1}\\
\frac{\partial v_{2,0}}{\partial \xi} + \frac{\lambda \theta}{2} \bar{v}_{2,0} + \mu v_{2,0} &= 0 \label{v20 equation 1}\\
\frac{\partial v_{1,1}}{\partial \xi} + \mu v_{1,1} &= - \frac{\partial v_{1,0}}{\partial \eta} - \frac{\theta \hat{\lambda}}{4} \bar{v}_{2,0} - \frac{\hat{\mu}}{2} \left( v_{1,0} + v_{2,0} \right) \label{v11 equation 1}\\
\frac{\partial v_{2,1}}{\partial \xi} + \frac{\lambda \theta}{2} \bar{v}_{2,1} + \mu v_{2,1} &= - \frac{ \partial v_{2,0}}{\partial \eta} + \frac{\lambda \theta}{2} \left( \Delta_1 \frac{\partial \bar{v}_{2,0}}{\partial \xi} + \Delta_{\text{cr}} \frac{\partial \bar{v}_{2,0}}{\partial \eta} \right) \label{v21 equation 1}\\
&- \frac{\theta \hat{\lambda}}{4} \bar{v}_{2,0} - \frac{\hat{\mu}}{2} \left( v_{1,0} + v_{2,0} \right) - \frac{\lambda \hat{\theta}}{4} \left( \bar{v}_{1,0} + \bar{v_{2,0}} \right) \nonumber\end{aligned}$$
It is easy to check that $$v_{1,0} = \tilde{c}(\eta) \exp(- \mu \xi) \hspace{5mm} \text{ and } \hspace{5mm} v_{2,0} = A(\eta) \cos(\omega_{\text{cr}} \xi) + B \sin(\omega_{\text{cr}} \xi)$$ solve Equations \[v10 equation 1\] and \[v20 equation 1\], respectively, and we can rearrange Equation \[v20 equation 1\] and use our expression for $v_{2,0}$ to observe that $$\bar{v}_{2,0} = - \frac{2}{\lambda \theta} \left[ \frac{\partial v_{2,0}}{\partial \xi} + \mu v_{2,0} \right] = \frac{2}{\lambda \theta} \left[ -(\mu A + \omega_{\text{cr}} B) \cos(\omega_{\text{cr}} \xi) + (\omega_{\text{cr}} A - \mu B) \sin(\omega_{\text{cr}} \xi) \right].$$ Thus, we have the following expressions for terms in Equations \[v11 equation 1\] and \[v21 equation 1\]
$$\begin{aligned}
\frac{\partial v_{1,0}}{\partial \eta} &= \tilde{c}' \exp(-\mu \xi)\\
\frac{\partial v_{2,0}}{\partial \eta} &= A' \cos(\omega_{\text{cr}} \xi) + B' \sin(\omega_{\text{cr}} \xi)\\
\frac{\partial \bar{v}_{2,0}}{\partial \eta} &= \frac{2}{\theta \lambda} \left[ -(\mu A' + \omega_{\text{cr}} B') \cos(\omega_{\text{cr}} \xi) + (\omega_{\text{cr}} A' - \mu B') \sin(\omega_{\text{cr}} \xi) \right]\\
\frac{\partial \bar{v}_{2,0}}{\partial \xi} &= \frac{2 \omega_{\text{cr}}}{\theta \lambda} \left[ ( \omega_{\text{cr}} A - \mu B) \cos( \omega_{\text{cr}} \xi) + (\mu A + \omega_{\text{cr}} B) \sin( \omega_{\text{cr}} \xi) \right]\end{aligned}$$
and Equations \[v11 equation 1\] and \[v21 equation 1\] can respectively be rewritten as
$$\begin{aligned}
\frac{\partial v_{1,1}}{\partial \xi} + v_{1,1} &= \left(\tilde{c}'(\eta) - \frac{\hat{\mu}}{2} \tilde{c}(\eta)\right)\exp(- \mu \xi) + \cos(\omega_{\text{cr}} \xi) \left[ \frac{\hat{\lambda}}{2 \lambda} (\mu A(\eta) + \omega_{\text{cr}} B(\eta)) - \frac{\hat{\mu}}{2} A(\eta) \right] \nonumber\\
&+ \sin(\omega_{\text{cr}} \xi) \left[ - \frac{\hat{\lambda}}{2 \lambda} (\omega_{\text{cr}} A(\eta) - \mu B(\eta)) - \frac{\hat{\mu}}{2} B(\eta) \right] \label{v11 equation 2}\\
\frac{\partial v_{2,1}}{\partial \xi} &+ \frac{\lambda \theta}{2} \bar{v}_{2,1} + \mu v_{2,1} = - \frac{\hat{\mu}}{2} \tilde{c} \exp(- \mu \xi) \nonumber\\
&+ \cos(\omega_{\text{cr}} \xi) \Bigg[A'(\eta)(- \mu \Delta_{\text{cr}} - 1) + B'(\eta) (- \omega_{\text{cr}} \Delta_{\text{cr}}) \nonumber \\
&+ A(\eta) \left(\omega_{\text{cr}}^2 \Delta_1 + \frac{\mu \hat{\lambda}}{2 \lambda} - \frac{\hat{\mu}}{2} + \frac{\mu \hat{\theta}}{2 \theta}\right) + B(\eta)\left(- \mu \omega_{\text{cr}} \Delta_1 + \frac{\omega_{\text{cr}} \hat{\lambda}}{2 \lambda} + \frac{\omega_{\text{cr}} \hat{\theta}}{2 \theta}\right) \Bigg] \nonumber\\
&+ \sin(\omega_{\text{cr}} \xi) \Bigg[ A'(\eta)\left( \omega_{\text{cr}} \Delta_{\text{cr}} \right) + B'(\eta) (-\mu \Delta_{\text{cr}} - 1) \nonumber\\
&+ A(\eta) \left( \mu \omega_{\text{cr}} \Delta_1 - \frac{\omega_{\text{cr}} \hat{\lambda}}{2 \lambda} - \frac{\omega_{\text{cr}} \hat{\theta}}{2 \theta} \right) + B(\eta) \left( \omega_{\text{cr}}^2 \Delta_1 + \frac{\mu \hat{\lambda}}{2 \lambda} - \frac{\hat{\mu}}{2} + \frac{\mu \hat{\theta}}{2 \theta} \right) \Bigg] . \label{v21 equation 2}\end{aligned}$$
We observe that the general homogeneous solutions for $v_{1,1}$ and $v_{2,1}$ are the same as the general homogeneous solutions for $v_{1,0}$ and $v_{2,0}$, respectively. In both Equations \[v11 equation 2\] and \[v21 equation 2\], there are terms present in the inhomogeneous part that are not linearly independent of the corresponding homogeneous solution. It is easy to see, by the method of undetermined coefficients for example (which introduces a factor of $\xi$ on to terms in the particular solution that correspond to terms in the inhomogeneity that are linearly dependent with a homogeneous solution), that such terms will give rise to secular terms in the particular solutions to each equation. We want to set terms in the inhomogeneities that introduce secular solutions equal to zero because our asymptotic expansions would otherwise become invalid for large time as the series would no longer be asymptotic when $\xi = O(\frac{1}{\epsilon})$, for example, at which point $O(\epsilon)$ terms in the series would become $O(1)$. Equating the coefficients of these terms in the inhomogeneities equal to zero yields the following equations.
$$\begin{aligned}
\frac{d \tilde{c}}{d \eta} &= \frac{\hat{\mu}}{2} \tilde{c} \label{secular equation 1}\\
\frac{d A}{d \eta} &= K_1 A(\eta) + K_2 B(\eta) \label{secular equation 2}\\
\frac{d B}{d \eta} &= K_3 A(\eta) + K_4 B(\eta) \label{secular equation 3}\end{aligned}$$
Solving Equation \[secular equation 1\] gives us that $\tilde{c}(\eta) = \tilde{k} \exp(\frac{\hat{\mu}}{2} \eta)$ and therefore $v_{1,0} = \tilde{k} \exp(\frac{\hat{\mu}}{2} \eta - \mu \xi)$ which decays to $0$ for sufficiently small $\epsilon$. We observe that the system of Equations \[secular equation 2\]-\[secular equation 3\] is in the form $$c_1 A' + c_2 B' + c_3 A + c_4 B = 0$$ $$-c_2 A' + c_1 B' - c_4 A + c_3 B = 0$$ where $$c_1 = - \mu \Delta_{\text{cr}} - 1, \hspace{5mm} c_2 = - \omega_{\text{cr}} \Delta_{\text{cr}}, \hspace{5mm} c_3 = \omega_{\text{cr}}^2 \Delta_1 + \frac{\mu \hat{\lambda}}{2 \lambda} - \frac{\hat{\mu}}{2} + \frac{\mu \hat{\theta}}{2 \theta}, \hspace{5mm} c_4 = - \mu \omega_{\text{cr}} \Delta_1 + \frac{\omega_{\text{cr}} \hat{\lambda}}{2 \lambda} + \frac{\omega_{\text{cr}} \hat{\theta}}{2 \theta}.$$ This tells us that
$$\begin{aligned}
K_1 &= K_4 = \frac{-(c_1 c_3 + c_2 c_4)}{c_1^2 + c_2^2}\\
K_2 &= - K_3 = \frac{c_2 c_3 - c_1 c_4}{c_1^2 + c_2^2}.\end{aligned}$$
So, we have the linear system $$\begin{aligned}
\begin{bmatrix}
\frac{d A}{d \eta} \\
\frac{d B}{d \eta}
\end{bmatrix} = \begin{bmatrix}
K_1 & - K_3 \\
K_3 & K_1
\end{bmatrix} \begin{bmatrix}
A\\
B
\end{bmatrix}. \label{slow flow system}\end{aligned}$$
Recall that $$v_{2,0} = A(\eta) \cos(\omega_{\text{cr}} \xi) + B(\eta) \sin(\omega_{\text{cr}} \xi)$$ so that $A(\eta)$ and $B(\eta)$ represent the amplitudes of each term in $v_{2,0}$. Thus, the equilibrium point $A(\eta) = B(\eta) = 0$ of this linear system corresponds to when $v_{2,0} = 0$ and it also corresponds to when the sinusoidal terms in the inhomogeneity in \[v11 equation 2\] are equal to zero which would make $v_{1,1}$ decay for sufficiently small $\epsilon$. Because of this, the stability of the equilibrium point $(A,B) = (0,0)$ to \[slow flow system\] corresponds to the stability of the DDE system given in Equations \[perturbed equation 1\]-\[perturbed equation 2\]>. Thus, our problem reduces to analyzing the stability of a linear system.
Now we define the following matrix $$K = \begin{bmatrix}
K_1 & - K_3 \\
K_3 & K_1
\end{bmatrix}.$$
Note that since we assumed $\lambda \theta > 2 \mu$, we have that each entry of $K$ is real. To analyze the stability of Equation \[slow flow system\], we need to determine whether the real parts of the eigenvalues of $K$ are positive or negative. However, keep in mind that the entries of $K$ depend on $\Delta_1$, so if we can find conditions on what the value of $\Delta_1$ must be in order for the real parts of the eigenvalues of $K$ to change sign, then we’ll essentially have found an approximation (up to $O(\epsilon)$ terms) for the critical value of $\Delta$ for which the stability of our DDE system given in Equations \[perturbed equation 1\]-\[perturbed equation 2\] changes and a Hopf bifurcation occurs. In particular, we note the special structure of this matrix $K$ (it is actually the matrix representation of the complex number $K_1 + i K_3$) and see that it has eigenvalues $K_1 \pm i K_3$. Thus, $K_1$ is the real part of both of the eigenvalues of $K$, so we want to find conditions on $\Delta_1$ under which $K_1$ is positive or negative. We see that $\text{sgn}(K_1) = -\text{sgn}(c_1 c_3 + c_2 c_4) $.
$$\begin{aligned}
c_1 c_3 + c_2 c_4 &= ( - \mu \Delta_{\text{cr}} - 1) \left( \omega_{\text{cr}}^2 \Delta_1 + \frac{\mu \hat{\lambda}}{2 \lambda} - \frac{\hat{\mu}}{2} + \frac{\mu \hat{\theta}}{2 \theta} \right) + (- \omega_{\text{cr}} \Delta_{\text{cr}}) \left( - \mu \omega_{\text{cr}} \Delta_1 + \frac{\omega_{\text{cr}} \hat{\lambda}}{2 \lambda} + \frac{\omega_{\text{cr}} \hat{\theta}}{2 \theta} \right)\\
&= - \omega_{\text{cr}}^2 \Delta_1 - \left[ \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \lambda} \hat{\lambda} - \frac{1 + \mu \Delta_{\text{cr}}}{2} \hat{\mu} + \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \theta} \hat{\theta} \right]\end{aligned}$$
So we see that $K_1 < 0$ when $$\Delta_1 < - \left( \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \lambda \omega_{\text{cr}}^2} \hat{\lambda} - \frac{1 + \mu \Delta_{\text{cr}}}{2 \omega_{\text{cr}}^2} \hat{\mu} + \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \theta \omega_{\text{cr}}^2} \hat{\theta} \right)$$ and $K_1 > 0$ when $$\Delta_1 > - \left( \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \lambda \omega_{\text{cr}}^2} \hat{\lambda} - \frac{1 + \mu \Delta_{\text{cr}}}{2 \omega_{\text{cr}}^2} \hat{\mu} + \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \theta \omega_{\text{cr}}^2} \hat{\theta} \right)$$ and since $\Delta = \Delta_{\text{cr}} + \epsilon \Delta_1 + O(\epsilon^2)$, we see that the critical value of $\Delta$ for which the stability of our perturbed DDE system given in Equations \[perturbed equation 1\]-\[perturbed equation 2\] changes is $$\Delta_{\text{mod}} = \Delta_{\text{cr}} - \epsilon \left( \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \lambda \omega_{\text{cr}}^2} \hat{\lambda} - \frac{1 + \mu \Delta_{\text{cr}}}{2 \omega_{\text{cr}}^2} \hat{\mu} + \frac{\mu + \Delta_{\text{cr}}(\mu^2 + \omega_{\text{cr}}^2)}{2 \theta \omega_{\text{cr}}^2} \hat{\theta} \right) + O(\epsilon^2).$$
An observation we can make that more clearly relates this expression of $\Delta_{\text{mod}}$ to the form of $\Delta_{\text{cr}}$ given in @novitzky2019nonlinear is that $$\Delta_{\text{mod}} = \frac{\arccos \left( \frac{ -2 \left( \mu + \frac{\hat{\mu} \epsilon}{2} \right) }{ \left( \lambda + \frac{\hat{\lambda} \epsilon}{2} \right) \left( \theta + \frac{\hat{\theta} \epsilon}{2} \right) } \right)}{\omega_{\text{mod}}} + O(\epsilon^2) \hspace{5mm} \text{ where }$$ $$\omega_{\text{mod}} = \frac{1}{2} \sqrt{ \left( \lambda + \frac{\hat{\lambda} \epsilon}{2} \right)^2 \left( \theta + \frac{\hat{\theta} \epsilon}{2} \right)^2 - 4 \left( \mu + \frac{\hat{\mu} \epsilon}{2} \right)^2 }$$ which can be seen by Taylor expanding about $\epsilon = 0$. To give some intuition regarding how this observation was made, consider the case where we only perturb the arrival rate $\lambda$. Linearizing the system, neglecting $O(\epsilon^2)$ terms, we’ll obtain equations \[u1 equation 45\] and \[u2 equation 46\], but with $\hat{\theta} = \hat{\mu} = 0$.
$$\begin{aligned}
\overset{\bullet}{u}_1(t) &= \frac{(\lambda + \hat{\lambda} \epsilon) \theta}{4}\left[ u_2(t - \Delta) - u_1(t - \Delta) \right] - \mu u_1(t) \\
\overset{\bullet}{u}_2(t) &= \frac{\lambda \theta}{4} [u_1(t- \Delta) - u_2(t - \Delta)] - \mu u_2(t) \end{aligned}$$
Applying the transformation $v_1(t) = u_1(t) + u_2(t)$ and $v_2(t) = u_1(t) - u_2(t)$ gives us equations \[v1 equation 1\] and \[v2 equation 1\] except with $\hat{\theta} = \hat{\mu} = 0$.
$$\begin{aligned}
\overset{\bullet}{v}_1(t) + \mu v_1(t) &= - \frac{\theta \hat{\lambda}}{4}\epsilon v_2(t - \Delta) \\
\overset{\bullet}{v}_2(t) + \left( \frac{\lambda \theta}{2} + \frac{\theta \hat{\lambda}}{4} \epsilon \right) v_2(t - \Delta) + \mu v_2(t) &= 0\end{aligned}$$
We see that the coupling that was present in equations \[v1 equation 1\] and \[v2 equation 1\] has been simplified. We see that the homogeneous solution to the equation for $v_1(t)$ decays with time. Thus, if $v_2(t)$ is stable, then the particular solution for the $v_1(t)$ equation will be stable and if $v_2(t)$ is unstable then the system is unstable. Because of this, we can restrict our attention to the equation for $v_2(t).$ Letting $v_2(t) = e^{rt}$ gives us the characteristic equation $$r + C e^{- r \Delta} + \mu = 0$$ where $C = \frac{\lambda \theta}{2} + \frac{\theta \hat{\lambda}}{4} \epsilon $. The system is stable when $r$ has negative real part and it is unstable when $r$ has positive real part, so the change in stability occurs when $r$ crosses the imaginary axis, so we let $r = i \omega$ for some real $\omega$. Collecting real and imaginary parts gives us the equations
$$\begin{aligned}
\sin(\omega \Delta) &= \frac{\omega}{C}\\
\cos(\omega \Delta) &= - \frac{\mu}{C}\end{aligned}$$
and since $\cos^2(\omega \Delta) + \sin^2(\omega \Delta) = 1$, we are able to get $$\Delta_{\text{mod}} = \frac{\arccos \left( - \frac{\mu}{C} \right) }{\omega_{\text{mod}}}, \hspace{5mm} \omega_{\text{mod}} = \sqrt{C^2 - \mu^2}.$$ We see that $$C = \frac{\lambda \theta}{2} + \frac{\theta \hat{\lambda}}{4} \epsilon = \frac{\theta \left( \lambda + \frac{\hat{\lambda}}{2} \epsilon \right)}{2}$$ so we get $$\Delta_{\text{mod}} = \frac{2 \arccos \left( - \frac{2 \mu}{ \left( \lambda + \frac{\hat{\lambda} \epsilon }{2} \right) \theta } \right)}{\sqrt{ \left( \lambda + \frac{\hat{\lambda}}{2} \epsilon \right)^2 \theta^2 - 4 \mu^2 }}.$$
While this isn’t a rigorous approach to obtaining the expression we got for $\Delta_{\text{mod}}$ with all of the parameters perturbed as the coupling in the general case causes complications, it should at least give some intuition for why we considered the expression above. The perturbations to the parameters end up being divided by $2$ in each case due to the transformation from $u_1$ and $u_2$ to $v_1$ and $v_2$. This also has to do with the fact that the perturbation terms for the $\lambda$ and $\mu$ cases only appear in a single equation in equations \[v1 equation 1\] and \[v2 equation 1\], so when forming the equation for $v_2(t)$, these terms do not get the factor of $2$ that terms that were in both equations (but differed by a factor of -1) got. Also, even though the $\theta$ perturbation is in both equations, it is multiplying a $u_1$ term in both cases which transforms to $$u_1 = \frac{v_1 + v_2}{2}$$ which introduces a factor of $\frac{1}{2}.$
Another important observation to make is that our expression for $\Delta_{\text{mod}}$ appears to not depend on $\hat{\alpha}$ up to first order. However, if we collect $O(\epsilon^2)$ terms when linearizing the system, we can see the contributions from $\hat{\alpha}$. We illustrate this in Theorem \[delta\_mod\_alpha\_theorem\] by considering the special case where $\hat{\lambda} = \hat{\mu} = \hat{\theta} = 0$ for ease of calculation.
\[delta\_mod\_alpha\_theorem\] When $\hat{\lambda} = \hat{\mu} = \hat{\theta} = 0$, we have that $$\Delta_{\text{mod}} = \Delta_{\text{cr}} + \frac{4 \mu^3 + \mu^2 \lambda^2 \theta^2 \Delta_{\text{cr}}}{4 \omega_{\text{cr}}^2 (\lambda \theta + 2 \mu)^2} \epsilon^2 \hat{\alpha}^2 + O(\epsilon^3).$$
It can be shown by a calculation similar to the one in Section \[sec\_equilibrium\] that the equilibrium point in this special case is $$(q_1^*, q_2^*) = \left( \frac{\lambda}{2 \mu} + a_1 \epsilon + a_2 \epsilon^2 + O(\epsilon^3), \frac{\lambda}{2 \mu} + b_1 \epsilon + b_2 \epsilon^2 + O(\epsilon^3) \right)$$ where $$a_1 = \frac{\lambda}{2 (\lambda \theta + 2 \mu)}, \hspace{5mm} b_1 = \frac{- \lambda}{2 (\lambda \theta + 2 \mu)}$$ and $a_2 = b_2 = 0$. Linearizing about this equlibrium point and neglecting $O(\epsilon^3)$ terms gives us the following linear system.
$$\begin{aligned}
\overset{\bullet}{u}_1(t) &= \left( \frac{\lambda \theta}{4} - \frac{1}{16} \left[ (a_1 - b_1)^2 \lambda \theta^3 + 2 (b_1 - a_1) \lambda \theta^2 \hat{\alpha} + \lambda \theta \hat{\alpha}^2 \right] \epsilon^2 \right) [u_2(t - \Delta) - u_1(t - \Delta)] - \mu u_1(t)\\
\overset{\bullet}{u}_2(t) &= \left( \frac{\lambda \theta}{4} - \frac{1}{16} \left[ (a_1 - b_1)^2 \lambda \theta^3 + 2 (b_1 - a_1) \lambda \theta^2 \hat{\alpha} + \lambda \theta \hat{\alpha}^2 \right] \epsilon^2 \right) [u_1(t - \Delta) - u_2(t - \Delta)] - \mu u_2(t)\\\end{aligned}$$
Using the transformation $$v_1(t) = u_1(t) + u_2(t), \hspace{5mm} v_2(t) = u_1(t) - u_2(t)$$ gives us the system
$$\begin{aligned}
\overset{\bullet}{v}_1(t) &+ \mu v_1(t) = 0\\
\overset{\bullet}{v}_2(t) &+ \left( \frac{\lambda \theta}{2} - \frac{1}{8} \left[ (a_1 - b_1)^2 \lambda \theta^3 + 2 (b_1 - a_1) \lambda \theta^2 \hat{\alpha} + \lambda \theta \hat{\alpha}^2 \right] \epsilon^2 \right) v_2(t - \Delta) + \mu v_2(t) = 0\end{aligned}$$
We see that $v_1(t)$ decays with time and thus we restrict our analysis to (3.20). Letting $v_2(t) = e^{rt}$, we get the characteristic equation $$r + D e^{- r \Delta} + \mu = 0$$ where we let $D = \left( \frac{\lambda \theta}{2} - \frac{1}{8} \left[ (a_1 - b_1)^2 \lambda \theta^3 + 2 (b_1 - a_1) \lambda \theta^2 \hat{\alpha} + \lambda \theta \hat{\alpha}^2 \right] \epsilon^2 \right)$ for ease of notation. If $r$ has negative real part, then we have stability and we have instability when $r$ has positive real part. Thus, the change in stability occurs when $r$ crosses the imaginary axis, that is when $r = i \omega$ for some real $\omega$. Letting $r = i \omega$ and collecting real and imaginary parts gives us the following two equations
$$\begin{aligned}
\sin(\omega \Delta) &= \frac{\omega}{D}\\
\cos(\omega \Delta) &= - \frac{\mu}{D}\end{aligned}$$
so that, using the fact that $\cos^2(\omega \Delta) + \sin^2(\omega \Delta) = 1$, we get $$\Delta_{\text{mod}} = \frac{\arccos \left( - \frac{\mu}{D} \right)}{\omega_{\text{mod}}}, \hspace{5mm} \omega_{\text{mod}} = \sqrt{D^2 - \mu^2}.$$ Taylor expanding $\Delta_{\text{mod}}$ about $\epsilon = 0$ gives us the result $$\Delta_{\text{mod}} = \Delta_{\text{cr}} + \frac{4 \mu^3 + \mu^2 \lambda^2 \theta^2 \Delta_{\text{cr}}}{4 \omega_{\text{cr}}^2 (\lambda \theta + 2 \mu)^2} \epsilon^2 \hat{\alpha}^2 + O(\epsilon^3)$$ when $\hat{\lambda} = \hat{\mu} = \hat{\theta} = 0.$
Numerical Verification of Hopf Bifurcation
------------------------------------------
Below we show plots of queue length versus time for various cases to demonstrate the bifurcation in $\Delta$. In each figure below, we consider our system with various parameters either being perturbed or not perturbed from symmetry. In each case, we consider having a delay $.05$ below and $.05$ above the corresponding critical delay value. In each case, we see that the queue length amplitudes decay to equilibrium values when the delay $\Delta$ is below the critical value. When we increase the delay to be above the critical delay, we see oscillations increase and approach a fixed amplitude forming a limit cycle. This suggests that we have a Hopf bifurcation at the critical delay. This observation prompts us to consider the amplitudes of limit cycles in Section \[sec\_amplitude\].
In Figure \[Fig7\], we consider the symmetric case and we see that the amplitudes of the queues oscillate and decay when the delay is below the critical delta and approach some limiting amplitude when the delay is above the critical delta. In Figure \[Fig8\], we consider the case where only the arrival rate $\lambda$ is perturbed positively. In this case, we see that increasing the arrival rate in one of the queues causes the critical delay to be less than the critical delay in the symmetric case. In Figure \[Fig9\], the service rate $\mu$ is the only perturbed parameter and we are able to see that increasing the service rate in one of the queues leads to an increase in the critical delay. These observations tell us that increasing the inflow of customers in one queue makes the system more susceptible to oscillations caused by delayed information whereas increasing the service rate in one of the queues helps to mitigate this issue. In Figure \[Fig10\], the only perturbed parameter is $\theta$ and we see that increasing the value of $\theta$ corresponding to one of the queues causes a decrease in the critical delay. We note that increasing the value of the $\theta$ corresponding to one of the queues increases the number of arrivals into that queue and thus it makes sense that it impacts the critical delay in the same direction that perturbing the arrival rate does. In Figure \[Fig11\], the only perturbed parameter is $\alpha$. We see that the critical delay is roughly the same as the critical delay in the symmetric case which isn’t surprising based on the result of Theorem \[delta\_mod\_alpha\_theorem\] which says that perturbing $\alpha$ only affects the value of the critical delay if we include $O(\epsilon^2)$ terms. Figure \[Fig12\] is an example where all four of the parameters we have discussed were perturbed from symmetry. The impact that perturbing all four of these parameters has on the critical delay will depend on how much each parameter is perturbed by.
![$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 0$, $\lambda = 10,\mu=1, \theta=1, \alpha=0$, $\Delta_{\text{mod}} = \Delta_{\text{cr}} \approx .3617$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig7"}](./Code/Paper_Figures/Fig_sym2.pdf "fig:") ![$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 0$, $\lambda = 10,\mu=1, \theta=1, \alpha=0$, $\Delta_{\text{mod}} = \Delta_{\text{cr}} \approx .3617$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig7"}](./Code/Paper_Figures/Fig_sym1.pdf "fig:")
![$\hat{\lambda} = 1, \hat{\mu} = \hat{\theta} = \hat{\alpha} = 0$, $\Delta_{\text{mod}} \approx .3596$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig8"}](./Code/Paper_Figures/Fig_1.pdf "fig:") ![$\hat{\lambda} = 1, \hat{\mu} = \hat{\theta} = \hat{\alpha} = 0$, $\Delta_{\text{mod}} \approx .3596$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig8"}](./Code/Paper_Figures/Fig_5.pdf "fig:")
![$\hat{\mu} = 1, \hat{\lambda} = \hat{\theta} = \hat{\alpha} = 0$, $\Delta_{\text{mod}} \approx .3646$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig9"}](./Code/Paper_Figures/Fig_2.pdf "fig:") ![$\hat{\mu} = 1, \hat{\lambda} = \hat{\theta} = \hat{\alpha} = 0$, $\Delta_{\text{mod}} \approx .3646$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig9"}](./Code/Paper_Figures/Fig_6.pdf "fig:")
![$\hat{\theta} = 1, \hat{\lambda} = \hat{\mu} = \hat{\alpha} = 0$, $\Delta_{\text{mod}} \approx .3408$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig10"}](./Code/Paper_Figures/Fig_3.pdf "fig:") ![$\hat{\theta} = 1, \hat{\lambda} = \hat{\mu} = \hat{\alpha} = 0$, $\Delta_{\text{mod}} \approx .3408$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig10"}](./Code/Paper_Figures/Fig_7.pdf "fig:")
![$\hat{\alpha} = 1, \hat{\lambda} = \hat{\mu} = \hat{\theta} = 0$, $\Delta_{\text{mod}} \approx .3617$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig11"}](./Code/Paper_Figures/Fig_4.pdf "fig:") ![$\hat{\alpha} = 1, \hat{\lambda} = \hat{\mu} = \hat{\theta} = 0$, $\Delta_{\text{mod}} \approx .3617$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig11"}](./Code/Paper_Figures/Fig_8.pdf "fig:")
![$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 1$, $\Delta_{\text{mod}} \approx .3416$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig12"}](./Code/Paper_Figures/Fig_general2.pdf "fig:") ![$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 1$, $\Delta_{\text{mod}} \approx .3416$ $\epsilon = .1, \lambda = 10, \mu=1, \theta=1, \alpha=0$\
On $[-\Delta, 0]$, $q_1 = 4.99$, $q_2 = 5.01$, Left: $\Delta = \Delta_{\text{mod}} - .05$, Right: $\Delta = \Delta_{\text{mod}} + .05$[]{data-label="Fig12"}](./Code/Paper_Figures/Fig_general1.pdf "fig:")
Amplitude of Limit Cycle {#sec_amplitude}
========================
In the previous section, we observed that increasing the delay past the critical value $\Delta_{\text{mod}}$ causes oscillations in the queue lengths and ultimately gives rise to a limit cycle. In this section, we aim to approximate the amplitude of the limit cycle when the delay is close to $\Delta_{\text{mod}}$. To do this we will resort to using Lindstedt’s method.
Using Lindstedt’s method, we obtain the following approximation, $\tilde{A}$, of the amplitude of limit cycles near the critical delay $$\tilde{A} = \sqrt{\Delta - \Delta_{\text{mod}}} \ \sqrt{ \frac{8 (\bar{\lambda}^2 \bar{\theta}^2 - 4 \bar{\mu}^2)^2}{2 \bar{\lambda}^2 \bar{\mu} \theta^2 \bar{\theta}^2 - 8 \bar{\mu}^3 \theta^2 + \left[ \lambda \theta^2 \bar{\lambda} \bar{\theta} + \frac{4 \lambda \theta^2}{\bar{\lambda} \bar{\theta}} \bar{\mu}^2 (\theta - 1) \right] \sqrt{\bar{\lambda}^2 \bar{\theta}^2 - 4 \bar{\mu}^2} \arccos \left(- \frac{2 \bar{\mu}}{\bar{\lambda} \bar{\theta}} \right) } }.$$
We will expand the system of Equations \[perturbed equation 1\]-\[perturbed equation 2\] about the approximate equilibrium point $$(q_1^*, q_2^*) = \left(\frac{\lambda}{2 \mu} + a\epsilon + O(\epsilon^2), \frac{\lambda}{2 \mu} + b \epsilon + O(\epsilon^2)\right)$$ where $a$ and $b$ are as defined in Theorem \[equilibrium theorem\]. However, unlike the equilibrium and stability calculations, we need to Taylor expand to third order in order to find the amplitude. By Taylor expanding to third order (cubic) and dropping $O(\epsilon^2)$ terms leaves us with the following cubic system of DDEs
$$\begin{aligned}
\overset{\bullet}{u}_1(t) &= -(\mu + \hat{\mu} \epsilon) u_1(t) - \frac{1}{4} \left( \lambda \theta + (\lambda \hat{\theta} + \hat{\lambda} \theta) \epsilon \right) u_1(t - \Delta) + \frac{1}{4} \left( \lambda \theta + \theta \hat{\lambda} \epsilon \right)u_2(t - \Delta)\\ &+ \frac{1}{32 \mu} (\lambda^2 \theta^2 \hat{\theta} + 2 \mu (a-b) \lambda \theta^3 - 2 \mu \hat{\alpha} \lambda \theta^2) \epsilon (u_1(t- \Delta) - u_2(t - \Delta))^2\\ &+ \frac{1}{48} \left( \lambda \theta^3 + (3 \lambda \theta^2 \hat{\theta} + \theta^3 \hat{\lambda}) \epsilon \right) u_1^3(t-\Delta) - \frac{1}{48} \left( \lambda \theta^3 + \theta^3 \hat{\lambda} \epsilon \right) u_2^3(t - \Delta)\\ &- \frac{1}{16} (\lambda \theta^3 + (2 \lambda \theta^2 \hat{\theta} + \theta^3 \hat{\lambda})\epsilon) u_1^2(t - \Delta) u_2(t-\Delta) + \frac{1}{16} (\lambda \theta^3 + (\lambda \theta^2 \hat{\theta} + \theta^3 \hat{\lambda}) \epsilon) u_1(t - \Delta) u_2^2(t-\Delta)\\
\overset{\bullet}{u}_2(t) & = - \mu u_2(t) + \frac{1}{4} (\lambda \theta + \lambda \hat{\theta} \epsilon) u_1(t-\Delta) - \frac{1}{4} \lambda \theta u_2(t-\Delta)\\ &+ \frac{1}{32 \mu} (\lambda^2 \theta^2 \hat{\theta} + 2 \mu (a-b) \lambda \theta^3 - 2 \mu \hat{\alpha} \lambda \theta^2) \epsilon (u_1^2(t-\Delta) + 2 u_1(t-\Delta) u_2(t-\Delta) - u_2^2(t-\Delta)) \\ &- \frac{1}{48} (\lambda \theta^2 + 3 \lambda \theta^2 \hat{\theta} \epsilon) u_1^3(t - \Delta) + \frac{1}{48} \lambda \theta^3 u_2^3(t - \Delta) \\ &+ \frac{1}{16} (\lambda \theta^3 + 2 \lambda \theta^2 \hat{\theta} \epsilon) u_1^2(t - \Delta) u_2(t - \Delta) - \frac{1}{16} (\lambda \theta^3 + \lambda \theta^2 \hat{\theta} \epsilon) u_1(t - \Delta) u_2^2(t - \Delta).\end{aligned}$$
As we did in the linear case, we make the change of variables $$v_1(t) = u_1(t) + u_2(t), \hspace{5mm} v_2(t) = u_1(t) - u_2(t)$$ and we let $$v_1(t) = v_{1,0}(t) + \epsilon v_{1,1}(t) + O(\epsilon^2)$$ $$v_2(t) = v_{2,0}(t) + \epsilon v_{2,1}(t) + O(\epsilon^2).$$ Collecting $O(1)$ terms gives us the following two equations
$$\begin{aligned}
\overset{\bullet}{v}_{1,0}(t) + \mu v_{1,0}(t) &= 0 \label{v10 equation 3}\\
\overset{\bullet}{v}_{2,0}(t) + \frac{\lambda \theta}{2} v_{2,0}(t-\Delta) - \frac{\lambda \theta^3}{24} v_{2,0}^3(t-\Delta)+ \mu v_{2,0}(t) &= 0 \label{v20 equation 3}\end{aligned}$$
and collecting $O(\epsilon)$ terms gives us
$$\begin{aligned}
\overset{\bullet}{v}_{1,1}(t) &+ \mu v_{1,1}(t) = - \frac{\theta \hat{\lambda}}{4} v_{2,0}(t - \Delta) + \frac{\theta^3 \hat{\lambda}}{48} v_{2,0}^3(t - \Delta) - \frac{\hat{\mu}}{2} (v_{1,0}(t) + v_{2,0}(t) ) \label{v11 equation} \\
\overset{\bullet}{v}_{2,1}(t) &+ \frac{\lambda \theta}{2} v_{2,1}(t - \Delta) - \frac{\lambda \theta^3}{8} v_{2,0}^2(t - \Delta) v_{2,1}(t - \Delta) + \mu v_{2,1} = - \left( \frac{\theta \hat{\lambda}}{4} + \frac{\lambda \hat{\theta}}{4} \right) v_{2,0}(t - \Delta) \nonumber \\ &- \frac{\lambda \hat{\theta}}{4} v_{1,0}(t - \Delta) + \left( \frac{2 \lambda \theta^3 \mu (a - b) + \lambda^2 \theta^2 \hat{\theta} - 2 \lambda \theta^2 \mu \hat{\alpha}}{16 \mu} \right) v_{2,0}^2(t - \Delta) \nonumber \\ &+ \frac{\theta^3 \hat{\lambda}}{48} v_{2,0}^3(t -\Delta) + \frac{\lambda \theta^2 \hat{\theta}}{16} [ v_{1,0}(t - \Delta) v_{2,0}^2(t - \Delta) + v_{2,0}^3(t - \Delta) ] - \frac{\hat{\mu}}{2}(v_{1,0}(t) + v_{2,0}(t)).\end{aligned}$$
We observe that we can directly solve Equation \[v10 equation 3\] and its solution is given by $v_{1,0}(t) = \hat{c} \exp(- \mu t)$, for some constant $\hat{c}$, meaning $$v_{1}(t) = u_1(t) + u_2(t) = \hat{c} \exp(- \mu t) + O(\epsilon)$$ so that for small $\epsilon$ and large time, we have that $$u_1(t) \approx - u_2(t)$$ which tells us that the amplitudes of the queue lengths are approximately symmetric (which is expected given that we’re perturbing a symmetric model) and thus $$v_{2,0}(t) = u_1(t) - u_2(t) + O(\epsilon) \approx 2 u_1(t)$$ gives us approximately twice the amplitude of the limit cycle. Because of this, we narrow our interest to Equation \[v20 equation 3\]. Since we are interested in the amplitudes of limit cycles near the bifurcation point, we are working under the assumption that $\Delta - \Delta_{\text{mod}}$ is small. Letting $\Delta_{0} = \Delta_{\text{mod}}$ and $\omega_0 = \omega_{\text{mod}}$, we use the following transformations.
$$\tau = \omega t, \hspace{5mm} v_{2,0}(t) = \sqrt{\epsilon} v(t)$$ $$v(t) = v_0(t) + \epsilon v_1(t) + \cdots, \hspace{5mm} \Delta = \Delta_0 + \epsilon \Delta_1 + \cdots, \hspace{5mm} \omega = \omega_0 + \epsilon \omega_1 + \cdots$$ Matching powers of $\epsilon$ and dropping higher ordered terms, we get two equations.
$$\begin{aligned}
\omega_0 v_0'(\tau) + \frac{\lambda \theta}{2} v_0(\tau - \omega_0 \Delta_0) + \mu v_0(\tau) &= 0 \label{eqn_2} \\
\omega_0 v_1'(\tau) + \frac{\lambda \theta}{2} v_1(\tau - \omega_0 \Delta_0) + \mu v_1(\tau) &= - \omega_1 v_0'(\tau) \nonumber \\ &+ \frac{\lambda \theta}{2}(\omega_0 \Delta_1 + \omega_1 \Delta_0) v_0'(\tau - \omega_0 \Delta_0) \nonumber \\ &+ \frac{\lambda \theta^3}{24} v_0^3(\tau - \omega_0 \Delta_0) \label{lindstedt inhomogeneous equation 1}\end{aligned}$$
Noting that $v_0(\tau) = A \sin(\tau)$ satisfies Equation \[eqn\_2\] and that the homogeneous form of Equation \[lindstedt inhomogeneous equation 1\] is the same as that of Equation \[eqn\_2\], we substitute in $v_0(\tau) = A \sin(\tau)$ into the inhomogeneity of Equation \[lindstedt inhomogeneous equation 1\] and set the terms that would introduce secular terms in the particular solution equal to $0$. Doing this yields a system of two equations and two unknowns: $A$ and $\omega_1$. Solving for these unknowns, we obtain $$A = \sqrt{ \frac{8 \Delta_1 (\bar{\lambda}^2 \bar{\theta}^2 - 4 \bar{\mu}^2)^2}{2 \bar{\lambda}^2 \bar{\mu} \theta^2 \bar{\theta}^2 - 8 \bar{\mu}^3 \theta^2 + \left[ \lambda \theta^2 \bar{\lambda} \bar{\theta} + \frac{4 \lambda \theta^2}{\bar{\lambda} \bar{\theta}} \bar{\mu}^2 (\theta - 1) \right] \sqrt{\bar{\lambda}^2 \bar{\theta}^2 - 4 \bar{\mu}^2} \arccos \left(- \frac{2 \bar{\mu}}{\bar{\lambda} \bar{\theta}} \right) } }$$ and $$\omega_1 = - \frac{1}{\Delta_{0}} \left( \omega_0 \Delta_1 + \frac{A^2 \theta^2 \cos(\omega_0 \Delta_0)}{16 \sin(\omega_0 \Delta_0)} \right)$$ where
$$\bar{\lambda} := \lambda + \frac{\hat{\lambda} \epsilon}{2}, \hspace{5mm} \bar{\mu} := \mu + \frac{\hat{\mu} \epsilon}{2}, \hspace{5mm} \bar{\theta} := \theta + \frac{\hat{\theta} \epsilon}{2}.$$ We assume that $\Delta - \Delta_0 \approx \epsilon$ which implies that $\Delta_1 \approx 1$ and since $v_{2,0}(t) = \sqrt{\epsilon} v(t)$, our approximation of the amplitude, $\tilde{A}$, is $$\tilde{A} = \sqrt{\Delta - \Delta_{\text{mod}}} \sqrt{ \frac{8 (\bar{\lambda}^2 \bar{\theta}^2 - 4 \bar{\mu}^2)^2}{2 \bar{\lambda}^2 \bar{\mu} \theta^2 \bar{\theta}^2 - 8 \bar{\mu}^3 \theta^2 + \left[ \lambda \theta^2 \bar{\lambda} \bar{\theta} + \frac{4 \lambda \theta^2}{\bar{\lambda} \bar{\theta}} \bar{\mu}^2 (\theta - 1) \right] \sqrt{\bar{\lambda}^2 \bar{\theta}^2 - 4 \bar{\mu}^2} \arccos \left(- \frac{2 \bar{\mu}}{\bar{\lambda} \bar{\theta}} \right) } }.$$
We now demonstrate numerically how well this approximation matches the actual amplitude of the limit cycle. In each figure below, we will plot the queue lengths against time and approximate the maximum and minimum values of each queue length by adding or subtracting $\frac{1}{2} \tilde{A}$ from the equilibrium for each queue. The cycle lines are the approximations corresponding to $q_1$ and the dashed green lines are the approximations corresponding to $q_2$.
![\
$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 1, \epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=1$, on $[-\Delta, 0]$ $q_1 = 4.99$ and $q_2 = 5.01$\
Left: $\Delta - \Delta_{\text{mod}} = .05$, $ \text{Amplitudes: } q_1 \approx 1.3593, q_2 \approx 1.3524$, $\text{Approximation } \approx 1.4562$\
Right: $\Delta - \Delta_{\text{mod}} \approx .1$ $ \text{Amplitudes: } q_1 \approx 1.9576, q_2 \approx 1.9503$, $\text{Approximation } \approx 2.0594$[]{data-label="Fig13"}](./Code/Paper_Figures/amplitude_05_fig_2.pdf "fig:") ![\
$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 1, \epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=1$, on $[-\Delta, 0]$ $q_1 = 4.99$ and $q_2 = 5.01$\
Left: $\Delta - \Delta_{\text{mod}} = .05$, $ \text{Amplitudes: } q_1 \approx 1.3593, q_2 \approx 1.3524$, $\text{Approximation } \approx 1.4562$\
Right: $\Delta - \Delta_{\text{mod}} \approx .1$ $ \text{Amplitudes: } q_1 \approx 1.9576, q_2 \approx 1.9503$, $\text{Approximation } \approx 2.0594$[]{data-label="Fig13"}](./Code/Paper_Figures/amplitude_1_fig_2.pdf "fig:")
![\
$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 1, \epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=1$, on $[-\Delta, 0]$ $q_1 = 4.99$ and $q_2 = 5.01$\
Left: $\Delta - \Delta_{\text{mod}} = .15$, $ \text{Amplitudes: } q_1 \approx 2.4265, q_2 \approx 2.4208$, $\text{Approximation } \approx 2.5222$\
Right: $\Delta - \Delta_{\text{mod}} \approx .2$, $\text{Amplitudes: } q_1 \approx 2.8292, q_2 \approx 2.8268$, $\text{Approximation } \approx 2.9124$[]{data-label="Fig14"}](./Code/Paper_Figures/amplitude_15_fig_3.pdf "fig:") ![\
$\hat{\lambda} = \hat{\mu} = \hat{\theta} = \hat{\alpha} = 1, \epsilon = .1, \lambda=10, \mu=1, \theta=1, \alpha=1$, on $[-\Delta, 0]$ $q_1 = 4.99$ and $q_2 = 5.01$\
Left: $\Delta - \Delta_{\text{mod}} = .15$, $ \text{Amplitudes: } q_1 \approx 2.4265, q_2 \approx 2.4208$, $\text{Approximation } \approx 2.5222$\
Right: $\Delta - \Delta_{\text{mod}} \approx .2$, $\text{Amplitudes: } q_1 \approx 2.8292, q_2 \approx 2.8268$, $\text{Approximation } \approx 2.9124$[]{data-label="Fig14"}](./Code/Paper_Figures/amplitude_2_fig_2.pdf "fig:")
In Figure \[Fig13\] and Figure \[Fig14\], all four of the parameters are perturbed with $\epsilon = .1$ and we consider the cases when $\Delta - \Delta_{\text{mod}}$ is equal to .05, .1, .15, and .2. As we increase the delay past the critical delay, we see the amplitude increase as our amplitude approximation would expect. Indeed, our approximation of the amplitude is proportional to $\sqrt{\Delta - \Delta_{\text{mod}}}$, so increasing the difference between the delay and the critical delay will result in an increase in the approximation of the amplitude. Keep in mind that our approximation of the amplitude is really only an $O(1)$ approximation as our calculation was based off of Equation \[v20 equation 3\] and did not rely on the equations obtained by collecting $O(\epsilon)$ terms, which made the analysis more manageable. Consequently, it is not particularly surprising that we see a noticeable amount of error between the actual amplitude and our approximation of the amplitude. The error seems to be around .1 for all four cases.
Conclusion and Future Research {#sec_conclusion}
==============================
In this paper, we analyze a two-dimensional fluid model that incorporates customer choice which depends on delayed queue length information. This model is different from those considered in previous literature because of the asymmetry we introduced by perturbing four of the model parameters corresponding to one of the two queues. Analyzing this model allows us to explore the impact that breaking the symmetry has on the dynamics of the queueing system. We see how perturbing different model parameters can have different effects on the system’s equilibrium, which we find a first-order approximation for. We consider the stability of this equilibrium to derive a first-order approximation for the critical delay at which the system exhibits a change in stability. Numerical experiments suggest that a Hopf bifurcation occurs at this critical delay as a limit cycle appears to be born when the delay is increased past the critical delay value we derived. We employ Lindstedt’s method to get an $O(1)$ approximation for the amplitude of limit cycles near the bifurcation point.
There are several extensions that could be made to this work. One extension would be to consider a system generalized to have $N > 2$ queues where parameters corresponding to each queue have different perturbations so that no two queues in the system are symmetric to each other. Another potential extension would be to consider different choice functions or to simply change the information that the choice model depends on. Such analysis could provide a better understanding of how providing customers with different types of information affects the dynamics of the system. One could also consider a queueing system with time-varying arrival rates, as in @pender2018analysis, but with asymmetry introduced to the system. We plan on exploring some of these extensions in future work.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the Center for Applied Mathematics at Cornell University for sponsoring Philip Doldo’s research. Finally, we acknowledge the gracious support of the National Science Foundation (NSF) for Jamol Pender’s Career Award CMMI \# 1751975.
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---
author:
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Krzysztof Kutak\
Universiteit Antwerpen, Groenenborgerlaan 171, Antwerpen, Belgium\
E-mail:
title: 'Saturation effects in QCD from linear transport equation.'
---
Introduction
============
Perturbative Quantum Chromodynamics (pQCD) at high energies can be formulated in coordinate space in the dipole picture [@Mueller]. If we in particular focus on Deep Inelastic Scattering the scattering process can be described in this picture as interaction of virtual photon which has just enough energy to dissociate into a ’color dipole’ with the hadronic target carrying most of the total energy. The interaction process is described here by the dipole-nucleus scattering amplitude. This amplitude can be modeled and we will focus here on the Golec-Biernat Wüsthoff saturation model [@GBW] which includes saturation effects. It was motivated by requirements that at the high energy limit of QCD the total cross section for hadronic processes should obey unitarity requirements. At present there are much more sophisticated approaches to introduce these requirements in a description of scatterings at high energies [@JalilianMarian:1997jx; @JalilianMarian:1997gr; @JalilianMarian:1997dw; @Bal; @Kov; @Avsar:2006jy; @BartKut]. However, one can still ask the question if there is any dynamics behind the GBW model or to put it differently is there any equation to which formula proposed by Golec-Biernat and Wüsthoff is a solution? And what is the role of the initial conditions? In this article we report on answer to these questions provided in [@Kutak:2009zk].\
GBW model and a transport equation
==================================
The GBW amplitude following form GBW cross section and related to it by $\sigma(x,r)=2\int d^2b N(x,r,b)$ reads (here we are interested in the original formulation): N(x,r,b)=(b\_0-b)\[eq:GBW98\] where $b$ is the impact parameter of the collision defined as distance between center of the proton with radius $b_0$ and center of a dipole scattering on it, $r$ is a transversal size of the dipole, $x$ is the Bjorken variable, $R_0(x)\!=\!\frac{1}{Q_0}\left(\frac{x}{x_0}\right)^{\lambda/2}$ is the so called saturation radius and its inverse defines saturation scale, $Q_s(x)\!=\!1/R_0(x)$ and $x_0$, $\lambda$ are free parameters. This amplitude saturates for large dipoles $r\!\gg\!2R_0$ and exhibits geometrical scaling which has been confirmed by data [@Stasto:2000er].
Transport equation for unintegrated gluon density
-------------------------------------------------
The dipole amplitude (\[eq:GBW98\]) can be related to the unintegrated gluon density which convoluted with the $k_T$ dependent off-shell matrix elements allows to calculate observables in the high energy limit of QCD. This relation is the following (after assumption that the dipole is much smaller than the target): f(x,k\^2,b)=k\^4\^2\_k(-i[k]{}) \[eq:ugd\] where $r$ and $k$ are two-dimensional vectors in transversal plane of the collision and $r\equiv|\bold r|$, $k\equiv|\bold k|$\
Performing this transformation we obtain the known result [@GBF]: f(x,k\^2,b)=(b\_0-b)R\_0\^2(x)k\^4\[eq:ugd2\] Now motivated by the fact that function $f(x,k^2,b)$ exhibits a maximum both as a function of $x$ for fixed $k^2$ and as a function of $k^2$ for fixed $x$, we differentiate $f(x,k^2,b)$ with respect to $x$ and $f(x,k^2,b)/k^2$ with respect to $k^2$. We obtain: \_x f(x,k\^2,b)= \[eq:poch1\] \_[k\^2]{} = \[eq:poch2\] Dividing eqn. (\[eq:poch1\]) by (\[eq:poch2\]) and rearranging the terms and defining ${\cal F}(x,k^2,b)=f(x,k^2,b)/k^2$, $Y=\ln x_0/x$, $L=\ln k^2/Q_0^2$ we obtain: \_Y [F]{}(Y,L,b)+\_L [F]{}(Y,L,b)=0 \[eq:trans\] which is the first order linear wave equation also known as the transport equation. As it is linear it cannot generate saturation dynamically but it can propagate well the initial condition leading to a successful phenomenology [@GBW]. It describes the change (wave) in the particle distribution flowing into and out of the phase space volume with velocity $\lambda$. This wave propagates in one direction. The quantity ${\cal F}(x,k^2,b)$ gains here the interpretation of a number density of gluons with momentum fraction $x$ with the transversal momentum $k^2$ at distance $b$ from the center of the proton. The general solution of (\[eq:trans\]) can be found by the method of characteristics and is given by: (Y,L,b)=[F]{}\_0(L-Y,b) \[eq:sol\] One can go back from (\[eq:trans\]) to (\[eq:ugd2\]) using following initial condition at $x=x_0$: (x=x\_0,k\^2,b)= (b\_0-b)k\^2(-k\^2)\
\[eq:gbwini\] This initial condition has saturation built in, since the gluon density vanishes for small $k^2$. Knowing the properties of the linear first order partial differential equation we see that the property of saturation of GBW was a consequence of the wave solution which relates $x$ and $k^2$ supplemented by initial conditions with saturation built in. We also see that the [*critical line*]{} of the GBW saturation model visualizing, the dependence of the saturation scale on $x$, $Q_s(x)=Q_0\left(\frac{x_0}{x}\right)^{\lambda/2}$ is in fact from the mathematical point of view the characteristics of the transport equation.
Transport equation for the dipole amplitude in momentum space
-------------------------------------------------------------
Similar investigations can be repeated for the momentum space representation of the dipole amplitude $N(x,r,b)$ which we denote by $\phi(x,k^2,b)$. (x,k\^2,b)=(-i[k]{}) \[eq:ugd\] A nonlinear pQCD evolution equation like the Balitksy-Kovchegov (BK) equation written for $\phi$ (in large target approximation) takes quite simple form and can be related directly to the statistical formulation of the high energy limit of QCD (see [@SM2] and references therein). Applying this transformation to (\[eq:GBW98\]) we proceed with differentiation similarly as before and we obtain: \_Y(Y,L,b)+\_L (Y,L,b)=0 which is, as before, the transport equation.
Relation to pQCD
----------------
It is tempting to investigate the relation between found transport equation and the high energy pQCD evolution equations like [@Bal; @Kov; @BartKut]. Let us focus here in particular on the form of the BK equation in large cylindrical target approximation for the dipole amplitude in momentum space for which the nonlinear term is just a simple local quadratic expression. The BK equation for the dipole amplitude in the momentum space reads: \_Y(Y,k\^2,b)=(-)(Y,k\^2,b)-\^2(Y,k\^2,b) \[eq:BKdip\] where $\overline\alpha=\frac{N_c\alpha_s}{2\pi}$ and $\chi(\gamma)=2\psi(1)-\psi(\gamma)-\psi(1-\gamma)$ is the characteristic function of the BFKL kernel which allows for emission of dipoles and therefore drives the rise of the amplitude. The role of the nonlinear term is roughly to allow for multiple scatterings of dipoles which contributes with negative sign and slows down the rise of the amplitude. This equation provides unitarization of the dipole amplitude [@BKphen] for fixed impact parameter and admits traveling wave solution in the diffusion approximation[@SM1]. The analytic solution of (\[eq:BKdip\]) within the diffusion approximation relying on expanding the kernel of (\[eq:BKdip\]) up to second order and mapping it to the Fisher-Kolmogorov equation has been obtained by Munier and Peschanski [@MP3]. It reads: (Y,k\^2,b)=(b\_0-b) () ()\^[\_c-1]{} \[eq:solBK1\] where $\gamma_c=0.373$ and $Q_s^2(Y)$ is emergent saturation scale given by: Q\_s\^2(Y)=Q\_0\^2e\^[-|’(\_c) Y-Y - +[O]{}(1/Y)]{} \[eq:satscal\] By inspection we see that (\[eq:solBK1\]) does not obey the transport equation. The problem is caused by the diffusion term. However, we can consider the asymptotic regime called “front interior” [@MP3], region where transverse momenta $k$ is close to the saturation scale $Q_s(Y)$ and rapidity $Y$ is large and where the condition $\ln^2\left(\frac{k^2}{Q_s^2(Y)}\right)/2\overline\alpha\chi''(\gamma_c)Y\!<\!\!\!<\!1$ is satisfied. In this regime (\[eq:solBK1\]) simplifies and after taking derivatives as in the previous sections we obtain the following wave equation: \_Y (Y,L,b)+\_[BK]{}\_L (Y,L,b)=0 \[eq:transBK\] where $\lambda_{BK}=\partial \log Q_s^2(Y)/\partial Y$. In the limit where $\lambda_{BK}$ does not depends on energy [@IJM] we obtain: \_[BK]{}=-’(\_c) \[eq:asymlimit\]
Conclusions
===========
In this note we have shown that the GBW saturation model is the exact solution of a one-dimensional linear transport equation of the form (\[eq:trans\]). We conclude that since (\[eq:trans\]) is a linear equation the saturation property has to be provided in the initial condition. We found that for the GBW model this equation is universal for the unintegrated gluon density $f(x,k^2,b)$ and the dipole amplitude in momentum space $\phi(x,k^2,b)$ but the details of the shape of the wave depends on the initial condition which is different for each of them. We also studied the relation of the transport equation to the BK equation in the diffusion approximation. We have shown that in the region of phase space where diffusion and splitting processes are of the same order as the nonlinear term, the GBW model is consistent with the BK equation.
[99]{} A.H. Mueller, B. Patel Nucl.Phys. B425 (1994) 471-488 K. Golec-Biernat and M. Wüsthoff, [*Phys. Rev.*]{} [**D59**]{} (1999) 014017; J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, Nucl. Phys. B [**504**]{}, 415 (1997). J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, “The Wilson renormalization group for low x physics: Towards the high Phys. Rev. D [**59**]{}, 014014 (1999). J. Jalilian-Marian, A. Kovner and H. Weigert, “The Wilson renormalization group for low x physics: Gluon evolution at Phys. Rev. D [**59**]{}, 014015 (1999). I. I. Balitsky, Nucl. Phys. [**B463**]{} (1996) 99; Phys. Rev. Lett. [**81**]{} (1998) 2024; [ *Phys. Rev.*]{} [**D60**]{} (1999) 014020; Phys. Lett. [**B518**]{} (2001) 235; [*Nucl.Phys.*]{} [**A692**]{} (2001) 583. Y. V. Kovchegov, [*Phys.Rev*]{} [**D60**]{} (1999) 034008 E. Avsar, G. Gustafson and L. Lonnblad, JHEP [**0701**]{}, 012 (2007). J. Bartels, K. Kutak 521 K. Kutak, Phys. Lett. B [**675**]{} (2009) 332
A. M. Stasto, K. J. Golec-Biernat and J. Kwiecinski, ‘ Phys. Rev. Lett. [**86**]{}, 596 (2001). s. Lett. B [**675**]{} (2009) 332\[arXiv:0903.3521 \[hep-ph\]\].
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abstract: 'The abstract should appear at the top of the left-hand column of text, about 0.5 inch (12 mm) below the title area and no more than 3.125 inches (80 mm) in length. Leave a 0.5 inch (12 mm) space between the end of the abstract and the beginning of the main text. The abstract should contain about 100 to 150 words, and should be identical to the abstract text submitted electronically along with the paper cover sheet. All manuscripts must be in English, printed in black ink.'
address: 'Author Affiliation(s)'
bibliography:
- 'strings.bib'
- 'refs.bib'
title: AUTHOR GUIDELINES FOR ICIP 2017 PROCEEDINGS MANUSCRIPTS
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One, two, three, four, five
Introduction {#sec:intro}
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REFERENCES {#sec:ref}
==========
List and number all bibliographical references at the end of the paper. The references can be numbered in alphabetic order or in order of appearance in the document. When referring to them in the text, type the corresponding reference number in square brackets as shown at the end of this sentence [@C2]. An additional final page (the fifth page, in most cases) is allowed, but must contain only references to the prior literature.
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abstract: 'By employing a recently constructed hyperon-nucleon potential the equation of state of $\beta$-equilibrated and charge neutral nucleonic matter is calculated. The hyperon-nucleon potential is a low-momentum potential which is obtained within a renormalization group framework. Based on the Hartree-Fock approximation at zero temperature the densities at which hyperons appear in neutron stars are estimated. For several different bare hyperon-nucleon potentials and a wide range of nuclear matter parameters it is found that hyperons in neutron stars are always present. These findings have profound consequences for the mass and radius of neutron stars.'
author:
- 'H. apo'
- 'B.-J. Schaefer'
- 'J. Wambach'
title: On the appearance of hyperons in neutron stars
---
Introduction {#sec:intro}
============
Neutron stars (NS) are compact objects with interior densities of several times normal nuclear density. The precise and detailed structure and composition of the inner core of a NS is not known at present. Several possibilities such as mixed phases of quark and nuclear matter, kaon or pion condensates or color superconducting quark phases are under debate.
In the present work the influence of hyperons with strangeness $S=-1$ ($\Lambda$, $\Sigma^-$, $\Sigma^0$ and $\Sigma^+$) on the composition and structure of a NS is investigated. In this context the central and essential quantity to be analyzed is the equation of state (EoS). The EoS determines various NS observables such as the mass range or the mass-radius relation of the star. The composition and structure of neutron and also of protoneutron stars have been investigated in detail with a wide range of EoS for dense nuclear matter [@Prakash:1996xs; @Strobel:1999vn]. The emergence of hyperons for increasing nucleon densities has been suggested in the pioneering work of [@1960SvA.....4..187A]. Since then the impact of hyperons on dense matter has been studied extensively with different approaches, see e.g. [@Pandharipande:1971up; @Glendenning:1984jr; @Keil:1995hw; @Schaffner:1995th; @Prakash:1996xs; @Baldo:1999rq; @Vidana:2000ew] Unfortunately, the details of the hyperon-nucleon ($YN$) interaction and even more of the hyperon-hyperon ($YY$) interaction are known only poorly. The limited amount of available experimental data enables the construction of many different potentials. For example, the Nijmegen group has proposed six different potentials which all describe the low-energy data such as phase shifts equally well, see e.g. [@rijken].
We wish to explore the differences between the available $YN$ interactions and their influences on the appearance of hyperons in neutron stars. For this purpose the hyperon $YN$ threshold densities are calculated. Since the $\Sigma^0$ and $\Sigma^+$ hyperons are heavier than the $\Lambda$ and $\Sigma^-$ hyperons they typically appear later. Hence, we will focus on the $\Lambda$ and $\Sigma^-$ hyperons in the following.
The paper is organized as follows: In Sec. \[sec:EoS\] the EoS of dense matter including hyperons are calculated. For the pure nucleonic part of the EoS a parameterization is used. A central quantity which enters the EoS is the single-particle potential. The derivation of the single-particle potential for hyperons is given in Sec. \[sec:SPP\]. The requirements which are necessary for equilibrium are discussed in Sec. \[sec:beta\] and the results are collected in Sec. \[sec:Threshold densities\]. The next section Sec. \[sec:structure\] shows the calculations of neutron stars with and without hyperons. Finally, the work is summarized and conclusions are drawn in Sec. \[sec:summary\].
Single-particle potentials {#sec:SPP}
==========================
Single-particle potentials are a useful and important tool to determine the density at which hyperons begin to appear in baryonic matter. They can be obtained from an effective low-momentum $YN$ potential $\ensuremath{V_\text{low k}}$ [@schaefer; @wagner] in the Hartree-Fock approximation. Details concerning the derivation of the $YN$ single-particle potentials in this approximation can be found in [@Djapo:2008qv]. With these single-particle potentials the chemical potentials and particle energies in $\beta$-equilibrated matter can be calculated. This allows to compute the threshold densities for the appearance of a given hyperon species and to establish the concentrations of all particles in dense matter at a given density. In the process we will also determine the EoS for the mixture of leptons and baryons (electrostatic interactions are neglected because their energies are orders of magnitude smaller than the other interaction energies).
Our many-body scheme employs several “bare” $YN$ interactions as input for the $\ensuremath{V_\text{low k}}$ calculation: the original Nijmegen soft core model NSC89 [@nsc89], a series of new soft core Nijmegen models NSC97a-f [@rijken], a recent model proposed by the Jülich group J04 [@juelich] and chiral effective field theory (${\chi\text{EFT} }$) [@poli]. The first three models are formulated in the conventional meson-exchange (OBE) framework, while the ${\chi\text{EFT} }$ is based on chiral perturbation theory (for recent reviews see e.g. [@bedaque-2002-52; @epelbaum-2006-57; @Furnstahl:2008df]). In the ${\chi\text{EFT} }$ approach a cutoff $\Lambda_\ensuremath{{\chi\text{EFT} }}$ enters which we fix to $\Lambda_\ensuremath{{\chi\text{EFT} }}=600$ MeV and label the results obtained by ${\chi\text{EFT} }$600.
In general, the effective low-momentum $YN$ interaction, the $\ensuremath{V_\text{low k}}$ for hyperons, is obtained by solving a renormalization group equation. The starting point for the construction of the $\ensuremath{V_\text{low k}}$ is the half-on-shell $T$-matrix. An effective low-momentum $\ensuremath{T_\text{low
k}}$-matrix is then obtained from a non-relativistic Lippmann-Schwinger equation in momentum space by introducing a momentum cutoff $\Lambda$ in the kernel. Simultaneously, the bare potential is replaced with the corresponding low-momentum potential $\ensuremath{V_\text{low k}}$, $$\begin{aligned}
& &T_{\text{low }k, y' y}^{\alpha'\alpha}(q',q;q^2)=
V_{\text{low }k, y' y}^{\alpha'\alpha}(q',q)+ \nonumber \\
&&\frac{2}{\pi}
\sum_{\beta, z} P\!\!\int\limits_0^{\Lambda}\!\!dl\; l^2
\frac{V_{\text{low }k, y' z}^{\alpha'\beta}(q',l)
T_{\text{low }k, z
y}^{\beta\alpha}(l,q;q^2)}{E_{y}(q)-E_{z}(l)}\ .
\label{eq:vlowk} \end{aligned}$$ The on-shell energy is denoted by $q^2$ while $q'$, $q$ are relative momenta between a hyperon and nucleon. The labels $y$, $y'$, $z$ indicate the particle channels, and $\alpha$, $\alpha'$, $\beta$ denote the partial waves, e.g. $\alpha=LSJ$ where $L$ is the angular momentum, $J$ the total momentum and $S$ the spin. In Eq. (\[eq:vlowk\]) the energies in the denominator are given by $$\begin{aligned}
E_{y}(q)&=&M_{y}+\frac{q^2}{2\mu_{y}},
\label{eq:ene} \end{aligned}$$ with the reduced mass $\mu_{y}=M_Y M_N/M_y$, where $M_y$ is the total mass of the hyperon-nucleon system, $M_{y}=M_Y+M_N$. Finally, the effective low-momentum $\ensuremath{V_\text{low k}}$ is defined by the requirement that the $T$-matrices are equivalent for all momenta below the cutoff $\Lambda$. Details for several bare $YN$ potentials within the RG framework can be found in Refs. [@schaefer; @wagner].
![Density dependence of $\Lambda$ single-particle potentials for various hyperon-nucleon interactions in symmetric nuclear matter.[]{data-label="fig:Lrho"}](Lrhodraft.eps){width="10."}
From the different effective $\ensuremath{V_\text{low k}}$ interactions we calculate the single-particle potential $U_{b}(p)$ of a baryon $b \in \{ p,n, \Lambda, \Sigma^-, \Sigma^0, \Sigma^+, \Xi^-, \Xi^0 \}$ with three-momentum $p = |\vec p |$. In general, it is defined as the diagonal part in spin and flavor space of the proper self-energy for the corresponding single-particle Green’s function. In the Hartree-Fock (HF) approximation for a uniform system it represents the first-order interaction energy of the baryon with the filled Fermi sea. It is evaluated as the diagonal elements of the low-momentum potential matrix, $V_{y}^{\alpha} (q)$, where an evident short-hand labeling for the diagonal elements has been introduced (cf. Eq. (\[eq:vlowk\])). Note, that the relative momentum is given by $q=\left|\vec{p}-\vec{p}'\right|$ and the particle channel index is given by $y=bb'$. In the HF approximation the single-particle potential has two contributions: the (direct) Hartree- and the (exchange) Fock-term [@Walecka] $$\begin{aligned}
U_b(p)\!=\!\!\!\sum_{\vec{p}'\alpha b'}\!\!\left(
\left.V^{\alpha}_{y}(q)\right|_{\rm direct}\!\!+
(-1)^{L+S} \left.V^{\alpha}_{y}(q)\right|_{\rm exchange}\right)\ .
\label{plane}\end{aligned}$$ In this expression the diagonal elements of the nucleon-nucleon ($NN$) interaction is also included. In principle, an effective low-momentum potential for the $NN$ interaction is also known but we will use for the $NN$-sector a parametric Ansatz to be discussed later (see Eq. (\[eq:ParEoS\])).
![Same as in Fig. \[fig:Lrho\] but for the $\Sigma^-$ hyperon.[]{data-label="fig:Smrho"}](Smrhodraft.eps){width="10."}
The density dependence for several $\Lambda$ potentials at rest in symmetric nuclear matter (no hyperons present) is shown in Fig. \[fig:Lrho\]. The square represents the generally excepted empirical depth of $U_{\Lambda}(p=0)\approx -30\;\text{MeV}$. While most of the potentials used can reproduce this value, the Jülich potential (J04) yields a stronger binding while the old Nijmegen potential (NSC89) underestimates the binding. All other potentials agree up to the saturation density. However, with increasing density, the differences grow, leading to different bindings at rest. This will have consequences for the predictions of the $\Lambda$ hyperon concentration in dense nuclear matter. A comparison with other works, [@schulze], [@vidana1] and [@rijken] shows some differences but mostly yields similar results.
Fig. \[fig:Smrho\] shows the density dependence of several $\Sigma^-$ potentials at rest in symmetric nuclear matter similar to Fig. \[fig:Lrho\]. No agreement of the various $U_{\Sigma^-}$ potentials over the density range considered is seen. Compared with a $G$-matrix calculation a stronger binding for the $\Sigma^-$ single-particle potential is obtained. In Ref. [@Djapo:2008qv] further details concerning the hyperon single-particle potentials can be found.
For constructing the total energy/particle one has to distinguish which part of the total single-particle potential comes from the in-medium nucleon interaction and which from the hyperons. For this purpose, the single-particle potential can be split into a nucleonic and hyperonic contribution $$\begin{aligned}
U_b(p)=U^N_b(p)+U^Y_b(p)\ ,
\label{eq:split}\end{aligned}$$ where the first one, $U^N_{b}(p)$, denotes the baryon interacting with a nucleon and the second one, $U^Y_{b}(p)$, means the baryon interaction with a hyperon. Note, that the baryon can either be a nucleon or a hyperon. Accordingly, the nucleonic contribution is calculated from Eq. (\[plane\]) via $$\begin{aligned}
U^N_b(p)=\!\!\!\!\!\sum_{\begin{array}{c} \scriptstyle \vec{p}' \alpha \\
\scriptstyle b'={p,n} \end{array}}\!\!\!\!\!\left(
\left.V^{\alpha}_{y}(q)\right|_{\rm direct}\!\!+
(-1)^{L+S} \left.V^{\alpha}_{y}(q)\right|_{\rm exchange}\right)
\label{eq:Nb}\end{aligned}$$ and analogously for the hyperonic contribution.
Equation of State {#sec:EoS}
=================
By means of the single-particle potential the total energy per particle, $E/A$, can be easily calculated. It is given by the total energy of a baryon of mass $M_{b}$ and its kinetic and potential energy $U_b (p)$ divided by the total baryon number density $\rho_B$: $$\begin{aligned}
\label{E/A}
E/A&=&\frac{2}{\rho_B}\sum_{b}\int\limits_{0}^{k_{F_b}}\frac{d^3 p}{(2\pi)^3}
\left(M_{b}+\frac{p^2}{2M_{b}}+\frac{1}{2}U_{b}(p)\right).\end{aligned}$$ The potential $U_{b}(p)$ describes the average field which acts on these baryons due to their interaction with the medium. The baryon Fermi momentum $k_{F_b}$ is given by $$\begin{aligned}
k^3_{F_b}=3\pi^2 x_b \rho_B\end{aligned}$$ with the baryon fraction ratio $x_b=\rho_b/\rho_B$ for baryon $b$.
It is well-known that non-relativistic many-body calculations, based on purely two-body forces, fail to reproduce the properties of nuclear matter at saturation density. This is also the case in the present work. In order to proceed we replace the purely nucleonic contributions (without the influence of the hyperons) by an analytic parameterization developed by Heiselberg and Hjort-Jensen [@Heiselberg:1999mq]. This replacement makes the study of hyperons more robust since the $NN$ sector can be controlled more easily.
From Eq. (\[E/A\]) the purely nucleonic contribution to the energy per particle reads $$\begin{aligned}
E_{NN}/A_N\!=\!\frac{2}{\rho_N}\!\sum_{N}\int\limits_{0}^{k_{F_N}}\!
\frac{d^3
p}{(2\pi)^3}\!\left(\!M_N+\frac{p^2}{2M_N}+\frac{1}{2}U^N_N(p)\!\right)
\label{nucleonic}\end{aligned}$$ where the single-particle potential $U_N^N (p)$ only contains the nucleonic contribution, cf. Eq. (\[eq:Nb\]). However, instead of using the Hartree-Fock expression we employ the following parameterization $$\begin{aligned}
E_{NN}/A_N =M_N-E_0u\frac{u-2-\delta}{1+u\delta}+S_0u^{\gamma}(1-2x_p)^2
\label{eq:ParEoS}\end{aligned}$$ where $u=\rho_N/\rho_0$ denotes the ratio of the total nucleonic density $\rho_N=(x_p+x_n)\rho_B$ to the nuclear saturation density $\rho_0 =0.16\;\text{fm}^{-3}$. The corresponding proton and neutron fraction are denoted by $x_p$ and $x_n$. The parameters $E_0, \delta, S_0$ are related to properties of nuclear matter at saturation density, i.e. $E_0$ is the binding energy per nucleon at saturation density while $S_0$ and $\delta$ are connected to the symmetry energy and incompressibility, respectively.
As mentioned in the previous section, cf. Eq. (\[eq:split\]), we can separated the potential contribution of nucleons into one coming from the interaction with other nucleons $U^N_N(p)$, and one coming from the interaction with hyperons $U^Y_N(p)$. The latter does not contribute to the purely nucleonic EoS when we consider symmetric matter only. However, in the next section, when we investigate hyperons such terms are considered.
The parameterization (\[eq:ParEoS\]) is fitted to the energy per particle in symmetric matter obtained from variational calculations with the Argonne $V_{18}$ nucleon-nucleon interaction including three-body forces and relativistic boost corrections [@Akmal:1998cf]. The best fit parameters are $E_0=-15.8\;\text{MeV}$, $S_0=32\;\text{MeV}$, $\gamma=0.6$ and $\delta=0.2$. The EoS from Ref. [@Akmal:1998cf] is considered as one of the most reliable ones. In this way possible uncertainties coming from the nucleonic EoS are minimized.
The symmetry energy in dense matter is defined as $$\begin{aligned}
a_t=\left.\frac{1}{8} \frac{\partial^2 E/A}{\partial
x_p^2}\right|_{\rho_B=\rho_0}\ .\end{aligned}$$ Since there are no hyperons at saturation density we can use Eq. (\[eq:ParEoS\]) directly to obtain $a_t=S_0$. The incompressibility is given by $$\begin{aligned}
K_0=\left.9\rho^2 \frac{\partial^2 E/A}{\partial
\rho^2}\right|_{\rho_B=\rho_0} \end{aligned}$$ from which we obtain the relation $K_0=-18E_0/(1+\delta)$.
To study the impact of the softness of the EoS and the effects of the symmetry energy we vary $K_0$ and $a_t$ in a broader interval. From experimental constraints the values for $K_0$ range between $200\;\text{MeV}$ and $300\;\text{MeV}$ and those for $a_t$ between $28\;\text{MeV}$ and $36\;\text{MeV}$, see Ref. [@Lattimer:2006xb] and references therein. We vary $K_0$ and $a_t$ within these limits to study the effects of both parameters on the appearance and concentrations of hyperons in dense matter. While they do not influence the particle concentrations directly they modify the composition of the matter indirectly by changing the available energy. In this way the point at which hyperons will appear is affected. Results for various values of $K_0$ are shown in Fig. \[PEoS\] (in symmetric matter the energy per particle is only sensitive to the incompressibility). In all cases the saturation point is at $E/A=-16$ MeV. The parameter range allows us to classify the nucleonic EoS as a stiffer ($K_0=300$ MeV) or a softer ($K_0=200$ MeV) one.
![Parametric EoS in symmetric nuclear matter as obtained from the parameterization of Ref. [@Heiselberg:1999mq].[]{data-label="PEoS"}](PEoSsym.eps){width="8."}
In addition, $K_0$ directly influences the maximum allowed mass of a neutron star. By increasing $K_0$ the energy of the system is increased and as a consequence, more and more hyperons can be produced. This in turn will decrease the allowed maximum mass of a neutron star. Such a nontrivial connection creates a conundrum: if we use a stiffer nucleonic EoS by increasing $K_0$ we then allow for higher hyperon concentrations which softens the total EoS.
By means of Eq. (\[nucleonic\]) we can split the total energy per particle, Eq. (\[E/A\]), into a purely nucleonic part and a remainder $E'/A$ via $$\begin{aligned}
E/A=\frac{\rho_N}{\rho_B}E_{NN}/A+E'/A
\label{E/A2}\end{aligned}$$ with the rest $$\begin{aligned}
E'/A&=&\frac{2}{\rho_B}\sum_{N}\int\limits_{0}^{k_{F_N}}\frac{d^3 p}{(2\pi)^3}
\frac{1}{2}U^Y_N(p)\nonumber\\&+&
\frac{2}{\rho_B}\sum_{Y}\int\limits_{0}^{k_{F_Y}}\frac{d^3 p}{(2\pi)^3}
\left(M_Y+\frac{p^2}{2M_Y}\right.\nonumber\\&+&\left.\frac{1}{2}U^N_Y(p)
+\frac{1}{2}U^Y_Y(p)\right)\ .
\label{rest}\end{aligned}$$
In symmetric matter, which is composed only of nucleons, $E'/A$ vanishes, but with this separation we can calculate the total $E/A$ for arbitrary hyperon concentrations. In the following we will calculate the EoS including hyperons by determining their concentrations in $\beta$-equilibrium.
$\beta$-equilibrium
===================
Composition of matter {#sec:beta}
---------------------
In order to determine the threshold densities for hyperons their concentrations are needed. These are fixed by charge neutrality and $\beta$-equilibrium. The latter refers to the equilibrium under the weak interaction decays $$\begin{aligned}
B_1\rightarrow B_2+l+\bar{\nu}_l\end{aligned}$$ where $B_1$ and $B_2$ denote the baryons, $l \in \{ e^-, \mu^-, \tau^- \}$ the negatively charged leptons and $\bar{\nu}_l$ the corresponding neutrinos. In the case when the neutrinos are not trapped in the star (i.e. $\mu_{\nu}=0$) these requirements amount to $$\begin{aligned}
\label{eq:q}
0 &=&
\sum_{b}(\rho^{(+)}_{b}-\rho^{(-)}_{b})+\sum_l(\rho^{(+)}_{l}-\rho^{(-)}_{l}) \end{aligned}$$ for the charge neutrality and $$\begin{aligned}
\mu_{\Xi^-}=\mu_{\Sigma^-}&=&\mu_n+\mu_e,\label{eq:sne}\\
\mu_{\Xi^0}=\mu_{\Lambda}=\mu_{\Sigma^0}&=&\mu_n,\label{eq:ln}\\
\mu_{\Sigma^+}=\mu_p&=&\mu_n-\mu_e,\label{eq:sp}\end{aligned}$$ for the chemical potentials. The densities of positively and negatively charged baryons and leptons are denoted by $\rho^{(\pm)}_b$ and $\rho^{(\pm)}_l$, respectively. The chemical potentials $\mu$ are labeled by the corresponding particles. In the absence of neutrinos all lepton and antilepton chemical potentials are equal. In addition to the electrons, muons are also present. The $\tau$-lepton does not appear since it is too heavy.
![Density ratios for different particles for a “soft” nucleonic EoS as a function of the baryon density using the ${\chi\text{EFT}}$600 model.[]{data-label="200.32.60"}](x200.32.60.eps){width="9"}
At zero temperature, the chemical potential of a fermion system is equal to its Fermi energy. For relativistic noninteracting leptons it is given by $$\begin{aligned}
\mu_l=\sqrt{m_l^2 + k_{F_l}^2}= \sqrt{m_l^2+ \left(3\pi^2
x_l\rho\right)^\frac{2}{3}}\ ,\end{aligned}$$ with the corresponding lepton density ratio $x_l = \rho_l/\rho_L$. The total lepton density $\rho_L$ is the sum over all three leptons. For nonrelativistic interacting baryons, the chemical potential for species $b$ reads $$\begin{aligned}
\mu_{b}=M_{b}+\frac{k_{F_b}^2}{2M_{b}}+U_{b}(k_{F_b})\ .
\label{eq:munr}\end{aligned}$$
For a given total baryon density $\rho_B$ the equations (\[eq:q\])-(\[eq:sp\]) govern the composition of matter, i.e. the baryonic and leptonic concentrations. The corresponding solution is referred to as $\beta$-stable matter.
For the sake of consistency we now have to treat the nucleonic part of the chemical potential $\mu_N$ in the same way as the corresponding energy per particle. Since the chemical potential can be obtained as a derivative of the energy density $\epsilon$ and is related to the energy per particle via $\epsilon=\rho_B E/A$, we use the definition $$\begin{aligned}
\mu_b=\frac{\partial \epsilon}{\partial \rho_b}\ ,\end{aligned}$$ to have the appropriate replacement in the nucleonic chemical potential. Finally, we arrive at the expression $$\begin{aligned}
\mu_N=\frac{\partial \epsilon_{NN}}{\partial \rho_N}+U^Y_N(k_{F_Y}),
\label{eq:mupara}\end{aligned}$$ where we have effectively replaced $M_N+\frac{k^2_{F_N}}{2M_N}+U^N_N(k_{F_N})$ of Eq. (\[eq:munr\]) with the derivative $\partial \epsilon_{NN}/\partial \rho_N$. In this way the parameterization Eq. (\[eq:ParEoS\]) enters in the nucleonic part of the chemical potential.
![Same as Fig. \[200.32.60\] but for “stiff” nucleonic EoS using the NSC97f model.[]{data-label="300.32.Nf"}](x300.32.Nf.eps){width="9"}
Since we are only parameterizing the nucleonic sector, no such replacement is necessary for the hyperons. However, since we have neglected the $YY$ interaction $U^Y_Y(k_{F_Y})$ is zero and Eq. (\[eq:munr\]) reduces to $$\begin{aligned}
\mu_Y=M_Y+\frac{k^2_{F_Y}}{2M_Y}+U^N_Y(k_{F_Y})\ .\end{aligned}$$
For the determination of the particle concentration the single-particle potential in equilibrium is used. For hyperons below the threshold density it is given by $U_Y(p=0)$, similar to the symmetric matter case. Above the threshold density it depends on composition and density. In Fig. \[fig:usmpf\] the density dependence of $U_{\Sigma^-}(k_{F_{\Sigma^-}})$ in $\beta$-equilibrium for two different incompressibilities $K_0$ is shown. In the figure a kink in the curves appears at the point where the hyperons appear.
Another observation is the relative ordering and the magnitudes which resemble those of the single-particle potentials at zero momentum in symmetric matter as shown in Fig. \[fig:Lrho\]. Essentially, the NSC97a, NCS97c, NSC97f and J04 interactions are still slightly attractive while the NSC89 and $\chi$EFT600 remain repulsive. Similar observations hold for the $\Lambda$ system. A new structure in form of a second inflection point emerges due to the appearance of the $\Sigma^-$ hyperon.
![Density dependence of $U_{\Sigma^-} (k_{F_{\Sigma^-}})$ for $\beta$-equilibrated matter. Upper panel: $K_0=200$ MeV, lower panel: $K_0=300$ MeV.[]{data-label="fig:usmpf"}](usmpf200.32.eps "fig:"){width="9."}\
![Density dependence of $U_{\Sigma^-} (k_{F_{\Sigma^-}})$ for $\beta$-equilibrated matter. Upper panel: $K_0=200$ MeV, lower panel: $K_0=300$ MeV.[]{data-label="fig:usmpf"}](usmpf300.32.eps "fig:"){width="9."}
A better indicator at which densities hyperons start to appear is given by the concentrations of all particles and is displayed in Figs. \[200.32.60\] and \[300.32.Nf\]. In Fig. \[200.32.60\] a “soft” nucleonic EoS is used in combination with an attractive $\Lambda N$ and a very repulsive $\Sigma N$ interaction implemented by the ${\chi\text{EFT} }$600 model. In contrast in Fig. \[300.32.Nf\] a “stiff” EoS is used represented by the NSC97f model which has a similar $\Lambda N$ interaction compared to the ${\chi\text{EFT} }$600 model but also an attractive $\Sigma
N$ interaction. This difference already leads to very different density profiles. While in Fig. \[200.32.60\] the $\Lambda$ hyperon is the first one which appears and no $\Sigma^-$ hyperons are present, the $\Sigma^-$ hyperon appears first in Fig. \[300.32.Nf\].
One should note that with the appearance of the $\Sigma^-$ hyperon the density of the negatively charged leptons starts to drop immediately. This is because their role in the charge neutrality condition, Eq. (\[eq:q\]), is now being taken over by the $\Sigma^-$. Similarly, the appearance of the $\Lambda$ hyperon will accelerate the disappearance of neutrons since both are neutral particles.
![Various EoS for $\beta$-equilibrated matter for two different $K_0$ as a function of density. Upper panel: soft EoS, lower panel: stiff EoS.[]{data-label="bEoS"}](eos200.32.eps "fig:"){width="9."}\
![Various EoS for $\beta$-equilibrated matter for two different $K_0$ as a function of density. Upper panel: soft EoS, lower panel: stiff EoS.[]{data-label="bEoS"}](eos300.32.eps "fig:"){width="9."}
Once the composition of the matter has been determined by demanding $\beta$-equilibrium we can calculate the energy per particle. For this purpose, we cannot use Eq. (\[E/A\]), but have to use Eqs. (\[E/A2\]) and (\[eq:ParEoS\]). The result is presented in Fig. \[bEoS\] where the energy per particle in $\beta$-stable matter is shown as a function of the density for different $YN$ models. The symmetry energy is fixed to $a_t = 32$ MeV while the incompressibility is set to $K_0=200$ MeV (upper panel in the figure) and to $K_0=300$ MeV (lower panel). In addition, the EoS with hyperons is compared with the purely nucleonic one.
One easily observes the onset of the hyperon appearance at the point at which the curves start to deviate. As expected the differences between the various $YN$ interactions do not modify the EoS for very small densities. In the range between $(2 - 3)\rho_0$, all EoS’s are similar to each other. However, for increasing densities the influence of hyperons becomes more significant resulting in rather different EoS’s. This concerns not only the magnitudes of the different energies per particle but also their slopes at higher densities. These variations will lead to differences in the pressure and finally to significant changes in the possible maximum mass of a neutron star.
Threshold densities {#sec:Threshold densities}
-------------------
{width="18."}\
{width="18."}
The appearance of a given hyperon species is determined by increasing the density for fixed $K_0$ and $a_t$. The resulting threshold densities for the $\Sigma^-$ hyperon for certain $K_0$ and $a_t$ are collected in Fig. \[fig:stre\] for six different $YN$ interactions. Similarly, the threshold densities for the $\Lambda$ hyperon are shown in Fig. \[ltre\]. From these figures one sees how the single-particle potentials for various $YN$ interactions modify the threshold densities. In this way, the properties of the $YN$ interaction in Fig. \[fig:Lrho\] and Fig. \[fig:Smrho\] can be attributed to the hyperon appearances.
From Fig. \[fig:stre\] one concludes that the $\Sigma^-$ hyperon appears between $1.4 \rho_0$ and $2.4\rho_0$ with the exception of the ${\chi\text{EFT} }$600 model. For almost all $YN$ interactions used in the present study the $\Sigma^-$ is the first hyperon which will appear even though the $\Lambda$ hyperon is the lighter one. The reason is that the heavier mass of the $\Sigma^-$ is offset due to the presence of the $e^-$ chemical potential, cf. Eq. (\[eq:sne\]). In general, heavier and more positively charged particles appear later. In the case of the $\Sigma^-$, compared to the $\Lambda$, the effect caused by the electric charge dominates the one coming from the mass in almost all cases. For the $\Sigma^-$ hyperon a further modification caused by the electric charge, is the influence of $a_t$ on the threshold density because the electron chemical potential is modified by the symmetry energy. Thus, the decrease of the threshold densities due to the increase of $K_0$ is analogous to the increase due to $a_t$.
For the $\Lambda$ hyperon the range of threshold densities is between densities from $1.7 \rho_0$ to $4.5 \rho_0$ depending on the choice of $K_0$, $a_t$ and the $YN$ interaction used, cf. Fig. \[ltre\]. The influence of $K_0$ on the threshold density for this hyperon is larger than the one from $a_t$. This is reasonable since $K_0$ controls the rate of the energy increase with the density more directly, while $a_t$ affects only the details of the $\beta$-equilibrium. One clearly recognizes in Fig. \[ltre\] that the $\Lambda$ appears earlier for larger incompressibilities. Thus, in general we see that for increasing $K_0$ the threshold densities decrease for both hyperons.
{width="18."}\
{width="18."}
In contrast to the previous $K_0$ and $a_t$ discussion, the influence of the single-particle potentials on the threshold densities is harder to analyze.
The threshold densities for the $\Sigma^-$ are largest for the ${\chi\text{EFT} }$600 interaction which yields the most repulsive $\Sigma^-$ single-particle potential. In general, hyperons will appear earlier for a more attractive single-particle potential. This becomes obvious from Eq. (\[eq:munr\]): the chemical potential decreases for a more negative $U_b(k_{F_b})$ and, consequently, the threshold density will also decrease. Thus, the most repulsive single-particle potential like the one for the ${\chi\text{EFT} }$600 leads to the largest threshold density. For the $\Lambda$ hyperon the threshold densities are smallest for the most attractive single-particle potential obtained with the J04 model, cf. Fig. \[ltre\]. On the other hand, they are largest for the most repulsive NSC89 interaction. For the NSC97f interaction, which is between these extremes, the $\Lambda$ threshold densities are very close to those of the most repulsive NSC89 one, cf. Fig. \[ltre\]. This stems from the appearance of the $\Sigma^-$ hyperon. The effect is caused by the slowdown of the increase of the neutron chemical potential and is further related to the rapid increase of the $\Sigma^-$ density just after its appearance, cf. Fig. \[300.32.Nf\]. Basically, the slowdown occurs as soon as a new hyperon appears because most of the energy is used for its creation. Once the concentration of the hyperon has reached a plateau, the neutron chemical potential resumes its increase until a further hyperon might appear. Thus, the appearance of the first hyperon shifts the threshold density of the next hyperon towards higher values. This effect explains why the threshold densities of the $\Lambda$ are so similar for the NSC97f and NSC89 interactions. It also makes clear why the $\Lambda$ threshold densities for the ${\chi\text{EFT}
}$600 interaction are smaller than those of the NSC97a, NSC97c and NSC97f interactions even though their $\Lambda$ single-particle potentials are almost the same, cf. Fig. \[fig:Lrho\]. In the case of the J04 model the delay mechanism described above becomes very interesting. For this $YN$ interaction the $\Lambda$ and $\Sigma^-$ hyperon appear almost at the same density. In this case the neutron chemical potential stagnates but the $\Lambda$ and the $\Sigma^-$ single-particle potentials are attractive enough to compensate for this.
To summarize this section strangeness appears around $\sim 2\rho_0$ for all $YN$ models and parameter sets used. Note, that the appearance of the first hyperon, whether it is the $\Sigma^-$ or the $\Lambda$, cannot be altered by taking into account $YY$ interactions which have been neglected in this work. The present study in terms of the broad parameter ranges as well as the multitude of $YN$ interaction models reveals that strangeness in the interior of neutron stars cannot be ignored. Similar conclusions are obtained in the Brueckner-Hartree-Fock theory [@Baldo:1998hd].
Structure of neutron stars {#sec:structure}
==========================
In this section we analyze the effect of the EoS including hyperons on neutron stars. We focus on non-rotating stars, ignoring any changes, caused by the rotation. For a given EoS, the mass-radius relation of a NS can be determined by solving the familiar Tolman-Oppenheimer-Volkoff equation (TOV) [@Oppenheimer:1939ne]. To describe the outer crust and atmosphere of the star i.e., the region of subnuclear matter densities for very small baryon densities below $\rho_B < 0.001\;\textrm{fm}^{-3}$, we have used the EoS of Baym, Pethick, and Sutherland [@Baym:1971pw], which relies on properties of heavy nuclei. For densities between $0.001\;\textrm{fm}^{-3} \leq \rho_B \leq 0.08\;\textrm{fm}^{-3}$,i.e. for the inner crust, we have used the EoS of Negele and Vautherin [@Negele:1971vb], who have performed Hartree-Fock calculations of the nuclear crust composition. Details on crust properties can be found e.g. in [@Pethick:1995di; @Blaschke:2001uj], while recent state-of-the-art approaches are discussed in [@Ruester:2005fm].
In Fig. \[fig:mR\] the mass-radius relation of a NS for a soft EoS (left panel) and for a stiff EoS (right panel) is shown. The symmetry energy $a_t = 32$ MeV is kept fixed in both calculations and the resulting mass-radius relation without any strangeness is also added for comparison.
As can be seen from the figure the appearance of hyperons reduces the NS mass drastically compared with the pure $NN$ case. Even for larger values of the incompressibility, i.e. $K_0=300$ MeV, the maximum mass, obtained with all $YN$ interactions used, is still below the largest precisely known and measured NS masses $1.44\;M_{\odot}$ of the Hulse-Taylor binary pulsar. This is not an unusual result and is also seen in other, related works such as e.g. [@Vidana:2000ew; @SchaffnerBielich:2002ki; @SchaffnerBielich:2007yq; @Mornas:2004vs]. In general, any inclusion of further degrees of freedom to the nucleons will reduce the NS mass.
Due to large uncertainties in the high density behavior of the symmetry energy we investigate its influence on the NS masses as follows: The parameter $\gamma$ in Eq. (\[eq:ParEoS\]) determines the symmetry energy changes, i.e., how asy-stiff or asy-soft the EoS is. The value of $\gamma = 0.6$, used so far, represents a asy-soft system [@Wolter:2008zw]. Above saturation density we change the value of this parameter and use $\gamma=1$ while keeping all other parameters fixed representing a asy-stiff system. The effect of the $\gamma$ modification is shown in Fig. \[fig:m2R\].
Only small changes in the mass-radius relation are obtained when hyperons are included. In most cases the mass difference is $~0.1
M_\odot$. An increasing $\gamma$ leads to an increase in the proton concentration which in turn leads to an increase in the $\Sigma^-$ concentration. This largely cancels any energy gain from an increased symmetry energy term that could possibly increase the mass of a neutron star. The biggest mass change is seen for $K_0=200$ MeV with the $\chi$EFT600 interaction since this model does not contain any $\Sigma^-$ hyperons. However, even in this case the difference is below $0.2 M_\odot$ which leads to a maximum mass below $1.2 M_\odot$. A more noticeable change can be observed for the radius. A radius shift of 1 km towards larger radii is seen for all curves. Furthermore, if we in addition vary $a_t$ at saturation density between 28 and 36 MeV, a less pronounced effect on the NS mass is visible as compared to the variation of $\gamma$.
To complete our investigation of hyperon effects on the neutron star EoS we also need to consider modifications induced by the $YY$ and $\Xi N$ interactions. In order to evaluate the effects of the $S=-2$ sector we have to construct $YY$ and $\Xi N$ $\ensuremath{V_\text{low
k}}$ interactions, along the lines discussed earlier for the $S=-1$ sector. Obviously, since there are more particles to consider, the situation complicates considerably from a numerical point of view. Unfortunately, unlike the $S=-1$ sector where we had several different “bare” interactions from which we constructed the $\ensuremath{V_\text{low k}}$ potentials, in the $S=-2$ sector we only have one, namely the NSC97 interaction. We also note that the inclusion of the $S=-2$ sector will not influence the appearance of the first hyperon. Hence we have neglected the $S=-2$ up to this point.
Fig. \[fig:mYYR\] shows the mass-radius relation of a NS with different $YN$, $YY$ and $\Xi N$ interactions is presented. The effect of the inclusion of the $S=-2$ sector is rather marginal. As is visible in Fig. \[fig:mYYR\] the maximum masses are lower than in the previous cases which is reasonable since a further degree of freedom, the $\Xi$ particle, is added. However we note that the $NSC97$ $YY$ interaction is attractive which is the reason for the decrease of the allowed maximum NS masses, see e.g. [@Vidana:2000ew]. A repulsive $YY$ interaction would have an opposite effect as shown in [@Mornas:2004vs].
It is also interesting to point out that for the $\chi$EFT600 interaction, where the $\Sigma^-$ hyperons appears late, if at all, it is the $\Xi$ hyperons which takes its pace and influences the maximum mass of a NS [@Glendenning:2000jx; @Balberg:1998ug]. For all other models in which the $\Sigma^-$ appears earlier than the $\Xi$ their influence on the NS mass is marginal.
Summary and conclusions {#sec:summary}
=======================
The main intention of the present work was to study the consequences of available hyperon-nucleon interactions on the composition of neutrons stars and the maximum masses. The analysis was performed in the framework of the renormalisation group improved $\ensuremath{V_\text{low k}}$ interaction deduced from available bare potentials. Since the experimental data base is very limited, these potentials are not well constrained, in contrast to the nucleon-nucleon case.
We have determined the threshold densities for the appearence of hyperons in $\beta$-equilibrated neutron star matter to lowest order in a loop expansion. To explore the sensitivity to available $YN$ potentials we have constructed single-particle potentials for the $\Lambda$ and $\Sigma^-$ hyperon in lowest order and deduced the energy per particle. We have replaced the pure nucleonic contribution to the energy per particle by an analytic parameterization. This replacement enables us to vary the incompressibility $K_0$ and symmetry energy $a_t$ of the purely nucleonic EoS and to investigate their influence on the threshold densities of hyperons. The composition of $\beta$-stable matter has been determined by the requirement of charge neutrality and $\beta$-equilibrium. The corresponding threshold densities for various values of $K_0$ and $a_t$ were evaluated. The most important conclusion is that a more attractive single-particle potential will decrease the chemical potential and thus decrease the threshold density of the corresponding hyperon.
We have found that, irrespective of the $YN$ interactions, incompressibility and symmetry parameter used, hyperons will appear in dense neutron star matter at densities around $\sim 2\rho_0$. This inevitably leads to a significant softening of the EoS which in turn results in smaller maximum masses of a neutron star compared to a purely nucleonic EoS. Notably, the predicted maximum masses are well below the observed value of $1.4\;M_{\odot}$, an outcome also known from other works, e.g. [@Vidana:2000ew; @SchaffnerBielich:2002ki; @SchaffnerBielich:2007yq; @Mornas:2004vs]. This poses a serious problem.
The softening of the EoS due to hyperons cannot be circumvented by stiffening the nucleonic EoS, i.e., by increasing $K_0$, since this will cause hyperons to appear earlier. Changing the high-density behavior of the symmetry energy dependence or including the $S=-2$ sector does not alter this conclusion either. For more details about the $S=-2$ sector, in particular $\Xi$ hyperons in dense baryonic matter see e.g. [@Wang:2005vg; @Pal:1999sq; @Huber:1997mg; @Glendenning:1982nc]. This can only mean, that correlations beyond the one-loop level could be important to stiffen the hyperon contributions to the EoS. This, however, is not sufficient as Brueckner-Hartree-Fock calculations indicate [@Baldo:1998hd; @Baldo:1999rq]. As has been known for a long time from non-relativistic nuclear many-body theory, three-body interactions are crucial to yield a stiff nucleonic EoS. Repulsive three-body forces may also play a role in the hyperon sector. There is, however, little empirical information available at present. There is also the possibility of an early onset of quark-hadron transition to cold quark matter. This might also stiffen the high-density equation of state.
Clearly, our results pose significant restrictions on any reasonable equations of state employed in the study of neutron star matter. With the prediction of a low onset of hyperon appearance it becomes mandatory to seriously consider strangeness with respect to neutron stars. Eventhough our predictions for the maximum masses of neutron stars are too low, the treatment of hyperons in neutron stars is necessary and any approach to dense matter must address this issue.
Acknowledgments {#sec:Acknowledgements .unnumbered}
===============
H has been partially supported by the Helmholtz Gemeinschaft under grant VH-VI-041 and by the Helmholtz Research School for Quark Matter Studies. BJS acknowledges support by the BMBF under grant 06DA123 and JW from the Extreme Matter Institute within the Helmholtz Alliance ’Cosmic Matter in the Laboratory’. This work was also supported in parts by the grant SFB634 of the Deutsche Forschungsgemeinschaft. We thank R. Alkofer and M. Oertel for a critical reading of the manuscript.
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---
author:
- |
R. B. MacKenzie\
Groupe de physique des particules, Laboratoire René-J.-A.-Lévesque, Université de Montréal, C. P. 6128, succ. centre-ville, Montréal, Québec, Canada, H3C 3J7\
E-mail:
- |
M. B. Paranjape[^1]\
Departamento de F[í]{}sica Te[ó]{}rica, Facultad de Ciencias, Universidad de Zaragoza, 50009, Zaragoza, España\
E-mail:
title: 'From Q-Walls to Q-Balls '
---
Introduction
============
Non-topological solitons were first described in the mid-70’s [@fls] in the context of scalar field models with two fields, symmetry breaking and a conserved charge. They reappeared in a somewhat different guise in the mid 80’s in theories without spontaneous symmetry breaking as considered by Coleman [@c] and the name $Q$-balls was invented. In the mid 90’s they have re-surfaced in connection with models of scalar fields where the potential has flat or quasi-flat directions, which is generic in models of supersymmetry breaking, as described by Kusenko [@k].
The original non-topological solitons and -balls had an energy which behaved as E\~Q for large conserved charge $Q$. If $\mu $ is less than the mass of a perturbative, charged excitation of unit charge, then the -balls are clearly stable against decay into perturbative charged particles. Indeed, Coleman [@c] defined balls as the minimal energy classical configurations of fixed charge and was able to show their absolute, classical stability. However, it was soon realized that linear terms induced via quantum corrections in the effective potential would lead to the decay of these -balls [@ccgm].
The recently-described -balls do not suffer this fate. They have an intriguing energy-charge dispersion, $E\sim Q^{(3/4)}$ for large in 3 spatial dimensions. Hence these -balls are arbitrarily tightly bound compared to the energy of an aggregate of perturbative quanta of equal charge. We note that such an asymptotic energy-charge dispersion with the non-integer power $3/4$ also arises in another context. For solitons in models with non-trivial topological Hopf windings [@nf], there is an energy bound, which is not attained, but behaves as $Q^{(3/4)}$. It would be interesting to find a model where a “Hopf soliton", perhaps of the -ball, type can exactly saturate such a bound. Indeed, we will find below exactly solvable -balls which do saturate a similar bound in 1 spatial dimension.
In this paper we investigate extended objects of the -ball type. These are the solitons of one- and two-dimensional models embedded in three dimensions, forming -walls and -strings, respectively. Similar structures have been studied recently in [@bs; @pv] both analytically and using numerical simulations. The cases discussed concern -balls which have an energy-charge dispersion that grows linearly with the charge, $E\sim\mu Q$. It is found that collisions of such -balls can give rise to -string like states. In [@pv] stable -ring solitons are described. Here stability is imbued via an additional topological twisting of the phase as one goes around the ring. We find our study is complementary to these studies.
In the next section, we study -solitons in arbitrary spatial dimension $d$. The energetics of these objects are interesting: the energy $E_d$ of a soliton in $d$ dimensions depends on its charge $Q_d$ as $E_d\sim Q_d^{d/(d+1)}$. The lower the dimension, the more energetically favourable the -soliton over a conventional configuration of charge $Q_d$. At first glance, this would seem to indicate that extended objects in three dimension would be energetically preferred over -balls. However, as we shall see, this turns out not to be the case.
In Section 3, we consider embedded lower-dimensional -solitons in three dimensions. The energy $E_2$ and charge $Q_2$ of a two-dimensional soliton, for instance, are the energy and charge per unit length of the corresponding -string. The energy of a -string of length $L$ and of fixed three-dimensional charge varies as $L^{1/3}$. Thus the tension varies as $L^{-2/3}$, reducing rapidly with length. However, the energy per unit length also varies as $L^{-2/3}$, so the natural time scale for oscillations of -strings is equivalent to that of topological strings (whose energy per unit length and tension are independent of $L$).
In Section 4 we study the stability of -strings or walls against “beading”, [*i.e.,*]{} the formation along the length of the string (or surface of the wall) of clumps of higher charge (and energy) per unit length (or surface) which would ultimately result in the formation of a one-dimensional chain (or two-dimensional sheet) of non-interacting balls.
We study the stability of walls, in particular, since there the analysis is much simpler, essentially because the one-dimensional soliton can be found analytically for certain potentials. In studying small oscillations about such a solution, the eigenvalue equation determining the frequency of these oscillations can be related to a Schrödinger equation in supersymmetric quantum mechanics. Applying known results from that theatre enables us to identify a single unstable mode for the fluctuation, two zero-modes (one of which corresponds to translation of the wall) and a spectrum of stable modes. The unstable mode can have an arbitrarily large characteristic time, so the wall can be extremely long-lived. We expect the same behaviour for strings, but have not yet done the analysis in that case.
-balls in any dimension
=======================
Consider a charged scalar field in $d$ spatial dimensions, the Lagrangian governing its behaviour is =d\^dx( [12]{}\_\_a \^\_a -V(||)) where $a=1,2$, $|\phi |=\sqrt{\phi_1^2+\phi_2^2}$ and we use a simple $SO(2)$ global gauge invariance whose conserved charge corresponds to . The conserved charge is normalized as Q=d\^dx\_[ab]{}\_a\_t\_b. The scalar potential, inspired by supersymmetric models, satisfies V(||)= where $\Lambda$ is a constant which is determined by the scale of supersymmetry breaking. The minimum of the potential occurs at $|\phi |=0$ and the charge symmetry is unbroken. -ball solutions are found with the ansatz \_1 +i\_2 = e\^[it]{}(r) where $\omega$ is a constant, $r$ is the radial coordinate and $\varphi (r)$ is real. We immediately obtain Q=d\^dx \^2 (r). The profile taken for $\varphi (r)$ satisfies the asymptotic conditions (r) = where $\phi_0$ is taken large enough so that the asymptotic behaviour of the potential is valid. Hence the -ball interpolates from large non-vacuum values around $\phi_0$ in its center to $\varphi =0$, the perturbative vacuum, at infinity. The existence of stable solutions to the equations of motion for $\varphi (r)$ in 3 spatial dimensions is the content of the previous papers [@c; @c2] and is easiest seen by the classic continuity arguments of Coleman [@c]. Here we re-do the analysis in arbitrary spatial dimensions $d$.
The equations of motion for $\varphi (r)$ are \^2(r) -V\^((r))+\^2(r)=0\[eqom\] which, given the form of the potential, become |\^2(r) +\^2(r)&=&0r0\^2(r)-m\^2(r)&=&0r using the assumed asymptotics of the potential. The solutions of these equations are immediate, but dimension dependent. For small $r$ we get spherical Bessel functions of order 0 in 3 dimensions, ordinary Bessel functions of order 0 in 2 dimensions, and trigonometric functions in 1 dimension, with argument $\omega r$. For large $r$ the argument simply changes to $imr$, which is imaginary.
$d=3$
-----
The asymptotic solutions are (r)=\[as\] where $\phi_0$ is a constant that is fixed by the total charge of the solution, $\varphi_0$ is a dependent constant and $j_0(\omega r)$ and $k_0(m r)$ are the spherical Bessel function of order 0 of real and imaginary argument respectively. The asymptotic solutions interpolate from one to the other in a relatively small region, and it is difficult to obtain analytic solutions, except in the 1 dimensional case. However, the continuity argument of Coleman easily establishes the existence of a solution.
The idea is to first observe that the equation of motion \[eqom\] can be interpreted to describe the classical motion of a “particle" in one dimension of position $\varphi$, in the potential $-V(\varphi ) +{1\over 2}\omega^2\varphi^2$ and in the presence of a specific frictional term that is time dependent coming from the Laplacian, and where $r$ plays the role of time. Indeed the equation of motion is equivalently written as (r) -V\^((r)) +\^2 (r)=-[2r]{}(r) where $\dot\varphi (r) =\partial_r\varphi (r)$. The potential is negative for small $\varphi$, and is approximately a downward opening parabola of curvature $-m^2$, whose maximum passes through the origin at $\varphi (r) =0$. It turns over, after achieving a minimum (negative) value, into an upward opening parabola of curvature $\omega^2$ for large enough values of $\varphi (r)$. Clearly, eventually the potential becomes positive. Now if we start with an initial value $\phi_0$ for which $V(\phi_0)$ is negative, the “particle" will never rise up high enough to reach the origin, even in infinite time, and will perform damped oscillations till it settles down to the global minimum value of $-V(\varphi ) +{1\over 2}\omega^2\varphi^2$. On the other hand if we start with a large initial value of $\phi_0$, then since the “motion" is governed arbitrarily well by the spherical Bessel function $j_0(\omega r)$, the “particle" will attain a given value close to the point where the full potential $-V(\varphi ) +{1\over 2}\omega^2\varphi^2$ vanishes, in a fixed time, independent of the initial value $\phi_0$ which is of the order of $1/\omega$. The velocity at this point will be proportional to $\phi_0$. Hence the subsequent evolution can be arranged to commence with an arbitrarily high initial velocity by choosing $\phi_0$ larger and larger. It is clear then, that there exists an initial value for $\phi_0$ so that the “particle"will arrive at this point with sufficient velocity that the subsequent motion will rise up and go over the maximum at $\varphi=0$ in finite time. Hence, by continuity, there exists an initial value for $\phi_0$ which will exactly achieve the maximum as time goes to infinity, [*i.e.,*]{} for $r\rightarrow\infty$. Such a configuration is the solution and it will give the appropriate interpolation between the two asymptotic solutions \[as\]. Clearly the value of $\varphi_0$ will be governed by the value of $\phi_0$ which in turn will be fixed by the value $\omega$. Hence the total charge can be thought of as a function of $\omega$ alone, or conversely all of the parameters depend only on the total charge .
The energy of such a configuration can be approximated as |E&&\_[r<1/]{} -.7cm[d\^3x ( \^2\^2(r) +|(r)|\^2 +)]{}&=&\_[r<1/]{} -.7cm[d\^3x ( 2\^2\^2(r) +)]{}= [\_0\^2]{} +[\^3]{} where we neglect the contributions from the surface and the exterior of the -ball, and where $\alpha$ and $\beta$ are positive constants. This expression is of course reasonable only for -balls with large charge . Using Q=d\^3 x \^2(r)\_[r<1/]{} -.7cm[d\^3 x\_0\^2j\_0\^2(r)]{}=[\_0\^2\^2]{} where $\gamma$ is another positive constant, we get E + [\^3]{} Imposing $dE/d\omega =0$ yields =[3\^4]{} which gives |&\~& Q\^[-1/4]{}\_0 &\~& Q\^[1/4]{} E&\~& Q\^[3/4]{} .
$d=2$
-----
Here the solution for small $r$ is $\phi_0J_0(\omega r)$ where $J_0(\omega r)$ is the ordinary Bessel function of order 0, and for large $r$ the solution is $\varphi_0K_0(mr)$, where $K_0(mr)$ is the Bessel function of order zero of imaginary argument. The energy is then approximately given by |E\_2&&\_[r<1/]{} -.7cm[d\^2x ( \^2\^2(r) +|(r)|\^2 +)]{}&=& [Q\_2]{} +[\^2]{} which yields upon imposing $dE_2/d\omega =0$ |&\~& Q\_2\^[-1/3]{}\_0 &\~& Q\_2\^[1/3]{} E\_2&\~& Q\_2\^[2/3]{} . Here $E_2$ and $Q_2$ are the appropriate 2 dimensional energy and charge respectively, which in terms of 3 dimensional energy and charge are the energy per unit length and the charge per unit length.
$d=1$ {#2.3}
-----
Here the equation of motion can be reduced to quadratures, and it can be integrated exactly in many cases. It is more convenient to use the normal spatial coordinate $x\in [-\infty ,\infty ]$ rather than the radial coordinate $r$. The equation of motion for $\varphi (x)$ where $\phi_1(x)+i\phi_2(x) =e^{i\omega t}\varphi (x)$ is -V\^((x)) +\^2(x) = 0 which yields \_[\_0]{}\^[(x)]{}[d]{} =\_[x\_0]{}\^[x]{}dx.\[quadrature\] Here $x_0$ gives the position of the -ball and the $\pm$ is chosen depending on whether $x> x_0$ or $x< x_0$. The square root should not become imaginary thus we must have that $V(\varphi ) >\omega^2\varphi^2$. Since for large values of $\varphi$ the potential becomes flat and the required inequality is violated, there is a fixed maximum value for $\phi_0$ given by the condition $V(\phi_0 ) =\omega^2\phi_0^2$. There is no “friction" term coming from the Lapalcian in 1 dimension, hence conservation of energy dictates that this value of $\phi_0$ is the starting point for $\varphi_0(x)$. Small $\omega$ implies large values of $\phi_0$ which gives approximately that $\phi_0^2\approx{\Lambda /\omega^2}$ that is $\phi_0\sim 1/\omega$. The quadrature \[quadrature\] is not particularly useful in the case of a general potential $V(\varphi (x))$. The solution for $x\approx x_0$ is $\varphi (x)=\phi_0\cos ( \omega (x-x_0))$ while for $|x-x_0|>>\pi /2\omega$ is $\varphi (\pm x)=\varphi_0 e^{\mp mx}$. The charge is given by Q\_1=dx\^2 (x)\_0\^2\_[x\_0-(/2)]{}\^[x\_0+(/2)]{}-1.2cm[dx \^2((x-x\_0)) ]{} =[\_0\^22]{} while the energy is |E\_1&=&dx ( \^2\^2 (x) +(\_x(x))\^2 + V ((x) )) &=&dx( 2\^2\^2 +V((x) ) -(x) V\^((x) ))&& \_[|x-x\_0|<(/2)]{} [-1.8truecm dx( 2\^2\^2 +)]{}&=&2\^2\_0\^2( \_[|x-x\_0|<(/2)]{}[-1.8truecm dx\^2((x-x\_0))]{}) +&=&[2\_0\^22]{}+. Imposing =0 yields \_0\^2 -[\^2]{}=[2Q\_1]{}-[\^2]{}=0 implying \^2=[2 Q\_1]{}. This gives, |&\~& Q\_1\^[-1/2]{}\_0 &\~& Q\_1\^[1/2]{} E\_2&\~& Q\_1\^[1/2]{} .
Arbitrary spatial dimension
---------------------------
Here we record the corresponding formulae in arbitrary spatial dimension $d$. The $Q_d$-ball solution will be to the appropriate generalization of the free spherical wave in its centre with argument $\omega r$ while from its edge it will evolve into the exponentially decaying type solution of mass $m$ outside, [*i.e.,*]{} the same function but of imaginary argument $imr$. One easily finds |Q\_d&&[\_d\_0\^2\^d]{}= [\_d\_0\^2\^[(d-1)]{}]{}E\_d&&\_d \_0\^2\^[(2-d)]{}+[\_d\^d]{}&=&[\_d Q\_d\^[(d-1)]{}\_d]{} \^[(2-d)]{}+[\_d\^d]{}&=&[\_d Q\_d\_d]{} +[\_d\^d]{} where $\alpha_d , \beta_d$ and $\gamma_d$ are the corresponding constants in $d$ dimensions. Imposing $dE_d/d\omega =0$ yields in a straightforward manner |&\~& Q\_d\^[-1/(d+1)]{}\_0 &\~& Q\_d\^[1/(d+1)]{} E\_d&\~& Q\_d\^[d/(d+1)]{} .
Energetics of -strings and -walls
=================================
Suppose initial conditions have created a large -wall or -string in 3 dimensions. We will observe here that such structures have interesting energetics. We address the question of stability of such objects in the next section.
-strings
--------
We define -strings as linear configurations in 3 dimensions whose cross section corresponds to a 2 dimensional $Q_2$-ball, with a fixed charge per unit length $Q_2$. Such a configuration will simply terminate at two end points or close on itself yielding a closed -string configuration. We will consider for convenience the second situation, of a closed -string of length $L$. We will assume that $L$ is sufficiently long so that gradient energies due to the gradual bending that is necessary to close the -string are negligible. The relevant conserved charge of a closed -string then is given by Q=Q\_2L while the energy is given by E=E\_2L. Replacing for $E_2\sim Q_2^{2/3}=(Q/L)^{2/3}$ we get E\~Q\_2\^[2/3]{}L=Q\^[2/3]{}L\^[1/3]{}. Hence a -string of total charge satisfies an energy-charge dispersion relation E\_[Q[-string]{}]{}\~Q\^[2/3]{}. This does not mean that the -string is even more tightly bound than a -ball with the same charge, which satisfies $E\sim Q^{3/4}$. Overall scaling of the -string always lowers its energy. The energy-length dispersion is E\_[Q[-string]{}]{}\~L\^[1/3]{} so that as the length decreases, so does the energy. Indeed the -string will shrink and reduce its energy until its length is comparable to its width, at which point the description in terms of a -string is no longer valid.
The energy-length dispersion is however, remarkable. The restoring force which tends to cause the -string to shrink F\_[Q[-string]{}]{}=-[dEdL]{}\~-[13]{}[Q\^[2/3]{}L\^[2/3]{}]{} disappears as $L$ gets very large. This is in contradistinction to the case of a topological string, such as a cosmic string or vortex loop, where the energy-length dispersion is $E_{\rm top.-string}=\mu L$, where $\mu$ is the mass per unit length, giving a constant restoring force. Nevertheless, the time scale for shrinking of a -string is not markedly different, for a large -string the mass per unit length also decreases accordingly with its length, compensating for the reduced restoring force.
The time scale for shrinking of a topological string is governed by the effective Lagrangian \_[top.-string]{}=[12]{}L[L\^24\^2]{}-L =( [LL\^28\^2]{}-L) while for a -string we have \_[Q-string]{}=[12]{}( [Q\_2\^[2/3]{}L\^[2/3]{}]{}) L[L\^24\^2]{}-( [Q\_2\^[2/3]{}L\^[2/3]{}]{}) L =Q\_2\^[2/3]{} ( [L\^[1/3]{}L\^28\^2]{}-L\^[1/3]{}) . We see that neither $\mu$ nor $\alpha Q_2^{2/3}$ play a role in the dynamics. Conservation of energy for the topological string gives +L=L\_0 while for the -string -L\^[1/3]{}=L\_0\^[1/3]{}, where $L_0$ is the initial length in each case. With the change of variables $L=v^2$ for the topological string and $L=u^6$ for the -string the equations become easily integrable, yielding the time for contraction $T_{\rm top.-string}=L_0/4\sqrt 2$ and $T_{Q\rm -string}=45L_0/32\sqrt 2$.
We will see in the next section that the -string and the -wall are both unstable to beading into arrays of smaller -balls. If the -string were to spontaneously decay into a linear array of $N$, -balls each of charge $q$, we have Q=Nq from conservation of charge, while the energy of such a final state is E=q\^[3/4]{}N=Q\^[3/4]{}N\^[1/4]{}. Evidently such a state has higher energy than the absolute minimum of energy in the sector of charge , given by the -ball of charge , ($N=1$), which has energy $E=\beta Q^{3/4}$. But it will be an intermediate state with less energy than the -string if (adopting the notation of Section 2.4 to avoid confusion) \_3Q\^[3/4]{}N\^[1/4]{}<\_2 Q\^[2/3]{}L\^[1/3]{}. This implies N<([\_2\_3]{})\^4[L\^[4/3]{}Q\^[1/3]{}]{} which gives $N/L<(\beta_2/\beta_3 )^4Q_2^{-1/3}$. As the size of each smaller -ball is approximately $R\sim L/N$ thus $R>(\beta_3/\beta_2 )^4Q_2^{1/3}$, [*i.e.,*]{} the -string will be stable against decay into a chain of $N$, -balls if the size of each -ball is smaller than $Q_2^{1/3}\sim 1/\omega$, which is just the width of the original -string.
-walls
------
We can repeat the above analysis for the case of a closed -wall of total 3 dimensional charge and with a surface area $L^2$. Then Q=L\^2Q\_1 and E\_[Q-wall]{}=L\^2E\_1 =L\^2\_1Q\_1\^[1/2]{}=\_1L\^2 (Q/L\^2)\^[(1/2)]{}=\_1 Q\^[(1/2)]{}L. This is to be compared with the energy of a topological domain wall, E\_[top.-wall]{}=L\^2 where $\mu$ is the mass per unit area. For a spherical surface the effective Lagrangian describing the dynamics of a spherical topological domain wall is \_[top.-wall]{}=[12]{}L\^2[L\^24]{}-L\^2 while for a spherical -wall we get \_[Q-wall]{}=[12]{}\_1Q\^[1/2]{}L[L\^24]{} -\_1Q\^[1/2]{}L. These yield the conservation law for the topological domain wall L\^2L\^2 +L\^2=L\_0\^2 while for the -wall LL\^2 +L=L\_0, where $L_0$ is the initial value of $L$. These are easily integrable and yield the time for contraction to be T\_[top.-wall]{}=[L\_02]{} and T\_[Q-wall]{}= for the topological domain wall and for the -wall respectively. Again we see that the time for contraction is about the same in the two cases.
If the -wall spontaneously beads up into a locally planar array of $N$ smaller -balls of charge $q$, we have Q=Nq and E=N\_3 q\^[3/4]{}=\_3Q\^[3/4]{}N\^[1/4]{}. Then energy considerations dictate \_3N\^[1/4]{}Q\^[3/4]{}<\_1Q\^[1/2]{}L, ([\_3\_1]{})\^4Q\_1\^[1/2]{}<([L\^2N]{})\^[1/2]{} which says the size of each small -ball must be greater than the width of the -wall.
Stability
=========
The general rate of decay is governed by the stability analysis of the fluctuations about the -wall or -string configuration. The total Lagrangian is expanded to second order in the fluctuations about the soliton configuration. The first order terms vanish because the soliton is a solution of the full 3 dimensional equations of motion. The second order truncation yields a generalized normal mode problem. Real frequencies lead to oscillatory behaviour while imaginary frequencies yield exponential growth or decay. In the latter case, the lifetime is taken to the inverse of the magnitude of the imaginary frequency, which is the time scale for the instability leading to beading. We will analyze the problem of the stability of a -wall for a specific potential which leads to an analytically accessible problem.
Analytical -walls {#aqw}
-----------------
As we have seen in section \[2.3\], the expression for a -ball in 1 dimension can be brought to quadrature. For many choices of the potential the resulting integral can be computed analytically [@khare]. The stability analysis of the corresponding -wall is related to the elegant theory of exactly solvable quantum mechanical systems, supersymmetric quantum mechanics and shape invariant potentials [@cks]. Consider the potential V()=\^2( [52]{}\^2 -2[\^3\_0]{}) .\[potential\] Then -V()+[12]{}\^2\^2=0 implies \^2=0=\_0. The non-trivial zero is fixed at $\phi_0$ and the -ball solution will start from this value for $\varphi$ at its centre. The equation of motion is -\^\_0(x)-\^2\_0(x) +V\^(\_0(x))=0 which becomes -\^\_0(x) +\^2(4\_0(x)-6[\_0\^2(x)\_0]{})=0.\[varphi0\] The quadrature \[quadrature\] is integrable in this case, with solution \_0(x)=\_0[sech]{}\^2 (x). \[solution\] It is remarkable that the stability problem around such a -wall is related to the corresponding suspersymmetric quantum mechanical system, as we will now show.
Second-order Lagrangian
-----------------------
The stability analysis proceeds with the second-order expansion of the total Lagrangian about the -wall configuration. We will consider an infinite planar -wall, which then gives that \[solution\] is an exact solution of the full 3 dimensional equations. This will be an arbitrarily close approximation for a large but finite closed -wall surface. The second-order Lagrangian for $\phi (\vec x,t)=e^{i\omega t}\varphi (\vec x,t)=e^{i\omega
t}(\varphi_0(x)+\varphi_1 (\vec x,t) +i\varphi_2(\vec x,t))$ is \^[\[2\]]{}=d\^3 x [12]{}(\_t\_a-\_[ab]{}\_b) (\_t\_a-\_[ab]{}\_b)+ [12]{}\_i\_a\_i\_a -[12]{}V\^(\_0)\_1\^2-[12]{}[V\^(\_0)\_0]{}\_2\^2 where $a,b=1,2$. Arbitrary variations $\varphi_1$ and $\varphi_2$ are permitted from the point of view of charge conservation. All spatial Fourier components in the orthogonal directions will correspond to displacement of charge along the wall in a periodic fashion. Charge conservation constraints will only apply to the zero frequency Fourier component, that is constant along the wall, and here the variations will have to be constrained. But this component corresponds to fluctuations that are effectively 1 dimensional, depending only on the coordinate normal to the -wall configuration. We know that the -wall is by definition stable under such fluctuations, hence we will not be concerned with these fluctuations. The equations of motion for $\varphi_a$ are |\_1-2\_2-\^2\_1-\^2 \_1 +V\^(\_0)\_1&=&0\_2+2\_1-\^2\_2-\^2 \_2 +[V\^(\_0)\_0]{}\_2&=&0. Replacing $\varphi_a(\vec x,t)\rightarrow e^{-i(\Omega t+k_1 y+k_2z) }
\varphi_a(x)$ and writing the resulting equations in matrix form yields =(\^2 -|k|\^2) which is a transcendental equation for $\Omega$.
The operator on the LHS plays the role of a Hamiltonian whose eigenvalues determine if a mode is oscillatory or exponential. The Hamiltonian is a hermitean operator for real $\Omega$, hence its spectrum is real. It is easy to see that all non-negative eigenvalues of the Hamiltonian at $\Omega =0$ will give rise to oscillatory modes of the fluctuation spectrum. Only negative modes at $\Omega =0$, for small enough values of $|\vec k|$ can yield exponential modes, but even these become oscillatory modes for large enough $|\vec k|$. The non-standard aspect of this stability problem is that the Hamiltonian also directly depends upon the fluctuation frequency $\Omega$. We must find the spectral curves $E(\Omega )$ as functions of $\Omega$ and then look for solutions of the equation $E(\Omega ) = \Omega^2-|\vec k|^2$. The RHS is an upward opening parabola for real $\Omega$ with intercept at $-|\vec k|^2$. The spectral curve of any eigenvalue of the Hamiltonian that is non-negative at $\Omega =0$ will intersect with the parabola at two distinct points. This is because for large enough $\Omega$ the spectral curves are simply asymptotic to the linear functions $E(\Omega )=\pm 2\omega\Omega $. It is only the eigenvalues of the Hamiltonian that are negative at $\Omega=0$ which can yield exponentially decaying solutions. We will see that there is exactly one such mode for this Hamiltonian with the choice of the scalar potential taken in \[potential\].
Spectrum of the Hamiltonian and supersymmetric quantum mechanics
----------------------------------------------------------------
The Hamiltonian for $\Omega =0$ is given by == . There is an obvious exact zero energy level, \^-== since $\varphi_0$ satisfies the equation \[varphi0\]. Remarkably the mode \^+==\[zeromode\] is also an exact zero mode, essentially the corresponding eigenvalue equation is the derivative of the equation of motion for $\varphi_0(x)$ given by equation \[varphi0\]. The notation will become clear below. This is, of course, not just a coincidence. The two sub-Hamiltonians which comprise ${\cal H}$ are actually supersymmetric partners and their spectra are paired except for the lowest energy mode. Indeed since \[zeromode\] is the derivative of $\varphi_0(x)$ it has a node, hence it cannot be the lowest energy eigenvalue of ${\cal H}_+$, and there has to be a negative energy mode.
### Stability equations and supersymmetric quantum mechanics[@gj] {#431}
Consider the class of supersymmetric quantum mechanics models determined by the super potential =n(x). Then the corresponding supersymmetric partner Hamiltonians are given by \_=[W]{}\^2\^=n\^2\^2-n(nn)\^2[sech]{}\^2x The two Hamiltonians ${\cal H}_\pm$ are given in terms of \_+ = A\^A\_-=AA\^. where A=[ddx]{} +[W]{}A\^=-[ddx]{} +[W]{} which gives |[H]{}\_+ &=&-[d\^2dx\^2]{} +n\^2\^2-n(n+1)\^2[sech]{}\^2x\_- &=&-[d\^2dx\^2]{} +n\^2\^2-n(n-1)\^2[sech]{}\^2x. The ground state is the wave function annihilated by $A$ A\_[-1]{}=0 which integrates easily to \_[-1]{}(x)=\_[-1]{}(0)[sech]{}\^nx. Then, if ${\cal H}_+\psi_+ =E_+\psi_+$ then ${\cal H}_-(A\psi_+) =E_+(A\psi_+)$ which shows that all energy levels are (supersymmetrically) paired except for the level annihilated by $A$. Equivalently if ${\cal H}_-\psi_-= E_-\psi_-$ the ${\cal H}_+(A^\dagger\psi_-) =E_-(A^\dagger\psi_-)$. This pairing is equivalent to the former except that there is no state annihilated by $A^\dagger$. To find all the levels of ${\cal H}_\pm$ we proceed by changing the value of $n\rightarrow n-1$. This process continues until we reach $n=1$ and the Hamiltonian ${\cal H}_-|_{n=1}=-(d^2/dx^2)+\omega^2$ for which we know the spectrum to be free, plane waves. The ground states of all intermediate Hamiltonians are given by \~[sech]{}\^mxm=1,2,,n. Then applying the appropriate strings of $A^\dagger_m$ (using an obvious notation) generates all of the bound states of ${\cal H}_\pm $ as shown in detail in [@cks]. The interesting point is that we can easily find the potential $V(\varphi_0)$ that is necessary to generate this system of supersymmetric Hamiltonians for the stability problem of the -wall. Indeed, we impose that |[V\^(\_0)\_0]{}&=&\^2n(n-1)[sech]{}\^2x +B\[consistency1\] V\^(\_0)&=& \^2n(n+1)[sech]{}\^2x +C\[consistency2\] where $B,C$ are constants to be adjusted. This yields ( [V\^(\_0)\_0]{} -B)= \^2n [sech]{}\^2x =[1n+1]{}( V\^(\_0) -C) . This equation is equivalent to -([n+1n-1]{})\_0\^[-( [n+1n-1]{})-1]{}V\^(\_0) +\_0\^[-( [n+1n-1]{})]{}V\^(\_0) +( [ B(n+1)-C(n-1)n-1]{}) \_0\^[-( [n+1n-1]{})]{}=0 after multiplying by an integrating factor. Then this equation is easily integrated to V(\_0) = [D\_0\^[( [n+1n-1]{})+1]{}]{}+( [ B(n+1)-C(n-1)4]{}) \_0\^2 +G for some integration constants $D,G$. Now imposing consistency with equations \[consistency1\] yields =D\_0\^[( [2n-1]{})]{}+( [ B(n+1)-C(n-1)2]{})=\^2n(n-1)[sech]{}\^2x +B. Taking $D=\omega^2 n(n-1)$ yields \_0=[sech]{}\^[n-1]{}x =B On the other hand from \[consistency2\] V\^(\_0)=D( [n+1n-1]{}) \_0\^[-( [n+1n-1]{})-1]{}+( [ B(n+1)-C(n-1)2]{}) =\^2n(n+1)[sech]{}\^2x +C hence taking $D=\omega^2 n(n-1)$ as before is consistent since D( [n+1n-1]{}) =\^2 n(n+1) as required. The constants however must satisfy (along with the equation from before) |[ B(n+1)-C(n-1)2]{} &=&B&=&C. These are two homogeneous linear equations for $B,C$ which only have the trivial solution except if the system is dependent. It is indeed, and we find the solution is simply $B=C$. We see that the potential is a polynomial for only the cases $n=2,3$.
Stability analysis for analytical -walls
----------------------------------------
The system studied in section \[aqw\] uses V\^(\_0(x) )=( -6\^2 [\_0(x)\_0]{}+5\^2)\_0(x) hence replacing $\varphi_0(x)=\phi_0{\rm sech}^2\omega x$ yields =( 5\^2-6\^2 [\_0(x)\_0]{})= 5\^2-6\^2[sech]{}\^2x while V\^(\_0(x) )=( 5\^2-12\^2 [\_0(x)\_0]{})=5\^2-12\^2[sech]{}\^2x. Thus the system corresponds to the case $n=3$ above with $B=C=5\omega^2$. The corresponding stability equation is | &=&(\^2 -|k|\^2)&&.\[eigenvalue\] The spectrum of this Hamiltonian at $\Omega=0$ corresponds to a single negative energy bound state with $E=-5\omega^2$ \_[-5\^2]{}\^+= two zero energy bound states with $E=0$ \_[0]{}\^+= \_[0]{}\^-= and two positive energy bound states with $E=3\omega^2$ \_[3\^2]{}\^+= \_[3\^2]{}\^-= and a continuum starting at $E> 4\omega^2$. All these modes are found in a straightforward manner using the general ideas schematized in section \[431\], for more details see [@cks].
The eigenvalue problem \[eigenvalue\] has solutions that are best expressed graphically. The spectral curve $E(\Omega)$ of each eigenvalue of the Hamiltonian on the LHS of \[eigenvalue\] must intersect the parabola $\Omega^2 -|\vec k|^2$. The eigenvalue problem for this Hamiltonian can be written algebraically as (H\_0+H\_1\_3+2\_2 )=E()where $\sigma_i$ are the usual Pauli matrices. The spectrum is doubly degenerate for each non-negative mode at $\Omega=0$ due to the underlying supersymmetry. Clearly (H\_0+H\_1\_3-2\_2 )\_3=E()\_3hence the spectrum is even in $\Omega$. However we can without loss of generality restrict $\Omega >0$ because $\varphi_i$ are in principle real: reversing the sign of $\Omega\rightarrow -\Omega $ does not generate an independent set of solutions.
We see clearly that the non-negative modes satisfy $E(\Omega )> \Omega^2 -|\vec k|^2$ for $\Omega\approx 0$, but as $\Omega\rightarrow\infty$ these spectral curves asymptote to $E(\Omega )\rightarrow \pm 2\omega\Omega$, the spectrum of the operator 2\_2. These are linear functions of $\Omega$ which for sufficiently large $\Omega$ always satisfy $\Omega^2 -|\vec k|^2>2\omega\Omega$. Hence by continuity each spectral curve must intersect at least once with the parabola, giving rise to a real solution for $\Omega$, and hence an oscillatory mode in the fluctuation spectrum. We speculate that this occurs exactly once for the spectral curve of each non-negative eigenvalue.
Only the negative mode needs to be treated differently. Here, at $\Omega=0$ and for $|\vec k|^2$ small enough, the negative mode is already below the parabola and hence there is generically no real solution for $\Omega$. We cannot compute the solution analytically, but a perturbative approach is fruitful. We can compute the spectral curve to second order in perturbation theory (the first order change vanishes). We get |E()&=& -5\^2 -4\^2\^2\_[n=1]{}\^ + &=&-5\^2 -\^2 B\^2 +o(\^4 ) where $B^2$ is a calculable positive real number that is independent of $\omega$. It is not illuminating to compute $B^2$, it is a number of order 1. Then the equation to be satisfied is -5\^2-\^2 B\^2 =\^2 -|k|\^2 that is \^2 =[-5 \^2 +|k|\^21+B\^2]{}. For $|\vec k|^2<5 \omega^2$ this gives $\Omega^2 <0$, [*i.e.,*]{} the solution for $\Omega $ is imaginary and the corresponding mode is unstable. Indeed = i hence as $|\vec k|^2\rightarrow 5 \omega^2$ from above, we get $\Omega\rightarrow 0$. The lifetime is defined as $T=1/|\Omega | $ hence for small values of $|\vec k|$ we get $T\sim (1/\omega )$. But as $|\vec k|^2\rightarrow 5 \omega^2$ we get $T\rightarrow\infty$. Thus the very long wavelength spatial Fourier modes decay with lifetime of the order of $1/\omega$ while as the wavelength $1/|\vec k|$ approaches $1/5\omega$, which is of the order of the thickness of the -wall, the lifetime becomes arbitrarily long, the modes become quasi stable. For larger $|\vec k|$ (shorter wavelength) the modes are of course strictly stable. Analysis of related, exactly solvable simple models such as $H(\Omega )\psi=(k^2 +\alpha\sigma_3+\Omega\sigma_2)\psi=\Omega^2 \psi$ yields the same kind of behavior, generically. Hence we can conclude that the fluctuation spectrum corresponds to stable oscillatory modes except for one degree of freedom. The corresponding mode is unstable only if the wavelength of the spatial Fourier mode is larger than the width of the -wall. Furthermore, the lifetime of the unstable mode is larger than $1/\omega\sim Q_1^{(1/2)}=(Q/L^2)^{(1/2)}$, which clearly can be taken to be arbitrarily large. Hence we conclude that -wall, or equally well a -string can be arbitrarily long-lived.
Discussion and conclusion
=========================
In this paper we have studied lower-dimensional -balls and extended objects based on them in models with a scalar field potential with a flat direction for large field values. The energy of a -ball of charge $Q_d$ in $d$ space dimensions varies as $Q_d^{(d/d+1)}$. Thus, as three-dimensional -balls become more and more energetically advantageous with increasing charge, the same is true (only more so) in two or in one dimension.
The extended configurations, called -strings and -walls, also have interesting energetics. A string of fixed charge and of varying length $L$, for example, has energy $E\sim L^{1/3}$, indicating a rapidly decreasing tension as the length is increased. The time scale of oscillations turns out to be of the same order as that for a topological string.
These strings and walls have a different instability not shared by topological strings: namely, density fluctuations along the string or wall can form, causing a sort of beading: the extended object would evolve into a one- or two-dimensional array of smaller -balls.
This instability was examined in detail for -walls, where supersymmetric quantum mechanics enables a rather detailed analysis. Only modes corresponding to long-wavelength fluctuations are unstable, and the time scale of growth of these fluctuations can be rather large (compared to the characteristic time scale of the -wall).
It is not clear whether or why such objects would form in the early Universe. (Unlike conventional cosmic strings, there is no topological impetus for their formation.) But if they did form, they could be long-lived, and could help contribute to structure formation in the Universe. Such a possibility merits further investigation.
We thank NSERC of Canada for financial support.
[999]{} T.D. Lee, G. C. Wick, ; R. Friedberg, T.D. Lee, A. Sirlin, , , ; R. Friedberg, T.D. Lee, . S. Coleman, , . A. Kusenko, , ; G. Dvali, A. Kusenko, M. Shaposhnikov, ; A. Kusenko, M. Shaposhnikov, ; A.Kusenko, M. Shaposhnikov, P.G. Tinyakov, . A. Cohen, S. Coleman, H. Georgi, A. Manohar, . A. F. Vakulenko, L. V. Kapitanski, [*Dokl. Akadem. Nauk USSR*]{}, [**248**]{} (1979) 210; L. Faddeev, A. J. Niemi, [*Nature*]{} [**387**]{} (1997) 58; L. Faddeev, A. J. Niemi, . S. Coleman, V. Glaser, A. Martin, . C.N. Kumar, A. Khare, [*J. Phys.*]{}[**A20**]{} (1987) [L1219]{}. F. Cooper, A. Khare, U. Sukhatme, and references therein. We understand that the results of this section were also obtained by J. Goldstone and R. L. Jaffe and J. Goldstone and R. Jackiw, as cited in [@z]. B. Zwiebach, ; J. Minahan, B. Zwiebach, . R. Battye, P. Sutcliffe, . M. Axenides, E. Floratos, S. Komineas, L. Perivolaropoulos, ; M. Axenides, S. Komineas, L. Perivolaropoulos, M. Floratos, .
[^1]: permanent address: Groupe de physique des particules, Laboratoire René-J.-A.-Lévesque, Université de Montréal, C. P. 6128, succ. centre-ville, Montréal, Québec, Canada, H3C 3J7
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abstract: 'Standard big bang nucleosynthesis (BBNS) promises accurate predictions of the primordial abundances of deuterium, helium-3, helium-4 and lithium-7 as a function of a single parameter. Previous measurements have nearly always been interpreted as confirmation of the model ([@cop95]). Here we present a measurement of the deuterium to hydrogen ratio (D/H) in a newly discovered high redshift metal-poor gas cloud at redshift $z=2.504$. This confirms our earlier measurement of D/H ([@tyt96]), and together they give the first accurate measurement of the primordial D abundance, and a ten-fold improvement in the accuracy of the cosmological density of ordinary matter. This is a high density, with most ordinary matter unaccounted or dark, which is too high to agree with measurements of the primordial abundances of helium-4 and lithium-7. Since the D/H measurement is apparently simple, direct, accurate and highly sensitive, we propose that helium requires a systematic correction, and that population II stars have less than the primordial abundance of $^7$Li. Alternatively, there is no concordance between the light element abundances, and the simple model of the big bang must be incomplete and lacking physics, or wrong.'
author:
- 'SCOTT BURLES & DAVID TYTLER'
title: Cosmological Deuterium Abundance and the Baryon Density of the Universe
---
In the standard big bang model the primordial abundances of the light elements depend on the unknown value of the cosmological baryon (ordinary matter) to photon ratio, $\eta$ ([@cop95],[@wag67]–[@sar96]). A measurement of primordial D/H, which is very sensitive to $\eta$, leads to precise predictions for the abundances of the other light elements, which can be compared with observations to test the model.
Unfortunately D is destroyed inside stars, which then eject gas with H but no D. As more gas is processed and ejected, the value of D/H in the interstellar medium of our Galaxy drops further below primordial. We look for primordial D/H in quasar absorption systems (QAS) ([@ada76],[@web91]), some of which sample gas at early epochs, and in intergalactic space where there are too few stars to destroy significant amounts of D.
The absorption system at $z=2.504$ towards QSO 1009+2956 was from 40 QSO spectra which we obtained at Lick Observatory because it showed a steep Lyman limit break and weak metal lines, which indicate a low $b$-value suitable for the detection of deuterium. Of the 15 QSOs which met these criteria and were subsequently observed with the 10-m W. M. Keck telescope, only 2 have yielded deuterium measurements ([@this]). We now describe a measurement of D/H, in this new QAS, which agrees with our previous low value, but not with measurements of a high D/H ([@son94],[@car94]). Either BBNS was inhomogeneous ([@jed95]) and D/H varies spatially, or more likely, the spectra showing high D/H were contaminated.
In Figure 1 we show nine lines in the absorption system at $z=2.504$ towards QSO 1009+2956. All six metal absorption lines are best fit by two components separated by 11 km s$^{-1}$, using the parameters given in Table 1. The dotted lines show the expected positions of the H and D lines for the gas at the well determined redshifts of these components. The two components adequately fit the blue side of the Lyman features, but an extra H component at $v_{rel}= +40$ is seen on the red side which is not significant in our measurement of D/H.
The D Lyman-$\alpha$ absorption is unsaturated and has a well determined total column density, N(D I). However the amount of D in the red component is not well determined because it is blended with both the blue D and H components. If we do not include the red D component, systematic under-absorption would occur at that velocity, so we set (D/H)$_{red}$=(D/H)$_{blue}$=(D/H)$_{total}$.
All lines from the same gas will have the same turbulent velocity dispersions $b_{tur}$, but the thermal widths will depend on the mass of each ion: $b_{therm} = 0.128\sqrt{Tm_p/m_{ion}}$ , where $m_p$ is the proton mass. We use the $b$-values of Si, C, and H in Table 1 to obtain the temperatures and $b_{tur}$ of the two components. These values are consistent with a single $b_{tur}$ for all lines, and allow us to calculate that the red component of D should have $b$(D I) $=\sqrt{b_{therm}^2 + b_{tur}^2}$ = 15.4 which we use as a constraint on the fit because we cannot measure this value from the D line. The widths of the H and D lines are dominated by thermal motions, so the lines are accurately fit by Voigt profiles ([@tyt96],[@lev96]). A simultaneous fit to the Lyman $\alpha$, $\beta$, and $\gamma$ lines in the Keck spectrum and the Lyman continuum absorption in the Lick spectrum (Figure 2) gives D/H = $3.0 \, ^{+0.6}_{-0.5} \times 10^{-5}$ ($1\sigma$ random photon and fitting errors).
A systematic increase in D/H comes from the chance superposition of weak H absorption at the expected position of D. To estimate this, we made noise-less model spectra with D and H lines at the redshifts given by the metal lines. We used the $b$ values and N(H I) from Table 1, but we vary N(D I). For each N(D I) we made $10^6$ spectra, and to each of these we added random lines to simulate the forest, using the known distribution of $z$, N(H I) and $b$-values for forest lines ([@hu95]). We then calculated the likelihood that the data came from the model spectra for each D/H. Assuming that all values of D/H are equally likely [*a priori*]{}, the expected or mean D/H is Log (D/H)$_{exp} = -4.60 \, _{-0.04}^{+0.02}$, where the errors are the standard deviations of the likelihood distribution above and below the mean. The correction from the Monte Carlo simulations changes our value from a formal upper limit to a measurement of D/H. Additional systematic errors in D/H from fitting the continuum level are estimated to be $\Delta$Log (D/H) = 0.06. We varied the continuum by 2% near (Figure 1), and by 10% near the Lyman continuum (Figure 2), and for each combination we re-fit the spectra. Including the uncertainties due to systematics, we find $$Log \left({D \over H}\right)
= -4.60 \pm 0.08 \pm 0.06$$ where the errors represent the $1\sigma$ random error followed by the systematic errors from the continuum level.
The blue side of the blue D line appears unblended, and is fit with $b$(DI) = 15.7 $\pm$ 2.1 km s$^{-1}$. This is consistent the value of $b = 13.5 \pm 0.5$ predicted from the $b_{therm}$ and $T$ given by the metal and H lines. Less than 2% of H lines have $b$-values this small or smaller ([@hu95]), so it is unlikely that this line is strongly contaminated with H. Metal lines can be this narrow, although they are usually narrower with $b < 10$ km s$^{-1}$. The D feature is unlikely to be a metal line because metal lines of this strength are nearly always accompanied by other lines, and the spectra has been searched for these lines. The [*a posteriori*]{} probability of a random H line with $b< 17.8$ and $N > 12.6$ cm$^{-2}$, to account for $>0.5$ of the D line, and redshift within 20 of that of the metal lines is $< 8 \times 10^{-4}$. A high correlation of “satellite" components on these velocity scales could increase the probability by as much as a factor of 3 ([@web91],[@hu95]).
The residual flux below the Lyman edge gives an accurate measure of all H I in this velocity region (Figure 2, [@ste90]). There are no other lines within 5000 of $z=2.50$ which have H I column densities $>10^{16}$ cm$^{-2}$, so that all of the Lyman continuum absorption must be produced by gas in the $z=2.504$ absorption system. The blue side of the , and lines are best fit if all of this H I is near the two velocity components which are seen in the metals. There could be be additional H at velocities between the metal lines and $+40$ , provided this gas has very low metal abundances (\[C/H\] $< -3.5$). However, nearly all known QAS with large N(H I) have metal abundances \[C/H\] $> -3$.
The column densities of the metals and neutral hydrogen give the metallicity and neutral fraction of the gas. Following Donahue & Shull ([@don91]), we model the system as an optically thin gas ionized by a typical QSO photoionizing spectrum given by Mathews & Ferland. We estimate the ionization parameter (the ratio of the number of photons with energies above one Rydberg to the number of atoms): U $\equiv n_\gamma/n_p \approx 10^{-2.8}$, which gives the metal abundances shown in Table 1. For a photoionization model of low metallicity components, this corresponds to a neutral hydrogen fraction of H I/H $\approx 10^{-2.5}$.
The measured D/H is consistent with normal Galactic chemical evolution ([@tyt96],[@edm94]); the destruction of D in known populations of stars can account for our D/H, that in the per-solar nebula, and the current ISM D/H. In the ISM, D/H $= 1.6 \pm 0.1 \times 10^{-5}$ ([@lin95]), for \[O/H\] = -0.25. If the destruction of D is proportional to \[O/H\], then we would expect 0.005 of the D would be destroyed for \[O/H\] $<-2.5$ ([@ste95]).
We find \[C/Si\] $\simeq -0.3$ in both components which is characteristic of low metallicity stars in the halo of our Galaxy. This suggests that the C and Si were created in “normal" supernovae. If some additional astrophysical processes destroys D, it must do so without producing more C and Si than we see, or other elements which we would have seen if they were made, and without changing the usual C/Si ratio. If primordial D/H were $24 \times 10^{-5}$ ([@son94],[@car94]) then 87% of D must be destroyed in Q1009$+$2956 and 90% in Q1937$-$8118. The level of destruction would be large and similar for \[C/H\] $= -3.0 $ to $ -2.2$, but small for Q0014+8118 with \[C/H\] $< -3.5$. Redshift is apparently not a factor.
The QAS towards Q1937-1009 in which we previouly measured a low D/H is extremely similar to Q1009+2956, except that N(H I) is larger, and the redshift is higher ([@tyt96]). We used the Lyman series lines up to 19, and the absence of flux in the Lyman continuum to constrain the N(H I), and we obtained smaller errors: Log D/H $= -4.64 \pm 0.06 \pm 0.06$.
The best estimate of the primordial D/H is obtained by taking the average of our two measurements weighted by the squares of their random errors: $$Log \left({D \over H}\right) = -4.62 \pm 0.05 \pm 0.06,$$ where the first error is the random error on the weighted mean, and the second is the larger of the systematic errors from the continuum level uncertainty. We do not add these systematic errors because they have a similar origin and the two measurments agree to within the random errors: Prob$(\chi^2 \geq 0.18) = 0.67$. In linear units, the mean is $$\left({D \over H}\right) = 2.4 \pm 0.3 \pm 0.3 \times 10^{-5}$$
In Table 2 we list all published measurements of D/H in QAS. Only the two discussed above are measurements. The others were initially presented as limits or possible detections, and they will all be biased to higher than true D/H because they do not include corrections for weak H at the position of D.
The QAS towards Q0014+8118 differs from our two in several ways. Since no metal lines were detected, the velocity structure of the cloud could be determined only by median filtering of the higher-order lines in the forest ([@son94]). The Lyman-$\alpha$ feature was considerably more complex: five components were required for an adequate fit (instead of three), with two components within 30 of the deuterium absorption line ([@car94]). The neutral hydrogen column density of the component where deuterium is measured, Log N(H I) = 16.74, is 4 and 10 times lower than the column densities in our two QAS, which reduces the sensitivity to low D/H.
Rugers & Hogan reanalyzed the published spectra of Q0014+8118, and determined that the deuterium feature is better fit with two very narrow components separated by 21 ([@rug96]). They claim that it is very unlikely that there are two narrow lines by chance at the expected position of D, but we are not convinced because there are no metal lines to constrain the velocities and their model is a poor fit to the data in many places ([@poorfit]). These data remain consistent with this D line being contaminated ([@ste94]).
Towards QSO 1202-0725, Wampler et al. ([@wam96]) find $D/H < 15 \times 10^{-5}$ at a redshift $z=4.672$. This QAS has high metallicity, \[O/H\] = 0.3, and does not look suitable for inferences of primordial D/H. Towards QSO 0420-3851, Carswell et al. ([@car96]) find a lower limit of $D/H > 2 \times 10^{-5}$ at $z=3.086$. This QAS also has a high metallicity, \[O/H\] = -1.0. The D I column is fairly well-determined, but the H I is high, N(H I) $> 10^{18}$ cm$^{-2}$, and is very uncertain. All data in Table 2 are consistent with our low D/H value, but our data are inconsistent with high D/H, unless D/H is distributed inhomogeneously ([@jed95]).
We now discuss the baryon density implied by our low D/H measurements. In Figure 3 we show how D/H fits into the standard cosmological framework. For the remainder of this paper, we present the implications of the new D/H measurements on these quantities. The current cosmological density of baryons, $\rho_b$, is given by $$\eta = 6.4 \, _{-0.4}^{+0.5} \, ^{+0.6}_{-0.5} \, ^{+0.3}_{-0.3}
\times 10^{-10},$$ where the third error is the $1\sigma$ from nucleosynthesis predictions ([@sar96]). The density of photons from the Cosmic Microwave Background ([@mat94]), $n_\gamma = 411 \, cm^{-3}$ gives: $$\rho_b \equiv \eta \, n_\gamma \, m_p = 4.4 \,
\pm 0.3 \, \pm 0.4 \, \pm 0.2
\times 10^{-31} \, g \, cm^{-3},$$ and as a fraction of the current critical density, $\rho_c = 3 H_0^2/8 \pi G$, $$\Omega_b \equiv {{\rho_b} \over {\rho_c}}
= 0.024 \, \pm 0.002 \, \pm 0.002 \, \pm 0.001 \, h^{-2}$$ where the Hubble constant, $H_0 = 100 \, h$ Mpc$^{-1}$. Since the observed density of visible baryons in stars and hot gas is about $\Omega_{LUM} = 0.003$ ([@per92]), most baryons (about 94% for $h=0.7$) are unaccounted.
In Figures 4–6, we present the predicted adundances of $^4$He, $^3$He and $^7$Li relative to hydrogen ([@sar96]). We show values of $\eta$ in the shaded region which are consistent with our D/H. The usually accepted estimates of primordial $^4$He come from measurements of low-metallicity extragalactic H II regions. Even modest chemical production can significantly change the abundance of $^4$He, so the primordial value is inferred by extrapolating $^4$He measurements to zero metallicity ([@pag92]–[@thu96]). Pagel et al. reported an inferred primordial mass fraction of $^4$He, $Y_p = 0.228 \pm 0.005$, with a 95% upper bound of 0.242 including systematic errors ([@pag92]). Recently, Thuan et al. found $Y_p = 0.241 \pm 0.003$ ([@thu96]).
These $^4$He measurements are not consistent with our D/H, which predicts $$Y_p = 0.249 \pm 0.001 \, \pm 0.001 \, \pm 0.001.$$ To quantify the discordance of D/H with the $^4$He measurements mentioned above, we form the weighted mean $\eta$ using our mean D/H and either of the $^4$He values. We weight with the random errors alone, where we now include nuclear uncertainties in quadrature in our D/H error. We find Prob($\chi^2 > 5.2) = 0.02$ for the low $Y_p$, and Prob($\chi^2 > 3.0) = 0.08$ for the high $Y_p$. If BBNS is correct, this disagreement shows that there must be systematic errors which should be identified and corrected. Reducing our D/H by its systematic error, gives Prob($\chi^2 > 3.7) = 0.05$ for the low $Y_p$ shifted up by its systematic error, and Prob($\chi^2 > 2.7) = 0.1$ for the high $Y_p$, which does not have a quoted systematic error. Even with the allowances of current systematic error estimates, D/H and Y$_p$ fail to predict the same value of $\eta$ with BBNS.
$^3$He is more complex and poorly understood because it is both created and destroyed in stars. The ratio of the D mass fraction in the ISM to primordial D is $X_{D(ISM)}/X_{D0} = 0.67 \pm 0.09$ (random, not systematic error). If $^3$He undergoes the same amount of destruction, then its primordial abundance would be 1/0.67 times that in the ISM, which is an upper limit because $^3$He is also created (e.g. primordial D is burned in stars to make $^3$He). Low mass stars may produce $^3$He copiously ([@roo76]), as is suggested by its high abundance in the ejecta of planetary nebulae ([@roo92]). If this is the case, only upper limits on $^3$He, not D + $^3$He, can be correctly applied to cosmology. However the observed abundance in Galactic H II regions is 5 – 20 times lower than expected ([@gal95]), for unknown reasons, perhaps because some of the $^3$He made in low mass stars is later destroyed, or because observations are made in H II regions which contain the ejecta of high mass stars which destroy $^3$He ([@oli95b]).
The abundance of $^7$Li in stars in the disk of our Galaxy (population I) spans three orders of magnitude, because Li is both made (e.g. by cosmic ray spallation in the interstellar medium, and in novea and supernovae) and destroyed, but warm metal-poor halo (population II) stars have similar “plateau" abundances of $^7$Li in their atmospheres. These measurements ([@spi82]-[@tho94]), shown in Figure 6, are often taken as the primordial abundance, but recent data show that this is not justified. Stars with very similar temperatures and metallicities, subgiant stars in the globular cluster M92, and turnoff stars in the halo, have significantly different $^7Li$ abundances ([@rya96]), perhaps because of different amounts of $^7$Li depletion. Ryan et al. have found that the $^7$Li abundance depends on both temperature and metallicity in ways which were not predicted and are not understood. If we are to obtain a primordial $^7$Li abundance we must either (1) understand why its abundance varies from star to star, and learn to make quantitative predictions of the level of depletion, or (2) make measurements in relatively unprocessed gas. Our D/H measurements imply that $^7$Li in population II stars has been depleted by about 0.5 dex, which may be difficult to reconcile with the near constancy of $^7$Li in the warm halo stars.
We conclude that our D/H measurements are probably the first measurements of a primordial abundance ratio of any elements, and that they give about an order of magnitude improvement in the accuracy of estimates of $\rho_b$.
Where are the baryons? The MACHO collaboration recently announced detection of baryonic dark matter in the halo of our Galaxy from 7 microlensing events towards the Large Magellanic Cloud ([@pra96]). The most likely mass of the MACHOs is 0.3 - 0.5 solar masses, from the durations of the events and the velocities implied by the model of the halo of our Galaxy. This corresponds to a standard halo model mass of 1.6$\times 10^{11}\,M_\odot$, and a large fraction of the total mass. If the halos in all galaxies are entirely baryonic MACHOS, they have $\Omega_{MACHO} > 0.13$ (90% confidence, with prefered values exceeding 0.3 for h=0.75 [@zar94]). This is more than $\Omega_b = 0.043 \, \pm 0.009$, which is predicted from our two measurements, so $<0.5$ (prefered value 0.16) of halos are baryonic. The MACHO detections strongly imply a high $\Omega_b$: if halos are 50% baryonic MACHOS then $\Omega_b > 0.065$ (90% confidence for h=0.75), enough to account for all baryons.
X-ray observations reveal that baryons in the form of hot gas contribute a fraction $f_x = (0.05 - 0.14) \, h^{-3/2}$ of the total mass of clusters of galaxies ([@whi93]). The remaining mass is some combination of unseen dark baryons (e.g. MACHOS) and any other non-baryonic dark matter, such as massive supersymmetric particles. The baryon fraction in clusters is then $f_b \geq f_x$. Since galaxy clusters are the largest bound structures in the universe, their $f_b$ should provide good estimates of the cosmological value, so the total mass density of the universe is then: $$\Omega_{total} = \Omega_b /f_b \leq (0.14 -- 0.58) \, h^{-1/2},$$ where we used $\Omega_b$ from D/H and the inequality allows for $f_b \geq f_x$ ([@afootnote]). Although our D/H measurements give a very high $\Omega_b$, it is not high enough to permit $\Omega_{total}=1$ from matter: we have $\Omega_{total} < 0.72$ from $\Omega_b =0.029 \, h^{-2}$, $h=0.6$, and $f_b = f_x$. Either the universe is open, as is now allowed by the theory of inflationary cosmology $\Omega_{total} \leq 1$ ([@lin96]), or there is a cosmological constant.
We are forced to conclude, given the present state of primordial abundance measurements, that D/H gives the only reliable constraints on $\eta$ and $\Omega_b$. The practice of finding consensus among all the light elements should be heavily weighted to D/H, because it is most sensitive to $\eta$, and because of the apparent absence of systematic uncertainties arising from destruction and creation. The discordance with other light elements does not demand alternative models to the standard big-bang, rather it begs for all measurements of primordial abundances to be made in similar pristine sites. The search for more deuterium measurements in QAS will continue, and at the current rate, $\Omega_b\,h^2$ is likely to be known to 5% by the end of the millenium.
[cccc]{} Ion & Blue Component & Red Component & Total H I & N = 17.36 $\pm$ 0.09 & N = 16.78 $\pm$ 0.11 & N = 17.46 $\pm$ 0.05 & b = 18.8 $\pm$ 0.5 & b = 21.9 $\pm$ 4.1 & D I & N = 12.84 $\pm$ 0.09 & N = 12.26 & N = 12.94 $\pm$ 0.06 & b = 15.7 $\pm$ 2.1 & b = 15.4 & Si III & N = 12.81 $\pm$ 0.06 & N = 12.55 $\pm$ 0.04 & N = 13.00 $\pm$ 0.03 & b = 4.9 $\pm$ 0.4 & b = 4.8 $\pm$ 0.8 & Si IV & N = 12.50 $\pm$ 0.03 & N = 12.05 $\pm$ 0.08 & N = 12.63 $\pm$ 0.02 & b = 4.9 $\pm$ 0.6 & b = 3.9 $\pm$ 1.5 & C II & N = 12.46 $\pm$ 0.12 & N = 12.18 $\pm$ 0.18 & N = 12.64 $\pm$ 0.08 & b = 7.0 $\pm$ 3.2 & b = 3.7 $\pm$ 4.2 & C IV & N = 12.81 $\pm$ 0.04 & N = 12.56 $\pm$ 0.06 & N = 13.00 $\pm$ 0.03 & b = 5.4 $\pm$ 0.6 & b = 5.6 $\pm$ 1.2 & T($10^4$ K) & 2.1 $\pm$ 0.1 & 2.4 $\pm$ 0.7 & b$_{tur}$ & 3.2 $\pm$ 0.4 & 2.3 $\pm$ 1.4 & & $-2.9$ & $-2.8$ & ... & $-2.5$ & $-2.6$ & ... & $-0.4$ & $-0.2$ & ...
[ccccccc]{} QSO & $z_{abs}$ & D/H$\times 10^5$ & 1$\sigma$ & logN(H I) & \[C/H\] & Reference ISM & 0.0 & 1.6 & 0.1 & 18.2 & ... & [@lin95] 1009+2956 & 2.504 & 2.5 & 0.5 & 17.46 & –2.9 & This Paper 1937$-$1009 & 3.572 & 2.3 & 0.3 & 17.94 & $-$2.2, $-$3.0 & [@tyt96] 0014+8118 & 3.320 & $\leq 19-25$ & ... & 16.7 & $<-$3.5 & [@son94],[@car94],[@rug96] 1202$-$0725 & 4.672 & $\leq 15$ & ... & 16.7 & ... & [@wam96] 0420$-$3851 & 3.086 & $\geq 2$ & ... & $\geq 18$ & -1.0 & [@car96]
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Figure Captions
===============
Figure 1: Velocity plots of Lyman $\alpha, \beta, \gamma$ (left), and all the metal lines (right) in the absorption system towards QSO 1009+2956 ($z_{em}=2.616$, V=16). Zero velocity corresponds to the redshift $z = 2.503571$, of the blue component. The vertical dotted lines also show the red component at 2.503704. The histogram represents the observed counts in each pixel normalized to the quasar continuum. The smooth curve show the best fitting Voigt profiles convolved with the instrumental resolution ([@voigt]). The Lyman lines and Si III (1206) are in the Lyman alpha forest region where there is additional absorption which we do not fit.
On the nights of December 27, 28 1995, we obtained 5.9 hours of spectra with the HIRES echelle spectrograph on the 10-m W. M. Keck Telescope ([@vog94]). Two exposures of 2.5 and 2 hours covered 3540 – 5530 Å, while a third 1.4 hour covered 3165 – 4370 Å. A 1.14 arcsec slit produced spectra with resolution of 8 kms$^{-1}$. Each exposure was accompanied by dark, quartz lamp, and Throium-Argon arclamp exposures. A standard star was observed to trace the echelle orders and remove the blaze response in the spectra. We used our standard data reduction ([@tyt96]).
Figure 2: The Lick Spectrum of QSO 1009+2956 shows the Lyman Limit due to the absorber at $z=2.504$. On November 28, 1995 we used the Kast spectrograph on the Lick 3-m telescope to obtain 1.9 hours of integration covering 3100 Å- 5950 Å. The spectra were calibrated to vacuum heliocentric wavelengths, and optimally extracted.
The smooth line shows the convolved Voigt proflies of the higher order Lyman lines calculated with the parameters in Table 1. The optical depth, measured by the ratio of flux blueward and redward of 3200 Å, constrains the total column density of neutral hydrogen.
Figure 3: A representation of the flow of information in the standard cosmological model. Boxes show measurements, and circles show theories and derived quantities. Most references to quantities shown are discussed in the text. The range for the Hubble Constant, $H_0$, is taken from Mould et al. ([@mou95]). When two numbers are shown, one is for $h=0.98$ and the other is $h=0.64$. When $\Omega_b$ is used, we add or subtract all three errors to enlarge the range.
Figure 4: The BBNS predicted primordial abundances of D and $^3$He as a function of $\eta$ and $\Omega_b\,h^2$ ([@sar96]). The lower two rectangles are defined by the $1\sigma$ random plus systematic errors on D/H towards Q1009+2956 and Q1937–1009. The shaded region is from our mean D/H value, with $1\sigma$ random errors from the quadratic sum of our random error plus the $\sigma$ nuclear error. We then add on the systematic error. The upper limit shows D/H towards Q0014+8118.
Figure 5: As Figure 4 but for $Y_p$, the mass fraction of $^4$He. Dashed rectangles show the bounds (1$\sigma$ statistical plus systematic errors) from recent measurements of $^4$He ([@pag92],[@thu96]). The lack of intersection with the shaded region illustrates the inconsistency of D/H with Helium-4 measurements. The lower rectangle represents the estimates of $^4$He deduced by Pagel et als., and it includes in its range a systematic underestimation of $Y_p$ of 0.004.
Figure 6: As Figure 4 but $^7$Li/H The Spite and Thorburn $^7$Li plateau from measurements of population II stars are shown as dashed and dot-dashed lines respectively. ([@spi82],[@tho94]). Significant depletion of the surface $^7$Li abundance is the likely source of the discordances. ([@rya96]). The dotted line shows upper limits from lithium measurements in population I stars.
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author:
- 'Thierry Dumas, Franck Galpin, and Philippe Bordes [^1]'
bibliography:
- 'iterative\_training\_of.bib'
title: Iterative training of neural networks for intra prediction
---
[^1]: The authors are with Interdigital, 975 avenue des Champs Blancs, 35576, Cesson-S[é]{}vign[é]{}, France (e-mails: [[email protected]]([email protected]), [[email protected]]([email protected]), and [[email protected]]([email protected])).
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abstract: 'We investigate a competition of tendencies towards ferromagnetic and incommensurate order in two–dimensional fermionic systems within functional renormalization group technique using temperature as a scale parameter. We assume that the Fermi surface (FS) is substantially curved, lies in the vicinity of van Hove singularity points and perform an account of the self–energy corrections. It is shown that the character of magnetic fluctuations is strongly asymmetric with respect to the Fermi level position relatively to van Hove singularity (VHS). For the Fermi level above VHS we find at low temperatures dominant incommensurate magnetic fluctuations, while below VHS level we find indications for ferromagnetic ground state. In agreement with the Mermin–Wagner theorem, at finite temperatures and in small magnetic fields we obtain rather small magnetization, which appears to be a power–law function of magnetic field. It is found that the FS curvature is slightly increased by correlation effects, and the renormalized bandwidth decreases at sufficiently low temperatures.'
author:
- 'P. A. Igoshev$^{a,b}$, V. Yu. Irkhin$^{a}$, and A. A. Katanin$^{a,b}$'
title: 'Magnetic fluctuations and self–energy effects in two–dimensional itinerant systems with van Hove singularity of electronic spectrum'
---
Introduction
============
During last two decades, the problem of magnetic fluctuations in itinerant–electron compounds attracts substantial interest in connection with physics of layered systems. In copper–oxide high–temperature superconductors the importance of magnetic fluctuations is seen from the proximity of antiferromagnetically ordered region to the region of superconducting state with pronounced incommensurate magnetic fluctuations[@La; @Keimer]. Another impressive example of peculiar magnetic properties of low–dimensional systems is provided by the family of layered strontium ruthenates, which are pure metallic and incidentally have a very complex magnetic behavior.
Neutron scattering [@Sidis] of the one–layer paramagnetic metal Sr$_{2}$RuO$_{4}$ (which at low temperatures $T\lesssim 1.5$ K becomes an unconventional, most likely $p$–wave, superconductor[@Ru_review; @Sc_review]) reveals an incommensurate character of magnetic fluctuation spectrum, with the largest contribution corresponding to the wave vector $\mathbf{Q}_{s}\sim (2/3)(\pi,\pi)$ and smaller one with the wave vector $\mathbf{Q}_{l}\sim (1/3)(\pi,\pi)$. It was argued[@Mazin] that incommensurate short–wave magnetic fluctuations characterized by the wavevector $\mathbf{Q}_{s}$ are expected to originate from $\alpha $ and $\beta $ bands (produced by $d_{xz}$ an $d_{yz}$ orbitals and generating quasi–one–dimensional sheets of FS), while the long–wave magnetic fluctuations with the wavevector $\mathbf{Q}_{l}$ originate most likely from the $\gamma$ band, produced by $d_{xy}$ orbitals and generating quasi–two–dimensional sheet of FS. Moreover, magnetic properties of this compound can be controlled by the substitution effect: being doped by lantanium, Sr$_{2-y}$La$_{y}$RuO$_{4}$ exhibits enhancement of magnetic susceptibility. This enhancement is likely related to the position of the Fermi level of $\gamma$ band: The fit of the tight–binding spectrum to experimentally determined Fermi surfaces allows *to expect* the onset of the ferromagnetic order at $y=0.27$, related to the elevation of the Fermi level of the $\gamma$ band towards a van Hove singularity (VHS). However, no ferromagnetic order was found for this doping (Ref. ), despite the VHS at the Fermi level, revealed by ARPES [@TransverseVH].
Bilayer ruthenate compound Sr$_{3}$Ru$_{2}$O$_{7}$ also provides a number of interesting physical properties, in particular incommensurate magnetic fluctuations [@Capogna] and metamagnetism [@Multi-Ru]; the latter was assumed to be related to the proximity of Fermi level to VHS[@Binz; @Yuki]. The effect of substitution of Sr by Ca in (Sr$_{1-x}$Ca$_{x}$)$_{3}$Ru$_{2}$O$_{7}$ was studied in Ref. . Alhtough the Wilson ratio approaches $\sim 700$ for $x=0.2$, and the system becomes nearly ferromagnetic in the temperature interval ($3\lesssim T\lesssim 10$ K), further lowering of temperature does not result in the long–range ferromagnetic ordering, but instead forces the system to freeze into a spin–glass state. Therefore, an important problem related to the magnetic properties of metallic two–dimensional compounds is revealing the conditions, for which the ferromagnetic order, which is observed to be suppressed experimentally, is stable.
From the theoretical point of view, magnetic properties of layered systems are closely related to their electronic structure. It is justified by ARPES experiments and first principle calculations of above–mentioned compounds that the electronic spectrum is quasi–two–dimensional and can be fit using nearest ($t$) and next–nearest neighbour ($t^{\prime }$) hoppings. To investigate an interplay of different magnetic states in itinerant layered systems it is convenient to consider the two–dimensional (2D) one–band Hubbard model. Despite the fact that this model is strongly simplified as compared to the situation in real compounds, it catches an important dependence of ground–state ordering on the form of the electronic spectrum, in particular on the $t^{\prime }/t$ ratio and band filling. On the other hand, the electronic spectrum containing nearest and next–nearest neighbor hoppings retains VHS which is often present in the density of states (DOS) of real compounds.
A large enough value of DOS at the Fermi level, occuring due to VHS, leads to ferromagnetic ground state according to the Stoner theory[@Penn]. This approach, however, neglects magnetic fluctuations, in particular, of incommensurate type. A generalization of the Stoner theory to treat the spin spiral instability and Néel antiferromagnetism reveals that in a major part of the phase diagram the spiral phase is even more preferable than the ferromagnetic one [@Schulz; @OurIC]. In particular, the phases with the wave vectors $\mathbf{Q}=(Q,Q)$ and $\mathbf{Q}=(Q,\pi)$ were found to compete strongly with the ferromagnetic phase[@OurIC]. The critical value of the Hubbard interaction $U_{\mathrm{c}}$ for the ferromagnetic instability is smallest in the region with the Fermi level below VHS, where the competition of ferromagnetic and $(Q,\pi)$ phase is observed. On the other hand, in the vicinity and well above VHS the ferromagnetic phase is to a great extent suppressed by the incommensurate $(Q,Q)$ phase.
To investigate the stability of ferromagnetic order with respect to long–wave magnetic fluctuations in itinerant 2D electronic systems at moderate Coulomb interaction, the *spin-fermion* model was recently used [@QS]. The static character of magnetic fluctuations in 2D systems with magnetically ordered ground state allowed to determine the boundaries of ferromagnetic region and the most important magnetic instability, competing with ferromagnetism. It was argued that the region of stability of ferromagnetism shrinks substantially, especially at VH filling, due to competition with incommensurate magnetic fluctuations, in qualitative agreement with mean–field[@OurIC] and variational results [@Okabe].
In the limit of large Coulomb repulsion the stability of saturated ferromagnetic ground state with respect to other magnetic structures was investigated starting from 1960s. In particular, Nagaoka has found that saturated ferromagnetic state is unstable with respect to the spin–wave excitation with the wave vector $\mathbf{Q}=(\pi,\pi)$, which implies that it competes with commensurate Néel antiferromagnetic order [@Nagaoka]. This result was later supplemented with variational principle results indicating that at least in part of the phase diagram the wave vector of the corresponding magnetic instability is incommensurate[@Okabe].
Dynamical mean field theory (DMFT) provides a tool for unbiased account of the local electronic correlation effects and determination of the favorable type of ground–state magnetic order for arbitrary interaction strength. Within DMFT the magnetic phase diagram for the Hubbard model was investigated on a Bethe [@Pruschke1; @Pruschke2] and square [@Kotliar] lattices with nearest and next–nearest neighbor hoppings. For small $t'$ away from half filling and sufficiently large Coulomb interaction ferromagnetic, paramagnetic and incommensurate magnetic (the region where commensurate solutions of the DMFT equations do not converge) phases were found. At large enough $t'\lesssim t/2$ the ferromagnetism is found to be dominating in a broad region of parameters [@Pruschke2].
The non–local electronic correlations for saturated ferromagnetic state can be treat in particular in the Kanamori’s T–matrix approximation (which is exact in the limit of small carrier concentration), applied to this problem in Ref. . However, this approach does not account for contributions from the other channels of electron interaction, which are important at not too small concentrations. The possibility of incommensurate magnetic ordering is also missed by this approximation.
To study the interplay of magnetic and electronic properties and their effect on the possibility of ferromagnetic instability, we focus here on the functional remormalization group (fRG) technique, which is a powerful tool for treatment of correlation effects (see Ref. for the introduction in the application to the Hubbard model). In the pioneer study of Ref. , the use of the Polchinski’s form of fRG technique allowed to construct the phase diagram of the nearest–neighbor hopping Hubbard model ($\mu-T$ plane). The momentum cut–off scheme of the Wick–ordered version of fRG equations was employed to obtain the ground–state instability type at small $t^{\prime }/t$ in Ref. . Later the temperature [@Salmhofer; @Kampf; @Katanin] and Hubbard interaction [@Honerkamp] were used as flow parameters in one–particle irreducible (1PI) investigations of the $t-t'$ Hubbard model. Main result of these studies is that the small nearest–neighbor–hopping $(t^{\prime }/t\leq0.2)$ favors the antiferromagnetic instability at VH filling, moderate one $(0.2\leq t^{\prime }/t\leq 0.35)$, the $d$–wave supeconducting instability, rather large $t^{\prime }/t>0.35$ corresponds to ferromagnetic instability. Away from van Hove band filling the competition of antiferromagnetic and superconducting instabilities is obtained at small $t^{\prime }/t$, while at larger values $t^{\prime }/t\gtrsim 0.35$ the competition of ferromagnetic instability and $p$–wave superconductivity is observed [@Salmhofer; @Kampf; @Honerkamp].
The essential shortcoming of the above reviewed approaches [@Salmhofer; @Honerkamp; @Halboth; @Kampf] is that these do not consider the self–energy corrections to the electronic Green function. Hence, important non–trivial effects of renormalization of the electronic spectrum in the vicinity of magnetic phase transition (or in the regime with strong magnetic fluctuations) can be missed, provided that the Fermi level is near VHS. On the other hand, spin–dependent self–energy corrections provide the mechanism of the response to magnetic field, which is crucial in the context of invesigation of ferromagnetic instability. Note that only antiferromagnetic and superconducting phases were considered previously within the combination of fRG and mean–field approach (Ref. ) and within the fRG approach in the symmetry broken phase [@Lauscher]. Considering ferromagnetic instability poses a problem of scale–dependent FS, which was theoretically elaborated only in the self–adjusting Polchinski and Wick–ordered schemes, proposed in Ref. .
In this paper we present a study of the evolution of magnetic and electronic properties with decreasing temperature within 1PI fRG in zero and small finite magnetic field. We treat accurately the FS problem, including movement of projecting points and momentum dependence of the self–energy. The plan of the paper is the following. In Sec. \[model\_diagram\] we introduce the model and review earlier approaches to ferromagnetic instability at weak and intermediate coupling. In Sec. \[SE\_fRG\] we introduce the details of our novel fRG approach to take into account the fluctuation effects, retaining the electronic self–energy. In Sec. \[Results\] we present and discuss the numerical results. In Sec. \[Conclusions\] we present the conclusions.
The model and earlier approaches {#model_diagram}
================================
We consider the Hubbard model with the action $$\mathcal{S}=\beta \sum_{k\sigma }(-\mathrm{i}\nu_{n}+\epsilon_{\mathbf{k}}-\mu -h\sigma )c_{k\sigma }^{+}c_{k\sigma }+\frac{\beta
U}{4N}\sum_{k_{1}k_{2}k_{3}k_{4}\sigma \sigma ^{\prime }}c_{k_{1}\sigma
}^{+}c_{k_{2}\sigma ^{\prime }}^{+}c_{k_{3}\sigma ^{\prime }}c_{k_{4}\sigma
}\delta_{k_{1}+k_{2},k_{3}+k_{4}},
\label{Action}$$where $\beta=1/T$ is inverse temperature, $N$ is the number of lattice sites, $h$ is a magnetic field directed along $z$ axis, measured in units of Bohr magneton, $\epsilon_{\mathbf{k}}$ is an electronic spectrum, $\mu$ is the chemical potential, $U $ is the Hubbard on–site interaction, $\delta$ is the Kronecker $\delta$–symbol. The sums in Eq. (\[Action\]) are taken over 4–vectors $k=(\mathrm{i}\nu_{n},\mathbf{k}),$ where $\nu_{n}=\pi(2n+1)T$ are the fermionic Matsubara frequencies ($n\in \mathbb{Z}$). We consider electronic dispersion on the square lattice $$\epsilon_{\mathbf{k}}=-2t(\cos k_{x}+\cos k_{x})+4t^{\prime }(\cos
k_{x}\cos k_{y}+1),
\label{bare_ek}$$where $t,t^{\prime}>0$ are hopping parameters. Such a form of the spectrum corresponds to VHS at zero energy. We fix the ratio $t^{\prime}/t=0.45$.
To gain insight into possible types of magnetic order, we consider the magnetic susceptibility $\chi_{\mathbf{q}}(\omega)$ in paramagnetic state [@Moriya], $$\chi_{\mathbf{q}}^{-1}(\omega)=\phi_{\mathbf{q}}^{-1}(\omega)-U,
\label{RPA}$$ where $\phi_{\mathbf{q}}(\omega )$ is the susceptibility, irreducible in particle–hole channel. The paramagnetic state is unstable with respect to the formation of incommensurate state with the wave vector $\mathbf{Q}$, provided that the maximum of irreducible susceptibility $\phi_{\mathbf{q}}(0)$ is at $\mathbf{q=Q}$ and $\chi_{\mathbf{Q}}(0)$ diverges. A critical interaction for stability of such an incommensurate magnetic state $U_{\rm c}=\phi_{\mathbf{Q}}^{-1}(0)$. For the sake of simplicity let us consider the random phase approximation (RPA) where $\phi_{\mathbf{q}}(\omega)$ coincides with the non–interacting spin susceptibility $$\chi^0_{\mathbf{q}}({{\rm i}}\omega_n)=-\frac{T}{N}\sum_{\mathbf{k}\nu
_{n}}G_{\mathbf{k}}^{0}(\mathrm{i}\nu_{n})G_{\mathbf{k}+\mathbf{q}}^{0}(\mathrm{i}\nu_{n}+\mathrm{i}\omega_{n}),
\label{Sus0}$$ where $G_{\mathbf{k}}^{0}(\mathrm{i}\nu_{n})=(\mathrm{i}\nu_{n}-\epsilon_{\mathbf{k}}+\mu)^{-1}$ is the non–interacting electronic Green function. In Fig. \[Sus\] we present zero–temperature momentum profile of static non–interacting magnetic susceptibility $\chi^0_{\mathbf{q}}(\omega=0)$. Below VH filling the competition between incommensurate magnetic structure with large wave vector and a variety of incommensurate instabilities with small magnetic wave vectors is observed (see Fig. \[Sus\]a). On the other hand, above VH filling, the ferromagnetic instability competes with the long–wave incommensurate magnetic instability [@QS](see Fig. \[Sus\]b). This consideration shows that for large enough $t'$ the tendency to incommensurate magnetic ordering originates from the geometry of the FS.
![Momentum dependence of the static magnetic susceptibility of free electrons ${\chi^0_{\mathbf{q}}(\omega=0)}$, ${t^{\prime}=0.45t}$: a) Fermi level below VH filling (${\mu=-0.05t}$); b) Fermi level above VH filling (${\mu=0.05t}$)[]{data-label="Sus"}](below "fig:")![Momentum dependence of the static magnetic susceptibility of free electrons ${\chi^0_{\mathbf{q}}(\omega=0)}$, ${t^{\prime}=0.45t}$: a) Fermi level below VH filling (${\mu=-0.05t}$); b) Fermi level above VH filling (${\mu=0.05t}$)[]{data-label="Sus"}](above "fig:")
The competition of ferromagnetism and incommensurate magnetic ordering beyond mean–field approximation was discussed within the quasistatic approach (QSA) in Ref. . Under the assumption of the ground state to be ferromagnetically ordered, it was found that magnetic fluctuations can reconstruct the momentum dependence of the susceptibility only at finite electronic interaction, stabilizing the ferromagnetic ground state with respect to incommensurate magnetic fluctuations. Above the critical value of the interaction, the maximum of $\phi_{\mathbf{q}}(0)$ is reached at $\mathbf{q=0}$. The corresponding result for the boundary separating the ferromagnetic and $(Q,Q)$ incommensurate instabilities is presented below in Fig. \[Summary\].
The effort of direct account of the electron correlation effects was performed in temperature–flow fRG study[@Salmhofer; @Kampf] ($\Sigma=0$–fRG). While the problem of ferromagnetism formation was investigated in zero magnetic field, the self–energy renormalization was neglected, which makes the FS scale independent and the chemical potential not renormalized. Formal consequences of this assumption are discussed in Sec. \[Relation\_prev\].
Functional renormalization group with self–energy corrections {#SE_fRG}
=============================================================
Present study is an extension of the study of Ref. . Contrary to previous approaches, we (i) do not neglect the self–energy corrections $\Sigma$ to the electronic Green’s function $G$ and account for their momentum dependence, which results in the renormalization of spectrum parameters $t$ and $t^{\prime }$ and moving FS, (ii) partially take into account momentum dependence of the vertex inside the patches, which allows to account for the scale dependence of the Fermi surfaces $\mathcal{F}_{\sigma }(s)$, corresponding to different spin projections $\sigma=\pm1$.
fRG equations {#fRG_equations}
-------------
We use the 1PI fRG equations [@fRG] in the truncation of Ref. $$\dot{\Gamma}=\left. \Gamma \ast \frac{d}{ds}(G_sG_s)\ast \Gamma \right\vert_{\mathrm{pp}}+\left. \Gamma \ast \frac{d}{ds}(G_sG_s)\ast \Gamma \right\vert_{\mathrm{ph}}+\left. \Gamma \ast \frac{d}{ds}(G_sG_s)\ast \Gamma \right\vert_{\mathrm{ph1}}, \label{fRG_symbol1}$$ $$\dot{\Sigma}=\Gamma \ast S,
\label{fRG_symbol2}$$ where $\Gamma $ is the 1PI 4–vertex function (we refer to it as a vertex below), pp, ph, and ph1 denote particle–particle and two independent particle–hole channels, $\Sigma $ is the self–energy, $S=-G_s^2dG^{-1}_{0,s}/ds$ is the single–scale propagator, $G_{0,s}$ and $G_s$ are appropriately rescaled non–interacting and interacting Green’s functions, momentum arguments and spin indices are omitted for brevity. The asteriscs denote the summation over internal 4–momentum and spin indices corresponding to specified channel, dots denote the derivatives, taken with respect to the scale parameter $s$.
If FM order parameter or magnetic field are directed along $z$ axis, the SU(2) symmetry is broken. The remaining axial symmetry results in the self–energy which is diagonal in spin indices $\sigma,\sigma^{\prime}$ $$\Sigma_{\sigma \sigma ^{\prime }}(k)=\Sigma_{k\sigma}\delta_{\sigma\sigma ^{\prime}},$$and the vertex function components are nonzero provided that $\sigma_1+\sigma_2=\sigma_3+\sigma_4$. We consider the vertex component $\Gamma_{\sigma_{1}\sigma_{2};\sigma _{1}\sigma _{2}}$ (for brevity we denote it as $\Gamma _{\sigma _{1}\sigma _{2}}$), the others can be obtained using exact relation $$\Gamma_{\sigma_{1}\sigma_{2};\sigma_{3}\sigma_{4}}(k_{1},k_{2};k_{3},k_{4})=-\Gamma_{\sigma _{2}\sigma _{1};\sigma
_{3}\sigma_{4}}(k_{2},k_{1};k_{3},k_{4})=\Gamma_{\sigma_{2}\sigma_{1};\sigma_{4}\sigma_{3}}(k_{2},k_{1};k_{4},k_{3}).$$ Another useful property, which is a consequence of the symmetry with respect to the combination of time–reversal transformation and $\sigma\rightarrow-\sigma$ transformation, is $$\Gamma_{\sigma _{1}\sigma _{2};\sigma _{3}\sigma_{4}}(k_{1},k_{2};k_{3},k_{4})=\bar{\Gamma}_{\sigma_{3}\sigma_{4};\sigma
_{1}\sigma _{2}}(k_{3},k_{4};k_{1},k_{2}).$$
Neglecting the frequency dependence of the vertex and self–energy the fRG equations can be written in the following explicit form: $$\begin{gathered}
\label{DGamma}
\dot{\Gamma}_{\sigma _{1}\sigma_{2}}(\mathbf{k}_{1},\mathbf{k}_{2};\mathbf{k}_{3},\mathbf{k}_{4})=\\
=(1-\delta_{\sigma_{1}\sigma _{2}}/2)\frac1{N}\sum_{\mathbf{p}}\Gamma _{\sigma
_{1}\sigma _{2}}(\mathbf{k}_{1},\mathbf{k}_{2};\mathbf{p},\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{p})\mathcal{L}_{\sigma _{1}\sigma _{2}}^{\mathrm{pp}}(\mathbf{p},\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{p})\Gamma _{\sigma
_{1}\sigma _{2}}(\mathbf{p},\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{p};\mathbf{k}_{3},\mathbf{k}_{4}) \\
-\frac1{N}\sum_{\mathbf{p}\sigma }\Gamma _{\sigma _{1}\sigma }(\mathbf{k}_{1},\mathbf{p};\mathbf{k}_{3},\mathbf{k}_{1}-\mathbf{k}_{3}+\mathbf{p})\mathcal{L}_{\sigma \sigma }^{\mathrm{ph}}(\mathbf{p},\mathbf{k}_{1}-\mathbf{k}_{3}+\mathbf{p})\Gamma _{\sigma \sigma _{2}}(\mathbf{k}_{1}-\mathbf{k}_{3}+\mathbf{p},\mathbf{k}_{2};\mathbf{p},\mathbf{k}_{4}) \\
+\delta_{\sigma _{1}\sigma _{2}}\frac1{N}\sum_{\mathbf{p}\sigma }\Gamma _{\sigma
_{1}\sigma }(\mathbf{k}_{1},\mathbf{k}_{2}-\mathbf{k}_{3}+\mathbf{p};\mathbf{k}_{4},\mathbf{p})\mathcal{L}_{\sigma \sigma }^{\mathrm{ph}}(\mathbf{p},\mathbf{k}_{2}-\mathbf{k}_{3}+\mathbf{p})\Gamma _{\sigma \sigma _{2}}(\mathbf{p},\mathbf{k}_{2};\mathbf{k}_{2}-\mathbf{k}_{3}+\mathbf{p},\mathbf{k}_{3})] \\
+(1-\delta _{\sigma _{1}\sigma _{2}})\frac1{N}\sum_{\mathbf{p}}\Gamma _{\sigma
_{1}\sigma _{2}}(\mathbf{k}_{1},\mathbf{k}_{2}-\mathbf{k}_{3}+\mathbf{p};\mathbf{p},\mathbf{k}_{4})\mathcal{L}_{\sigma _{1}\sigma _{2}}^{\mathrm{ph}}(\mathbf{p},\mathbf{k}_{2}-\mathbf{k}_{3}+\mathbf{p})\Gamma _{\sigma
_{1}\sigma _{2}}(\mathbf{p},\mathbf{k}_{2};\mathbf{k}_{3},\mathbf{k}_{2}-\mathbf{k}_{3}+\mathbf{p})\end{gathered}$$ $$\dot{\Sigma}_{\mathbf{k}\sigma }=\frac{1}{2N}\sum_{\mathbf{p}\sigma ^{\prime
}}\Gamma _{\sigma \sigma ^{\prime }}(\mathbf{k},\mathbf{p};\mathbf{k},\mathbf{p})\left[ f_{\mathbf{p}\sigma ^{\prime }}+(2\varepsilon _{\mathbf{p}\sigma ^{\prime }}-\Sigma _{\mathbf{p}\sigma ^{\prime }})f_{\mathbf{p}\sigma
^{\prime }}^{\prime }\right] -\dot{\mu}\sum_{\mathbf{p}\sigma ^{\prime
}}\Gamma _{\sigma \sigma ^{\prime }}(\mathbf{k},\mathbf{p};\mathbf{k},\mathbf{p})f_{\mathbf{p}\sigma ^{\prime }}^{\prime }-\Sigma _{\mathbf{k}\sigma }/2, \label{DSigma}$$where $$\begin{gathered}
\mathcal{L}_{\sigma \sigma ^{\prime }}^{\mathrm{pp}}(\mathbf{k},\mathbf{k}^{\prime })=\left[ \frac{\varepsilon _{\mathbf{k}\sigma }f_{\mathbf{k}\sigma
}^{\prime }+\varepsilon _{\mathbf{k}^{\prime }\sigma ^{\prime }}f_{\mathbf{k}^{\prime }\sigma ^{\prime }}^{\prime }}{\varepsilon _{\mathbf{k}\sigma
}+\varepsilon _{\mathbf{k}^{\prime }\sigma ^{\prime }}}+\frac{(\dot{\Sigma}_{\mathbf{k}\sigma }-\dot{\mu})f_{\mathbf{k}\sigma }^{\prime }+(\dot{\Sigma}_{\mathbf{k}^{\prime }\sigma ^{\prime }}-\dot{\mu})f_{\mathbf{k}^{\prime
}\sigma ^{\prime }}^{\prime }}{\varepsilon _{\mathbf{k}\sigma }+\varepsilon
_{\mathbf{k}^{\prime }\sigma ^{\prime }}}\right. \label{Lpp} \\
-\left. \frac{(f_{\mathbf{k}\sigma }+f_{\mathbf{k}^{\prime }\sigma ^{\prime
}}-1)(\dot{\Sigma}_{\mathbf{k}\sigma }+\dot{\Sigma}_{\mathbf{k}^{\prime
}\sigma ^{\prime }}-2\dot{\mu})}{(\varepsilon _{\mathbf{k}\sigma
}+\varepsilon _{\mathbf{k}^{\prime }\sigma ^{\prime }})^{2}}\right],\end{gathered}$$$$\label{Lph}
\mathcal{L}_{\sigma \sigma ^{\prime }}^{\mathrm{ph}}(\mathbf{k},\mathbf{k}^{\prime })=-\left[ \frac{\varepsilon _{\mathbf{k}\sigma }f_{\mathbf{k}\sigma }^{\prime }-\varepsilon _{\mathbf{k}^{\prime }\sigma ^{\prime }}f_{\mathbf{k}^{\prime }\sigma ^{\prime }}^{\prime }}{\varepsilon _{\mathbf{k}\sigma }-\varepsilon _{\mathbf{k}^{\prime }\sigma ^{\prime }}}+\frac{(\dot{\Sigma}_{\mathbf{k}\sigma }-\dot{\mu})f_{\mathbf{k}\sigma }^{\prime }-(\dot{\Sigma}_{\mathbf{k}^{\prime }\sigma ^{\prime }}-\dot{\mu})f_{\mathbf{k}^{\prime }\sigma ^{\prime }}^{\prime }}{\varepsilon _{\mathbf{k}\sigma
}-\varepsilon _{\mathbf{k}^{\prime }\sigma ^{\prime }}}-\frac{(f_{\mathbf{k}\sigma }-f_{\mathbf{k}^{\prime }\sigma ^{\prime }})(\dot{\Sigma}_{\mathbf{k}\sigma }-\dot{\Sigma}_{\mathbf{k}^{\prime }\sigma ^{\prime }})}{(\varepsilon
_{\mathbf{k}\sigma }-\varepsilon _{\mathbf{k}^{\prime }\sigma ^{\prime
}})^{2}}\right].$$We have introduced the renormalized electronic spectrum $$\varepsilon _{\mathbf{k}\sigma }=\epsilon_{\mathbf{k}}-\mu-h\sigma+\Sigma_{\mathbf{k}\sigma},
\label{renorm_e}$$and $f_{\mathbf{p}\sigma }\equiv f(\varepsilon _{\mathbf{p}\sigma })=\frac{1}{2}(1-\tanh (\beta \varepsilon _{\mathbf{p}\sigma }/2)).$
Below we consider temperature cutoff [@Salmhofer] with $s=\log(t/T)$. The single–scale propagator has a form $$S_{k\sigma }^{T}=-G_{k\sigma }^{2}\left( \frac{\mathrm{i}\nu _{n}-\epsilon_{\mathbf{k}}-h\sigma+\mu}{2T^{1/2}}+T^{1/2}\frac{d\mu }{dT}\right)
\label{S_T}$$ where $G_{k\sigma}=(\mathrm{i}\nu_{n}-\varepsilon _{\mathbf{k}\sigma })^{-1}$. In the present study we choose temperature independent bare chemical potential ($d\mu/dT=0$), although the temperature dependence of $\mu$ can be adjusted, e. g., to keep the number of particles fixed[@Comment]. The equations (\[DGamma\]) and (\[DSigma\]) are basic for 1PI fRG approach since they can, in principle, determine the renormalization of the electronic spectrum and interaction at any temperature. This system of equations should be supplemented by initial conditions at $s=-\infty(T=+\infty )$. In the infinite–temperature limit correlation effects are absent, and one obtains for the self–energy $\Sigma_{\mathbf{k}\sigma }=U/2$ and interaction $\Gamma_{\sigma_{1}\sigma_{2}}(k_{1},k_{2};k_{3},k_{4})=U(1-\delta_{\sigma_{1}\sigma_{2}}).$
The fRG equations (\[DGamma\]),(\[DSigma\]) form the system of ordinary *functional* differential equations. To solve them numerically, one has to introduce some approximation procedure to reduce the considered system to a finite system of ordinary equations. Below we present the procedure of numerical solution in details.
The self–energy ansatz and Fermi surface
----------------------------------------
The first step is the use of common patching scheme [@Salmhofer] to avoid dealing with system of functional equations. However, we do not neglect the momentum dependence of the self–energy (and vertices) inside the patches, in particular Eq. (\[DSigma\]) is fully employed. \[self-energy\_ansatz\] We assume that the self–energy has the form $$\label{se_ansatz}
\Sigma_{\mathbf{k}\sigma }=-2\delta t_{\sigma }(\cos k_{x}+\cos
k_{y})+4\delta t_{\sigma }^{\prime }\cos k_{x}\cos k_{y}+\Sigma_{\sigma}.$$where the parameters $\delta t_{\sigma },\delta t_{\sigma }^{\prime}$ and $\Sigma_{\sigma}$ are determined as follows. While solving numerically the system of fRG equations, at each step we calculate the values of $\dot{\Sigma}_{\mathbf{k}\sigma }$ on two sets of ($\sigma$–dependent) projecting points (PPs, see for details Appendix). For considered $\mathbf{k}\in$ PPs the linear regression procedure is used, $$\dot{\Sigma}_{\mathbf{k}\sigma }\rightarrow -2\delta \dot{t}_{\sigma }(\cos
k_{x}+\cos k_{y})+4\delta \dot{t}_{\sigma }^{\prime }\cos k_{x}\cos k_{y}+\dot{\Sigma}_{\sigma }.$$In this way we determine the flow of unknown quantities $\delta t_{\sigma},\delta t_{\sigma }^{\prime }$ and $\Sigma_{\sigma}$ in Eqs. (\[DGamma\]),(\[DSigma\]).
Such a choice efficiently reduces the number of variables and retains VHS points of the renormalized spectrum. Note that $\Sigma_{\sigma}$ does not depend on $\mathbf{k}$ and renormalizes the chemical potential $\mu ;\delta t_{\sigma }$ and $\delta t_{\sigma}^{\prime}$ contribute to the change of momentum dependence of the spectrum ($\delta t_{\sigma }$ corresponds to the bandwidth renormalization).
Therefore, it is convenient to represent the renormalized spectrum (\[renorm\_e\]) in the form $$\varepsilon_{\mathbf{k}\sigma }=-2t_{\mathrm{eff,\sigma }}(\cos k_{x}+\cos
k_{x})+4t_{\mathrm{eff,\sigma }}^{\prime }(\cos k_{x}\cos k_{y}+1)-\mu _{\mathrm{eff,\sigma }},$$ where $$\mu_{\mathrm{eff,}\sigma }=\mu-\Sigma_{\sigma }+4\delta t_{\sigma
}^{\prime }+h\sigma ,\;t_{\mathrm{eff,\sigma }}=t+\delta t_{\sigma }\;t_{\mathrm{eff,\sigma }}^{\prime }=t^{\prime }+\delta t_{\sigma }^{\prime}.
\label{mu_eff}$$The scale–dependent effective chemical potential $\mu _{\mathrm{eff,}\sigma}$ results in Fermi surface, calculated with the renormalized spectrum parameters $t_{\mathrm{eff,\sigma }},t_{\mathrm{eff,\sigma }}^{\prime }$. Applying present method one should be careful in determining the geometry of the FS. If $t_{\mathrm{eff,\sigma }}^{\prime }/t_{\mathrm{eff,\sigma }}<1/2,$ the bottom of the band is at the energy $w_{\sigma }=-4t_{\mathrm{eff,\sigma }}+8t_{\mathrm{eff,\sigma }}^{\prime }$ and the FS is singly connected; however, if $t_{\mathrm{eff,\sigma }}^{\prime }/t_{\mathrm{eff,\sigma }}\geq 1/2$ the bottom of the band is zero. In this case if $\mu_{\mathrm{eff,\sigma }}\geq w_{\sigma },$ the FS is singly connected and the patching scheme of Ref. can be used, while for $\mu _{\mathrm{eff,\sigma}}<w_{\sigma}$, the Fermi surface consists of two disconnected parts and the patching scheme should be chosen differently.
Despite that the self–energy renormalization are included in the present study, we neglect incoherent contributions to the Green functions. Due to this renormalization, the actual Fermi level $\mu_{\mathrm{eff}}$ is determined by the combination of the bare spectrum and the self–energy parameters (\[mu\_eff\]) at the end of the flow.
The vertex ansatz {#vertex_ansatz}
-----------------
The vertex function $\Gamma $ is represented by its values at the current FS. Since we always trace the renormalization of $\Sigma _{\mathbf{k}\sigma }$ we have to take into account moving of the Fermi surface during fRG flow. PPs of the current FS are changing during the flow and the discrete (projected) vertex function derivative acquires an additional contribution corresponding to this movement, $$\frac{d\Gamma}{ds}=\frac{\partial\Gamma}{\partial s}+\frac{\partial \Gamma}{\partial k_{\mathrm{PP}}}\frac{dk_{\mathrm{PP}}}{ds}.$$ We denote symbolically the derivatives with respect to PPs ($k_{PP}$) as ${\partial }/{\partial k_{\mathrm{PP}}}.$
To take into account this momentum dependence of $\Gamma$ we assume that, apart from the position of external legs in certain patches, the vertex function depends on momenta $\mathbf{k}_{i}$ through the renormalized energies $\varepsilon _{\mathbf{k}_{i}\sigma }$ and linearize the latter dependence. Let the momenta of external legs $\mathbf{k}_{1}^{\mathrm{c}},\mathbf{k}_{2}^{\mathrm{c}},\mathbf{k}_{3}^{\mathrm{c}}$ be on the current FS and consider vertex with all momenta $\mathbf{k}_{i}$ belonging to the main set of PPs and three vertices with two momenta $\mathbf{k}_{i}$ belonging to the main set of PPs (see Appendix), and one belonging to the auxiliary set. Therefore we have 4 possibilities for the choice of $\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}$, and obtain a system of four linear equations (we use $\varepsilon _{\mathbf{k}_{i}^{\mathrm{c}}\sigma_{i}}=0$) $$\label{v_ansatz}
\Gamma_{\sigma \sigma ^{\prime }}(\mathbf{k}_{1}^{\mathrm{c}},\mathbf{k}_{2}^{\mathrm{c}};\mathbf{k}_{3}^{\mathrm{c}}))+\sum_{i}\partial _{i}\Gamma_{\sigma\sigma^{\prime }}(\mathbf{k}_{1}^{\mathrm{c}},\mathbf{k}_{2}^{\mathrm{c}};\mathbf{k}_{3}^{\mathrm{c}}))\varepsilon _{\mathbf{k}_{i}\sigma_{i}}=\Gamma _{\sigma \sigma ^{\prime }}(\mathbf{k}_{1},\mathbf{k}_{2};\mathbf{k}_{3}),$$ with unknown quantities $\Gamma _{\sigma \sigma ^{\prime }}(\mathbf{k}_{1}^{\mathrm{c}},\mathbf{k}_{2}^{\mathrm{c}};\mathbf{k}_{3}^{\mathrm{c}}))$ and $\partial _{i}\Gamma _{\sigma \sigma ^{\prime }}(\mathbf{k}_{1}^{\mathrm{c}},\mathbf{k}_{2}^{\mathrm{c}};\mathbf{k}_{3}^{\mathrm{c}})$. Solving the system (\[v\_ansatz\]) one determines the vertex on the current FS.
Relation to previous approaches {#Relation_prev}
-------------------------------
In zero magnetic field, neglecting the momentum dependence of the self–energy (i. e. with $\delta t=\delta t^{\prime}=0$), one can demand the chemical potential to absorb all the corrections to the self–energy in Eq. (\[DGamma\]), which is possible since the electronic spectrum enters the Eq. (\[DGamma\]) through $\varepsilon_{\mathbf{k}\sigma}$ only. On the other hand, Eq. (\[DSigma\]), being reformulated with $\mu ^{\prime}=\mu-\Sigma$, reduces to $\dot{\mu}^{\prime }=0$. Hence, in this case, $\mu $ is replaced by a constant $\mu ^{\prime}$. The important consequence of this approximation is that FS is fixed within this ansatz, since it is determined solely by the scale–independent parameter $\mu ^{\prime}$.
If the field is non–zero, the self–energy corrections cannot be absorbed into the chemical potential even when the renormalization of the hopping parameters is neglected. This, in turn, results in moving of spin FSs during the flow, so that projected vertex acquires a contribution from this moving. As described in detail in Sect. \[vertex\_ansatz\], we account for this contribution in the self–consistent numerical scheme of treatment of Eqs. (\[DGamma\]), (\[DSigma\]), and (\[v\_ansatz\]).
Temperature dependence of the renormalized parameters {#Results}
=====================================================
In this Section we present and discuss numerical results of the present fRG approach accounting for the self–energy corrections in zero (Sec. \[h=0\]) and finite magnetic field (Sec. \[h<>0\]). Afterwards, in Sec. \[Phase\_diagram\] we present our results for the phase diagram.
Results in zero magnetic field {#h=0}
------------------------------
In this subsection we consider the results of fRG calculations in the spin symmetric phase for $U/t=3$ and 4. We choose the bare chemical potential $\mu$ in such a way that at the end of the flow the renormalized position of the Fermi level $\mu_{\mathrm{eff}}$, determined by Eq. (\[mu\_eff\]), lies in the vicinity of VHS. It is clear physically and verified numerically in our investigation that the magnetic response is suppressed for the Fermi level well separated from VHS.
Starting from an infinite–temperature limit the position of the Fermi level first tends to increase from its high–temperature Hartree value $\mu-U/2$ to almost low–temperature Hartree value $\bar\mu=\mu-U\int_{-4t+8t'}^{\bar\mu}\rho(\varepsilon) d\varepsilon$, $\rho$ being the bare DOS. At even lower temperatures $\mu$ decreases due to correlation effects. The obtained low–temperature scale dependence of $\mu_{\mathrm{eff}}$ for different bare chemical potentials $\mu$ is shown in Fig. \[fermi\_fig\]a ($U/t=3$) and Fig. \[fermi\_fig\]b ($U/t=4$). We stop the flow, when the effective interaction becomes too large ($\gtrsim2.5W$, where $W=8t$ is bare bandwidth). The corresponding scale $s_{\rm min}$ yields the minimal temperature which is available within the flow, $T_{\rm min}=t\exp(-s_{\rm min})$. In the following we parametrize the initial chemical potential of flow by the Hartree Fermi level $\bar\mu$.
![Temperature dependence of the renormalized Fermi level $\mu_{\mathrm{eff}}$ at $h=0$: a) $U=3t$, b) $U=4t$. Chemical potential values $\bar\mu$ (see text) are shown by numbers near the plots.[]{data-label="fermi_fig"}](fermiR3 "fig:"){width=".5\textwidth"}![Temperature dependence of the renormalized Fermi level $\mu_{\mathrm{eff}}$ at $h=0$: a) $U=3t$, b) $U=4t$. Chemical potential values $\bar\mu$ (see text) are shown by numbers near the plots.[]{data-label="fermi_fig"}](fermiR4 "fig:"){width=".5\textwidth"}
The cases $U=3t$ and $U=4t$ are somewhat different due to the absence of the region of saturation of $\mu_{\rm eff}(s)$ dependence for $U=3t$, although the dependence $\mu_{\rm eff}(s)$ becomes weak at the end of the flow. Below we consider the renormalized Fermi level $\mu_{\rm eff}$ in the saturation region (if it exists) as a representative parameter which characterizes the flow, since the renormalization of other parameters of the electronic spectrum is not too strong. The charge response is slightly suppressed in the vicinity of van Hove filling, for the case $\mu_{\rm eff}<0$ more than for $\mu_{\rm eff}>0$. Non–monotonous behavior of $\mu_{\mathrm{eff}}(s)$ slightly above VHS ($\bar\mu=1.0t$) for $U=4t$, which consequences are discussed below, is worthy of notice.
![Effective (renormalized) hopping parameters: a) $t_{\mathrm{eff}}/t$, b) $t_{\mathrm{eff}}^{\prime }/t_{\mathrm{eff}}$ for $U=4t$. Numbers correspond to the value of chemical potential $\bar\mu$[]{data-label="spectrum_fig"}](spectrum_t "fig:"){width=".5\textwidth"}![Effective (renormalized) hopping parameters: a) $t_{\mathrm{eff}}/t$, b) $t_{\mathrm{eff}}^{\prime }/t_{\mathrm{eff}}$ for $U=4t$. Numbers correspond to the value of chemical potential $\bar\mu$[]{data-label="spectrum_fig"}](spectrum_t1 "fig:"){width=".5\textwidth"}
In our scheme the effective chemical potential renormalization has substantial contribution from $t$ and $t^{\prime }$ renormalizations (see Eq. (\[mu\_eff\])). In Fig. \[spectrum\_fig\] we present for instance $t_{\mathrm{eff}}/t$ and $t_{\mathrm{eff}}^{\prime }/t_{\mathrm{eff}}$ plots for $U=4t$. While the bandwidth ($t_{\mathrm{eff}}/t$) is somewhat reduced at the end of the flow well below VHS, indicating prominent correlation effects; for $\mu_{\rm eff}$ above VHS $t_{\rm eff}/t$ first increases with decreasing temperature (which implies that correlation effects in this regime are not substantial) and then decreases at lower temperatures, which is related to enhancement of the vertex in the vicinity of magnetic instability. On the other hand, the ratio $t^{\prime }/t$ is not strongly renormalized (this means that the self–energy effects do not change substantially the curvative of the Fermi surface) and monotonously increases towards the value 1/2 as $\bar\mu$ increases, which corresponds to flattening of the electronic dispersion. The complete treatment of the case $\bar\mu \geq 1.4t$, where $t_{\mathrm{eff}}^{\prime}/t_{\mathrm{eff}}$ exceeds $1/2$ during the flow, corresponding to a change of FS geometry (see for details Sec. \[self-energy\_ansatz\]), requires special consideration and is beyond the scope of the present study.
The type of leading magnetic instability can be inferred from the behavior of vertices. Let us consider the scale profiles of maximal vertex. We consider maximal total vertex $\Gamma_{\uparrow \downarrow }^{\mathrm{max}}$ (the maximum is taken over all possible combinations of momenta) and maximal exchange vertex $\Gamma_{\uparrow\downarrow}^{\mathrm{max,E}}$ (the maximum is taken over all combinations of momenta with $\mathbf{k}_{2}=\mathbf{k}_{3}$). The ferromagnetic instability is accompanied by coincidence of $\Gamma_{\uparrow\downarrow}^{\mathrm{max}}$ and $\Gamma_{\uparrow \downarrow }^{\mathrm{max,E}}$ in the vicinity of transition point where both the values diverge, reflecting an instability of the paramagnetic state with respect to zero–momentum collective spin excitations. Therefore, the criterion for commensurate magnetic fluctuations has the form $$\label{commensurability}
\Delta\Gamma_{\rm max}\equiv\Gamma_{\uparrow \downarrow }^{\mathrm{max}}-\Gamma_{\uparrow\downarrow}^{\mathrm{max,E}}=0$$ and yields an information about the type of leading instability.
![The scale profiles of $\Gamma^{\mathrm{max}}_{\uparrow\downarrow}$ (solid lines) and $\Gamma^{\mathrm{max, E}}_{\uparrow\downarrow}$ (dashed lines), see text, in the case $h=0$ a) $U=3t$, b) $U=4t$ for different choices of the chemical potential $\bar\mu$ (shown by numbers) yielding Fermi level below VHS ($\mu_{\mathrm{eff}}<0$)[]{data-label="vertex_3"}](3vMAX_part1 "fig:"){width="50.00000%"}![The scale profiles of $\Gamma^{\mathrm{max}}_{\uparrow\downarrow}$ (solid lines) and $\Gamma^{\mathrm{max, E}}_{\uparrow\downarrow}$ (dashed lines), see text, in the case $h=0$ a) $U=3t$, b) $U=4t$ for different choices of the chemical potential $\bar\mu$ (shown by numbers) yielding Fermi level below VHS ($\mu_{\mathrm{eff}}<0$)[]{data-label="vertex_3"}](4vMAX_part1 "fig:"){width="50.00000%"}
![The same as in Fig. \[vertex\_3\] for Fermi level being above VHS ($\mu_{\mathrm{eff}}>0$)[]{data-label="vertex_4"}](3vMAX_part2 "fig:"){width="50.00000%"}![The same as in Fig. \[vertex\_3\] for Fermi level being above VHS ($\mu_{\mathrm{eff}}>0$)[]{data-label="vertex_4"}](4vMAX_part2 "fig:"){width="50.00000%"}
The scale dependences of $\Gamma_{\uparrow \downarrow}^{\mathrm{max}}$ and $\Gamma_{\uparrow \downarrow }^{\mathrm{max,E}}$ for different $\bar\mu$ are presented in Fig. \[vertex\_3\] ($\mu_{\rm eff}<0$) and Fig. \[vertex\_4\] ($\mu_{\rm eff}>0$). Let us consider first the case of Fermi level below VHS, $\mu_{\mathrm{eff}}<0$. For both the cases $U=3t$ (Fig. \[vertex\_3\]a) and $U=4t$ (Fig. \[vertex\_3\]b) $\Gamma _{\uparrow \downarrow }^{\mathrm{max}}$ is diverging, and $\Delta\Gamma_{\rm max}/\Gamma _{\uparrow \downarrow }^{\mathrm{max}}$ vanishes or is very small at the end of the flow. Therefore, magnetic fluctuations are predominantly ferromagnetic at the end of the flow. On the other hand, $\Delta\Gamma_{\rm max}$ increases as $\mu_{\rm eff}$ approaches VHS: for both $U=3t$ and $U=4t$ $\Delta\Gamma_{\rm max}$ is the largest in the case $\bar\mu=0.8t$. This means that in the vicinity of VHS the ferromagnetic fluctuations hardly dominate over incommensurate ones.
For $\mu_{\mathrm{eff}}>0$ (see Fig. \[vertex\_4\]), the difference $\Delta\Gamma_{\rm max}$ is non–zero up to the lowest temperatures where $\Gamma_{\rm max}$ diverges. Moreover, the ratio $\Delta\Gamma_{\rm max}/\Gamma_{\uparrow\downarrow }^{\mathrm{max}}$ increases as $\mu$ increases. We interpret this as that the incommensurate fluctuations are dominating over the ferromagnetic ones. In the case $\bar\mu=1.0t$, where the temperature dependence of $\mu_{\rm eff}(s)$ is strongly non–monotonous (see Fig. \[fermi\_fig\]b), which is possibly related with invalidity of ansatz (\[se\_ansatz\]),(\[v\_ansatz\]) in this regime and causes the shift of the point of vertex diverging to substantially lower temperatures. This case should be considered more correcly in further elaborated studies.
In the next subsection we introduce a small magnetic field to investigate magnetic properties of the system.
Magnetization in finite magnetic field {#h<>0}
--------------------------------------
In this subsection we supplement the picture in zero magnetic field considered above by the results for magnetic response in finite magnetic field. In case $h>0$ we have $\mu_{\mathrm{eff\uparrow}}\neq\mu_{\mathrm{eff\downarrow}}$ due to spin dependence of the spectrum parameters $t_{\mathrm{eff,\sigma}},t_{\mathrm{eff,\sigma}}^{\prime}$ and $\Sigma_{\sigma}$ and, to a small extent, due to the presence of magnetic field. At low scales (high temperatures) the spectrum parameters do not depend on the spin projection substantially and $\mu_{\mathrm{\rm eff\uparrow}}-\mu_{\mathrm{\rm eff\downarrow }}\approx 2h$, but with increasing the scale (lowering temperature) the strong spin dependence can be realized which is a manifestation of exchange enhancement.
The strengh of magnetic response is characterized by the magnetization $m$ which can be easily calculated using current parametes of the electronic spectrum $\delta t_{\sigma},\delta t_{\sigma}',\Sigma_{\sigma}$ (we remind that the frequency dependence of the self–energy is neglected), $$\label{magnetization_def}
m=\frac1{2N}\sum_{\mathbf{k}}\left(f_{\mathbf{k}\uparrow}-f_{\mathbf{k}\downarrow}\right).$$
![ The logarithmic plots of the dependence $m(h)$ and its fit to $m\propto h^\alpha$. The chemical potentials $\bar\mu$ and fitted $\alpha$ are shown near the plots.[]{data-label="log"}](logarifmic)
![The magnetization scale profiles in the field $h=10^{-4}t$ for different chemical potential $\bar\mu$ specified by numbers: a) $U=3t$, b) $U=4t$[]{data-label="m-s"}](m-comp){width="100.00000%"}
![The scale profiles of ratios $m/h$ (a,c,e) and $m/h^{1/3}$ (b,d,f) for $U=4t$ at different small magnetic fields: $h/t=(5,10,20,40)\cdot 10^{-5}$ and different bare chemical potentials $\bar\mu$, which values are shown explicitly[]{data-label="U4_mh_fig"}](mh_multi_new){width="100.00000%"}
Fig. \[m-s\] shows the comparison of the scale profiles of magnetization $m$ for different bare chemical potentials. For $\mu_{\rm eff}$ above VHS we find slightly negative $m$ at the end of the flow for both the cases $U=3t$ and $U=4t$. This is possibly related to an influence of incommensurate magnetic fluctuations above VHS, as conjectured from the results in zero magnetic field. Below VHS, where dominating ferromagnetic fluctuations were observed in the absence of magnetic field, the magnetic response to magnetic field becomes positive and considerable, especially in the vicinity of VHS: the maximal value of $m$ at the end of the flow increases, but the temperature of sharp increase of the magnetization becomes lower as $\mu$ increases. Note, that the absolute values of magnetization are rather small ($m\ll2n$) at the end of the flow. To verify the fulfillment of Mernin–Wagner theorem, we fit the data for magnetization taken at lowest temperatures of the flow to $m\propto h^\alpha$ dependence (see Fig. \[log\]), which gives $\alpha\in(0.46,0.83)$ depending on the chemical potential $\mu$. In the vicinity of VHS level ($\bar\mu=1.0t,1.1t$) for $U=4t$ the magnetization is somewhat larger at the end of the flow, in particular $\alpha=0.17$ at $\bar\mu=1.1t$. Additionally, in this case magnetization tends to saturate at lowest temperatures with increasing the magnetic field (not shown). For these chemical potentials the renormalized Fermi level lies very near VHS and the Mermin–Wagner theorem may not be fulfilled to a good accuracy. On the other hand, these cases are on the borderline between the regions of strong ferromagnetic and incommensurate fluctuations. The temperature dependence of the exponent $\alpha$ is of particular interest, since it allows to investigate the magnetic properties of the system while entering the region of strong magnetic fluctuations. It is obvious that at high temperatures $m\propto h$ ($\alpha=1$), but at low temperatures this relation can be violated due to fluctuations. To determine the temperature of the crossover into the regime with strong ferromagnetic fluctuations, we adopt the criterion $\alpha=1/3$, dictated by the mean–field value of the critical exponent at the conventional magnetic phase transition. We denote the crossover scale as $s^*$ (the corresponding temperature is $T^*=t\exp(-s^*)$).
Fig. \[U4\_mh\_fig\] shows the comparative (with respect to the magnetic field) representation of $m/h$ and $m/h^{1/3}$ scale profiles for different positions of Fermi level below VHS in the case $U=4t$. In cases $\bar\mu=0.4t,0.6t$ the crossover scale is reached within the flow and lies near its end, while in the case $\bar\mu=0.9t$ it is beyond the flow and is obtained by an extrapolation. Therefore, increasing $\bar\mu$ tends to decrease $T^*$. We do not find a well–resolved crossover scale at $U=3t$, but we note that the system is close to it at the end of the flow in the case $\bar\mu=0.4t$, where the magnetic fluctuations tend to be commensurate. This means that at $U=3t$ crossover to ferromagnetic ordering can be realized only at extremely low temperatures which are not available within the present approach. For the case $\mu_{\mathrm{eff}}>0$ and both $U=3t$ and $U=4t$ we find a relatively weak response which becomes negative close to the end of the flow, as discussed above.
The distribution of temperatures $T_{\rm min}$ and $T^*$ for different chemical potentials yields a natural connection between zero–field ($T_{\rm min}$) and finite–field ($T^*$) results. We find that both $T_{\rm min}$ and $T^*$ decrease as the system approaches VHS, which is possibly an effect of incommensurate magnetic fluctuations and temperature smearing of VHS (above van Hove filling DOS falls down faster than below it); below VHS we obtain $T^*<T_{\rm min}$. However, in the case $\bar\mu=1.1t$ we find $T^*>T_{\rm min}$, which is possibly connected to the abovediscussed violation of the Mermin–Wagner theorem in a close vicinity above VHS.
Phase diagram {#Phase_diagram}
-------------
, $\Sigma=0$–fRG [@Kampf] (solid lines), QSA [@QS] (long–dashed lines) and present SE fRG (symbols) approaches, see text. The boundary lines corresponding to $\Sigma=0$–fRG and QSA approaches separate ferromagnetic and paramagnetic regions below van Hove filling and ferromagnetic and incommensurate magnetic phases above van Hove filling. []{data-label="Summary"}](fRG_diagram)
In this Section we summarize the results obtained in zero (Sect. \[h=0\]) and finite (\[h<>0\]) magnetic field and compare our results with the results of previous approaches. The phase diagram constructed in terms of renormalized Fermi level $\mu_{\rm eff}$ (see discussion in Sec. \[h=0\]) and Coulomb interaction $U$ is shown in Fig. \[Summary\].
In the case $\mu_{\rm eff}<0$ the large region on the phase diagram is found where the zero–field magnetic fluctuations are predominantly commensurate. In finite magnetic field, the system exhibits some indications of ferromagnetic order in the ground state: at $U=3t$ a crossover from paramagnetic to ferromagnetic order is observed well beyond the flow, while at $U=4t$ the crossover to ferromagnetic state is well–resolved in the close vicinity of the end of the flow. In the case $\mu_{\rm eff}>0$ (excluding the region of near vicinity of Fermi level to VHS) we do not find commensurate magnetic fluctuations to be dominating in zero magnetic field: they are competely replaced by incommensurate flucuations with corresponding maximal vertex $\Gamma_{\uparrow \downarrow}^{\mathrm{max}}$ being diverging. This conclusion agrees with the result in finite magnetic field, where the magnetization $m$ becomes slightly negative at low temperatures. Therefore we conclude that ferromagnetic order is suppressed by an incommensurate magnetic fluctuations if the Fermi level is above VHS. The case of $\mu_{\rm eff}$ very near but above VHS is worthy of special attention: in the case $U=3t$ we find incommensurate magnetic fluctuations in zero magnetic field and no indication of ferromagnetic ordering in finite field. However, very near VHS at $U=4t$ we find almost commensurate magnetic fluctuations in zero magnetic field and ferromagnetic–like behavior in finite magnetic field. At the same time, in this case $\mu_{\rm eff}$ depends non–monotonously on $\mu$ and the magnetization on the value of magnetic field. These states are shown by “?” symbol in Fig. \[Summary\]. The mean–field and quasistatic approximation (MFA and QSA) boundary lines calculated in Refs. and are shown on the phase diagram in Fig. \[Summary\] for comparison (the effective chemical potential in MFA $\mu_{\rm eff}^{\rm MF}=\mu-Un/2$ and in QSA it is determined from the condition for electronic density $n(\mu_{\text{eff}})=n_{0}(\mu),$ where $n_{0}(\mu)$ is the number of particles in the non–interacting model). The critical interaction for stability of ferromagnetism within the present approach $U_{\mathrm{c}}>3t$ is somewhat larger then the QSA and the mean–field resuts ($U_{\mathrm{c}}(\mu_{\mathrm{eff}}=0)\approx2t$), which naturally suggests that the account for the electronic correlations results in an enhancement of $U_{\mathrm{c}}$. The results of the present study partly agree with those calculated within fRG approach without self–energy effects[@Kampf] ($\Sigma=0$–fRG approach, see Fig. \[Summary\]), obtained by studying the temperature dependences of magnetic and superconducting susceptibilities in zero magnetic field. The lower threshold for ferromagnetism $U_{\mathrm{c}}$ obtained in the present study, appears to be finite contrary to $\Sigma=0$–fRG. At the same time, away from VHS these two approaches qualitatively agree.
The comparison of different approaches MFA, QSA, $\Sigma=0$–fRG, SE fRG shows a step–by–step restriction of the size of ferromagnetic region. Ferromagnetism is practically absent for $\mu_{\rm eff}>0$ within SE fRG, but not restricted with respect to previous approaches in the case $\mu_{\rm eff}<0$. Above VHS ferromagnetism is destroyed by well–resolved incommensurate magnetic fluctuations, while below VHS quantum commensurate fluctuations dominate.
Conclusions {#Conclusions}
===========
In this study we present a fRG treatment of magnetic order in the Hubbard model, controlled by the Fermi level being in the vicinity of van Hove singularity (VHS) and on–site Coulomb interaction. The introduced version of fRG accounts for self–energy corrections, which implies a proper account of the Fermi surface moving as the temperature decreases. The following aspects of this problem are investigated: the renormalization of vertices in zero magnetic field, spin splitting in finite field, finite–temperature behavior and the electronic spectrum renormalization.
We find that magnetic properties of the system are substantially asymmetric with respect to the Fermi level position relatively to VHS. In particular, ferromagnetic ordering is strongly suppressed above VHS due to competition with incommesurate magnetic fluctuations. Below VHS we find precurors of ferromagnetic ordering at low temperatures and not too small interaction (of order of half bandwidth). The temperature of the crossover to strong ferromagnetic fluctuations decreases as the Fermi level elevates towards VHS.
Correlation effects do not change substantially the form of electronic spectrum which is characterized by $t'/t$. However, the renormalized ratio $t'/t$ monotonously increases as the Fermi level rises towards VHS. When the Fermi level is at VHS we find that the ansatz used for the self–energy momentum dependence is too crude to catch delicate features of the renormalization.
The results of the present approach improve the results of the mean–field approximation[@OurIC] and quasistatic approach[@QS] and agree with previous fRG study [@Kampf] for the position of the Fermi level away from VHS. The Mermin–Wagner theorem is also shown to be fulfilled to a good accuracy in the present approach in the most part of the phase diagram. The magnetic field dependence of magnetization demonstrates power–law behavior with the exponents $\alpha\in(0.62,0.83)$. The observed region of ferromagnetism for the Fermi level above VHS is rather narrow. This explains qualitatively the magnetic behavior of one– and two–layer ruthenates (which have, after corresponding particle–hole transformation, $\mu_{\rm eff}>0$). The non–Fermi–liquid behavior of doped Sr$_2$RuO$_4$[@Kikugawa] can be possibly related to the non–monotonous temperature dependence of the Fermi level near VHS in the present approach, which neglects non–quasiparticle contributions. These contributions may therefore be important for the Fermi level near VHS (see, e. g. Ref. ), which is the subject of future study. The results obtained demonstrate an important role of many–electron renormalizations of electron spectrum, in particular of chemical potential, in the presence of VHS. This fact can be crucial for the criterium and properties of weak itinerant ferromagnetism. Another important point is that peculiarities of ferromagnetic ordering related to the presence of VHS cannot be even qualitatively captured by any elaborate method based on DOS consideration only.
The obtained finite–temperature picture, in particular, the relation of the character of magnetic fluctuations to the position of the Fermi level and absence of the long–range order (which is stated by the Mermin–Warger theorem) should be possibly supplemented by a zero-temperature study of this problem. Investigation of the effect of finite (but not too small) magnetic field and possibility of the metamagnetic transitions is also of interest. Another unsolved problem is a unified description of weak and strong ferromagnets, which can receive new insights from weak–coupling investigations of the Hubbard model.
We are grateful to W. Metzner and H. Yamase for stimulating discussions. This work was supported in part by Partnership Program of the Max–Planck Society, the project ’Quantum Physics of Condensed Matter’ of Presidium of Russian Academy of Science and President Program of Scientific Schools Support 4711.2010.2.
Projecting points {#PP_ansatz}
=================
To parametrize the momentum dependence of the self–energy and vertex, we introduce a set of points in the Brillouin zone (below we refer to them as projecting points, PPs), so that the values of functions at these points represent the function. In prevous studies [@Salmhofer] this set of the points was chosen as an intersection of lines of constant angles (located at the center of patches) with FS which was assumed to be fixed. Since we account for momentum dependence of the self–energy and vertex more accurately, we supplement this set of PPs at FS by a corresponding set in the vicinity of FS by choosing additional points belonging to ”shifted“ FS and call it auxiliary set of PPs. However, since moving of FS is continuous process, we apply discrete Runge–Kutt procedure (RKP) to solve the system of differential equations numerically, which causes some complications in definition of PP.
At any discrete step of RKP we introduce the PPs of the following types: (i) main PPs, which belong to FS, determined by the chemical potential $\mu$, and the self–energy $\Sigma_{\mathbf{k}\sigma}$ at the beginning of the step. (ii) auxiliary PPs which are detemined analogously to main PPs, with $\mu$ being shifted by $\delta \mu_{\sigma}$ for different spin projections, and $\delta\mu_{\sigma}$ is determined by typical shift of momentum–independent part of self–energy $\Sigma _{\sigma }$ at previous step. (iii) current PPs which are determined by current $\Sigma _{\mathbf{k}\sigma }$ within the step. Note, that at the beginning of the step main PPs coincide with current ones. Introduction of the auxiliary PPs set is needed since, due to definition of main PPs set, the functions chosen to represent momentum dependence of $\Sigma $ ($\cos k_{x}+\cos k_{y},\cos k_{x}\cos k_{y},1$) are linearly dependent at the constant energy set (see Section \[self-energy\_ansatz\]), which does not allow to use them for linear regression of $\Sigma$–derivatives. Therefore, an additional set of PPs is needed; the calculation of the vertex near current FS allows to expand the vertex linearly beyond simple projecting ansatz (see detail explaination in Section \[vertex\_ansatz\]), which was used in earlier studies.
RKP makes its step using the vertex and self–energy derivatives calculated at the intermediate (current) value of argument $s$ at fixed (during the step) sets of main and auxiliary PPs. From this we extract derivatives $\dot{\delta}t_{\sigma}$, $\dot{\delta}t_{\sigma }^{\prime }$ and $\dot{\Sigma}_{\sigma }$, see Section \[self-energy\_ansatz\]. However, the right–hand side of fRG equations contains *current* self–energy function and vertex function, projected on the current FS. RKP allows to calculate this current self–energy function and use it to determine current FS and corresponding current PPs set. Then we apply the procedure of Sect. \[vertex\_ansatz\] accounting for the influence of the FS moving on the current value of projected vertex.
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|
---
abstract: 'We derive properties of general universal embezzling families for bipartite embezzlement protocols, where any pure state can be converted to any other without communication, but in the presence of the embezzling family. Using this framework, we exhibit various families inequivalent to that proposed by van Dam and Hayden. We suggest a possible improvement and present detail numerical analysis.'
author:
- 'Debbie Leung[^1]'
- 'Bingjie Wang[^2]'
date: '6 July, 2014'
title: '**Characteristics of Universal Embezzling Families**'
---
Introduction {#primer}
============
We begin by defining bipartite [*quantum state embezzlement*]{} between Alice and Bob. Let ${\ensuremath{\left|\varphi\right\rangle}}$ and ${\ensuremath{\left|\mu\right\rangle}}$ be bipartite quantum states; embezzlement of ${\ensuremath{\left|\varphi\right\rangle}}$ from ${\ensuremath{\left|\mu\right\rangle}}$ is the transformation ${\ensuremath{\left|\mu\right\rangle}} \mapsto
{\ensuremath{\left|\mu\right\rangle}}{\ensuremath{\left|\varphi\right\rangle}}$ using only local operations. Operationally, Alice and Bob share ${\ensuremath{\left|\mu\right\rangle}}$ and, without further communication, “embezzle” a shared ${\ensuremath{\left|\varphi\right\rangle}}$.
Pure bipartite entangled states, their interconversions, and their applications in quantum information processing tasks have been well-studied. Axiomatically, entanglement, as a quantum correlation, does not increase without communication, rendering exact embezzlement impossible for a general ${\ensuremath{\left|\varphi\right\rangle}}$. Surprisingly, van Dam and Hayden [@DH02] showed embezzlement can be approximated, with arbitrary precision, as the dimension of ${\ensuremath{\left|\mu\right\rangle}}$ grows. Furthermore, arbitrary ${\ensuremath{\left|\varphi\right\rangle}}$ can be embezzled from the same ${\ensuremath{\left|\mu\right\rangle}}$. We call such a sequence of states [$\left|\mu(n)\right\rangle$]{} a [ *universal embezzling family*]{}.
Embezzlement has found interesting applications. It enables remote parties to share an arbitrary state on demand without communication (see for example [@V-private]). Furthermore, embezzlement hides the existence or the disappearance of a quantum state from any external observer. Thus embezzlement is used in the noisy channel simulation in the original [@QRSTa] and an alternative [@QRSTb] proof of the quantum reverse Shannon theorem. Finally, in [@LTW08], embezzlement motivates a game for which no finite amount of entanglement suffices in an optimal strategy, and provides proofs that some natural classes of quantum operations are not topologically closed.
The results in [@DH02] have been extended in several ways. An alternative embezzling family for any number of parties is proposed in [@LTW08]. This family also achieves better approximation for a given dimension of $|\mu\>$ for non-universal embezzlement (a method attributed to [@HS-private]). Reference [@V-private] provides an embezzlement protocol that is robust against discrepancy between the descriptions of ${\ensuremath{\left|\varphi\right\rangle}}$ available to Alice and Bob. There are many unresolved questions concerning embezzlement. In the multiparty setting, the only known universal multiparty embezzlement family is an $\epsilon$-net of the non-universal embezzlement states [@LTW08]; perhaps more efficient universal families exist. In the bipartite setting, the family in [@DH02] is not known to be optimal, but it has been elusive to find an optimality proof or a better family. Likewise, there may be a lower dimensional resource state for the robust protocol in [@V-private]. Very few universal embezzling families are known, and finite size effect or the computational complexity of embezzlement is hardly studied.
In this paper, we focus on the bipartite setting. We derive conditions for universal embezzlement, and exhibit a countably infinite number of inequivalent families. We conjecture a universal embezzling family based on our findings, and provide numerical evidence for the improvement in efficiency.
During the preparation of this manuscript, we learned of the result by Dinur, Steurer, and Vidick reported in [@V-private], and another on-going study of embezzlement by Haagerup, Scholz, and Werner [@HSW-private].
Canonical Form for Embezzlement {#canonical-form-for-embezzlement .unnumbered}
-------------------------------
Any pure bipartite state has a Schmidt decomposition (see for example [@NC00]). Since the parties can perform local unitary operations, without loss of generality, ${\ensuremath{\left|\mu\right\rangle}} = \sum_{i=1}^{ \tilde{n}} \mu_i |i\>_{A_1}
|i\>_{B_1}$ and ${\ensuremath{\left|\varphi\right\rangle}} = \sum_{j=1}^{m} \varphi_j |j\>_{A_2}
|j\>_{B_2}$ where $\{{\ensuremath{\left|i\right\rangle}}\}_{i=1}^{\tilde{n}}$, $\{|j\>\}_{j=1}^{m}$ are orthonormal bases for Alice’s systems $A_1,
A_2$, and for Bob’s systems $B_1, B_2$. Furthermore, $\mu_i$, $\varphi_i$ can be chosen non-negative and decreasing, with $\sum
\mu_i^2 = 1$ and $\sum \varphi_i^2 = 1$ so [$\left|\mu\right\rangle$]{} and [$\left|\varphi\right\rangle$]{} are normalized. We refer to $\tilde{n}$ as the Schmidt rank and the $\mu_i$s as Schmidt coefficients of [$\left|\mu\right\rangle$]{}; the same terminology holds for the Schmidt decomposition of any bipartite state.
In this canonical form, there is an exchange symmetry between Alice and Bob. Furthermore, any quantum operation can be implemented as an isometry, $U$, with possibly larger output space. In embezzlement, the actual output state is $U \otimes U {\ensuremath{\left|\mu\right\rangle}}$.
Measure of success and optimal strategy {#measure-of-success-and-optimal-strategy .unnumbered}
---------------------------------------
One measure of the precision of the embezzlement protocol is the fidelity. The fidelity between two pure states [$\left|\varphi\right\rangle$]{} and [$\left|\psi\right\rangle$]{} is given by $F({\ensuremath{\left|\varphi\right\rangle}},{\ensuremath{\left|\psi\right\rangle}}) = |\<\varphi|\psi\>|$ (see [@NC00]). From [@Vidal-Jonathan-Nielsen-00], it follows that the fidelity between the output $U \otimes U {\ensuremath{\left|\mu\right\rangle}}$ and the target ${\ensuremath{\left|\mu\right\rangle}}
{\ensuremath{\left|\varphi\right\rangle}}$ is optimized by the isometry $U$ taking $A_1 \mapsto A_1A_2$ (likewise for Bob) that simply permutes the basis states, such that ${\ensuremath{\left|\omega\right\rangle}} {:}= U^\dagger \otimes U^\dagger {\ensuremath{\left|\mu\right\rangle}}
{\ensuremath{\left|\varphi\right\rangle}}$ has decreasing Schmidt coefficients. The optimal fidelity is $\max_U \<\mu| \<\varphi| \, (U \otimes U
{\ensuremath{\left|\mu\right\rangle}}) = \<\omega| \, (|\mu\> \otimes |1\>|1\>)$. The state $|\mu\> \otimes |1\>|1\>$ has Schmidt coefficients $\mu_i$ followed by zeros. We denote it by the equivalent state $|\mu\>$ throughout.
In this paper, we only consider embezzlement protocols that involve permutation of the basis states. We often consider “optimal embezzlement” as described above. Given a universal embezzling family, we focus on a subsequence [$\left|\mu(n)\right\rangle$]{} indexed by the local dimension $n = \tilde{n}$.
Intuitively, a state $|\mu\>$ is useful for universal embezzlement if its Schmidt coefficients $\mu_i's$ has high fidelity with respect to $\{\mu_i \varphi_j\}_{ij}$ for any valid $\{\varphi_j\}$.
General vs regular embezzling families {#general-vs-regular-embezzling-families .unnumbered}
--------------------------------------
The most general embezzling family has the form $${\ensuremath{\left|\mu(n)\right\rangle}} = \sum_{i=1}^n \mu(i, n){\ensuremath{\left|i\right\rangle}}{\ensuremath{\left|i\right\rangle}}$$ where for each $n$, $\mu(i, n)$ is decreasing with $i$ and $\sum_{i=1}^n \mu(i, n)^2 = 1$. An interesting special case concerns embezzling families whose Schmidt coefficients are generated by decreasing functions of one variable $i$, $f : \N \mapsto \R^+$. They are given by $${\ensuremath{\left|\mu(f,n)\right\rangle}} = \frac{1}{\sqrt{C(f,n)}}\sum_{i=1}^n f(i){\ensuremath{\left|i\right\rangle}}{\ensuremath{\left|i\right\rangle}} \,.$$ where $C(f, n) = \sum_{i=1}^n f(i)^2$ so [$\left|\mu(f,n)\right\rangle$]{} is normalized. We call these universal embezzling families “regular”. They are a direct generalization of the universal embezzling family proposed in [@DH02]: $${\ensuremath{\left|\mu(f_{dh}, n)\right\rangle}} = \frac{1}{\sqrt{C(f_{dh},n)}}
\sum_{i=1}^{n} \frac{1}{\sqrt{i}}{\ensuremath{\left|i\right\rangle}}{\ensuremath{\left|i\right\rangle}}$$ where $f_{dh}(x) = 1/\sqrt{x}$.
Properties of Embezzling Families {#general}
=================================
In this section, we present necessary conditions and sufficient conditions for a sequence, [$\left|\mu(n)\right\rangle$]{}, to be a universal embezzling family.
First, for universal embezzlement, it suffices to be able to embezzle any Schmidt rank 2 state. We first introduce a lemma stating that embezzlement of different Schmidt rank $m$ states can be done in superposition. This result is a simple generalization of both embezzlement and coherent state exchange [@LTW08].
\[embinsuperposition\] Suppose it is possible to embezzle any [$\left|\varphi\right\rangle$]{} with Schmidt rank $m$ using $|\mu\>$ with fidelity at least $F$ (see Section \[primer\]), then the following transformation $$\sum_{j=1}^{k} \alpha_j |\mu\> |jj\> \rightarrow
\sum_{j=1}^k \alpha_j |\mu\> |\varphi_j\>$$ can be performed with fidelity at least $F$ without communication, for any $\alpha_j$’s satifying $\sum_{j=1}^k |\alpha_j|^2 = 1$ and for each $|\varphi_j\>$ of the form $$ |\varphi_j\> = \sum_{l=1}^m \varphi_{j,l} |m(j{-}1){+}l\>|m(j{-}1){+}l\> \,
~\text{with}~ \sum_{l=1}^m |\varphi_{j,l}|^2 = 1
\,.
$$
The given embezzlement property, as specified in Section \[primer\], implies that $\forall j, \exists U_j$ such that $F(U_j \otimes U_j |\mu\> |11\>, |\mu\> \sum_{l=1}^{m} \varphi_{j,l} |ll\>)
\geq F$. Modifying the input and output bases gives a $\widetilde{U}_j$ such that $F(\widetilde{U}_j \otimes \widetilde{U}_j |\mu\> |jj\>, |\mu\>
|\varphi_j\>) \geq F$. Further define $\widetilde{U}_j |\xi\> |j'\> =
0$ for all $|\xi\>$ whenever $j' \neq j$. So, $U = \sum_j
\widetilde{U}_j$ is an isometry satisfying: $$\left[ \sum_{j'=1}^k \alpha_{j'}^* \<\mu| \<\varphi_{j'}| \right]
\left[ U \otimes U \sum_{j=1}^{k} \alpha_j |\mu\> |jj\> \right]
=
\left[ \sum_{j'=1}^k \alpha_{j'}^* \<\mu| \<\varphi_{j'}| \right]
\left[ \sum_{j=1}^k \alpha_j \widetilde{U}_j \otimes
\widetilde{U}_j |\mu\> |jj\> \right]
\geq F \,.$$
We now analyze embezzlement of general states by recursively embezzling Schmidt rank $2$ states while reusing the embezzlement state. To do so, we use two facts concerning the [*trace distance*]{} between two density matrices $\sigma_{1,2}$ of equal dimension, defined as $T(\sigma_1, \sigma_2) {:}{=} \, \frac{1}{2} \| \sigma_1 -
\sigma_2 \|_1$ where $\|\cdot\|_1$ denotes the Schatten $1$-norm. First, for two pure states, $T({\ensuremath{\left|\sigma_1\right\rangle}},{\ensuremath{\left|\sigma_2\right\rangle}})^2 +
F({\ensuremath{\left|\sigma_1\right\rangle}}, {\ensuremath{\left|\sigma_2\right\rangle}})^2 = 1$. Second, the trace distance is nonincreasing under any quantum operation and is subadditive. (See [@Ruskai94; @FuchsvdG99; @NC00] for detail.) In particular, if $F(|\sigma\>, U|\sigma_1\>) \geq F_1$ and $F(|\sigma_1\>,|\sigma_2\>) \geq F_2$, then, $$\label{subaddtrdist}
\sqrt{1-F(|\sigma\>,U|\sigma_2\>)^2} = T(|\sigma\>, U|\sigma_2\>) \leq
T(|\sigma\>, U|\sigma_1\>) + T(|\sigma_1\>, |\sigma_2\>)
\leq \sqrt{1-F_1^2} + \sqrt{1-F_2^2} \,,$$ which bounds the performance of substituting $|\sigma_1\>$ by $|\sigma_2\>$ in any operation $U$.
\[only2qubits\] Suppose it is possible to embezzle any Schmidt rank $2$ state from $|\mu\>$ with fidelity at least $F$. Then, embezzlement of any Schmidt rank $m$ state $|\varphi\>$ can be achieved with fidelity at least $F_m$ where $1-F_m^2 \leq \lceil \log_2 m \rceil^2
(1-F^2)$.
It suffices to prove the theorem for $m=2^l$ for $l \in \N$ via induction on $l$. The base case $l=1$ is given. Assume, for some $k$, for any state [$\left|\phi\right\rangle$]{} with Schmidt rank at most $2^k$, there exists an isometry $V$, such that $F_k = F(V \otimes
V{\ensuremath{\left|\mu\right\rangle}},{\ensuremath{\left|\mu\right\rangle}}{\ensuremath{\left|\phi\right\rangle}})$ satisfies $1 - F_k^2 \leq k^2 (1 -
F^2)$.
It remains to show that any ${\ensuremath{\left|\varphi\right\rangle}} = \sum_{i=1}^m \varphi_i
{\ensuremath{\left|i\right\rangle}}{\ensuremath{\left|i\right\rangle}}$ with $m = 2^{k+1}$ can be embezzled with the desired fidelity. To do so, let $\alpha_j^2 = \varphi_{2j-1}^2 + \varphi_{2j}^2$ and $|\varphi_j\> = \alpha_j^{-1} (\varphi_{2j{-}1}|2j{-}1\>|2j{-}1\> +
\varphi_{2j}|2j\>|2j\>)$ for $j=1, 2, 3, \cdots, 2^k$. Apply the induction hypothesis; so ${\ensuremath{\left|\phi\right\rangle}} = \sum_{j=1}^{m/2} \alpha_j {\ensuremath{\left|jj\right\rangle}}$ can be embezzled with fidelity at least $F_k$ with some isometry $V$. In addition, from Lemma \[embinsuperposition\], $|\mu\>{\ensuremath{\left|\phi\right\rangle}} \rightarrow |\mu\>
(\sum_{j=1}^{m/2} \alpha_j |\varphi_j\>) = |\mu\>|\varphi\>$ can be performed with fidelity at least $F$. Finally, using Eq. (\[subaddtrdist\]), we evaluate the fidelity of composing these two steps by taking $|\sigma\> = |\mu\>|\varphi\>$, $|\sigma_1\> = |\mu\> {\ensuremath{\left|\phi\right\rangle}}$, and ${\ensuremath{\left|\sigma_2\right\rangle}} = V \otimes V |\mu\>$. This yields $\sqrt{1-F_{k+1}^2} \leq
\sqrt{1-F^2} + \sqrt{1-F_k^2} \leq
\sqrt{1-F^2} + \sqrt{k^2(1 - F^2)}
=(k + 1)\sqrt{1 - F^2}.$
[*Remark.*]{} Due to Lemma \[only2qubits\], we take ${\ensuremath{\left|\varphi\right\rangle}}
= \alpha{\ensuremath{\left|00\right\rangle}} + \beta{\ensuremath{\left|11\right\rangle}}$ unless otherwise stated. In [$\left|\omega\right\rangle$]{}, the Schmidt coefficients either have the form $\alpha
\mu(i, n)$ or $\beta \mu(i, n)$ which we will refer to as $\alpha$ and $\beta$ terms respectively.
Our next observation implies the divergence of the normalization factor $C(f,n)$ for regular embezzling families.
\[divergentcn\] If [$\left|\mu(n)\right\rangle$]{} is a universal embezzling family, then $\mu(1,n) \rightarrow 0$ as $n \rightarrow \infty$. In particular, for regular universal embezzling families, $C(f, n)\rightarrow
\infty$ as $n \rightarrow \infty$.
Let $F$ be the fidelity of the embezzlement protocol, minimized over $|\varphi\>$. Lower bound $1-F$ by considering specifically $|\varphi\> = (|11\>+|22\>) / \sqrt{2}$: $$1 - {\operatorname{F}}({\ensuremath{\left|\mu(n)\right\rangle}}, {\ensuremath{\left|\omega\right\rangle}}) =
1 - \sum_i \mu_i \omega_i =
\frac{1}{2} \sum_{i=1}^{2n} (\mu_i - \omega_i)^2
\geq \frac{1}{2} (\mu_1 - \omega_1)^2 =
\frac{1}{2} \left(1 - \frac{1}{\sqrt{2}}\right)^2 \mu_1^2 \,,$$ where we use the shorthard $\mu_i$ for $\mu(i,n)$, $\omega_i$’s are the Schmidt coefficients of ${\ensuremath{\left|\omega\right\rangle}}$ in decreasing order, and $\omega_1 = \mu_1/\sqrt{2}$. Since $\mu_1 > 0$ (else [$\left|\mu(n)\right\rangle$]{} cannot be a valid quantum state), $F \rightarrow 1$ implies $\mu(1,n) \rightarrow 0$ as $n \rightarrow \infty$.
For regular families [$\left|\mu(f,n)\right\rangle$]{}, $\mu(1,n) = f(1)/\sqrt{C(f,n)}$, so $C(f, n)\rightarrow \infty$ as $n \rightarrow \infty$.
Note that [$\left|\mu(f, n)\right\rangle$]{} $=$ [$\left|\mu(cf, n)\right\rangle$]{} for any constant $c$. Thus, we consider the [*order*]{} of a regular universal embezzling family defined as follows: a universal embezzling family has [*order*]{} $g$ if and only if $C(f,n)=\Theta(g)$, [*e.g.*]{}, [$\left|\mu(f_{dh},n)\right\rangle$]{} has order $\ln n$. Lemma \[divergentcn\] shows that the “misalignment” of the first terms of [$\left|\mu(n)\right\rangle$]{} and [$\left|\omega(n)\right\rangle$]{} has to be corrected by a divergent order.
The next lemma gives a sufficient condition in terms of the asymptotic behavior of the ratio $\rho({\ensuremath{\left|\varphi\right\rangle}}, f, i) \, {:}{=} \,
\omega_i/\mu_i$. First, given ${\ensuremath{\left|\varphi\right\rangle}} = \sum_{j=1}^m \varphi_j
|j\>|j\>$ and $f$, we explain how to make this ratio well-defined for all $i \in \N$. Fix an arbitrary $n$ and let $\mu(i,n) = f(i)/\sqrt{C(f,n)}$ for $i=1,\cdots,n$. Let $\omega(i,n)$ be the $i$-th largest element in $S_n = \{ \mu(i,n) \varphi_j \}$. Define $\rho({\ensuremath{\left|\varphi\right\rangle}}, f, i)$ to be $\omega(i,n)/\mu(i,n)$ for $i=1,\cdots,n$. Note that the $\sqrt{C(f,n)}$ factors cancel out in the ratios. Furthermore, let $n'>n$ and define $\omega(i,n')/\mu(i,n')$ for $i =
1,\cdots, n'$ similarly. The first $n$ ratios coincide with $\omega(i,n)/\mu(i,n)$ because the $n$ largest terms in $S_{n'} = \{
\mu(i,n') \varphi_j \}$ are labeled by the same $(i,j)$’s as those in $S_n$.
\[convergenttl\] Let $f : \N \mapsto \R^+$ be a decreasing function with $C(f, n)\rightarrow\infty$. If $\forall{\ensuremath{\left|\varphi\right\rangle}}$, $\rho({\ensuremath{\left|\varphi\right\rangle}},f,i)
\rightarrow 1$, then ${\ensuremath{\left|\mu(f, n)\right\rangle}}$ forms a regular universal embezzling family.
Since $\rho \rightarrow 1$, given any $\eps > 0$, $\exists n_\eps$ such that $(1-\eps)\mu_i < \omega_i < (1+\eps)\mu_i$ for all $i > n_\eps$. Thus $$\begin{aligned}
{\operatorname{F}}({\ensuremath{\left|\mu(f, n)\right\rangle}}, {\ensuremath{\left|\omega\right\rangle}}) & = \sum_{i=1}^n\mu_i\omega_i \; = \;
\sum_{i=1}^{n_\eps}\mu_i\omega_i \; + \! \! \sum_{i={n_\eps}+1}^n\mu_i\omega_i
\; \; > \sum_{i={n_\eps}+1}^n\mu_i\omega_i
\\
& > (1-\eps)\sum_{i={n_\eps}+1}^n \mu_i^2
\; > \; (1-\eps) - \sum_{i=1}^{n_\eps} \mu_i^2
\; > \; (1-\eps) - \frac{C(f,n_\eps)}{C(f,n)} \,.\end{aligned}$$ Since $n_\eps$ does not depend on $n$, and $C(f, n) \rightarrow
\infty$, ${\operatorname{F}}({\ensuremath{\left|\mu(f, n)\right\rangle}}, {\ensuremath{\left|\omega\right\rangle}}) \rightarrow 1$. Thus, [$\left|\mu(f, n)\right\rangle$]{} forms a universal embezzling family. In fact, $1-F < \eps + C(f,n_\eps) / C(f,n)$.
We note on the side that Lemma \[convergenttl\] does not have a natural converse. Universal embezzling families may exist with infinitely many but intermittent violations of the condition $\rho({\ensuremath{\left|\varphi\right\rangle}}, f, i) \approx 1$.
Variations on [$\left|\mu(f_{dh}, n)\right\rangle$]{}
=====================================================
In this section and the next, we focus on regular universal embezzling families. We consider the “simplest” variation from $f_{dh}$, which is $f = g / \sqrt{x}$. This construction can be used in two ways to yield a universal embezzling family.
\[h\_construction\] Let $h: \N \rightarrow \R^+$. If, $C(f, n) \rightarrow \infty$, $f = h / \sqrt{x}$ is decreasing, and $h(kx + c) / h(x) \rightarrow 1$ as $x \rightarrow \infty$ for any constant $k \in \N$, $c \in \N \bigcup \{0\}$, then, [$\left|\mu(f, n)\right\rangle$]{} forms a universal embezzling family.
First, if $h(kx + c) / h(x) \rightarrow 1$ as $x \rightarrow \infty$, for any constants $k \in \N$, $c \in \N \bigcup \{0\}$, then, $h(k_1x + c_1) / h(k_2x + c_2) \rightarrow 1$ as $x \rightarrow \infty$ for any constants $k_1, k_2 \in \N$ and $c_1, c_2 \in \N \bigcup \{0\}$. This follows from the quotient rule $$\lim_{x \rightarrow \infty}
\frac{h(k_1x + c_1)}{h(k_2x + c_2)} =
\frac{ \lim_{x \rightarrow \infty} \frac{h(k_1x + c_1)}{h(x)} }
{ \lim_{x \rightarrow \infty} \frac{h(k_2x + c_2)}{h(x)} }
= 1 \,.$$
Following Lemma \[only2qubits\], consider ${\ensuremath{\left|\varphi\right\rangle}} = \alpha{\ensuremath{\left|11\right\rangle}} + \beta
{\ensuremath{\left|22\right\rangle}}$. Let $z = (\alpha / \beta)^2$. Recall that the optimal fidelity is achieved with decreasing Schmidt coefficients $\omega_i$ for $|\omega\>$. Here, we consider a particular ordering of Schmidt coefficients, ${\ensuremath{\left|\widetilde{\omega}\right\rangle}}$, which can be suboptimal. Then, any lower bound on ${\operatorname{F}}({\ensuremath{\left|\mu(f,n)\right\rangle}},
{\ensuremath{\left|\widetilde{\omega}\right\rangle}})$ also applies to ${\operatorname{F}}({\ensuremath{\left|\mu(f,n)\right\rangle}}, {\ensuremath{\left|\omega\right\rangle}})$.
First, suppose $z = p/q \in \Q$. Call the $p$ largest $\alpha$-terms (see remark to Lemma \[only2qubits\]) the first $\alpha$-block, the next $p$ largest $\alpha$-terms the second $\alpha$-block, and so on. Define the $\beta$-blocks similarly, but with block size $q$ instead. Construct ${\ensuremath{\left|\widetilde{\omega}\right\rangle}}$ such that the $l$-th block of $p+q$ terms comes from the $l$-th $\alpha$- and $\beta$-blocks. In other words, for $l(p+q)+1 \leq i \leq (l+1)(p+q)$: $$\widetilde{\omega}_i = \begin{cases} \alpha f(lp + C_1) / C(f, n)
\text{ or } \\
\beta f(lq + C_2) / C(f, n) \end{cases}$$ where $1 \leq C_1 \leq p$ and $1 \leq C_2 \leq q$. Now consider $\widetilde{
\omega}_i / \mu_i$ where $i = l(p+q) + C$ for any $0 \leq C \leq p+q$. If $\widetilde{\omega}_i$ is an $\alpha$-term, then $$\frac{\widetilde{\omega}_i}{\mu_i} = \alpha \sqrt{\frac{l(p+q) + C}{lp + C_1}}
\cdot\frac{h(lp + C_1)}{h( \,l (p{+}q) + C)} \,.$$ As $i \rightarrow \infty$, $l \rightarrow \infty$, $h(lp + C_1) / h(l(\,p{+}q)+C) \rightarrow 1$, so $\widetilde{\omega}_i / \mu_i \rightarrow \alpha \sqrt{(p+q)/p} = 1$. If $\widetilde {\omega}_i$ is a $\beta$-term, with a similar argument, $\widetilde\omega_i / \mu_i \rightarrow \beta \sqrt{(p+q)/q} = 1$. Then, by Lemma \[convergenttl\], ${\operatorname{F}}({\ensuremath{\left|\mu(f,n)\right\rangle}}, {\ensuremath{\left|\widetilde{\omega}\right\rangle}})\rightarrow1$.
If $z \not\in\Q$, the above proof applied to rational approximations of $z$ provides the desired result. More specifically, if $z = (\alpha / \beta)^2 \not\in\Q$, $\forall\delta > 0$, $\exists z^\prime = p/q \in \Q$ such that $$\label{bound}
\left(\frac{\alpha}{\beta}\right)^2 - \delta < \frac{p}{q} <
\left(\frac{\alpha}{\beta}\right)^2 + \delta \,.$$ The previous argument shows that $\widetilde{\omega}_i/\mu_i$ tends to either $\alpha\sqrt{(p+q) / p}$ or $\beta\sqrt{(p+q) / q}$. Eliminating $p/q$ in these expression using (\[bound\]) gives: $$1 - \frac{\delta\beta^4}{\alpha^2 + \delta\beta^2} < \alpha\sqrt{\frac{p+q}{p}}
< 1 + \frac{\delta\beta^4}{\alpha^2 + \delta\beta^2} ~~\text{ and }~~
1 - \delta\beta^2 < \beta\sqrt{\frac{p+q}{q}} < 1 + \delta\beta^2$$ and both quantities tend to $1$ as $\delta \rightarrow 0$.
\[g\_construction\] Let $g: \N \mapsto \R^+$ be an increasing function such that $f = g /
\sqrt{x}$ is decreasing. If, in addition, $\forall m \in \N, C(f, n/m) / C(f, n)
\rightarrow 1$ as $n \rightarrow \infty$, then [$\left|\mu(f,n)\right\rangle$]{} forms a universal embezzling family.
This proof derives heavily from [@DH02].
Claim: $\forall j \,, ~\omega_j \leq \mu_j$. Let $N(t) =
|\{l: \mu_t < \omega_l\}|$. The claim is equivalent to $N(t) < t$ as $\{\omega_l\}$ is decreasing. Since $\omega_l = \varphi_i f(j) / C(f,n)$ for some $i,j$, we let $N_i^t = | \{j: \mu_t < \varphi_i f(j) / C(f,n)\}|$. Now, $$\mu_t < \varphi_i \frac{f(j)}{C(f,n)} ~~\Leftrightarrow~~
f(t) < \varphi_i f(j) ~~\Leftrightarrow~~
\frac{jg(t)^2}{tg(j)^2} < \varphi_i^2 \,.$$ We can infer that $t \leq j$ since the middle inequality implies $f(j) < f(t)$ and $f$ is decreasing. Then, the last inequality and the monotonicity of $g$ imply that $j < \varphi_i^2 t$, so $N_i^t < \varphi_i^2 t$ and $N(t) = \sum_i N_i^t < t$ (recall the normalization $\sum_i \varphi_i^2 = 1$). Finally, $$\label{g_fid}
{\operatorname{F}}(|\mu(f,n)\>,{\ensuremath{\left|\omega\right\rangle}}) = \sum_{i=1}^n \mu_i \omega_i
\geq \sum_{i=1}^n \omega_i^2
\geq \sum_{j=1}^{\lfloor n/m \rfloor}\sum_{i=1}^m
\frac{\varphi_i^2 f(j)^2}{C(f,n)} =
\frac{C(f, \lfloor n/m \rfloor )}{C(f, n)} \rightarrow 1$$ where the last inequality comes from replacing the sum with possibly fewer and smaller terms.
Lemma \[g\_construction\] states that $f$ can fall off slower than $f
= 1 / \sqrt{x}$ as long as $C(f, n/m) / C(f, n) \rightarrow 1$.
New classes of regular universal embezzling families
====================================================
Now we present two sequences of regular universal embezzling families using Lemmas \[h\_construction\] and \[g\_construction\]. First, define $\lambda(x) = \ln (x + e)$ and its n-fold composition: $\lambda^0(x)=x$, $\lambda^1(x)=\ln(x+e)$, $\lambda^2(x)=\ln(\ln(x+e) + e)$, and so on.
Now define the $G$ and $H$ functions of class $r$ as: $$G_r(x) = \frac{1}{\sqrt x} \prod_{s=1}^r \sqrt{\lambda^s(x)}$$ $$H_r(x) = \frac{1}{\sqrt x} \prod_{s=1}^r \frac{1}{\sqrt{\lambda^s(x)}}$$
For every $r$, we will see that [$\left|\mu(G_r, n)\right\rangle$]{} and [$\left|\mu(H_r, n)\right\rangle$]{} have different orders and are universal embezzling families. Therefore, the number of orders for regular universal embezzling families is infinite.
To estimate $C(H_r, n)$, we use integral approximations: $$\frac{d}{dx} \lambda^{r+1}(x) =
\prod_{s=0}^{r} {1 \over \lambda^s(x) + e} \approx
H_r(x)^2
\Rightarrow \sum_{i=1}^n H_r(i)^2 \approx
\int_{1}^n H_r(x)^2dx \approx
\lambda^{r+1}(n)$$ Thus, the order of [$\left|\mu(H_r, n)\right\rangle$]{} is $\lambda^{r{+}1}(n) \approx
\ln^{r{+}1}(n)$ for large $n$.
For $C(G_1, n)$, we apply integral approximations and the inequality $G_1(x)^2 \leq (\ln(x{-}e))/(x{-}e)$ for $x \geq 5$ to obtain: $$\label{g_estimate}
\int_1^n \frac{\ln(x+e)}{x+e} < \int_1^n \frac{\ln(x+e)}{x} \approx
\sum_{i=1}^n G_1(i)^2 \leq \sum_{i=1}^5 G_1(i)^2
+ \int_5^n \frac{\ln(x-e)}{x-e} \,.$$ The integrals are all well approximated by $(\ln n)^2/2$. Thus $C(G_1,n) = \Theta[(\ln n)^2]$. For general $C(G_r, n)$, there is no simple approximation, but we can show that subsequent orders are progressively “higher.” First, $$C(G_{r{+}1},n) = \sum_{i=1}^n G_{r{+}1}(i)^2
= \sum_{i=1}^n \lambda^{r{+}1}(i) \, G_{r}(i)^2
\geq \sum_{i=1}^n G_{r}(i)^2 = C(G_{r},n) \,.
\label{eq:Gorder}$$ We show by contradiction that $C(G_{r+1},n) \not = \Theta[C(G_r, n)]$. If so, there are constants $\kappa,n_0$, such that $\forall n > n_0$, $C(G_{r+1}, n) \leq \kappa C(G_r, n)$. Pick $n_1 > n_0$ so that $\lambda^{r{+}1}(n_1) \geq 3 \kappa$, and $n_2 > n_1$ such that $\sum_{i=1}^{n_1} G_r(i)^2 \leq \sum_{i=n_1{+}1}^{n_2} G_r(i)^2$. Now, $$\kappa C(G_r,n_2)
\leq
2 \kappa \sum_{i=n_1{+}1}^{n_2} G_r(i)^2
\leq
\frac{2}{3} \lambda^{r{+}1}(n_1) \sum_{i=n_1{+}1}^{n_2} G_r(i)^2
\leq
\frac{2}{3} \sum_{i=n_1{+}1}^{n_2} G_r(i)^2 \lambda^{r{+}1}(i)^2
\leq
\frac{2}{3} C(G_{r{+}1},n_2)$$ a contradiction.
Embezzling Properties of $|\mu(G_r,n)\>$ {#embezzling-properties-of-mug_rn .unnumbered}
----------------------------------------
First, we sketch that $G_r(x)$ is decreasing. Let $t(x)=\lambda(x)/\sqrt{x}$. Then, $t(x)$ is decreasing because its first derivative has the same sign as $\theta(x) = 2x - (x+e) \ln (x+e)$, and $\forall x>0$, $\theta(x) < 0$ because its first derivative is negative and $\theta(0) < 0$. Therefore, $\lambda(x+1) / \sqrt{x + 1} <
\lambda(x) / \sqrt{x}$ and $\lambda(x+1)/\lambda(x) < \sqrt{x+1} / \sqrt{x}$ for $x > 0$. Repeating this result yields: $\lambda^2(x+1)/\lambda^2(x) <
\sqrt{\lambda(x+1)}/ \sqrt{\lambda(x)} < [(x + 1) / x]^{1/4}$, etc. Now: $$\frac{G_r(x+1)^2}{G_r(x)^2} =
\frac{x}{x+1}\prod_{i=1}^r\frac{\lambda^i(x+1)}{\lambda^i(x)} <
\frac{x}{x+1}\prod_{i=1}^r\left[\frac{x+1}{x}\right]^{1/2^i} < 1$$ so the positive functions $G_r$ are all decreasing.
Second, $\forall r \geq 1, C(G_r, n)$ diverges (see Eq. (\[eq:Gorder\])).
We can establish that [$\left|\mu(G_1, n)\right\rangle$]{} forms a universal embezzling family using Lemma \[g\_construction\], by using the estimate (\[g\_estimate\]) to conclude that $$\frac{C(G_1, n/m)}{C(G_1, n)} \sim \left(1 - \frac{\ln m}{\ln n}\right)^2 \,.$$ However, the lower bound for fidelity of embezzlement by [$\left|\mu(G_1, n)\right\rangle$]{} is no better than that of [$\left|\mu(f_{dh}, n)\right\rangle$]{}, despite Lemma \[convergenttl\] (recall: $1 - F < \eps + C(f, N_\eps) / C(f, n)$) and the higher order of $G_1$.
For other $G_r$, we will show that Lemma \[h\_construction\] applies. We first show by induction that $\forall s \in \N$, $\lambda^s(k x + c) /
\lambda^s(x) \rightarrow 1$ for any constants $k, c$
For $s=1$: $$\frac{\lambda^1(kx + c)}{\lambda^1(x)}
= \frac{\ln(x + c/k) + \ln k}
{\ln(x)} \rightarrow 1 \,.
\label{requals1}$$
For $s \geq 2$, both $\lambda^{s{-}1}(kx + c) \rightarrow \infty$ and $\lambda^{s{-}1}(x) \rightarrow \infty$. By induction hypothesis, their ratio tends to $1$. Thus, the proven base case (\[requals1\]) implies $\lambda^s(kx + c) / \lambda^s(x) = \lambda^1( \lambda^{s{-}1}
(kx + c)) / \lambda^1( \lambda^{s{-}1}(x)) \rightarrow 1$.
Then, for any class $r$, by the limit rule for products and the continuity of $\sqrt{\cdot}$ and $1/\cdot$ over the range of interest, both $\prod_{s=1}^{r}\sqrt{\lambda^s(x)}$ and $\prod_{s=1}^r1/\sqrt{\lambda^s(x)}$ satisfy the condition in Lemma \[h\_construction\]. Thus, all $3$ conditions in Lemma \[h\_construction\] holds for $G_r$ and $|\mu(G_r,n)\>$ forms a universal embezzling family.
Embezzling Properties of $|\mu(H_r,n)\>$ {#embezzling-properties-of-muh_rn .unnumbered}
----------------------------------------
$H_r$ is obviously decreasing $\forall r$. We have already shown that $\prod_{s=1}^r
1/\sqrt{\lambda^s(x)}$ satisfies the condition in Lemma \[h\_construction\] and $C(H_r, n) \rightarrow \infty$ from our estimate of $C(H_r, n)$. Therefore, by Lemma \[h\_construction\], [$\left|\mu(H_r, n)\right\rangle$]{} forms a universal embezzling family.
However, [$\left|\mu(f_{dh}, n)\right\rangle$]{} performs better when embezzling any entangled state. This follows from Lemma \[divergentcn\] and the fact $C(H_r, n) / C(f_{dh},n) \rightarrow 0$ as $n \rightarrow \infty$.
Entanglement of [$\left|\mu(f_{dh}, n)\right\rangle$]{}, [$\left|\mu(G_1, n)\right\rangle$]{}, and [$\left|\mu(H_1, n)\right\rangle$]{} {#entanglement-of-leftmuf_dh-nrightrangle-leftmug_1-nrightrangle-and-leftmuh_1-nrightrangle .unnumbered}
---------------------------------------------------------------------------------------------------------------------------------------
Another metric of embezzlement efficiency is the amount of entanglement required in creating [$\left|\mu\right\rangle$]{}. For the original embezzling family proposed in [@DH02], using integral approximations: $$\text{Ent}({\ensuremath{\left|\mu(f_{dh}, n)\right\rangle}}) = -\sum_{i=1}^n \mu_i^2 \log_2(\mu_i^2)
\approx -\sum_{i=1}^n \frac{1}{i}\frac{1}{\ln n} \log_2
\frac{1}{i}\frac{1}{\ln n} \,.$$ Simplifying the above and using integral approximations, the leading term of $\text{Ent}({\ensuremath{\left|\mu(f_{dh}, n)\right\rangle}})$ is $(\log_2 n)/2$.
Similarly, we can estimate $\text{Ent}({\ensuremath{\left|\mu(G_1, x)\right\rangle}})$. We use $C(G_1,n) \approx (\ln n)^2/2$ and the approximation $\lambda(x) \approx \ln x$ to conclude that $$\text{Ent}({\ensuremath{\left|\mu(G_1, n)\right\rangle}})
\approx -\sum_{i=1}^n \frac{\ln i}{i}\frac{2}{(\ln n)^2} \log_2
\frac{\ln i}{i}\frac{2}{(\ln n)^2}
\approx \frac{2}{3} \log_2 n$$ where the last estimate concerns only the lead term and uses integral approximations.
Finally, we use $C(H_1,n) \approx \lambda^2(n) \approx \ln \ln n$ to estimate $\text{Ent}({\ensuremath{\left|\mu(H_1, n)\right\rangle}})$ which is $\approx (\log_2 n) /
(\ln \ln n)$.
For a fixed Schmidt rank, $\text{Ent}(|\mu(G_1,n)\>)$ and $\text{Ent}(|\mu(f_{dh},n)\>)$ are of the same order. Meanwhile, $\text{Ent}(|\mu(H_1,n)\>) \ll \text{Ent}(|\mu(f_{dh},n)\>)$. However, if one fixes the precision, a higher Schmidt rank is needed to embezzle using $H_1$ than $f_{dh}$.
Outperforming [$\left|\mu(f_{dh}, n)\right\rangle$]{}?!
=======================================================
In the previous sections, we examine regular families that do not have order $\ln n$. There are interesting sequences that are not regular. One such sequence is presented in [@LTW08] (due to [@HS-private]): $${\ensuremath{\left|\mu(n)\right\rangle}} = \sqrt{\frac{2}{N+1}} \sum_{k=1}^N
\sin\left(\frac{k\pi}{N{+}1}\right)
{\ensuremath{\left|00\right\rangle}}^{\otimes k} {\ensuremath{\left|\varphi\right\rangle}}^{\otimes N - k + 1}
\label{eq:aram-peter}$$ where $n = 2^N$. This sequence enables the embezzling of the specific state ${\ensuremath{\left|\varphi\right\rangle}}$ with fidelity at least $1-\pi^2/2N^2 = 1-\pi^2/2
(\log_2 n)^2$, a marked improvement over the provable lower bound $1-O(1/\log_2 n)$ of the fidelity achieved by [$\left|\mu(f_{dh}, n)\right\rangle$]{}. The sequence in (\[eq:aram-peter\]) also saturates an upper bound of the fidelity proved in [@DH02]. However, if ${\ensuremath{\left|\varphi\right\rangle}} = ({\ensuremath{\left|11\right\rangle}} + {\ensuremath{\left|22\right\rangle}})
/ \sqrt{2}$ and Alice and Bob want to embezzle ${\ensuremath{\left|\varphi'\right\rangle}} = \alpha {\ensuremath{\left|11\right\rangle}} + \beta {\ensuremath{\left|22\right\rangle}}$, the fidelity $\rightarrow (\alpha+\beta)/\sqrt{2}$ as $N \rightarrow \infty$ which is bounded away from $1$ when ${\ensuremath{\left|\varphi'\right\rangle}} \neq {\ensuremath{\left|\varphi\right\rangle}}$.
Instead, we propose the following. Let $gh$ be defined, for fixed $n$, and for $x \in \N, 1 \leq x \leq n$ as: $$gh(x) = \begin{cases} H_1(1) \text{ when } x = 1 \\
H_1(x) \text{ when } C(gh, x - 1) \geq \ln(x) \\
G_1(x) \text{ when } C(gh, x - 1) < \ln(x) \,. \end{cases}$$ Then define $GH(x)$ for $x \in \N, 1 \leq x \leq n$ as $gh(x)$ with elements in decreasing order (the $n$ dependence is implicit here) and designate ${\ensuremath{\left|\mu(n)\right\rangle}} = \sum_{i=1}^n GH(i){\ensuremath{\left|i\right\rangle}}{\ensuremath{\left|i\right\rangle}}$. Due to the limited dependence on $n$, we can still define $C(GH, n)$ as before, and it differs from $\ln n$ by at most $G_1(n)^2$ or $H_1(n)^2$, but both $G_1(x)$ and $H_1(x) \rightarrow 0$ as $x
\rightarrow \infty$. Therefore, $C(GH, n) \rightarrow \ln n$ as $n
\rightarrow \infty$.
The precise performance of ${\ensuremath{\left|\mu(n)\right\rangle}}$ as a universal embezzling family is hard to analyse. So, we [*numerically*]{} evaluate the optimal fidelity (see Section \[primer\]) of embezzling three sample states: ${\ensuremath{\left|\varphi_{+}\right\rangle}} = (2{\ensuremath{\left|00\right\rangle}} + {\ensuremath{\left|11\right\rangle}}) / \sqrt{5}$, ${\ensuremath{\left|\varphi_{*}\right\rangle}} = (\sqrt{\pi {-} 1}{\ensuremath{\left|00\right\rangle}} + {\ensuremath{\left|11\right\rangle}}) / \sqrt{\pi}$, and ${\ensuremath{\left|\varphi_{\circ}\right\rangle}} = ({\ensuremath{\left|00\right\rangle}} + {\ensuremath{\left|11\right\rangle}}) / \sqrt{2}$, using ${\ensuremath{\left|\mu(n)\right\rangle}}$ for $n = 2^N$, $N=3, \cdots, 33$. For comparison, we also perform numerical optimization for the fidelity of embezzlement using ${\ensuremath{\left|\mu(f_{dh}, n)\right\rangle}}$.
Figure \[fig:num\] summarizes the result.
All calculations are done in IEEE double-precision. The main source of inaccuracy in the numerical optimization is the accumulation of machine truncation errors in the calculation of $C(GH, n)$. We directly calculate $C(GH, n)$ for $3 \leq N \leq 26$ and approximate $C(GH, n)$ by $\ln n$ for $18 \leq N \leq 33$. The two methods yield optimal fidelities differing by less than $2 \times 10^{-6}$ for $18 \leq N \leq 26$.
A quick inspection of Figure \[fig:num\] suggests that ${\ensuremath{\left|\mu(n)\right\rangle}}$ is indeed a universal embezzling family. Furthermore, ${\ensuremath{\left|\mu(n)\right\rangle}}$ outperforms ${\ensuremath{\left|\mu(f_{dh}, n)\right\rangle}}$ for the specific cases studied.
We extrapolate the data to try to understand the asymptotic behavior of ${\ensuremath{\left|\mu(n)\right\rangle}}$. The least square fits to the optimal fidelities to embezzle ${\ensuremath{\left|\varphi_{+}\right\rangle}}, {\ensuremath{\left|\varphi_{*}\right\rangle}}, {\ensuremath{\left|\varphi_{\circ}\right\rangle}}$ using ${\ensuremath{\left|\mu(n)\right\rangle}}$ are: $$\begin{aligned}
F_{+} & = 0.9980 - 0.0759/N - 0.6358/N^2
\nonumber
\\
F_{*} & = 0.9976 - 0.1395/N - 0.6691/N^2
\nonumber
\\
F_{\circ} & = 0.9974 - 0.1971/N - 0.6862/N^2 \,.
\nonumber\end{aligned}$$ The fitting parameters are insensitive to the method used to generate $C(GH, n)$. When fitting the data for $N_0 \leq N \leq 33$, the fitting parameters are slightly sensitive to $N_0$. We show the fits for $N_0 = 10$, when the constant term is smallest, the magnitude for the coefficients of the $1/N$ and $1/N^2$ terms are smallest and largest respectively. For $N_0$ ranging from $5$ to $20$, the constant can increase by $0.001$, the magnitude of the second coefficients can increase by $0.03$, that of the third coefficient can decrease by $0.3$. We cannot conclude convincingly whether $F \rightarrow 1$ as $N \rightarrow \infty$.
The corresponding fits for the embezzling family ${\ensuremath{\left|\mu(f_{dh},n)\right\rangle}}$ for $N_0 = 10$ are: $$\begin{aligned}
F_{+} & = 0.999982 - 0.377165/N + 0.282380/N^2
\nonumber
\\
F_{*} & = 0.999970 - 0.484107 / N + 0.359519 / N^2
\nonumber
\\
F_{\circ} & = 0.999960 - 0.565744 / N + 0.418400 / N^2
\nonumber\end{aligned}$$ When $N_0$ ranges from $5$ to $20$, the constant can increase by $0.0001$, the magnitudes of the second and third coefficients can increase by $0.01$ and $0.1$.
From the various fits, ${\ensuremath{\left|\mu(f_{dh},n)\right\rangle}}$ starts to outperform ${\ensuremath{\left|\mu(n)\right\rangle}}$ when $N \approx 140-160$.
We note on the side that [@DH02] provides lower and upper bounds on the optimal fidelity of embezzlement using ${\ensuremath{\left|\mu(f_{dh}, n)\right\rangle}}$. We present the actual optimal performance (numerically) for small $N$ that may be of interest elsewhere.
Discussions
===========
We have provided necessary conditions and sufficient conditions for universal embezzling in the bipartite setting. We exhibit an infinite number of inequivalent families, present a family that outperforms that proposed in [@DH02] for small $N$, but the latter [ *appears*]{} optimal asymptotically based on our numerics. Our work does not resolve whether there is a regular or general universal embezzling family achieving fidelity $1-O(1/(\log_2 n)^2)$. We hope our results are a step towards answering some of these questions.
Acknowledgements
================
We thank Wim van Dam, Aram Harrow, Patrick Hayden, Thomas Vidick, and Volkher Scholz for their generous sharing of research results concerning embezzlement, and Thomas Vidick for very useful comments on an earlier version of the manuscript. The fidelity of embezzling ${\ensuremath{\left|\varphi'\right\rangle}}$ from (\[eq:aram-peter\]) was calculated by Michal Kotowski.
[BDH[[$^{+}$]{}]{}09]{}
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[^1]: Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada.
[^2]: University of Cambridge, Cambridge, Cambridgeshire, United Kingdom.
|
---
author:
- Taro Langner
- Andreas Östling
- Lukas Maldonis
- Albin Karlsson
- Daniel Olmo
- Dag Lindgren
- Andreas Wallin
- Lowe Lundin
- Robin Strand
- Hkan Ahlström
- Joel Kullberg
bibliography:
- 'references.bib'
title: 'Kidney segmentation in neck-to-knee body MRI of 40,000 UK Biobank participants'
---
Introduction
============
The UK Biobank studies more than half a million volunteers participants, collecting health data on medical records, blood and urine samples, lifestyle, and genetics.[@sudlow_uk_2015] Together with the vast range of metadata and a long-term follow-up, medical images are acquired for a subgroup of 100,000 participants. Of these, 10,000 are also scheduled to attend a repeat scan at a second, later imaging visit. The protocols include Magnetic Resonance Imaging (MRI) of the brain, heart, pancreas and liver, but also neck-to-knee body MRI[@west_feasibility_2016] which combines vast amounts of anatomical information in a single comprehensive 6-minute scan, which covers the all tissue of the kidneys in overlapping imaging stations. The human kidney plays a vital role in the filtration of blood, secretion of hormones, and regulation of blood pressure. Its shape and function are impacted by genetic factors, but also underlie natural variation based on sex, body size, and age.[@emamian1993kidney] In addition to congenital anomalies such as renal fusion, or horseshoe kidneys,[@glodny2009kidney] and autosomal dominant polycystic kidney disease (ADPKD),[@sharma2017kidney] morphological changes with associated medical complications can arise from factors such as chronic kidney disease, hypertension, [@hoy2008nephron] and diabetes. [@rossing2002risk] Kidney volume as a biomarker is therefore of clinical interest for diagnostics, monitoring of disease progression, and medical hypothesis testing. With the extensive image data available in the UK Biobank, non-invasive, image-based assessments of kidney volume could provide a substantial sample size of these measurements.
In clinical practice, kidney volume is often approximated with a rotational ellipsoid model based on kidney width, depth, and length as obtained by sonography.[@bakker1998vitro] However, validation with water displacement methods has shown that ellipsoid models can underestimate kidney volume by up to 29%,[@cheong2007normal] or 25% even with MRI.[@bakker1998vitro] As an alternative, measurements by voxel count, or disc-summation, can be obtained by delineation of three-dimensional voxels which correspond to kidney tissue in volumetric medical images. When obtained from MRI, these segmentation-based measurements have been found to show no significant deviation to those determined by water displacement.[@bakker1998vitro] When applied to the UK Biobank, manual segmentation is no longer feasible, however, as even a typical processing time of ten minutes per subject would amount to tens of thousands of man-hours for the UK Biobank cohort as a whole. A rich body of literature has been devoted to computer-aided segmentation techniques of the kidneys and other visceral organs in volumetric medical imaging data. For the kidney in particular, various approaches have been proposed predominantly for image data from Computed Tomography (CT), including techniques based on statistical shape models and region growing [@lin2006computer], graph cuts [@ali2007graph], and deformable boundaries [@shehata20183d]. Contemporary benchmark challenges are increasingly dominated by machine learning techniques such as deep learning with convolutional neural networks, as seen in the MICCAI *2019 Kidney and Kidney Tumor Segmentation (KiTS19)* [@heller2019state] and with CT image data, in which similar approaches have also been proposed for measurements of total volume in subjects with ADPKD. [@sharma2017automatic] Fully convolutional networks for semantic image segmentation [@long2015fully] range from architectures such as the U-Net with 2D data[@ronneberger2015u] to 2.5D[@han2017automatic] and 3D techniques,[@isensee2019attempt] which are able to learn the task of segmenting specific image structures from reference data in training. In the UK Biobank, related approaches have already been applied for segmentation of cardiac MRI of up to 5,000 subjects [@zheng20183; @bai2018automated] and large-scale segmentation of this data has been conducted with other methods as well, such as sparse active shape models on up to 20,000 subjects. [@attar2019quantitative]
The purpose of this work is to propose, validate, and apply a segmentation pipeline for automated quantification of parenchymal kidney tissue in UK Biobank neck-to-knee body MRI. A neural network based on a 2.5D U-Net variation is evaluated in cross-validation and applied for inference to all 40,000 subjects with available MRI data, resulting in measurements of healthy tissue volume of the left and right kidney, as well as their mutual distance. Potential failure cases and other outliers are identified with algorithmic quality ratings, with a large number of anomalies such as renal fusion and polycystic cases being highlighted for scrutiny. We are not aware of any existing kidney volume measurements within the UK Biobank, or any other study with MRI-based measurements of kidney volume at a comparable sample scale. The obtained values and code samples can be shared as return data and made available for further research.
The following contributions are made:
- Segmentation pipeline with accurate and efficient 2.5D U-Net variation
- Deep regression control
- Quality controlled results
Methods
=======
A neural network was trained for semantic segmentation of two-dimensional, axial slices of the UK Biobank neck-to-knee body MRI. Manually created reference segmentations of parenchymal kidney volume in 122 subjects were used to train and validate the network, and to make design choices regarding data pre-processing and hyperparameter selection. The resulting network configuration was then embedded in a processing pipeline and applied in inference to the entire cohort, with algorithmic quality ratings flagging suspected failure cases for exclusion. A schematic overview over the experiments is shown in Fig.\[fig\_overview\].
![Among all UK Biobank subjects, two subsets *A* and *B* were manually segmented. A neural network was evaluated in two cross-validation experiments on these and applied for inference to all remaining subjects. After excluding 5% of results in two quality control stages, about 37,500 measurements remain as final result.[]{data-label="fig_overview"}](overview-eps-converted-to.pdf){width="\textwidth"}
![Segmented parenchymal tissue of right (red) and left (blue) kidney in MRI of a male subject.[]{data-label="fig_segmented"}](5666132_labeled.png){width="\textwidth"}
UK Biobank data
---------------
At the time of writing, UK Biobank neck-to-knee body MRI of 40,264 participants has been released. Subjects were recruited by letter from the National Health Service and scanned at three different imaging centres in Great Britain with a Siemens Aera 1.5T device, using a dual-echo protocol that acquired overlapping images in six stations covering the body from neck to knee within about 6 minutes with TR = 6.69, TE = 2.39/4.77 ms, and flip angle 10deg.[@west_feasibility_2016] The reconstructed, volumetric station images encode voxel-wise intensity values with a separate water and fat signal (UK Biobank field 20201-2.0). The head, arms, and lower legs extend outside of the field of view and are often distorted near the image borders.
The kidneys are typically located in the second and third imaging station, each of which were acquired in a 17s breath-hold with typical dimensions of voxels of mm. In this work, those subjects with image artefacts in these two stations, such as water-fat swaps, background noise, metal objects, but also non-standard poses, misalignment in the scanner, and corrupted data were excluded after visual inspection of mean intensity projections,[@Langner2020] leaving 39,560 subjects. At scan time, these men and women (52% female) were 44 to 82 (mean 64) years old, with BMI 14 to 62 (mean 27) $kg/m^2$ and a 95% majority of self-reported White British ethnicity.
-TODO: Shorten the following Due to their overlapping field of view, the two adjacent imaging stations may both contain the same part of the kidneys. To take this into account, all validation metrics and measurements in this work are based on a fused representation in which the intensity values and segmentation labels of both stations were combined by interpolation along their mutual overlap.[@Langner2019] The neural network does not depend on image fusion, however, since it is applied to two-dimensional, axial slices that are entirely contained within a given station. The network is therefore directly applied to the station image data after discarding the outermost three slices, which are strongly affected by artefacts such as signal loss and folding. Image fusion is only performed subsequently, on the station image data and combined slices of pixel-wise labels assigned by the network. The image fusion can reconstruct a combined, physically consistent field of view but provides no guarantee for anatomical coherence. The same structure may still be distorted or occur twice in the fused image due to motion during image acquisition. Motion artefacts must therefore be expected to affect even the fused images, with changes in pose and deformations due to the separate breath-holds during which the images were acquired. As described in the following sections, the pipeline takes several measures to quantify these effects.
### Reference segmentations
Three operators created reference segmentations by marking all voxels corresponding to healthy, parenchymal kidney tissue in the water signal of the second and third imaging station. The segmented tissue corresponds to the cortex and medulla, both of which appear with high MRI water signal intensity. Based on their lower signal intensities cysts, the calyces, ureters, and major vessels were excluded. These reference segmentations were used to train and validate the neural network. For the final measurements, the left and right kidney were separated by subsequent post-processing with an algorithmic connected component analysis. An example for a segmented MRI slice is seen in Fig.\[fig\_segmented\].
Dataset *A* consists of 64 subjects selected by random sampling stratified by age, gender, and weight,[@ostling2019automated] whose water signal images were manually segmented in the software SmartPaint[@malmberg2017smartpaint] by an experienced image analyst. The consistency of these segmentations was evaluated on a subset of 5 subjects, which were segmented repeatedly by the operator for a blinded assessment of intra-operator variability.
Dataset *B* contains another 64 subjects, with no overlap to dataset *A*, with 33 cases segmented by the second and 31 cases by the third operator, also segmented in SmartPaint. Instead of using a fully manual procedure, this dataset was segmented by correcting the proposals generated by a preliminary inference network trained on dataset *A*. The 64 candidates were selected among the most challenging cases based on an algorithmic quality rating for segmentation smoothness which is described in more detail further below. As a result, subjects with morphological anomalies are over-represented in this dataset, with several pathological cases that are challenging to segment even for human operators. To determine inter-operator variability, these two operators also segmented the same subset of 5 subjects from dataset *A* for which the intra-operator variability was previously determined.
Neural network configuration
----------------------------
A fully convolutional neural network was trained for semantic image segmentation of axial slices. The underlying architecture is a 2.5D variation of the U-Net [@ronneberger2015u] with a VGG11 encoder pretrained on ImageNet,[@ternaus] extended with ResNet-style short skip connections. The 2.5D input formatting combines three adjacent slices to form one sample, providing additional volumetric information to the network. Similar techniques with five slices [@han2017automatic] ranked among the most successful contributions for segmentation of liver tissue in the 2017 MICCAI Liver Tumor Segmentation (LiTS) Benchmark Challenge. [@bilic2019liver]
For network training, the three outermost axial slices of each station were removed due to excessive artefacts, such as signal loss and folding. Each remaining axial slice was individually normalized after clipping of the brightest one percent of values for stability. No image fusion was performed at this stage. The network assigned pixel-wise labels to two-dimensional, axial slices based on a 2.5D input sample is formed by a stack of three adjacent slices from the water signal volume. In addition to the target slice, one additional slice is extracted from above and below each, using a periodic border condition. For an evenly divisible size, the slices were symmetrically zero-padded to form a stack of $224 \times 192 \times 3$ pixels. Each of these stacks forms one input sample for the network, which predicts a two-dimensional segmentation for the central slice in the format of $224 \times 192$ pixels. The network architecture was trained for 80,000 iterations with a pixel-wise cross-entropy loss, batch size one, and online augmentation with randomized, elastic deformations.[@ostling2019automated] Using the Adam optimizer, a learning rate of $0.0001$ is maintained until iteration 60,000 and then lowered by factor 10 for improved stability. After reverting the slice padding, the segmented slices with pixel-wise labels can be stacked to obtain voxel-wise labels for an entire input station. A GitHub implementation is linked in the supplementary material.
### Training data
Three experiments were conducted with this neural network configuration. For training and validation of the network, both dataset *A* and *B* were available, with image data of 64 subjects each. The samples of dataset *B*, however, were selected among the most challenging, including some pathological cases and are largely based on refined segmentations originally proposed by the network itself. Using these samples for validation would yield results that are not representative for the UK Biobank cohort as a whole and dataset *B* was therefore never used for validation. Six of its 64 cases were furthermore excluded due to excessive morphological anomalies, tumours, cysts and congenital renal fusion where both kidneys are interconnected and form a single structure. Thus, 58 cases of dataset *B* remained for further use. A single-operator validation was performed by conducting a classical 8-fold cross-validation on the 64 cases of dataset *A*. This dataset was consequently split on subject level into 8 subsets of even size, for each of which in turn segmentations were predicted by a network instance trained on data of all remaining 7 subsets. Each network instance was thereby trained on data of 56 subjects, corresponding to about 4,250 labelled slices. This single-operator cross-validation aims to quantify how well the operator of dataset *A* can be emulated on a representative sample of the UK Biobank.
Secondly, the main validation experiment quantifies the benefit of access to both datasets (*A* $\cup$ *B*) by repeating the cross-validation with the exact same splits, but with samples of dataset *B* added for training only. In this way, the network instance for each split used the same validation subjects as before, but was trained on both the remaining 56 cases of dataset *A* and all well-formatted 58 cases of dataset *B* combined, for a total of 114 subjects, or 8,650 labelled slices for training. The network thereby learns a compromise in segmentation style between all operators, with validation results that are representative for the actual inference pipeline.
Finally, the network was applied for inference itself to all those subjects with no reference data. It was trained on a combined dataset of the 64 cases of datasets *A* and the well-formatted 58 cases of dataset *B*, for a total of 122 cases with about 9,250 labelled slices in total.
Measurements
------------
The second and third stations of a given subject were labelled by the neural network and subsequently fused into a single, combined volume for both the water signal and voxel-wise labels each, by resampling to a common voxel grid and interpolation along the overlapping areas. A kidney volume measurement was obtained from these fused segmentation images by summing up the number of voxels labelled as kidney tissue, scaled with the physical voxel dimensions. Post-processing extracted the two largest connected components individually, which are assumed to be the left and right kidney, identified by the relative position of their centres of mass. The latter also enables a measurement of the relative position and euclidean distance between both kidneys.
Validation metrics
------------------
When validating the network output against known reference segmentations, the segmentation quality was evaluated with the Sørensen–Dice coefficient, or Dice score, and Jaccard index. To avoid averaging with empty imaging stations, these metrics were only calculated after fusing the image stations for a given subject. All measurements were likewise only derived after image fusion, and evaluated with several complementary metrics. Averaging the absolute differences between predicted value and reference for all subjects yields a mean absolute error (MAE). In addition to this value, a relative error measurement is reported for a better sense of scale. Dividing the absolute differences on a per-case basis by the true measurement value, estimated here as the mean between prediction and reference, before averaging, results in a symmetric mean absolute percentage error (SMAPE). For a single example case with true volume of 250cm$^3$, an absolute difference of 25cm$^3$ would thereby amount to a SMAPE of 10%. Instead of estimating the true value as the mean of prediction and reference, an alternative would be to simply use the reference value directly. However, the known high variation between references created by different operators suggests that the chosen symmetrical definition may be more robust. In addition to these metrics, the quality of fit between predicted values and reference can be quantified with the coefficient of determination (R$^2$), whereas error bounds are estimated by the 95% limits of agreement (LoA).
Algorithmic quality controls
----------------------------
When applying the network in inference to those cases with no existing reference measurements, the aforementioned validation metrics can not be calculated. Exhaustive quality control by manual inspection is likewise hardly feasible at this scale. The evaluation during inference is therefore based on algorithmic quality ratings as simple indicators for outliers and potential failure cases. While ratings of high quality provide no guarantee for correct results, low ratings can help to identify those cases that are likely to contain anomalies or potential segmentation failures. The distribution of ratings were examined in two separate control stages, after each of which the most severe outliers were flagged for exclusion (see supplementary material for details).
The first stage of quality controls evaluates the image quality with an *image fusion rating*, *segmentation fusion rating*, and *location rating*. Even a hypothetically perfect segmentation can result in faulty measurements if parts of the kidney are not contained in the image at all or occur redundantly due to motion. The agreement between both stations in their overlapping area is therefore examined, both for the MRI water signal and their segmented labels. Large differences indicate bad anatomical alignment between the imaging stations, leading to low quality ratings. Additionally, the relative offset along the longitudinal axis between the centre line of the fused subject volume and the centre of mass of all segmented voxels is penalized. Low values for this rating indicate that the kidneys are located at the top or bottom edge, possibly extending beyond the field of view. The second quality control stage rates the segmentation quality with a *segmentation smoothness rating* and *scrap volume rating*, examining the smoothness of the segmented volume along the longitudinal axis and the share of voxels which are not part of either of the two largest connected components. The slice-wise segmentation by the 2.5D neural network may encounter failure cases where entirely disconnected islands of tissue are spuriously segmented or excluded due to their position and local appearance. Low ratings indicate atypical shapes for the segmented kidney tissue which may require further scrutiny.
Assuming that the measurements obtained by the proposed segmentation pipeline are objective and accurate, a high level of agreement should be possible with other, radically different methodologies. The inferred measurements were therefore provided to an alternative machine learning technique, which was trained to emulate them in cross-validation. A close agreement does not provide any guarantee that the proposed measurements are semantically correct, since both techniques could in principle be affected by the same confounding factors. Individual outliers with low agreement, however, could identify failure cases of the proposed system that would require additional scrutiny.
The alternative method chosen for this purpose is a framework for image-based regression with convolutional neural networks for biometry [@Langner2020]. This technique can emulate several measurements, including volumes of muscle and adipose tissue, from two-dimensional projections of the UK Biobank neck-to-knee body MRI. Due to performing a regression, it only requires numerical values and does not depend on direct access to ground truth segmentation images. In this work, the regression framework was trained without any further changes in 10-fold cross-validation to emulate the measurements of the segmentation pipeline. The quality of fit depends in part on the performance of the regression framework itself, which needs to analyse the kidneys in the two-dimensional representation of the MRI data. To our knowledge, no previous attempt has been made to use it for kidney biometry, so that this limitation can not yet be quantified. A second limiting factor, which is of chief interest in this work, is the consistency of the measurements that form the ground truth data. Since the regression framework can not be expected to learn a correct prediction of reference values that are unsystematic, erratic outliers, large errors for individual samples in cross-validation indicate potential flaws in the reference data. These outliers can then be traced back to potential failure cases of the proposed segmentation pipeline, which supplied the reference data to the regression framework.
All those subjects with measurements that remained after exclusion with the algorithmic quality ratings were available for regression network training. However, the MRI data for some of these subjects contains artefacts outside of the two stations used in the kidney segmentation pipeline, such as water-fat swaps in the thighs and metal artefacts in the knees. Since the regression network uses a 2d representation of the entire scan, 677 affected subjects were therefore excluded from the regression analysis.
Results
=======
Network validation
------------------
In both validation experiments, the neural network reached a Dice score of 0.956 on the 64 subjects dataset *A*. The main result, in which the 58 selected subjects of dataset *B* were added for training, measured combined kidney volume with an average error of 10 cm$^3$, or 3.8%. These values are slightly worse than those achieved by the single-operator cross-validation, and the LoA indicate systematic oversegmentation by the network relative to the operator of dataset *A*, similar to the operators who supplied the training data for dataset *B*. Table \[tab\_validation\] and Fig. \[fig\_cv\_64\] show more validation metrics for these results, together with the variability between human operators for context. Additional detail is given in the Supplementary Tables \[tab\_supp\_jaccard\], \[tab\_supp\_cv\] and \[tab\_supp\_operators\], with the corresponding Jaccard indices and individual measurements of left and right kidney for both network and operators.
![Main validation result for 64 subjects of dataset *A*, with images from dataset *B* added for training. The diagonal line in the scatter plot on the left represents a hypothetical perfect result, whereas the dashed lines in the Bland-Altman plot on the right give the 95% Limits of Agreement. When compared to the reference, the network appears to emulate a tendency towards oversegmentation which is also seen in Table \[tab\_validation\] for the operators who provided reference segmentations for *B*.[]{data-label="fig_cv_64"}](cv_total_volume_64-eps-converted-to.pdf){width="\textwidth"}
[l|rc|rrcc]{} & N & Dice & MAE & SMAPE & R$^2$ & LoA\
**Network validation** &&&&&&\
Main result & 64 & 0.956 & 10 cm$^3$ & 3.8% & 0.950 & (-26 to 14 cm$^3$)\
Single-operator & 64 & 0.956 & 9 cm$^3$ & 3.4% & 0.950 & (-22 to 23 cm$^3$)\
&&&&&&\
**Human variability** &&&&&&\
Intra-operator & 5 & 0.962 & 6 cm$^3$ & 2.6% & 0.994 & (-4 to 13 cm$^3$)\
Inter-operator & 5 & 0.920 & 27 cm$^3$ & 10.0% & 0.839 & (-59 to 5 cm$^3$)\
\
\
\
\
\
Inference
---------
The inference network generated measurements for all those 39,432 subjects lacking reference segmentations. Only a small number of these cases exhibited disjunct or fragmented segmentations, with the *scrap volume* rating indicating that, on average, only about 1 in 900 voxels were not part of the two largest connected components segmented for the given subject (about half of a preliminary run trained on dataset *A* only). Low quality ratings are concentrated in a small subgroup of subjects, as shown in Figure \[fig\_exclusion\_dist\], which were isolated in the following quality control stages.
Based on the algorithmic ratings for image quality, the top one percent of worst *location cost* and *image fusion cost* as well as the top two percent of worst *segmentation fusion cost* were flagged for exclusion in a first control stage. Due to their mutual overlap, the subjects marked in this way amount to about 3.6% of all cases, many of which show signs of severe motion artefacts or misalignments of the field of view. Some of these cases were trivially re-included, having been flagged by the *location cost* for proximity to the image borders while being too small to extend beyond them. Next, the algorithmic ratings for segmentation quality were examined for the remaining subjects As the top one percent of worst *segmentation smoothness cost* and worst *scrap volume cost*, another 1.8% of subjects were flagged in this step. More cases with motion were caught at this stage, as well as genuine failure cases in which the network mistakenly segmented parts of the spleen or liver, but also cases of fragmentation caused by severe cystic formations. In total, 5% of subjects were ultimately excluded, with representative cases shown in Supplementary Fig. \[fig\_supp\_excluded\]. Many of these cases contain pathological anomalies. In turn, perhaps up to a third of them could potentially be re-included without any corrections, but were not considered any further in this work.
![Inferred UK Biobank parenchymal kidney volume (left + right) in cm$^3$ for 17,846 male and 19,622 female subjects.[]{data-label="fig_inference_hist"}](histogram_by_gender-eps-converted-to.pdf){width="\textwidth"}
[l|cl@[0.1cm]{}c|cr@[0.2cm]{}l]{}
Property & mean $\pm$ SD & \[min, & max\] & median & (10%, & 90%)\
**Combined** &&&&&&\
male & 282 $\pm$ 51 & \[91, & 586\]& 277 & (221, & 348)\
female & 224 $\pm$ 40& \[76, & 499\] & 220 & (177, & 276)\
&&&&&&\
**Left** &&&&&&\
male & 143 $\pm$ 29& \[0, & 408\] & 141 & (110, & 178)\
female & 114 $\pm$ 22 & \[0, & 304\] & 112 & (88, & 141)\
&&&&&&\
**Right** &&&&&&\
male & 139 $\pm$ 28& \[0, & 408\] & 137 & (108, & 173)\
female & 110 $\pm$ 22 & \[0, & 268\] & 108 & (86, & 138)\
\
\
After these exclusions, 37,468 subjects remain, with 17,846 men and 19,622 women. Disjunct scrap volume occurs in about 20% of these subjects, but amounts to only 1 in 2,200 segmented voxels on average and never exceeds a share of 2.5% for any individual. Outliers with unusually high or low volumes were inspected as potential failure cases, but were found to be plausible measurements of subjects with missing kidneys, unilateral hypertrophy/atrophy, or were associated with outliers of body size. An in-depth medical analysis of the resulting measurements is beyond the scope of this work and remains to be explored in the future. However, as a brief summary, Fig. \[fig\_inference\_hist\] shows the distribution of measured combined kidney volume, with additional statistics given in Table \[tab\_inference\_results\], and further detail on the offset between kidneys in Supplementary Table \[tab\_inference\_offset\].
![Distribution of algorithmic quality ratings, sorted separately for each rating. High values of the cost terms indicate low image quality. In stage one (top row) and stage two (bottom row) of quality controls, the highlighted top one or two percent of subjects were accordingly flagged for exclusion as potential failure cases.[]{data-label="fig_exclusion_dist"}](exclude_z_cost-eps-converted-to.pdf "fig:"){width="33.00000%"} ![Distribution of algorithmic quality ratings, sorted separately for each rating. High values of the cost terms indicate low image quality. In stage one (top row) and stage two (bottom row) of quality controls, the highlighted top one or two percent of subjects were accordingly flagged for exclusion as potential failure cases.[]{data-label="fig_exclusion_dist"}](exclude_img_fuse_cost-eps-converted-to.pdf "fig:"){width="33.00000%"} ![Distribution of algorithmic quality ratings, sorted separately for each rating. High values of the cost terms indicate low image quality. In stage one (top row) and stage two (bottom row) of quality controls, the highlighted top one or two percent of subjects were accordingly flagged for exclusion as potential failure cases.[]{data-label="fig_exclusion_dist"}](exclude_seg_fuse_cost-eps-converted-to.pdf "fig:"){width="33.00000%"}
![Distribution of algorithmic quality ratings, sorted separately for each rating. High values of the cost terms indicate low image quality. In stage one (top row) and stage two (bottom row) of quality controls, the highlighted top one or two percent of subjects were accordingly flagged for exclusion as potential failure cases.[]{data-label="fig_exclusion_dist"}](exclude_seg_smoothness_cost-eps-converted-to.pdf "fig:"){width="33.00000%"} ![Distribution of algorithmic quality ratings, sorted separately for each rating. High values of the cost terms indicate low image quality. In stage one (top row) and stage two (bottom row) of quality controls, the highlighted top one or two percent of subjects were accordingly flagged for exclusion as potential failure cases.[]{data-label="fig_exclusion_dist"}](exclude_scrap_volume_cost-eps-converted-to.pdf "fig:"){width="33.00000%"}
Runtimes and memory requirements
--------------------------------
Training the 2.5D U-Net on an Nvidia RTX 2080 Ti 11GB GPU for 80,000 iterations required about 30 minutes per split, or about 3.5 hours for the entire 8-fold cross-validation. The MRI data for water and fat signal was stored in DICOM format on an encrypted USB-SSD, amounting to 750 GB for 40,000 subjects. The inference pipeline loaded and processed individual scan volumes from this drive. Despite an efficient GPU implementation, the image fusion formed the bulk of processing time during inference, amounting to almost 30 hours for all 40,000 subjects.
Discussion
==========
As main validation result, the proposed measurements for total kidney volume agree with the reference for a mean error of , or 3.8%, and Dice score 0.956. The assessment of human performance showed slightly superior results for blinded repeat segmentation by a single operator, with mean error , or 2.6%, and Dice score 0.962, whereas the variability between different human operators was more than twice as large as the network error. When applied to the entire cohort, around 37,500 subjects yielded volume measurements with no signs of potential measurement failure, whereas another 5% require further controls.
Only healthy parenchymal tissue was segmented, including cortex and medulla while excluding the renal pelvis, calyces, ureters, major vessels and cysts. The measurements obtained by the inference therefore differ from those typically used for the tracking of conditions such as ADPKD, which may nonetheless benefit from the identification of pathological outliers in this work. These cases are highly concentrated in the 5% of subjects flagged by the algorithmic quality controls, which also helped to identify about 40 suspected cases of renal fusion. With median total kidney volumes of for men and for women, the volumes acquired by the proposed pipeline are smaller than those typically reported in the medical literature, especially when more than just parenchymal tissue is selected. In comparison, a previous study of 150 men and women reported volumes that were about 35% larger, based on a disc-summation method in MRI that excluded the renal pelvis and vasculature, with further validation by a water displacement method.[@cheong2007normal] Another study of 1,852 men and women yielded volumes that were about 20% larger, based on a disc-summation method on manual delineations in MRI that excluded cysts and large vessels.[@roseman2017clinical] Values similar to those obtained in this work occur only in their reported lower quartile range of measurements. More similar values were obtained by previous studies that also focussed on the renal parenchyma, segmenting cortex and medulla only. For segmentations in CT of 1,344 men and women, the renal parenchyma was reported to be about 8% larger in a subgroup with similar mean age to this work.[@wang2014age] In yet another study with MRI of 50 men and women with renovascular disease, the reported volumes were only about 3% larger, based on manual segmentations and voxel count measurements in MRI, with only cortical and medullary tissue being included [@gandy2007clinical]. In terms of methodology, kidney segmentations with Dice scores of up to 0.974 have been reported in the literature for benchmark challenges involving neural networks on CT data.[@isensee2019attempt; @heller2019state] Reaching comparable quality on the UK Biobank neck-to-knee body MRI may not be technically feasible, as the given images are of lower resolution and even repeat segmentation by human operators yielded lower consistency in this work. With no fixed image contrast, such as the Hounsfield units in CT, an objectively consistent placement of the kidney outline in MRI is more challenging. Nonetheless, it is possible that a 3D network architecture could reach superior performance. Future work may explore this potential, but will have to account for the massively increased runtime requirements for 3D architectures. Based on the reported runtimes[@isensee2019attempt], a 3D network may require up to an entire day for training as opposed to the 30 minutes for the 2.5 architecture used in this work, and a similar factor may apply to the inference. Competitive results have also been reported for other approaches that do not utilize neural networks. A recently published approach with appearance-guided deformable boundaries reached a mean Dice score of 0.95 with a 9.5% percentage error for total kidney volume in abdominal diffusion MRI of 72 men and women[@shehata20183d]. Whereas these metrics are similar to the validation results reached in this work, it is worth noting that their reported runtime would also amount to a total of almost two months for inference on 40,000 subjects as compared to the two days required by the proposed segmentation pipeline. In an older technique based on adaptive region growing in CT of 30 subjects, comparable quality was only reached in the best case, with a mean Dice score of 0.88[@lin2006computer]. A more recent work with a multi-atlas technique reported a mean Dice of 0.952 for kidney segmentation in CT of 22 subjects[@yang2014automatic]. The latter may not be directly comparable to the proposed pipeline however, as a rather convex segmentation style and about double the image resolution available in the UK Biobank was used. Another previous study on CT of ADPKD, segmented with fully convolutional neural networks similar to the one proposed in this work, reported a mean Dice score of 0.86 for three data of three different studies.[@sharma2017automatic]
When training the network, dataset *B* was provided for additional guidance on the most challenging morphology. Nonetheless, severe anomalies such as renal fusion and suspected ADPKD were excluded and are thereby effectively accepted as failure cases of the proposed pipeline. This design decision was motivated by the concern that the network may learn a compromise, allowing for better results on these outliers while simultaneously performing worse on the majority of typical cases. The benefit of dataset *B* is not immediately clear from the validation results of Table \[tab\_validation\], where the main result is actually outperformed by the cross-validation using only single-operator data. This is likely a side-effect of validating on references created by the operator of dataset *A* only. The individual segmentation styles of the two operators of dataset *B* show a tendency towards oversegmentation relative to the operator of dataset *A*, marking about 10% more volume. This tendency is emulated by the network trained on the combined datasets, which learned a compromise of segmentation styles. This compromise achieves a slightly lower agreement with the references of dataset *A*, but nonetheless appears more robust. Preliminary inference runs, which were trained on dataset *A* only, produced up to three times more *scrap volume* than the main configuration presented here.
The 2.5D U-Net was able to correctly segment one subject with a missing right kidney in dataset *A* during cross-validation, even though no comparable case existed in the training data. It is possible that the two-dimensional input format may have enabled the network to learn unilateral segmentation based on individual slices containing tissue of only one kidney, with the other being further above or below. Although no rigorous comparison was attempted here, the 2.5D modifications to the original U-Net [@ronneberger2015u] are estimated to accelerate training by about 25% and increase the Dice score by about 0.02.
Several limitations apply. The neural networks trained in this work can only be expected to show comparable performance on future MRI with the same imaging protocol, type of MRI device, and subject demographics. When applied to data of other studies, new training data may be required. The given MRI data is arguably not optimal for kidney volumetry, being originally intended for body composition analysis.[@west_feasibility_2016] With kidney tissue being typically contained in two breath-hold imaging stations, the measurement error is potentially compounded by artefacts such as motion and other factors that are not represented in the validation metrics. Even though the algorithmic quality ratings can be expected to identify the worst cases, actual correction may be possible with registration techniques for image mosaicing[@ceranka2018registration], which were not attempted here. Similarly, those cases excluded by the location rating could be trivially recovered by including the adjacent imaging stations in the pipeline. Another limitation is the degree to which the algorithmic quality ratings themselves are automated. With intuitive, rule-based scores, they provide a high level of control over the exclusions and successfully identify the most severe outliers and failure cases. While the need for manual controls is thereby vastly reduced, the study of their distributions and choice of percentiles does require human intervention and would ideally be automated entirely. No guarantee is provided that all failure cases are exhaustively identified, or that the excluded cases are indeed inadequate. The conservative criteria chosen in this work exclude 5% of cases, which nonetheless translates to about 2,000 subjects for further inspection, many of which are presumably of acceptable quality. The current post-processing steps may furthermore yield misleading results in rare cases where severe cystic formations fragment the healthy kidney tissue such that objective delineation is no longer possible. In these cases, the two largest connected components may occur on the same side, leading to implausible distance and unilateral volume measurements.
The inference of high-quality measurements will therefore remain a continuous effort. A considerable advantage is posed by the high speed of the proposed pipeline, simplifying future coverage of newly released scans and repeated inference runs. With the latter, differently trained segmentation networks could potentially be applied for inference, with the model variation serving as a proxy for prediction uncertainty. Similarly, reference segmentations that segregate cortex and medulla, include or even isolate cysts may enable the inference of new, dedicated measurements to complement those obtained in this work. With new post-processing steps it would also be possible to provide measurements of kidney length, width, and depth so that ellipsoid volumes could be studied. More elaborate quality controls could furthermore rely on independent shape models or atlas segmentations, which have been previously used for large-scale quality controls of UK Biobank cardiovascular MRI segmentation in a variation of the concept of reverse classification accuracy.[@robinson2019automated]
While the collection of metadata and MRI acquisition by the UK Biobank are still in progress, the obtained measurements can already be provided as return data and used for further research. Whereas the currently available blood biochemistry and urine assays predate the MRI by several years, various fields on body size and composition are already available for association studies. The latter are often based on semi-automated processing of the same neck-to-knee body MRI as used in this work and do not yet cover all released subjects. Recent work on image-based regression with neural networks for biometry[@Langner2020] can nonetheless provide accurate approximations months or years before full coverage by the reference methods is achieved, so that many associations could already be studied. Likewise, genetic information is readily available and future work may also examine the repeat imaging visit as planned for another 10,000 subjects. The proposed pipeline is expected to successfully process these images without any need for changes, and the resulting measurements could enable further study of longitudinal effects and disease outcomes associated with changes in parenchymal kidney volume.
Conclusion
==========
The proposed kidney segmentation pipeline generates fast, accurate, and objective delineations with close agreement to human operators. It was applied to all available UK Biobank neck-to-knee body MRI, with only 5% of results showing signs of potential measurement failure. Similar performance is expected for future UK Biobank releases, with the remaining results already forming a substantial sample of left and right parenchymal kidney volume measurements that can be shared for further medical research.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by a research grant from the Swedish Heart- Lung Foundation and the Swedish Research Council (2016-01040, 2019-04756) and used the UK Biobank resource under application no. 14237. Thanks to Lisa Jarl and Paul Hockings for their advice.
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
T.L. wrote the main manuscript text and conducted the experiments, A.Ö. conducted an early precursor of the single-operator cross-validation[@ostling2019automated], L.M. provided the reference segmentations of dataset *A*, A.K. and D.O. provided the reference segmentations of dataset *B*, D.L, A.W., and L.L. implemented the 2.5D network architecture in a related project, R.S. and J.K supervised the project and data access from the UK Biobank, H.A. provided medical expertise. All authors read and approved the final manuscript.
Additional information {#additional-information .unnumbered}
======================
Supplementary Material
======================
The following material provides additional detail on the experiments and results, with code samples and further documentation on GitHub: <https://github.com/tarolangner/ukb_segmentation>
Validation details
------------------
Jaccard indices for the validation experiments are given in Supplementary Table \[tab\_supp\_jaccard\]. More detail, including separate evaluations of left and right kidney volumes, as well as distance, are given in Supplementary Table \[tab\_supp\_cv\] for the network and Supplementary Table \[tab\_supp\_operators\] for human operator variability.
[l|rrr]{} & N & Jaccard index&\
**Network validation** & & &\
Main result & 64 & 0.916&\
Single-operator & 64 & 0.917&\
& &\
**Human variability** & &&\
Intra-operator & 5 & 0.927 &\
Inter-operator & 5 & 0.852&\
\
\
\
[l|cc|cc]{}
& &\
& Main result & Single-operator & Intra-operator & Inter-operator\
Jaccard index & 0.916 & 0.917 & 0.927 & 0.852\
N & 64 & 64 & 5 & 5\
\
\
\
[l|rrrc|rrrc]{} & &\
Property & MAE & SMAPE & R$^2$ & LoA & MAE & SMAPE & R$^2$ & LoA\
Volume, total & 9.6 cm$^3$ & 3.77% & 0.950 & (-25.6 to 13.9 cm$^3$) & 8.7 cm$^3$ & 3.38% & 0.951 & (-22.1 to 23.2 cm$^3$)\
Volume, left & 5.4 cm$^3$ & 4.09% & 0.954 & (-15.2 to 8.9 cm$^3$) & 5.0 cm$^3$ & 3.66% & 0.950 & (-13.8 to 14.6 cm$^3$)\
Volume, right & 4.5 cm$^3$ & 6.62% & 0.968 & (-12.2 to 6.9 cm$^3$) & 4.1 cm$^3$ & 6.42% & 0.971 & (-9.9 to 10.8 cm$^3$)\
Distance\* & 0.3 mm & 0.25% & 1.000 & (-0.8 to 0.9 mm) & 0.3 mm & 0.24% & 1.000 & (-1.0 to 0.8 mm)\
\
\
\
[l|rcrc|rcrc]{} & &\
Property & MAE & SMAPE & R$^2$ & LoA & MAE & SMAPE & R$^2$ & LoA\
Volume, total & 5.6 cm$^3$ & 2.56% & 0.994 & (-4.5 to 13.4 cm$^3$) & 27.0 cm$^3$ & 10.23% & 0.839 & (-59.0 to 5.0 cm$^3$)\
Volume, left & 2.0 cm$^3$ & 1.93% & 0.996 & (-2.1 to 5.4 cm$^3$) & 13.0 cm$^3$ & 10.28% & 0.870 & (-25.1 to -0.92 cm$^3$)\
Volume, right & 3.6 cm$^3$ & 3.15% & 0.990 & (-2.8 to 8.3 cm$^3$)& 14.0 cm$^3$ & 10.18% & 0.798 & (-34.4 to 6.3 cm$^3$)\
Distance & 0.2 mm & 0.14% & 1.000 & (-0.6 to 0.4 mm) & 0.4 mm & 0.29% & 1.000 & (0.1 to 0.7 mm)\
\
\
\
Algorithmic quality ratings
---------------------------
This section describes the individual algorithmic quality ratings in detail. Supplementary Fig. \[fig\_exclusion\_dist\] shows their distribution in the inference run, with example images for failure cases shown in Supplementary Fig. \[fig\_supp\_excluded\].
When the segmentation yields more than just two connected components, the sum of left and right individual volume may be exceeded by the measured total segmented volume. The share of this superfluous *scrap volume* is an important indicator for potential segmentation failure, as spurious segmentation of disjunct areas is a common failure case of many neural network architectures. This effect can also occur in anomalous cases with severe cystic formations that fragment the kidney tissue. In turn, less than two connected components can occur when only one kidney is segmented. This, too, may not necessarily reflect a failure of the network, as prior surgical removal and genetic variations such as fused horseshoe kidneys may arguably lead to only one volume of healthy kidney tissue existing in the image. Anomaly of the imaging process due to undetected water-fat swaps, motion, and non-standard image contrast also affect this behaviour.
In image fusion, the *segmentation fusion cost* is defined as the sum of absolute differences in the overlap of two adjacent stations, based on their voxel-wise binary labels. The *image fusion cost* uses the same term on the voxel-wise water signal intensities and normalizes the sum with the range of existing intensity values to take into account variations in contrast. High values of these terms indicate that the anatomy in both stations does not line up smoothly due to motion or other artefacts.
Another potential problem is misalignment of the subject, so that the kidneys are not fully contained in the second and third imaging station. The *positioning cost* rating quantifies the offset of the kidney centres of mass from the centre line of the image along the longitudinal axis. High values consequently identify those subjects with segmented kidney volume that is placed too high or too low on the bed, potentially reaching beyond the field of view.
The *segmentation smoothness* of the fused binary labels is then rated by summing up the absolute differences between the segmentation volume to a copy of itself, which is shifted by one voxel along the longitudinal axis. Any sudden change in segmentation between adjacent axial slices is thereby penalized, and high values indicate spurious gaps or islands created by the network.
![Four characteristic failure cases with segmented right (red), left (blue), and disjunct scrap (green) kidney labels overlaid in coronal view for an outlier in vertical location (top left), genuine segmentation failure, where the network marked parts of the liver (top right), failure due to motion during imaging, despite an arguably correct segmentation (bottom left), and severe fragmentation of healthy tissue by cysts (bottom right).[]{data-label="fig_supp_excluded"}](5887293_labeled.png "fig:"){width="45.00000%"} ![Four characteristic failure cases with segmented right (red), left (blue), and disjunct scrap (green) kidney labels overlaid in coronal view for an outlier in vertical location (top left), genuine segmentation failure, where the network marked parts of the liver (top right), failure due to motion during imaging, despite an arguably correct segmentation (bottom left), and severe fragmentation of healthy tissue by cysts (bottom right).[]{data-label="fig_supp_excluded"}](4120936_labeled.png "fig:"){width="45.00000%"} ![Four characteristic failure cases with segmented right (red), left (blue), and disjunct scrap (green) kidney labels overlaid in coronal view for an outlier in vertical location (top left), genuine segmentation failure, where the network marked parts of the liver (top right), failure due to motion during imaging, despite an arguably correct segmentation (bottom left), and severe fragmentation of healthy tissue by cysts (bottom right).[]{data-label="fig_supp_excluded"}](1396290_labeled.png "fig:"){width="45.00000%"} ![Four characteristic failure cases with segmented right (red), left (blue), and disjunct scrap (green) kidney labels overlaid in coronal view for an outlier in vertical location (top left), genuine segmentation failure, where the network marked parts of the liver (top right), failure due to motion during imaging, despite an arguably correct segmentation (bottom left), and severe fragmentation of healthy tissue by cysts (bottom right).[]{data-label="fig_supp_excluded"}](1303217_labeled.png "fig:"){width="45.00000%"}
Kidney offsets
--------------
The left and right kidney are identified as the two largest connected components in the segmented volume. Based on the centre of mass, their relative position in the human body can be described, with euclidean distance values given in Supplementary Table \[tab\_inference\_offset\].
[l|cl@[0.1cm]{}c|rr@[0.2cm]{}l]{}
Property & mean $\pm$ SD & \[min, & max\] & median & (10%, & 90%)\
**Distance** &&&&&&\
male & 153 $\pm$ 15 & \[0, & 211\] & 153 & (137, & 171)\
female & 127 $\pm$ 12 & \[0, & 215\] & 127 & (114, & 142)\
&&&&&&\
**Offset X** &&&&&&\
male & 152 $\pm$ 15 & \[0, & 208\] & 152 & (135, & 171)\
female & 125 $\pm$ 12 & \[0, & 180\] & 125 & (112, & 139)\
&&&&&&\
**Offset Y** &&&&&&\
male & 2 $\pm$ 8 & \[-70, & 47\] & 2 & (-7, & 11)\
female & 5 $\pm$ 7 & \[-41, & 47\] & 4 & (-4, & 13)\
&&&&&&\
**Offset Z** &&&&&&\
male & 5 $\pm$ 15 & \[-88, &95\] & 5 & (-14, &25)\
female & 16 $\pm$ 16 & \[-190, & 118\] & 16 & (-4, & 36)\
\
|
---
abstract: 'Ferrofluid heating by an external alternating field is studied based on the rigid dipole model, where the magnetization of each particle in a fluid is supposed to be firmly fixed in the crystal lattice. Equations of motion, employing the Newton’s second law for rotational motion, the condition of rigid body rotation, and the assumption that the friction torque is proportional to angular velocity, are used. This oversimplification permits us to expand the model easily: to take into account the thermal noise and inter-particle interaction that allows to estimate from unified positions the role of thermal activation and dipole interaction in the heating process. Our studies are conducted in three stages. The exact expressions for the average power loss of a single particle are obtained within the dynamical approximation. Then, in the stochastic case, the power loss of a single particle is estimated analytically using the Fokker-Planck equation and numerically using the effective Langevin equation. Finally, the power loss for the particle ensemble is obtained using the molecular dynamics method. Here, the local dipole fields are calculated approximately based on the Barnes-Hut algorithm. The revealed trends in the behaviour of both a single particle and the particle ensemble suggest the way of choosing the conditions for obtaining the maximum heating efficiency. The competitiveness character of the inter-particle interaction and thermal noise is investigated in details. Two situations, when the thermal noise rectifies the power loss reduction caused by the interaction, are described. The first of them is related to the complete destruction of dense clusters at high noise intensity. The second one is originated from the rare switching of the particles in clusters due to thermal activation, when the noise intensity is relatively weak. In this way, the constructive role of noise appears in the system.'
author:
- 'T. V. Lyutyy'
- 'V. V. Reva'
bibliography:
- 'Lyutyy\_Ukraine\_PowerLoss\_FNP\_Ensemble\_Revised.bib'
title: 'Energy dissipation of rigid dipoles in a viscous fluid under the action of a time-periodic field: the influence of thermal bath and dipole interaction'
---
INTRODUCTION {#Int}
============
The ferrofluid [@Rosensweig1985Ferrohydrodynamics; @0038-5670-17-2-R02] response to external fields is a key factor for its applications. Firstly, in this regard, an attention should be given to high-performance microwave absorbers designed in a complex manner, which are under intense investigation now [@C4TA01681E; @doi:10.5185/amlett.2015.5807; @C5TA02381E; @Varshney2017; @doi:10.1002/9781119208549.ch13]. Secondly, a special attention deserves magnetic fluid hyperthermia method for cancer treatment [@0022-3727-36-13-201; @JORDAN1999413; @andr2006magnetism]. In both examples mentioned above, the energy of a periodic external field is transformed into the thermal one due to viscous rotation of the ferrofluid particles and damped precession of the magnetization within the particles crystal lattice. The heating process is often described within the quasi-equilibrium assumption and theory of linear response to an alternating field [@Rosensweig2002370]. Despite the simplicity and obviousness of this concept, its application domain is narrow. As evidenced from various experiments [@andr2006magnetism] including the recent [@0022-3727-49-29-295001; @doi:10.1109/TMAG.2016.2516645; @doi:10.1063/1.4974803; @Arteagacardona2016636; @doi:10.1021/acs.molpharmaceut.5b00866; @doi:10.1021/acsnano.7b01762; @doi:10.2147/IJN.S141072; @doi:10.1166/sam.2017.2948; @doi:10.1021/acsnano.7b01762], strong deviations from the analytical predictions occur. In a wide sense, these differences are originated from the features of individual dynamics aroused as a consequence of coupled mechanical and magnetic motions of each particle and structure of a ferrofluid aroused as a consequence of inter-particle interaction.
The analytical description of the driven coupled motion of a magnetic particle in a fluid with its magnetic moment is an interesting and complicated problem. Since the base model equations were introduced by Cebers [@Cebers1975], the consistent investigation of the particle response to periodic fields is absent up to now (see Introduction in Ref. [@Lyutyy201887] for details). Because of computational difficulties, the simplified approaches are widely used to these purposes. The first approach is the fixed particle model [@PhysRevE.86.061404; @PhysRevE.93.012607], within which the whole particle is supposed to be locked into the solid matrix. The second one is the rigid dipole (RD) model, within which the particle magnetization is supposed to be locked into the crystal lattice. The last approach is very fruitful, well corresponding to the real anisotropic particle of a radius more than $20 \textrm{nm}$ and not very high frequencies ($~ 10^3 - 10^6 \textrm{MHz}$) of external fields [@andr2006magnetism]. It is interesting to note that in recent studies of the coupled dynamics [@PhysRevB.95.104430], a large amount of the results was obtained for the RD limit.
Due to small sizes (several tens of nanometers) of the particles, which are contained in a ferrofluid, thermal bath is the first factor defining the response to external fields. Hence, the description of the individual particle motion should be stochastic. To this end, the Langevin and Fokker-Planck formalisms are developed. Since the basic concept was discussed in [@0038-5670-17-2-R02; @Raikher_1994], numerous investigations devoted to the RD response to external fields have been conducted. Although, this response was also treated in terms of the complex magnetic susceptibility [@PhysRevE.63.011504; @Raikher2011; @PhysRevE.83.021401; @0953-8984-15-23-313; @PhysRevE.82.046310; @SotoAquino201546], it can strongly differ from the prediction of the above mentioned quasi-equilibrium linear model presented in [@Rosensweig2002370]. The attention was also paid to the important problem of energy dissipation, which is characterised by the average power loss. Despite the scientific relevance of the results presented, they are miscellaneous and do not permit to estimate from unified positions the influence of all system parameters on the heating process.
Due to the long range character of the inter-particle dipole interaction, the dipole fields are the second factor defining the ferrofluid response to external fields. Hence, we encounter the need to solve a many-body problem. To this end, a few approximate approaches are used, and there is no general solution for this. The most analytical techniques are based on different modifications of the mean field concept [@OGRADY1983958; @MOROZOV199051; @BUYEVICH1992276; @PhysRevE.64.041405; @PhysRevE.79.021407], see more results in survey [@0034-4885-67-10-R01]. Unfortunately, this approach does not account the nearest neighbor correlations and the possible structure formation discussed in [@PhysRevE.53.2509; @PhysRevE.57.4535; @PhysRevLett.110.148306]. Moreover, the mean field approach is hard to apply when a periodic external field acts. The observed in [@OGRADY1983958; @MOROZOV199051; @BUYEVICH1992276; @PhysRevE.64.041405; @PhysRevE.79.021407] growth of the quasi-static susceptibility caused by the interaction does not imply the increase in the imaginary part of the complex susceptibility that is confirmed by the measurements of the specific absorption rate [@0022-3727-49-29-295001; @doi:10.1109/TMAG.2016.2516645; @doi:10.1063/1.4974803; @Arteagacardona2016636; @doi:10.1021/acs.molpharmaceut.5b00866; @doi:10.1021/acsnano.7b01762; @doi:10.2147/IJN.S141072; @doi:10.1166/sam.2017.2948; @doi:10.1021/acsnano.7b01762]. In order to calculate correctly the distribution of the local dipole fields, the numerical simulation is demanded.
There are two numerical techniques, namely Monte Carlo (MC) and molecular dynamics (MD) methods [@Gould2007Simulation; @Rapaport2004MD; @Haile1997MD]. Despite easy implementation and low consumption of computational resources, the MC method is useless in the case of time-dependent excitations. Hence, this technique is applied for the investigation of the equilibrium and structure properties [@0022-3727-13-7-003; @BRADBURY1986745; @PhysRevE.59.2424; @PhysRevE.71.061203; @PhysRevLett.110.148306]. The MD method is based on the integration of the coupled Langevin equations for each RD in the ensemble. This technique has stronger requirements to both the computational equipment and program code, but it is free of restrictions, which are intrinsic to the MC method. Therefore, the MD method is widely used for the description of the properties of ferrofluids. Thus, the ferrofluid structure and initial susceptibility were studied in the works [@PhysRevE.66.021405; @PhysRevE.75.061405]; work [@TANYGIN20124006] reports the numerical results of the self-organization and phase transitions in the aggregated structures; the size distribution impact, dynamical and structural effects were studied in [@PhysRevE.88.042315; @PhysRevE.92.012306]; at last, the magnetic susceptibility spectra and relaxation properties were treated numerically in [@PhysRevE.93.063117; @C5SM02679B]. At the same time, the influence of the dipole interaction on the power loss was investigated using the model based on the Landau-Lifshitz equation, where internal damping precession of the magnetic moment is only taken into account [@PhysRevB.85.045435; @PhysRevB.87.174419; @PhysRevB.89.014403; @PhysRevB.90.214421]. This approach is valid under some circumstances, but it is used primarily because of the simpler equations of motion. Therefore, the role of the dipole interaction in the energy dissipation of RD placed into a viscous liquid remains unclear.
The aim of the present study is in the following. On the one hand, we need to decompose the impact of the regular component, thermal excitation, and collective behaviour on the response of a ferrofluid to a periodic field. On the other hand, we need to trace the synergy of these factors in the ferrofluid heating in order to get insight about the possibility to control and choose the optimal parameters. The final goal defines the methodology of our analysis and the article structure. Firstly, we exploit the purely dynamical approach and find out the possible analytical solutions of the equations of forced rotational motion for a single RD in a viscous fluid. Secondly, we investigate the statistical properties of a single RD interacting with both external periodic field and thermal bath. Finally, we apply the MD method for the description of the joint motion of the particle ensemble. Within this framework, the Barnes-Hut algorithm [@Barnes-Hut-Nature1986] and CUDA technology [@Sanders2011CUDA] for facilitation of the local dipole fields calculation are utilized successfully.
DESCRIPTION OF THE MODEL {#Desc}
========================
We consider a ferromagnetic particle of radius $R$, uniform mass density $D$, and magnetization $M$ placed in a fluid of viscosity $\eta$. This particle rotates in the fluid under a magnetic field $\mathbf{H} = \mathbf{H}(t)$, which in a general case consists of the external and dipole parts $\mathbf{H} = \mathbf{H}^{ext} + \mathbf{H}^{dip}$. Further the following assumptions are used. First, the exchange interaction between the atomic magnetic moments is assumed to be dominant. Therefore, the magnitude $|\mathbf{M}| = M$ of the particle magnetization $\mathbf{M}$ can be considered as a constant parameter. Second, the particle radius is assumed to be small enough (less than a few tens of nanometers), and the nonuniform distribution of magnetization is energetically unfavorable. Therefore, the single-domain state, which is characterised by $\mathbf{M} = \mathbf{M}(t)$, takes place. And third, the uniaxial anisotropy magnetic field is assumed to be strong enough, and the particle magnetization is directed along this field. Therefore, $\mathbf{M}$ is locked rigidly into the particle body. In fact, the above assumptions are the ground of the RD model. In the simplest way, the rotational dynamics of RD is described firstly in [@doi:10.1063/1.1671698]. If the rotation of the fluid surrounding the particle is neglected, then the particle dynamics is governed by a pair of coupled equations
\[eqs1\] $$\begin{aligned}
\dot{\mathbf{M}} &=& \boldsymbol{\upomega} \times \mathbf{M},
\label{eq_a}
\\[6pt]
J\dot{\boldsymbol{\upomega}} &=& V\mathbf{M} \times \mathbf{H} - 6\eta V \boldsymbol{\upomega}.
\label{eq_b}\end{aligned}$$
Here, $\boldsymbol{\upomega} = \boldsymbol{\upomega}(t)$ is the angular velocity of the particle, the overdot denotes the time derivative, $J = (2/5)D VR^{2}$ is the moment of inertia of the particle, $V = (4/3)\pi R^{3}$ is the particle volume (we associate the hydrodynamic volume of the particle with its own volume), and the cross denotes the vector product.
In our study we neglect the hydrodynamical interaction. Such approach is widely used, despite it may be oversimplification in some cases as well as the rigid dipole approximation. However, the results of [@doi:10.1021/jp8113978] suggests that the fluid motion can be not critical for structural properties of ferromagnetic particle ensemble. The other reason to use the simple model lies in our intention to utilize the common and inexpensive facilities for further simulations.
Then, we assume that the particle is driven by an external time-periodic field of the following types:
\[eq:ext\_fields\] $$\begin{aligned}
\mathbf{H}^{ext}\!&=&\!H_{m}\big[\mathbf{e}_{x} \cos(\Omega t)\!+\!\mathbf{e}_{y}\varrho \sin(\Omega t)\big]\!+\!\mathbf{e}_{z} H_{0z}, \nonumber \\
\label{eq:ext_fields_cp}
\\[6pt]
\mathbf{H}^{ext}\!&=&\!\mathbf{e}_{z} H_{m}\cos(\Omega t),
\label{eq:ext_fields_lp}\end{aligned}$$
where $\mathbf{e}_x$, $\mathbf{e}_y$, $\mathbf{e}_z$ are the unit vectors of the Cartesian coordinates, $H_{m}$ is the field amplitude, $\Omega$ is the field frequency, $H_{0z}$ is the static field applied along the $oz$ axis, and $\varrho$ is the factor, which determines the polarization type ($-1 \leq \varrho \leq 1$). The basic concept of the model is sketched in Fig. \[fig:Model\]a for a circularly-polarized field (Eq. (\[eq:ext\_fields\_cp\])) and in Fig. \[fig:Model\]b for a linearly-polarized field ($\varrho = 1$, Eq. (\[eq:ext\_fields\_lp\])).
![(Color online) Schematic representation of our model for the single-particle case. Spherical rigid dipole, its magnetic moment, spherical and Cartesian coordinate systems, and external fields acting on the system are depicted. Plot a: the circularly-polarized field (\[eq:ext\_fields\_cp\]) is applied. Plot b: the linearly-polarized field (\[eq:ext\_fields\_cp\]) is applied.[]{data-label="fig:Model"}](Fig01a.eps "fig:"){width="0.6\linewidth"} ![(Color online) Schematic representation of our model for the single-particle case. Spherical rigid dipole, its magnetic moment, spherical and Cartesian coordinate systems, and external fields acting on the system are depicted. Plot a: the circularly-polarized field (\[eq:ext\_fields\_cp\]) is applied. Plot b: the linearly-polarized field (\[eq:ext\_fields\_cp\]) is applied.[]{data-label="fig:Model"}](Fig01b.eps "fig:"){width="0.6\linewidth"}
Equations of motion for Brownian rotation of a single particle
--------------------------------------------------------------
In the case when the particle size is sufficiently small, the left-hand side of Eq. (\[eq\_b\]), i.e. the inertia term $J\boldsymbol{\upomega}$, can be neglected for reasonable frequencies. Using this massless approximation and assuming that a random torque $\boldsymbol{\upxi} = \boldsymbol{\upxi}(t)$, which is generated by the thermal bath, is also applied to the particle, one can write $$\boldsymbol{\upomega} = \frac{1}{6\eta}\mathbf{M} \times \mathbf{H} + \frac{1} {6\eta V} \boldsymbol{\upxi}.
\label{omega}$$ Substituting the last relation into Eq. (\[eq\_a\]), we obtain the equation $$\dot{\mathbf{M}} = - \frac{1}{6\eta}\mathbf{M} \times (\mathbf{M} \times \mathbf{H}) -
\frac{1}{6\eta V} \mathbf{M} \times \boldsymbol{\upxi},
\label{eq:Eq_of_motion_base}$$ which describes the stochastic rotation of particles in a viscous fluid. Since the particle magnetization $M$ is constant in time, for further calculations it is reasonable to rewrite Eq. (\[eq:Eq\_of\_motion\_base\]) in the spherical coordinates $\mathbf{M} = \mathbf{e}_x M \sin\theta \cos\varphi + \mathbf{e}_y M \sin\theta \sin\varphi + \mathbf{e}_z M \cos\theta$ (see Fig. \[fig:Model\])
\[eq:Langevin\_angular\_base\] $$\begin{aligned}
\displaystyle \dot{\theta} &=& \frac{1}{\tau_{1}}
\left(h_x \sin\theta\cos\varphi + h_y \sin\theta\sin\varphi\ + h_z \cos\theta \right) \cot\theta\! -\nonumber
\\
\displaystyle &-& \frac{1}{\tau_{1}}\frac{h_z}{\sin\theta} +\!\sqrt{\frac{2}{\tau_{2}}}\left(\zeta_{y}\cos\varphi - \zeta_{x} \sin\varphi\right), \label{eq:Langevin_angular_base_a}
\\ [10pt]
\displaystyle \dot{\varphi} &=& \frac{1}{\tau_{1}\sin^{2}\theta}\left(h_{y}\cos\varphi - h_{x} \sin\varphi\right) -\nonumber
\\
&-& \sqrt{\frac{2}{\tau_{2}}}\big[(\zeta_{x}\cos \varphi + \zeta_{y}\sin\varphi)\cot\theta - \zeta_{z}\big].
\label{eq:Langevin_angular_base_b}\end{aligned}$$
Here, $\boldsymbol{\upzeta} = \left(12\eta V k_{\mathrm{B}}T\right)^{-1/2} \boldsymbol{\upxi}$ is the rescaled random torque, $k_\mathrm{B}$ is the Boltzmann constant, $T$ is the absolute temperature, $\tau_{1} = 6\eta/M^{2}$ and $\tau_{2} = 6\eta V/(k_\mathrm{B}T)$ are the characteristic times of the particle rotation induced by the external magnetic field and thermal torque, respectively, $\mathbf{h} = \mathbf{H}/M$ is the reduced field. The Cartesian components $\zeta_{\nu}$ ($\nu =x,y,z$) of $\boldsymbol{\upzeta}$ are assumed to be independent Gaussian white noises with zero means, $\langle \zeta_{\nu} \rangle =0$, and correlation functions $\langle \zeta_{\nu}(t) \zeta_{\nu}(t') \rangle = \Delta \delta(t-t')$, where $\langle \cdot \rangle$ denotes averaging over all realizations of the Wiener processes $W_{\nu}(t)$ producing noises $\zeta_{\nu}$, $\Delta$ is the dimensionless noise intensity, and $\delta(t)$ is the Dirac delta-function.
An important feature of the Langevin equations (\[eq:Langevin\_angular\_base\]) is that noises $\zeta_{\nu}$ are multiplicative ones, i.e. they are multiplied by functions of the angles $\theta$ and $\varphi$. Therefore, the properties of the particle motion can depend on the interpretation of the noises that, in turn, can complicate further processing. To overcome these difficulties, we consider the problem of stochastic rotation of the particle from the statistical point of view. Following [@PhysRevE.92.042312], the Fokker-Planck equation, which corresponds to Eqs. (\[eq:Langevin\_angular\_base\]), is written as $$\begin{aligned}
\frac{\partial P}{\partial t} &+& \nonumber \\
&+& \frac{1}{\tau_{1}} \frac{\partial}{\partial \theta}\bigg[
\big( h_x \cos\varphi + h_y \sin\varphi \big)\cos\theta - h_z \sin\theta + \nonumber \\
&+& \frac{\cot\theta}{\kappa} \bigg]P + \frac{1}{\tau_{1}\sin^{2}\theta} \frac{\partial}{\partial\varphi}
\big( h_y \cos\varphi - h_x \sin\varphi \big) P - \nonumber \\
&-& \frac{1}{\tau_{2}} \frac{\partial^{2}P}{\partial \theta^{2}}-\frac{1}{\tau_{2}} \frac{1}{\sin^{2}\theta} \frac{\partial^{2}P}{\partial \varphi^{2}} = 0,
\label{eq:FP}\end{aligned}$$ where $P=P(\theta,\varphi,t)$ is the time-dependent probability density function for the rotational states of the particle, $\kappa=\tau_2/\tau_1 = M^2 V / k_{\mathrm{B}}T$ is the relationship between the magnetic and thermal energies, which shows the relative contribution of thermal fluctuations. The latter parameter is the most useful for the analysis of the role of temperature in the particle behavior including energy absorption of the external time-periodic field. By the direct substitution, it is easy to show that the following *effective* Langevin equations
\[eq:Langevin\_angular\_eff\] $$\begin{aligned}
\displaystyle \dot{\theta} &=& \frac{1}{\tau_{1}} \big( h_x \cos\varphi + h_y \sin\varphi \big)\cos\theta -
\frac{1}{\tau_{1}} h_z \sin\theta + \nonumber \\
\displaystyle&+&\frac{1}{\tau_{2}}\cot \theta + \sqrt{\frac{2}{\tau_{2}}}\,\mu_{1}, \label{eq:Langevin_angular_eff_a}
\\ [10pt]
\displaystyle \dot{\varphi} &=& \frac{1}{\tau_{1}}\big( h_y \cos\varphi - h_x \sin\varphi \big)\frac{1}{\sin\theta} + \nonumber \\
\displaystyle &+& \sqrt{\frac{2}{\tau_{2}}} \frac{1}{\sin\theta}\, \mu_{2}, \label{eq:Langevin_angular_eff_b}\end{aligned}$$
where $\mu_{i} = \mu_{i}(t)$ ($i = 1,2$) are the independent Gaussian white noises with zero means, $\langle \mu_{i}(t) \rangle =0$, and delta correlation functions, $\langle \mu_{i}(t)\mu_{i}(t') \rangle = \delta(t-t')$, are equal in the statistical sense to the initial equations of motion (\[eq:Langevin\_angular\_base\]). At the same time, equations (\[eq:Langevin\_angular\_eff\]), in contrast to Eqs. (\[eq:Langevin\_angular\_base\]), do not contain the terms, where the regular functions of angular arguments are multiplied by the noises responsible for fluctuations of the same argument. In other words, only Eq. (\[eq:Langevin\_angular\_eff\_b\]) formally contains the multiplicative noise, but here the noise responsible for the fluctuations of the azimuthal angle $\varphi$ is combined with the function of the polar angle $\theta$. Since $\mu_{i}$ are represented by the Gaussian white noises, which are interpreted in the Stratonovich sense, the use of Eqs. (\[eq:Langevin\_angular\_eff\]) instead of Eqs. (\[eq:Langevin\_angular\_base\]) in the numerical simulations is more convenient because of simpler and faster algorithms. It is especially important in a view of the ensemble simulation.
Introducing the dimensionless time $\tilde{t} = t/\tau_{1}$, one can write the system of the *reduced effective* Langevin equations (\[eq:Langevin\_angular\_eff\]) as follows
\[eq:Langevin\_angular\_eff\_red\] $$\begin{aligned}
\displaystyle \frac{d\theta}{d\tilde{t}} &=& \big( h_x \cos\varphi + h_y \sin\varphi \big)\cos\theta -
h_z \sin\theta + \nonumber\\
&+& \frac{1}{\kappa}\! \cot\theta + \!\sqrt{\frac{2}{\kappa}} \tilde{\mu}_{1},\label{eq:Langevin_angular_eff_red_a}
\\ [10pt]
\displaystyle \frac{d\varphi}{d\tilde{t}} &=& \frac{1}{\sin\theta}\big( h_y \cos\varphi - h_x \sin\varphi \big)-\nonumber\\
&-& \displaystyle \sqrt{\frac{2}{\kappa}} \frac{1}{\sin\theta}\, \tilde{\mu}_{2},\label{eq:Langevin_angular_eff_red_b}\end{aligned}$$
where $\tilde{\mu}_{i} = \tilde{\mu}_{i}(\tilde{t}) = \sqrt{\tau_{1}}\, \mu_{i}( \tilde{t} \tau_{1})$ ($i=1,2$) are the dimensionless Gaussian white noises with $\langle \tilde{\mu}_{i}(\tilde{t}) \rangle =0$ and $\langle \tilde{\mu}_{i} (\tilde{t}) \tilde{\mu}_{i} (\tilde{t}') \rangle = \delta( \tilde{t} - \tilde{t}')$. It is this system of equations, which is the most convenient for both the analytical and numerical treatment, especially for the large ensemble simulation.
Simulation of the interacting particle ensemble
-----------------------------------------------
The interaction is a very important factor that defines the dynamics and properties of a ferrofluid. The problem should to be considered at the following two stages. On the one hand, the resulting dipole fields acting on each particle can influence considerably the rotational dynamics and, in particular, the energy absorbed from the external field. On the other hand, the dipole interaction due to the attractive character induces the cluster structure of the ensemble, that, in turn, impacts the dipole field distribution. The exact analytical description of the particle ensemble driven by time-periodic fields in a viscous liquid is most likely impossible. Therefore, a numerical simulation is demanded. As stated in the introduction, there are two approaches to simulate the ferromagnetic particle ensembles, namely, the MC and MD methods (see [@Gould2007Simulation; @Rapaport2004MD; @Haile1997MD] for details). The last approach is more suitable for high-performance simulation in real time, but it requires large processing power. In this regard, the optimal form of the basic equations, which describe the dynamics of a single particle, plays the key role. And the effective equations (\[eq:Langevin\_angular\_eff\_red\]) are fit enough for these purposes.
In our approach we expand the model developed above to the case of an ensemble. Thus, here we consider an ensemble of equal spherical uniform ferromagnetic uniaxial particles with the parameters stated above. The rotational motion of the particles is described by the effective stochastic equations similar to Eqs. (\[eq:Langevin\_angular\_eff\_red\]), which are complemented by the standard equations of translational motion [@PhysRevE.66.021405; @Polyakov20131483] written for the massless case as
\[eq:Langevin\_angular\_eff\_red\_k\] $$\begin{aligned}
\displaystyle \frac{d\theta_k}{d\tilde{t}} &=& \big( h_{kx} \cos\varphi_k + h_{ky} \sin\varphi_k \big)\cos\theta_k -
h_{kz} \sin\theta_k + \nonumber\\
&+& \frac{1}{\kappa}\! \cot\theta_k + \!\sqrt{\frac{2}{\kappa}} \tilde{\mu}_{k1},\label{eq:Langevin_angular_eff_red_k_a}
\\ [10pt]
\displaystyle \frac{d\varphi_k}{d\tilde{t}} &=& \frac{1}{\sin\theta_k}\big( h_{ky} \cos\varphi_k - h_{kx} \sin\varphi_k \big) -\nonumber\\
&-& \displaystyle \sqrt{\frac{2}{\kappa}}\frac{1}{\sin\theta_k}\, \tilde{\mu}_{k2},\label{eq:Langevin_angular_eff_red_k_b}
\\ [10pt]
\displaystyle \frac{d{\boldsymbol{\rho}}_k}{d\tilde{t}} &=& \frac{16 \pi}{9} (\mathbf{f}^{dip}_k + \mathbf{f}^{sr}_k) + \sqrt{\frac{8}{3 \kappa}}\tilde{\mu}_{k3},\label{eq:Langevin_angular_eff_red_k_c}\end{aligned}$$
where $\boldsymbol{\rho}_k$ is the vector defining the reduced (here, the particle radius $R$ is the distance unit) coordinates of the given particle, $k$ is the index number of the given particle in the ensemble, $\mathbf{h}_k = \mathbf{h}_{k}^{dip} + \mathbf{h}^{ext}$ is the resulting dimensionless field acting on the $k$-th particle, which consists of the external uniform part ($\mathbf{h}^{ext} = \mathbf{H}^{ext}/M$) and the resulting reduced dipole field $$\mathbf{h}_{k}^{dip} = \sum_{j = 1, j \neq k}^{N}{\frac {4 \pi}{3} \frac {3 \boldsymbol{\rho}_{kj} (\mathbf{u}_j \boldsymbol{\rho}_{kj}) -
\mathbf{u}_j \boldsymbol{\rho}_{kj} ^{\, 2}} {\rho_{kj} ^{\, 5}}},
\label{eq:h_dip_red}$$ where $\boldsymbol{\rho}_{kj}$ is the vector joining two particles measured in $R$ units, $\mathbf{u}_j = \mathbf{M}_j/M$ is the reduced magnetic moment of the $j$-th particle; $N$ is the total number of particles.
There are two forces acting on each particle which should be taken into account. Firstly, $\mathbf{f}^{dip}_k$ is the force acting on the $k$-th particle and arising from its dipole interaction with all other particles in the ensemble. Secondly, $\mathbf{f}^{sr}_k$ is the force arising from an anti-aggregation coating, which is widely used in real ferrofluids to prevent the particle aggregation. To represent the explicit form of $\mathbf{f}^{dip}_k$, we applied the standard definition $\mathbf{f}^{dip}_k = \left(\mathbf{u}_k\bigtriangledown_k\right)\mathbf{h}^{dip}_k$. The coating provides the particles repulsion primarily. There are numerous examples in literature, when the force produced by this coating is modeled using the Lennard-Jones potential. This type of potential is the most suitable because it leads to the equilibrium states and prevents the infinite spread of the particles even if the dipole interaction is weak. $\mathbf{f}^{sr}_k = - \bigtriangledown_k W_k$, where $W_k = 4\varepsilon \sum_{j = 1, j \neq k}^{N}{\left[\left(\sigma/\rho_{kj}\right)^{12} - \left(\sigma/\rho_{kj}\right)^6\right]}$. Here, $\sigma$ is the parameter defining the equilibrium distance between two particles, and $\varepsilon$ is the parameter defining the potential barrier depth. Finally, we rewrite the above forces acting on the particle as
\[eq:Langevin\_angular\_eff\_forces\] $$\begin{aligned}
\mathbf{f}_k^{dip} \!&=&\! \sum_{j = 1, j \neq k}^{N} \Biggl[
3 \frac {\boldsymbol{\rho}_{kj} (\mathbf{u}_j \mathbf{u}_k) + \mathbf{u}_k (\mathbf{u}_j \boldsymbol{\rho}_{kj}) +
\mathbf{u}_j (\mathbf{u}_k \boldsymbol{\rho}_{kj})} {\rho_{kj}^{\, 5}} - \nonumber
\\
\!&-&\! 15 \frac {\boldsymbol{\rho}_{kj} (\mathbf{u}_k \boldsymbol{\rho}_{kj})(\mathbf{u}_j \boldsymbol{\rho}_{kj})}
{\rho_{kj}^{\, 7}} \Biggl], \label{eq:Langevin_angular_eff_forces_a}
\\
\mathbf f_k^{sr} \!&=&\! 24 \varepsilon \sum_{j = 1, j \neq k}^{N}
\frac {\boldsymbol{\rho}_{kj}} {\boldsymbol{\rho}_{kj} ^{\, 2}}
\left[ {\left( \frac{\sigma}{\rho_{kj}} \right)} ^{12}
- {\left (\frac{\sigma}{\rho_{kj}} \right) } ^6 \right]. \label{eq:Langevin_angular_eff_forces_b}\end{aligned}$$
The dipole field calculation is the most computational power consuming part of the numerical algorithm. This is the main factor determining the optimal balance between the computational time ($\mathcal{T}_{sim}$), ensemble size ($N$), and equipment used. The exact direct calculation of the dipole fields induced by all particles is characterised by the square relationship between time and size ($\mathcal{T}_{sim} \sim N^2$). Instead of the cumbersome exact calculation, two approximations are used. The first approximation is the so-called fast multipole method [@GREENGARD280] that provides the performance $\mathcal{T}_{sim} \sim N$, and the second one is the Barnes-Hut algorithm [@Barnes-Hut-Nature1986] that provides the performance $\mathcal{T}_{sim} \sim N\log N$. Despite the better facilitation, the fast multipole method does not calculate the neighbour correlations with a good accuracy. Therefore, in our calculation we have utilized the Barnes-Hut approach. Its main idea consists in the averaging of the fields generated by the particles, which are far enough from the particle under consideration, and, in contrary, the fields generated by the nearest particles are calculated exactly.
Another important feature of our numerical approach is the use of computing capabilities of video cards. This gives an excellent possibility of high-performance calculations on common PC. The video card graphics processing units designed for displaying real time video can be adapted for general-purpose computing. The so-called CUDA technology, which was unveiled by Nvidia company [@Sanders2011CUDA], provides us with the convenient tools for this. Nowadays, many scientific problems can be solved in an inexpensive way and without special facilities like clusters or supercomputers. The collective dynamics of the particle ensembles with the long-range dipole interaction is a suitable problem to demonstrate the abilities of CUDA. The details of the used simulation technique are explained in [@Polyakov20131483].
The power loss: definitions and calculation technique
-----------------------------------------------------
The dynamics of a particle in a viscous fluid is accompanied by the dissipation of magnetic energy in an external field. We introduce the power loss $Q$, i.e. the magnetic energy dissipation per unit time, in a standard way using the variation of the magnetic energy $\delta W$, which is associated with the magnetic moment increment $\delta \mathbf{M}$ in the external field $\mathbf{H}^{ext}$. Within the assumption that all energy changes are transformed into the irreversible losses, one can write $\delta Q = \mathbf{H}^{ext}\delta \mathbf{M}$. In the simplest noiseless single-particle case, the resulting $Q$ value was obtained by averaging over time $$Q = \lim_{\tau \to \infty} \frac{1}{\tau} \int_{0}^{\tau} \mathbf{H}^{ext} \frac{\partial\mathbf{M}}{\partial t}dt.
\label{eq:def_Q}$$ In the reduced form $\widetilde{Q} = Q/(M^2\tau^{-1}_{1})$, which is the quantity of our main interest, the power loss can be written in the form $$\widetilde{Q} = \lim_{\widetilde{\tau} \to \infty} \frac{1}{\widetilde{\tau}}
\int_{0}^{\widetilde{\tau}} \mathbf{h}^{ext}
\frac{\partial\mathbf{u}}{\partial \widetilde{t}} d\widetilde{t},
\label{eq:def_Q_red}$$ where $\mathbf{u} = \mathbf{M}/M$ is the unit vector of the particle magnetization. It is reasonable to underline here that in the simplest cases of periodic forced motion of $M$, the integration in Eq. (\[eq:def\_Q\_red\]) can be conducted over the reduced field period ($\widetilde{\mathcal{T}} = \mathcal{T}/\tau_1$) only, $$\widetilde{Q} = \frac{1}{\widetilde{\mathcal{T}}}
\int_{0}^{\widetilde{\mathcal{T}}} {\mathbf{h}^{ext}\frac{\partial\mathbf{u}}{\partial\widetilde{t}}} d\widetilde{t}.
\label{eq:def_Q_red_1}$$
In the stochastic case, we need to perform averaging over all the angular states taking into account the probability of each of them. And here, the reduced power loss is calculated as follows $$\widetilde{Q} = \lim_{\widetilde{\tau} \to \infty} \int_{\pi}^{0}d\theta \int_{2\pi}^{0}d\varphi P(\theta, \varphi, \widetilde{t}) \int_{0}^{\widetilde{\tau}}\mathbf{h}^{ext} \frac{\partial\mathbf{u}} {\partial\widetilde{t}} d\widetilde{t}.
\label{eq:def_Q_stoch_red}$$ Since $\mathbf{u}$ is the stochastic function, the integration in Eq. (\[eq:def\_Q\_stoch\_red\]) cannot be carried out in a way, which is common for regular functions. The main difficulty here is in the interpretation of time derivative of $\mathbf{u}$. To avoid this, let us use the well-known approach of integration by parts $\int^{b}_{a} {u(x)v'(x)dx} = \left[U(x)V(x)\right]^{b}_{a} - \int^{b}_{a} {U'(x)V(x)dx}$. Then, we neglect the possible nonlinear effects, such as chaotic [@PhysRevB.91.054425] or quasi-periodic [@0953-8984-21-39-396002] modes, which occur in the internal magnetic dynamics in the fixed particle. From the numerical solution of equations of motion (\[eq:Langevin\_angular\_base\]) for the noiseless case, and equations of motion (\[eq:Langevin\_angular\_eff\]) for nonzero temperature, the following conclusions can be done. Firstly, these modes can be generated in a narrow frequency domain, when $\Omega \sim 1/\tau_1$. Secondly, in our case the effects caused by these modes are suppressed by thermal noise on the large time scale. Therefore, we suppose that $\left[\mathbf{u}\mathbf{h}\right]^{0}_{\widetilde{\mathcal{T}}} = 0$, and for the further calculations we use the relationship $$\widetilde{Q} = - \lim_{\widetilde{\tau} \to \infty} \int_{\pi}^{0}d\theta \int_{2\pi}^{0}d\varphi P(\theta, \varphi, \widetilde{t}) \int_{0}^{\widetilde{\tau}} {\mathbf{u}\frac{\partial\mathbf{h}^{ext}} {\partial\widetilde{t}} d\widetilde{t}}.
\label{eq:def_Q_stoch_red_1}$$
If the probability density $ P(\theta, \varphi, \widetilde{t})$ is the known function of period $\widetilde{\mathcal{T}}$, the integration in Eq. (\[eq:def\_Q\_stoch\_red\_1\]) can be performed over this period $$\widetilde{Q} = - \frac{1} {\widetilde{\mathcal{T}}} \int_{\pi}^{0}d\theta \int_{2\pi}^{0}d\varphi P(\theta, \varphi, \widetilde{t}) \int_{0}^{\widetilde{\mathcal{T}}} \mathbf{u} \frac{\partial\mathbf{h}^{ext}} {\partial\widetilde{t}} d\widetilde{t}.
\label{eq:def_Q_stoch_red_2}$$
For the numerical simulation, the integration in Eq. (\[eq:def\_Q\_stoch\_red\_1\]) is replaced by the summation, and the corresponding difference scheme is used. Based on the spherical representation of the reduced magnetic moment $\mathbf{u} = \mathbf{e}_x\sin\theta \cos\varphi + \mathbf{e}_y\sin\theta \sin\varphi + \mathbf{e}_z\cos\theta$ and Eq. (\[eq:def\_Q\_stoch\_red\_1\]), the explicit form of this difference scheme can be written as $$\begin{aligned}
\widetilde{Q} &=& \frac{-1} {N_1 N_2} \sum^{N_1 N_2}_{i = 1} \bigg[\sin\theta(\widetilde{t}_i) \cos\varphi(\widetilde{t}_i) \frac {\partial h^{ext}_x(\widetilde{t}_i)} {\partial \widetilde{t}} + \nonumber\\
&+& \sin\theta(\widetilde{t}_i) \sin\varphi(\widetilde{t}_i)
\frac {\partial h^{ext}_y(\widetilde{t}_i)}{\partial \widetilde{t}} + \cos\theta(\widetilde{t}_i)
\frac {\partial h^{ext}_z(\widetilde{t}_i)}{\partial \widetilde{t}}\bigg]\Delta \widetilde{t}, \nonumber\\
\label{eq:def_Q_stoch_red_num}\end{aligned}$$ where $N_1 = \widetilde{\mathcal{T}}/\Delta \widetilde{t}$ is the number of time steps on the external field period, $N_2 = \widetilde{\mathcal{T}}_{sim}/\widetilde{\mathcal{T}}$ is the number of periods, during which the simulation is carried out, $\Delta \widetilde{t}$ is the time increment, which is constant in the simulation.
Finally, in the case of the interacting ensemble composed of $N$ particles and simulated on the basis of Eqs. (\[eq:Langevin\_angular\_eff\_red\_k\]), we need to conduct the additional averaging over all particles in the ensemble. We update the technique used in Eq. (\[eq:def\_Q\_stoch\_red\_num\]), and the resulting expression applied for calculation of the power loss in the interacting ensemble has the following form: $$\begin{aligned}
\displaystyle \widetilde{Q}\!&=&\!\frac{-1} {N_1 N_2 N} \sum^{N}_{k = 1} \sum^{N_1 N_2}_{i = 1} \bigg[\sin\theta_k(\widetilde{t}_i) \cos\varphi_k(\widetilde{t}_i) \frac {\partial h^{ext}_{kx}(\widetilde{t}_i)} {\partial \widetilde{t}}\!+ \nonumber\\
\!&+&\!\displaystyle \sin\theta_k(\widetilde{t}_i) \sin\varphi_k(\widetilde{t}_i)
\frac {\partial h^{ext}_{ky}(\widetilde{t}_i)}{\partial \widetilde{t}}\!+\!\cos\theta(\widetilde{t}_i)
\frac {\partial h^{ext}_{kz}(\widetilde{t}_i)}{\partial \widetilde{t}}\bigg]\Delta \widetilde{t}. \nonumber\\
\label{eq:def_Q_stoch_red_num_1}\end{aligned}$$ We suppose that the decrease in the external field energy is simultaneously compensated from the external field source. Moreover, we do not take into account the energy increments arising from the changes of the dipole field $\mathbf{h}_{k}^{dip}$. The increase in the energy of the given particle with changing dipole field is accompanied by the same decrease in the energy of other particles, which are the sources of this dipole field. In other words, the dipole field can transfer energy from one particle to another, but cannot produce the additional power loss.
The details of the numerical calculation in the present paper are the following. To simulate the single-particle stochastic dynamics, the system of equations (\[eq:Langevin\_angular\_eff\_red\]) was solved by the second-order Runge-Kutta method with the time quantification step of $\Delta \widetilde{t} = 0.005 \widetilde{\mathcal{T}} $ in the range of $N_2 = 1000$ reduced field periods for each point of the plot. To simulate the behaviour of the interacting ensemble, the system of equations (\[eq:Langevin\_angular\_eff\_red\_k\]) was solved in the same way with the time quantification step of $\Delta \widetilde{t} = 0.005 \widetilde{\mathcal{T}} $ in the range of $N_2 = 1000$ reduced field periods for each point of the plot for $N = 4096$ number of particles. The values of other system parameters are specified below. The video-cards Nvidia GeForce 450 GTS and Nvidia GeForce 650 GTS Ti were used for our simulation. The program code was realized using C++ language and Eclipse development environment.
RESULTS AND DISCUSSION {#Res}
======================
Single-particle noiseless case
------------------------------
In the simplest case of a diluted ferrofluid, when each particle is far enough from its neighbours and when the thermal energy is much smaller than the magnetic one ($\kappa \gg 1$), the equations of motion (\[eq:Langevin\_angular\_base\]) can be solved exactly in some specific cases. These results have a direct practical value. Firstly, they can be applied to the power loss calculation under the mentioned circumstances. Secondly, they establish the limit values for the stochastic and interacting cases. The last, but not least, is the methodological relevance of the results obtained. The presence of the analytical solutions in the simplest case lets us verify the processing methods of more complicated cases. Despite some results for the regular dynamics of a rigid dipole in a viscous fluid were obtained earlier (see Refs. [@Raikher2011; @PhysRevE.83.021401; @Lyutyy201887]), it is reasonable to systematize and generalize all of them here.
### Motion in the linearly-polarized field
Let us consider the linearly-polarized field action firstly. Since the field given by Eq. (\[eq:ext\_fields\_lp\]) oscillates along the $oz$-axis only, the azimuthal angle $\varphi$ will remain constant and $\dot{\varphi}=0$. Therefore, Eq. (\[eq:Langevin\_angular\_base\_b\]) can be neglected. Substituting Eq. (\[eq:ext\_fields\_lp\]) into Eq. (\[eq:Langevin\_angular\_base\_a\]) within the assumption $\kappa \rightarrow \infty$, we obtain a rather simple differential equation $$\displaystyle \frac{d\theta}{d\widetilde{t}} = - h_{m}\sin\theta \cos(\widetilde{\Omega}\tilde{t}).
\label{eq:dinam_lp_base_eq}$$ Eq. (\[eq:dinam\_lp\_base\_eq\]) can be integrated directly, and the expression describing the particle spherical motion is written as $$\tan (\theta/2) = \tan (\theta_0/2)\exp\left[-\dfrac{h_{m}}{\widetilde{\Omega}}\sin(\widetilde{\Omega}\tilde{t})\right],\\
\label{eq:dinam_lp_solution}$$ where $\theta_0$ is the initial polar angle of vector $\mathbf{u}$, $h_{m} = H_{m}/M$ is the reduced field amplitude. As obvious from Eq. (\[eq:dinam\_lp\_solution\]), the scale of particle oscillations is very sensitive to the ratio $h_{m}/\widetilde{\Omega}$. When $h_{m}/\widetilde{\Omega} \gg 1$, the particle reorientation along the external field is performed fast enough, and the particle is practically immobilized during the most part of the field period. In contrary, when $h_{m}/\widetilde{\Omega} \ll 1$, only small oscillations are performed around the initial position, which is defined by $\theta_0$.
Using (\[eq:def\_Q\_red\_1\]), the average value of the power loss under the linearly-polarized field action can be written in the form $$\widetilde{Q} = \dfrac{\widetilde{\Omega}^{2}h_{m}}{2\pi} \int_{0}^{\widetilde{\mathcal{T}}} d\widetilde{t} \tanh \left[ \dfrac{h_{m}}{\widetilde{\Omega}}\sin(\widetilde{\Omega} \tilde{t}) - \dfrac{x_0}{2}\right] \sin (\widetilde{\Omega} \tilde{t}),
\label{eq:Q_lin}$$ where $x_0 = \ln \left(\tan^2 (\theta_0/2\right))$ is the constant defined by the initial state of $\mathbf{u}$. Despite the exact integration is impossible here, the value of the power loss given by Eq. (\[eq:Q\_lin\]) can be evaluated approximately in the limits of low and high frequencies. Thus, when $h_{m}/\widetilde{\Omega} \gg 1$, the power loss demonstrates a linear behaviour with respect to the field frequency and amplitude, $\widetilde{Q}|_{h_{m}/\widetilde{\Omega}\rightarrow \infty} \rightarrow h_{m}\widetilde{\Omega}/\pi$. Another feature of this asymptotics is an independence on the initial position of $\mathbf{u}$ that is explained as follows. During the field period, the particle has enough time to perform two reorientations along with the external field of the type Eq. (\[eq:ext\_fields\_lp\]) from any initial position. Therefore, the stationary mode is not sensitive to $\theta_0$. The high-frequency asymptotics ($h_{m}/\widetilde{\Omega} \ll 1$) is characterized by an independence on the field frequency, $\widetilde{Q}|_{h_{m}/\widetilde{\Omega}\rightarrow 0} \rightarrow 0.5 h_{m}^2 \cosh^{-2}x_0$. The independency on the frequency explained as follows. When the condition $h_{m}/\widetilde{\Omega}\rightarrow 0$ holds, the oscillations is small. The power loss is defined by the the oscillation amplitude and frequency. The latter is equal to the external field frequency, while the former is inversely to it. And the field frequency increase leads to the proportional decrease in the $\mathbf{u}$ oscillation amplitude that compensates the power loss increase.
### Precession in the circularly-polarized field
Then we consider the circularly-polarized field action. Using again the condition $\kappa \rightarrow \infty$ for the noiseless assumption together with the representation of the circularly-polarized field Eq. (\[eq:ext\_fields\_cp\]), we transform Eqs. (\[eq:Langevin\_angular\_base\]) into the set of differential equations $$\begin{array}{ll}
\displaystyle \frac{d\theta}{d\widetilde{t}} = h_{m}\cos\theta \cos(\widetilde{\Omega}\tilde{t}-\varphi) - h_{0z} \sin\theta,
\\ [10pt]
\displaystyle \frac{d\varphi}{d\widetilde{t}} = h_{m}\frac{\sin(\widetilde{\Omega}\tilde{t}-\varphi)}{\sin\theta},
\end{array}
\label{eq:dinam_cp_base_eq}$$ where $h_{0z} = H_{0z}/M$. One of the possible modes of motion is the precession of the vector $\mathbf{u}$ along with the external field $\mathbf{h}^{ext}$. In this case, the solution of Eqs. (\[eq:dinam\_cp\_base\_eq\]) is given in the form of $\varphi = \varrho \widetilde{\Omega}\tilde{t} -\Phi$ and $\theta = \Theta$. The direct substitution of the las formulas into Eqs. (\[eq:dinam\_cp\_base\_eq\]) permits us to write the set of algebraical equations $$\begin{array}{ll}
h_{m}\cos\Theta \cos\Phi - h_{0z} \sin\Theta = 0,
\\ [10pt]
\widetilde{\Omega} \sin \Theta - h_{m}\sin\Phi = 0,
\end{array}
\label{eq:dinam_cp_solution}$$ the solution of which exhaustively describes the precessional mode. It is important to note that this mode remains stable, when $h_{0z} \neq 0$ or when $h_{0z} = 0$ and $h_{m}\widetilde{\Omega} < 1$. The straightforward calculations using Eqs. (\[eq:dinam\_cp\_solution\]) and Eq. (\[eq:def\_Q\_red\_1\]) yield the following value of the power loss: $$\widetilde{Q} = \widetilde{\Omega}^{2} \sin^{2}\Theta.
\label{eq:Q_cp}$$ In the case of the static field absence ($h_{0z} = 0$), vector $\mathbf{u}$ rotates in the $xoy$ plane and Eq. (\[eq:Q\_cp\]) is reduced into the expression $\widetilde{Q} = \widetilde{\Omega}^{2}$. It is notably that the power loss here does not depend on the field amplitude $h_{m}$.
### Small oscillations around the initial position of $\mathbf{u}$
Finally, we consider the limit case when the vector $\mathbf{u}$ performes rotational oscillations in a small vicinity around its initial position defined by the angles $\theta_0$ and $ \varphi_0$ (see Fig. \[fig:Model\_new\_coord\]). This situation takes place for a small enough ratio of the field amplitude and frequency ($h_{m}/\widetilde{\Omega} \ll 1$). Then, we suppose that the external field is defined as Eq. (\[eq:ext\_fields\_cp\]), but in addition we assume that $h_{0z} = 0$ and $-1 < \varrho < 1$ that includes the linear, elliptical, and circular polarization of $\mathbf{h}^{ext}$. The solution of the basic equations (\[eq:Eq\_of\_motion\_base\]) in the noiseless limit can be found in the linear approximation. The linearization procedure used here is similar to that reported in [@PhysRevB.91.054425] and consists in the following. We introduce the primed coordinate system $x' y' z'$, which is rotated by the angles $\theta_0$ and $\varphi_0$ with respect to the laboratory system $xyz$. In this new coordinate system, the vector $\mathbf{u}$ can be written in the linear approximation as
![(Color online) Schematic representation of our model in the case of small oscillations of the single particle. The further analysis is performed in the primed coordinate system rotated by angles $\varphi_0$, $\theta_0$, which define the initial position of the particle. As an example, the external field is supposed to be circularly-polarized, but the consideration remains valid for other polarizations[]{data-label="fig:Model_new_coord"}](Fig02.eps){width="0.6\linewidth"}
$$\mathbf{u}=\mathbf{e}_{x'}u_{x'}+\mathbf{e}_{y'}u_{y'}+\mathbf{e}_{z'},
\label{eq:m_lin_gen_sol}$$
where $\mathbf{e}_{x'}, \mathbf{e}_{y'}, \mathbf{e}_{z'}$ are the unit vectors of the coordinate system $x' y' z'$. The external field $\mathbf{h}^{ext}$ in the primed coordinate system $x' y' z'$ can be represented using the known rotation matrix as follows $$\begin{array}{lcl}
{\mathbf{h}^{ext}}' = \mathbf{C}\cdot
\left(
\begin{array}{c}
h_{m} \cos(\widetilde{\Omega}\tilde{t}) \\
\varrho h_{m} \sin(\widetilde{\Omega}\tilde{t})\\
0 \\
\end{array}
\right),
\label{eq:h_C}
\end{array}$$ $${\mathbf{C}} =
\left(
\begin{array}{lcr}
\cos\theta_0 \cos\varphi_0 & \cos\theta_0 \sin\varphi_0 & - \sin\theta_0 \\
-\sin\varphi_0 & \cos\varphi_0 & 0 \\
\sin\theta_0 \cos\varphi_0 & \sin\theta_0 \sin\varphi_0 & \cos\theta_0 \\
\end{array}
\right),
\label{eq:C}$$ Substituting Eq. (\[eq:h\_C\]) into Eq. (\[eq:Eq\_of\_motion\_base\]) within the assumption that $ u_{x'}, u_{y'} \sim h_{m}$ and neglecting all the terms, which contain $h_{m}$ in any power greater than one, we derive the linearized system of equations for $\mathbf{u}$ in the form $$\begin{array}{l}
\dfrac{d u_{x'}}{d\tilde{t}}= \cos\theta_0 \cos\varphi_0\ cos(\widetilde{\Omega}\tilde{t}) + \varrho\!\cos\theta_0 \sin\varphi_0 \sin (\widetilde{\Omega}\tilde{t}),\\
[2pt]
\dfrac{d u_{y'}}{d\tilde{t}}= - \sin\varphi_0 \cos(\widetilde{\Omega}\tilde{t}) + \varrho\!\cos\varphi_0 \sin(\widetilde{\Omega}\tilde{t}).\\
\end{array}
\label{eq:Eq_of_motion_base_lin}$$ Then, we use the standard trigonometric representation of the solution of Eqs. (\[eq:Eq\_of\_motion\_base\_lin\]) $$\begin{array}{lcl}
u_{x'}= a\cos(\widetilde{\Omega}\tilde{t}) + b\sin(\widetilde{\Omega}\tilde{t}), \\
%\nonumber
[2pt]
u_{y'}= c\cos(\widetilde{\Omega}\tilde{t}) + d\sin(\widetilde{\Omega}\tilde{t}). \\
\end{array}
\label{eq:Eq_of_motion_base_lin_sol}$$ After direct substitution of Eqs. (\[eq:Eq\_of\_motion\_base\_lin\_sol\]) into Eqs. (\[eq:Eq\_of\_motion\_base\_lin\]), one can easily obtain the unknown constants $$\begin{array}{lcl}
a \!\!&=&\!\! h_{m}\widetilde{\Omega}^{-1} \sin \varphi_{0}, \\
[2pt]
b \!\!&=&\!\! h_{m}\widetilde{\Omega}^{-1} \cos \theta_{0} \cos \varphi_{0}, \\
[2pt]
c \!\!&=&\!\! - h_{m}\widetilde{\Omega}^{-1} \cos \varphi_{0}, \\
[2pt]
d \!\!&=&\!\! h_{m}\widetilde{\Omega}^{-1} \cos \theta_{0} \sin \varphi_{0}. \\
\end{array}
\label{eq:Eq_of_motion_base_lin_sol_coef}$$ And at last, we integrate directly Eq. (\[eq:def\_Q\_red\_1\]) substituting Eqs. (\[eq:Eq\_of\_motion\_base\_lin\_sol\]) and Eqs. (\[eq:Eq\_of\_motion\_base\_lin\_sol\_coef\]) and obtain the short expression for the desired power loss $$\widetilde{Q} = 0.5 h_{m}^2 D,
\label{eq:Q_small_osc}$$ where $$D = \cos^2 \theta_0(\cos^2 \varphi_0 + \varrho^2\sin^2 \varphi_0) + \varrho^2\cos^2 \varphi_0 + \sin^2 \varphi_0.
\label{eq:D}$$ It is natural that Eq. (\[eq:Q\_lin\]) coincides with the high-frequency asymptotic of Eq. (\[eq:Q\_small\_osc\]) up to a constant.
As an intermediate conclusion we want to emphasize that the results obtained in the dynamical approximation set the limit values for the power loss derived in other approximations. But, as it will be shown below, these estimations for high frequencies can have a practical meaning. Then, we need to underline that the expressions found in the dynamical approximation depend strongly on the initial conditions for all cases, excluding the uniform precession mode under the circularly-polarized field action. And the main feature is that the frequency behaviour of the power loss is not similar to the analogous obtained for the case of magnetic dynamics inside the uniaxial particle, which is fixed rigidly in the solid matrix [@PhysRevB.91.054425]. Firstly, for low frequencies the dependencies $\widetilde{Q}(\widetilde{\Omega})$ are quite different for various types of field polarization. When the circularly-polarized field is applied, then $\widetilde{Q} = \widetilde{\Omega}^2$, while $\widetilde{Q} \sim \widetilde{\Omega} h_{m}$ if the linearly-polarized field is applied. Secondly, for high frequencies $\widetilde{Q}(\widetilde{\Omega})$ demonstrates the same saturated character for all polarization types and tends to non-zero constants, but its values for the circularly-polarized field are, at least, two times larger than for the linearly-polarized one depending on the initial position of $\mathbf{u}$.
A single particle in the thermal bath
-------------------------------------
Obviously that thermal fluctuations blur the rotational trajectories of the particle and suppress its response to the external field. This is a reason of the power loss decay with temperature. And the rate of this decay is of great interest from both the theoretical and practical viewpoints. To this end, the direct integration of the equations of motion, which are stochastic here, is not suitable. Therefore, the analytical estimations are conducted statistically using the probability density function and Fokker-Planck formalism, see Eq. (\[eq:FP\]). Because of the difficulties in the exact integration of Eq. (\[eq:FP\]) in the case of the time-periodic field action, its solution is often sought in different approximations, such as the effective field approximation [@Raikher_1994], where the form of the distribution corresponds to that of the distribution in a static field or to the steady-state solution in the linear approximation in $\kappa \widetilde{\Omega}$ [@PhysRevE.92.042312]. It is remarkable that in the case of the linearly-polarized field Eq. (\[eq:ext\_fields\_lp\]) action, the Fokker-Planck equation (\[eq:FP\]) can be found exactly in the form of series [@0953-8984-15-23-313]. Here we summarize all the results in the context of the power loss problem and confirm them numerically based on Eqs. (\[eq:Langevin\_angular\_eff\_red\]).
### Random motion in the linearly-polarized field
Firstly let us consider the case, when the external field oscillates along the $oz$-axis, see Eq. (\[eq:ext\_fields\_lp\]). Due to the symmetry reasons, we suppose that the probability density function $P$ depends on the polar angle $\theta$ only. Then, following [@0953-8984-15-23-313], we present $P$ in the form of $P = P(\theta, \tilde{t}) = \sin\theta f(\tilde{t})$ that, in turn, allows to transform the Fokker-Planck equation (\[eq:FP\]) into $$\begin{aligned}
\dfrac {df}{d\tilde{t}} &=& \nonumber \\
&=& \dfrac {1}{\kappa} \bigg[\dfrac {1}{\sin\theta} \dfrac {\partial}{\partial\theta}\bigg(\sin\theta\dfrac {\partial f}{\partial\theta} + f\kappa h_{m} \sin^2\theta\cos(\widetilde{\Omega} \tilde{t})\bigg)\bigg].\nonumber \\
\label{eq:f_lp_eq}\end{aligned}$$ To simplify the further calculations, we use the designation $\cos\theta = x$. Taking into account the latter, we write finally $$\begin{aligned}
\dfrac {df}{d\tilde{t}} \!&=&\! \nonumber \\
\!&=&\! \dfrac {1}{\kappa} \dfrac {\partial}{\partial x}\bigg[(1-x^2)\dfrac {\partial f}{\partial x} - \kappa h_{m} \cot(\widetilde{\Omega} \tilde{t})(1-x^2)f \bigg].\nonumber \\
\label{eq:f_lp_eq_1}\end{aligned}$$ We underline that equation (\[eq:f\_lp\_eq\_1\]) coincides entirely with the corresponding expression in [@0953-8984-15-23-313]. Its solution was obtained by expansion in Legendre polynomials and harmonics $$\begin{aligned}
f \!&=&\! \nonumber \\
\!&=&\! 0.5\!+\!\sum^{\infty}_{\ell=0} {\bigg[ \sum^{\infty}_{n=0}{A_{\ell n}\cos(n\widetilde{\Omega} \tilde{t})}\!+\! \sum^{\infty}_{n=0}{B_{\ell n}\sin(n\widetilde{\Omega} \tilde{t})} \bigg]}\mathcal{P}_{\ell}(x),\nonumber \\
\label{eq:f_sol}\end{aligned}$$ where $\mathcal{P}_{\ell}(x)$ are the Legendre polynomials, $n$, $\ell$ are the whole numbers. Direct substitution of Eq. (\[eq:f\_sol\]) into Eq. (\[eq:f\_lp\_eq\_1\]) lets us to derive the algebraic set of equations, which yields the unknown coefficients $A_{\ell n}$, $B_{\ell n}$ $$\begin{aligned}
-n\widetilde{\Omega} A_{\ell n} \!&=&\! \nonumber \\
\!&=&\!\dfrac{1}{\kappa}\bigg[\!-\!\ell(\ell + 1) B_{\ell n}\!+\!\dfrac{\kappa h_{m}}{2}\bigg( \dfrac{\ell(\ell+1)}{2\ell-1}\big(B_{\ell-1,n-1}\!+\! \nonumber \\ \!&+&\!B_{\ell-1,n+1}\big)-\dfrac{\ell(\ell+1)}{2l+3}\big(B_{\ell+1,n-1}\!+\!B_{\ell+1,n+1}\big)\bigg)\bigg],
\nonumber \\
\label{eq:A_ln}\end{aligned}$$ $$\begin{aligned}
n\widetilde{\Omega} B_{\ell n} \!&=&\! \nonumber \\
\!&=&\! \dfrac{1}{\kappa}\bigg[\!-\!\ell(\ell + 1) A_{\ell n}\!+\!\nonumber \\
\!&+&\! \dfrac{\kappa h_{m}}{2}\bigg( \dfrac{\ell(\ell+1)}{2\ell-1}\big((1\!+\!\delta_{n \ell})A_{\ell-1,n-1} + A_{\ell-1,n+1}\big)\!- \nonumber \\
\!&-&\! \dfrac{\ell(\ell+1\!)}{2\ell+3}\big((1+\delta_{n \ell})A_{\ell+1,n-1}\!+\!A_{\ell+1,n+1}\big)\bigg)\bigg],
\nonumber \\
\label{eq:B_ln}\end{aligned}$$ when $n \geq 1$ or $\ell \geq 1$. Other cases are defined as follows $$A_{\ell 0} = \dfrac{\kappa h_{m}}{2}\bigg[ \dfrac{1}{2\ell-1}A_{\ell - 1, 1} - \dfrac{1}{2\ell+3}A_{\ell + 1, 1}\bigg]
\label{eq:A_l0_ln}$$ for $\ell \geq 1$, $A_{00} = 0.5$, $A_{0n} = 0$, $B_{\ell 0} = 0$, $B_{0n} = 0$, and, finally, $A_{\ell n} = 0$, $B_{\ell n} = 0$, when $\ell + n$ is odd.
The power loss $\widetilde{\Omega}$ in the case of the linearly-polarized field action was obtained by the direct substitution of Eq. (\[eq:f\_sol\]) (taking into account that $P = P(\theta, \tilde{t}) = \sin\theta f(\tilde{t})$) into Eq. (\[eq:def\_Q\_stoch\_red\_2\]). After performing all integration procedures, the ultimate expression has a rather simple form $$\widetilde{Q} = \dfrac{1}{3} h_{m} \widetilde{\Omega} B_{11},
\label{eq:Q_stoch_lp}$$ which correlates with the noiseless analogue, see the analysis of Eq. (\[eq:Q\_lin\]). The noise influence is hidden in the parameter $B_{11}$, which can be analyzed only in a numerical way. As seen from Fig. \[fig:onepart\_res\_lp\_B\_11\], the dependencies $B_{11} (\kappa)$, which were obtained through the numerical treatment of Eqs. (\[eq:A\_ln\])-(\[eq:A\_l0\_ln\]), are saturated for large frequencies. Then, for small $\kappa$, the values of $B_{11}$ increase rapidly at large frequencies, while for large $\kappa$ these trends are changed to the opposite. This fact suggests about the advantage of the high-frequency fields use, when the temperature is high. Finally, $B_{11} (\kappa)$ grows with the field amplitude, especially for big $\kappa$. Therefore, the sensitivity of $B_{11}$ to $\kappa$ increases with $h_{m}$ and $\widetilde{\Omega}$.
![(Color online) The numerically obtained dependencies of the coefficient $B_{11}$ in Eqs. (\[eq:B\_ln\]) on the parameter $\kappa$, which represents the relationship between the magnetic and thermal energies. The saturated character and complex frequency behaviour are in focus[]{data-label="fig:onepart_res_lp_B_11"}](Fig03.eps){width="1.0\linewidth"}
To confirm the analytical findings, the set of simulation runs has been performed on the basis of the numerical integration of Eqs. (\[eq:Langevin\_angular\_eff\_red\]). As we can see from Fig. \[fig:onepart\_res\_lp\_num\], the analytical and numerical results are in a good agreement in a wide range of parameters. Then, the dependencies are qualitatively similar to the noiseless limit, and the difference decreases with $\kappa$. At last, this difference is more pronounced for small frequencies and vanishes for large ones.
![(Color online) The frequency dependencies of the power loss for different values of the relationship between the magnetic and thermal energies while the linearly-polarized field is applied. The value of the field amplitude is chosen as $h_{m} = 0.05$. The numerical and analytical predictions are in a good agreement. The saturated character of the behaviour and the correspondence with the noise-free results are highlighted[]{data-label="fig:onepart_res_lp_num"}](Fig04.eps){width="1.0\linewidth"}
### Random precession in the circularly-polarized field
In the case of the circularly-polarized field action, the approximate solution of the Fokker-Planck equation (\[eq:FP\]) is grounded on the synchronous (in the average sense) rotation of $\mathbf{u}$ along with $\mathbf{h}^{ext}$ [@PhysRevE.92.042312]. The transition from the azimuthal angle $\varphi$ to the lag angle $\psi = \varphi - \widetilde{\Omega} \tilde{t}$ permits us to represent the steady-state solution $P_{\mathrm{st}}$ of the Fokker-Planck equation (\[eq:FP\]) as a function of two variables $P_{\mathrm{st}} =P_{\mathrm{st}} (\theta, \psi)$. Since $\partial P_{\mathrm{st}}/\partial \tilde{t} = \widetilde{\Omega} \partial P_{\mathrm{st}}/\partial \psi$, we can find the equation for the steady-state probability density $P_{\mathrm{st}}$ directly from Eq. (\[eq:FP\]) $$\begin{aligned}
\widetilde{\Omega}\dfrac{\partial P_{\mathrm{st}}}{\partial\psi} &+& \nonumber \\
&+& \frac{\partial}{\partial \theta}\bigg[h_{m}\cos\psi\cos\theta - h_z \sin\theta + \frac{\cot\theta}{\kappa}\bigg]P_{\mathrm{st}} + \nonumber \\
&+& \sin^{2}\theta \frac{\partial}{\partial\psi}h_{m}\sin\psi P_{\mathrm{st}} - \frac{1}{\kappa}\frac{\partial^{2}P_{\mathrm{st}}}{\partial \theta^{2}} - \nonumber \\
&-& \frac{1}{\kappa}\frac{1}{\sin^{2}\theta} \frac{\partial^{2}P_{\mathrm{st}}}{\partial\psi^{2}} = 0.
\label{eq:FPst}\end{aligned}$$ Following [@PhysRevE.92.042312] and assuming that $\kappa \widetilde{\Omega} \ll 1$, the steady-state probability density $P_{\mathrm{st}}$ is represented in the linear approximation in $\kappa \widetilde{\Omega}$ $$P_{\mathrm{st}} = (1 + \kappa \widetilde{\Omega} F)P_{0},
\label{eq:P_st}$$ where $$\begin{aligned}
P_{0} \!&=&\! \frac{1}{Z} \sin\theta \exp \left[\kappa h_{m}(\sin\theta \cos\psi\!-\!h_{z}\cos\theta)\right],\nonumber \\
Z \!&=&\! \int_{0}^{\pi}\!d\Theta\!\int_{0}^{2\pi}\!d\psi \sin \theta \exp\left[\kappa h_{m}(\sin\theta \cos\psi\!-\!h_{z}\cos\theta)\right].\nonumber \\
\label{eq:P_0}\end{aligned}$$ In what follows, we restrict ourselves to the case, when $h_{z} = 0$ and $\kappa h_{m} \ll 1$. Then, using Eqs. (\[eq:P\_0\]) and (\[eq:FPst\]), it is easy to show that in the first-order approximation in $\kappa h_{m}$ the unknown function $F$ is governed by the equation $$\frac{1}{\sin\theta} \frac{\partial} {\partial\theta}\bigg(\!\sin\theta\frac{\partial F}{\partial\theta}\bigg)
+ \frac{1}{\sin^{2}\theta} \frac{\partial^{2} F}{\partial \psi^{2}} = - \kappa h_{m}\sin\theta\sin\psi.
\label{eq:F_eq}$$ The solution of this equation is simple enough $$F = 0.5 \kappa h_{m} \sin\theta \sin\psi.
\label{eq:F_sol}$$ Taking into account that up to quadratic order in $\kappa h_{m}$, the normalization constant in Eqs. (\[eq:P\_0\]) can be written as $Z = 4\pi (1 + \kappa^{2}h^{2}/6)$ and $$\begin{aligned}
P_{0} &=& \frac{\sin\theta}{4\pi}\! \Big[1 + \kappa h_{m}\sin\theta \cos\psi - \nonumber \\
&-& \frac{\kappa^{2}h_{m}^{2}}{6}\left(1 - 3\sin^{2}\theta \cos^{2}\psi\right) \Big]\!,
\label{eq:P_0_1}\end{aligned}$$ from Eq. (\[eq:P\_st\]) one immediately gets $$\begin{aligned}
P_{\mathrm{st}} &=& \nonumber \\
\!&=&\! \dfrac{\sin\theta}{4 \pi} \bigg[1\!+\!\kappa h_{m} \sin\theta \cos\psi\!-\!\dfrac{\kappa^2 h_{m}^2}{6}\big(1\!-\!\nonumber \\
\!&-&\!3 \sin^2\theta \cos^2\psi\big)\bigg]\!+\!\dfrac{1}{8 \pi} \kappa^2 h_{m} \widetilde{\Omega} \sin^2\theta \sin\psi. \nonumber \\
\label{eq:P_st_fin}\end{aligned}$$
Finally, the power loss $\widetilde{\Omega}$ in the case of the circularly-polarized field action is also obtained by the direct substitution of Eq. (\[eq:P\_st\_fin\]) into Eq. (\[eq:def\_Q\_stoch\_red\_2\]). After all, we derive the following formula $$\widetilde{Q} = \dfrac{1}{6} h_{m}^2\widetilde{\Omega}^2 \kappa^2.
\label{eq:Q_stoch_cp}$$ The main feature of the last expression is in the quadratic dependence on $h_{m}$, which is not typical for the noiseless case, see Eq. (\[eq:Q\_cp\]). Then, attention deserves the quadratic dependence on $\kappa$ that denotes the power loss decay on temperature as $T^2$. Here, we underline that Eq. (\[eq:Q\_stoch\_cp\]) is not applicable for small intensities of thermal noise, when $\kappa \gg 1$.
![(Color online) The comparison of the analytical and numerical results while the circularly-polarized field is applied. The value of the relationship between the magnetic and thermal energies is chosen as $\kappa = 5$, the value of the field amplitude is chosen as $h_{m} = 0.01$. The theory gives larger values, and the difference increases with frequency[]{data-label="fig:onepart_res_cp_theor"}](Fig05.eps){width="1.0\linewidth"}
![(Color online) The frequency dependencies of the power loss for different values of the relationship between the magnetic and thermal energies while the circularly-polarized field is applied. The value of the field amplitude is chosen as $h_{m} = 0.05$. The behaviour is similar to the case of the linearly-polarized field action (see Fig. \[fig:onepart\_res\_lp\_num\]), but the values are approximately two times larger[]{data-label="fig:onepart_res_cp_num"}](Fig06.eps){width="1.0\linewidth"}
To verify our analytical findings and see the domain of applicability of our results, the set of simulation runs has been performed on the basis of the numerical integration of Eq. (\[eq:Langevin\_angular\_eff\_red\]). As we can see from Fig. \[fig:onepart\_res\_cp\_theor\], the analytical and numerical results correlate well for practically interesting values of the noise intensities and external field parameters. Thus, for $\kappa = 5$ and $h_{m} = 0.01$, Eq. (\[eq:Q\_stoch\_cp\]) gives feasible results for the field frequencies up to $\widetilde{\Omega} \sim 0.1$. The study of the power loss in the whole acceptable range of parameters can be performed only numerically. As expected, when the ratio of the magnetic and thermal energies $\kappa$ grows, the power loss tends to the noiseless limit values. As seen from Fig. \[fig:onepart\_res\_cp\_theor\], for small frequencies the difference between the power loss values grows nonlinearly on $\kappa$. However, this difference decreases on $\widetilde{\Omega}$ and is small enough for $\widetilde{\Omega} \sim 1$. At high frequencies $\widetilde{Q}$ tends to a constant, which is proportional to $h_{m}$ in a wide range of $\kappa$. Finally, comparing Fig. \[fig:onepart\_res\_cp\_num\] and Fig. \[fig:onepart\_res\_lp\_num\] one can conclude that the dependencies obtained for the circularly- and linearly-polarized fields are qualitatively similar. In accordance with the noiseless results Eq. (\[eq:Q\_small\_osc\]), the high-frequency limit value for the circularly-polarized field is approximately two times larger than for the linearly-polarized one. Here, the two-fold difference is actual for the entire frequency range that is a consequence of the thermal bath presence and is not typical for the noiseless case. The integrated results for different $\kappa$ and $h_{m}$ were illustrated in Fig. \[fig:onepart\_res\_cp\_num\_3d\], where all mentioned trends and features are shown in one set. As seen from Fig. \[fig:onepart\_res\_cp\_num\_3d\]a, for small frequencies ($\widetilde{\Omega} = 0.05$ in the figure) the character of the dependencies is in a good agreement with Eq. (\[eq:Q\_stoch\_cp\]), where the power loss is proportional to $\kappa^2$ and $h_{m}^2$. In contrast, as follows from Fig. \[fig:onepart\_res\_cp\_num\_3d\]b, for large frequencies ($\widetilde{\Omega} = 1$ in the figure) the dependence on $\kappa$ becomes weak enough, while the nonlinear increase on $h_{m}$ remains.
![(Color online) The dependencies of the power loss on the field frequency and the relationship between the magnetic and thermal energies while the circularly-polarized field is applied. Plot a) demonstrates a strong nonlinear decrease in the power loss on both parameters, when the frequency is low (the value of the field frequency is chosen as $\widetilde{\Omega} = 0.05$). Plot b) demonstrates a weak dependence of the power loss on the relationship between the magnetic and thermal energies, when the frequency is high (the value of the field frequency is chosen as $\widetilde{\Omega} = 1$)[]{data-label="fig:onepart_res_cp_num_3d"}](Fig07a.eps "fig:"){width="1.0\linewidth"} ![(Color online) The dependencies of the power loss on the field frequency and the relationship between the magnetic and thermal energies while the circularly-polarized field is applied. Plot a) demonstrates a strong nonlinear decrease in the power loss on both parameters, when the frequency is low (the value of the field frequency is chosen as $\widetilde{\Omega} = 0.05$). Plot b) demonstrates a weak dependence of the power loss on the relationship between the magnetic and thermal energies, when the frequency is high (the value of the field frequency is chosen as $\widetilde{\Omega} = 1$)[]{data-label="fig:onepart_res_cp_num_3d"}](Fig07b.eps "fig:"){width="1.0\linewidth"}
As a preliminary summary, we want to underline the following. Thermal fluctuations lead to decay of the power loss, but the character of the dependencies $\widetilde{Q}(\widetilde{\Omega})$ remains similar to the noiseless limit. The difference caused by thermal noise decays on the field frequency. The results presented for the circularly-polarized field correlate qualitatively with the results obtained in [@Raikher2011; @PhysRevE.83.021401], while for the linearly-polarized field we, in fact, briefly repeated the results presented in [@0953-8984-15-23-313]. But we conducted our investigation for all these cases in a uniform and rigorous manner, which is based on the Fokker-Planck equation Eq. (\[eq:FP\]) and power loss determination Eq. (\[eq:def\_Q\]) for analytical estimations. In turn, the explicit form of the Langevin equation Eq. (\[eq:Langevin\_angular\_eff\_red\]) is used for the numerical simulation, which gives us a deep understanding of the role of thermal fluctuations in absorption of an alternating field by rigid dipoles.
Influence of the inter-particle interaction
-------------------------------------------
The inter-particle interaction can essentially impact the response to an external field, and accounting of this interaction is important in a view of real ferrofluid applications. Even for relatively small volume fractions (for example, 1 $\%$), due to the repulsion caused by the surfactant covering of each particle and the long-range dipole interaction between particles, the behaviour of each particle will be different from the single-particle approximation outlined above. The dipole interaction intends to join particles into clusters. Such structures are extremely undesirable for hyperthermia by reason of further metabolism and excretion. To prevent this clustering, the particles are coated with a surfactant providing repulsion. Competition between the above mentioned interactions can modify the specific power loss of each particle in a wide range, that is in focus of this section.
The inter-particle interaction, by and large, increases the magnetic energy, and there are two consequences of this fact. Firstly, the regular component of motion becomes strong due to the interaction, and the stochastic component, in contrast, is suppressed. At a glance, such suppression can result in the power loss increase. Secondly, the interaction fixes the particle magnetic moments, and the response to an external field becomes poor. This trend has led to the power loss decrease that was observed in recent experiments [@0022-3727-49-29-295001; @doi:10.1109/TMAG.2016.2516645; @doi:10.1063/1.4974803; @Arteagacardona2016636; @doi:10.1021/acs.molpharmaceut.5b00866; @doi:10.1021/acsnano.7b01762; @doi:10.2147/IJN.S141072; @doi:10.1166/sam.2017.2948; @doi:10.1021/acsnano.7b01762]. Nevertheless, the mentioned experiments do not describe all possible situations and inspire future experimental and numerical investigations.
As stated above, the interaction leads to the cluster formation, when strong dipole fields hold each particle that obstructs further translational and rotational motions. At the same time, the clusters are an origin of a few phenomena, which modify essentially the power loss value. Firstly, each particle tries to reduce its energy and get the equilibrium or quasi-equilibrium state caused by the interaction. In particular, such states are generated by the dipole field, which tries to align the particle magnetic moments along the defined directions. Due to thermal fluctuations, the magnetic moment together with the particle can perform the transition or switching between these states. Such switchings change the magnetic alignment in the clusters and lead to frustrations. Under some circumstances, this can result in the power loss increase. Secondly, large enough fluctuations can destroy the clusters completely and the particle response to an external field becomes better. Here, the power loss also increases, and we interpret this as the constructive role of thermal noise. We have studied in-depth all the above phenomena including the influence of the system parameters on their conditions of occurrence. In this regard, the volume fraction, noise intensity, and surfactant characterizations are the most interesting.
To explain the mentioned phenomena, let us consider the mechanisms of cluster formation in detail. From a position of the minimal magnetostatic energy, the particles should be closer to each other. Also, the magnetic moment of each particle should be directed along the resulting dipole field, which is generated by other particles. Since the magnetic lines of force are closed curves, two trends take place. Firstly, the particle magnetic moments try to be aligned along one direction, and this leads to the chain-like cluster formation. Secondly, the chain fragments tend to be arranged in the antiparallel way and attract each other forming the antiferromagnetic structure. To prevent such agglomeration, each particle is covered with a special surfactant, which provides steric repulsion. The competition between the dipole attraction and steric repulsion can lead to quite different results. We need to underline that since the magnitude of magnetization is important for the performance of the hyperthermia method, it is reasonable to synthesize particles with magnetization as large as possible. As a consequence, the intensity of the dipole interaction should increase, and the clusters will become denser. Therefore, the actuality of interaction accounting will become larger.
![(Color online) Ensemble simulation results: the frequency dependencies of the power loss for different values of the particle volume fractions while the circularly-polarized field is applied. The value of the field amplitude is chosen as $h_{m} = 0.05$, the value of the relationship between the magnetic and thermal energies is chosen as $\kappa = 10$, the value of the potential barrier depth is chosen as $\varepsilon = 0.04$, the dimensionless equilibrium distance between two particles is chosen as $\sigma = 2.25 $. The particles are aggregated into chine-like structures[]{data-label="fig:interact_vol_frac_nonclust"}](Fig08.eps){width="0.9\linewidth"}
Various system parameters influence the cluster formation process differently, and this impact can be ambiguous. As a rule, increase in the volume fraction leads to the particles agglomeration that, in turn, results in the power loss decrease (see Fig. \[fig:interact\_vol\_frac\_nonclust\]). We tend to explain this by different cluster types. When the volume fraction is small, the chain-like structures are formed. The particles in such structures interact weakly, therefore, they are more sensitive to an external field. For a larger volume fraction, the short chain fragments join each other forming denser structures with stronger interaction. And these aggregated structures have a weak response to an external field. But there are some exceptions from this trend. Firstly, when the noise intensity is small enough, the role of the particle concentration can be negligible. This happens because the formed clusters for different volume fractions have the similar structure and remain stable. Secondly, the power loss can increase with the volume fraction, as seen from Fig. \[fig:interact\_vol\_frac\] by the following reasons. When the inter-particle interaction in the clusters is strong, remagnetization of the whole cluster requires a stronger field, and, as a consequence, the hysteresis loop widens. It is confirmed by the fact that the curve for 5$\%$ in Fig. \[fig:interact\_vol\_frac\] exceeds the curve for 3$\%$. This effect disappears for larger frequencies, when the clusters do not reverse completely during the field period. Here, strong interaction suppresses the response of each particle, and, in contrary, the power loss becomes smaller in comparison with the case of lower volume fractions. The ambiguous character of the power loss dependence on the particles concentration has an experimental confirmation n the measurement of specific absorbtion rate of the iron oxide nanoparticles, dispersed in agar [@doi:10.1007/s11051-015-2921-9] (See Fig. 4, page 3).
![(Color online) Ensemble simulation results: the frequency dependencies of the power loss for different values of the particle volume fractions while the circularly-polarized field is applied. The value of the field amplitude is chosen as $h_{m} = 0.05$, the value of the relationship between the magnetic and thermal energies is chosen as $\kappa = 25$, the value of the potential barrier depth is chosen as $\varepsilon = 0.04$, the dimensionless equilibrium distance between two particles is chosen as $\sigma = 2.25 $. The particles are aggregated into dense clusters, consists of a chain fragments[]{data-label="fig:interact_vol_frac"}](Fig09.eps){width="0.9\linewidth"}
Thermal fluctuations break the order with increasing temperature, and this can affect on the ferrofluid response differently. A very interesting effect occurs under some circumstances, when the magnetic energy of a particle is larger than the thermal one, but not so large to exclude the essential fluctuations during the field period. In this case, each particle in a cluster is in the quasi-equilibrium state provided by the resulting dipole field. The particle magnetic moment fluctuates most of the time in the vicinity of one of these states. Due to rare, but strong fluctuations, the particle can transit from one state to another. Such phenomenon is similar to the relaxation of the magnetic moment in the fixed uniaxial particle described in [@PhysRev.130.1677] or to the field-induced switching in the same particle considered in [@PhysRevLett.97.227202]. The transition process proceeds fast enough, but during it each particle is frustrated and characterized by a high energy in an external field. Generally, this leads to the power loss increase, especially for high frequencies, when the time of one transition becomes comparable with the field period. Such increment of the energy dissipation requires a number of conditions, since a lot of factors impact the ratio between the thermal and deterministic energies, namely, noise intensity, surfactant parameters, and cluster types. In Fig. \[fig:interact\_switch\_clust\], the power loss increase by thermal fluctuations is reflected in the behaviour of the curve for $\kappa = 25$. As seen, the curves for $\kappa = 25$ and $\kappa = 40$ coincide for small frequencies. But while the frequency increases, the difference between the curves for $\kappa = 25$ and $\kappa = 40$ becomes larger. As expected, for frequencies more than $\widetilde{\Omega} = 5$ the power loss values for $\kappa = 25$ will be larger even than for $\kappa = 10$. In this sense, this effect can be more productive than the cluster decomposition described below.
![(Color online) Ensemble simulation results: the frequency dependencies of the power loss for different values of the relationship between the magnetic and thermal energies while the circularly-polarized field is applied. The value of the field amplitude is chosen as $h_{m} = 0.05$, the value of the volume fraction is chosen as 1$\%$, the value of the potential barrier depth is chosen as $\varepsilon = 0.04$, the dimensionless equilibrium distance between two particles is chosen as $\sigma = 2.25 $. The origin of the unusual curve behaviour for $\kappa = 25$ consists in the particles switchings between the quasi-equilibrium states, which are constituted by the local dipole fields[]{data-label="fig:interact_switch_clust"}](Fig10.eps){width="0.9\linewidth"}
Despite the noise suppresses the response of each particle to an external field, for the interacting ensemble it can lead to quite different results. The particles in dense clusters are strongly bonded and weakly exposed to an external field. But if the thermal energy is comparable with the magnetic one, thermal fluctuations make the particles free that completely prevents the cluster formation. In this way, thermal fluctuations increase the particle response to an external periodic field and, correspondingly, the energy absorbed from this field. We tend to interpret this as the constructive role of noise. The results of the set of simulations confirming this phenomenon are depicted in Fig. \[fig:interact\_construct\_noise\_role\]. Firstly, for large noise intensities ($\kappa = 10$) the particle aggregation does not occur, and the phenomenon is extremely pronounced. Secondly, for smaller noise intensities the clusters, nevertheless, are partially formed, but they can be destructed depending on other parameters. As seen, at low frequencies the curve for $\kappa = 25$ almost coincides with the curve for $\kappa = 40$, and both of them are located far above the curve for $\kappa = 10$. In both these cases, the similar clusters are formed, and they are not completely broken by thermal noise. But for high frequencies, when the field-induced oscillations of the particles promote the destruction of clusters, the differences between the mentioned curves increases.
![(Color online) Ensemble simulation results: the frequency dependencies of the power loss for different values of the relationship between the magnetic and thermal energies while the circularly-polarized field is applied. The value of the field amplitude is chosen as $h_{m} = 0.05$, the value of the volume fraction is chosen as 3$\%$, the value of the potential barrier depth is chosen as $\varepsilon = 0.02$, the dimensionless equilibrium distance between two particles is chosen as $\sigma = 2.25 $. The curves positions suggest that the clusters are broken completely, when the noise intensity increases[]{data-label="fig:interact_construct_noise_role"}](Fig11.eps){width="0.9\linewidth"}
In summary, we point out that the inter-particle interaction as well as thermal noise suppresses the particle response to an external field. At the same time, both these factors can compete with each other, and the resulting power loss in a periodic external field can vary widely. The cluster formation due to the dipole interaction with further cluster transformations and destruction due to thermal bath and external periodic field actions plays the crucial role here. The main practical result is that the temperature increase promotes the energy absorbtion. There are two reasons for this: the switchings of the particles in clusters due to fluctuations that temporary frustrates the magnetic order, and the complete decomposition of the clusters by noise that increases the particle response to an external field.
Conclusions
===========
We present the comprehensive study of the interaction of identical spherical uniform particles of magnetization $M$, which are placed in a fluid of viscosity $\eta$, with a time-periodic external magnetic field of amplitude $H_m$ and frequency $\Omega$. We imply that the magnetization is locked in the crystal lattice by high magnetic anisotropy. This approach is also the so-called rigid dipole model. The main efforts are devoted to the environmental heating that is a result of the interaction of the external field with the particles. We characterize this heating by the dimensionless power loss $\widetilde{Q}$. The investigation is conducted in three stages. Firstly, we have performed the consideration of the single-particle noise-free or dynamical approximation. Secondly, we have accounted both the external field and thermal bath using the Langevin and Fokker-Planck equations. Finally, the inter-particle interaction effects have been investigated numerically using the Barnes-Hut algorithm [@Barnes-Hut-Nature1986] and CUDA technology [@Sanders2011CUDA]. We summarize our findings as follows.
In the dynamical approximation, the dependencies $\widetilde{Q}(\widetilde{\Omega})$ at $\widetilde{\Omega} \ll 1$ are quite different depending on the external field polarization type. This occurs because the rotational trajectories of the particle and its response to the external fields with various polarizations are different enough for low frequencies. Thus, $\widetilde{Q} = \widetilde{\Omega}^2$ (the dependence on the field amplitude is absent) for the circularly-polarized field and $\widetilde{Q} \sim \widetilde{\Omega} h_{m}$ for the linearly-polarized one. This is a purely dynamical effect, and since it is not observed in the presence of thermal bath, there is no any reason to speak about its practical meaning.
The dependencies $\widetilde{Q}(\widetilde{\Omega})$ have a saturated character and at $\widetilde{\Omega} \ll 1$ tend to the constant, which is proportional to $h_{m}^{2}$. The values of $\widetilde{Q}$ for the circularly-polarized field are, at least, two times bigger than for the linearly-polarized one. In contrast to the low-frequency case, the main trends here remain, when thermal fluctuations are present. Therefore, the results of the dynamical approximation for high frequencies of the external field can be applicable to a real particle in a fluid.
Thermal fluctuations reduce the power loss in a nonlinear way at small frequencies. Thus, for the circularly-polarized external field this reduction is realized as $\kappa^2$ ($\kappa$ is the relationship between the magnetic and thermal energies), when the conditions $\kappa h_{m} \ll 1$, $\kappa \widetilde{\Omega} \ll 1$ hold. At the same time, for large enough frequencies ($\widetilde{\Omega} \sim 1$), the difference between the power loss values at various noise intensities, which is defined by $\kappa$, becomes less pronounced. At last, when $\widetilde{\Omega} \ll 1$, the results are almost indistinguishable from the dynamical approximation in a wide range of $\kappa$. This behaviour needs to be accounted to improve the performance of the magnetic fluid hyperthermia method.
The power loss values for the circularly-polarized field approximately two times exceed the values for the linearly-polarized one in the whole frequency range. This is obvious for the quasi-equilibrium approximation described in [@Rosensweig2002370] and follows from our analysis of the noise-free limit for high frequencies. But it is rather unexpected in a light of low-frequency behaviour in the dynamical approximation described above. This fact suggests that thermal fluctuations change qualitatively the particle behaviour.
The interaction between the particles has a strong impact on the ensemble susceptibility to the external periodic fields. Because of the cluster formation, each particle is under the action of the strong enough resulting dipole field that complicates its response. Evidently, this results in a considerable decay of the power loss that is in agreement with the experiments. At the same time, the different cluster types are possible, and even small changes of the field, particle or fluid parameters can cause the quite different structure of the ensemble. These effects are especially actual for low frequencies, when the clusters inverse their magnetization completely during the field period. This leads to the strong difference between the $\widetilde{Q}$ values in comparison with the results of the single-particle case. This difference reduces with the field frequency, since for high frequencies each particle performs oscillations around its initial position without full inversion of the magnetization. Therefore, the interaction impact is not critical, when $\widetilde{\Omega} \gg 1$, excluding the specific situation described below.
However, the inter-particle interaction and thermal noise are the competing factors. We have revealed two situations, when thermal noise leads to the power loss increase in the presence of inter-particle interaction. We associate these phenomena with the constructive role of noise. In both mentioned cases, the resulting values of $\widetilde{Q}$ can be almost equal to the value for the single-particle approximation. The first of them occurs, when the noise intensity is large. The clusters are destroyed here that results in the $\widetilde{Q}$ increase as a consequence of the better response of the particles to the external field. And under certain other conditions, larger noise intensity corresponds to larger power loss values. The second of them occurs, when the noise intensity is not so large, and the magnetic energy is comparable with the thermal one. Here, the fluctuations partially blur the particles order in the clusters that also leads to the significant increase in $\widetilde{Q}$ for high field frequencies. We have explained this by rare switchings of the particles in the clusters between the quasi-equilibrium states formed by the resulting dipole fields. The switching process is performed through the frustrated states characterised by high energies. Exactly the high energy consumption in these mentioned states causes the observed increase in the power loss.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
The authors express appreciation to S. I. Denisov and Yu. S. Bystrik for the valuable comments and discussion. The authors are also grateful to the Ministry of Education and Science of Ukraine for partial financial support under Grant No. 0116U002622.
References {#references .unnumbered}
==========
|
---
author:
- 'N. Lützgendorf'
- 'M. Kissler-Patig'
- 'E. Noyola'
- 'B. Jalali'
- 'P. T. de Zeeuw'
- 'K. Gebhardt'
- 'H. Baumgardt'
bibliography:
- 'ref.bib'
date: 'Received Febrary 1, 2011; accepted July 20, 2011'
title: 'Kinematic signature of an intermediate-mass black hole in the globular cluster NGC 6388. [^1]'
---
[Intermediate-mass black holes (IMBHs) are of interest in a wide range of astrophysical fields. In particular, the possibility of finding them at the centers of globular clusters has recently drawn attention. IMBHs became detectable since the quality of observational data sets, particularly those obtained with HST and with high resolution ground based spectrographs, advanced to the point where it is possible to measure velocity dispersions at a spatial resolution comparable to the size of the gravitational sphere of influence for plausible IMBH masses.]{} [We present results from ground based VLT/FLAMES spectroscopy in combination with HST data for the globular cluster NGC 6388. The aim of this work is to probe whether this massive cluster hosts an intermediate-mass black hole at its center and to compare the results with the expected value predicted by the $M_{\bullet} - \sigma$ scaling relation.]{} [The spectroscopic data, containing integral field unit measurements, provide kinematic signatures in the center of the cluster while the photometric data give information of the stellar density. Together, these data sets are compared to dynamical models and present evidence of an additional compact dark mass at the center: a black hole.]{} [Using analytical Jeans models in combination with various Monte Carlo simulations to estimate the errors, we derive (with $68 \% $ confidence limits) a best fit black-hole mass of $ (17 \pm 9) \times 10^3 M_{\odot}$ and a global mass-to-light ratio of $M/L_V = (1.6 \pm 0.3) \ M_{\odot}/L_{\odot}$.]{}
Introduction
============
For a long time, only two mass ranges of black holes were known. On the one hand, we have stellar mass black holes, which are remnants of massive stars, and can be observed in binary systems. On the other hand, there are supermassive black holes at the centers of galaxies, some of them accreting at their Eddington limit and producing the brightest objects known (quasars). It has been demonstrated that supermassive black holes show a tight correlation between their mass and the velocity dispersion of the galaxy in which they reside [e.g. @ferrarese_2000; @gebhardt_2000; @gultekin_2009]. Extrapolating this relation to the lower velocity dispersions of globular clusters, with $\sigma \sim 10 - 20$ [$\mathrm{km\, s^{-1}\, }$]{}, predicts central black holes in these objects with masses of $10^3 - 10^4 M_{\odot}$.
Due to the small amount of gas and dust in globular clusters, the accretion efficiency of a potential black hole at the center is expected to be low. Therefore, the detection of IMBHs at the centers of globular clusters through X-ray and radio emissions is challenging [@miller_2002; @maccarone_2008]. Nevertheless, there is another way to detect IMBHs in globular clusters: exploring the kinematics of these systems in the central regions. This method, proposed forty years ago [@bahcall_1976; @wyller_1970], has long been limited by the quality of observational datasets, since it requires velocity dispersion measurements at a spatial resolution comparable to the size of the gravitational sphere of influence for plausible IMBH masses ($1 - 2 ''$ for large Galactic globular clusters). However, with existence of the Hubble Space Telescope (HST) and with high spatial resolution ground based integral-field spectrographs, the search for IMBHs was revitalized.
[@gebhardt_1997; @gebhardt_2000] and [@gerssen_2002] claimed the detection of a black hole of $(3.2 \pm 2.2) \times 10^3 M_{\odot}$ in the globular cluster M15 from photometric and kinematic observations. After more investigations this cluster no longer appears as a strong IMBH candidate [e.g. @dull_1997; @baumgardt_2003a; @baumgardt_2005; @bosch_2006], but new detections of IMBH candidates in other clusters followed. [@gebhardt_2002; @gebhardt_2005] used the velocity dispersion measured from integrated light near the center of the M31 cluster G1 to argue for the presence of a $(1.8 \pm 0.5) \times 10^4 M_{\odot}$ dark mass at the cluster center. The possible presence of an IMBH in G1 gained further credence with the detection of weak X-ray and radio emission from the cluster center [@pooly_2006; @kong_2007; @ulvestad_2007]. Also, the globular cluster $\omega$ Centauri (NGC 5139) has been proposed to host a black hole at its center [@noyola_2008; @noyola_2010]. The authors measured the velocity-dispersion profile with an integral field unit and used orbit based dynamical models to analyze the data. [@vdm_2010] studied the same object using proper motions from HST images. They found less compelling evidence for a central black hole, but more importantly, they found a location for the center that differs from previous measurements. Both G1 and $\omega$ Centauri have been suggested to be stripped nuclei of dwarf galaxies [@freeman_1993; @meylan_2001] and therefore may not be the best representatives of globular clusters. The key motivation of this work is to probe more globular clusters for the presence of IMBHs.
Further evidence for the existence of IMBHs is the discovery of ultra luminous X-ray sources at non-nuclear locations in starburst galaxies [e.g. @fabbiano_1989; @colbert_1999; @matsumoto_2001; @fabiano_2001]. The brightest of these compact objects (with $L \sim 10^{41} \ \mathrm{erg}\, \mathrm{s}^{-1}$) imply masses larger than $10^3 M_{\odot}$ assuming accretion at the Eddington limit. Several realistic formation scenarios of black holes in globular clusters have been developed. The two main formation theories are: a) IMBHs would be Population III stellar remnants [@madau_2001], or b) they would form in a runaway merging of young stars in sufficiently dense clusters [@zwart_2004; @gurkan_2004; @freitag_2006]. In addition [@miller_2002] presented scenarios for the capture of clusters by their host galaxies and accretion in the galactic disk in order to explain the observed bright X-ray sources.
Our goal in this paper is to study the globular cluster NGC 6388. This cluster is located $11.6$ kpc away from the Sun, in the outer bulge of our Galaxy. [@white_1972] assigned it a high metallicity after studying its color magnitude diagram. Later, [@illingworth_1974] determined the dynamical mass of the cluster: with $\sim 1.3 \times 10^6 M_{\odot}$ it belongs to the most massive clusters in the Milky Way. In addition, its high central velocity dispersion of $18.9$ [$\mathrm{km\, s^{-1}\, }$]{}[@pryor_1993], and assuming a black-hole mass correlation with velocity dispersion, make NGC 6388 a good candidate for detecting an intermediate-mass black hole. Besides the kinematic properties, the photometric characteristics are also quite interesting. @noyola_2006 [hereafter NG06] found a shallow cusp in the central region of the surface brightness profile of NGC 6388. N-body simulations showed that this is expected for a cluster hosting an intermediate-mass black hole [@baumgardt_2005]. For that reason, [@lanzoni_2007] investigated the projected density profile and the central surface brightness profile with a combination of HST high-resolution and ground-based wide-field observations. They found the observed profiles are well reproduced by a multimass, isotropic, spherical King model, with an added central black hole with a mass of $\sim 5.7 \times 10^3 M_{\odot}$. Also, the work of [@miocchi_2007] suggests the presence of an intermediate-mass black hole in NGC 6388 as a possible explanation for the extended horizontal branch.
Another interesting fact about NGC 6388 is that it appears to contain multiple stellar populations [e.g. @yoon_2000; @piotto_2008]. This fact makes the scenario of merging clusters plausible, leading to potentially complicated dynamics in the center. Recently, X-ray observations for NGC 6388 showed no significant signatures for an accreting IMBH [@cseh_2010]. But as already mentioned, this does not rule out a quiescent black hole. In summary, this cluster was chosen as it presents many interesting features. We measured the kinematics of the central regions, allowing us to probe the result of [@lanzoni_2007] with a different method, taking the kinematic properties as well as the photometric properties into account.
The basic approach of this work is to first study the light distribution of the cluster. Photometric analysis, such as the determination of the cluster center and the measurement of a surface brightness profile, is described in section \[phot\]. De-projecting this profile gives an estimation of the gravitational potential produced by the visible mass. The next step is to study the dynamics of the cluster. Section \[spec\] gives an overlook of the FLAMES observations and data reduction and section \[kin\] describes the analysis of the spectroscopic data. With the resulting velocity-dispersion profile, it is possible to estimate the actual dynamical mass. The next step is to compare the data to dynamical Jeans models (section \[jeans\]). These models take the light profile and predict a velocity-dispersion profile, which is scaled to the data in order to obtain the mass-to-light ratio. This is done for models containing different black-hole masses until the best fit to the observed profile is found. In section \[con\] we summarize our results, list our conclusions and give an outlook for further studies.
[l l l]{} Parameter & Value & Reference\
RA (J200) & $17\mathrm{h} \ 36\mathrm{m} \ 17\mathrm{s}$ & NG\
DEC (J200) & $-44^{\circ}\ 44' \ 08''$ & NG\
Galactic Longitude l & $345.56 °$ &H\
Galactic Latitude b & $-6.74 °$ &H\
Distance from the Sun $R_{\mathrm{SUN}}$ & $11.6 \ \mathrm{kpc}$ & M\
Core Radius $r_c$ & $7.2''$ & L\
Central Concentration c & $1.8$ & L\
Heliocentric Radial Velocity V$_r$ & $81.2 \pm 1.2 \ \mathrm{km}/\mathrm{s}$ & H\
Central Velocity Dispersion $\sigma$ & $18.9 \ \mathrm{km}/\mathrm{s}$ & PM\
Age & $ (11.5 \pm 1.5) \ \mathrm{Gyr}$ & M\
Metallicity $[\mathrm{Fe}/\mathrm{H}]$ & $-0.6 \ \mathrm{dex}$ & H\
Integrated Spectral Type & G2 &H\
Reddening E(B-V) & $0.38 $ & M\
Absolute Visual Magnitude $M_{Vt}$ & $-9.42 \ \mathrm{mag}$ & H\
Photometry {#phot}
==========
The photometric data, retrieved from the archive, were obtained with the Advanced Camera of Surveys (ACS) of the Hubble Space Telescope (HST) in the Wide-Field Channel (under the HST program SNAP-9821, PI: B.J. Pritzl) between October 2003 and June 2004. This data set is composed of six B (F435W), V (F555W) and I (F814W) images with exposure times of 11, 7 and 3s, respectively. It gives a complete coverage of the central region of the cluster out to a radius of $110''$. The data were calibrated, geometrically corrected and dither-combined as retrieved from the European HST-Archive (ST-ECF, Space Telescope European Coordinating Facility[^2]).
Color magnitude diagram (CMD) of NGC 6388 {#sec_cmd}
-----------------------------------------
The CMD was obtained using the programs *daophot II*, *allstars* and *allframes* by P. Stetson, applied to our HST image. For a detailed documentation of these routines see [@stetson_1987]. These programs were especially developed for photometry in crowded fields and therefore ideally suited for the analysis of globular clusters. The routines *find*, *phot* and *psf* identify the stars, perform aperture photometry and compute the average point spread function (PSF) over the image, respectively.
Once the PSF has been defined, the next step is to group the neighboring stars to apply the multiple-profile-fitting routine simultaneously by the task *allstars*. Afterwards, the *find* task is applied again to find, in the star-subtracted image, stars which were not found in the first run. The entire procedure was performed on the V- and I-band images independently. As a next step the programs *daomaster* and *allframes* were used to combine both images and to create the final catalog containing all stars, their positions and magnitudes in the two bands. At the end, we calibrated the final catalog to the Johnson magnitude system by following the steps described in [@sirianni_2005].
In order to get better quality at the faint end of the CMD, one final step was applied to the catalog. The program *separation* [@stetson_2003] computes a separation index for every star in a catalog. This index is calculated by, first, evaluating a local surface brightness at the position of a given star. Second, the local surface brightness is compared to the sum of the surface brightnesses produced by the PSF of all the other stars in the field at the position of the centroid of that star. The ratio of these two surface brightness values expressed in magnitudes determines the separation index. Thus, the stars could be selected considering background-light contamination and not only by magnitude. Figure \[cmd\] shows the final CMD of stars with a separation index $\ge 5$ overplotted with the positions of the brightest stars in the ARGUS pointing and the spectroscopic template star (see section \[spec\]).
![The color-magnitude diagram of NGC 6388. Overplotted are the brightest stars identified in the ARGUS field of view (red circles), and the template star used (star symbol).[]{data-label="cmd"}](cmd_only.ps){width="50.00000%"}
Center of the cluster {#phot_center}
---------------------
Using the star catalog generated with *daophot*, the center of the cluster can be determined. Precise knowledge of the cluster center is important since the shape of the surface brightness and the angular averaged line-of-sight velocity distribution (LOSVD) profiles depend on the position of that center. Using the wrong center typically produces a shallower inner profile. For example determining the center for $\omega$ Centauri is not trivial [@noyola_2006; @noyola_2010; @vdm_2010]. Fortunately, NGC 6388 is not as extended and has a steeper light profile than $\omega$ Centauri so that the center is easier to determine. NG06 determined the center of this cluster to be at $\alpha=17:36:17.18, \ \delta=-44:44:07.83 \ (J2000)$, with an uncertainty of $0.5''$, by minimizing the standard deviation of star counts in eight segments of a circle. In view of the discrepancies about the center location for $\omega$ Centauri, we decided to recompute the center of NGC 6388 and to evaluate how precisely the center can be determined. NG06’s center on our I-band image was used as a first guess for the following methods to determine the center in the ACS images, i.e. our reference frame for the spectroscopy.
{width="\textwidth"}
The first method is a simple star count as described in [@47tuc]. The catalog generated with *daophot* (see section \[sec\_cmd\]) contains 88,406 stars. In a field of $4'' \times 4''$ a grid of trial centers was created, using a grid spacing of 2 ACS pixels ($0.1''$). Around each trial center a circle of 300 pixels ($15''$) was considered and divided into 16 wedges as shown in Figure \[center\]. The stars in each wedge were counted and compared to the opposite wedge. The differences in the total number of stars between two opposing wedges were summed for all 8 wedge pairs. The coordinates that minimized the difference defined a first guess of the center of the cluster. This center was refined as described in the next sections. We compared our result to the center obtained by NG06. The two centers are only $0.32''$ apart and thus coincide within the error bars of $0.5''$ (as determined by NG06 performing artificial image tests).
The second method that we used is also described in [@47tuc]. Instead of comparing the total number in pairs of opposite wedges, a cumulative radial distribution for each wedge in 4 bins was generated. This time 8 wedges were used instead of 16 to avoid too large stochastic errors due to insufficient numbers of stars in each bin. The bins were placed at equidistant radii. Again the absolute value of the integrated difference between the radial distributions in any two opposing wedges was calculated and the minimum used to determine the new center of the cluster. The so derived center lies within $0.1''$ of the one derived with the previous method and within $0.3''$ of NG06’s location.
We present a last method in which the light of the stars instead of their number, is considered. Similar to the first method, 16 wedges without any radial bins were generated, but this time not the stars were counted but the luminosities of the stars were summed up in each wedge. In order to avoid a bias by the contamination of a few bright stars, only stars from the lower giant branch and the horizontal branch (between $m_V = 15$ and $m_V = 19$) were used. This approach reproduces NG06’s center within the error bars ($0.28''$) as well. The method is illustrated in Figure \[center\]. Shown are the grid of trial centers, the 16 wedges we applied to each of them, and the final contour plot.
For the subsequent analysis, we used the center derived from our ACS catalog. To estimate the error of our center determination we took the scatter from the three different methods as well as an additional test where only eight wedges where used. For these eight wedges we repeated the routine by only using the cardinal wedges and the semi-cardinal wedges separately. For this dataset (HST J8ON08OYQ, also used as position reference coordinates) we derived a final position of the center of:
$$(x_c,y_c) = (2075.5, 2548.5) \pm (3.1,2.1) \ \mathrm{pixel}$$
\[p3\] $$\begin{aligned}
\alpha &=& 17:36:17.441, \ \Delta\alpha = 0.2'' \ \mathrm{(J2000)} \\ \label{p4}
\delta &=& -44:43:57.33, \ \Delta\delta = 0.1'' \end{aligned}$$ \[p4\]
All the derived centers lie within a few tenths of an arcsecond ($\sim 10^{-2}$ pc) radius. However, it must be considered that all the methods are using the same catalog derived by DAOPHOT. This catalog most likely suffers from incompleteness since the bright stars are covering the fainter stars underneath. This could bias the center towards the bright stars even if they are excluded in the counts. Our center agrees within the error bars with the one determined in NG06 on WFPC2 images. It also coincides with the center found by [@lanzoni_2007] which, according to them, lies $\sim 0.5''$ northwest of the NG06 center and therefor closer to ours.
Surface brightness profile {#phot_sb}
--------------------------
The last step of the photometric analysis was to obtain the surface brightness profile. This is required as an input for the Jeans models described in the following section. To obtain the profile, a simple method of star counts in combination with an integrated light measurement from the ACS image was applied. The fluxes of all stars brighter than $m_V = 18$ were summed in radial bins of equal width (50 pix) around the center and divided by the area of each bin. Since there are no stars in the gap between the two ACS CCD chips this area was subtracted from each affected bin.
In addition, the integrated light for stars fainter than $m_V = 18$ was measured directly from the HST image. Using the same radial bins as in the star count method, we measured the statistical distribution of counts per pixel excluding regions with stars with $m_V < 18$. After trying different methods to derive the “average" of the distribution of counts, we applied a simple average that takes into account the faintest pixels i.e. stars. At the end, the flux per pixel was transformed back into magnitude per square arcseconds and added to the star counts profile.
We compared our profile with [@trager_1995], [@lanzoni_2007] and NG06. The latter was derived by measuring the integrated light using a bi-weight estimator while [@lanzoni_2007] derived their points by taking the average of the counts per pixel in each bin. We were able to reproduce the shape in the outer regions, but due to the method and data that we used our profile in the innermost region has a high uncertainty. The errors on our profile were obtained by Poisson statistics of the number of stars in each bin. With a linear fit inside the core radius ($\sim 7''$) we derived a logarithmic slope of $\beta = 0.7 \pm 0.2$ (where $\beta$ corresponds to $\mu_V \sim \log r^{\, \beta}$ with the surface brightness $\mu_V$), which results in a slope of the surface luminosity density $I(r) \varpropto r^{\, \alpha}$ of $\alpha = -0.28 \pm 0.08$. This value is steeper (but consistent within the errors) than the slope of $\alpha =-0.13 \pm 0.07$ derived by NG06 and also consistent with the slope derived by [@lanzoni_2007] of $\alpha=-0.23 \pm 0.02$.
To derive the SB profile, we used a calibrated catalog but we also combined this with measurements that we obtained from counts per pixel. It is possible that the transformations from counts into magnitudes do not match the calibrations of the catalog and therefore would cause an offset in our profile. The fact that we used all the light in the image, including the background light could also shift the profile to higher magnitudes than the intrinsic brightness. For that reason, we scaled our profile to the outer parts of the profile by [@trager_1995]. This profile was obtained with photometrically calibrated data and provides a good reference for the absolute values of the profile. The final result of the surface brightness profile is shown in Figure \[sb\_fit\].
[c c c c]{}
$\log r$ & $V$ &$\Delta V_{l}$ & $\Delta V_{h}$\
\[arcsec\] & $[\mbox{mag} / \mbox{arcsec}^2]$ &$ [\mbox{mag} / \mbox{arcsec}^2]$ & $[\mbox{mag} / \mbox{arcsec}^2]$\
- 0.40 & 14.00 & 0.52 & 0.35\
- 0.12 & 14.53 & 0.52 & 0.35\
0.18 & 14.31 & 0.21 & 0.18\
0.40 & 14.74 & 0.16 & 0.14\
0.60 & 14.78 & 0.10 & 0.09\
0.81 & 15.18 & 0.07 & 0.07\
1.06 & 15.62 & 0.05 & 0.05\
1.22 & 16.21 & 0.05 & 0.05\
1.33 & 16.68 & 0.06 & 0.06\
1.42 & 17.18 & 0.07 & 0.07\
1.50 & 17.50 & 0.07 & 0.07\
1.56 & 17.91 & 0.08 & 0.07\
1.62 & 18.20 & 0.08 & 0.08\
1.67 & 18.54 & 0.09 & 0.09\
1.71 & 18.74 & 0.10 & 0.09\
1.75 & 18.80 & 0.09 & 0.09\
1.79 & 19.17 & 0.11 & 0.10\
1.82 & 19.33 & 0.11 & 0.10\
1.85 & 19.61 & 0.13 & 0.11\
1.88 & 19.44 & 0.10 & 0.10\
![The surface brightness profile of NGC 6388. The profile shows a clear cusp for the inner 10 arcseconds. Also shown is the MGE fit (solid line) which was used to parametrize our profile (see section \[models\].)[]{data-label="sb_fit"}](SB_fit.eps){width="9"}
Spectroscopy {#spec}
============
Observations
------------
The spectroscopic data were observed with the GIRAFFE spectrograph of the FLAMES (Fiber Large Array Multi Element Spectrograph) instrument at the Very Large Telescope (VLT) in ARGUS (Large Integral Field Unit) and IFU (Integral Field Unit) mode. The set contains spectra from the center (ARGUS) and the outer regions (IFU) for the globular cluster NGC 6388. The observations were performed during two nights (2009-06-14/15). The ARGUS unit was set to the 1 : 1 magnification scale (pixel size: $0.3 ''$, $14 \times 22$ pixel array) and pointed to three different positions (exposure times of the pointings: A: $3 \times 480 \, \mathrm{s} + 3 \times 1500 \, \mathrm{s}$, B: $3 \times 1500 \, \mathrm{s}$, C: $1 \times 1500 \, \mathrm{s}$) to cover the sphere of influence of the potential black hole (see Figure \[recon\]). The IFU fibres were placed around the core radius of the cluster to obtain velocity dispersions in the outer regions.
The kinematics were obtained from the analysis of the Calcium Triplet ($\sim 850 \, \mathrm{nm}$) which is a strong feature in the spectra. The expected velocity dispersions lie in the range 5-20 [$\mathrm{km\, s^{-1}\, }$]{}and need to be measured with an accuracy of 1-2 [$\mathrm{km\, s^{-1}\, }$]{}. This implied using a spectral resolution around $10000$, available in the low spectral resolution mode set-up LR8 ($820-940 \, \mathrm{nm}, \, \mathrm{R} = 10400$).
![The positions of the three ARGUS pointings (A, B and C) reconstructed on the HST/ACS image.[]{data-label="recon"}](reconstructed_all.eps){width="45.00000%"}
Data reduction
--------------
We reduced the spectroscopic data with the GIRAFFE pipeline programmed by the European Southern Observatory (ESO). This pipeline consists of five recipes which are briefly described below.
First, a master bias frame was created by the recipe *gimasterbias* from a set of raw bias frames. Next, a master dark was produced by the recipe *gimasterdark* which corrected each input dark frame for the bias and scaled it to an exposure time of 1 second. The recipe *gimasterflat* was responsible for the detection of the spectra on a fiber flat-field image for a given fiber setup. *Gimasterflat* located the fibers, determined the parameters of the fiber profile by fitting an analytical model of this profile to the flat-field data and created an extracted flat-field image. This image was later used to apply corrections for pixel-to-pixel variations and fiber-to-fiber transmission.
The pipeline recipe *giwavecalibration* computed a dispersion solution and a slit geometry table for the fiber setup in use. This was done by extracting the spectra from the bias corrected ThAr lamp frame, using the fiber localization (obtained through the flat-field), selecting the calibration lines from the line catalog, and predicting the positions of the ThAr lines on the detector using an existing dispersion solution as an initial guess. The quality of the calibration was checked in two ways. The first one was by measuring the centroid of one dominating sky emission line in all 14 sky spectra and the second was a cross-correlation in Fourier-space of all sky spectra.
Both methods did not show any systematic shift of the spectra and resulted in an RMS of $0.03$ Å, which corresponds to a velocity of $1$ [$\mathrm{km\, s^{-1}\, }$]{}. The last recipe, *giscience* combined all calibrations and extracted the final science spectra. A simple sum along the slit was applied as extraction method. The input observations were averaged and created a reduced science frame, the extracted and rebinned spectra frame. At the end, the recipe also produced a reconstructed image of the respective field of view of the IFU and the ARGUS observations.
The most important parts of our reduction are the sky subtraction and an accurate wavelength calibration to avoid artificial line broadenings due to incorrect line subtractions. The program we used was developed by Mike Irwin and described in [@battaglia_2008]. To subtract the sky, the program first combined all (14) sky fibers using a 3-sigma clipping algorithm and computed an average sky spectrum. Using a combination of median and boxcar, it then split the continuum and the sky-line components in order to create a sky-line template mask. For the object spectra, the same method of splitting the continuum from the lines was applied. In the process, the sky-lines were masked out to get a more accurate definition of the continuum. Afterwards, the sky-line mask and the line-only object spectra were compared to find the optimum scale factor for the sky spectrum in order to subtract the sky-lines from the object spectra. The continuum was added back to the object spectra and the sky continuum subtracted by the same scaling factor as obtained for the lines (assuming that lines and continuum have the same scaling factor).
After applying the sky subtraction program, the last step in the reduction of the data was to remove the cosmic rays from the spectra. This was done using the program LA-Cosmic developed by [@Lacos] and based on a Laplacian edge detection. In order to avoid bright stars dominating the averaged spectra when they were combined, we applied a simple normalization by fitting a spline to the continuum and dividing by it.
For the small IFUs, we applied the same reduction steps as for the large integral-field unit, except for the cosmic ray removal, since LA-Cosmic did not give a satisfying result. However, we were able to average all exposures applying a sigma clipping method in order to remove the cosmic rays, due to the fact that these pointings were not dithered.
![Combined spectra of the first, third and sixth bin overplotted by their best fit (red line). Due to the binning, the spectra show a high signal-to-noise ratio.[]{data-label="model_point"}](spec_all.eps){width="50.00000%"}
{width="\textwidth"}
{width="\textwidth"}
Kinematics {#kin}
==========
This section describes how we created a velocity map in order to check for rotation or other peculiar kinematic signals and how we derived a velocity-dispersion profile. This profile is then used to fit analytic models, described in the next section.
Velocity map
------------
The reconstructed images of the three ARGUS pointings were matched to the HST image. With this information, the pointings were stitched together and each spectrum correlated to one position in the field of view. The resulting catalog of spectra and their coordinates allowed us to combine spectra in different bins. The combined pointing contains 24 $\times$ 29 spaxels.
For each spaxel, the penalized pixel-fitting (pPXF) program developed by [@cappellari_2004] was used to derive the kinematics in that region. Figure \[argus\] shows a) the field of view of the three pointings on the HST image, b) the field convolved with a Gaussian and resampled to the resolution of the ARGUS array and c) the reconstructed and combined ARGUS image. The corresponding velocity map is shown in Figure \[vel\]. For every ARGUS pointing, a velocity map was derived before combining. The C pointing is a bit more noisy due to the fact that it only had one exposure. To check for systematic wavelength offsets we compared the derived velocities of the different pointings and exposures at overlapping spaxels. We found offsets of up to 1 $\mathrm{km\, s.^{-1}\, }$ We checked for systematic wavelength offsets between the pointings by cross-correlating their averaged sky spectra. The rms of the shifts measures 0.006 pixel (0.0018 Å) which corresponds to a velocity shift of $\sim$ 0.1 $\mathrm{km\, s.^{-1}\, }$ We conclude that the velocity offsets are not due to uncertainties in the wavelength calibration and do not cause problems for our further analysis. Instead, the offsets are explained as follows:
i\) Different pointings were dithered about half an ARGUS spaxel. This means that in every pointing a slightly different combination of stars contributes to the different spaxel, which could cause a different velocity measurement. ii) Errors in the reconstruction of the pointing positions could also cause velocity offsets between two positions. If the spaxels do not point at the exact same position we measure, similar to point i), different velocities. iii) Not all pointings had the same exposure time. This causes different signal-to-noise ratios for different pointings. This could also explain some variations in the velocity measurements with the pPXF routine. Note that for the velocity dispersion measurements, we grouped the individual spectra in large annuli. For that reason, the position offsets do not cause a problem for the further analysis as the “error" in position is not propagated into the velocity dispersion measurement as a systematic shift.
We used a star from the central pointing (pointing A) as a velocity template. This has the advantage that it went through the same instrumental set-up as well as the same reduction steps as all other spectra and therefore the same sampling and wavelength calibration. The green circle in the HST image in Figure \[argus\] (left panel) and the blue star in the color magnitude diagram (Figure \[cmd\]) marks the star which was used. We also identified the brightest stars form the pointing in the CMD to make sure that the template and other dominating stars are not foreground stars (see Figure \[cmd\]). In order to derive an absolute velocity scale, the line shifts of the templates were measured by fitting a Gaussian to each line and deriving the centroid. This was compared with the values of the Calcium Triplet in a rest frame and the average shift was calculated. We then corrected the radial velocity for the heliocentric reference frame.
Shot noise corrections \[snoise\]
---------------------------------
An obvious feature of the velocity map in Figure \[vel\] is the bright blue spot in the middle left of the pointing which seems to dominate the blue (approaching) part of the map. We investigated whether we see a real rotation around the center or just one or a few bright stars with a peculiar velocity dominating their environment and with their light contaminating neighboring spaxels. Therefore, we tested the influence of each star on the adjacent spaxels. This allowed us to test if the derived velocity dispersion is representative of the entire population at a given radius or whether it is biased by a low number of stars, i.e. dominated by shot noise.
[c c c c c]{}
$\log r$ & V & $\rm V_{\rm{RMS}}$& $h3$& $h4$\
\[arcsec\] & $[\mbox{km} / \mbox{s}]$ &$[\mbox{km} / \mbox{s}]$ & &\
$-0.05$ & $-1.6 \pm 2.9$ & $23.3 \pm 3.1$ & $-0.05 \pm 0.16$ & $-0.16 \pm 0.13$\
$0.26$ & $-2.2 \pm 0.6$ & $25.7 \pm 1.8$ & $-0.07 \pm 0.01$ & $-0.10 \pm 0.01$\
$0.43$ & $-0.1 \pm 0.6$ & $24.0 \pm 1.3$ & $-0.08 \pm 0.02$ & $-0.07 \pm 0.02$\
$0.56$ & $ 2.9 \pm 0.6$ & $22.2 \pm 1.1$ & $-0.09 \pm 0.02$ & $-0.09 \pm 0.02$\
$0.65$ & $ 6.4 \pm 0.6$ & $20.5 \pm 0.9$ & $-0.08 \pm 0.02$ & $-0.07 \pm 0.02$\
$0.78$ & $ 4.6 \pm 0.8$ & $18.6 \pm 0.8$ & $-0.03 \pm 0.04$ & $-0.01 \pm 0.03$\
$1.04$ & $ 3.5 \pm 0.8$ & $16.8 \pm 2.9$ & $-0.12 \pm 0.03$ & $-0.14 \pm 0.02$\
$1.26$ & $ 6.7 \pm 1.0$ & $13.6 \pm 2.9$ & $-0.07 \pm 0.04$ & $-0.03 \pm 0.04$\
To perform this test, we considered our photometric catalog (described in section \[phot\]) for the field of view covered by the ARGUS pointings. At every position of a star in the catalog, a two dimensional Gaussian was modeled with a standard deviation set to the seeing of the ground based observations ($\mathrm{FWHM}=0.9''$) and scaled to the total flux of the star. The next step was to measure the absolute amount and fraction of light that each star contributes to the surrounding spaxels. After computing these values for every star in the pointing, we had the following information for each spaxel: a) how many stars contribute to the light of that spaxel, and b) which fraction of the total light is contributed by each star, i.e. we determined whether the spectrum in a given spaxel was dominated by one or a few stars. The test showed that most of the spaxels contained meaningful contributions by more than 10 stars. Some spaxels, however, were dominated by a single star contributing more than 50 % to the spaxel’s light. For this reason, the contribution in percent of the brightest star was also derived by the program. We found out that the blue area in the left side of the velocity map of Figure \[vel\] is not due to a single star. In fact this area in the velocity map corresponds to a group of at least 10 stars moving with 10 - 40 [$\mathrm{km\, s^{-1}\, }$]{}with respect to the cluster systemic velocity.
Velocity-dispersion profile
---------------------------
To derive a radial velocity-dispersion profile for the stellar population of NGC 6388 it is necessary to bin the spectra accordingly. We divided the pointing into six independent angular bins, each of them with the same width of three ARGUS spaxels $(0.9'', \, 0.04 \, \mathrm{pc})$. To check the effect of the distribution of the bins on the final result, we tried different combinations of bins and bin distances as well as overlapping bins. We found no change in the global shape of the profile when using different bins. Therefore the choice of the bins was not critical, but in order to make an accurate error estimation, independent bins are more useful.
In each bin, all spectra of all exposures where combined with a sigma clipping algorithm to remove any remaining cosmic rays. Velocity and velocity-dispersion profiles were computed using the pPXF method applied to the binned spectra. The velocity map in Figure \[vel\] shows the dynamics in the cluster center. A rotation-like or a shearing signature is visible, but it seems to be a very local phenomenon (within $3'', \, 0.15 \ \mathrm{pc}$). In earlier attempts, we split the pointing in two halves (tilted line in figure \[vel\]) in order check for consistency and symmetry on both sides. The velocity dispersion was then derived separately for the binned spectra on each side. Both sides show a rise in the velocity-dispersion profile but differ in their shape and absolute values from each other. However, treating both sides separately would not properly take into account a possible rotation at the center and therefore not exactly represent the observed data. We decided to use the total radial profile over the 360 degree bins and measure the second moment $\rm V_{\rm{RMS}}=\sqrt{\rm V_{\rm{rot}}^2+\sigma^2}$, with the rotation velocity $\rm V_{\rm{rot}}$ and the velocity dispersion $\sigma$. The reason why we chose to analyze V$_{\rm{RMS}}$ instead of $\sigma$ is twofold. First, the Jeans models require the V$_{\rm{RMS}}$ rather than the pure velocity dispersion as an input. Second, the broadening of the line we measured represents V$_{\rm{RMS}}$ and not $\sigma$. The velocity dispersion can be obtained by subtracting the rotation velocity from this quantity. Determining the rotation is difficult due to the large shot noise introduced by the small number of spatial elements in the central region. The second moment, however, is robustly measured since we average over all angles. For simplicity we refer to the V$_{\rm{RMS}}$ profile as the velocity-dispersion profile in our study.
In addition to the central pointings, we derived kinematics for regions further out using the small IFU measurements, which were scattered at larger radii around the cluster. Unfortunately, the surface brightness of the cluster drops quickly with radius resulting in a low signal-to-noise ratio for the IFUs most distant from the center. These could therefore not be used for further analysis. Only the two innermost positions showed reasonable signal, so that these two pointings could be used as two separated data points at $11''$ and $18''$ radius, respectively. The disadvantage of the small IFU fields (20 spaxels for a total field-of-view of $3'' \times 2''$) is that these values are very affected by shot noise since only a few stars fall into the small field-of-view. Consequently, these points show larger errors than the rest of the profile.
Further, we estimated the radial velocity of the cluster in the heliocentric reference frame. We combined all spectra in the pointing and measured the velocity relative to the velocity of the template. This value was then corrected for the motion of the template and the heliocentric velocity and results in a value of $\rm V_r = (80.6 \pm 0.5)$ [$\mathrm{km\, s^{-1}\, }$]{}which is in good agreement with the value from [@harris_1996] $\rm V_r = (81.2 \pm 1.2)$ [$\mathrm{km\, s^{-1}\, }$]{}.
We ran Monte Carlo simulations to estimate the error on V$_{\rm{RMS}}$ due to shot noise. From the routine described in section \[snoise\], we knew exactly how many stars are contributing (*nStars*) and how many spaxels are summed (*nSpax*) in each bin. We took all stars detected in each bin and their corresponding magnitude. Each of the stars was then assigned a velocity chosen from a Gaussian velocity distribution with a fixed dispersion. We used our template spectrum and created *nSpax* spaxel by averaging *nStars* flux weighted and velocity shifted spectra. The resulting spaxels were then normalized, combined and the kinematics measured with pPXF (as for the original data). After 1000 realizations for each bin, we obtained the shot noise errors from the spread of the measured velocity dispersions. For the outer IFU pointings, we extrapolated the surface density to larger radii and performed the same Monte Carlo simulations as for the inner pointing, with random magnitudes drawn from the magnitude distribution in this region. The errors for the velocity and the higher moments were derived by applying Monte Carlo simulations to the spectrum itself. This was done by repeating the measurement for 100 different realizations of data by adding noise to the original spectra. [see @cappellari_2004 section 3.4]
The resulting profile is displayed in Figure \[models\]. The plot shows the second moment of the velocity distribution V$_{\rm{RMS}}$. Except for the innermost point, a clear rise of the profile towards the center is visible. The highest point reaches more than 25 [$\mathrm{km\, s^{-1}\, }$]{}before it drops quickly below 20 [$\mathrm{km\, s^{-1}\, }$]{}at larger radii. In table \[tab\_moments\] we list the results of the kinematic measurements. The first column lists the radii of the bins. The following columns show the central velocities of each bin in the reference frame of the cluster, the second moment V$_{\rm{RMS}}$ as well as higher velocity moments h3 and h4. It is conspicuous that all h4 moments tend to have negative values. This hints at a lack of radial orbits in the central region of the globular cluster.
For general purposes we also determined the effective velocity dispersion $\sigma_e$ as described in [@gultekin_2009]. This is defined by:
$$\sigma_e^2 = \frac{\int_{0}^{R_e} \rm V_{\rm{RMS}}^2 \, I(r) \, dr}{\int_{0}^{R_e} I(r) \, dr}$$
\[sig\_e\]
Where $R_e$ stands for the half light radius ($\sim 40''$) and $I(r)$ the surface brightness profile. We extrapolated our kinematic data to the half light radius since our furthest out data point only reaches a radius of $18 ''$. This results in an effective velocity dispersion of $\sigma_e= (18.9 \pm 0.3)$ [$\mathrm{km\, s^{-1}\, }$]{}which is in perfect agreement with the value derived by [@pryor_1993] listed in table \[tab\_6388\].
{width="\textwidth"}
Dynamical models \[jeans\]
==========================
The main goal of this work is to compare the derived kinematics and light profile with a set of simple analytical models in order to test whether NGC 6388 is likely to host an intermediate-mass black hole in its center. We used Jeans models as implemented and described in [@cappellari_2008].
Isotropic and anisotropic models
--------------------------------
The first input for the Jeans models is the surface brightness profile in order to estimate the 3-D density distribution in the cluster. Given the fact that the density can only be observed in a projected way, the profiles have to be deprojected. One way of doing this is by applying the multi-Gaussian expansion (MGE) method developed by [@emsellem_1994]. The basic approach of this method is to parametrize the projected surface brightness with a sum of Gaussians since the deprojection of a Gaussian function results again in a Gaussian.
To apply this parametrization and to compare results of the Jeans equation to our data, we used the Jeans Anisotropic MGE (JAM) dynamical models for stellar kinematics of spherical and axisymmetric galaxies, as well as the multi-Gaussian expansion implementation developed by [@cappellari_2002; @cappellari_2008]. The IDL routines provided by M. Cappellari[^3] enable the modeling of the surface brightness profile and fitting the observed velocity data and the mass-to-light ratio at the same time. We used our surface brightness profile in combination with a spherical model with different values of anisotropy and constant $M/L_V$ setups along the radius of the cluster.
In Figure \[models\], we plot the isotropic model data comparison for our velocity-dispersion profile. The JAM program does not actually fit the model to the data but rather calculates the shape of the second moment curve from the surface brightness profile, deconvolves the profile with the PSF of the IFU observations, and then scales it to an average value of the kinematic data in order to derive the $M/L_V$. This explains why all trial models meet in one point. The thick black line marks the model with the lowest $\chi^2$ value and therefore the best fit.
With the given surface brightness profile and without a central black hole, the models predict a drop in velocity dispersion towards the central region. As a final result, we used the $\chi^2$ statistics of the fit to estimate an error. This results in: $M_{\bullet} = (18 \pm 6) \times 10^3 M_{\odot}$ and $M/L_V = (1.7 \pm 0.2) \ M_{\odot}/L_{\odot}$. Figure \[chi\_cont\] shows the contour plot of $\chi^2$ as a function of black-hole mass and mass-to-light ratio over a grid of isotropic models (black points). The contours represent $\Delta \chi^2= 1.0, 2.7, 4.0$ and $6.6$ which correspond to a confidence of $68, 90, 95$ and $99$ percent for 1 degree of freedom (marginalized). This implies that for an isotropic model and this specific surface brightness profile the models predict a black hole of at least $M_{\bullet} = 5 \times 10^3 M_{\odot}$ with a confidence of 99 percent.
We also tested whether anisotropic Jeans models would result in a significantly better fit. To do this, we repeated the model fitting with $\beta \neq 0$ values. We found a decrease of $\chi^2$ for a rising $\beta$ down to $\chi^2=4.21$ for $\beta=0.5$ with a lower black-hole mass of $M_{\bullet}= 5.3 \times 10^3 M_{\odot}$. However, this requires a very high anisotropy of $\beta = 1 - \overline{\rm V_{\theta}^2} / \overline{\rm V_{r}^2}=0.5$ which is expected to be unstable in the center of a relaxed globular cluster such as NGC 6388. The anisotropy will be discussed in more detail in section \[sec\_aniso\].
Error estimation
----------------
In section \[phot\_sb\] we introduced the different surface brightness profiles for NGC 6388. The [@trager_1995] profile contains the most data points and extends to large radii ($\sim 4'$). However, the inner regions (important for our dynamical purposes) are not as well sampled by the ground based observations used by Trager et al. The profile by NG06 was derived by measuring integrated light on a WFPC2 image using a bi-weight estimator and covers the inner regions of the cluster very well. To calibrate the profiles to the correct absolute values NG06 also adjusted it to the Trager et al. profile. The third profile was obtained by [@lanzoni_2007] by computing the average of the counts per pixel in each bin of their ACS/HRC images. Since it was not clear in the paper how they calibrated their photometry, this profile was also shifted (by a small amount of $0.09 \, \mathrm{mag}$) to overlap with the Trager et al. profile. As a test, we used each of these profiles as an input for the Jeans models and found similar black-hole masses varying from $M_{\bullet} = (25 \pm 6) \times 10^3 M_{\odot}$ for the [@trager_1995] profile to $M_{\bullet} = (11 \pm 5) \times 10^3 M_{\odot}$ for the [@lanzoni_2007] data. The result from the profile provided by NG06 is identical to the fit of our profile in black-hole mass and slightly lower in the mass-to-light ratio ($M/L_V = (1.5 \pm 0.2) \ M_{\odot}/L_{\odot}$). Again we find the lowest $\chi^2$ value (and therefore the best fit) for an anisotropic model of $\beta=0.5$. In this case the observations can be explained without any black hole for all profiles. As mentioned in the previous section this rather unlikely configuration will be discussed in more detail in the next subsection.
From our studies of the surface brightness profiles, we see that they yield similar, but not identical results. The shape of the SB profile predicts the shape of the velocity-dispersion profile. Therefore, it is crucial to test the effect of variation of the surface brightness profile in the inner regions. To perform this, we run Monte Carlo simulations on the six innermost points of the profile. We used 1000 runs and a range of $\beta$ between $-0.5$ and $0.5$. The results are displayed in figure \[montecarlo\]. The black points mark the mean value of the derived black-hole masses. The shaded contours represent the $68, 90$ and $95$ percent confidence limits. The mean black-hole mass decreases at higher $\beta$ values since a radial anisotropy can mimic the effect of a central black hole. Nevertheless, for an isotropic case, we have a black hole detection within a mass range of $ 17_{-7}^{+8} \times 10^3 M_{\odot}$ and a global mass-to-light ratio of $M/L_V = 1.6_{-0.2}^{+0.2} \ M_{\odot}/L_{\odot}$ (errors are the 68 percent confidence limits). We derived the total error (resulting from kinematic and photometric data) by again, running Monte Carlo simulations and varying both, the velocity-dispersion profile and the surface brightness profile at the same time. As a final result for the uncertainties of our method we derived $\delta M_{\bullet} = 9 \times 10^3 M_{\odot}, \, \delta M/L_V = 0.3 \ M_{\odot}/L_{\odot} $ . This shows, that a significant percentage of the error ($ \sim 75 \%$) in the context of isotropic Jeans models results from the uncertainties of the surface brightness profile.
![The contours of the $\chi^2$ as a function of black-hole mass and mass-to-light ratio. Each point represents a particular isotropic model. The plotted contours are $\Delta \chi^2 = 1.0, 2.7, 4.0$ and $6.6$ implying a confidence of $68, 90, 95$ and $99$ percent.[]{data-label="chi_cont"}](chi_cont.eps){width="50.00000%"}
Anisotropy in relaxed globular clusters \[sec\_aniso\]
------------------------------------------------------
In the previous sections we have shown, that the strong significance for a black hole with isotropic models vanishes when using anisotropic models. The anisotropic Jeans models show a better fit for a model with much lower black-hole masses or without a black hole requiring an anisotropy of $\beta=0.5$ or higher. Thus, it is necessary to investigate the probability of such a kinematic configuration in our cluster. We run N-body simulations for $30000$ equal-mass stars, distributed according to a Plummer model. The initial anisotropy was created by a cold collapse, which resulted in a cluster that was slightly anisotropic in its core and had increasing anisotropy further out rising to $\beta=0.9$ outside the half-mass radius. Furthermore, we took the tidal field of the galaxy into account by letting the cluster move on a circular orbit around a point-mass galaxy. The ratio between tidal radius and core radius was set to 1:10 as it is the case for most globular clusters as well as for NGC 6388. We then let the cluster evolve over a time scale of several relaxations times and calculated the anisotropy for snapshots of the model.
Figure \[aniso\] shows the result of these tests. The different colors represent different areas in the cluster. Important for our case are the kinematics inside the half-light radius (curves with $< 50 \%$ light/mass enclosed in figure \[aniso\]). The system relaxes very quickly in the central regions. After six relaxation times the inner regions of the cluster are almost isotropic with $\beta$ values below $0.2$. The relaxation time of NGC 6388 is of order $10^9 yrs$ [at the half-mass radius, @harris_1996] which implies that the cluster is about $10$ relaxation times old. This shows, that high anisotropies (such as the the ones needed from our best fit models) in NGC 6388 are not stable and would have vanished over a short time scale. Therefore, we limit our discussion to the results from our isotropic models.
Remaining concerns \[concerns\]
-------------------------------
In our analysis we used the assumption of a constant $M/L_V$ for the entire cluster. In reality, this ratio can vary with radius and it may not be well described with a single value if the cluster is mass segregated. The rise in the velocity-dispersion profile could then be mimicked by a dark remnant cluster at the center. Such a scenario is expected in core collapsed clusters which display surface brightness profiles with logarithmic slopes of -0.8 or higher [@baumgardt_2005]. We looked at the deprojected light-density and mass-density profiles resulting from the surface brightness profile and the velocity-dispersion profile. The mass-density profile from the kinematics drops with $\varpropto r^{-2}$ which would be expected from a core collapsed cluster. But the light profile is shallow and does not support that hypotheses: with a concentration of $c=1.7$ and a logarithmic slope of the surface brightness profile of $\alpha = -0.28 \pm 0.08$ NGC 6388 does not show any signs of core collapse in the distribution of the visible stars. We compared these profiles with simulated core collapsed or mass segregated clusters [described in @baumgardt_2003] and were not able to find a good agreement with either shape of the light profile or slope of the mass density of the cluster. We currently consider the presence of a dark remnant cluster unlikely but we will perform detailed n-body simulations with a variable M/L in the near future.
Summary and conclusions {#con}
=======================
We derived the mass of a potential intermediate-mass black hole at the center of the globular cluster NGC 6388 by analyzing spectroscopic and photometric data. With a set of HST images, the photometric center of the cluster was redetermined and the result of NG06 confirmed. Furthermore, a color magnitude diagram as well as a surface brightness profile, built from a combination of star counts and integrated light, were produced. The spectra from the ground-based integral-field unit ARGUS were reduced and analyzed in order to create a velocity map and a velocity-dispersion profile. In the velocity map, we found signatures of rotation or at least complex dynamics in the inner three arcseconds (0.15 pc) of the cluster. We derived a velocity-dispersion profile by summing all spectra into radial bins and applying a penalized pixel fitting method.
Using the surface brightness profile as an input for spherical Jeans equations, a model velocity-dispersion profile was obtained. We ran several isotropic models with different black-hole masses and scaled them to the observed data in order to measure the mass-to-light ratio. Using $\chi^2$ statistics, we were able to find the model which fits the observed data best. We run Monte Carlo simulations on the inner points of the surface brightness profile in order to estimate the errors resulting from the particular choice of a light profile. From this, we determined the black-hole mass and the mass-to-light ratio as well as the scatter of these two results. The final results, with $68 \%$ confidence limits are: A black-hole mass of $ (17 \pm 9) \times 10^3 M_{\odot}$ and a global mass-to-light ratio of $M/L_V = (1.6 \pm 0.3) \ M_{\odot}/L_{\odot}$. In addition, we run anisotropic models. The confidence of black hole detection is decreasing rapidly with increasing radial anisotropy. However, using N-body simulations, we found that in a relaxed cluster such as NGC 6388 an anisotropy higher than $\beta=0.2$ inside the half mass radius will be erased within a few relaxation times.
With respect to the black-hole mass estimate of $5.7 \times 10^3 \ M_{\odot}$ by [@lanzoni_2007], which they derived from photometry alone, our derived black-hole mass is larger by a factor of three with our surface brightness profile profile. Using their light profile, we obtained a black-hole mass of $M_{\bullet} = (11 \pm 5) \times 10^3 M_{\odot}$ which includes their value within one sigma error. [@cseh_2010] obtained deep radio observations of the inner regions of NGC 6388 and discussed different accretion scenarios for a possible black hole. They found an upper limit for the black-hole mass of $\sim 1500 M_{\odot}$ since no radio source was detected at the location of any Chandra X-ray sources. However, inserting our black-hole mass in equation 5 of [@cseh_2010] and using their assumptions as well as the Chandra X-ray luminosity of the central region, results in minimal but possible accretion rates and conversion efficiency ($\varepsilon \eta \sim 10^{-7}$). Considering the fact that supermassive black holes have masses not much higher than $0.2$ % of the mass of their host systems [@marconi_2003; @haering_2004], our mass with $0.9 \%$ seems to be higher than the predictions for larger systems. Globular clusters in contrast lose much of their mass during their evolution. This could naturally result in higher values of black-hole mass - host system mass ratios. Due to the complicated dynamics, the $1 \sigma$ uncertainties are 40 % for our black-hole mass and 10 % for the derived $M/L_V$. Figure \[m\_sigma\] shows the position of NGC 6388 in the black-hole mass velocity dispersion relation. With our derived effective velocity dispersion of $\sigma_e = (18.9 \pm 0.3)$ [$\mathrm{km\, s^{-1}\, }$]{}the results for NGC 6388 seem to coincide with the prediction made by the $M_{\bullet} - \sigma$ relation.
The mass-to-light ratio of $M/L_V = (1.7 \pm 0.3) \, M_{\odot}/L_{\odot}$ derived in this work is consistent within the error bars with the dynamical results of @McLaughlin_2005 [$M/L_V\sim 1.8 $] but slightly lower than the value predicted by stellar population models. According also to [@McLaughlin_2005], we would expect a value of $M/L_V = (2.55 \pm 0.28) \, M_{\odot}/L_{\odot}$ for $\mathrm{[Fe/H]}=-0.6 \, \mathrm{dex}$ and $(13 \pm 2) \, \mathrm{Gyr}$. @baumgardt_2003 have shown that dynamical evolution of star clusters causes a depletion of low-mass stars from the cluster. Since these contribute little light, the $M/L_V$ value drops as the cluster evolves. Dynamical evolution could therefore explain part of the discrepancy between theoretical and observed $M/L_V$ values.
Summarizing, it can be said that the globular cluster NGC 6388 shows a variety of interesting features in its photometric properties (e.g. the extended horizontal branch) as well as in its kinematic properties (e.g. the high central velocity dispersion or the rotation like signature in the velocity map). In this work we investigated the possibility of the existence of an intermediate-mass black hole at the center. With simple analytical models we were already able to reproduce the shape of the observed velocity-dispersion profile very well if the cluster hosts an IMBH. Future work would be to compute detailed N-body simulations of NGC 6388 in order to verify whether a dark cluster of remnants at the center would be an alternative. Additionally, it is crucial to obtain kinematic data at larger radii to constrain the mass-to-light ratio and the models further out. Also proper motions for the central regions would further constrain the black hole hypothesis.
The study of black holes in globular clusters is currently limited to a handful of studies. For this reason, it is necessary to probe a large sample of globular clusters for central massive objects in order to move the field of IMBHs from “whether" to “under which circumstances do" globular clusters host them. Thus tying this field to the one of nuclear clusters and super-massive black holes. From the experience we gathered with NGC 6388, we can say that the dynamics of globular clusters is not as simple as commonly assumed and by searching for intermediate-mass black holes, we will also be able to get a deeper insight into the dynamics of globular clusters in general.
![The $M_{\bullet} - \sigma$ relation for galaxies at the high mass range (filled circles) and the first globular clusters with potential black hole detection (filled stars). The slope of the line $\log (M_{\bullet}/M_{\odot}) = \alpha + \beta \, \log ( \sigma / 200 \mathrm{km\, s^{-1}\, })$ with $(\alpha, \beta) = (8.12 \pm 0.08, 4.24 \pm 0.41)$ was taken from [@gultekin_2009]. The mass of the black holes of $\omega$ Centauri, G1 and the upper limits of M15 and 47 Tuc were obtained by [@noyola_2010], [@gebhardt_2005], [@bosch_2006] and [@47tuc], respectively. Overplotted is the result of NGC 6388 with our derived effective velocity dispersion $\sigma_e = 18.9$ [$\mathrm{km\, s^{-1}\, }$]{}.[]{data-label="m_sigma"}](M_sigma.eps){width="50.00000%"}
This research was supported by the DFG cluster of excellence Origin and Structure of the Universe (www.universe-cluster.de). We also thank Nadine Neumayer for constructive feedback and inspiring discussions, Annalisa Calamida for her support in the photometric analysis and Giuseppina Battaglia for helping with the sky subtraction. H.B. acknowledges support from the Australian Research Council through Future Fellowship grant FT0991052. We thank the anonymous referee for constructive comments and encouraging further analysis concerning anisotropy and error estimation.
[^1]: Based on observations collected at the European Organization for Astronomical Research in the Southern Hemisphere, Chile (083.D-0444).
[^2]: Based on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre (CADC/NRC/CSA).
[^3]: Available at http://purl.org/cappellari/idl
|
---
abstract: |
Consider a connected orientable surface $S$ of infinite topological type, i.e. with infinitely-generated fundamental group.
Our main purpose is to give a description of the geometric structure of an arbitrary subgraph of the arc graph of $S$, subject to some rather general conditions. As special cases, we recover the main results of J. Bavard [@Bavard] and Aramayona-Fossas-Parlier [@AFP].
In the second part of the paper, we obtain a number of results on the geometry of connected, $\Mod(S)$-invariant subgraphs of the curve graph of $S$, in the case when the space of ends of $S$ is homeomorphic to a Cantor set.
author:
- 'Javier Aramayona & Ferrán Valdez'
date: 'May 2016. Revised version: October 2016'
title: 'On the geometry of graphs associated to infinite-type surfaces'
---
[^1]
Introduction
============
There has been a recent surge of activity around mapping class groups of infinite-type surfaces, i.e. with infinitely-generated fundamental group. The motivation for studying these groups stems from several places, as we now briefly describe.
First, infinite-type surfaces appear as inverse limits of surfaces of finite type. In particular, infinite-type mapping class groups are useful in the study of asymptotic and/or stable properties of their finite-type counterparts. This is the approach taken by Funar-Kapoudjian [@FK1], where the authors identify the homology of an infinite-type mapping class group with the stable homology of the mapping class groups of its finite-type subsurfaces.
In a related direction, a number of well-known groups appear as subgroups of the mapping class group of infinite-type surfaces. For instance, Funar-Kapoudjian [@FK2] realized Thompson’s group $T$ as a topologically-defined subgroup of the mapping class group of a certain infinitely-punctured sphere.
A third piece of motivation for studying mapping class groups of infinite-type surfaces comes from dynamics, as explained by D. Calegari in [@Calegari]. More concretely, let $S$ be a closed surface, $P\subset S$ a finite subset, and consider the group $\mathrm{Homeo}(S, P)$ of those homeomorphisms of $S$ that preserve $P$ setwise. Let $G < \mathrm{Homeo}(S, P)$ be a subgroup that acts freely on $S-P$. Then $G$ admits a natural homeomorphism to $\Mod(S - K, P)$, where $K$ is either a finite set or a Cantor set. See [@Calegari] for more details.
Combinatorial models
--------------------
A large number of problems about mapping class groups of finite-type surfaces may be understood through the various simplicial complexes built from curves and/or arcs on surfaces. Notable examples of these are the [*curve graph*]{} ${\mathcal{C}}(S)$ and the [*arc graph*]{} ${\mathcal{A}}(S)$; see Section \[sec:defcurves\] for definitions. When $S$ has finite type, a useful feature of these complexes is that, with respect to their standard path-metric, they are hyperbolic spaces of infinite diameter; see [@MS] and [@MM1], respectively.
In sharp contrast, in the case of an infinite-type surface these complexes often have finite diameter; see Section \[sec:defcurves\]. This obstacle was first overcome by J. Bavard [@Bavard] in the particular case when $S$ is a sphere minus a Cantor set and an isolated point. Indeed, she proved that a certain subgraph of the arc graph is hyperbolic and has infinite diameter, and used this to construct non-trivial quasi-morphisms from the mapping class group of the plane minus a Cantor set. Subsequently, Aramayona-Fossas-Parlier [@AFP] have produced similar graphs for arbitrary surfaces, subject to certain conditions on the set of punctures of $S$. However, the definition of these subgraphs is surprisingly subtle, and small variations in the definition may produce graphs that have finite diameter or are not hyperbolic.
### Arc graphs
Our main goal is to give a unified description of the possible geometric structures of an arbitrary subgraph of the arc graph of an infinite-type surface, subject to some rather natural conditions on the given graph. First, we will require that it be [*sufficiently invariant*]{}, that is invariant under $\Mod(S,P)$, for some (possibly empty) finite set $P$ of punctures. In addition, we will assume that every such graph satisfies the [*projection property*]{}. This property is needed only for technical reasons, and thus we refer the reader to Section \[sec:arcproofs\] for details. However, we stress that this restriction is easy to check, and often automatically satisfied, once one is given an explicit subgraph ${\mathcal{G}}(S)$ of ${\mathcal{A}}(S)$. This is the case with the graphs considered in [@AFP] and [@Bavard]; see Remark \[rem:projection\] below.
Before we state our result, recall from [@Schleimer] that a [*witness*]{}[^2] of a subgraph ${\mathcal{G}}(S)$ of $\mathcal{A}(S)$ is an essential subsurface $Y$ of $S$ such that every vertex of ${\mathcal{G}}(S)$ intersects $Y$ essentially. Given a witness $Y$, we denote by ${\mathcal{G}}(Y)$ the subgraph of ${\mathcal{G}}(S)$ spanned by those vertices of ${\mathcal{G}}(S)$ that are entirely contained in $Y$. We will prove:
Let $S$ be a connected orientable surface of infinite type, and ${\mathcal{G}}(S)$ a connected, sufficiently invariant subgraph of ${\mathcal{A}}(S)$ with the projection property.
1. If every witness of ${\mathcal{G}}(S)$ has infinitely many punctures, then ${\mathcal{G}}(S)$ has finite diameter.
2. Otherwise, ${\mathcal{G}}(S)$ has infinite diameter. Moreover:
1. If every two witnesses of ${\mathcal{G}}(S)$ intersect, then ${\mathcal{G}}(S)$ is hyperbolic if and only if ${\mathcal{G}}(Y)$ is uniformly hyperbolic, for every finite-type witness $Y$.
2. If ${\mathcal{G}}(S)$ has two disjoint witnesses of finite type, then it is not hyperbolic.
\[thm:mainhyp\]
We stress that part (2b) of Theorem \[thm:mainhyp\] is merely a manifestation of Schleimer’s [*Disjoint Witnesses Principle*]{} [@Schleimer; @MS], although we have included a proof in Section \[sec:arcproofs\] for completeness. In addition, we remark that once one is given an [*explicit*]{} subgraph ${\mathcal{G}}(S)$ of ${\mathcal{A}}(S)$, it is trivial to decide what the witnesses of ${\mathcal{G}}(S)$ are and, in particular, where ${\mathcal{G}}(S)$ falls in the description offered by Theorem \[thm:mainhyp\]; see the various corollaries below. Finally, we will see in Section \[sec:arcproofs\] that the assumptions that ${\mathcal{G}}(S)$ has the projection property will not be used in the proof of part (1) of Theorem \[thm:mainhyp\], and thus that part holds in slightly more generality; this remark will be useful for the various corollaries of Theorem \[thm:mainhyp\], see below.
As a special case of Theorem \[thm:mainhyp\], we recover the main result of Aramayona-Fossas-Parlier [@AFP]; see Section \[sec:arcproofs\] for the necessary definitions:
Let $S$ be a connected orientable surface, and $P$ a non-empty finite set of isolated punctures. Then, the relative arc graph ${\mathcal{A}}(S,P)\subset {\mathcal{A}}(S)$ is hyperbolic and has infinite diameter. \[cor:afp\]
Once again, we stress that this result was first proved by Bavard [@Bavard] in the special case when $S$ is a sphere minus a Cantor set and one isolated puncture.
We will see in Corollary \[cor:Pnonisolated\] in Section \[sec:arcproofs\] that, on the other hand, if $P$ contains a puncture that is not isolated, Theorem \[thm:mainhyp\] implies that ${\mathcal{A}}(S,P)$ has finite diameter. More drastically, if $S$ has no isolated punctures at all, then there are no geometrically interesting $\Mod(S)$-invariant subgraphs of ${\mathcal{A}}(S)$:
Let $S$ be a connected orientable surface with at least one puncture. If $S$ has no isolated punctures, then any connected $\Mod(S)$-invariant subgraph of ${\mathcal{A}}(S)$ has finite diameter. \[cor:cantor\]
See Section \[sec:arcproofs\] for some further consequences of Theorem \[thm:mainhyp\].
### Curve graphs
In the light of Corollary \[cor:cantor\], there are no geometrically interesting $\Mod(S)$-invariant subgraphs of ${\mathcal{A}}(S)$ if $S$ is a punctured surface with no isolated punctures. With this motivation we are going to study $\Mod(S)$-invariant subgraphs of the curve graph ${\mathcal{C}}(S)$ instead. We will restrict our attention to the case when $S$ has no isolated ends; as we will see, the situation heavily depends on whether $S$ has finite or infinite genus. Before going any further, we note that the case when $S$ has isolated ends is covered in the recent preprint [@DFV]; see Remark \[rem:dfv\] below.
Before we state our results, we denote by $\Nonsep(S)$ the [*non-separating curve graph*]{} of $S$, namely the subgraph of ${\mathcal{C}}(S)$ spanned by all non-separating curves. Further, let $\Nonsep^*(S)$ be the [*augmented*]{} nonseparating curve graph of $S$, whose vertices are all nonseparating curves on $S$ together with those curves that cut off a disk containing every puncture of $S$. Finally, denote by $\Outer(S)$ the subgraph of ${\mathcal{C}}(S)$ spanned by all the [*outer curves*]{} on $S$, namely those curves which cut off a disk containing some, but not all, punctures of $S$. See Section \[sec:defcurves\] for further definitions.
We start with the case when the genus of $S$ is finite:
Let $S$ a connected orientable surface of infinite type, with finite genus and no isolated punctures. Then, a $\Mod(S)$-invariant subgraph ${\mathcal{G}}(S)\subset {\mathcal{C}}(S)$ has infinite diameter if and only if ${\mathcal{G}}(S) \cap \Outer(S) = \emptyset$. Moreover, in this case:
1. If ${\mathcal{G}}(S) \cap \Nonsep(S) = \emptyset$ then ${\mathcal{G}}(S)$ is not hyperbolic.
2. If ${\mathcal{G}}(S) \cap \Nonsep(S) \ne \emptyset$ then ${\mathcal{G}}(S)$ is quasi-isometric to $\Nonsep(S)$ or $\Nonsep^*(S)$.
\[thm:nonsep\]
The classification of infinite-type surfaces, stated as Theorem \[thm:class\] in Section \[sec:ends\], tells us that, under the hypotheses of the theorem, $S$ is homeomorphic to a closed surface with a Cantor set removed.
As an immediate consequence of Theorem \[thm:nonsep\], we get that if $S$ has genus 0 then any connected, $\Mod(S)$-invariant subgraph of ${\mathcal{C}}(S)$ has finite diameter; compare with Corollary \[cor:cantor\] above.
In the light of Theorem \[thm:nonsep\], a natural problem is to decide whether $\Nonsep(S)$ (resp. $\Nonsep^*(S)$) is hyperbolic for $S$ a surface of genus $g$ and with infinitely many punctures. As we will see in Proposition \[prop:nonsepwitnesses\] below, the answer is positive if and only if $\Nonsep(S)$ (resp. $\Nonsep^*(S_{g,n})$) is hyperbolic [*uniformly*]{} in $n$; compare with part (2a) of Theorem \[thm:mainhyp\] above. We remark that $\Nonsep(S_{g,n})$ is known to be hyperbolic by the work of Masur-Schleimer [@MS] and Hamensdädt [@Ham], although the hyperbolicity constant may well depend on $S$. Similary, $\Nonsep^*(S_{g,n})$ is conjecturally hyperbolic by the work of Masur-Schleimer [@MS], since every two of its witnesses intersect, see Example \[example\] in Section \[sec:defcurves\]; on the other hand, even if this were the case, the hyperbolicity constant may well depend on $S$, again.
Next, we deal with the case when the genus of $S$ is infinite:
Let $S$ be a connected orientable surface of infinite genus and no isolated ends. If ${\mathcal{G}}(S)$ is a $\Mod(S)$-invariant subgraph of ${\mathcal{C}}(S)$, then $\diam({\mathcal{G}}(S)) = 2$. \[thm:curvediam2\]
If $S$ has a finite number $\ge 5$ of isolated ends, Durham-Fanoni-Vlamis [@DFV] have constructed $\Mod(S)$-invariant subgraphs of ${\mathcal{C}}(S)$ that are hyperbolic and have infinite diameter. \[rem:dfv\]
The plan of the paper is as follows. Section \[sec:metricdefs\] provides the necessary background on $\delta$-hyperbolic spaces and quasi-isometries. In Section \[sec:ends\] we recall some facts about the space of ends of a surface. In Section \[sec:defcurves\] we briefly introduce mapping class groups and some of the combinatorial complexes one can associate to a surface. In Section \[sec:arcproofs\] we prove Theorem \[thm:mainhyp\] and discuss some of its consequences. Finally, Section \[sec:curveproofs\] contains the proofs of Theorems \[thm:nonsep\] and \[thm:curvediam2\], together with some open questions.
[**Acknowledgements.**]{} This project stemmed out of discussions with Juliette Bavard, and the authors are indebted to her for sharing her ideas and enthusiasm. We want to thank LAISLA and CONACYT’s Red temática Matemáticas y Desarrollo for its support. This work started with a visit of the first named author to the UNAM (Morelia). He would like to thank the Centro de Ciencias Matemáticas for its warm hospitality. He also thanks Brian Bowditch, Saul Schleimer for conversations. Finally, the authors are grateful to Federica Fanoni and Nick Vlamis for discussions and for pointing out several errors in an earlier version of this draft.
Hyperbolic metric spaces {#sec:metricdefs}
========================
We briefly recall some notions on large-scale geometry that will be used in the sequel. For a thorough discussion, see [@GH].
Let $X$ be a geodesic metric space. We say that $X$ is [*$\delta$-hyperbolic*]{} if there exists $\delta \ge 0$ such that every triangle $T\subset X$ is [*$\delta$-thin*]{}: there exists a point $c\in X$ at distance at most $\delta$ from every side of $T$.
We will simply say that a geodesic metric space is [*hyperbolic*]{} if it is $\delta$-hyperbolic for some $\delta \ge 0$.
Let $(X,d_X), (Y,d_Y)$ be two geodesic metric spaces. We say that a map $f:(X,d_X) \to (Y, d_Y)$ is a [*quasi-isometric embedding*]{} if there exist $\lambda \ge 1$ and $C\ge 0$ such that $$\frac{1}{\lambda} d_X(x,x') - C \le d_Y(f(x),f(x')) \le \lambda d_X(x,x') + C,$$ for all $x,x' \in X$. We say that $f$ is a [*quasi-isometry*]{} if, in addition to $(1)$, there exists $D\ge 0$ such that $Y$ is contained in the $D$-neighbourhood of $f(X)$. More concretely, for all $y \in Y$ there exists $x \in X$ with $d_Y(y , f(x)) \le D$.
We say that two spaces are [*quasi-isometric*]{} if there exists a quasi-isometry between them. The following is well-known:
Suppose that two geodesic metric spaces $X,Y$ are quasi-isometric to each other. Then $X$ is hyperbolic if and only if $Y$ is hyperbolic.
The ends of a surface {#sec:ends}
=====================
Let $S$ be a connected orientable surface, possibly of infinite topological type. We will briefly recall the definition of the [*space of ends*]{} of $S$, and refer the reader to [@Raymond] and [@Richards] for a more thorough discussion on the space of ends of topological spaces and surfaces respectively.
An [*exiting sequence*]{} is a collection $U_1 \supseteq U_2 \supseteq \ldots$ of connected open subsets of $S$, such that:
1. $U_n$ is not relatively compact, for any $n$;
2. The boundary of $U_n$ is compact, for all $n$;
3. Any relatively compact subset of $S$ is disjoint from $U_n$, for all but finitely many $n$.
We deem two exiting sequences to be [*equivalent*]{} if every element of the first sequence is contained in some element of the second, and vice-versa. An [*end*]{} of $S$ is defined as an equivalence class of exiting sequences, and we write ${{\text{Ends}}}(S)$ for the set of ends of $S$. We put a topology on ${{\text{Ends}}}(S)$ by specifying the following basis: given a subset $U \subset S$ with compact boundary, let $U^*$ be the set of all ends of $S$ that have a representative exiting sequence that is eventually contained in $U$.
The following theorem is a special case of Theorem 1.5 in [@Raymond]:
Let $S$ be a connected orientable surface. Then ${{\text{Ends}}}(S)$ is totally disconnected, separable, and compact; in particular, it is a subset of a Cantor set. \[thm:ends\]
We now proceed to describe the classification theorem for connected orientable surfaces of infinite type [@Richards]. Before this, we need some notation. Say that an end of $S$ is [*planar*]{} if it has a representative exiting sequence whose elements are eventually planar; otherwise it is said to be [*non-planar*]{}. We denote by ${{\text{Ends}}}^p(S)$ and ${{\text{Ends}}}^n(S)$, respectively, the subspaces of planar and non-planar ends of $S$. Clearly ${{\text{Ends}}}(S)={{\text{Ends}}}^p(S)\sqcup{{\text{Ends}}}^n(S)$. In [@Richards], Richards proved:
Let $S_1$ and $S_2$ be two connected orientable surfaces. Then $S_1$ and $S_2$ are homeomorphic if and only if they have the same genus, and ${{\text{Ends}}}^n(S_1)\subset{{\text{Ends}}}(S_1)$ is homeomorphic to ${{\text{Ends}}}^n(S_2)\subset{{\text{Ends}}}(S_2))$ as nested topological spaces. That is, there exist a homeomorphism $h:{{\text{Ends}}}(S_1)\to{{\text{Ends}}}(S_2)$ whose restriction to ${{\text{Ends}}}^n(S_1)$ defines a homeomorphism between ${{\text{Ends}}}^n(S_1)$ and ${{\text{Ends}}}^n(S_2)$. \[thm:class\]
We remark that this theorem was later extended by Prishlyak and Mischenko [@PM] to surfaces with non-empty boundary.
Arcs, curves, and witnesses {#sec:defcurves}
===========================
In this section we will introduce the necessary definitions about arcs and curves that appear in our results. Throughout, let $S$ be a connected, orientable surface of infinite topological type. Let $\Pi$ be a (possibly empty) set of marked points on $S$, which we feel free to regard as marked points, punctures, or (planar) ends of $S$.
Mapping class group
-------------------
The mapping class group $\Mod(S)$ is the group of self-homeomorphisms of $S$ that preserve $\Pi$ setwise, up to isotopy preserving $\Pi$ setwise. Given a (possibly empty) finite subset $P$ of $\Pi$, we define $\Mod(S,P)$ to be the subgroup of $\Mod(S)$ whose every element preserves $P$ setwise. Observe that $\Mod(S,\emptyset) = \Mod(S)$.
Arcs and curves
---------------
By a [*curve*]{} on $S$ we mean the isotopy class of a simple closed curve on $S$ that does not bound a disk with at most one puncture. An [*arc*]{} on $S$ is the isotopy class of a simple arc on $S$ with both endpoints in $\Pi$.
The [*arc and curve graph*]{} ${\mathcal{AC}}(S)$ of $S$ is the simplicial graph whose vertices are all arcs and curves on $S$, and where two vertices are adjacent in ${\mathcal{AC}}(S)$ if they have disjoint representatives on $S$. As is often the case, we turn ${\mathcal{AC}}(S)$ into a geodesic metric space by declaring the length of each edge to be 1.
Observe that $\Mod(S)$ acts on ${\mathcal{AC}}(S)$ by isometries. As mentioned in the introduction, we will concentrate in subgraphs of ${\mathcal{A}}(S)$ that are invariant under big subgroups of $\Mod(S)$. More concretely, we have the following definition:
We say that a subgraph ${\mathcal{G}}(S)$ of ${\mathcal{AC}}(S)$ is [*sufficiently invariant*]{} if there exists a (possibly empty) subset $P$ of $\Pi$ such that $\Mod(S,P)$ acts on ${\mathcal{G}}(S)$.
We will be interested in various standard $\Mod(S)$-invariant subgraphs of ${\mathcal{AC}}(S)$, whose definition we now recall.
The [*arc graph*]{} ${\mathcal{A}}(S)$ is the subgraph of ${\mathcal{AC}}(S)$ spanned by all vertices of ${\mathcal{AC}}(S)$ that correspond to arcs on $S$; note that ${\mathcal{A}}(S)=\emptyset$ if and only if $\Pi=\emptyset$. Observe that if $S$ has infinitely many punctures then ${\mathcal{A}}(S)$ has finite diameter.
Similarly, the [*curve graph*]{} ${\mathcal{C}}(S)$ is the subgraph spanned by those vertices that correspond to curves on $S$. Note that ${\mathcal{C}}(S)$ has diameter 2 for every surface of infinite type.
The [*nonseparating curve graph*]{} $\Nonsep(S)$ is the subgraph of ${\mathcal{C}}(S)$ spanned by all nonseparating curves on $S$. A related graph is the [*augmented nonseparating curve graph*]{} $\Nonsep^*(S)$, whose vertices are curves that either do not separate $S$, or else bound a disk containing every puncture of $S$. Note that these graphs have diameter 2 if $S$ has infinite genus.
Finally, the [*outer curve graph*]{} $\Outer(S)$ is the subgraph of ${\mathcal{C}}(S)$ spanned by those curves $\alpha$ that bound a disk with punctures on $S$, and such that both components of $S-\alpha$ contain at least one puncture of $S$. Observe that $\Outer(S) = \emptyset$ if $S$ is closed or has exactly one puncture, and that $\Outer(S)$ has finite diameter if $S$ has infinitely many punctures.
As the reader may suspect at this point, these observations constitute the main source of inspiration behind the statements of Theorems \[thm:nonsep\] and \[thm:curvediam2\].
Witnesses
---------
Let $S$ be a connected orientable surface of infinite type, and ${\mathcal{G}}(S)$ a connected subgraph of ${\mathcal{AC}}(S)$. As mentioned in the introduction, we will use the following notion, originally due to Schleimer [@Schleimer]:
A [*witness*]{} of ${\mathcal{G}}(S)$ is an essential subsurface $Y\subset S$ such that every vertex of ${\mathcal{G}}(S)$ intersects $Y$ essentially.
Observe that if $Y$ is a witness of ${\mathcal{G}}(S)$ and $Z$ is a subsurface of $S$ such that $Y \subset Z$, then $Z$ is also a witness.
For the sake of concreteness, let $S$ be a connected orientable surface of finite genus $g$, possibly with infinitely many punctures.
1. If ${\mathcal{G}}(S) = {\mathcal{A}}(S)$, then $Y \subset S$ is a witness if and only if $Y$ contains every puncture of $S$.
2. If ${\mathcal{G}}(S) = {\mathcal{C}}(S)$, then $Y\subset S$ is a witness if and only if $Y=S$.
3. If ${\mathcal{G}}(S) = \Nonsep(S)$, then $Y\subset S$ is a witness if and only if $Y$ has genus $g$.
4. Let ${\mathcal{G}}(S) = \Nonsep^*(S)$, and suppose $S$ has at least two punctures so that $\Nonsep^*(S) \ne \Nonsep(S)$. Then $Y\subset S$ is a witness if and only if $Y$ has genus $g$ and at least one puncture.
\[example\]
Subgraphs of the arc graph {#sec:arcproofs}
==========================
In this section we give a proof of Theorem \[thm:mainhyp\]. The main tool is the following variant of Masur-Minsky’s [*subsurface projections*]{} [@MM2]:
[**Subsurface projections.**]{} Let $Y$ be a witness of ${\mathcal{G}}(S)$, and suppose $Y$ is not homeomorphic to an annulus. There is a natural projection $$\pi_Y:{\mathcal{G}}(S) \to {\mathcal{A}}(Y)$$ defined by setting $\pi_Y(v)$ to be any connected component of $v \cap Y$. In particular, $\pi_Y(v)=v$ for every $v \subset Y$; in other words, the restriction of $\pi_Y$ to ${\mathcal{G}}(Y)$ is the identity. Observe that the definition of $\pi_Y$ involves a choice, but any two such choices are disjoint and therefore at distance at most 1 in ${\mathcal{A}}(Y)$. The same argument gives:
Let $S$ be a surface and $Y$ an essential subsurface not homeomorphic to an annulus. If $u,v$ are disjoint arcs which intersect $Y$ essentially, then $\pi_Y(u)$ and $\pi_Y(v)$ are disjoint. \[lem:lip\]
For technical reasons, which will become apparent in the proof of Lemma \[lem:qi\] below, we will be interested in subgraphs of ${\mathcal{A}}(S)$ for which the subsurface projections defined above satisfy the following property:
We say that a subgraph ${\mathcal{G}}(S) \subset {\mathcal{A}}(S)$ has the [*projection property*]{} if, for every finite-type witness $Y$ of ${\mathcal{G}}(S)$, the graphs $\pi_Y({\mathcal{G}}(S))$ and ${\mathcal{G}}(Y)$ are quasi-isometric via a quasi-isometry that is the identity on ${\mathcal{G}}(Y)$ and whose constants do not depend on $Y$.
As mentioned in the introduction, we remark that deciding whether a given [*explicit*]{} subgraph of ${\mathcal{A}}(S)$ has the projection property is normally easy to check; see Remark \[rem:projection\] below.
The following lemma, which is a small variation of Corollary 4.2 in [@AFP], is the main ingredient in the proof of Theorem \[thm:mainhyp\]. We note that this is the sole instance in which we will make use of the assumption that ${\mathcal{G}}(S)$ has the projection property.
Let $S$ be a surface of infinite type, and ${\mathcal{G}}(S) \subset {\mathcal{A}}(S)$ a connected subgraph with the projection property. Then, for every finite-type witness $Y$ of ${\mathcal{G}}(S)$, the subgraph ${\mathcal{G}}(Y)$ is uniformly quasi-isometrically embedded in ${\mathcal{G}}(S)$. \[lem:qi\]
Let $u,v$ be arbitrary vertices of ${\mathcal{G}}(Y)$. First, observe that since ${\mathcal{G}}(Y) \subset {\mathcal{G}}(S)$, we have $$d_{{\mathcal{G}}(S)}(u,v) \le d_{{\mathcal{G}}(Y)}(u,v).$$ To show a reverse coarse inequality, we proceed as follows. Consider a geodesic $\gamma \subset {\mathcal{G}}(S)$ between $u$ and $v$. The projected path $\pi_Y(\gamma)$ is a path in $\pi_Y({\mathcal{G}}(S))$ between $u= \pi_Y(u)$ and $v=\pi_Y(v)$, and $${\rm length}_{\pi_Y({\mathcal{G}}(S))}(\pi_Y(\gamma)) \le {\rm length}_{{\mathcal{G}}(S)}(\gamma),$$ by Lemma \[lem:lip\]. In particular, $$d_{\pi_Y({\mathcal{G}}(S))}(u,v) \le d_{{\mathcal{G}}(S)}(u,v).$$ Since ${\mathcal{G}}(S)$ has the projection property, there exist constants $L \ge 1$ and $C \ge 0$ (which depend only on $S$) such that $$d_{{\mathcal{G}}(Y)}(u,v) \le L \cdot d_{\pi_Y({\mathcal{G}}(S))}(u,v) + C,$$ and thus the result follows by combining the above two inequalities.
We are now ready to prove Theorem \[thm:mainhyp\].
Let $S$ be a connected, orientable surface of infinite type, and denote by $\Pi$ the set of marked points of $S$. Let ${\mathcal{G}}(S)$ be a connected subgraph of ${\mathcal{A}}(S)$ with the projection property, and invariant under $\Mod(S, P)$ for some $P \subset \Pi$ finite (possibly empty).
We first prove part (1); in fact, we will show that the diameter of ${\mathcal{G}}(S)$ is at most 4. Let $u,v$ be two arbitrary distinct vertices of ${\mathcal{G}}(S)$. We first claim that there exists $w\in {\mathcal{G}}(S)$ that intersects both $u$ and $v$ a finite number of times. To see this, observe that if $u$ and $v$ have no endpoints in common, then their intersection number is finite and thus we may take $w=u$. Suppose now that $u$ and $v$ share two distinct endpoints $p, p' \in \Pi$. Then there exists an element $h$ in the subgroup of $\Mod(S,P)$ whose every element fixes $p$ and $p'$, such that $w=h(u)$ intersects both $u$ and $v$ a finite number of times, as desired. The rest of cases are dealt with in a similar fashion. This finishes the proof of the claim.
Continuing with the proof, we now claim that there is a vertex $z\in {\mathcal{G}}(S)$ that is disjoint from $v$ and $w$. Indeed, consider the surface $F(v,w)$ filled by $v$ and $w$, which has finite type since $v$ and $w$ intersect finitely many times. Since every witness of ${\mathcal{G}}(S)$ has infinitely many punctures, by assumption, we deduce that $F(v,w)$ is not a witness, and therefore there exists a vertex $z \in {\mathcal{G}}(S)$ that does not intersect $F(v,w)$. Using the same reasoning, there exists a vertex $z' \in {\mathcal{G}}(S)$ that is disjoint from $u$ and $w$. Thus, $$u \to z' \to w \to z \to v$$ is a path of length at most 4 in ${\mathcal{G}}(S)$ between $u$ and $v$, as desired.
We now proceed to prove part (2), arguing along similar lines to [@AFP]. To show that ${\mathcal{G}}(S)$ has infinite diameter we proceed as follows. By assumption, there exists a witness $Y$ of ${\mathcal{G}}(S)$ with finitely many punctures. After replacing $Y$ by a finite-type surface containing every puncture of $Y$, we may assume that $Y$ has finite type and $\Mod(Y)$ contains a pseudo-Anosov. Essentially by Luo’s argument proving that the curve graph of a finite-type surface has infinite diameter (see the comment after Proposition 3.6 of [@MM1]), we deduce that ${\mathcal{G}}(Y)$ has infinite diameter. Since ${\mathcal{G}}(Y)$ is quasi-isometrically embedded in ${\mathcal{G}}(S)$, by Lemma \[lem:qi\], it follows that ${\mathcal{G}}(S)$ has infinite diameter, as desired.
Next, we establish part (2a) Assume that every two witnesses of ${\mathcal{G}}(S)$ intersect, and suppose first that there exists $\delta= \delta(S)$ such that ${\mathcal{G}}(Y)$ is $\delta$-hyperbolic, for every finite-type witness $Y$. We will prove that ${\mathcal{G}}(S)$ is $\delta$-hyperbolic. To this end, consider a geodesic triangle $T \subset {\mathcal{G}}(S)$, and let $Z$ be a witness of ${\mathcal{G}}(S)$ containing every vertex of $T$, so that $T$ may be viewed as a triangle in ${\mathcal{G}}(Z)$. First, if $Z$ has infinitely many punctures then ${\mathcal{G}}(Z)$ has diameter $\le 4$, by the proof of part (1). Therefore $T$ has a $4$-center in ${\mathcal{G}}(Z)$, and thus also in ${\mathcal{G}}(S)$, as desired. Assume now that $Z$ has finitely many punctures; in this case, again up to replacing $Z$ by a connected, finite-type surface containing every puncture of $Z$, we may in fact assume that $ Z$ is connected and has finite type. Since ${\mathcal{G}}(Z)$ is $\delta$-hyperbolic, by assumption, $T$ has a $\delta$-center in $G(Z)$, and thus also in ${\mathcal{G}}(S)$. Since $T$ is arbitrary and uniformly thin, we obtain that ${\mathcal{G}}(S)$ is hyperbolic, as claimed.
Using a very similar argument to the one just given, we also deduce that the hyperbolicity of ${\mathcal{G}}(S)$ implies that of ${\mathcal{G}}(Y)$, for every finite-type witness $Y$ of ${\mathcal{G}}(S)$. This finishes the proof of part (2a).
It remains to show part (2b). Assume that ${\mathcal{G}}(S)$ has two disjoint witnesses $Y,Z\subset S$, each of finite type. As remarked above, after enlarging $Y$ and/or $Z$ if necessary we may assume that ${\mathcal{G}}(Y)$ and ${\mathcal{G}}(Z)$ have infinite diameter. Since $Y$ and $Z$ are witnesses, the projection maps $\pi_Y$ and $\pi_Z$ are well-defined. Therefore there is a projection map $$\pi: {\mathcal{G}}(S) \to {\mathcal{A}}(Y) \times {\mathcal{A}}(Z)$$ which is simply the map $\pi_Y \times \pi_Z$. Using this projection and the same arguments as in the proof of Lemma \[lem:qi\], the fact that ${\mathcal{G}}(S)$ has the projection property implies that ${\mathcal{G}}(S)$ contains a quasi-isometrically embedded copy of ${\mathcal{G}}(Y) \times {\mathcal{G}}(Z)$. By choosing a bi-infinite quasi-geodesic in ${\mathcal{G}}(Y)$ and in ${\mathcal{G}}(Z)$, we obtain ${\mathcal{G}}(S)$ contains a quasi-isometrically embedded copy of $\mathbb{Z}^2$, as claimed. This finishes the proof of part (2b), and hence of Theorem \[thm:mainhyp\].
As mentioned in the introduction, the proof of part (1) of Theorem \[thm:mainhyp\] does not use that ${\mathcal{G}}(S)$ has the projection property; this will be crucial for Corollaries \[cor:cantor\] and \[cor:Pnonisolated\] below. \[rem:outofjailfree\]
Consequences
------------
We proceed to discuss some of the consequences of Theorem \[thm:mainhyp\] mentioned in the introduction, starting with Corollary \[cor:afp\]. Before doing so, we need some definitions from [@AFP]. Let $\Pi$ be the set of marked points of $S$, where we assume that $\Pi\ne \emptyset$. As always, we will feel free to view the elements of $\Pi$ as marked points, punctures, or (planar) ends of $S$.
We say that a marked point $p \in \Pi$ is [*isolated*]{} if it is isolated in $\Pi$, where the latter is equipped with the subspace topology (here we are viewing $\Pi$ as a set of marked points on $S$). Let $P \subset \Pi$ be a non-empty finite subset of marked points on $S$. Define ${\mathcal{A}}(S,P)$ as the subgraph of ${\mathcal{A}}(S)$ spanned by those arcs that have at least one endpoint in $P$. Note that $\Mod(S,P)$ acts on ${\mathcal{A}}(S,P)$, and hence ${\mathcal{A}}(S,P)$ is sufficiently invariant.
The graphs ${\mathcal{A}}(S,P)$ have the projection property: if $Y$ is a finite-type witness of $S$ then $\pi_Y({\mathcal{A}}(S,P))$ is uniformly quasi-isometric to ${\mathcal{A}}(Y, P)$, which is ${\mathcal{G}}(Y)$ for ${\mathcal{G}}(S) = {\mathcal{A}}(S,P)$. The proof that both graphs are quasi-isometric boils down to the fact that, for $v \in {\mathcal{A}}(S,P)$, there is at least one component of $v \cap Y$ that has an endpoint in $P$, which we can use to define a subsurface projection map with nice properties. \[rem:projection\]
We are now in a position to prove Corollary \[cor:afp\]:
Since $P$ is finite and every puncture is isolated, there exists a witness containing only finitely many punctures (any finite-type surface containing $P$ will do). Now, part (2) of Theorem \[thm:mainhyp\] applies with ${\mathcal{G}}(S) = {\mathcal{A}}(S,P)$, and thus ${\mathcal{A}}(S,P)$ has infinite diameter. Moreover, if $Y$ is a finite-type witness of ${\mathcal{A}}(S,P)$ then ${\mathcal{G}}(Y) = {\mathcal{A}}(Y,P)$, which is $7$-hyperbolic by [@HPW].
We now prove Corollary \[cor:cantor\]:
Let $S$ be as in the statement, and ${\mathcal{G}}(S)$ be a connected, $\Mod(S)$-invariant subgraph of ${\mathcal{A}}(S)$. As $S$ has no isolated punctures, using for instance the classification theorem for infinite-type surfaces [@Richards] we deduce that there exists an infinite sequence of distinct and pairwise disjoint vertices of ${\mathcal{G}}(S)$ such that every two distinct arcs in the sequence have no endpoints in common. In particular, every witness of ${\mathcal{G}}(S)$ must have an infinite number of punctures, and so the result follows from part (1) of Theorem \[thm:mainhyp\].
In addition, we recover the following observation due to Bavard (stated as Proposition 3.5 of [@AFP]):
Suppose $P \subset \Pi$ contains a puncture that is not isolated. Then ${\mathcal{A}}(S,P)$ has finite diameter. \[cor:Pnonisolated\]
Finally, one could define ${\mathcal{A}}(S,P,Q)$ to be, for disjoint finite subsets $P,Q$ of isolated punctures, the subgraph of ${\mathcal{A}}(S)$ spanned by those arcs that have one endpoint in $P$ and the other in $Q$. In this situation we have the following result, also due to Bavard (unpublished):
The graph ${\mathcal{A}}(S,P,Q)$ is not hyperbolic.
Observe that $Y$ is a witness of ${\mathcal{A}}(S,P,Q)$ if and only if it contains $P$ or $Q$. In particular, there are two disjoint witnesses of finite type, and part (2b) of Theorem \[thm:mainhyp\] applies.
Subgraphs of the curve graph {#sec:curveproofs}
============================
In this section we deal with connected, $\Mod(S)$-invariant subgraphs of the curve graph, proving Theorems \[thm:nonsep\] and \[thm:curvediam2\]. As mentioned in the introduction, we restrict our attention to the case when $S$ has no isolated ends, which in turn implies that ${{\text{Ends}}}(S)$ is homeomorphic to a Cantor set, by the classification theorem for infinite-type surfaces [@Richards] described in Section \[sec:ends\].
We first prove Theorem \[thm:nonsep\]. The arguments we will use are similar in spirit to those used in the previous section, but adapted to this particular setting.
Let $S$ be a connected, orientable surface of infinite type, with finite genus and no isolated ends. Let ${\mathcal{G}}(S)$ be a connected, $\Mod(S)$-invariant subgraph of ${\mathcal{C}}(S)$.
Suppose first that ${\mathcal{G}}(S) \cap \Outer(S) \ne \emptyset$. We want to conclude that $\diam({\mathcal{G}}(S)) = 2$. To this end, let $\alpha$ and $\beta$ be arbitrary vertices of ${\mathcal{G}}(S)$. If $\alpha$ and $\beta$ are disjoint, there is nothing to prove, so assume that $i(\alpha, \beta) \ne 0$. Let $F(\alpha,\beta)$ be the subsurface of $S$ filled by $\alpha$ and $\beta$, which has finite topological type since $\alpha$ and $\beta$ are compact. Therefore, there exists a connected component $Y$ of $S - F(\alpha, \beta)$ that has infinitely many punctures. Now, the fact that ${{\text{Ends}}}(S)$ is a Cantor set and the classification theorem for infinite-type surfaces, together imply that $\Mod(S)$ acts transitively on $\Outer(S)$. Thus there exists $h \in \Mod(S)$ and $\gamma \in \Outer(S)$ such that $h(\gamma) \subset Y$. In particular, $h(\gamma)$ is disjoint from both $\alpha$ and $\beta$ and hence $d_{{\mathcal{G}}(S)}(\alpha, \beta) =2$.
Hence from now on, we assume that ${\mathcal{G}}(S) \cap \Outer(S) = \emptyset$. Suppose first that, in addition, ${\mathcal{G}}(S) \cap \Nonsep(S) = \emptyset$, and so every element of ${\mathcal{G}}(S)$ is a curve that either separates $S$ into two surfaces of positive genus, or cuts off a disk containing every puncture of $S$. We claim that ${\mathcal{G}}(S)$ has two disjoint witnesses, and thus fails to be hyperbolic. To construct these witnesses, consider a multicurve $M$ consisting of $\rm{genus}(S)+1$ non-separating curves on $S$ such that $S- M= W_1 \sqcup W_2$, with $W_i$ a surface of genus 0 for $i=1,2$ and containing at least one puncture of $S$. By construction, $W_1$ and $W_2$ are witnesses for ${\mathcal{G}}(S)$. Let $P_i$ be the finite subset of punctures of $W_i$ coming from the elements of $M$. Using subsurface projections as in the previous section gives a quasi-isometric embedding $${\mathcal{A}}(W_1, P_1) \times {\mathcal{A}}(W_2,P_2) \to {\mathcal{G}}(S),$$ thus obtaining a quasi-isometrically embedded copy of $\mathbb{Z}^2$ inside ${\mathcal{G}}(S)$. In particular, ${\mathcal{G}}(S)$ is not hyperbolic and has infinite diameter.
Hence, from now on we assume that ${\mathcal{G}}(S) \cap \Nonsep(S) \ne \emptyset$, which in particular implies that $\Nonsep(S) \subset {\mathcal{G}}(S)$, since $\Mod(S)$ acts on ${\mathcal{G}}(S)$. There are two cases to consider:
[**Case I.**]{} [*No vertex of ${\mathcal{G}}(S)$ bounds a disk with punctures.* ]{}
In this case, we claim:
[**Claim.**]{} The inclusion map $\Nonsep(S) \hookrightarrow {\mathcal{G}}(S)$ is a quasi-isometry.
We begin by showing that the inclusion map is a quasi-isometric embedding. In fact, more is true: we will prove that, given $\alpha,\beta\in \Nonsep(S)$ and a geodesic $\sigma$ in ${\mathcal{G}}(S)$ between them, we can modify $\sigma$ to a geodesic $\sigma'$ in $\Nonsep(S)$ of the same length. (We remark that this argument is contained in the proof that the nonseparating curve complex is connected; see Theorem 4.4 of [@FM].) Let $\gamma\in \sigma$ be a curve in ${\mathcal{G}}(S) - \Nonsep(S)$. By hypothesis, $S - \gamma = Y \cup Z$, where $Y$ and $Z$ both have positive genus. Let $\gamma_L$ and $\gamma_R$ be the vertices of $\sigma$ preceding (resp. following) $\gamma$. The assumption that $\sigma$ is geodesic implies that either $\gamma_L,\gamma_R \subset Y$ or $\gamma_L,\gamma_R \subset Z$; suppose for the sake of concreteness that we are in the former case. Since $Z$ has positive genus, it contains a nonseparating curve $\gamma'$ which, by construction, is disjoint from $\gamma_L$ and $\gamma_R$. Replacing $\gamma$ by $\gamma'$ on $\sigma$ produces a geodesic in ${\mathcal{G}}(S)$ with a strictly smaller number of separating curves.
At this point, we know that the inclusion map $\Nonsep(S) \hookrightarrow {\mathcal{G}}(S)$ is a (quasi-)isometric embedding. To see that it is a quasi-isometry, observe that every element of ${\mathcal{G}}(S)$ is at distance at most 1 from an element of $\Nonsep(S)$. This finishes the proof of the claim.
[**Case II.**]{} [*There is a vertex of ${\mathcal{G}}(S)$ which bounds a disk with punctures.* ]{}
Since ${\mathcal{G}}(S) \cap \Outer(S) = \emptyset$, we get an inclusion $\Nonsep^*(S) \subset {\mathcal{G}}(S)$. Using the same arguments as in the previous claim, we obtain:
[**Fact.**]{} The inclusion map $\Nonsep^*(S) \hookrightarrow {\mathcal{G}}(S)$ is a quasi-isometry.
In the light of the claims above, in order to finish the proof of the theorem it suffices to show:
[**Claim.**]{} The graphs $\Nonsep(S)$ and $\Nonsep^*(S)$ have infinite diameter.
We prove the result for $\Nonsep(S)$, as the case of $\Nonsep^*(S)$ is totally analogous.
In a similar fashion to what we did in the previous section, we are going to prove that, for every finite-type witness $Y$, the subgraph $\Nonsep(Y)$ is quasi-isometrically embedded in $\Nonsep(S)$; once this has been done, the claim will follow since $\Nonsep(Y)$ has infinite diameter, which again may be deduced using Luo’s argument showing that the curve graph has infinite diameter; see Proposition 3.6 of [@MM1].
In this direction, let $Y$ be a finite-type witness of $\Nonsep(S)$; in other words, $Y$ is a finite-type subsurface of $S$ of the same genus as $S$, see Example \[example\] above. Let ${\mathcal{A}}(Y, \partial Y)$ be the subgraph of ${\mathcal{A}}(Y)$ spanned by those vertices that have both endpoints on $\partial Y$. Similarly, let $\mathcal{A}\Nonsep(Y)$ be the subgraph of $\mathcal{AC}(Y)$ spanned by the vertices of $\Nonsep(Y) \cup {\mathcal{A}}(Y, \partial Y)$. The inclusion map $$\Nonsep(Y)\hookrightarrow \mathcal{A}\Nonsep(Y)$$ is a quasi-isometry, where the constants do not depend on $Y$; to see this, one may use the standard argument to show that the embedding of $C(Y)$ into ${\mathcal{AC}}(Y)$ is a uniform quasi-isometry (see for instance Exercise 3.15 of [@Schleimer]). Now, as in the previous section there is a subsurface projection $$\pi_Y: \Nonsep(S) \to \mathcal{A}\Nonsep(Y)$$ that associates, to an element of $\Nonsep(S)$, its intersection with $Y$. Using an analogous reasoning to that of Lemma \[lem:qi\], we obtain that $\Nonsep(Y)$ is uniformly quasi-isometrically embedded in $\Nonsep(S)$, as desired. This finishes the proof of the claim, and thus that of Theorem \[thm:nonsep\].
The graphs $\Nonsep(S)$ and $\Nonsep^*(S)$ have an intriguing geometric structure. Indeed, using a small variation of the proof of Theorem \[thm:mainhyp\], we obtain:
Let $S$ be a connected surface of finite genus $g$ and with infinitely many punctures. Then $\Nonsep(S)$ (resp. $\Nonsep^*(S)$) is hyperbolic if and only if $\Nonsep(S_{g,n})$ (resp. $\Nonsep^*(S_{g,n})$) is hyperbolic uniformly in $n$. \[prop:nonsepwitnesses\]
In the light of Example \[example\], the finite-type witnesses of $\Nonsep(S)$ and $\Nonsep^*(S)$ are precisely the subsurfaces of the form $S_{g,n}$; compare with part (3) of Theorem \[thm:mainhyp\].
Again, we argue only for $\Nonsep(S)$, as the other case is very similar. Let $T$ be a geodesic triangle in $\Nonsep(S)$. Since $T$ has finitely many vertices and curves are compact, there exists a finite-type subsurface $Y$ of $S$ that contains every element of $T$. Thus we can view $T$ as a geodesic triangle in $\Nonsep(Y)$. If $\Nonsep(S_{g,n})$ is hyperbolic uniformly in $n$, there is $\delta = \delta(g)$ such that $T$ has a $\delta$-center $\alpha \in \Nonsep(Y)$ (with respect to the distance function in $\Nonsep(Y)$). In particular, $\alpha$ is at distance at most $\delta$ from the sides of $T$, where distance is measured in $\Nonsep(Y)$, and hence is a $\delta$-centre for $T$ in $\Nonsep(S)$. Thus, $\Nonsep(S)$ is $\delta$-hyperbolic.
The other direction is analogous.
As mentioned in the introduction, it is known that $\Nonsep(S_{g,n})$ is hyperbolic [@Ham; @MS], but in principle the hyperbolicity constant may well depend on $n$. Similarly, $\Nonsep^*(S_{g,n})$ is conjecturally hyperbolic by Masur-Schleimer’s principle that every two witnesses intersect [@MS], but even in this case the hyperbolicity constant could again depend on $n$. Thus we ask:
For fixed $g$, are $\Nonsep(S)$ and $\Nonsep^*(S_{g,n})$ hyperbolic uniformly in $n$? More generally, are they hyperbolic uniformly in both $g$ and $n$?
Finally, we prove Theorem \[thm:curvediam2\]:
Let $S$ be a connected orientable surface of infinite genus with no isolated ends; in other words, ${{\text{Ends}}}(S)$ is homeomorphic to a Cantor set. Consider a $\Mod(S)$-invariant subgraph ${\mathcal{G}}(S)$ of ${\mathcal{C}}(S)$. Let $\alpha$ and $\beta$ be arbitrary vertices of ${\mathcal{G}}(S)$, noting again that the subsurface $F(\alpha,\beta)$ filled by them has finite type. Using the classification theorem for infinite-type surfaces, there exists $h \in \Mod(S)$ such that $h(\alpha) \subset S - F(\alpha, \beta)$, thus giving a path of length 2 in ${\mathcal{G}}(S)$ between $\alpha$ and $\beta$.
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Departamento de Matemáticas, Universidad Autónoma de Madrid & Instituto de Ciencias Matemáticas, CSIC. `[email protected]`
Centro de Ciencias Matemáticas, UNAM (Morelia). `[email protected]`
[^1]: The first author was partially funded by grants RYC-2013-13008 and MTM2015-67781-P. The second author was supported by PAPIIT projects IN100115, IN103411 and IB100212.
[^2]: This definition is due to Schleimer [@Schleimer], who referred to witnesses as [*holes*]{}. The word “witness" has been suggested to us by S. Schleimer.
|
---
abstract: 'Using the Katz-Arinkin algorithm we give a complete classification of irreducible rigid irregular connections on a punctured ${\mathbb{P}}^1_{\mathbb{C}}$ having differential Galois group $G_2$, the exceptional simple algebraic group. In addition to hypergeometric systems and their Kummer pull-backs we construct families of $G_2$-connections which are not of these types.'
author:
- Konstantin Jakob
bibliography:
- 'mybib.bib'
title: 'The Classification of Rigid Irregular $G_2$-Connections'
---
Introduction
============
The study of rigid differential systems in one dimension, also called systems without accessory parameters, is a classic topic in complex analysis and geometry that dates back as far as Riemann. Consider a complex linear differential equation $$\partial_z^d y+q_{d-1}(z)\partial_z^{d-1}y+...+q_1(z)\partial_z y+q_0(z)y=0$$ with rational functions $q_i\in {\mathbb{C}}(z)$. The poles of the $q_i$ are called singularities of the equation. Given a fundamental solution matrix $F$ of the above equation, one can analytically continue $F$ along simple loops $\gamma_j$ around the singularities $z_j$. This process yields another fundamental solution $F_j$ for every loop $\gamma_j$ and therefore there is a linear relation $F=F_jM(\gamma_j)$. This defines a representation $$\rho:\pi^{\textup{top}}_1({\mathbb{P}}^1,z_0)\rightarrow {\textup{GL}}_d({\mathbb{C}})$$ called the monodromy representation of the equation. In the case that the singularities of the equation are regular (a notion that is made precise in Section \[rigloc\]), the image of $\rho$ is a Zariski dense subgroup of the differential Galois group of the equation. In this case we say that such a differential system is *rigid*, if it is determined up to isomorphism by the local monodromy at the singularities alone. In other words rigid systems admit no non-trivial isomonodromic deformations. Perhaps the most famous example of such a system is the Gaussian hypergeometric equation $$z(z-1)y''+(\gamma-(\alpha+\beta+1)z)y'-\alpha\beta y=0$$ where $\alpha, \beta$ and $\gamma\in {\mathbb{C}}$ are complex parameters. This system has three regular singularities at $0,1$ and $\infty$.
In [@Ka96], Katz gave a thorough analysis of rigid systems on a punctured ${\mathbb{P}}^1$, proving that any such system arises from a local system of rank one through an operation called middle convolution and through twists with systems of rank one. One of his key observations was the following. Let $f(t)=t^{\alpha-\gamma}(1-t)^{\gamma-\beta-1}$ and $g(t)=t^{-\alpha}$. Then the additive convolution $$\int_0^1 f(t)g(z-t)dt=\int_0^1 t^{\alpha-\gamma}(1-t)^{\gamma-\beta-1}(z-t)^{-\alpha} dt$$ is a solution of the hypergeometric equation. Additionally, the function $g$ is a solution of $ty'+\alpha y=0$ and can be interpreted as representing a Kummer system $({\mathcal{O}},d+\alpha dt)$ on ${\mathbb{G}_m}$. On the other hand $f$ should represent some rigid local system, so that the hypergeometric system arises as a convolution of a rigid system with a Kummer system.
Dettweiler and Reiter used Katz’s existence algorithm for rigid local systems in [@Dett10] to give a complete classification of irreducible local systems of rank $7$ on a punctured ${\mathbb{P}}^1$ whose monodromy is dense in the simple algebraic group $G_2$. In particular they derived the existence of motives for motivated cycles that have motivic Galois group of type $G_2$.
In this paper we extend the classification of Dettweiler and Reiter in the complex setting to connections on a punctured ${\mathbb{P}}^1$ with irregular singularities and of differential Galois group $G_2$ and with slopes at most $1$. There are several reasons for assuming this bound. Since twists with a rank one connection preserve rigidity, the slopes of rigid systems are a priori unbounded. Still, most known examples of rigid connections of type $G_2$ and of connections of similar type have their slopes bounded by $1$. This includes for example the Frenkel-Gross connection from [@Fr09] and generalized hypergeometric modules as studied in [@Ka90].
The second reason is of a technical nature. Without the bound on the slopes the invariants governing an irregular singularity are much harder to control. We will see in Section \[G2conn\] what this means in a more precise sense.
Famous examples of such connections are given by generalized hypergeometric systems given by differential operators of the form $$\textup{Hyp}(P,Q)=P(\theta)+zQ(\theta)$$ where $P,Q\in {\mathbb{C}}[z]$ are polynomials, ${\mathbb{C}}[z]\langle\partial_z\rangle$ is the Weyl-algebra and $\theta = z\partial_z\in {\mathbb{C}}[z]\langle\partial_z\rangle$. Operators of this type have been studied in detail by Katz in $\cite{Ka90}$ and many results about their differential Galois groups are known, in particular about the case of differential Galois group $G_2$. Still, explicit occurrences of $G_2$ as the differential Galois group of a connection which is neither hypergeometric nor a pull-back of a hypergeometric system have not been found as of now. In this paper we construct such examples using a generalization of Katz’s existence algorithm due to Arinkin and Deligne. In [@Arinkin10], Arinkin proved that given a connection with irregular singularities which is rigid (meaning in this case that it is determined up to isomorphism by its formal type), you can extend Katz’s algorithm by including the Fourier-Laplace transform of ${\mathcal{D}}$-modules which is defined as pull-back by the Fourier automorphism $$F:{\mathbb{C}}[\tau]\langle\partial_\tau\rangle \rightarrow {\mathbb{C}}[z]\langle\partial_z\rangle$$ given by $F(\tau)=\partial_z$ and $F(\partial_\tau)=-z$. The key ingredient in this proof was provided by Bloch and Esnault in $\cite{BlochEsnault04}$ where they prove that Fourier transform preserves rigidity. To achieve this result, they defined the local Fourier transform on ${\mathbb{C}(\!(t)\!)}$-connections modelled after the local Fourier transform in positive characteristic defined by Laumon in [@Laumon87]. We note that since everything in the complex setting is modeled after the positive characteristic case, there is an analogue of the extended algorithm in that setting.
This paper is organized as follows. In Section \[rigloc\] we give the definition of rigidity, introduce a criterion to identify irreducible rigid connections and recall facts about formal connections over ${\mathbb{C}(\!(t)\!)}$. In particular we introduce the refined Levelt-Turrittin decomposition of ${\mathbb{C}(\!(t)\!)}$-connections into so called elementary modules $${\textup{El}}(\rho,\varphi,R)=\rho_+({\mathscr{E}}^\varphi\otimes R)$$ where $\rho$ is a ramification of some degree $p$ and ${\mathscr{E}}^\varphi =({\mathbb{C}(\!(t)\!)},d+d\varphi)$ for some $\varphi\in {\mathbb{C}(\!(t)\!)}$ and some regular connection $R$. From this we give invariants classifying the formal system of a connection.
In Section \[KA-alg\] we recall the operations needed for the Katz-Arinkin algorithm and give results about the change of the local invariants under Fourier transform and middle convolution. These results rely heavily on the formal stationary phase formula of López (cf. [@Lo04]) and explicit computations of the local Fourier transform of elementary modules by Sabbah in [@Sa08].
In Section \[tannaka\] we review the Tannakian theory for connections on a curve and for connections over ${\mathbb{C}(\!(t)\!)}$, providing tools to classify connections with prescribed differential Galois group and to compute some invariants of such connections.
Section \[G2conn\] is dedicated to the study of the local and global structure of rigid irregular irreducible connections with differential Galois group $G_2$. This provides a rough classification of these connections, in particular yielding the result that any such connection has at most two singularities and can therefore be seen as a connection on ${\mathbb{G}_m}$.
In Section \[classif\] we prove the main result of this paper which is the following theorem. For the exact notation we refer to the last section.
Let $\alpha_1,\alpha_2,\lambda,x,y,z\in {\mathbb{C}}^*$ such that $\lambda^2\neq 1, \alpha_1\neq \pm \alpha_2, z^4\neq 1$ and such that $x,y, xy$ and their inverses are pairwise different and let $\varepsilon$ be a primitive third root of unity. Every formal type occuring in the following list is exhibited by some irreducible rigid connection of rank $7$ on ${\mathbb{G}_m}$ with differential Galois group $G_2$.
[ c c ]{} $0$ & $\infty$\
\
$({\mathbf{J}}(3),{\mathbf{J}}(3),1)$ &
-----------------------------------------------------
${\textup{El}}( 2,\alpha_1,(\lambda,\lambda^{-1}))$
$\oplus\, {\textup{El}}(2,2\alpha_1,1) \oplus (-1)$
-----------------------------------------------------
\
\[15pt\] $(-{\mathbf{J}}(2),-{\mathbf{J}}(2),E_3) $ &
-----------------------------------------------------
${\textup{El}}( 2,\alpha_1,(\lambda,\lambda^{-1}))$
$\oplus\, {\textup{El}}(2,2\alpha_1,1)\oplus (-1)$
-----------------------------------------------------
\
\[15pt\] $(xE_2,x^{-1}E_2,E_3)$ &
-----------------------------------------------------
${\textup{El}}( 2,\alpha_1,(\lambda,\lambda^{-1}))$
$\oplus\, {\textup{El}}(2,2\alpha_1,1)\oplus (-1)$
-----------------------------------------------------
\
\[15pt\]\
$({\mathbf{J}}(3),{\mathbf{J}}(2), {\mathbf{J}}(2))$ &
------------------------------------------------------------------
${\textup{El}}(2,\alpha_1,1) \oplus {\textup{El}}(2,\alpha_2,1)$
$\oplus\,{\textup{El}}(2,\alpha_1+\alpha_2,1) \oplus (-1)$
------------------------------------------------------------------
\
\[15pt\]\
$(iE_2,-iE_2,-E_2,1)$ &
-------------------------------------------------
${\textup{El}}(3,\alpha_1,1)$
$\oplus\,{\textup{El}}(3,-\alpha_1,1)\oplus(1)$
-------------------------------------------------
\
\[15pt\]\
${\mathbf{J}}(7)$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\] $(\varepsilon{\mathbf{J}}(3), \varepsilon^{-1}{\mathbf{J}}(3),1)$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\] $(z{\mathbf{J}}(2), z^{-1}{\mathbf{J}}(2), z^2, z^{-2},1)$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\] $(x{\mathbf{J}}(2),x^{-1}{\mathbf{J}}(2), {\mathbf{J}}(3))$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\] $(x,y,xy,(xy)^{-1},y^{-1},x^{-1}, 1)$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\]
Conversely, the above list exhausts all possible formal types of irreducible rigid irregular $G_2$-connections on open subsets of ${\mathbb{P}}^1$ with slopes at most $1$.
This provides a complete classification of irreducible rigid connections with differential Galois group $G_2$, in particular providing the aforementioned non-hypergeometric examples of such systems. We will discuss which systems arise as pullbacks in the final section after proving the main theorem.
**Acknowledgements.** The author would like to thank Michael Dettweiler for his support during the writing of this article and Stefan Reiter for various fruitful conversations concerning systems of type $G_2$.
Rigidity and local data {#rigloc}
=======================
Let $X$ be a smooth connected complex curve and denote by ${\textup{D.E.}}(X)$ the category of connections on $X$ as in [@Ka87 1.1.]. By a connection we mean a locally free ${\mathcal{O}}_X$-module ${\mathscr{E}}$ of finite rank equipped with a connection $$\nabla: {\mathscr{E}}\rightarrow {\mathscr{E}}\otimes \Omega^1_{X/{\mathbb{C}}}.$$ Let ${\overline{X}}$ be the smooth compactification of $X$ and for any $x\in{\overline{X}}-X$ let $t$ be a local coordinate at $x$. The completion of the local ring of ${\overline{X}}$ at $x$ can be identified with ${\mathbb{C}(\!(t)\!)}$. We define $\Psi_x({\mathscr{E}})={\mathbb{C}(\!(t)\!)}\otimes {\mathscr{E}}$ to be the restriction of ${\mathscr{E}}$ to the formal punctured disk around $x$ and call the collection of isomorphism classes $$\{[\Psi_x({\mathscr{E}})]\}_{x\in {\overline{X}}}$$ the *formal type* of ${\mathscr{E}}$, cf. [@Arinkin10 2.1.]. Note that $\Psi_x({\mathscr{E}})$ is trivial whenever $x\in X$, so the formal type of ${\mathscr{E}}$ is actually determined by the rank $h({\mathscr{E}})$ of ${\mathscr{E}}$ and the family $\{[\Psi_x({\mathscr{E}})]\}_{x\in {\overline{X}}-X}$. We call a connection ${\mathscr{E}}$ *rigid* if it is determined up to isomorphism by its formal type.
Any $\Psi_x({\mathscr{E}})$ obtained in this way is a ${\mathbb{C}(\!(t)\!)}$-connection, by which we mean a finite dimensional ${\mathbb{C}(\!(t)\!)}$-vector space admitting an action of the differential operator ring ${\mathbb{C}(\!(t)\!)}\langle\partial_t\rangle$. Its dimension will be called the *rank* of the connection. The category of ${\mathbb{C}(\!(t)\!)}$-connections is denoted by $\textup{D.E.}({\mathbb{C}(\!(t)\!)})$. Recall that any ${\mathbb{C}(\!(t)\!)}$-connection $E$ can be decomposed as $$E=\bigoplus_{y\in {\mathbb{Q}}_{\ge 0}} E(y)$$ where only finitely many $E(y)$ are non-zero and where ${\textup{rk}}(E(y))\cdot y\in {\mathbb{Z}}_{\ge 0}$. Any $y\in {\mathbb{Q}}_{\ge 0}$ with $E(y)\neq 0$ is called a *slope* of $E$, see [@Ka87 2.3.4.].
The slopes can be computed in the following way. One proves that $E$ is isomorphic to a connection of the form $${\mathbb{C}(\!(t)\!)}\langle\partial_t\rangle / (L)$$ for some operator $L\in {\mathbb{C}(\!(t)\!)}\langle\partial_t\rangle$ where $(L)$ denotes the left-ideal generated by $L$. To $L$ we can associate its Newton polygon $N(L)$ and the slopes of $E$ are given by the slopes of the boundary of $N(L)$. This is independent of the choice of $L$.
The *irregularity* of $E$ is given by ${\textup{irr}}(E):=\sum_y {\textup{rk}}(E(y))\cdot y$. It vanishes if and only if every slope is zero. In this case we call $E$ *regular singular*. Note that if $E\cong {\mathbb{C}(\!(t)\!)}\langle\partial_t\rangle / (L)$, then $E$ is regular singular if and only if the Newton polygon of $L$ is a quadrant. Given a connection ${\mathscr{E}}$ on a curve $X$ as above, its irregularity at $x\in {\overline{X}}-X$ is defined as ${\textup{irr}}_x({\mathscr{E}})={\textup{irr}}(\Psi_x({\mathscr{E}}))$.
A refinement of the slope decomposition is given as follows. Let $\rho\in u{\mathbb{C}}[[u]]$ and regard $\rho$ as a covering of the formal disc ${\textup{Spec\,}}{\mathbb{C}}[[t]]$ through the map ${\mathbb{C}}[[t]]\rightarrow {\mathbb{C}}[[u]], t\mapsto \rho(u)$. For $\varphi\in {\mathbb{C}(\!(u)\!)}$ and $R$ a regular singular ${\mathbb{C}(\!(u)\!)}$-connection, we define the associated *elementary module* to be $${\textup{El}}(\rho,\varphi,R):=\rho_+({\mathscr{E}}^\varphi\otimes R)$$ where $\rho_+$ denotes the push-forward and ${\mathscr{E}}^\varphi=({\mathbb{C}(\!(u)\!)},d+d\varphi)$. We denote by $p$ the order of $\rho$ and by $q=q(\varphi)$ the order of the pole of $\varphi$ (if $\varphi\in {\mathbb{C}}[[u]]$ we let $q=0$). The Levelt-Turrittin theorem (or rather a variant due to [@Sa08 3.3]) states the following. Given any ${\mathbb{C}(\!(t)\!)}$-connection $E$ there is a finite set $\Phi\subset {\mathbb{C}(\!(t)\!)}$ such that $$E\cong \bigoplus_{\varphi\in\Phi}{\textup{El}}(\rho_\varphi,\varphi,R_\varphi ).$$ We denote by $p=p(\varphi)$ the order of $\rho_\varphi$. This decomposition is called *minimal* if no $\rho_1,\rho_2$ and $\varphi_1$ exist such that $\rho_\varphi=\rho_1\circ\rho_2$ and $\varphi=\varphi_1\circ\rho_2$ and if for $\varphi,\psi\in\Phi$ with $p(\varphi)=p(\psi)$ there is no $p$-th root of unity $\zeta$ such that $\varphi=\psi\circ \mu_\zeta$ where $\mu_\zeta$ denotes multiplication by $\zeta$. In this case the above decomposition is unique.
For later use we will collect some facts about elementary modules in the following proposition.
\[emprop\] Let ${\textup{El}}(\rho,\varphi,R)$ and ${\textup{El}}(\nu,\psi,S)$ be elementary modules. The following holds.
The dual of ${\textup{El}}(\rho,\varphi,R)$ is given as ${\textup{El}}(\rho,-\varphi,R^*)$ where $R^*$ denotes the dual connection of $R$.
Let $p$ be the degree of $\rho$, $r$ the rank of $R$ and $(t^{(p-1)r/2})$ the connection $({\mathbb{C}(\!(t)\!)}, d+((p-1)r/2)dt/t)$. The determinant connection $\det{\textup{El}}(\rho,\varphi,R)$ is isomorphic to ${\mathscr{E}}^{r\textup{Tr}\varphi}\otimes \det(R)\otimes (t^{(p-1)r/2}).$
Suppose $\rho(u)=\nu(u)=u^p$. Then ${\textup{El}}(\rho,\varphi,R)\cong {\textup{El}}(\nu,\psi,S)$ if and only if there exists $\zeta$ with $\zeta^p=1$ and $\psi\circ\mu_\zeta\equiv \varphi \mod {\mathbb{C}}[[u]]$ and $R\cong S$ where $\mu_\zeta$ denotes multiplication by $\zeta$.
More generally, suppose the degree of $\rho$ and the degree of $\nu$ are both $p$. Then ${\textup{El}}(\rho,\varphi,R)\cong {\textup{El}}(\nu,\psi,S)$ if and only if $R\cong S$ and there exists $\zeta$ with $\zeta^p=1$ and $\lambda_1,\lambda_2\in u{\mathbb{C}}[[u]]$ satisfying $\lambda_i'(0)\neq 0$ such that $\rho=\nu\circ \lambda_1$ and $$\varphi\equiv \psi\circ\lambda_1\circ (\lambda_2^{-1} \circ\mu_\zeta\circ\lambda_2)\mod{\mathbb{C}}[[u]].$$
We have $\rho^+\rho_+{\mathscr{E}}^\varphi\cong \bigoplus_{\zeta^p=1} {\mathscr{E}}^{\varphi\circ\mu_{\zeta}}$.
We will now restrict ourselves to the case of ${\overline{X}}={\mathbb{P}}^1$ and $X=U$ a non-empty open subset of ${\mathbb{P}}^1$. Let ${\textup{Hol}}({\mathbb{A}}^1)$ denote the category of holonomic $D$-modules on ${\mathbb{A}}^1$ and let $M$ be a holonomic module with singularities $x_1,...,x_r\in {\mathbb{A}}^1$ and $x_{r+1}=\infty$. In the following we will always assume that holonomic modules are not supported on a finite set of points. Restricting $M$ to the complement $U$ of its singularities turns $M$ into a connection on $U$ in the sense defined above. For any singularity $x_i$ let $t$ be a local coordinate at $x_i$ and define $$\begin{aligned}
\widehat{M}_{x_i} &= M\otimes{\mathbb{C}}[t], i=1,...,r \\
\widehat{M}_\infty &= \Psi_\infty(M). \\
\intertext{At the finite singularities there is a decomposition}
\widehat{M}_{x_i}&\cong \widehat{M}_{x_i}^{\textup{reg}}\oplus\widehat{M}_{x_i}^{\textup{irr}}\end{aligned}$$ into ${\mathbb{C}}[[t]]\langle\partial_t\rangle$-modules where $\widehat{M}_{x_i}^{\textup{reg}}$ is regular singular (meaning that the associated connection $\widehat{M}_{x_i}^{\textup{reg}}[t^{-1}]$ is regular singular) and $\widehat{M}_{x_i}^{\textup{irr}}$ is purely irregular. Additionally, $\widehat{M}_{x_i}^{\textup{irr}}$ admits the structure of a ${\mathbb{C}(\!(t)\!)}$-connection. Applying the refined Levelt-Turrittin yields a decomposition $$\widehat{M}_{x_i}^{\textup{irr}}\cong \bigoplus_{\varphi\in\Phi_i}{\textup{El}}(\rho_\varphi,\varphi,R_\varphi ).$$ To describe the formal type numerically we introduce for every singularity $x_i$ the *vanishing cycle local data* $$(^\varphi\nu_{x_i,\lambda,l}(M),\varphi\in\Phi_i, \lambda\in {\mathbb{C}}^*,l\in {\mathbb{Z}}_{\ge 0})$$ meaning that $\frac{1}{p(\varphi)}\cdot ^\varphi\!\nu_{x_i,\lambda,l}$ is the number of Jordan blocks of length $(l+1)$ with eigenvalue $\lambda$ of the monodromy of the regular singular connection $R_\varphi$ occuring in the Levelt-Turrittin decomposition of $M$ at $x_i$.
Recall that $\textup{D.E.}({\mathbb{C}(\!(t)\!)})$ is equivalent to the category of pairs $(V,T)$ where $V$ is a finite dimensional ${\mathbb{C}}$-vector space equipped with an automorphism $T\in GL(V)$. This equivalence is given by associating to a ${\mathbb{C}(\!(t)\!)}$-connection its space of nearby cycles equipped with the monodromy operator.
We say that a ${\mathbb{C}}[[t]]\langle\partial_t\rangle$-module $\widehat{M}$ is holonomic, if it is finitely generated and if its generic rank $\dim_{{\mathbb{C}(\!(t)\!)}}(M\otimes{\mathbb{C}(\!(t)\!)})$ is finite. The category of holonomic ${\mathbb{C}}[[t]]\langle\partial_t\rangle$-modules is denoted by ${\textup{Hol}}({\mathbb{C}}[[t]])$. The full subcategory ${\textup{Hol}}^{{\textup{reg}}}({\mathbb{C}}[[t]])$ of regular holonomic modules is equivalent to the category of tuples $$(V,W,\alpha,\beta)$$ where $\alpha\in {\textup{End}}(V,W)$ and $\beta\in {\textup{End}}(W,V)$ satisfy $\beta\circ\alpha+{\textup{id}}\in GL(V)$. This equivalence is given by assigning to $\widehat{M}$ the diagram $$\psi_t(\widehat{M})\rightleftarrows \phi_t(\widehat{M})$$ of nearby cycles and vanishing cycles with maps $$\begin{aligned}
{\textup{can}}:\psi_t(\widehat{M})&\rightarrow \phi_t(\widehat{M}), \\
{\textup{var}}:\phi_t(\widehat{M})&\rightarrow \psi_t(\widehat{M}).\end{aligned}$$ For details on the above two equivalences we refer to $\cite[Section 6]{Sa93}$.
If $\widehat{M}$ is a *minimal extension* at $t=0$, then ${\textup{can}}$ is surjective and ${\textup{var}}$ is injective, meaning that actually the vanishing cycles are given as the image of $N={\textup{var}}\circ{\textup{can}}$ which is the nilpotent part of the monodromy of $\widehat{M}$. In the following we will consider the *vanishing cycle local data* at the finite singularities given by $$\begin{aligned}
^\varphi\mu_{x_i,\lambda,l}(M)&=\ ^\varphi\nu_{x_i,\lambda,l}(M), &i=1,...,r, \lambda\neq 1 \textup{\, or \, } \varphi\neq 0, \\
^0\mu_{x_i,1,l}(M)&=\ ^0\nu_{x_i,1,l+1}(M), &i=1,...,r \end{aligned}$$ according to the shift in the monodromy filtration obtained by applying the nilpotent endomorphism $N$.
Suppose that $M$ is an irreducible holonomic $D$-module on ${\mathbb{A}}^1$. By [@Ka90 Corollary 2.9.6.1.], $M$ is a minimal extension at each of its finite singularities. Consider the following data:
1. The generic rank $h(M)$ of $M$,
2. the vanishing cycle local data $(^\varphi\mu_{x_i,\lambda,l}(M), i=1,...,r, \varphi\in \Phi_i,\lambda\in {\mathbb{C}}^*)$ and
3. the nearby cycle local data $(^\varphi\nu_{\infty,\lambda,l}(M), \varphi\in \Phi_{r+1},\lambda\in {\mathbb{C}}^*)$.
As $M$ is a minimal extension, these data completely determine $\widehat{M}_{x_i}$ for all $i=1,...,r+1$. Recall that $M$ is rigid if its isomorphism class is determined by its formal type. Therefore by the above discussion, these local data classify rigid irreducible $D$-modules on ${\mathbb{A}}^1$.
There is a criterion to identify rigid irreducible connections due to Katz in the case of regular singularities with a generalization by Bloch and Esnault in the case of irregular singularities.
Let ${\mathscr{E}}$ be an irreducible connection on $j:U\hookrightarrow {\mathbb{P}}^1$. Denote by $j_{!*}$ the middle extension functor. The connection ${\mathscr{E}}$ is rigid if and only if $$\chi({\mathbb{P}}^1,j_{!*}({\mathscr{E}nd}({\mathscr{E}})))=2$$ where $\chi$ denotes the Euler-de Rham characteristic.
For this reason, we set ${\textup{rig\,}}({\mathscr{E}})=\chi({\mathbb{P}}^1,j_{!*}({\mathscr{E}nd}({\mathscr{E}}))$ and call it the *index of rigidity*. Whenever ${\textup{rig\,}}({\mathscr{E}})=2$ we say that ${\mathscr{E}}$ is *cohomologically rigid*. The index of rigidity can be computed using local information only.
Let ${\mathscr{E}}$ be an irreducible connection on $j:U\hookrightarrow {\mathbb{P}}^1$ and let ${\mathbb{P}}^1-U={x_1,...,x_r}$. The index of rigidity of ${\mathscr{E}}$ is given as $${\textup{rig\,}}({\mathscr{E}})=(2-r)h({\mathscr{E}})^2-\sum_{i=1}^r {\textup{irr}}_{x_i}({\mathscr{E}nd}({\mathscr{E}}))+\sum_{i=1}^r \dim_{\mathbb{C}}{\textup{Soln}}_{x_i}({\mathscr{E}nd}({\mathscr{E}}))$$ where $h({\mathscr{E}})$ is the rank of ${\mathscr{E}}$ and ${\textup{Soln}}_{x_i}({\mathscr{E}nd}({\mathscr{E}}))$ is the space of horizontal sections of $\Psi_{x_i}({\mathscr{E}nd}({\mathscr{E}}))={\mathbb{C}(\!(t)\!)}\otimes{\mathscr{E}nd}({\mathscr{E}})$.
Given the formal type of ${\mathscr{E}}$ in its refined Levelt-Turrittin decomposition we can compute all invariants appearing in the above formula. We will first focus on the irregularity and come back to the computation of $\dim_{\mathbb{C}}{\textup{Soln}}_{x_i}({\mathscr{E}nd}({\mathscr{E}}))$ in Section \[tannaka\].
Let $E$ be a ${\mathbb{C}(\!(t)\!)}$-connection with minimal Levelt-Turrittin decomposition $$E=\bigoplus_i {\textup{El}}(\rho_i,\varphi_i,R_i).$$ Its endomorphism connection is then given by $$\label{endtensor} E\otimes E^*=\bigoplus_{i,j}{\textup{Hom}}({\textup{El}}(\rho_i,\varphi_i,R_i),{\textup{El}}(\rho_j,\varphi_j,R_j)).$$ As the irregularity of $E\otimes E^*={\textup{End}}(E)$ is given as sum over the slopes, it can be computed by combining this decomposition with the following proposition of Sabbah.
\[elmthom\] Let $\rho_i(u)=u^{p_i}, d=\gcd(p_1,p_2), p'_i=p_i/d$ and $\tilde{\rho}_i(w)=w^{p'_i}$. Consider the elementary connections ${\textup{El}}(\rho_i,\varphi_i,R_i), i=1,2$. We have $${\textup{Hom}}({\textup{El}}(\rho_1,\varphi_1,R_1),{\textup{El}}(\rho_2,\varphi_2,R_2))\cong \bigoplus_{k=0}^{d-1} {\textup{El}}([w\mapsto w^{p_1p_2/d}],\varphi^{(k)},R),$$ where $$\varphi^{(k)}(w)=\varphi_2(w^{p'_1})-\varphi_1((e^{\frac{2\pi ikd}{p_1p_2}}w)^{p'_2})$$ and $R=\tilde{\rho}_2^+R_1^*\otimes \tilde{\rho}_1^+R_2$.
Review of the Katz-Arinkin algorithm {#KA-alg}
====================================
We recall the various operations involved in the Arinkin algorithm as defined in [@Arinkin10]. Let $D_z={\mathbb{C}}[z]\langle \partial_z\rangle$ be the Weyl-algebra in one variable. The Fourier isomorphism is the map $$\begin{aligned}
F:D_\tau&\rightarrow D_z \\
\tau&\mapsto \partial_z \\
\partial_\tau&\mapsto -z.\end{aligned}$$ From now on we will always denote the Fourier coordinate by $\tau$ in the global setting. We will also use a subscript to indicate the coordinate on ${\mathbb{A}}^1$. Let $M$ be a $D_z$-module on ${\mathbb{A}}^1_z$. The *Fourier transform* of $M$ is $${\mathscr{F}}(M)=F^*(M).$$ Denote by $F^\vee: D_z\rightarrow D_\tau$ the same map as above with the roles of $z$ and $\tau$ reversed and let ${\mathscr{F}}^\vee=(F^\vee)^*$.
Observe that a module $M$ is holonomic if and only if ${\mathscr{F}}(M)$ is holonomic. The functor ${\mathscr{F}}$ therefore defines an equivalence $${\mathscr{F}}:{\textup{Hol}}({\mathbb{A}}^1_z)\rightarrow {\textup{Hol}}({\mathbb{A}}^1_\tau).$$ We have ${\mathscr{F}}^\vee\circ {\mathscr{F}}=\varepsilon^*$ where $\varepsilon$ is the automorphism of $D_z$ defined by $\varepsilon(z)=-z$ and $\varepsilon(\partial_z)=-\partial_z$.
Using the Fourier transform we define the middle convolution as follows. Let $\chi\in {\mathbb{C}}^*$ and let ${\mathcal{K}}_\chi$ be the connection on ${\mathbb{G}_m}$ associated to the character $\pi_1({\mathbb{G}_m},1)\rightarrow {\mathbb{C}}^\times$ defined by $\gamma \mapsto \chi$ where $\gamma$ is a generator. We call ${\mathcal{K}}_\chi$ a *Kummer sheaf*. Explicitly, ${\mathcal{K}}_\chi$ can be given as the trivial line bundle ${\mathcal{O}}_{{\mathbb{G}_m}}$ equipped with the connection $d+\alpha d/dz$ for any $\alpha\in {\mathbb{C}}$ such that $\exp(-2\pi i\alpha)=\chi$.
Let $i:{\mathbb{G}_m}\hookrightarrow {\mathbb{A}}^1$ be the inclusion. The *middle convolution* of a holonomic module $M$ with the Kummer sheaf ${\mathcal{K}}_\chi$ is defined as $${\textup{MC}}_\chi(M):= {\mathscr{F}}^{-1}(i_{!*}({\mathscr{F}}(M)\otimes {\mathcal{K}}_{\chi^{-1}}))$$ where ${\mathscr{F}}^{-1}$ denotes the inverse Fourier transform and $i_{!*}$ is the minimal extension. Note that ${\mathscr{F}}({\mathcal{K}}_{\chi})={\mathcal{K}}_{\chi^{-1}}$.
Given a connection ${\mathscr{E}}$ on an open subset $j:U\hookrightarrow {\mathbb{A}}^1$ we can apply the Fourier transform or the middle convolution to its minimal extension $j_{!*}{\mathscr{E}}$. We end up with a holonomic module on ${\mathbb{A}}^1$ which we can restrict in both cases to the complement of its singularities. This restriction is again a connection on some open subset of ${\mathbb{A}}^1$ and we denote it by ${\mathscr{F}}({\mathscr{E}})$ for the Fourier transform and ${\textup{MC}}_\chi({\mathscr{E}})$ for middle convolution. Whenever ${\mathscr{E}}$ is defined on an open subset $U\subset {\mathbb{P}}^1$ we can shrink $U$ such that $\infty\notin U$ and apply the above construction.
The Katz-Arinkin algorithm is given in the following theorem. It was proven in the case of regular singularities by Katz in [@Ka96] and in the case of irregular singularities by Arinkin in [@Arinkin10].
Let ${\mathscr{E}}$ be an irreducible connection on an open subset $U\subset {\mathbb{P}}^1$ and consider the following operations.
1. Twisting with a regular singular connection of rank one,
2. change of coordinate by a Möbius transformation,
3. Fourier transform and
4. middle convolution.
The connection ${\mathscr{E}}$ is rigid if and only if it can be reduced to a regular singular connection of rank one using a finite sequence of the above operations.
As middle convolution is itself a combination of Fourier transforms and twists the above statement holds even when omitting convolution. A crucial point in the proof of the above statement is the fact that all these operations preserve the index of rigidity. This was proven by Bloch and Esnault in [@BlochEsnault04 Theorem 4.3.] using the local Fourier transform which they defined in characteristic zero as an analogue to Laumon’s local Fourier transform from [@Laumon87].
Let $E$ be a ${\mathbb{C}(\!(t)\!)}$-connection, i.e. a finite dimensional ${\mathbb{C}(\!(t)\!)}$-vector space admitting an action of ${\mathbb{C}(\!(t)\!)}\langle\partial_t\rangle$. The *local Fourier transform* of $E$ from zero to infinity is obtained in the following way. Due to [@Ka87 Section 2.4.] there is an extension of $E$ to a connection ${\mathcal{M}}_E$ on ${\mathbb{G}_m}$ which has a regular singularity at infinity and whose formal stalk at zero is $E$. We define $${\mathscr{F}}^{(0,\infty)}(E):={\mathscr{F}}({\mathcal{M}}_E)\otimes_{{\mathbb{C}}[\tau]}{\mathbb{C}}(\!(\theta)\!)$$ where $\tau$ is the Fourier transform coordinate and $\theta=\tau^{-1}$. In a similar fashion define for $s\in {\mathbb{C}}^*$ transforms $${\mathscr{F}}^{(s,\infty)}(E)={\mathscr{E}}^{s/\theta}\otimes{\mathscr{F}}^{(0,\infty)}(E)$$ where ${\mathscr{E}}^{s/\theta}$ denotes again an exponential system. Recall that there also is a transform ${\mathscr{F}}^{(\infty,\infty)}$ which is of no interest to us, as it only applies to connections of slope larger than one. For details on this transform we refer to [@BlochEsnault04 Section 3.].
There are also transforms ${\mathscr{F}}^{(\infty,s)}$ which are inverse to ${\mathscr{F}}^{(s,\infty)}$, see [@Sa08 Section 1]. For the local Fourier transforms Sabbah computed explicitly how the elementary modules introduced in the first section behave. By abuse of notation we will write $${\mathscr{F}}^{(s,\infty)}(M)={\mathscr{F}}^{(s,\infty)}(\widehat{M}_s)$$ with $M$ being a holonomic $D$-module on ${\mathbb{A}}^1$. The most important tool for controlling the formal type under Fourier transform is the formal stationary phase formula of López.
Let $M$ be a holonomic $D$-module on ${\mathbb{A}}^1$ with finite singularities $\Sigma$. There is an isomorphism $$\widehat{{\mathscr{F}}(M)}_\infty \cong \bigoplus_{s\in \Sigma\cup\{\infty\}}{\mathscr{F}}^{(s,\infty)}(M).$$
Let $M$ be a holonomic ${\mathbb{C}}[[t]]\langle\partial_t\rangle$-module and choose an extension ${\mathcal{M}}$ as before. The formal type at infinity of the Fourier transform of this module is the local Fourier transform ${\mathscr{F}}^{(0,\infty)}(M)$. By \[Sabbah, 5.7.\], the local Fourier transform ${\mathscr{F}}^{(0,\infty)}(M)$ of a regular holonomic ${\mathbb{C}}[[t]]\langle\partial_t\rangle$-module $M$ is the connection associated to the space of vanishing cycles $(\phi_tM,T)$ where $T={\textup{id}}+{\textup{can}}\circ{\textup{var}}$.
\[explicitstatphase\] Let ${\textup{El}}(\rho, \varphi, R)$ be any elementary ${\mathbb{C}(\!(t)\!)}$-module with irregular connection. Recall that $${\textup{El}}(\rho,\varphi,R)=\rho_+({\mathscr{E}}^{\varphi}\otimes R)$$ and that $q=q(\varphi)$ is the order of the pole of $\varphi$ which is positive by assumption. Denote by $'$ the formal derivative and let $\widehat{\rho}=\frac{\rho'}{\varphi'}$, $\widehat{\varphi}=\varphi-\frac{\rho}{\rho'}\varphi'$, $L_q$ the rank one system with monodromy $(-1)^q$ and $\widehat{R}=R\otimes L_q$. The local Fourier transform of the elementary module is then given by $${\mathscr{F}}^{(0,\infty)}{\textup{El}}(\rho,\varphi,R)={\textup{El}}(\widehat{\rho},\widehat{\varphi},\widehat{R}).$$
In particular, we also have explicit descriptions $$\begin{aligned}
{\mathscr{F}}^{(s,\infty)}{\textup{El}}(\rho,\varphi,R)&\cong{\textup{El}}(\widehat{\rho},\widehat{\varphi}+s/(\theta\circ \widehat{\rho}),\widehat{R}) \\
{\mathscr{F}}^{(s,\infty)}(M)&\cong{\textup{El}}({\textup{id}},s/\theta,{\mathscr{F}}^{(0,\infty)}M) \\\end{aligned}$$ for $M$ a regular ${\mathbb{C}}[[t]]\langle\partial_t\rangle$-module.
Let ${\mathscr{L}}$ be a rigid irreducible connection on $U\subset {\mathbb{P}}^1$ all of whose slopes are at most $1$. Then in order to reduce the rank of ${\mathscr{L}}$ it suffices to twist with rank one connections whose slopes also do not exceed $1$.
The choice of the connection $\ell$ with which we have to twist in order to lower the rank is made explicit in the proof of [@Arinkin10 Theorem A]. Let $S={\mathbb{P}}^1-U$ be the set of singularities of ${\mathscr{L}}$. For each $x\in S$ we choose an irreducible subrepresentation $V_x$ of $\Psi_x({\mathscr{L}})$ such that $$\delta({\textup{End}}(\Psi_x({\mathscr{L}})))\ge \frac{{\textup{rk}}({\mathscr{L}})}{{\textup{rk}}(V_x)}\delta({\textup{Hom}}(V_x,\Psi_x({\mathscr{L}}))$$ where $\delta(E)={\textup{irr}}(E)+{\textup{rk}}(E)-\dim {\textup{Soln}}(E)$ for a formal connection $E$. Arinkin proves that either all $V_x$ are of rank one or if there is a $V_x$ of higher rank, there is exactly one such. In the first case $\ell$ is chosen so that $\Psi_x(\ell)$ is $V_x$ (up to a twist with a regular singular formal connection), meaning that all its slopes are at most $1$. In the second case, let $x_0$ be the unique singularity for which ${\textup{rk}}(V_{x_0})>1$. Then (up to a twist with a regular singular formal connection) Arinkin chooses $\ell$ in such a way that the slope of $${\textup{Hom}}(\Psi_{x_0}(\ell),V_\infty)$$ is fractional. This in done in the following way. By the Levelt-Turrittin-Theorem, $$V_\infty\cong \rho_*({\mathscr{E}}^\varphi \otimes \lambda)$$ for $\rho(u)=u^p$, $\lambda$ a regular singular connection of rank one and $\varphi$ a polynomial of the form $$\varphi(u)=\frac{a_{p}}{u^p}+...+\frac{a_1}{u}+a_0.$$ Then we have $${\mathscr{E}}^{\frac{-a_{p}}{t}}\otimes V_\infty \cong \rho_*({\mathscr{E}}^\varphi\otimes \lambda\otimes\rho^*{\mathscr{E}}^{\frac{-a_{p}}{t}})\cong \rho_*({\mathscr{E}}^{\varphi_{<a}}\otimes \lambda)$$ where $\varphi_{<p}(u)=\frac{a_{p-1}}{u^{p-1}}+...+\frac{a_1}{u}+a_0$. This connection has fractional slope $\frac{p-1}{p}<1$ and the connection ${\mathscr{E}}^{\frac{-a_{p}}{t}}$ we twisted with has slope $1$.
\[rigidslope\] Let $M$ be any irreducible rigid holonomic module on ${\mathbb{A}}^1$ all of whose slopes are at most one. Any non-zero slope of $M$ has numerator $1$.
The module $M$ is constructed using Fourier transform, twists with rank one connections and coordinate changes. Of these operations only Fourier transform and twisting has any impact on the slopes. By the above Lemma the systems with which we twist have slopes either $0$ or $1$. Therefore twisting preserves the property of the slope to have numerator $1$.
For the Fourier transform there are two possibilities. The first case is a transform ${\mathscr{F}}^{(0,\infty)}$ which produces a regular connection from a regular connection and which changes the ramification order from $p$ to $p+q$ and does not change the pole order in the case of an irregular module ${\textup{El}}(\rho,\varphi,R)$ with $p=p(\rho)$ and $q=q(\varphi)$. The second case is the transform ${\mathscr{F}}^{(s,\infty)}$ for $s\neq 0$ which changes the ramification order from $p$ to $p+q$ and the pole order from $q$ to $\max(q,p+q)=p+q$. So after applying ${\mathscr{F}}^{(s,\infty)}$ once, ${\mathscr{F}}^{(0,\infty)}$ only produces slopes of the form $\frac{p+q}{k(p+q)}$ where the $k$ counts the number of applications of ${\mathscr{F}}^{(0,\infty)}$. Hence they are always of the form $\frac{n}{kn}=\frac{1}{k}$ for $k,n\in {\mathbb{Z}}_{>0}$.
Let $M$ be as before a holonomic $D$-module with singularities $x_1,...,x_r$ and $x_{r+1}=\infty$ and denote by $y_1,...,y_s$ and $y_{s+1}=\infty$ the singularities of the Fourier transform of $M$. Write $$\widehat{M}_{x_i}=\widehat{M}_{x_i}^{{\textup{reg}}}\oplus \bigoplus_{\varphi\in \Phi_i}{\textup{El}}(\rho_\varphi,\varphi,R_\varphi).$$ This means we denote by $\Phi_i(M)$ (resp. $\Phi_j({\mathscr{F}}(M))$) be the indexing set of the refined Levelt-Turrittin decomposition of $M$ at $x_i$ (resp. of ${\mathscr{F}}(M)$ at $y_j$). The stationary phase formula gives a decomposition $$\Phi_{s+1}({\mathscr{F}}(M))=\coprod_{i=1}^{r+1}\Phi_{s+1}^{(i)}({\mathscr{F}}(M))$$ where $\Phi_{s+1}^{(i)}({\mathscr{F}}(M))$ is in bijection to $\Phi_i(M)$ through $$\Phi_i(M)\rightarrow \Phi_{s+1}^{(i)}({\mathscr{F}}(M)), \varphi \mapsto \widehat{\varphi}$$ with $\widehat{\varphi}$ given by Theorem \[explicitstatphase\]. Using the inverse Fourier transform we find a similar decomposition $$\Phi_{r+1}(M)=\coprod_{j=1}^{s+1}\Phi_{r+1}^{(j)}(M)$$ where $\Phi_{r+1}^{(j)}(M)$ is in bijection to $\Phi_j({\mathscr{F}}(M))$ through $\varphi\mapsto \widehat{\varphi}$. Recall that $\frac{1}{p(\varphi)}\cdot ^\varphi\!\nu_{x_i,\lambda,l}$ denotes the number of Jordan blocks of length $(l+1)$ with eigenvalue $\lambda$ of the monodromy of the connection $R_\varphi$ at the singularity $x_i$ and that $$^\varphi\!\mu_{x_i,1,l}(M)=^\varphi\!\nu_{x_i,1,l+1}(M), i=1,...,r$$ is the vanishing cycle local data. The above discussion yields the following bevaviour of nearby and vanishing cycle local data under Fourier transform: $$\begin{aligned}
p(\varphi_j) ^{\widehat{\varphi}_j}\!\mu_{y_j,\lambda,l}({\mathscr{F}}(M))&=p(\widehat{\varphi}_j) ^{\varphi_j}\!\nu_{\infty,(-1)^{q(\widehat{\varphi}_j)}\lambda,l}(M) \\
p(\varphi_i) ^{\widehat{\varphi}_i}\!\nu_{\infty,\lambda,l}({\mathscr{F}}(M))&= p(\widehat{\varphi}_i) ^{\varphi_i}\mu_{x_i,(-1)^{q(\varphi_i)}\lambda,l}(M).\end{aligned}$$
\[fourierrank\] Let $M$ be as above. The generic rank of its Fourier transform is given as $$h({\mathscr{F}}(M))=\sum_{i=1}^r \sum_{\varphi\in \Phi_i(M)} \frac{p(\widehat{\varphi})}{p(\varphi)}\, ^\varphi\!\mu_{x_i}(M)+\sum_{\varphi\in \Phi_{r+1}^{(s+1)} (M)} \frac{p(\widehat{\varphi})}{p(\varphi)}\nu_{x_{r+1}}(M)$$ where $^\varphi\mu_{x_i,\lambda}(M)=\sum_{l\ge 0} (l+1) ^\varphi\!\mu_{x_i,\lambda,l}(M)$ and omitting $\lambda$ means summing over all $\lambda \in {\mathbb{C}}^*$.
First note that the generic rank of any holonomic module $N$ can be computed as $$h(N)=\sum_{\lambda,\varphi,l}(l+1) ^\varphi\!\nu_{\infty,\lambda,l}(M).$$ Applying the formulae for the change of local data under Fourier transform and the decomposition of the $\Phi_i$ yields the claim.
Note that the second sum is always zero in our applications as it ranges over the $\varphi$ with slope $>1$ and by Lemma \[rigidslope\] the slopes of any rigid system are at most $1$.
Let $M$ be an irreducible holonomic module as before. Assume $M$ has a regular singularity at $\infty$ with scalar monodromy $\chi\cdot{\textup{id}}$. At the finite singularities the local data changes under middle convolution ${\textup{MC}}_\chi$, $\chi\neq 1$ as follows: $$^\varphi\mu_{x_i,\lambda,l}({\textup{MC}}_\chi(M))=\,^\varphi\mu_{x_i,\lambda/\chi^{p(\varphi)+q(\varphi)},l}(M).$$ At the infinity singularity ${\textup{MC}}_\chi(M)$ has scalar monodromy $\chi^{-1}\cdot{\textup{id}}$. Furthermore the rank of ${\textup{MC}}_\chi(M)$ is given by $$h({\textup{MC}}_\chi(M))=h({\mathscr{F}}(M))-h(M).$$
Given a holonomic module with arbitrary formal type at $\infty$ we can always arrange for it to have scalar monodromy at $\infty$ by adding a fake singularity and applying a change of coordinate.
The only thing missing is the change of local data under twists with rank one systems.
Let $M$ be a holonomic $D$-module with singularities $x_1,...,x_r,x_{r+1}$ where $x_{r+1}=\infty$ as before. Let ${\mathscr{L}}:={\mathscr{L}}_{(\lambda_1,...,\lambda_{r+1})}$ be the local system of rank one on ${\mathbb{P}}^1-\{x_1,...,x_{r+1}\}$ with monodromy $\lambda_i\ \in{\mathbb{C}}^*$ at $x_i$ (in particular $\lambda_1 \cdot ...\cdot \lambda_{r+1}=1$). Then we have $$^\varphi\nu_{x_{r+1},\lambda,l}(M\otimes{\mathscr{L}})=\,^\varphi\nu_{x_{r+1},\lambda/\lambda_{r+1}^{p(\varphi)},l}(M)$$ at the infinite singularity. At a finite singularity $x_i$ we have\
$^\varphi\mu_{x_i,\lambda,l}(M\otimes {\mathscr{L}})=\begin{cases} ^\varphi\mu_{x_i,\lambda/\lambda_{i}^{p(\varphi)},l}(M) \\
^0\mu_{x_i,1/\lambda,l+1}(M) \\
^0\mu_{x_i,1,l-1}(M) \\
h(M)-\mu_{x_i}(M)-^0\mu_{x_i,1,prim}(M)
\end{cases} $\
\
where $\mu_{x_i}(M)=\sum_\varphi \sum_\lambda \,^\varphi\mu_{x_i,\lambda}(M)$ and $^0\mu_{x_i,1,prim}(M) =\sum_{l\ge 0}\,^0\mu_{x_i,1,l}(M)$.
Tannakian formalism for connections {#tannaka}
===================================
Recall that if $X$ is a smooth connected complex curve, ${\textup{D.E.}}(X)$ is the category of connections on $X$. The usual notions of tensor product and internal hom turn ${\textup{D.E.}}(X)$ into a rigid abelian tensor category. Given any point $x\in X({\mathbb{C}})$, the functor associating to a connection ${\mathscr{E}}$ its fibre above $x$ defines a fibre functor ${\textup{D.E.}}(X)\rightarrow {\textup{Vect}}_{\mathbb{C}}$ where ${\textup{Vect}}_{\mathbb{C}}$ denotes the category of finite dimensional ${\mathbb{C}}$-vector spaces. This in turn endowes ${\textup{D.E.}}(X)$ with the structure of a neutral Tannakian category, cf. [@Ka87 1.1.].
Denote by $\omega_x$ the fibre functor associated to the point $x\in X({\mathbb{C}})$ and by ${\pi_1^{\textup{dR}}}(X/{\mathbb{C}},\omega_x)$ the affine ${\mathbb{C}}$-groupscheme ${\textup{Aut}}^{\otimes}(\omega_x)$. The functor $\omega_x$ induces an equivalence $${\textup{D.E.}}(X)\rightarrow {\textup{Rep}}_{\mathbb{C}}({\pi_1^{\textup{dR}}}(X/{\mathbb{C}},\omega_x))$$ of the category of connections on $X$ with the category of finite dimensional complex representations of ${\pi_1^{\textup{dR}}}(X/{\mathbb{C}},\omega_x)$. For a connection ${\mathscr{E}}$ on $X$ denote by $\rho_{\mathscr{E}}:{\pi_1^{\textup{dR}}}(X/{\mathbb{C}},\omega_x))\rightarrow {\textup{GL}}(\omega({\mathscr{E}}))$ the associated representation. The *differential Galois group* ${\textup{DGal}}({\mathscr{E}},\omega)$ of ${\mathscr{E}}$ can be defined as the image of $\rho_{\mathscr{E}}$. In this case ${\textup{DGal}}({\mathscr{E}},\omega)$ is a Zariski closed subgroup of ${\textup{GL}}(\omega_x({\mathscr{E}}))$.
For a reductive algebraic group $G$ we will refer to homomorphisms $${\pi_1^{\textup{dR}}}(X,x)\rightarrow G({\mathbb{C}})$$ as *$G$-connections*. In particular, if ${\mathscr{E}}$ is a rank $n$ connection on $X$ corresponding to $${\pi_1^{\textup{dR}}}(X,x)\rightarrow {\textup{GL}}_n({\mathbb{C}}),$$ $G\subset {\textup{GL}}_n$ and if ${\textup{DGal}}({\mathscr{E}})\subset G$ there’s a factorization $$\xymatrix{{\pi_1^{\textup{dR}}}(X,x) \ar[rr]\ar[dr]^\rho & & {\textup{GL}}_n({\mathbb{C}}) \\
& G({\mathbb{C}}) \ar@{^{(}->}[ur]
},$$ i.e. ${\mathscr{E}}$ is a $G$-connection. Therefore for a representation $\mu: G\rightarrow {\textup{GL}}(V)$ of $G$ we get a new connection $\mu({\mathscr{E}})$ corresponding to the representation $$\mu\circ \rho:{\pi_1^{\textup{dR}}}(X,x)\rightarrow {\textup{GL}}(V).$$ Recall that a connection ${\mathscr{E}}$ on an open subset $j:U\hookrightarrow {\mathbb{P}}^1$ is called cohomologically rigid if ${\textup{rig\,}}({\mathscr{E}})=2$. By [@Arinkin10 3.4.], ${\textup{rig\,}}({\mathscr{E}})\le 2$ for any irreducible connection ${\mathscr{E}}$, meaning that cohomological rigidity is equivalent to $$H^1({\mathbb{P}}^1,j_{!*}({\mathscr{E}nd}({\mathscr{E}}))=0.$$ For $G={\textup{DGal}}({\mathscr{E}})$ semi-simple, if the connection ${\mathscr{E}}$ is cohomologically rigid, then $$H^1({\mathbb{P}}^1,j_{!*}{\textup{Ad}}({\mathscr{E}}))=0$$ where ${\textup{Ad}}:G\rightarrow {\textup{GL}}(\mathfrak{g})$ denotes the adjoint representation of $G$, cf. also [@Fr09 Section 7].
From now on let $K={\mathbb{C}(\!(t)\!)}$. Recall that ${\textup{D.E.}}(K)$ is the category of connections over $K$, i.e. finite dimensional $K$-vector spaces admitting an action of $K\langle\partial_t\rangle$. Similar to the category ${\textup{D.E.}}(X)$ there are natural notions of tensor product and internal hom in ${\textup{D.E.}}(K)$ turning ${\textup{D.E.}}(K)$ into a rigid abelian tensor category. Let $E$ be a $K$-connection, consider its Katz extension ${\mathcal{M}}_E$ to a connection on ${\mathbb{G}_m}$ and denote by $({\mathcal{M}}_E)_x$ its fibre above $x$ for any point $x\in {\mathbb{G}_m}({\mathbb{C}})$. In [@Ka87 II. ,2.4.] Katz proves that $$\omega_x:E\mapsto ({\mathcal{M}}_E)_x$$ is a ${\mathbb{C}}$-valued fibre functor. If we fix any such functor $\omega$ on ${\textup{D.E.}}(K)$ it induces an equivalence $${\textup{D.E.}}(K)\rightarrow {\textup{Rep}}_{\mathbb{C}}(I)$$ for $I={\textup{Aut}}^{\otimes}(\omega)$. We will call $I$ the *local differential Galois group*. In analogy to the case of ${\textup{D.E.}}(U)$ we define the local differential Galois group ${\textup{DGal}}_{loc}(E,\omega)$ of a connection $E$ to be the image of the associated representation $\rho_E:I\rightarrow {\textup{GL}}(\omega(E))$.
We have the *upper numbering filtration* on $I$ which is a decreasing filtration defined in the following way. For any $y\in{\mathbb{R}}_{>0}$ let ${\textup{D.E.}}^{(<y)}(K)$ be the full subcategory of ${\textup{D.E.}}(K)$ consisting of connections with slopes $<y$ and denote by $\omega^y$ the restriction of $\omega$ to ${\textup{D.E.}}^{(<y)}(K)$. Dual to these subcategories there are faithfully flat homomorphisms $$I\rightarrow {\textup{Aut}}^{\otimes}(\omega^y)$$ whose kernels are closed normal subgroups of $I$. We denote them by $I^{(x)}$. This defines a decreasing filtration on $I$ with the property that for any connection $E$ with slopes $<y$ the kernel of its associated representation $\rho_E:I\rightarrow {\textup{GL}}(\omega(E))$ contains $I^{(x)}$.
Let $X$ be a smooth proper complex connected curve, $\Sigma$ a finite set of closed points of $X$ and $U=X-\Sigma$. For any connection ${\mathscr{E}}$ on $U$ and any $x\in \Sigma$ denote by ${\mathscr{E}}_x=K \otimes {\mathscr{E}}$ its formal type at $x$. The functor $$\begin{aligned}
\tilde{\omega}:{\textup{D.E.}}(U)&\rightarrow {\textup{Vect}}_{\mathbb{C}}\\
{\mathscr{E}}&\mapsto \omega({\mathscr{E}}_x) \end{aligned}$$ defines a fibre functor. We will write $I_x={\textup{DGal}}_{loc}({\mathscr{E}}_x,\omega)$ for the local differential Galois group of ${\mathscr{E}}$ at $x$. The formal type functor ${\textup{D.E.}}(U)\rightarrow {\textup{D.E.}}(K)$ induces a closed immersion ${\textup{DGal}}_{loc}({\mathscr{E}}_x,\omega)\hookrightarrow {\textup{DGal}}({\mathscr{E}},\tilde{\omega})$. Over ${\mathbb{C}}$ any two fibre functors on either category of connections are isomorphic and we will fix the above fibre functor and drop $\omega$ in the notation of the local and the global differential Galois group. Therefore we can consider ${\textup{DGal}}_{loc}({\mathscr{E}}_x)$ as a closed subgroup of ${\textup{DGal}}({\mathscr{E}})$. This will allow us to deduce information about the differential Galois group of a connection from its formal type at the singularities.
The local differential Galois group can also be defined in the following way. Let $E$ be a ${\mathbb{C}(\!(t)\!)}$-connection and $\langle E\rangle$ the full subcategory of objects which are finite direct sums of sub-quotients of objects $$E^{\otimes n}\otimes (E^*)^{\otimes m}, m,n\in {\mathbb{Z}}_{\ge 0}.$$ The restriction of any fibre functor $\omega$ of ${\textup{D.E.}}(X)$ to $\langle E\rangle$ turns $\langle E\rangle$ into a neutral Tannakian category. In particular we have ${\textup{Aut}}^\otimes(\omega{_{|_{\langle E\rangle}}})={\textup{DGal}}_{{\textup {loc}}}({\mathscr{E}},\omega)$. This construction can be made more concrete when defining the differential Galois group in the classical way as the automorphism group of a Picard-Vessiot field for the differential equation defined by the connection $E$, cf. [@Si09]. Let $L$ be such a Picard-Vessiot field for $E$. The equivalence $$S: \langle E\rangle \rightarrow {\textup{Rep}}({\textup{DGal}}_{{\textup {loc}}}(E)).$$ is given by assigning to an object $E'$ of $\langle E\rangle$ its horizontal sections after base change to $L$, i.e. $S(E')=\ker(\partial_t,L\otimes E')$. The differential Galois group acts on the kernel and $v\in S(E)$ is invariant under the action of $I={\textup{DGal}}_{\textup {loc}}(E)$ if and only if $v$ is a horizontal section of $E$. Therefore instead of ${\textup{Soln}}(E)$ we will sometimes abuse notation and will also write $E^I$.
We are now in the position to compute the dimension of the local solution space of a ${\mathbb{C}(\!(t)\!)}$-connection $E$ given its Levelt-Turrittin decomposition. Note that $\dim{\textup{Soln}}(E)=\dim{\textup{Soln}}(E^{\textup{reg}})$ as any connection which is purely irregular has no horizontal sections over ${\mathbb{C}(\!(t)\!)}$ (otherwise it would contain the trivial connection). If $E$ has minimal Levelt-Turrittin decomposition $E=\bigoplus_i {\textup{El}}(\rho_i,\varphi_i,R_i)$, Sabbah shows in [@Sa08 3.13.] that $$\begin{aligned}
\label{centdecomp}{\textup{End}}(E)^{{\textup{reg}}}=\bigoplus_i\rho_{i,+}{\textup{End}}(R_i).\end{aligned}$$ A regular ${\mathbb{C}(\!(u)\!)}$-connection $R$ is completely determined by its nearby cycles $(\psi_uR,T)$ with monodromy $T$. Its push-forward along any $\rho\in u{\mathbb{C}}[[u]]$ of degree $p$ corresponds to the pair $(\psi_uR\otimes{\mathbb{C}}^p,\rho_+T)$ with $\rho_+T$ given by the Kronecker product $T^{1/p}\otimes P_p$. Here $T^{1/p}$ is a $p$-th root of $T$ and $P_p$ is the cyclic permutation matrix on ${\mathbb{C}}^p$. Let $V_R$ be the representation associated to $R$. We have $$\dim{\textup{Soln}}(\rho_+R)=\dim V_{\rho_{+}R}^{I}=\dim \ker(\rho_+T-{\textup{id}})=\dim\ker(T-{\textup{id}}).$$ In particular $$\begin{aligned}
\label{centdim}\tag{Z}
\dim{\textup{Soln}}(\rho_+{\textup{End}}(R))&=\dim \ker(\rho_+{\textup{Ad}}(T)-{\textup{id}}) \\
&=\dim\ker({\textup{Ad}}(T)-{\textup{id}}) \\
&=\dim\textup{Z}(T) \end{aligned}$$ where $\textup{Z}(T)$ is the centraliser of $T$. Combining this with Formula \[centdecomp\] allows us to compute $\dim{\textup{Soln}}(E)$ for any connection $E$ provided we know its Levelt-Turrittin decomposition.
On connections of type $G_2$ {#G2conn}
============================
In this section we will restrict ourselves to irreducible rigid connections ${\mathscr{E}}$ on non-empty open subsets of ${\mathbb{P}}^1$ of rank $7$ with differential Galois group ${\textup{DGal}}({\mathscr{E}})=G_2$ (where we fix the embedding $G_2\subset SO(7)\subset{\textup{GL}}_7$) and all of whose slopes are at most $1$. Regarding the restriction on the slopes consider the following example.
Let $f\in {\mathbb{C}}[z]$ be a polynomial of degree $k$ which is prime to $6$. Then by [@Ka90 Theorem 2.10.6] the module $$M={\mathbb{C}[z]\langle\partial_z\rangle}/(L), \, L=\partial_z^7-f\partial_z-\frac{1}{2}f'$$ on ${\mathbb{A}}^1_z$ is irreducible and has differential Galois group $G_2$. It has one singularity at $\infty$ of slope $1+\frac{k}{6}$ and its formal type $V_\infty$ at $\infty$ decomposes into $$R\oplus V_\infty(\frac{6+k}{6})$$ where $R$ is regular singular of rank $1$ and $V:=V_\infty(\frac{6+k}{6})$ is irreducible of rank $6$. By the Levelt-Turrittin theorem $$V\cong {\textup{El}}(u^6,\varphi(u),R')$$ for some regular singular rank one connection $R'$ and $$\varphi(u)=\sum_{i=1}^{k+6}a_iu^{-i}$$ with $a_{k+6}\neq 0$. According to Proposition \[elmthom\] we have $${\textup{End}}(V_\infty)\cong \bigoplus_{\zeta\in \mu_6({\mathbb{C}})} {\textup{El}}(u^6, \varphi(u)-\varphi(\zeta u),\tilde{R}) \oplus\left( V\otimes R^\vee\right) \oplus \left(V^\vee \otimes R\right)\oplus {\mathbbm{1}}$$ where $\tilde{R}$ is regular of rank one and ${\mathbbm{1}}$ is the trivial connection. Whenever the coefficient of the degree $k+6$-term of $\varphi(u)-\varphi(\zeta u)$ does not vanish, the module $${\textup{El}}(u^6, \varphi(u)-\varphi(\zeta u),\tilde{R})$$ has irregularity $k+6$. Since $k$ is prime to $6$, $a_{k+6}-\zeta^k a_{k+6}=0$ if and only if $\zeta=1$. In this case the above module is regular. In total we have $${\textup{irr}}({\textup{End}}(V_\infty))=7(k+6).$$ Since $M$ only has one singularity at $\infty$ its index of rigidity is $${\textup{rig\,}}(M)=49-{\textup{irr}}_\infty(M)+\dim V^{I_\infty}=49-7(k+1)+2$$ where we used that $\dim V^{I_\infty}=2$ because $V_\infty$ is the direct sum of two irreducible modules. We find that ${\textup{rig\,}}(M)=2$ if and only if $k=1$. Therefore the above family of modules is rigid only for $k=1$. In this case it has slope $1+\frac{1}{6} > 1$. This suggests that rigidity combined with a differential Galois group of type $G_2$ should give bounds on the slopes, but it’s not clear how these could be obtained. Recall that the index of rigidity remains unchanged by twist with a rank one connection and hence after twisting $M$ in the case $k=1$ with the connection $({\mathbb{C}}[z],d-dz^{q})$ for $q>2$ would increase the slope to $q$. But the so-obtained connection will not be self-dual anymore, so it cannot be of type $G_2$.
As connections with regular singularities of this type have already been classified by Dettweiler and Reiter, we will from now on assume that every irreducible rigid $G_2$-connection has at least one irregular singularity. We give a first approximation to the complete classification theorem of Section \[classif\].
We will use the following notations. By $\rho_p$ we always denote the ramification $\rho_p(u)=u^p$, $R_k$ is a regular ${\mathbb{C}(\!(u)\!)}$-connection of rank $k$ and $\varphi_q$ is a rational function of pole order $q$ at zero. A regular connection $R$ on the formal disc ${\textup{Spec\,}}{\mathbb{C}(\!(u)\!)}$ is determined by its monodromy which can be given as a single matrix in Jordan canonical form. Let $M$ be an $n\times n$-matrix and $R$ the connection with monodromy $M$. We sometimes write $${\textup{El}}(\rho_p,\varphi_q,M)$$ for the elementary module $\rho_{p,+}({\mathscr{E}}^\varphi \otimes R)$. By $x{\mathbf{J}}(n)$ we denote a Jordan block of length $n$ with eigenvalue $x$, i.e. ${\mathbf{J}}(n)$ is a unipotent Jordan block of length $n$. Additionally, $E_n$ is the identity matrix of length $n$. We will write $$(x_1{\mathbf{J}}(n_1),...,x_s{\mathbf{J}}(n_s)$$ for a complex matrix in Jordan canonical form with eigenvalues $x_1,...,x_s$ and we will omit ${\mathbf{J}}(1)$.
**Local structure**. Recall from Lemma \[rigidslope\] that any slope of an irreducible rigid $G_2$-connection has numerator $1$. Additionally, a strong condition on the formal types is given by the self-duality which they have to satisfy. As stated in Proposition \[emprop\], the dual of an elementary connection ${\textup{El}}(\rho_p,\varphi_q,R)$ is $${\textup{El}}(\rho_p,-\varphi_q,R^*).$$
Let ${\mathscr{E}}$ be an irreducible rigid $G_2$-connection. The regular part of the formal type at any singularity $x$ of ${\mathscr{E}}$ is of dimension $1$, $3$ or $7$.
Denote by $E$ the formal type of ${\mathscr{E}}$ at $x$ and write $E=E^{\textup{reg}}\oplus E^{\textup{irr}}$. This corresponds to a representation $\rho=\rho^{\textup{reg}}\oplus\rho^{\textup{irr}}$ of the local differential Galois group $I$ at $x$. First note that this representation has to be self-dual. We will show that purely irregular ${\mathbb{C}(\!(t)\!)}$-connections of odd dimension are never self-dual. Let $E$ be such a connection and write $$E=\bigoplus {\textup{El}}(p_i,\varphi_i,R_i)$$ for its minimal Levelt-Turrittin decomposition in which all the $\varphi_i$ are not in ${\mathbb{C}}[[t]]$. For the dimension of $E$ to be odd, at least on of the elementary connections has to be odd dimensional, write ${\textup{El}}(p,\varphi,R)$ for that one. It’s dual cannot appear in the above decomposition, as the dimension would not be odd in that case. So it suffices to prove that ${\textup{El}}(p,\varphi,R)$ itself is not self-dual. A necessary condition for its self-duality is $$\varphi\circ\mu_{\zeta_p}\equiv -\varphi \mod {\mathbb{C}}[[u]].$$ Write $\varphi(u)=\sum_{i\ge -k}a_iu^i$ for some $k\in {\mathbb{Z}}_{\ge 0}$. The above condition translates to $$\sum_{i\ge -k}a_i\zeta_p^i+u^i\sum_{i\ge -k}+a_iu^i \in {\mathbb{C}}[[u]].$$ Since $\varphi$ is supposed to be not contained in ${\mathbb{C}}[[u]]$ there is an index $j<0$ such that $a_{j}\neq 0$. In this case we find that $a_j\zeta_p^j+a_j=0$, i.e. $\zeta_p^j=-1$. This can only hold if $p$ is even and in this case the dimension of ${\textup{El}}(p,\varphi,R)$ could not be odd.
Therefore the dimension of the regular part of $E$ has to be odd. Denote as before by $I^{(x)}$ the upper numbering filtration on $I$ and let $n=\dim E^{\textup{reg}}$. The smallest non-zero slope of $E$ is $1/6$, so we find $$\rho|_{I^{(1/6)}}= \mathbbm{1}^n\oplus \rho^{\textup{irr}}|_{I^{(1/6)}}$$ where $\mathbbm{1}$ denotes the trivial representation of rank one. In the case $n=5$, the image of $\rho$ contains elements of the form $(E_5,M)$ where $M$ is non-trivial. By Table 4 in [@Dett10] such elements do not occur in $G_2({\mathbb{C}})$.
The following proposition is a special case of Katz’s Main D.E. Theorem [@Ka90 2.8.1].
Let ${\mathscr{E}}$ be an irreducible rigid connection on $U\subset {\mathbb{P}}^1$ of rank $7$ with differential Galois group $G_2$. If at some point $x\in {\mathbb{P}}^1-U$ the highest slope of ${\mathscr{E}}$ is $a/b$ with $a>0$ and if it occurs with multiplicity $b$, then $b=6$.
We will later see that rigid $G_2$-connections necessarily have exactly two singularities which we can choose to be zero and infinity. By a criterion of Katz, any system satisfying the conditions of the above proposition will then necessarily be hypergeometric.
One of the main ingredients in the proof Katz’s Main D.E. Theorem is the use of representation theory through Tannakian formalism as presented in the previous section. Applying the above Proposition (and self-duality) yields the following possible list for the slopes and the respective dimensions in the slope decomposition.
[c c c]{} slopes & dimensions &\
\
$1$ & $4$\
\[3pt\] $1$ & $6$\
\[3pt\] $\frac{1}{2}, 1$ & $2, 2$\
\[3pt\] $\frac{1}{2}, 1$ & $2, 4$\
\[3pt\] $\frac{1}{2}, 1$ & $4, 2$\
\[3pt\] $\frac{1}{2}$ & $4$\
\[3pt\] $\frac{1}{2}$ & $6$\
\[3pt\] $\frac{1}{3}$ & $6$\
\[3pt\] $\frac{1}{4}, 1$ & $4, 2$\
\[3pt\] $\frac{1}{6}$ & $6$\
\[3pt\]
For an elementary module ${\textup{El}}(u^p,\varphi,R)$ with $\varphi\in {\mathbb{C}(\!(u)\!)}$ we would like to describe the possible $\varphi$ more concretely. We have the following Lemma.
\[exptorusspecial\] The pole order of any $\varphi \in {\mathbb{C}(\!(u)\!)}$ appearing in the Levelt-Turrittin decomposition into elementary modules of the formal type of a rigid irreducible connection of type $G_2$ with slopes at most $1$ can only be $1$ or $2$.
Suppose ${\textup{El}}(u^p,\varphi,R)$ appears in the formal type of such a system. Because the slopes are at most $1$ and all have numerator $1$, we have the following possibilities for $p$ and $q$ apart from $q=1$.
------------- --------- --
$q$ $p$
\[3pt\]
$2$ $2,4,6$
\[5pt\] $3$ $3,6$
\[5pt\] $4$ $4$
\[5pt\] $6$ $6$
\[5pt\]
------------- --------- --
Note that in the cases $(q,p)=(6,6)$, $(q,p)=(4,4)$ and $(q,p)=(2,6)$, the module ${\textup{El}}(u^p,\varphi,R)$ cannot be self-dual. Indeed that would mean that $\varphi(\zeta u)=-\varphi(\zeta)$. Write $v=u^{-1}$. If $a_q$ denotes the coefficient of $v^q$ then the above condition means that $$a_q(\zeta u)^q=-a_qu^q,$$ i.e. $\zeta^q=-1$. This is a contradiction in these cases. The formal type of a connection of type $G_2$ has to be self-dual and therefore in the case that $q$ is even, the dual of ${\textup{El}}(\rho,\varphi,R)$ also has to appear in the formal type. If $p=4$ or $p=6$ this contradicts the fact that the rank of the connection is $7$. We are therefore left with the following cases.
------------- -------- --
$q$ $p$
\[3pt\]
$2$ $2, 4$
\[5pt\] $3$ $3,6$
\[5pt\]
------------- -------- --
We analyze these cases separately. Suppose first we’re in the case that $q=3$ and $p=6$. Then ${\textup{El}}(u^6,\varphi,R)$ is at least six dimensional, so $\dim R=1$ and the module has to be self-dual already. The isomorphism class of ${\textup{El}}(u^p,\varphi,R)$ depends only on the class of $\varphi$ mod ${\mathbb{C}}[[u]]$, hence we think of $\varphi$ as a polynomial in $1/u$. We can then write $$\varphi(v)=a_3v^3+a_2v^2+a_1v$$ and self-duality implies that there is a $6$-th root of unity $\zeta$ such that $$a_3\zeta^3v^3=-a_3v^3.$$ Because $q=3$, $a_3\neq 0$ and we get that $\zeta^3=-1$. We have $a_2\zeta^2v^2=-a_2v^2$ implying that $a_2=0$. Therefore $\varphi$ is of the form $$\varphi(v)=a_3v^3+a_1v.$$ In order to rule out this case we will need the exponential torus of an elementary module. Consider the module $E={\textup{El}}(\sigma_p,\psi,L)$. Because of \[emprop\], 5 the *exponential torus* of $E$ is the subgroup $\mathcal{T}$ of $({\mathbb{C}}^*)^p=\{(t_1,...,t_p)\}$ defined by $\prod \nu_it_i=0,\nu_i\in{\mathbb{Z}}$ for any relation of the form $$\prod \exp(\psi\circ\mu_{\zeta_p^i})^{\nu_i}=1$$ satisfied by the $\psi\circ\mu_{\zeta_p^i}$, see for example [@Zoladek06 Section 11.22.]. The exponential torus can be considered as a subgroup of the local differential Galois group of $E$, i.e. $\mathcal{T}\subset G_2$ is a necessary condition for ${\textup{DGal}}_{\textup {loc}}(E)\subset G_2$.
We claim that the torus attached to ${\textup{El}}(\rho,\varphi,R)$ for $\varphi(v)=a_3v^3+a_1v$ is three-dimensional. As the rank of $G_2$ is $2$, this means that no elementary module of this form can appear in any formal type.
If $a_1=0$, by [@Sa08 Rem. 2.8.] we have $${\textup{El}}(u^6,a_3u^{-3},R)\cong {\textup{El}}(u^2,a_3u^{-1},(u^3)_*R)$$ hence actually $q=1$ in this case. We can therefore assume that $a_1\neq 0$. Let $\zeta_6$ be a primitive $6$-th root of unity. We have to compute all relations of the form $$\sum_{i=0}^5 k_i(a_3\zeta_6^{-3i}u^{-3}+a_1\zeta^{-i}u^{-1})=0, k_i\in {\mathbb{Z}}.$$ Equivalently, we find all relations $$0=\sum_{i=0}^5 k_i(a_1\zeta_6^{-i} u^2+a_3\zeta_6^{-3i})=\sum_{i=0}^5k_i(a_1\zeta_6^{-i} u^2+(-1)^ia_3).$$ First note that $$(a_1\zeta_6^{-i} u^2+a_3\zeta_6^{-3i})+(a_1\zeta_6^{-(i+3)} u^2+a_3\zeta_6^{-3(i+3)})=0$$ for $i=0,1,2$. Therefore any element in the exponential torus is of the form $$(x,y,z,x^{-1},y^{-1},z^{-1}).$$ It therefore suffices to prove that there are no further relations between the first three summands. Suppose there is a relation $$0=k(a_1 u^2+a_3)+l(-a_1\zeta_6^2u^2-a_3)+m(-a_1\zeta_6u^2+a_3)$$ with $k,l,m\in {\mathbb{Z}}$. We find that $k=l-m$ and as $a_1\neq 0$ we conclude $$\begin{aligned}
0&=&l-m-\zeta_6^2l-\zeta_6m=l-m+\zeta_6^2m-\zeta_6m-\zeta_6^2l-\zeta_6^2m \\
&=&(\zeta_6^2-\zeta_6)m+l-m-(l+m)\zeta_6^2 \\
&=&l-2m-(l+m)\zeta_6^2, \end{aligned}$$ using that $\zeta_6^2-\zeta_6=-1$. Therefore $l=-m$ and $-3m=0$, i.e. $m=0$. Finally, the exponential torus is given as $$\mathcal{T}=\{(x,y,z,x^{-1},y^{-1},z^{-1})\}\in ({\mathbb{C}}^*)^6$$ which is three-dimensional. Therefore a module of the above shape cannot appear in the formal type.
The case $q=3$ and $p=3$ works similarly. We have $$\varphi(v)=a_3v^3+a_2v^2+a_1v$$ and if $a_2=a_1=0$ as before we have $${\textup{El}}(u^3,a_3u^{-3},R)\cong {\textup{El}}(u,a_3u^{-1},(u^3)_*R).$$ We can therefore assume that either $a_2\neq 0 $ or $a_1\neq 0$. Let $\zeta_3$ be a primitive $3$-rd root of unity. We analyze the exponential torus attached to ${\textup{El}}(u^3,\varphi,R)$, i.e. we find all relations $$\sum_{i=1}^3k_i(a_3u^{-3}+a_2\zeta_2^{-2+}u^{-2}+a_1\zeta_3^{-i}u^{-1})=0.$$ This gives us the following system of equations $$\begin{aligned}
a_1(k_1\zeta_3^2+k_2\zeta_3+k_3)&=0 \\
a_2(k_1\zeta_3+k_2\zeta_3^2+k_3)a_2&=0 \\
a_3(k_1+k_2+k_3)&=0.\end{aligned}$$ As $a_3\neq 0$ we get $k_1=-(k_2+k_3)$. Now suppose that $a_1\neq 0$. We find that $$k_1\zeta_3^2+k_2\zeta_3+k_3=0$$ and we have $$k_1\zeta_3^2+k_2\zeta_3+k_3=-(k_2+k_3)\zeta_3^2+k_2\zeta_3+k_3=k_2(\zeta_3-\zeta_3^2)+k_3(1-\zeta_3^2).$$ Since $(\zeta_3-\zeta_3^2)=i\sqrt{3}$ and $1-\zeta_3^2=\frac{3}{2}+i\frac{\sqrt{3}}{2}$ we furthermore find that $$k_3\frac{3}{2}+i\sqrt{3}(\frac{1}{2}k_3+k_2)=0.$$ Hence $k_3=k_2=k_1=0$ and the exponential torus has to be three-dimensional. The case $a_2\neq 0$ is similar.
Finally we also exclude the case $q=2$ and $p=4$. We consider a module of the form $${\textup{El}}(u^4,a_2u^{-2}+a_1u^{-1},R).$$ Because of dimensional reasons, $R$ has dimension $1$ and the above module has to be self-dual. Since $q=2$ we have $a_2\neq 0$. Therefore for self-duality we have the condition $-a_2=\zeta^{-2}a_2$ from which it follows that $\zeta=\pm i$. In addition we also have $-a_1=\zeta^{-1}a_1$ which since $\zeta=\pm i$ can only be true if $a_1=0$. Finally as before we find $${\textup{El}}(u^4,a_2u^{-2}+a_1u^{-1},R)={\textup{El}}(u^4,a_2u^{-2},R)\cong {\textup{El}}(u^2, a_2u^{-1}, (u^2)_*R).$$ This concludes the proof.
We see that only the case $p=2$ and $q=2$ needs to be considered. The possible combinations of elementary modules in this case are either $$\label{spcase1}\tag{S1}{\textup{El}}(\rho_2,\varphi_2,R_1)\oplus {\textup{El}}(\rho_2,-\varphi_2,R_1^*)\oplus R_3$$ or $$\label{spcase2}\tag{S2}{\textup{El}}(\rho_2,\varphi_2,R_1)\oplus {\textup{El}}(\rho_2,-\varphi_2,R_1^*)\oplus {\textup{El}}(\rho_2,\varphi_1,R_1')\oplus R''_1.$$
We can compute the irregularity and the dimension of the solution space in these cases through the use of Proposition \[elmthom\] and Formula \[centdecomp\]. Using the formula $\dim{\textup{Soln}}(\rho_+{\textup{End}}(R))=\dim\textup{Z}(T)$ from the end of Section \[tannaka\] we find that $\dim{\textup{Soln}}\in \{5,7,11\}$ and using the formulae of Section \[rigloc\] we find that ${\textup{irr}}=20$ in the first case and $\dim{\textup{Soln}}=4$ and ${\textup{irr}}=39$ in the second case. Apart from these two special cases, all elementary modules appearing are of the form $${\textup{El}}(\rho_p,\frac{\alpha}{u}, R_k)$$ with $\alpha \in {\mathbb{C}}$. In this setting we can compute the dimension of the local solution space and its irregularity in the same way as we did for the two cases above. This yields the following table of possible combinations for the local invariants at irregular singularities.
[c c c c]{} \[locinvariants\] slopes & dimensions & $\dim{\textup{Soln}}({\mathscr{E}nd})$ & ${\textup{irr}}({\mathscr{E}nd})$\
\
$1$ & $4$ & $5,7,9,11,13,17$ & 32, 36\
\[3pt\] $1$ & $6$ & $7,9,11,13,15,19$ & 30, 38, 42\
\[3pt\] $\frac{1}{2}, 1$ & $2, 2$ & $7,9,11,13,15$ & 29\
\[3pt\] $\frac{1}{2}, 1$ & $2, 4$ & $4, 6,10$ & 37, 39\
\[3pt\] $\frac{1}{2}, 1$ & $4, 2$ & $5,7$ & 30, 32\
\[3pt\] $\frac{1}{2}$ & $4$ & $5,7,9,11,13$ & 16, 18\
\[3pt\] $\frac{1}{2}$ & $6$ & $4,6,10$ & 15, 19, 21\
\[3pt\] $\frac{1}{3}$ & $6$ & $3$ & 12, 14\
\[3pt\] $\frac{1}{4}, 1$ & $4, 2$ & $4$ & 27\
\[3pt\] $\frac{1}{6}$ & $6$ & $2$ & 7\
\[3pt\]
**Global structure**. Recall that the connection ${\mathscr{E}}$ is rigid if and only if ${\textup{rig\,}}({\mathscr{E}})=2$ where $${\textup{rig\,}}({\mathscr{E}})=\chi({\mathbb{P}}^1, j_{!*}({\mathscr{E}nd}({\mathscr{E}})))$$ is the index of rigidity. If we denote by $x_1,...,x_r$ the singularities of ${\mathscr{E}}$, the index of rigidity is given by $${\textup{rig\,}}({\mathscr{E}})=(2-r)49-\sum_{i=1}^r {\textup{irr}}_{x_i}({\mathscr{E}nd}({\mathscr{E}}))+\sum_{i=1}^r \dim_{\mathbb{C}}{\textup{Soln}}_{x_i}({\mathscr{E}nd}({\mathscr{E}})).$$
\[numbsings\] Let ${\mathscr{E}}$ be an irreducible rigid $G_2$-connection on $U\subset {\mathbb{P}}^1$ with singularities $x_1,...,x_r$ of slopes at most $1$. Then $2\le r\le 4$.
By Table 1 in [@Dett10] and by the table above we find that in any case $$\dim_{\mathbb{C}}{\textup{Soln}}_{x_i}({\mathscr{E}nd}({\mathscr{E}})) \le 29.$$ As ${\mathscr{E}}$ is rigid, we have $$2=(2-r)49-\sum_{i=1}^r {\textup{irr}}_{x_i}({\mathscr{E}nd}({\mathscr{E}}))+\sum_{i=1}^r \dim_{\mathbb{C}}{\textup{Soln}}_{x_i}({\mathscr{E}nd}({\mathscr{E}})).$$ Therefore we get $$2+(r-2)49+\sum_{i=1}^r {\textup{irr}}_{x_i}({\mathscr{E}nd}({\mathscr{E}})) \le 29r$$ and as ${\textup{irr}}_{x_i}({\mathscr{E}nd}({\mathscr{E}}))\ge 0$ we conclude $20r-96\le 0$. This cannot hold for $r\ge 5$. If $r=1$, the first equality above shows ${\textup{irr}}_{x_1}\ge 47$ which again cannot hold by the table above.
Let ${\mathscr{E}}$ be an irreducible rigid $G_2$-connection with singularities $x_1,...,x_r$ for $r\in \{2,3,4\}$. We define $R({\mathscr{E}})$ to be the tuple $$(s_1,...,s_r,z_1,...,z_r)\in {\mathbb{Z}}_{\ge 0}^{2r}$$ with $s_i={\textup{irr}}_{x_i}({\mathscr{E}nd}({\mathscr{E}}))$ and $z_i=\dim_{\mathbb{C}}{\textup{Soln}}_{x_i}({\mathscr{E}nd}({\mathscr{E}}))$. The necessary condition on $R({\mathscr{E}})$ for ${\mathscr{E}}$ to be rigid is $$2=(2-r)49-\sum_{i=1}^r s_i+\sum_{i=1}^r z_i.$$ This condition provides the following list of possible invariants in the cases $r=2$ and $r=3$. Additionally, one finds that no cases with $r=4$ appear.
[c]{} $r=3$\
\
$(0, 0, 16, 25, 29, 13)$\
$(0, 0, 16, 29, 29, 9)$\
$(0, 0, 18, 29, 29, 11)$\
[c c c]{} &$r=2$&\
\
$(0, 7, 7, 2)$ & $(0, 18, 13, 7)$ & $(0, 30, 25, 7)$\
$(0, 14, 13, 3)$ & $(0, 19, 11, 10)$ & $(0, 32, 25, 9)$\
$(0, 15, 7, 10)$ & $(0, 19, 17, 4)$ & $(0, 32, 29, 5)$\
$(0, 15, 11, 6)$ & $(0, 21, 13, 10)$ & $(0, 36, 25, 13)$\
$(0, 15, 13, 4)$ & $(0, 21, 17, 6)$ & $(0, 36, 29, 9)$\
$(0, 16, 7, 11)$ & $(0, 21, 19, 4)$ & $(0, 37, 29, 10)$\
$(0, 16, 9, 9)$ & $(0, 27, 25, 4)$ & $(0, 38, 25, 15)$\
$(0, 16, 11, 7)$ & $(0, 30, 13, 19)$ & $(0, 38, 29, 11)$\
$(0, 16, 13, 5)$ & $(0, 30, 17, 15)$ & $(0, 42, 29, 15)$\
$(0, 18, 9, 11)$ & $(0, 30, 19, 13)$ &\
Note that the two special cases (\[spcase1\]) and (\[spcase2\]) with $q=2$ do not appear. We can therefore classify the appearing elementary modules ${\textup{El}}(\rho_p,\varphi,R)$ by their ramification degree $p$, the coefficient $\alpha$ of $\varphi=\frac{\alpha}{u}$ and the monodromy of $R$. Now we can actually deal with the case $r=3$ by a case-by-case analysis using the Katz-Arinkin algorithm.
$\mathbf{(0, 0, 16, 25, 29, 13)}$. According to Table \[locinvariants\], the formal type at the irregular singularity has a $4$-dimensional part of slope $1/2$ and a $3$-dimensional regular part. In this case, the only possibility for the formal type is $${\textup{El}}(\rho_2,\alpha/u,\pm E_2)\oplus (\pm E_3).$$ Since $G_2\subset SO(7)$ this formal type has to have a trivial determinant. By \[emprop\], the regular part has to be $(E_3)$. Assume there exists a connection ${\mathscr{E}}$ on ${\mathbb{P}}^1-\{0,1,\infty\}$ with the above formal type at $\infty$ and local monodromy $(-E_4,E_3)$ and $({\mathbf{J}}(2),{\mathbf{J}}(2), E_3)$ at $0$ and $1$ respectively. The rank of the Fourier transform is computed according to Lemma \[fourierrank\] as $$h({\mathscr{F}}({\mathscr{E}}))=\,^0\mu_{0,-1,0}+\,^0\mu_{1,1,0}=\,^0\nu_{0,-1,0}+\,^0\nu_{1,1,1}=4+2=6.$$ After Fourier transform the formal type at $0$ will be of the form $${\textup{El}}(\rho_1,\widehat{\alpha}/u, \pm E_2)\oplus {\mathbf{J}}(2)^3$$ which has rank $8$. This is a contradiction and we can exclude this case.
$\mathbf{(0, 0, 16, 29, 29, 9)}$. The formal type has to be of the form $${\textup{El}}(\rho_2,\alpha/u,\pm E_2)\oplus (1,{\mathbf{J}}(2))$$ or of the form $${\textup{El}}(\rho_2,\alpha/u,\pm E_2)\oplus (-E_2,1)$$ and the same argument as above rules out both of these cases.
$\mathbf{(0, 0, 18, 29, 29, 11)}$. In this case the formal type at the irregular singularity is of the form $${\textup{El}}(\rho_2,\alpha/u,\pm 1)\oplus{\textup{El}}(\rho_2,\beta/u,\pm 1)\oplus (E_3)$$ with $\alpha\neq \beta,-\beta$ and we can again apply the same reasoning as in the first case.
We can therefore focus on the case $r=2$. A more thorough analysis of the shape of the elementary modules in question (applying the various criteria used up until now) shows that actually there are cases in which the irregularity $s_2$ does not occur with the local solution dimension $z_2$. After ruling these out we’re left with the following list of tuples $R({\mathscr{E}})$.
[c c c]{} &$r=2$&\
\
$(0, 7, 7, 2)$ & $(0, 16, 13, 5)$ & $(0, 32, 25, 9)$\
$(0, 14, 13, 3)$ & $(0, 18, 9, 11)$ & $(0, 32, 29, 5)$\
$(0, 15, 7, 10)$ & $(0, 18, 13, 7)$ & $(0, 36, 25, 13)$\
$(0, 15, 11, 6)$ & $(0, 19, 17, 4)$ & $(0, 36, 29, 9)$\
$(0, 15, 13, 4)$ & $(0, 21, 19, 4)$ & $(0, 37, 29, 10)$\
$(0, 16, 7, 11)$ & $(0, 27, 25, 4)$ & $(0, 38, 29, 11)$\
$(0, 16, 9, 9)$ & $(0, 30, 13, 19)$ &\
$(0, 16, 11, 7)$ & $(0, 30, 25, 7)$ &\
We would like to rule out further cases by computing the formal monodromy of the irregular formal type. For its definition in the general setting we refer to [@Mitschi96 Section 1]. We will describe how to compute the formal monodromy of an elementary connection ${\textup{El}}(\rho,\varphi,R)$ where $\rho$ has degree $p$ and $R$ is a regular connection. We can choose a connection $R^{1/p}$ such that $\rho^+R^{1/p}\cong R$ (this boils down to choosing a $p$-th root of the monodromy associated to $R$). Now $${\textup{El}}(\rho,\varphi,R)=\rho_+({\mathscr{E}}^\varphi\otimes \rho^+R^{1/p})\cong \rho_+{\mathscr{E}}^\varphi \otimes R^{1/p}$$ by the projection formula. Therefore by Proposition \[emprop\] (5), the differential equation associated to this elementary module has a formal solution of the form $$Y(t)=x^L e^{Q(t)}$$ where $x=t^p$, $Q(t)=\textup{diag}(\varphi(t), \varphi(\zeta_p t),...,\varphi(\zeta_p^{p-1}t))$ for a primitive $p$-th root of unity $\zeta_p$ and $L\in \textup{Mat}_n({\mathbb{C}})$. The formal monodromy $M$ is defined such that $YM$ is the solution obtained by formal counter-clockwise continuation of $Y$ around $0$, see [@vdP03 Chapter 3].
In the special case that $\varphi(t)=\alpha/t$ and $R$ is of rank one and corresponds to the monodromy $\lambda$, the formal monodromy is given as follows. Let $\lambda^{1/p}$ be a $p$-th root of $\lambda$ and choose $\mu$ such that $exp(2\pi i \mu)=\lambda^{1/p}$. The formal solution from above takes the form $$Y(t)=x^\mu e^{Q(t)}$$ and the action of the formal monodromy sends $Y(t)$ to $\lambda^{1/p}x^\mu e^{\tilde{Q}(t)}$ where $$\tilde{Q}(t)=\textup{diag}(\varphi(\zeta_pt), \varphi(\zeta_p^2 t),...,\varphi(\zeta_p^{p-1}t),\varphi(t)).$$ Therefore in addition to multiplication by $\lambda^{1/p}$ the formal monodromy permutes the basis of the solution space, i.e. $M=\lambda^{1/p}P_p$ where $P_p$ denotes as before the cyclic permutation matrix. We will compute one example to show how to apply this discussion.
$\mathbf{(0, 16, 9, 9).}$ The formal type at the irregular singularity has to be of the form $${\textup{El}}(\rho_2,\alpha,R)\oplus ({\mathbf{J}}(2), 1)$$ or of the form $${\textup{El}}(\rho_2,\alpha,R)\oplus (-E_2, 1)$$ where the connection $R$ corresponds to either $E_2$ or $-E_2$. In the first case we find that by the above discussion the formal monodromy is of the form $(E_2,-E_2,{\mathbf{J}}(2),1)$ or of the form $(iE_2,-iE_2,{\mathbf{J}}(2),1)$ both of which do not lie in $G_2({\mathbb{C}})$. In the second case suppose that there exists a connection ${\mathscr{E}}$ on ${\mathbb{G}_m}$ with the above formal type at $\infty$. The possibilities for the monodromy at $0$ are $(-{\mathbf{J}}(3),{\mathbf{J}}(3),-1)$, $(i{\mathbf{J}}(2),-i{\mathbf{J}}(2),-E_2,1)$ or $(x, -1, , -x, 1, -x^{-1},-1,x^{-1})$ where $x^4\neq 1$. In all these cases we compute $$h({\mathscr{F}}({\mathscr{E}}\otimes {\mathscr{L}}))=5$$ where ${\mathscr{L}}$ is the rank one system with monodromy $-1$ at $0$ and $\infty$. But the formal type at $0$ of ${\mathscr{F}}({\mathscr{E}}\otimes {\mathscr{L}})$ would be of rank $7$. Therefore this case cannot occur.
All cases apart from the ones in the following list can be excluded by a combination of all the criteria we’ve used so far. We obtain constraints on the formal type at $\infty$ and can apply the Katz-Arinkin algorithm to obtain contradictions.
[c]{} $r=2$\
\
$(0, 7, 7, 2)$\
$(0, 14, 13, 3)$\
$(0, 19, 17, 4)$\
$(0, 21, 19, 4)$\
Note that it might not suffice to simply apply one operation and compute the rank. We give an example of a case in which the computations are more complicated.
$\mathbf{(0,38,29,11)}$. The monodromy at $0$ is $({\mathbf{J}}(2),{\mathbf{J}}(2),E_3)$ and the formal type at $\infty$ has to be of the form $${\textup{El}}(\rho_1,\alpha,\lambda E_2)\oplus{\textup{El}}(\rho_1,-\alpha,\lambda^{-1} E_2)\oplus{\textup{El}}(\rho_1,2\alpha,\mu)\oplus{\textup{El}}(\rho_1,-2\alpha,\mu^{-1})\oplus (1).$$ Suppose there exists an irreducible connection ${\mathscr{E}}$ on ${\mathbb{G}_m}$ with this formal type. We will apply Fourier transforms, twists and middle convolution to the connection ${\mathscr{E}}$ to arrive at a contradiction.
Recall that ${\mathscr{F}}$ denotes the Fourier transform of connections and that ${\textup{MC}}_\chi$ is the middle convolution with respect to the Kummer sheaf ${\mathcal{K}}_\chi$. Let $\alpha_1,...,\alpha_r\in {\mathbb{C}}^*$ such that $\alpha_1\cdot ...\cdot \alpha_r=1$. We denote by ${\mathscr{L}}_{(\alpha_1,...,\alpha_{r+1})}$ the rank one connection on ${\mathbb{P}}^1-\{x_1,...,x_r\}$ with monodromy $\alpha_i$ at $x_i$. For ease of notation we will write $(\alpha_1,...,\alpha_r)\otimes -$ for the twist ${\mathscr{L}}_{(\alpha_1,...,\alpha_{r+1})}\otimes -$.
We compute the change of local data in the following scheme in which we write the operation used in the first column and the formal type at the singularities in the other columns.
The way the data changes is given by the explicit formulas of Section \[KA-alg\]. The $i$-th line is the result of applying the operation in the $(i-1)$-th line to the system in the $(i-1)$-th line. Writing $-$ in a column of a singularity means that this point is not singular.
In the last row we obtain a contradiction as the rank of the system is $1$, but its monodromy at $2\alpha$ resp. $-2\alpha$ is ${\mathbf{J}}(2)$.
In the next section we will construct irreducible rigid $G_2$-connections in the four cases that are left which leads to the proof of the complete classification theorem for irreducible rigid irregular $G_2$-connections.
Classification of irreducible irregular rigid $G_2$-connections {#classif}
===============================================================
To state our main result we will use the following notation. Similarly to the notation used before we will write $${\textup{El}}(p,\alpha,M)$$ for the elementary module $\rho_{p,+}({\mathscr{E}}^{\frac{\alpha}{u}}\otimes R)$ where $R$ is the connection on ${\textup{Spec\,}}{\mathbb{C}(\!(u)\!)}$ with monodromy $M$.
Let $\alpha_1,\alpha_2,\lambda,x,y,z\in {\mathbb{C}}^*$ such that $\lambda^2\neq 1, \alpha_1\neq \pm \alpha_2, z^4\neq 1$ and such that $x,y, xy$ and their inverses are pairwise different and let $\varepsilon$ be a primitive third root of unity. Every formal type occuring in the following list is exhibited by some irreducible rigid connection of rank $7$ on ${\mathbb{G}_m}$ with differential Galois group $G_2$.
[ c c ]{} $0$ & $\infty$\
\
$({\mathbf{J}}(3),{\mathbf{J}}(3),1)$ &
-----------------------------------------------------
${\textup{El}}( 2,\alpha_1,(\lambda,\lambda^{-1}))$
$\oplus\, {\textup{El}}(2,2\alpha_1,1) \oplus (-1)$
-----------------------------------------------------
\
\[15pt\] $(-{\mathbf{J}}(2),-{\mathbf{J}}(2),E_3) $ &
-----------------------------------------------------
${\textup{El}}( 2,\alpha_1,(\lambda,\lambda^{-1}))$
$\oplus\, {\textup{El}}(2,2\alpha_1,1)\oplus (-1)$
-----------------------------------------------------
\
\[15pt\] $(xE_2,x^{-1}E_2,E_3)$ &
-----------------------------------------------------
${\textup{El}}( 2,\alpha_1,(\lambda,\lambda^{-1}))$
$\oplus\, {\textup{El}}(2,2\alpha_1,1)\oplus (-1)$
-----------------------------------------------------
\
\[15pt\]\
$({\mathbf{J}}(3),{\mathbf{J}}(2), {\mathbf{J}}(2))$ &
------------------------------------------------------------------
${\textup{El}}(2,\alpha_1,1) \oplus {\textup{El}}(2,\alpha_2,1)$
$\oplus\,{\textup{El}}(2,\alpha_1+\alpha_2,1) \oplus (-1)$
------------------------------------------------------------------
\
\[15pt\]\
$(iE_2,-iE_2,-E_2,1)$ &
-------------------------------------------------
${\textup{El}}(3,\alpha_1,1)$
$\oplus\,{\textup{El}}(3,-\alpha_1,1)\oplus(1)$
-------------------------------------------------
\
\[15pt\]\
${\mathbf{J}}(7)$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\] $(\varepsilon{\mathbf{J}}(3), \varepsilon^{-1}{\mathbf{J}}(3),1)$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\] $(z{\mathbf{J}}(2), z^{-1}{\mathbf{J}}(2), z^2, z^{-2},1)$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\] $(x{\mathbf{J}}(2),x^{-1}{\mathbf{J}}(2), {\mathbf{J}}(3))$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\] $(x,y,xy,(xy)^{-1},y^{-1},x^{-1}, 1)$ & ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$\
\[10pt\]
Conversely, the above list exhausts all possible formal types of irreducible rigid irregular $G_2$-connections on open subsets of ${\mathbb{P}}^1$.
We give the construction for the four different cases. When varying the monodromy at zero in the same case, the construction is essentially the same up to twists with rank one systems. Therefore we only give the construction of the first system in every case. We denote these by ${\mathscr{E}}_1,...,{\mathscr{E}}_4$. Let $\mathscr{G}$ denote any operation on connections. We write $\mathscr{G}^k, k\in {\mathbb{Z}}_{>0}$, for its $k$-fold iteration.\
**Construction of ${\mathscr{E}}_1$.** Consider the connection ${\mathscr{L}}_1:={\mathscr{L}}_{(\lambda^{-1},-\lambda,\lambda^{-1},-\lambda)}$ on ${\mathbb{P}}^1-\{0,\frac{1}{4}\alpha_1^2, \alpha_1^2,\infty\}$ and the Möbius transform $\phi:{\mathbb{P}}^1\rightarrow {\mathbb{P}}^1, z\mapsto \frac{1}{z}$. Recall that ${\mathscr{F}}$ denotes the Fourier transform of connections. Our claim is that $${\mathscr{E}}_1:={\mathscr{F}}(\phi^*({\mathscr{F}}((1,-\lambda^{-1},1,-\lambda)\otimes {\textup{MC}}_{-\lambda}({\mathscr{L}}_1))))$$ has the formal type $({\mathbf{J}}(3),{\mathbf{J}}(3),1)$ at $0$ and $${\textup{El}}(2,\alpha_1,(\lambda,\lambda^{-1})) \oplus {\textup{El}}(2,2\alpha_1,1)\oplus (-1)$$ at $\infty$. Similar to before we compute the change of local data under the operations above in the following scheme.
By Proposition \[emprop\], 4, the connection $${\textup{El}}(\frac{4}{\alpha_1^2}u^2,\frac{\alpha_1^2}{2u}, (\lambda,\lambda^{-1}))\oplus {\textup{El}}(\frac{1}{\alpha_1^2}u^2,\frac{2\alpha_1^2}{u},1)\oplus (-1)$$ is isomorphic to $${\textup{El}}( 2,\alpha_1,(\lambda,\lambda^{-1})) \oplus {\textup{El}}(2,2\alpha_1,1)\oplus (-1).$$ This proves existence of the first type of connection.\
**Construction of ${\mathscr{E}}_2$.** For the second formal type at infinity, consider the connection ${\mathscr{L}}_2:={\mathscr{L}}_{(-1,-1,-1,-1,1)}$ on ${\mathbb{P}}^1-\{0,\frac{1}{4}\alpha_1^2,\frac{1}{4}\alpha_2^2,\frac{1}{4}(\alpha_1+\alpha_2)^2,\infty \}$. The connection $${\mathscr{E}}_2:={\mathscr{F}}(\phi^*{\mathscr{F}}({\mathscr{L}}_2)))$$ has the desired formal type $({\mathbf{J}}(3),{\mathbf{J}}(2), {\mathbf{J}}(2))$ at $0$ and $${\textup{El}}(2,\alpha_1,1) \oplus {\textup{El}}(2,\alpha_2,1)\oplus{\textup{El}}(2,\alpha_1+\alpha_2,1) \oplus (-1)$$ at $\infty$. The computation works the same way as before.\
**Construction of ${\mathscr{E}}_3$.** For the third type consider the connection $${\mathscr{L}}_3:= {\mathscr{L}}_{(-i,-\lambda,-\lambda^{-1},i)}$$ on ${\mathbb{P}}^1-\{0,\frac{1}{27}\alpha_1^3,-\frac{1}{27}\alpha_1^3,\infty \}$. The system $${\mathscr{E}}_4:={\mathscr{F}}(\phi^*((-1,-1)\otimes {\mathscr{F}}((i,-i)\otimes \phi^*({\mathscr{F}}((i,1,1,-i)\otimes {\textup{MC}}_i({\mathscr{L}}_3))))))$$ has the required formal type.\
**Construction of ${\mathscr{E}}_4$.** For the final type we consider ${\mathscr{L}}_4:={\mathscr{L}}_{(-1,-1,1)}$ on ${\mathbb{P}}^1-\{0,\frac{1}{6^6}\alpha_1^6,\infty \}$. The formal type ${\mathbf{J}}(7)$ at $0$ and ${\textup{El}}(6,\alpha_1,1)\oplus (-1)$ at $\infty$ is then exhibited by the connection $${\mathscr{E}}_4={\mathscr{F}}((\phi^*\circ {\mathscr{F}})^5({\mathscr{L}}_4)).$$ **The differential Galois groups.** We compute the differential Galois group $G$ of the above types using an argument of Katz from [@Ka90 §4.1.]. Note that all formal types are self-dual. Thus for $i=1,...,4$ we have that $$\Psi_x({\mathscr{E}}_i)\cong \Psi_x({\mathscr{E}}_i^*)$$ for $x=0,\infty$ and by rigidity we get ${\mathscr{E}}_i\cong {\mathscr{E}}_i^*$, i.e. all the above systems are globally self-dual. In addition the determinants are trivial meaning that actually $G\subset SO(7)$. We will focus first on the cases $i=1,2,3$. Let $G^0$ denote the identity component of $G$. By the proof of [@Ka12 25.2] there are now only three possibilities for $G^0$ which are $SO(7)$, $G_2$ or $SL(2)/\pm 1$. Since all these groups are their own normalizers in $SO(7)$ in all cases we find that $G=G^0$. We now only have to exclude the cases $G=SO(7)$ and $G=SL(2)/\pm 1$. First suppose that $G=SL(2)/\pm 1\cong SO(3)$.\
The group $SO(3)$ admits a faithful 3-dimensional representation $$\rho:SO(3)\rightarrow {\textup{GL}}(V).$$ Recall that $\rho({\mathscr{E}}_i)$ is the connection associated to the representation $${\pi_1^{\textup{dR}}}({\mathbb{G}_m},1)\rightarrow SO(3)\rightarrow {\textup{GL}}(V).$$ The connection $\rho({\mathscr{E}}_i)$ is a $3$-dimensional irreducible connection with slopes $\le 1/2$ at $\infty$ and which is regular singular at $0$. We have ${\textup{irr}}(\rho({\mathscr{E}}_i))\le 3/2$ and so either ${\textup{irr}}_\infty(\rho({\mathscr{E}}_i))=0$ or ${\textup{irr}}_\infty(\rho({\mathscr{E}}_i)))=1$. In the first case we have $${\textup{rig\,}}(\rho({\mathscr{E}}_i))=\dim ({\mathscr{E}nd}(\rho({\mathscr{E}}_i))^{I_0})+\dim ({\mathscr{E}nd}(\rho({\mathscr{E}}_i))^{I_\infty})\ge 6$$ which is a contradiction (recall that for any irreducible connection ${\mathscr{E}}$ on some open subset $U$ of ${\mathbb{P}}^1$ we always have ${\textup{rig\,}}({\mathscr{E}})\le 2$).\
In the second case, the formal type at $\infty$ of $\rho({\mathscr{E}}_i)$ has to be of the form $${\textup{El}}(2,\alpha,1)\oplus (-1)$$ and we compute $${\textup{rig\,}}(\rho({\mathscr{E}}_i))=\dim {\textup{End}}(\rho({\mathscr{E}}_i))^{I_0}+2-1\ge 4$$ which again yields a contradiction.\
Now we’re left with the cases $G=SO(7)$ and $G=G_2$. Recall that the third exterior power of the standard representation of $SO(7)$ is irreducible, so it suffices to prove that $G$ has a non-zero invariant in the third exterior power of its $7$-dimensional standard representation. In our case this amounts to finding horizontal sections of $\Lambda^3{\mathscr{E}}_i$ for $i=1,2,3$, i.e. we have to show that $H^0({\mathbb{G}_m}, \Lambda^3{\mathscr{E}}_i)\neq 0$ or equivalently by duality that $H^2_c({\mathbb{G}_m}, \Lambda^3 {\mathscr{E}}_i)\neq 0$. For this it suffices to prove that $$\chi({\mathbb{P}}^1,j_{!*}\Lambda^3{\mathscr{E}}_i)>0.$$ Recall that $$\chi({\mathbb{P}}^1,j_{!*}\Lambda^3{\mathscr{E}}_i)=\dim(\Lambda^3{\mathscr{E}}_i)^{I_0}+\dim(\Lambda^3{\mathscr{E}}_i)^{I_\infty}-{\textup{irr}}_\infty(\Lambda^3{\mathscr{E}}_i)$$ as $0$ is a regular singularity. These invariants can be computed using Sabbah’s formula for the determinant of elementary connections in Proposition \[emprop\], 3. For $i=1$, we have $$\begin{aligned}
&\, \Lambda^3({\textup{El}}(2,\alpha_1,\lambda)\oplus{\textup{El}}(2,\alpha_1,\lambda^{-1})\oplus({\textup{El}}(2,2\alpha_1,1)\oplus(-1)) \\
&=({\textup{El}}(2,\alpha_1,\lambda)\otimes \det{\textup{El}}(2,\alpha_1,\lambda^{-1})) \oplus (\det{\textup{El}}(2,\alpha_1,\lambda)\otimes{\textup{El}}(2,\alpha_1,\lambda^{-1})) \\
&\oplus (\det{\textup{El}}(2,\alpha_1,\lambda^{-1})\oplus({\textup{El}}(2,\alpha_1,\lambda)\otimes{\textup{El}}(2,\alpha_1,\lambda^{-1}))\oplus\det{\textup{El}}(2,\alpha_1,\lambda)) \\
&\otimes ((-1)\oplus{\textup{El}}(2,2\alpha_1,1)) \\
&\oplus({\textup{El}}(2,\alpha_1,\lambda^{-1})\oplus{\textup{El}}(2,\alpha_1,\lambda))\otimes (({\textup{El}}(2,2\alpha_1,1)\otimes (-1))\oplus \det({\textup{El}}(2,2\alpha_1,1)) \\
&\oplus(\det{\textup{El}}(2,2\alpha_1,1)\otimes (-1))\end{aligned}$$ As the slopes in our case are of the form $1/p$ with $p>1$ all occuring determinant connections are regular. Therefore the irregularity of this connection is $13$. Since $$\det{\textup{El}}(2,2\alpha_1,1)\otimes (-1) \cong (-1)\otimes (-1) \cong (1)$$ by Proposition \[emprop\], 3 we also have $\dim(\Lambda^3{\mathscr{E}}_1)^{I_\infty}\ge 1$. Finally we find that $$\chi({\mathbb{P}}^1,j_{!*}\Lambda^3{\mathscr{E}}_1)=13+\dim(\Lambda^3{\mathscr{E}}_1)^{I_\infty}-13\ge 1.$$ The second and thirds cases are completely analogous and we have $$\chi({\mathbb{P}}^1,j_{!*}\Lambda^3{\mathscr{E}}_2)=13+4-15=2$$ and $$\chi({\mathbb{P}}^1,j_{!*}\Lambda^3{\mathscr{E}}_3)=9+\dim(\Lambda^3{\mathscr{E}}_3)^{I_\infty}-{\textup{irr}}_\infty(\Lambda^3{\mathscr{E}}_3)\ge 9+2-10=1.$$ Therefore for $i=1,2$ we have ${\textup{DGal}}({\mathscr{E}}_i)= G_2$.
For the systems with formal type note that the systems in question have Euler characteristic $-1$ on ${\mathbb{G}_m}$ and therefore are hypergeometric by [@Ka90 Theorem 3.7.1]. By [@Ka90 4.1.] and [@Ka90 2.7.5.], the systems are Lie-irreducible, meaning that the associated representation of the identity component $G^0$ of $G$ is irreducible. We apply Katz’s Main DE Theorem [@Ka90 2.8.1.] to conclude that the derived group $[G^0,G^0]$ of the identity component of $G$ is $G_2$. Since $G_2$ is its own normalizer in $SO(7)$ it follows that $G=G_2$.\
**The above list exhausts all cases.** Let ${\mathscr{E}}$ be an irreducible irregular rigid $G_2$-connection, i.e. at some singularity the irregularity of ${\mathscr{E}}$ is positive. By the rough classification of Section \[G2conn\], the only possibilities for $R({\mathscr{E}})$ are $$\begin{aligned}
&(0, 7, 7, 2), \\
&(0, 14, 13, 3), \\
&(0, 19, 17, 4) \ \textup{or} \\
&(0, 21, 19, 4). \end{aligned}$$ Applying the same techniques as before, the only formal types left are those appearing in the above list together with one additional formal type which is given by the following table (here $\varepsilon$ denotes a primitive third root of unity).
[ c c ]{} $0$ & $\infty$\
\
$(\varepsilon E_3,\varepsilon^{-1} E_3,1)$ &
------------------------------------------------------------------
${\textup{El}}(2,\alpha_1,1) \oplus {\textup{El}}(2,\alpha_2,1)$
$\oplus\,{\textup{El}}(2,\alpha_1+\alpha_2,1) \oplus (-1)$
------------------------------------------------------------------
The connection $${\mathscr{E}}:={\mathscr{F}}((\varepsilon,\varepsilon^{-1})\otimes \phi^*({\mathscr{F}}((\varepsilon^{-1},1,1,1,\varepsilon)\otimes{\textup{MC}}_{\varepsilon^{-1}}({\mathscr{L}}_5))))$$ constructed from the rank one sheaf ${\mathscr{L}}_5:={\mathscr{L}}_{(-\varepsilon,-1,-1,-1,\varepsilon^{-1})}$ on $${\mathbb{P}}^1-\{0,\frac{1}{4}\alpha_1^2,\frac{1}{4}\alpha_2^2,\frac{1}{4}(\alpha_1+\alpha_2)^2,\infty \}$$ has the above formal type. We will prove by contradiction that ${\textup{DGal}}({\mathscr{E}})$ is not contained in $G_2$. Therefore suppose the contrary, i.e. ${\textup{DGal}}({\mathscr{E}})\subset G_2$. As we have seen before, the morphism $${\pi_1^{\textup{dR}}}({\mathbb{G}_m},1)\rightarrow {\textup{GL}}_7({\mathbb{C}})$$ corresponding to ${\mathscr{E}}$ factors through $G_2({\mathbb{C}})$. Denote by ${\textup{Ad}}$ the adjoint representation ${\textup{Ad}}:G_2\rightarrow \mathfrak{g}_2$. As ${\mathscr{E}}$ is rigid and irreducible by construction, we find that $$0=\dim H^1({\mathbb{P}}^1,j_{!*}{\textup{Ad}}({\mathscr{E}}))={\textup{irr}}_\infty({\textup{Ad}}({\mathscr{E}}))-\dim {\textup{Ad}}({\mathscr{E}})^{I_\infty}-\dim {\textup{Ad}}({\mathscr{E}})^{I_0}$$ and the same for the connection ${\mathscr{E}}_2$ we have constructed above. As the formal type at $\infty$ of ${\mathscr{E}}$ and ${\mathscr{E}}_2$ coincides, we find that $${\textup{irr}}_\infty({\textup{Ad}}({\mathscr{E}}))-\dim {\textup{Ad}}({\mathscr{E}})^{I_\infty}={\textup{irr}}_\infty({\textup{Ad}}({\mathscr{E}}_2))-\dim {\textup{Ad}}({\mathscr{E}}_2)^{I_\infty}$$ and in particular a necessary condition for both connections to have differential Galois group $G_2$ is $$\dim{\textup{Ad}}({\mathscr{E}})^{I_0}=\dim {\textup{Ad}}({\mathscr{E}}_2)^{I_0}.$$ These invariants are precisely the centraliser dimension of the local monodromy at $0$ of the connections in question. By Table 4 in [@Dett10], $\dim {\textup{Ad}}({\mathscr{E}}_2)^{I_0}=6$ and $\dim{\textup{Ad}}({\mathscr{E}})^{I_0}=8$ which yields a contradiction. Hence ${\textup{DGal}}({\mathscr{E}})$ is not contained in $G_2$.
Let ${\mathscr{E}}_4^5$ be the final system in the theorem with $x=\zeta_8$ and $y=\zeta_8^2$ and denote by $[q]:{\mathbb{G}_m}\rightarrow {\mathbb{G}_m}$ the morphism given by $z\mapsto z^q$. In this setting we find that $${\mathscr{E}}_3\cong [2]^*{\mathscr{E}}_4^5.$$ To see this we compute the pullback of the formal types. At the regular singularity, the pullback of the connection with monodromy $(\zeta_8,\zeta_8^2,\zeta_8^3,\zeta_8^5,\zeta_8^6,\zeta_8^7,1)$ has monodromy $(iE_2,-iE_2,-E_2,1)$. The pullback of ${\textup{El}}(6,\alpha_1, 1)\oplus(-1)$ is given due to [@Sa08 2.5 & 2.6] as $${\textup{El}}(3,\alpha,1)\oplus{\textup{El}}(3,\zeta_6^5\alpha,1)\oplus (1)\cong {\textup{El}}(3,\alpha,1)\oplus{\textup{El}}(3,-\alpha,1)\oplus (1),$$ since $\zeta_6^5\alpha=-\zeta_3^2\alpha$ and we can multiply by $\zeta_3$ to get $-\alpha$. By rigidity we get the desired isomorphism ${\mathscr{E}}_3\cong [2]^*{\mathscr{E}}_4^5$.
A similar analysis shows that systems in the second family ${\mathscr{E}}_2$ with formal type $${\textup{El}}(2,-\alpha_1,1)\oplus{\textup{El}}(2,\zeta_6^5\alpha_1,1)\oplus{\textup{El}}(2,\zeta_6^4,1)\oplus(-1)$$ at $\infty$ are pullbacks of the system ${\mathscr{E}}_4^4$, the second to last system of the theorem, with $x=\zeta_3$ under the map $[3]:{\mathbb{G}_m}\rightarrow{\mathbb{G}_m}$. Of course not every system in the family ${\mathscr{E}}_2$ is of this form and if they are not, they cannot be pullbacks of hypergeometrics (these would have to appear in the above list).
|
---
abstract: 'Variational Neural Machine Translation (VNMT) is an attractive framework for modeling the generation of target translations, conditioned not only on the source sentence but also on some latent random variables. The latent variable modeling may introduce useful statistical dependencies that can improve translation accuracy. Unfortunately, learning informative latent variables is non-trivial, as the latent space can be prohibitively large, and the latent codes are prone to be ignored by many translation models at training time. Previous works impose strong assumptions on the distribution of the latent code and limit the choice of the NMT architecture. In this paper, we propose to apply the VNMT framework to the state-of-the-art Transformer and introduce a more flexible approximate posterior based on normalizing flows. We demonstrate the efficacy of our proposal under both in-domain and out-of-domain conditions, significantly outperforming strong baselines.'
author:
- |
Hendra Setiawan Matthias Sperber Udhay Nallasamy Matthias Paulik\
Apple\
[{ hendra,sperber,udhay,mpaulik }@apple.com]{}
bibliography:
- 'anthology.bib'
- 'acl2020.bib'
title: Variational Neural Machine Translation with Normalizing Flows
---
Introduction
============
Translation is inherently ambiguous. For a given source sentence, there can be multiple plausible translations due to the author’s stylistic preference, domain, and other factors. On the one hand, the introduction of neural machine translation (NMT) has significantly advanced the field [@bahdanau+al-2014-nmt], continually producing state-of-the-art translation accuracy. On the other hand, the existing framework provides no explicit mechanisms to account for translation ambiguity.
Recently, there has been a growing interest in latent-variable NMT (LV-NMT) that seeks to incorporate latent random variables into NMT to account for the ambiguities mentioned above. For instance, incorporated latent codes to capture underlying global semantics of source sentences into NMT, while proposed fine-grained latent codes at the word level. The learned codes, while not straightforward to analyze linguistically, are shown empirically to improve accuracy. Nevertheless, the introduction of latent random variables complicates the parameter estimation of these models, as it now involves intractable inference. In practice, prior work resorted to imposing strong assumptions on the latent code distribution, potentially compromising accuracy.
In this paper, we focus on improving Variational NMT (VNMT) [@zhang-etal-2016-variational-neural]: a family of LV-NMT models that relies on the amortized variational method [@DBLP:journals/corr/KingmaW13] for inference. Our contributions are twofold. (1) We employ variational distributions based on normalizing flows [@rezende15], instead of uni-modal Gaussian. Normalizing flows can yield complex distributions that may better match the latent code’s true posterior. (2) We employ the Transformer architecture [@transformer], including *Transformer-Big*, as our VNMT’s generator network. We observed that the generator networks of most VNMT models belong to the RNN family that are relatively less powerful as a translation model than the Transformer.
We demonstrate the efficacy of our proposal on the German-English IWSLT’14 and English-German WMT’18 tasks, giving considerable improvements over strong non-latent Transformer baselines, and moderate improvements over Gaussian models. We further show that gains generalize to an out-of-domain condition and a simulated bimodal data condition.
VNMT with Normalizing Flows
===========================
**Background** Let $\boldsymbol{x}$ and $\boldsymbol{y}$ be a source sentence and its translation, drawn from a corpus $\mathcal{D}$. Our model seeks to find parameters $\theta$ that maximize the marginal of a latent-variable model $p_\theta(\boldsymbol{y},Z\mid \boldsymbol{x})$ where $Z \in \mathbb{R}^D$ is a sentence-level latent code similar to [@zhang-etal-2016-variational-neural]. VNMT models sidestep the marginalization by introducing variational distributions and seek to minimize this function (i.e., the Evidence Lower Bound or ELBO): $$\begin{aligned}
\sum_{(\boldsymbol{x},\boldsymbol{y})\in\mathcal{D}} \mathbb{E}_{q(Z\mid \boldsymbol{x}, \boldsymbol{y})} \left[\log p_\theta(\boldsymbol{y}\mid\boldsymbol{x},Z) \right] \nonumber
\\
- \text{KL}\left( q(Z\mid \boldsymbol{x}, \boldsymbol{y}) \mid\mid p(Z\mid \boldsymbol{x})\right), \label{eq_elbo1}\end{aligned}$$ where $q(Z\mid \boldsymbol{x}, \boldsymbol{y})$, $p(Z\mid \boldsymbol{x})$ are the variational posterior and prior distribution of the latent codes, while $p(\boldsymbol{y}\mid\boldsymbol{x},Z)$ is a *generator* that models the generation of the translation conditioned on the latent code[^1]. The ELBO is improved when the model learns a posterior distribution of latent codes that minimizes the reconstruction loss (the first term) while incurring a smaller amount of KL divergence penalty between the variational posterior and the prior (the second term).
The majority of VNMT models design their variational distributions to model unimodal distribution via isotropic Gaussians with diagonal covariance, which is the simplest form of prior and approximate posterior distribution. This assumption is computationally convenient because it permits a closed-form solution for computing the KL term and facilitates end-to-end gradient-based optimization via the re-parametrization trick [@rezende15]. However, such a simple distribution may not be expressive enough to approximate the true posterior distribution, which could be non-Gaussian, resulting in a loose gap between the ELBO and the true marginal likelihood. Therefore, we propose to employ more flexible posterior distributions in our VNMT model, while keeping the prior a Gaussian.\
\
**Normalizing Flows-based Posterior** proposed Normalizing Flows (NF) as a way to introduce a more flexible posterior to Variational Autoencoder (VAE). The basic idea is to draw a sample, $Z_0$, from a simple (e.g., Gaussian) probability distribution and to apply $K$ invertible parametric transformation functions $(f_k)$ called *flows* to transform the sample. The final latent code is given by $Z_K = f_K ( ... f_2 ( f_1(Z_0)) ... )$ whose probability density function, $q_\lambda(Z_K \mid \boldsymbol{x}, \boldsymbol{y})$, is defined via the change of variable theorem as follows: $$\begin{aligned}
q_0(Z_0 \mid \boldsymbol{x}, \boldsymbol{y}) \prod_{k=1}^K \left|\det\frac{\partial f_k(Z_{k-1}; \boldsymbol{\lambda}_k(\boldsymbol{x},\boldsymbol{y}))}{\partial Z_{k-1}}\right|^{-1},\end{aligned}$$ where $\lambda_k$ refers to the parameters of the $k$-th flow with $\lambda_0$ corresponds to the parameters of a base distribution. In practice, we can only consider transformations, whose determinants of Jacobians (the second term) are invertible and computationally tractable.
For our model, we consider several NFs, namely *planar flows* [@rezende15], *Sylvester flows* [@vdberg2018sylvester] and *affine coupling layer* [@realnvp45819], which have been successfully applied in computer vision tasks.
Planar flows (PF) applies this function: $$\begin{aligned}
f_k(Z;\lambda_k(\boldsymbol{x},\boldsymbol{y})) = Z + \mathbf{u} \cdot \mathrm{tanh}(\mathbf{w}^TZ + \mathbf{b}),\end{aligned}$$ where $\boldsymbol{\lambda}_k = \{\mathbf{u}, \mathbf{w} \in \mathbb{R}^D$, $\mathbf{b} \in \mathbb{R}\}$. Planar flows perform contraction or expansion to the direction perpendicular to the $(\mathbf{w}^TZ+ \mathbf{b})$ hyperplane.
Sylvester flows (SF) applies this function: $$\begin{aligned}
f_k(Z;\lambda_k(\boldsymbol{x},\boldsymbol{y})) = Z + \mathbf{A} \cdot \mathrm{tanh}(\mathbf{B}Z + \mathbf{b}),\end{aligned}$$ where $\boldsymbol{\lambda}_k = \{\mathbf{A}, \mathbf{B} \in \mathbb{R}^{M \times D}$, $\mathbf{b} \in \mathbb{R}^M\}$ and $M$ is the number of hidden units. Planar flows are a special case of Sylvester flows where $M=1$. In our experiments, we consider the orthogonal Sylvester flows [@vdberg2018sylvester], whose parameters are matrices with $M$ orthogonal columns.
Meanwhile, the affine coupling layer (CL) first splits $Z$ into $Z^{d_1}, Z^{d_2} \in \mathbb{R}^{D/2}$ and applies the following function: $$\begin{aligned}
f_k(Z^{d_1};\lambda_k(\boldsymbol{x},\boldsymbol{y})) & = Z^{d_1}, \\
f_k(Z^{d_2};\lambda_k(\boldsymbol{x},\boldsymbol{y},Z^{d_1})) & = Z^{d_2} \odot \exp(\boldsymbol{s}_k) + \boldsymbol{t}_k,\end{aligned}$$ where it applies identity transform to $Z^{d_1}$ and applies a scale-shift transform to $Z^{d_2}$ according to $\boldsymbol{\lambda}_k = \{ \boldsymbol{s}_k, \boldsymbol{t}_k\}$, which are conditioned on $Z^{d_1}$, $\boldsymbol{x}$ and $\boldsymbol{y}$. CL is less expressive than PF and SF, but both sampling and computing the probability of arbitrary samples are easier. In practice, we follow [@realnvp45819] to switch $Z^{d_1}$ and $Z^{d_2}$ alternately for subsequent flows.
As we adopt the amortized inference strategy, the parameters of these NFs are data-dependent. In our model, they are the output of 1-layer linear map with inputs that depend on $\boldsymbol{x}$ and $\boldsymbol{y}$. Also, as the introduction of normalizing flows no longer offers a simple closed-form solution, we modify the KL term in Eq. \[eq\_elbo1\] into: $$\begin{aligned}
\mathbb{E}_{q_\lambda(Z \mid \boldsymbol{x}, \boldsymbol{y})} \left[ \log q_\lambda(Z \mid \boldsymbol{x}, \boldsymbol{y}) - \log p_\psi(Z \mid \boldsymbol{x}) \right] \end{aligned}$$ where we estimate the expectation w.r.t. $q(Z_K{\mid}\boldsymbol{x};\boldsymbol{\lambda})$ via $L$ Monte-Carlo samples. We found that $L=1$ is sufficient, similar to [@zhang-etal-2016-variational-neural]. To address variable-length inputs, we use the average of the embeddings of the source and target tokens via a mean-pooling layer, i.e., $\text{meanpool}(\boldsymbol{x})$ and $\text{meanpool}(\boldsymbol{y})$ respectively.\
\
**Transformer-based Generator**We incorporate the latent code to the Transformer model by mixing the code into the output of the Transformer decoder’s last layer ($h_j$) as follows: $$\begin{aligned}
g_j &= \delta([h_j;Z]), \quad
h_j &= (1-g_j) * h_j + g_j * Z \nonumber\end{aligned}$$ where $g_j$ controls the latent code’s contribution, and $\delta(\cdot)$ is the sigmoid function. In the case of the dimension of the latent code ($D$) doesn’t match the dimension of $h_j$, we apply a linear projection layer. Our preliminary experiments suggest that Transformer is less likely to ignore the latent code in this approach compared to other approaches we explored, e.g., incorporating the latent code as the first generated token as used in [@zhang-etal-2016-variational-neural].\
\
**Prediction** Ultimately, we search for the most probable translation $(\boldsymbol{\hat{y}})$ given a source sentence $(\boldsymbol{x})$ through the evidence lower bound. However, sampling latent codes from the posterior distribution is not straightforward, since the posterior is conditioned on the sentence being predicted. suggests taking the prior’s mean as the latent code. Unfortunately, as our prior is a Gaussian distribution, this strategy can diminish the benefit of employing normalizing flows posterior.
explore two strategies, namely restricting the conditioning of the posterior to $\boldsymbol{x}$ alone (dropping $\boldsymbol{y}$) and introducing an auxiliary distribution, $r(Z{\mid}\boldsymbol{x})$, from which the latent codes are drawn. They found that the former is more accurate with the benefit of being simpler. This is confirmed by our preliminary experiments. We opt to adopt this strategy and use the mean of the posterior as the latent code at prediction time.\
\
**Mitigating Posterior Collapse**As reported by previous work, VNMT models are prone to posterior collapse, where the training fails to learn informative latent code as indicated by the value of KL term that vanishes to 0. This phenomenon is often attributed to the strong generator [@DBLP:conf/icml/AlemiPFDS018] employed by the models, in which case, the generator’s internal cells carry sufficient information to generate the translation. Significant research effort has been spent to weaken the generator network. Mitigating posterior collapse is crucial for our VNMT model as we employ the Transformer, an even stronger generator that comes with more direct connections between source and target sentences [@bahuleyan-etal-2018-variational].
To remedy these issues, we adopt the $\beta_C$-VAE [@prokhorov-etal-2019-importance] and compute the following modified KL term: $\beta \left| KL - C \right|$ where $\beta$ is the scaling factor while $C$ is a rate to control the KL magnitude. When $C>0$, the models are discouraged from ignoring the latent code. In our experiments, we set $C = 0.1$ and $\beta=1$. Additionally, we apply the standard practice of word dropping in our experiments.\
\
**Related Work**VNMT comes in two flavors. The first variant models the conditional probability akin to a translation model, while the second one models the joint probability of the source and target sentences. Our model adopts the first variant similar to [@zhang-etal-2016-variational-neural; @SuWXLHZ18; @DBLP:journals/corr/abs-1812-04405], while [@autoencodingvnmt; @NIPS2018_7409] adopt the second variant. The majority of VNMT models employ RNN-based generators and assume isotropic Gaussian distribution, except for [@mccarthy2019improved] and [@nfnmt]. The former employs the Transformer architecture but assumes a Gaussian posterior, while the latter employs the normalizing flows posterior (particularly planar flows) but uses an RNN-based generator. We combine more sophisticated normalizing flows and the more powerful Transformer architecture to produce state-of-the-art results.
Experimental Results
====================
**Experimental Setup**We integrate our proposal into the Fairseq toolkit [@ott2019fairseq; @gehring2016convenc; @gehring2017convs2s]. We report results on the IWSLT’14 German-English (De-En) and the WMT’18 English-German (En-De) tasks. For IWSLT’14, we replicate ’s setup with 160K training sentences and a 10K joint BPE vocabulary, while for WMT’18, we replicate ’s setup with 5.2M training sentences and a 32K joint BPE vocabulary. For WMT experiments, we report the accuracy using detokenized SacreBLEU [@post-2018-call] to facilitate fair comparison with other published results. Note that tokenized BLEU score is often higher depending on the tokenizer, thus not comparable. We apply KL annealing schedule and token dropout similar to [@bowman-etal-2016-generating], where we set the KL annealing to 80K updates and drop out 20% target tokens in the IWSLT and 10% in the WMT experiments.
The encoder and decoder of our Transformer generator have 6 blocks each. The number of attention heads, embedding dimension, and inner-layer dimensions are 4, 512, 1024 for IWSLT; and 16, 1024, 4096 for WMT. The WMT setup is often referred to as the *Transformer Big*. To our knowledge, these architectures represent the best configurations for our tasks. We set the latent dimension to $D=128$, which is projected using a 1-layer linear map to the embedding space. We report decoding results with beam=5. For WMT experiments, we set the length penalty to 0.6. For all experiments with NF-based posterior, we employ flows of length 4, following the results of our pilot study.\
\
**In-Domain Results**We present our IWSLT results in rows 1 to 6 of Table \[t\_result\]. The accuracy of the baseline Transformer model is reported in row (1), which matches the number reported by . In row (2), we report a static $Z$ experiment, where $Z=\text{meanpool}(\boldsymbol{x})$. We design this experiment to isolate the benefits of token dropping and utilizing average source embedding as context. As shown, the static $Z$ provides +0.8 BLEU point gain. In row (3), we report the accuracy of our VNMT baseline when the approximate posterior is a Gaussian, which is +1.3 BLEU point from baseline or +0.5 point from the static $Z$, suggesting the efficacy of latent-variable modeling. We then report the accuracy of different variants of our model in rows (4) to (6), where we replace the Gaussian posterior with a cascade of 4 PF, SF and CL, respectively. For SF, we report the result with $M=8$ orthogonal columns in row (5). As shown, these flows modestly add +0.2 to +0.3 points. It is worth noticing that the improvement introduces only around 5% additional parameters.
System \#params BLEU
---- ------------------------------ ---------- ---------- --
1 Transformer IWSLT 42.9M 34.5
2 + static $Z$ 42.9M 35.3
3 + $Z \sim$ Gaussian 43.6M 35.8
4 + $Z \sim$ 4 x PF *44.2M* *36.1*
5 + $Z \sim$ 4 x SF (M=8) 45.9M 36.0
6 + $Z \sim$ 4 x CL *44.3M* *36.1*
7 \(1) + distilled 42.9M 34.9
8 \(6) + distilled 44.3M **36.6**
9 [@edunov2018backtranslation] 29.0
10 Transformer Big 209.1M 28.9
11 + static $Z$ 209.1M 29.0
12 + $Z\sim$ Gaussian 210.5M 29.1
13 + $Z\sim$ 4 x PF 211.6M 29.3
14 $+ Z\sim$ 4 x SF (M=8) 215.3M *29.5*
15 $+ Z\sim$ 4 x CL 210.6M 29.2
16 \(10) + distilled 209.1M 29.2
17 \(14) + distilled 215.3M **29.9**
: The translation accuracy on the De-En IWSLT’14 task (rows 1-8), the En-De WMT’18 task (rows 10-17). Each task’s best results in the in-domain setting are *italicized*, while the results with added distilled data are in **bold**. \[t\_result\]
We report our WMT results that use the Transformer Big architecture in rows (10) to (15). For comparison, we quote the state-of-the-art result for this dataset from in row (9), where the SacreBLEU score is obtained from . As shown, our baseline result (row 10) is on par with the state-of-the-art result. The WMT results are consistent with the IWSLT experiments, where our models (rows 13-15) significantly outperform the baseline, even though they differ in terms of which normalizing flows perform the best. The gain over the VNMT baseline is slightly higher, perhaps because NF is more effective in larger datasets. In particular, we found that SF and PF perform better than CL, perhaps due to their simpler architecture, i.e., their posteriors are conditioned only on the source sentence, and their priors are uninformed Gaussian. Row (11) shows that the static $Z$’s gain is minimal. In row (14), our best VNMT outperforms the state-of-the-art Transformer Big model by +0.6 BLEU while adding only 3% additional parameters.\
\
**Simulated Bimodal Data** We conjecture that the gain partly comes from NF’s ability to capture non-Gaussian distribution. To investigate this, we artificially increase the modality of our training data, i.e., forcing all source sentences to have multiple translations. We perform the sequence-level knowledge distillation [@kim-rush-2016-sequence] with baseline systems as the teachers, creating additional data referred to as *distilled* data. We then train systems on this augmented training data, i.e., original + distilled data. Rows (7) and (16) show that the baseline systems benefit from the distilled data. Rows (8) and (17) show that our VNMT models gain more benefit, resulting in +2.1 and +0.9 BLEU points over non-latent baselines on IWSLT and WMT tasks respectively.\
\
**Simulated Out-of-Domain Condition**We investigate whether the in-domain improvement carries to out-of-domain test sets. To simulate an out-of-domain condition, we utilize our existing setup where the domain of the De-En IWSLT task is TED talks while the domain of the En-De WMT task is news articles. In particular, we invert the IWSLT De-En test set, and decode the English sentences using our baseline and best WMT En-De systems of rows (10) and (14). For this inverted set, the accuracy of our baseline system is 27.9, while the accuracy of our best system is 28.8, which is +0.9 points better. For reference, the accuracy of the Gaussian system in row (11) is 28.2 BLEU. While more rigorous out-of-domain experiments are needed, this result gives a strong indication that our model is relatively robust for this out-of-domain test set.\
\
Example 1 Source In ** book , ** presents 17 housing models for independent living in old age .
--------------------- ----------------------------------------------------------------------------------------------------------------------------------------------
Reference In ** Buch stellt ** 17 Wohnmodelle für ein selbstbestimmtes Wohnen im Alter vor .
Non-latent Baseline In ** Buch präsentiert ** 17 Wohnmodelle für ein unabhängiges Leben im Alter .
VNMT-G In ** Buch stellt die ** 17 Wohnmodelle für ein selbstbestimmtes Wohnen im Alter vor .
VNMT-NF In ** Buch präsentiert ** 17 Wohnmodelle für ein unabhängiges Leben im Alter .
Example 2 Source Even though ** earns S $ 3,000 ( $ 2,400 ) a month ** and her husband works as well , the monthly family income is insufficient , she says .
Reference Obwohl ** jeden Monat 3.000 Singapur-Dollar (ca 1.730 Euro ) ** verdiene –truncated–
Non-latent Baseline Obwohl pro Monat 3.000 S \$ ( 2.400 \$ ) verdient und auch ihr Mann arbeitet , ist das –truncated–
VNMT-G Obwohl ** jeden Monat 3.000 Singapur - Dollar ( ca 1.730 Euro ) ** –truncated–
VNMT-NF Obwohl ** S \$ 3.000 ( \$ 2.400 ) pro Monat ** verdient und ihr Mann auch –trunctated–
: Translation examples with different gender consistency. Inconsistent, consistent translations and source words are in **, **, ** respectively. []{data-label="t_example"}
**Translation Analysis** To better understand the effect of normalizing flows, we manually inspect our WMT outputs and showcase a few examples in Table \[t\_example\]. We compare the outputs of our best model that employs normalizing flows (VNMT-NF, row 14) with the baseline non-latent Transformer (row 10) and the baseline VNMT that employs Gaussian posterior (VNMT-G, row 12).
As shown, our VNMT model consistently improves upon gender consistency. In example 1, the translation of ** depends on the gender of its cataphora (**), which is feminine. While all systems translate the cataphora correctly to **, the baseline and VNMT-G translate the phrase to its masculine form. In contrast, the translation of our VNMT-NF produces the feminine translation, respecting the gender agreement. In example 2, only VNMT-NF and VNMT-G produce gender consistent translations.
Discussions and Conclusions
===========================
We present a Variational NMT model that outperforms a strong state-of-the-art non-latent NMT model. We show that the gain modestly comes from the introduction of a family of flexible distribution based on normalizing flows. We also demonstrate the robustness of our proposed model in an increased multimodality condition and on a simulated out-of-domain test set.
We plan to conduct a more in-depth investigation into actual multimodality condition with high-coverage sets of plausible translations. We conjecture that conditioning the posterior on the target sentences would be more beneficial. Also, we plan to consider more structured latent variables beyond modeling the sentence-level variation as well as to apply our VNMT model to more language pairs.
Word dropout
============
We investigate the effect of different dropout rate and summarize the results in Table \[t\_dropout\]. In particular, we take the VNMT baseline with Gaussian latent variable for IWSLT (row 3 in Table \[t\_result\]) and for WMT (row 12 in Table \[t\_result\]). As shown, word dropout is important for both setup but it is more so for IWSLT. It seems that tasks with low resources benefit more from word dropout. We also observe that above certain rate, word dropout hurts the performance.
Dropout rate 0.0 0.1 0.2 0.3
-------------- ------ ---------- ---------- ------
IWSLT 34.4 35.7 **35.8** 35.6
WMT 29.0 **29.1** 28.8 28.7
: Results of different dropout rate for IWSLT and WMT setup. The best results are in **bold**.[]{data-label="t_dropout"}
Latent Dimension
================
We report the results of varying the dimension of latent variable ($D$) in Table \[t\_latdim\]. For this study, we use the VNMT baseline with Gaussian latent variable in IWSLT condition (row 3 in Table \[t\_result\]) . Our experiments suggest that the latent dimension between 64 and 128 is optimal. The same conclusion holds for the WMT condition.
$D$ 8 16 32 64 128 256
------ ------ ------ ------ ------ ---------- ------ -- --
BLEU 35.6 35.5 35.4 35.7 **35.8** 35.4
: Results of different dropout rate for IWSLT. The best results are in **bold**.[]{data-label="t_latdim"}
Normalizing Flow Configuration
==============================
In the Experimental Results section, we report the accuracy for our models with 4 flows. In Table \[t\_nf\], we conduct experiments varying the number of flows for the IWSLT condition. Our baseline (num flows=0) is an NMT model with word dropout, which performs on par with the static $Z$ experiment reported in Table \[t\_result\]’s row 3. These results suggest that increasing the number of flows improves accuracy, but the gain diminishes after 4 flows. The results are consistent for all normalizing flows that we considered. We also conduct experiments with employing more flows, but unfortunately, we observe either unstable training or lower accuracy.
------- ---------- ---------- ---------- --
Num SF
Flows ($M$=8)
0
1 35.8 35.6 35.8
2 35.7 35.5 35.8
3 36.0 35.9 35.7
4 **36.1** 36.0 **36.1**
5 35.9 **36.1** 35.9
6 35.8 36.0 35.9
------- ---------- ---------- ---------- --
: Translation accuracy of VNMT models employing various number of flows in the IWSLT condition. The best results are in **bold**.[]{data-label="t_nf"}
In Table \[t\_ortho\], we conduct experiments varying the number of orthogonal columns ($M$) in our Sylvester normalizing flows (SF) experiments. As shown, increasing $M$ improves the accuracy up to $M=24$. We see no additional gain from employing more additional orthogonal columns beyond 24. In Table \[t\_result\], we report $M=8$, because it introduces the least number of additional parameters.
$M$ 2 4 8 16 24 32
------ ------ ------ ------ ------ ---------- ------
BLEU 35.7 35.5 36.0 36.0 **36.2** 35.9
: Results of different number of orthogonal columns for SF. The best results are in **bold**.[]{data-label="t_ortho"}
[^1]: In VAE terms, the posterior and prior distributions are referred to as the encoders, while the generator is referred to as the decoder. As these terms have other specific meaning in NMT, we avoid to use them in this paper.
|
---
author:
- 'Tomas Kulvicius$^{1,*}$, Sebastian Herzog$^{1}$, Minija Tamosiunaite $^{1,2}$ and Florentin Wörgötter$^{1}$[^1][^2][^3][^4][^5]'
title: '**Generation of Paths in a Maze using a Deep Network without Learning** '
---
[**Trajectory- or path-planning is a fundamental issue in a wide variety of applications. Here we show that it is possible to solve path planning for multiple start- and end-points highly efficiently with a network that consists only of max pooling layers, for which no network training is needed. Different from competing approaches, very large mazes containing more than half a billion nodes with dense obstacle configuration and several thousand path end-points can this way be solved in very short time on parallel hardware.**]{}
INTRODUCTION {#introduction .unnumbered}
============
Path planning is a prevalent problem that exists, for example, in traffic control to optimize traffic flow patterns, but also in robotics for finding the best way to a target or to plan an arm-hand trajectory when performing manipulation. Even protein-folding can be formulated as a path planning problem [@Amato2002] and additional relevant applications exist in other fields. In general, path planning is defined as the problem of finding a temporal sequence of valid states from an initial to a final state given some constraints [@Latombe2012].
In this contribution we are addressing path planning for multiple start- and end-points (i.e., multi-source multi-target) in large environments, like city maps. This problem relates to the single source shortest path (SSSP) problem. The task of SSSP is to find shortest paths in a graph between a vertex (node) and all other vertices such that the sum of the edges’ weights is minimised. Classical approaches to solve SSSP are Breadth First Search algorithm (BFS) [@Moore1959], Dijkstra’s algorithm [@Dijkstra1959], and the Bellman-Ford algorithm [@Bellman1958; @Ford1956]. BFS is suitable for unweighted graphs, i.e., is a uniform cost search and the obtained solution is optimal with respect to the number of nodes to travel from the source node to all other nodes. Dijsktra’s algorithm finds shortest paths in weighted graphs with positive weights, whereas the Bellman-Ford algorithm can also deal with graphs with negative weights. However, it is slower than Dijsktra’s algorithm. Another algorithm which finds shortest paths between all pairs of nodes in a graph is the Floyd–Warshall algorithm [@Floyd1962], however, running Dijkstra’s algorithm for each node is a better choice when considering sparse graphs. The advantage of all these methods is that they do not require learning and are parameter free methods, however, they can be computationally expensive.
Recently, an new approach, which also does not require learning has been proposed by [@Farias2019]. This method utilises GPU OpenGL shaders and is based on cone rasterization from sources and obstacle vertices. It can generate optimal path maps for multiple sources and outperforms other GPU based approaches [@Luo2010; @Merrill2012; @Wynters2013; @Kapadia2013; @Garcia2014]. However, as we will show later, this method is not very suitable in several cases, because it leads to relative long computational times.
Another class of algorithms is based on artificial neural networks ranging from bio-inspired approaches to deep learning methods. In the bio-inspired approaches [@Glasius1995; @Glasius1996; @Bin2004; @Yang2001; @Ni2017; @Rueckert2016], the environment is represented by a network with inhibitory (=obstacles) and excitatory (=free spaces) neurons arranged on a grid. Here, activity is propagated from the source neuron to the closest neurons and this procedure is repeated for many iterations until the activity is spread out across the whole network. Later, shortest paths can be found by following activity gradients. In principle, these networks are similar to the wavefront propagation algorithm [@Choset2005], which is a special case of the BFS algorithm. Although most of these these approaches (except [@Rueckert2016]) do not require network training, they are not parameter free. Also, as we will show later, the disadvantage of such algorithms is that the activity decreases exponentially [@Yang2001] and large mazes (e.g., larger than $500 \times 500$) can not be solved due to numerical precision problems.
Recently, deep learning approaches utilising deep multi-layer perceptrons (DMLP, [@Qureshi2018]), fully convolutional networks [@Perez2018; @Ariki2019; @Kulvicius2020], long short-term memory (LSTM) networks [@Bency2019], and deep reinforcement learning approaches [@Tai2017; @Panov2018] have been proposed for solving path finding problem, too. Most of these approaches (except [@Ariki2019] and [@Kulvicius2020]) deal with single path planning and/or consider relatively small environments (below $250 \times 250$). Moreover, these approaches require relatively large data sets (e.g., thousands of samples) as well as training to optimise network parameters.
In this paper we present a novel deep network, which consists of only max pooling layers (oMAP). The proposed method generates activity maps for single as well as for multiple sources to single but also multiple targets. It does not require training data such that learning is not necessary and, when following the activity gradient shortest paths are found. Furthermore, this approach can process very large environments on standard GPUs in very short time[^6].
![\[net\] Network architecture and algorithmic process. The network consists of many stacked identical max pooling layers with one filter of size $3 \times 3$, stride $1 \times 1$ and zero padding (no sub-sampling). Here, for graphical reasons, a wider filter is shown. The network receives a source map and an environment map (with obstacles, black) as inputs. The algorithm consists only of repeated max pooling, adding, and rectified-linear (ReLU) operations as shown in the figure.](network02_v18.png){width="1.0\linewidth"}
METHODS {#methods .unnumbered}
=======
**Input** {#input .unnumbered}
---------
The oMAP algorithm uses two binary images of size $m \times n$, an environment map $I_e$ and a source map $I_s$, with $s\ge 1$ sources, as an input.
For the source map, we set $I_s(i,j)=1$ at all source locations, otherwise we set $I_s(i,j)=0$.
The environment map $I_e$ represents an obstacle map where we set $I_e(i,j)=0$ if a grid cell is free (no obstacle) and $I_e(i,j)= -maxint$ if grid cell $(i,j)$ contains an obstacle. The choice of using here the numerically most negative integer ($-maxint$) is motivated by the algorithm for generating the activity map, described next.
**Activity Map Generation** {#activity-map-generation .unnumbered}
---------------------------
The oMAP network consists of $L$ identical max pooling layers $l_i$ (see Fig. \[net\]) with only one type of filter with size $3 \times 3$. We specifically use such a filter size in order to pass activity only to the nearest grid cells (similar to the wavefront expansion algorithm [@Choset2005]). Otherwise, in case of larger filters, activity could propagate also to grid cells, which are separated by obstacles.
We start with the source map $I_s$ and perform max pooling. Then we sum the resulting map with input maps $I_e$ plus $I_s$ into one intermediate-layer map. Note that the largest resulting grid cell value $v$ after this operation will be $v=n+1$ (see example in Figure \[map\_ex\]), where $n$ is the index of the current layer $l_n$. All source cells will obtain this value, hence $v_{source}=n+1$. Obstacle cells, on the other hand, obtain values, which remain negative and follow $v_{obst}<-maxint+n+1 < 0$.
Then we pass the resulting map through the standard rectified linear transfer function (ReLU). Because $v_{obst}< 0$ all grid values at obstacle locations remain at zero.
This is repeated for all layers until the last layer, which produces the final activity map $O$ of size $m \times n$ as an output.
Note that (different from, e.g., convolutional nets) we do not have any tunable weights, thus, no training is required.
![\[map\_ex\] Illustration of the generation of an activity map using oMAP. Black grid cells in the Environment Map stand for cell values of $-maxint$ and represent obstacles. Brown cell with value 1 in the Source Map denotes the source location. Activity at each layer after Add/ReLU operation (see Fig. \[net\]) is shown. ](map_generation_example_2_v18.png){width="0.99\linewidth"}
A graphical visualisation of the process of activity map generation using oMAP is shown in Fig. \[map\_ex\]. Here we used a maze of size $9 \times 9$ and placed one source in the middle. We show the obstacle map (black grid cells denote obstacles), the input activity (source map) and the activity of each max pooling layer after Add/ReLU operation. As we can see, after the first layer (see output $l_1$) activity is propagated from the source only to its neighbouring grid cells (except obstacles) which obtain values of $1$, while the activity at the source cell is increased by one to a value of $2$. From layer to layer, activity in the network grows and propagates to grid cells increasingly distant from the source. In this particular example, map generation is complete after nine layers.
### **Determining the number of layers $L$ and algorithmic complexity** {#determining-the-number-of-layers-l-and-algorithmic-complexity .unnumbered}
oMAP does not have any tunable variables and also the number of layers $L$, which is the only existing free parameter, can be unequivocally determined. It is identical to the maximal path length in an environment, which depends on the location of the source(s), the size of the environment and the distribution of obstacles. In general, however, the longest path is [*a priori*]{} unknown, but the structure of the oMAP algorithm allows determining $L$ during run-time. The algorithm can be run recursively adding layer after layer until the activity map does not contain any zero-values anymore (except at the obstacles). Thus, $L$ can be set using this procedure[^7].
The complexity of our algorithm is $\mathcal{O}(N \times L)$ where $N = m \times n$ is the number of grid cells in the map (corresponds to the number of $max$ operations per layer) and $L$ is the number of layers.
**Path Reconstruction** {#path-reconstruction .unnumbered}
-----------------------
A path from any given target location to the source (or closest source) can be found from the generated activity map $O$ by following the activity gradient; i.e., we start from the chosen target location and select a neighbouring grid cell out of its eight neighbors with maximum value and repeat this until reaching the source. Note that there can be cases of more than one neighboring cell with maximal value. In such a case, we chose the next cell randomly and we refer to this method as *simple path reconstruction*. Note that, due to the single step forward propagation by the max pooling method described above, it will not matter, which cell to choose, because following any of the resulting gradients will render the same number of steps back to the source.
Using map $l_9$ from Fig. \[map\_ex\], it is easy to see that this method can create all possible paths from any target back to the source in the middle.
Note that this method is similar to the wavefront expansion algorithm [@Choset2005], which is a special case of Breadth First Search (BFS, [@Moore1959]) and, thus, always renders optimal paths with respect to number of steps.
However, paths obtained by simple path reconstruction will be not necessarily optimal with respect to Euclidean distance. This is due to the fact, that all transitions (horizontal, vertical, diagonal) are weighted equally amounting to a uniform-cost search. To improve on this, we propose the [*Euclidean path reconstruction*]{} method, described next.
We first find a path, similar to above, by choosing a neighboring cell with maximum value, but now we only consider horizontal and vertical neighboring cells (’Manhattan’ transition). Thus, given the current path position $\{P_x(t)=i,P_y(t)=j\}$ the next step of the path is defined by $$\begin{aligned}
\{P_x(t+1),P_y(t+1)\} = \arg\max_{i,j} \{O(i,j-1),\\
\nonumber O(i,j+1),O(i-1,j),O(i+1,j)\}.\end{aligned}$$
We stop this path search as soon as the (nearest) source is reached. Then we straighten the path by removing intermediate points $P_{x,y}(t)$ if $||P_{x,y}(t-1)-P_{x,y}(t+1)|| = \sqrt{2}$, $t=2 \dots k-1$, where $k$ is the number of points in the path. Note that complexity of the path reconstruction procedure is $\mathcal{O}(k)$. The obtained path is now shortest with respect to the Euclidean distance.
oMAP was implemented using Tensorflow and Keras API[^8]. We used a PC with Intel Xeon Silver 4114 CPU (2.2GHz) and NVIDIA GTX 1080 Ti or NVIDIA Titan V GPU.
![\[dijkstra\_vs\_max\] **A, C)** Dijkstra’s algorithm versus **B, D)** oMAP. Activity maps (index 1) and reconstructed trajectories (index 2) are shown. Grid size is $100 \times 100$ in case A, B, and $101 \times 101$ in case C, D. For oMAP we used 100 layers for B and 450 layers for D. Blue trajectories represent optimal (shortest) paths following Euclidean path reconstruction and magenta trajectories represent non-optimal paths following simple path reconstruction. Green and red dots represent start- and end-points, respectively.](path_reconstruction_4_v18.png){width="1.0\linewidth"}
RESULTS {#results .unnumbered}
=======
**Generation of Activity Maps and Path Reconstruction for oMAP and other Algorithms** {#generation-of-activity-maps-and-path-reconstruction-for-omap-and-other-algorithms .unnumbered}
-------------------------------------------------------------------------------------
We compared our approach to several above mentioned algorithms, which do not require learning, i.e., Dijkstra’s algorithm [@Dijkstra1959], a biologically inspired neural network (shunting model) [@Yang2001] and a state-of-the-art algorithm based on OpenGL shaders [@Farias2019]. We assessed path optimality (shortest paths with respect to the Euclidean distance) as well as run-time. For the run-time comparison against OpenGL shaders we used the benchmark maps as in [@Farias2019].
### **oMAP versus Dijkstra** {#omap-versus-dijkstra .unnumbered}
First, we show qualitative results on the generation of avtivity maps with single and multiple sources using oMAP compared to Dijkstra’s algorithm. Examples of activity map generation and path reconstruction are shown in Fig \[dijkstra\_vs\_max\] in panels A, C for Dijkstra and in panels B, D for oMAP. Here we used two artificial environments, a relatively simple map with four obstacles (reproduced from [@Farias2019], panels A and B) and a complicated maze map (generated automatically, panels C and D). We show normalised activity maps between zero (dark blue) and maximum (dark red). We obtain a circular pattern when using Dijkstra (see panel A1) and a square pattern with oMAP (B1). This is due to the fact that Dijkstra uses a non-uniform cost (horizontal/vertical moves have a cost of $1$ and diagonal moves have a cost of $\sqrt{2}$), while for oMAP we use uniform cost (all moves have a cost of $1$).
In case of Dijkstra’s algorithm (panel A2) paths were reconstructed from the *visited nodes list* [@Dijkstra1959] and are of optimal Euclidean length, in spite of their ’wiggly’ appearance. On the other hand, maps generated by oMAP combined with simple path reconstruction (magenta paths in B2, D2) will lead to optimal paths with respect to the number of steps, but these paths are not optimal with respect to Euclidean distance. Euclidean path reconstruction (blue paths) solves this issue. Note that for this, one follows first (before straightening) the gradient only along Manhattan transitions. This will usually not lead to the same grid cell selection as for simple path reconstruction. Hence blue and magenta paths are independent of each other. This can be seen when comparing the reconstructed paths (see, for example, panel B2) from simple path reconstruction (magenta) with those from Euclidean path reconstruction (blue). The latter renders optimal paths, which are straighter than the ones from Dijkstra. Also for complex mazes, Euclidean reconstruction renders optimal paths (C, D), in this case identical to Dijkstra.
![\[multi\_source\] Examples of activity map generation using three (left) and nine (right) sources (white dots). Here we used a map from Moving AI benchmark [@MovingAI] (map resolution $767 \times 881$). ](example_multi_v18.png){width="1.0\linewidth"}
An example of activity map generation by oMAP using three and nine sources is shown in Fig. \[multi\_source\]. For this, we used a map from the Moving AI benchmark [@MovingAI] of resolution $767 \times 881$. We used 500 layers and 350 layers in case of three and nine sources, respectively. Results demonstrate that less layers are needed if the number of sources increases, since activity propagates from all sources at the same time, which, as a consequence, fills the complete map sooner.
### **oMAP versus Shunting Model** {#omap-versus-shunting-model .unnumbered}
Conceptually, our approach, as already discussed in the Introduction section, is similar to the path finding method using a biologically inspired neural network (shunting model, [@Yang2001]). Thus, we also compare our approach to this method. Results obtained on a map with a u-shape obstacle (a common benchmark for the evaluation of path finding methods; reproduced from [@Yang2001]) is shown in Fig. \[bio\_vs\_max\]. The central disadvantage of the shunting model is that large environments (e.g. above $500 \times 500$) cannot be addressed. Either neuronal activity drops very quickly when moving away from the source (see panel A) and soon reaches ’numerical-zero’. Or, for quite long run-times, activity can indeed be propagated to more distant locations but this easily leads to activity plateaus due to the nature of the model (see panel B). The authors comment on their model parameters [@Yang2001] but even after extended search in the parameters space, we were not able to arrive at a shunting model that could solve environments above $500 \times 500$. Thus, the shunting model is only applicable for relatively small environments. Moreover, the authors of that study state [@Yang2001] that their method generates optimal paths, which is not always the case. The paths in Fig. \[bio\_vs\_max\] A, B are non-optimal, due to the fact that activity of each neuron is computed as a weighted average activity of its nearest neighboring neurons, which may lead to sub-optimal paths in environments with obstacles.
![\[bio\_vs\_max\] **A, B)** Biologically inspired neural network [@Yang2001] versus **C)** oMAP with 100 layers. Grid size is $100 \times 100$. Greed and red dots represent start- and end-points, respectively. A 3D plot has been used to more clearly show the structure of the gradients.](bio_net_vs_max_net_v18.png){width="1.0\linewidth"}
In contrast to that, activity decreases linearly from the source when using oMAP, which allows generating activity maps for extremely large environments, using large enough $L$, and numerical precision problems do not exist, because we operate with integer numbers that grow maximally to the longest path length. As a consequence, the resulting paths are always optimal (Fig. \[bio\_vs\_max\] C).
### **oMAP versus OpenGL Shaders** {#omap-versus-opengl-shaders .unnumbered}
Possibly the most powerful, currently existing method that uses a wavefront propagation algorithm is described by [@Farias2019] and employs OpenGL shaders in a GPU implementation. This method will always produce optimal paths. Therefore, we chose to compare it to oMAP according to the run time of both algorithms (Fig. \[time\_OGLS\]). Note, however, that such across-implementation comparisons have to be taken with a grain of salt, because we cannot know how efficient the foreign implementation was.
For this, we used maps of size $1,000 \times 1,000$ with increasing number of obstacles as in [@Farias2019] (see obstacle configurations on the oMAP activity maps in Fig. \[time\_OGLS\], left). Note that in our case we set the source in the bottom-left corner, which is the worst case with respect to computer time. Results demonstrate that the run-time of the OpenGL shaders method depends on the number of obstacles in the scene and this method slows down non-linearly as the number of obstacles increases. By contrast, our method uses almost constant time and is faster than [@Farias2019] as soon as the number of obstacles increases. The slight run-time decrease observed for oMAP is due to the fact that the largest source-to-target distance in the map is decreasing as obstacles become smaller, and fewer layers are needed to create the map. Note that for oMAP run-time scales down by a factor of two if the source were located in the middle (best case). The different curves show the performance for different GPUs. Farias and Kallmann [@Farias2019] had used the older NVIDIA GTX 970 (blue). The performance of oMAP was also determined according to this hardware (black), but also when using the faster NVIDIA GTX 1080 Ti (red), which had been also used for all other experiments.
![\[time\_OGLS\] Run-time comparison between OpenGL shaders and oMAP for different types of hardware (NVIDIA GTX 970 vs. NVIDIA GTX 1080 Ti). Left: Activity maps generated using oMAP for the profile maps of size $1,000 \times 1,000$ reproduced from [@Farias2019] with 4, 16, 64, 196, and 400 obstacles. The number of layers for oMAP is given in parenthesis. The source was at the bottom left corner in all cases. Right: run-time comparison for these profiles. Data for the blue curve were taken from [@Farias2019].](res_time_OGLS_wProf_v18.pdf){width="1.0\linewidth"}
**Multi-source Multi-target Path Planning** {#multi-source-multi-target-path-planning .unnumbered}
-------------------------------------------
![\[res\_multi\] Multi-source - multi target path finding problem in a real taxi scenario. Top: Section of a Berlin city map ($332 \times 709$) with disks showing taxi positions obtained on March, 03, 2020 at 2:19 pm, which is a screenshot from the on-line application available at *https://www.taxi.de*. Middle: Multi-source activity map using oMAP (160 layers). Bottom: Shortest paths (blue lines) from customers (green dots) to taxis (red dots) are shown. Customer positions were defined manually.](berlin_taxi_v18.png){width="0.75\linewidth"}
Finally, we applied our method to a multi-source - multi-target task testing our network on a real taxi scenario in Berlin, where there are multiple taxi cabs and multiple customers. The task was to find the closest taxi cab for each customer. To be realistic as possible, we obtained taxi cab positions in the area of Berlin close to the train station at a specific moment in time using an on-line taxi application provided by <https://www.taxi.de>. The streets were extracted automatically using our own written program. Positions of the taxi cabs were set as sources (in total 33; taxi cabs which were very close to each other were marked with one source) and an activity map was generated using oMAP. Finally, eight customer positions were defined manually and the optimal paths were reconstructed from the activity map for each customer as described above. Results are shown in Fig. \[res\_multi\]. In the top panel we show taxi locations in Berlin close to the main train station from an internet-taxi-app at one given point in time. In the middle and the bottom panel we show the resulting activity map for 33 sources and the reconstructed shortest paths for eight customers to the closest taxi cabs, respectively. We used a network with 160 layers and it took only 20 $ms$ on NVIDIA GTX 1080 Ti to generate the activity map (resolution $332 \times 709$). The reconstructed paths (blue trajectories) show that the closest taxi cabs and the shortest paths were found for all eight customers.
**General Run-time Evaluation, Limitations and Practical Considerations** {#general-run-time-evaluation-limitations-and-practical-considerations .unnumbered}
-------------------------------------------------------------------------
### **Run-time** {#run-time .unnumbered}
Run-time evaluations evidently depend on the hardware used. Still, it is of interest to document where we stand given currently existing state of the art hardware. In the following we will therefore show that oMAP achieves remarkable performance when tested on an NVIDIA GTX 1080 Ti GPU. We define the linear grid size as $n$ and consider here square grids with $n^2$ grid cells (“nodes”). We used empty maps without obstacles because for oMAP the number of operations does not depend on the number of obstacles.
![\[time\_pro\] Run-time evaluation of oMAP. **A)** Run-time versus different number of nodes $n^2$ (linear grid size: $n$ = \[500; 1,000; 2,000; 3,000; 4,000\] for a fixed number of layers $L$. **B)** Run-time versus different number of layers ($L$=\[500; 1,000; 2,000; 3,000; 4,000\]) for a fixed linear grid size $n$. **C)** Run-time versus $n^3$ ($n$ = \[500; 1,000; 2,000; 3,000; 4,000\]) and different number of layers ($L$=\[0.5$\times n$, $n$, 1.5$\times n$, 2$\times n$\]). Slopes of the lines (from black to green) are: $1.5265\times10^{-10}$, $3.0593\times10^{-10}$, $4.6202\times10^{-10}$, $6.1493\times10^{-10}$.](res_time_ours_5_v18.pdf){width="1.0\linewidth"}
Panel A in Fig. \[time\_pro\] shows that run-time increases linearly against the number of nodes $n^2$ when keeping the number of layers constant. Similarly, linear growth is also observed when keeping the grid size $n$ constant and increasing $L$ (Fig. \[time\_pro\] B).
From above it is clear that more layers are needed when the grid gets bigger, because this leads to longer paths for which the network has to be increased. The shortest possible longest-path length is $0.5n$ (empty square grid, source in the middle[^9]). In panel C we, thus, consider the number of nodes $n^2$ together with a changeable number of layers $L$ and we let $L$ depend on the grid size for approximating the fact that in larger grids paths are longer. As expected from panels A and B these curves now linearly follow $n^3$ with slopes that increase for increasing $L$ also in a linear manner (for slope values see figure caption). From all this, an approximate equation for estimating the run-time for different grid sizes and different number of layers can be derived as $t_r(n,L)\approx 3.0751 \times 10^{-10} L \, n^2$.
![\[big\] Example of a huge grid with $n=26,000$ corresponding to 676,000,000 nodes. Panel 1 shows the full grid with 5,000 sources and panels 2 to 4 show magnifications to make the obstacles visible (panel 4). Colors encode activity as in Figure \[map\_ex\], blue=small and red=large values.](big_lr.png){width="1.0\linewidth"}
### **Limitations and Practical Considerations** {#limitations-and-practical-considerations .unnumbered}
The above run-time estimate holds on an NVIDIA GTX 1080 Ti GPU as long as oMAP runs essentially in forward mode without (too many) iterations $i$. Note, however, that the overhead of having to pass information iteratively back to the start of a new batch of layers will remain tiny if this happens just a few times.
As stated above, the number of required layers $L$ depends on the longest path. The theoretically existing longest path in any $n\times m$ grid is given as $\max(n,m) \times int((\min(n,m)+1)/2) + int(\min(n,m)/2)$, which is a path that meanders back and forth between two interleaved comb-like obstacle rows. In a square grid, this number can be approximated by $\frac{n^2}{2}$ and this number would have to be matched by $L$. From our experiments with real maps, we found, however, that this is a highly unrealistic situation and usually $1.5n<L<2n$ suffices.
For example, on this architecture, we could run oMAP in one forward pass for a remarkable linear grid size of $n_{max}=26,000$, equivalent to 676,000,000 nodes (see panel 1 in Fig. \[big\]) with maximal layer number of $L_{max}=3,450$ (on an NVIDIA GTX 1080 Ti GPU). This would correspond to a panel of $3.25\times3.25$ tiles when using 64 Megapixel images as individual “maps” with every pixel a grid cell. Run-time for this case was $692.3~s$. Interestingly, for this system $L_{max}$ does not depend on the grid-size.
Note that the number of required layers $L$ will decrease, and so does the run-time, as soon as more than one source exists. Figure \[big\] shows an example of a grid with $n=26,000$ and $5,000$ sources. This required $650$ layers and took only $147.4~s$ to run.
Some more examples that show the power and the limitation of oMAP are: if we assume $L=2 \, n$, then we can run systems with $n=1725$ in one forward pass in $\approx 3.1~s$. Under the same assumption ($L=2 \, n$) we would need $L=52,000$ for the maximal possible grid with $n=26,000$. This would require $i=15$ full iterations and a few more layers for iteration 16, resulting in a run time of $t_r\approx 10,800~s$ (3 hours).
CONCLUSION {#conclusion .unnumbered}
==========
We have presented a deep network for path finding in grid-like environments that does not need learning and runs very fast even for large environments with complex paths. It outperforms competing network approaches by a large margin and is easy to implement on standard GPUs, because of its simple structure. There are no free parameters except the number of layers $L$ for which, however, efficient approximations exist.
For example, in a square grid, the maximal path-length (worst case) can be approximated by $\frac{n^2}{2}$, which would have to be matched by $L$. However, realistically we found that $n<L<2 \, n$ was most often sufficient. The extreme case with highly meandering paths in Fig. \[dijkstra\_vs\_max\] D, which in practical, map-like situations is unlikely to exist, needed $L\approx 4.5 \, n$.
Furthermore, $L$ decreases, when more sources are introduced. Thus, the proposed oMAP approach has, in particular, high potential for multi-source multi-target applications in large environments.
APPENDIX {#appendix .unnumbered}
========
**Run-time Evaluation for Iterative versus One-shot Implementation** {#run-time-evaluation-for-iterative-versus-one-shot-implementation .unnumbered}
--------------------------------------------------------------------
We also compared the run-time of two different implementations of oMAP on two different GPUs, NVIDIA GTX 1080 Ti and NVIDIA Titan V. We compared two extreme cases, where in one case we ran oMAP with 2,000 max pooling layers for one iteration (one forward-pass) and in the other case we ran a single-layer oMAP network for 2,000 iterations recursively, i.e., the output was used as the next input. Results are shown in Table \[gpu\_comp\] where we can see that the former architecture is two orders of magnitude faster as compared to the latter, due to processing and passing data between the CPU and GPU, as compared to the case where all operations are processed solely on the GPU. Results also demonstrate that a significantly larger speed-up can be achieved with the multi-layer architecture as compared to the single-layer architecture when the faster GPU is used (2.72 times versus 1.07).
[|l|c|c|]{}
----------------------
Number of layers /
Number of iterations
----------------------
: \[gpu\_comp\] Run-time comparison between NVIDIA GTX 1080 Ti and NVIDIA Titan V for oMAP with a recursive versus a linear implementation. Grid size of $2,000 \times 2,000$ was used in all cases.
& **2,000 / 1** & **1 / 2,000**\
------------------------
Run-time (s) on
**NVIDIA GTX 1080 Ti**
------------------------
: \[gpu\_comp\] Run-time comparison between NVIDIA GTX 1080 Ti and NVIDIA Titan V for oMAP with a recursive versus a linear implementation. Grid size of $2,000 \times 2,000$ was used in all cases.
& 2.370 & 117.314\
--------------------
Runt-time (s) on
**NVIDIA Titan V**
--------------------
: \[gpu\_comp\] Run-time comparison between NVIDIA GTX 1080 Ti and NVIDIA Titan V for oMAP with a recursive versus a linear implementation. Grid size of $2,000 \times 2,000$ was used in all cases.
& 0.871 & 109.299\
[10]{}
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[^1]: The research leading to these results has received funding from the European Community’s H2020 Programme (Future and Emerging Technologies, FET) under grant agreement no. 732266, Plan4Act.
[^2]: $^{1}$T. Kulvicius, S. Herzog, M. Tamosiunaite and F. Wörgötter are with Department for Computational Neuroscience, University of Göttingen, 37073 Göttingen, Germany
[^3]: $^{2}$M. Tamosiunaite is also with the Faculty of Computer Science, Vytautas Mangnus University, Kaunas, Lithuania
[^4]: T.K. contributed to the conception and design of the work, development and implementation of the algorithm, data acquisition, analysis and interpretation of the data, and writing of the paper. S.H. contributed to the implementation of the algorithm, data acquisition, analysis and interpretation of the data. M.T. contributed to the analysis and interpretation of the data, and writing of the paper. F.W. contributed to the conception and design of the work, analysis and interpretation of the data, and writing of the paper.
[^5]: $^{*}$Correspondence should be sent to T.K. ([[email protected]]{})
[^6]: A preprint of this paper has been uploaded to <https://arxiv.org>
[^7]: Note, in the Appendix we will show that recursive running of oMAP is very slow due to a per-iteration required CPU-GPU handshake. Thus, for practical purposes, we did not run oMAP in recursive mode and determined $L$ instead by prima-vista estimating path complexity.
[^8]: The data and source code will be published online after acceptance.
[^9]: We are counting steps, where diagonal moves count the same as horizontal/vertical ones.
|
---
address:
- |
Nguyen Tu Cuong\
Institute of Mathematics\
18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam
- |
Tran Tuan Nam\
The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy\
and Ho Chi Minh University of Pedagogy\
280 An Duong Vuong, District 5, Ho Chi Minh City, Vietnam
author:
- Nguyen Tu Cuong
- Tran Tuan Nam
title: A local homology theory for linearly compact modules
---
[**Abstract.**]{} We introduce a local homology theory for linearly compact modules which is in some sense dual to the local cohomology theory of A. Grothendieck. Some basic properties such as the noetherianness, the vanishing and non-vanishing of local homology modules of linearly compact modules are proved. A duality theory between local homology and local cohomology modules of linearly compact modules is developed by using Matlis duality and Macdonald duality. As consequences of the duality theorem we obtain some generalizations of well-known results in the theory of local cohomology for semi-discrete linearly compact modules.
[*2000 Mathematics subject classification*]{}: 13D07, 13D45, 16E30.
[*Key words*]{}: linearly compact module, semi-discrete module, local homology, local cohomology.
1.5cm
Introduction {#S:intro}
============
Although the theory of local cohomology has been enveloped rapidly for the last 40 years and proved to be a very important tool in algebraic geometry and commutative algebra, not so much is known about the theory of local homology. First, E. Matlis in [@matthe], [@matthe2] studied the left derived functors $L^I_\bullet(-)$ of the $I-$adic completion functor $\Lambda_I(-)=\inlim(R/I^t\otimes_R -)$, where the ideal $I$ was generated by a regular sequence in a local noetherian ring $R$ and proved some duality between this functor and the local cohomology functor by using a duality which is called today the Matlis dual functor. Next, Simon in [@simsom] suggested to investigate the module $L^I_i(M)$ when $M$ is complete with respect to the $I$-adic topology. Later, J. P. C. Greenlees and J. P. May [@greder] using the homotopy colimit, or telescope, of the cochain of Koszul complexes to define so called local homology groups of a module $M$ by $$H_\bullet^I(M)=
H_\bullet(\text{Hom(Tel} K^\bullet (\underline{x}^t), M)),$$ where $\underline{x}$ is a finitely generated system of $I$ and they showed, under some condition on $\underline x$ which are automatically satisfied when $R$ is noetherian, that the left derived functors $L^I_\bullet(-)$ of the $I-$adic completion can be computed in terms of these local homology groups. Then came the work of L. Alonso Tarr' io, A. Jeremias L' opez and J. Lipman [@aloloc], they gave in that paper a sheafified derived-category generalization of Greenlees-May results for a quasi-compact separated scheme. Note that a strong connection between local cohomology and local homology was shown in [@aloloc] and [@greder]. Recently in [@cuothe], we defined the $i$-th [*local homology module*]{} $H^I_i(M)$ of an $R-$module $M$ with respect to the ideal $I$ by $$H^I_i(M)=\underset{t}{\underleftarrow{\lim}}
{\Tor}^R_i(R/I^t,M).$$ We also proved in [@cuothe] many basic properties of local homology modules and that $H^I_i(M)\cong L^I_i(M)$ when $M$ is artinian. Hence we can say that there exists a theory for the left derived functors $L^I_\bullet(-)$ of the $I-$adic completion (as the local homology functors) on the category of artinian modules over noetherian local rings parallel to the theory of local cohomology functors on the category of noetherian modules. However, while the local cohomology functors $H_I^\bullet(-)$ are still defined as the right derived functors of the $I$-torsion functor $\Gamma
_I(-)= \underset{t}{\underrightarrow{\lim}}\Hom _R(R/I^t,- )$ for not finitely generated modules, our definition of local homology module above may not coincide with $L^I_i(M)$ in this case. One of the most important reasons is that, even if the ring $R$ is noetherian, the $I$-adic completion functor $\Lambda_I(-)$ is neither left nor right exact on the category of all $R$-modules. Fortunately, it was shown by results of C. U. Jensen in [@jenles] that the inverse limit functors and therefore the local homology functors still have good behaviour on the category of linearly compact modules. The purpose of this paper is towards a local homology theory for linearly compact modules. It should be mentioned that the concept of linearly compact spaces was first introduced by Lefschetz [@lefalg] for vector spaces of infinite dimension and it was then generalized for modules by D. Zelinsky [@zellin] and I. G. Macdonald [@macdua]. It was also studied by other authors such as H. Leptin [@leflin], C. U. Jensen [@jenles], H. Zöschinger [@zoslin] $\ldots.$ The class of linearly compact modules is very large, it contains many important classes of modules such as the class of artinian modules, or the class of finitely generated modules over a complete ring.
The organization of our paper is as follows. In section \[S:mdcptt\] we recall the concepts of linearly compact and semi-discrete linearly compact modules by using the terminology of Macdonald [@macdua] and their basic facts . For any $R-$module $N$ and a $R-$module $M$ we show that there exists uniquely a topology induced by a free resolution of $N$ for $\Ext_R^i(N,M)$, and in addition $N$ is finitely generated, for $\Tor^R_i(N,M)$; moreover these modules are .
In section \[S:mddddp\] we present some basic properties of local homology modules of modules such as the local homology functor $H^I_i(-)$ is closed in the category of modules (Proposition \[P:dddpcpttlcptt\]). Proposition \[P:mdttdnvgrl\] shows that our definition of local homology modules can be identified with the definition of local homology modules of J. P. C. Greenlees and J. P. May [@greder 2.4] in the category of linearly compact modules.
In section \[S:ttvkttdddp\] we study the vanishing and non-vanishing of local homology modules. Let $M$ be a linearly compact $R-$module with $\Ndim M=d,$ then $H^I_i (M)=0$ for all $i> d$ (Theorem \[T:dlttdddpcp\]). It was proved in [@cuothe 4.8, 4.10] that $\Ndim M= \max \{ i\mid H^{\m}_i(M)\not=0\}$ if $M$ is an artinian module over a local ring $(R,\m)$, where $\Ndim
M$ is the noetherian dimension defined by N. R. Roberts [@robkru] (see also [@kirdim]). Unfortunately, as in a personal communication of H. Zöschinger, he gave us the existence of semi-discrete modules $K$ of noetherian dimension 1 such that $H^{\m}_i(K)=0$ for all non-negative integers $i$. However, we can prove in Theorem \[T:dlttdddpcpnrr\] that the above equality still holds for semi-discrete modules with $\Ndim M\not = 1,$ moreover $\Ndim
\Gamma_{\m}(M) = \max \big\{ i\mid H^{\m}_i(M)\not=0\big\}$ if $\Gamma_{\m}(M)\not= 0.$
In section \[S:tcntedddp\] we show that local homology modules $H^{\m}_i(M)$ of a semi-discrete linearly compact module $M$ over a noetherian local ring $(R,\m)$ are noetherian modules on the $\m-$adic completion $\widehat{R}$ of $R$ (Theorem \[T:dltcnotemddddp\]) On the other hand, for any ideal $I$, $H^I_d(M)$ is a noetherian $\Lambda_I(R)-$module provided $M$ is a semi-discrete linearly compact $R-$module with the Noetherian dimension $\Ndim M=d$ (Theorem \[T:dlmddddpnotetcd\]).
Section \[S:dnmdn\] is devoted to study duality. In this section $(R,\m)$ is a noetherian local ring and the topology on $R$ is the $\m-$adic topology. Let $E(R/\m)$ be the injective envelope of $R/\m$ and $M$ a $R-$module. Then the [*Macdonald dual*]{} $M^*$ of $M$ is defined by $M^*=Hom(M,E(R/\m))$ the set of continuous homomorphisms of $R-$modules. Note by Macdonald [@macdua 5.8] that $M$ is a semi-discrete module if and only if $D(M)=M^*$, where $D(M)=\Hom(M,E(R/\m))$ is the Matlis dual of $M$. The main result of this section is Theorem \[T:dldndddpvddddp\] which gives a duality between local cohomology modules and local homology modules.
In the last section, based on the duality theorem \[T:dldndddpvddddp\] and the properties of local homology modules in previous sections we can extend some well-known properties of local cohomology of finitely generated modules for semi-discrete modules.
In this paper, the terminology “isomorphism” means “algebraic isomorphism” and “topological isomorphism” means “algebraic isomorphism with the homomorphisms (and its inverse) are continuous”.
Linearly compact modules {#S:mdcptt}
=========================
First we recall the concept of linearly compact modules by using the terminology of I. G. Macdonald [@macdua] and some their basic properties. Let $M$ be a topological $R-$module. A [*nucleus*]{} of $M$ is a neighbourhood of the zero element of $M,$ and a [*nuclear base*]{} of $M$ is a base for the nuclei of $M.$ If $N$ is a submodule of $M$ which contains a nucleus then $N$ is open (and therefore closed) in $M$ and $M/N$ is discrete. $M$ is Hausdorff if and only if the intersection of all the nuclei of $M$ is $0.$ $M$ is said to be [*linearly topologized*]{} if $M$ has a nuclear base $\mathcal{M}$ consisting of submodules.
\[D:dnmdcptt\] A Hausdorff linearly topologized $R-$module $M$ is said to be [*linearly compact*]{} if $M$ has the following property: if $\mathcal{F}$ is a family of closed cosets (i.e., cosets of closed submodules) in $M$ which has the finite intersection property, then the cosets in $\mathcal{F}$ have a non-empty intersection.
It should be noted that an artinian $R-$module is linearly compact with the discrete topology (see [@macdua 3.10]).
\[R:rmcsmdtt\] Let $M$ be an $R-$module. If $\mathcal{M}$ is a family of submodules of $M$ satisfying the conditions:
\(i) For all $N_1,\ N_2 \in \mathcal{M}$ there is an $N_3\in \mathcal{M}$ such that $N_3\subseteq N_1\cap N_2,$
\(ii) For an element $x\in M$ and $N\in \mathcal{M}$ there is a nucleus $U$ of $R$ such that $Ux \subseteq N,$
then $\mathcal{M}$ is a base of a linear topology on $M$ (see [@macdua 2.1]).
The following properties of linearly compact modules are often used in this paper.
\[L:tcmdcptt4\] [(see [@macdua §3])]{} (i) Let $M$ be a $R-$module, $N$ a closed submodule of $M.$ Then $M$ is linearly compact if and only if $N$ and $M/N$ are linearly compact.
\(ii) Let $f: M\longrightarrow N$ be a continuous homomorphism of $R-$modules. If $M$ is linearly compact, then $f(M)$ is linearly compact and therefore $f$ is a closed map.
\(iii) If $\{M_i\}_{i\in I}$ is a family of linearly compact $R-$modules, then $\underset{i\in I}\prod M_i$ is linearly compact with the product topology.
\(iv) The inverse limit of a system of linearly compact $R-$modules and continuous homomorphisms is linearly compact with the obvious topology.
\[L:lnkhmdcptt\] [(see [@jenles 7.1])]{} Let $\{M_t\}$ be an inverse system of linearly compact modules with continuous homomorphisms. Then $\underset{t}{\underleftarrow{\lim}^i} M_t =0$ for all $i>0.$ Therefore, if $$0\longrightarrow
\{M_t\}\longrightarrow\{N_t\} \longrightarrow \{P_t\}
\longrightarrow 0$$ is a short exact sequence of inverse systems of $R-$modules, then the sequence of inverse limits $$0\longrightarrow \underset{t}{\underleftarrow{\lim}} M_t
\longrightarrow \underset{t}{\underleftarrow{\lim}} N_t
\longrightarrow \underset{t}{\underleftarrow{\lim}} P_t
\longrightarrow 0$$is exact.
Let $M$ be a linearly compact $R-$module and $F$ a free $R-$module with a base $\big\{ e_i\big\}_{i\in J}.$ We can define the topology on $\Hom_R(F,M)$ as the product topology via the isomorphism $\Hom_R(F,M)\cong M^J,$ where $M^J=\underset{i\in
J}\prod M_i$ with $M_i=M$ for all $i\in J.$ Then $\Hom_R(F,M)$ is a $R-$module by \[L:tcmdcptt4\] (iii). Moreover, if $h: F
\longrightarrow F'$ is a homomorphism of free $R-$modules, the induced homomorphism $h^*: \Hom_R(F',M)\longrightarrow
\Hom_R(F,M)$ is continuous by [@jenles 7.4]. Let now $${\bold F_{\bullet}}: \ \ldots\longrightarrow F_i \longrightarrow \ldots \longrightarrow F_1\longrightarrow F_0
\longrightarrow N \longrightarrow 0.$$ a free resolution of an $R$-module $N$. Then $\Ext_R^i(N,M)$ is a linearly topologized $R$-module with the quotient topology of $\Hom(F_i, M)$. This topology on $\Ext_R^i(N,M)$ is called the topology induced by the free resolution ${\bold F_{\bullet}}$ of $N$.
\[L:extcptt\] Let $M$ be a linearly compact $R-$module and $N$ an $R-$module. Then for all $i\geq 0,\ \Ext_R^i(N,M)$ is a linearly compact $R-$module with the topology induced by a free resolution of $N$ and this topology is independent of the choice of free resolutions of $N.$ Moreover, if $f: N\longrightarrow N'$ is a homomorphism of $R-$modules, then the induced homomorphism $
\Ext_R^i(N',M) \longrightarrow \Ext_R^i(N,M)$ is continuous.
Let ${\bold F_{\bullet}}$ be a free resolution of $N.$ It follows as about that $\Hom_R({\bold F_{\bullet}},M)$ is a complex of linearly compact modules with continuous homomorphisms. Therefore $\Ext^i_R(N,M)=H^i(\Hom_R({\bold F_{\bullet}},M))$ is by \[L:tcmdcptt4\] (i), (ii). Let now ${\bold G_{\bullet}}$ be a second free resolution of $N$. Then we get a quasi-isomorphism of complexes $\varphi_{\bullet} :
{\bold F_{\bullet}} \longrightarrow {\bold G_{\bullet}}$ lifting the identity map of $N$. Therefore the induced homomorphism $$\bar{\varphi}_i: H^i({\Hom}_R({\bold F_{\bullet}},M))\longrightarrow H^i({\Hom}_R({\bold G_{\bullet}},M))$$ is a topological isomorphism by [@jenles 7.4] and \[L:tcmdcptt4\] (i), (ii) for all $i$. Similarly we can prove for the rest statement
Let $N$ be a finitely generated $R-$module and $${\bold
F_{\bullet}}=\ \ldots\longrightarrow F_i \longrightarrow \ldots
\longrightarrow F_1\longrightarrow F_0 \longrightarrow N
\longrightarrow 0$$ a free resolution of $N$ with the finitely generated free modules. As above, we can define for a linearly compact module $M$ a topology on $\Tor^R_i(N,M)$ induced from the product topology of $F_i\otimes_R M$. Then by an argument analogous to that used for the proof of Lemma \[L:extcptt\], we get the following lemma.
\[L:torcptt\] Let $N$ be a finitely generated $R-$module and $M$ a linearly compact $R-$module. Then $\Tor^R_i(N,M)$ is a linearly compact $R-$module with the topology induced by a free resolution of $N$ (consisting of finitely generated free modules) and this topology is independent of the choice of free resolutions of $N.$ Moreover, if $f: N\longrightarrow N'$ is a homomorphism of finitely generated $R-$modules, then the induced homomorphism $\psi_{i,M} : \Tor^R_i(N,M) \longrightarrow \Tor^R_i(N',M)$ is continuous.
The next result is often used in the sequel.
\[L:ghnghtor\] Let $N$ be a finitely generated $R-$module and $\{M_t\}$ an inverse system of linearly compact $R-$modules with continuous homomorphisms. Then for all $i\geq 0,$ $\{\Tor^R_i
(N, M_t )\}$ forms an inverse system of linearly compact modules with continuous homomorphisms. Moreover, we have $${\Tor}^R_i (N, \underset{t}{\underleftarrow{\lim}} M_t )
\cong \underset{t}{\underleftarrow{\lim}} {\Tor}^R_i (N, M_t ).$$
Let ${\bold F_\bullet}$ be a free resolution of $N$ with finitely generated free $R-$modules. Since $\{M_t\}$ is an inverse system of linearly compact modules with continuous homomorphisms, $\{F_i\otimes_R M_t\}$ forms an inverse system of linearly compact modules with continuous homomorphisms for all $i\geq 0$ by \[L:tcmdcptt4\] (iii). Then $ \{\Tor^R_i (N, M_t
)\}$ forms an inverse system of linearly compact modules with continuous homomorphisms. Moreover $${\bold F_\bullet} \otimes_R \underset{t}{\underleftarrow{\lim}} M_t
\cong \underset{t}{\underleftarrow{\lim}}({\bold F_\bullet}
\otimes_R M_t ),$$since the inverse limit commutes with the direct product and $$H_i (\underset{t}{\underleftarrow{\lim}}({\bold F_\bullet} \otimes_R M_t ))
\cong\underset{t}{\underleftarrow{\lim}} H_i ({\bold F_\bullet}
\otimes_R M_t )$$by \[L:lnkhmdcptt\] and [@norani 6.1, Theorem 1]. This finishes the proof.
A $R-$module $M$ is called [*semi-discrete*]{} if every submodule of $M$ is closed. Thus a discrete $R-$module is semi-discrete. The class of semi-discrete linearly compact modules contains all artinian modules. Moreover, it also contains all finitely generated modules in case $R$ is a complete local noetherian ring (see [@macdua 7.3]). It should be mentioned here that our notions of linearly compact and semi-discrete modules follow Macdonald’s definitions in [@macdua]. Therefore the notion of linearly compact modules defined by H. Zöschinger in [@zoslin] is different to our notion of linearly compact modules, but it is coincident with the terminology of semi-discrete linearly compact modules in this paper.
Denote by $L(M)$ the sum of all artinian submodules of $M,$ we have the following properties of semi-discrete linearly compact modules.
\[L:lmcpttnrrat\] [(see [@zoslin 1 (L5)])]{} Let $M$ be a semi-discrete linearly compact $R-$module. Then $L(M)$ is an artinian module.
We now recall the concept of [*co-associated primes*]{} of a module (see [@chacop], [@yascoa], [@zoslin]). A prime ideal $\p$ is called [*co-associated*]{} to a non-zero $R-$module $M$ if there is an artinian homomorphic image $L$ of $M$ with $\p=\Ann_R L.$ The set of all co-associated primes to $M$ is denoted by $\Coass_R (M).$ $M$ is called $\p-$[*coprimary*]{} if $\Coass_R(M)=\big\{ \p\big\}.$ A module is called [*sum-irreducible*]{} if it can not be written as a sum of two proper submodules. A sum-irreducible module $M$ is $\p-$coprimary, where $\p=\big\{ x\in R/ x M\not= M \big\}$ (see [@chacop 2]).
\[L:cpttnrrcahh\] [(see [@zoslin 1 (L3,L4)])]{} Let $M$ be a semi-discrete linearly compact $R-$module. Then $M$ can be written as a finite sum of sum-irreducible modules and therefore the set $\Coass(M)$ is finite.
Local homology modules of linearly compact modules {#S:mddddp}
===================================================
Let $I$ be an ideal of $R,$ the [*i-th local homology*]{} module $H^I_i (M)$ of an $R-$module $M$ with respect to $I$ is defined by (see [@cuothe 3.1]) $$H^I_i (M) = \underset{t}{\underleftarrow{\lim}}{\Tor}^R_i (R/I^t , M).$$It is clear that $H^I_0(M)\cong \Lambda_I(M),$ in which $\Lambda_I(M)=\underset{t}{\underleftarrow{\lim}} M/I^tM$ the $I-$adic completion of $M.$
\[R:rmdndddp\] (i) As $I^t{\Tor}^R_i(R/I^t,M)=0,$ ${\Tor}^R_i(M/I^tM,N)$ has a natural structure as a module over the ring $R/I^t$ for all $t>0.$ Then $H^I_i(M) =
\underset{t}{\underleftarrow{\lim}}\Tor^R_i (R/I^t , M)$ has a natural structure as a module over the ring $\Lambda_I(R)=\inlim
R/I^t.$
\(ii) If $M$ is a finitely generated $R-$module, then $H^I_i(M)=0$ for all $i>0$ (see [@cuothe 3.2 (ii)]).
\[L:dddpvpksvdn\] [(see [@cuothe §3])]{} Let $I$ be an ideal generated by elements $ x_1, x_2,\ldots, x_r$ and $H_i(\underline{x}(t),M)$ the $i-$th Koszul homology module of $M$ with respect to the sequence $\underline{x}(t)=(x^t_1,\ldots,
x^t_r).$ Then for all $i\geq 0,$
\(i) $H^I_i (M) \cong \underset{t}{\underleftarrow{\lim}}
H_i (\underline{x}(t), M),$
\(ii) $H^I_i(M)$ is $I-$separated, it means that $\underset{t>0}\bigcap I^t H^I_i(M)=0.$
Let $M$ be a $R-$module. Then $\Tor^R_i(R/I^t,M)$ is also a $R-$module by the topology defined as in \[L:torcptt\], so we have an induced topology on the local homology module $H^I_i(M).$
\[P:dddpcpttlcptt\] Let $M$ be a linearly compact $R-$module. Then for all $i\geq 0,$ $H^I_i(M)$ is a linearly compact $R-$module.
It follows from \[L:torcptt\] that $\{\Tor^R_i(R/I^t,M)\}_t$ forms an inverse system of linearly compact modules with continuous homomorphisms. Hence $H^I_i(M)$ is also a linearly compact $R-$module by \[L:tcmdcptt4\] (iv).
The following proposition shows that local homology modules can be commuted with inverse limits of inverse systems of linearly compact $R-$modules with continuous homomorphisms.
\[P:dddpghghn\] Let $\{ M_s\}$ be an inverse system of linearly compact $R-$modules with the continuous homomorphisms. Then $$H^I_i(\underset{s}\varprojlim
M_s)\cong\underset{s}\varprojlim H^I_i( M_s).$$
Note that inverse limits are commuted. Therefore $$\begin{aligned}
H^I_i(\underset{s}\varprojlim M_s)&=\underset{t}\varprojlim {\Tor}^R_i(R/I^t,\underset{s}\varprojlim M_s)\\
&\cong \underset{t}\varprojlim\underset{s}\varprojlim {\Tor}^R_i(R/I^t, M_s)\\
&\cong \underset{s}\varprojlim\underset{t}\varprojlim
{\Tor}^R_i(R/I^t, M_s) =\underset{s}\varprojlim H^I_i(
M_s)\end{aligned}$$ by \[L:ghnghtor\].
Let $L^I_i$ be the $i-$th left derived functor of the $I-$adic completion functor $\Lambda_I.$ The next result shows that in case $M$ is linearly compact, the local homology module $H^I_i (M)$ is isomorphic to the module $L^I_i(M),$ thus our definition of local homology modules can be identified with the definition of J. P. C. Greenlees and J. P. May (see [@greder 2.4]).
\[P:mdttdnvgrl\] Let $M$ be a linearly compact $R-$module. Then $$H^I_i (M)\cong L_i^I (M)$$ for all $i\geq
0.$
For all $i\geq 0$ we have a short exact sequence by [@greder 1.1], $$0\longrightarrow
\underset{t}{\underleftarrow{\lim}^1}{\Tor}^R_{i+1}(R/I^t,M)\longrightarrow
L^I_i(M)\longrightarrow H^I_i(M)\longrightarrow 0.$$Moreover, it follows from \[L:torcptt\] that $\{\Tor^R_{i+1}(R/I^t,M)\}$ forms an inverse system of linearly compact modules with continuous homomorphisms. Hence, by \[L:lnkhmdcptt\] $$\underset{t}{\underleftarrow{\lim}^1}{\Tor}^R_{i+1} (R/I^t,M)=0$$ and the conclusion follows.
The following corollary is an immediate consequence of \[P:mdttdnvgrl\].
\[C:hqdkdddp\] Let $$0 \longrightarrow M' \longrightarrow M \longrightarrow M" \longrightarrow 0$$ be a short exact sequence of linearly compact modules. Then we have a long exact sequence of local homology modules $$\cdots \longrightarrow H^I_i (M') \longrightarrow H^I_i (M)\longrightarrow H^I_i (M")\longrightarrow \ $$ $$\cdots \longrightarrow H^I_0 (M') \longrightarrow H^I_0 (M)\longrightarrow H^I_0 (M")\longrightarrow 0.$$
The following theorem gives us a characterization of $I-$separated modules.
\[T:dldtmditach\] Let $M$ be a linearly compact $R-$module. The following statements are equivalent:
\(i) $M$ is $I-$separated, it means that $\underset{t>0}\bigcap I^t M =0.$
\(ii) $M$ is complete with respect to the $I-$adic topology, it means that $\Lambda_I(M) \cong M.$
\(iii) $H^I_0 (M) \cong M,\ H^I_i (M)= 0$ for all $i>0.$
To prove Theorem \[T:dldtmditach\], we need the two auxiliary lemmas. The first lemma shows that local homology modules $H^I_i
(M)$ are $\Lambda_I-$acyclic for all $i>0.$
\[L:dddpacylic\] Let $M$ be a linearly compact $R-$module. Then for all $j\geq 0,$ $$H^I_i (H^I_j (M)) \cong \begin{cases} H^I_j (M),& i= 0, \\
0, & i>0.\end{cases}$$
It follows from \[L:torcptt\] that $\big\{\Tor^R_j(R/I^t,
M)\big\}_t$ forms an inverse system of $R-$modules with the continuous homomorphisms. Then we have by \[P:dddpghghn\] and \[L:dddpvpksvdn\] (i), $$\begin{aligned}
H^I_i (H^I_j (M))
&= H^I_i(\underset{t}{\underleftarrow{\lim}}{\Tor}^R_j (R/I^t , M))\\
&\cong \underset{t}{\underleftarrow{\lim}} H^I_i({\Tor}^R_j (R/I^t , M)) \\
&\cong
\underset{t}{\underleftarrow{\lim}}\underset{s}{\underleftarrow{\lim}}
H_i (\underline{x}(s), {\Tor}^R_j (R/I^t , M)),\end{aligned}$$ in which $\underline{x} = (x_1 , \ldots ,x_r )$ is a system of generators of $I$ and $\underline{x}(s) = (x_1^s , \ldots ,x_r^s).$ Since $\underline{x}(s){\Tor}^R_j (R/I^t , M) = 0$ for all $s\geq
t,$ we get $$\underset{s}{\underleftarrow{\lim}} H_i (\underline{x}(s), {\Tor}^R_j (R/I^t , M)) \cong
\begin{cases} {\Tor}^R_j (R/I^t , M),& i=0,\\
0,& i>0.\end{cases}$$This finishes the proof.
\[L:dcddgiaomd\] Let $M$ be a linearly compact $R-$module. Then $$H^I_i (\underset{t>0}\bigcap I^t M) \cong \begin{cases} 0,& i=0,\\
H^I_i (M),& i>0.\end{cases}$$
From the short exact sequence of linearly compact $R-$modules $$0\longrightarrow I^tM \longrightarrow M\ \longrightarrow M/I^tM
\longrightarrow 0$$ for all $t>0$ we derive by \[L:lnkhmdcptt\] a short exact sequence of linearly compact $R-$modules $$0\longrightarrow \underset{t>0}\bigcap I^tM \longrightarrow M \longrightarrow \Lambda_I(M) \longrightarrow 0.$$ Hence we get a long exact sequence of local homology modules $$\cdots \longrightarrow H^I_{i+1} (\Lambda_I(M)) \longrightarrow H^I_i (\underset{t>0}\bigcap I^tM) \longrightarrow H^I_i (M) \longrightarrow H^I_i (\Lambda_I(M))\longrightarrow $$ $$\cdots \longrightarrow H^I_1 (\Lambda_I(M)) \longrightarrow H^I_0 (\underset{t>0}\bigcap I^tM) \longrightarrow H^I_0 (M) \longrightarrow H^I_0 (\Lambda_I(M))\longrightarrow 0.$$ The lemma now follows from \[L:dddpacylic\].
$(ii)\Leftrightarrow
(i)$ is clear from the short exact sequence $$0\longrightarrow \underset{t>0}\bigcap I^t M \longrightarrow M\longrightarrow \Lambda_I(M)\longrightarrow 0.$$ $(i) \Rightarrow (iii).$ We have $H^I_0
(M)\cong\Lambda_I(M) \cong M.$ Combining \[L:dcddgiaomd\] with (i) gives $H^I_i (M)\cong H^I_i (\underset{t>0}\bigcap I^t M) =0$ for all $i > 0.$
$(iii) \Rightarrow (ii)$ is trivial.
From Theorem \[T:dldtmditach\] we have the following criterion for a finitely generated module over a local noetherian ring to be linearly compact.
\[C:dkmdhhscpttvdd\] Let $(R,\m)$ be a local noetherian ring and $M$ a finitely generated $R-$module. Then $M$ is a $R-$module if and only if $M$ is complete with respect to the $\m-$adic topology.
Since $M$ is a finitely generated $R-$module, $M$ is $\m-$separated. Thus, if $M$ is a $R-$module, $\Lambda_{\m}(M)\cong M$ by \[T:dldtmditach\]. Conversely, if $M$ is complete in $\m-$adic topology, we have $M\cong \inlim
M/\m^t M.$ Therefore $M$ is a $R-$module by \[L:tcmdcptt4\] (iv), as $M/\m^t M$ are artinian $R-$modules for all $t>0.$
Vanishing and non-vanishing of local homology modules {#S:ttvkttdddp}
======================================================
Recall that $L(M)$ is the sum of all artianian submodules of $M$ and $\Soc(M)$ the socle of $M$ is the sum of all simple submodules of $M$. The $I-$torsion functor $\Gamma_I$ is defined by $\Gamma _I(M) = \underset{t>0}\cup ({0:_M} {I^t}).$ To prove the vanishing and non-vanishing theorems of local cohomology modules, we need the following lemmas.
\[L:dktdptdcqdddp\] Let $M$ be a semi-discrete linearly compact $R-$module. Then $H^I_0(M)=0$ if and only if $xM=M$ for some $x\in
I.$
By [@cuothe 2.5], $H^I_0(M)=0$ if and only if $IM=M.$ Hence the result follows from \[L:cpttnrrcahh\] and [@chacop 2.9].
\[L:dddpmdsb0tt\] Let $M$ be a semi-discrete linearly compact $R-$module and $\Soc(M)=0$. Then $$H^I_i(M)=0$$for all $i>0.$
Combining \[L:dcddgiaomd\] with \[L:dktdptdcqdddp\]we may assume, by replacing $M$ with $\underset{t>0}\bigcap I^tM,$ that there is an $x\in
I$ such that $x M=M.$ As $\Soc(M)=0,$ it follows from [@zoslin 1.6 (b)] that $0:_Mx=0.$ Thus we have an isomorphism $M\overset{x}\cong M.$ It induces an isomorphism $$H^I_i(M)\overset{x}\cong H^I_i(M)$$for all $i>0.$ By \[L:dddpvpksvdn\] (ii), we have $$H^I_i(M) = x H^I_i(M) = \underset{t>0}\bigcap x^t H^I_i(M)=0$$for all $i>0.$
\[L:lmbtcmdx\] Let $M$ be a semi-discrete linearly compact $R-$module. Then there are only finitely many distinct maximal ideals ${\m}_1, {\m}_2, \ldots, {\m}_n$ of $R$ such that $$L(M) = \underset{j=1}{\overset{n}\bigoplus} \Gamma_{\m_j}(M).$$
By \[L:lmcpttnrrat\], $L(M)$ is an artinian $R-$module. Thus, by virtue of [@shaame 1.4] there are finitely many distinct maximal ideals ${\m}_1, {\m}_2, \ldots, {\m}_n$ of $R$ such that $$L(M) = \underset{j=1}{\overset{n}\bigoplus}
\Gamma_{\m_j}(L(M))\subseteq \underset{j=1}{\overset{n}\bigoplus}
\Gamma_{\m_j}(M).$$ Therefore it remains to show that $
\Gamma_{\m}(M)$ is artinian for any maximal ideal $\m$ of $R$. Indeed, there is from [@zoslin Theorem] a short exact sequence $0 \longrightarrow N \longrightarrow M \longrightarrow A
\longrightarrow 0$, where $N$ is finitely generated and $A$ is artinian. Then we have an exact sequence $$0 \longrightarrow \Gamma_{\m}(N) \longrightarrow \Gamma_{\m}(M)
\longrightarrow \Gamma_{\m}(A).$$ Obviously, $\Gamma_{\m}(A)$ is an artinian $R-$module, $\Gamma_{\m}(N)$ is a finitely generated $R-$module annihilated by a power of $\m$, and hence it is of finite length. So $\Gamma_{\m}(M)$ is an artinian $R-$module as required.
\[L:dddpbtdddpmdmx\] Let $M$ be a semi-discrete linearly compact $R-$module. Then there are only finitely many distinct maximal ideals ${\m}_1, {\m}_2, \ldots, {\m}_n$ of $R$ such that $$H^I_i(M)\cong \underset{j=1}{\overset{n}\bigoplus} H^I_i(\Gamma_{\m_j}(M))$$for all $ i>0,$ and the following sequence is exact $$0\longrightarrow \underset{j=1}{\overset{n}\bigoplus} H^I_0(\Gamma_{\m_j}(M))\longrightarrow H^I_0(M)\longrightarrow
H^I_0(M/\underset{j=1}{\overset{n}\bigoplus}\Gamma_{\m_j}(M))\longrightarrow
0.$$
The short exact sequence of linearly compact $R-$modules$$0\longrightarrow L(M)\longrightarrow
M\longrightarrow M/L(M)\longrightarrow 0$$gives rise to a long exact sequence of local homology modules $$\ldots\longrightarrow H^I_{i+1}(M/L(M))\longrightarrow H^I_i(L(M))\longrightarrow H^I_i(M)\longrightarrow H^I_i(M/L(M))\longrightarrow\ldots.$$ By \[L:dddpmdsb0tt\], $H^I_i(M/L(M))= 0$ for all $i>0,$ as $\Soc(M/L(M))=0.$ Then we get $H^I_i(M)\cong H^I_i(L(M))$ for all $i>0$ and the short exact sequence $$0\longrightarrow H^I_0(L(M))\longrightarrow H^I_0(M)\longrightarrow H^I_0(M/L(M))\longrightarrow 0.$$ Now the conclusion follows from \[L:lmbtcmdx\].
We have an immediate consequence of \[L:lmbtcmdx\] and \[L:dddpbtdddpmdmx\] for the local case.
\[C:dddpmbdddpgamamvdp\] Let $(R,\m)$ be a local noetherian ring and $M$ a semi-discrete linearly compact $R-$module. Then $$L(M)= \Gamma_{\m}(M), \ \ H^I_i(M)\cong H^I_i(\Gamma_{\m}(M))$$for all $ i>0,$ and the following sequence is exact $$0\longrightarrow H^I_0(\Gamma_{\m}(M))\longrightarrow H^I_0(M)\longrightarrow
H^I_0(M/\Gamma_{\m}(M))\longrightarrow 0.$$
We now recall the concept of [*Noetherian dimension*]{} of an $R-$module $M$ denoted by $\Ndim M.$ Note that the notion of Noetherian dimension was introduced first by R. N. Roberts [@robkru] by the name Krull dimension. Later, D. Kirby [@kirdim] changed this terminology of Roberts and refereed to [*Noetherian dimension*]{} to avoid confusion with well-know Krull dimension of finitely generated modules. Let $M$ be an $R-$module. When $M=0$ we put $\Ndim M = -1.$ Then by induction, for any ordinal $\alpha,$ we put $\Ndim M = \alpha$ when (i) $\Ndim M < \alpha$ is false, and (ii) for every ascending chain $M_0 \subseteq M_1 \subseteq \ldots$ of submodules of $M,$ there exists a positive integer $m_0 $ such that $\Ndim(M_{m+1} /M_m )<
\alpha$ for all $m \geq m_0$. Thus $M$ is non-zero and finitely generated if and only if $\Ndim M = 0.$ If $0 \longrightarrow M"
\longrightarrow M\longrightarrow M'\longrightarrow 0$ is a short exact sequence of $R-$modules, then $\Ndim M= \max\{ \Ndim M",
\Ndim M'\}.$
\[R:cntrmarcpnrr\] (i) In case $M$ is an artinian module, $\Ndim M
<\infty$ (see [@robkru]). More general, if $M$ is a semi-discrete module, there is a short exact sequence $0
\longrightarrow N \longrightarrow M \longrightarrow A
\longrightarrow 0$ where $N$ is finitely generated and $A$ is artinian (see [@zoslin Theorem])). Hence $\Ndim M = \max\{
\Ndim N, \Ndim A\} < \infty.$
\(ii) If $M$ is an artinian $R-$module or more general, a semi-discrete $R-$module, then $\Ndim M \leqslant \max\{\dim
R/\p \mid \p \in \Coass(M)\}.$ Especially, if $M$ is an artinian module over a complete local noetherian ring $(R,\m),$ $\Ndim M
= \max\{\dim R/\p \mid \p \in \Coass(M)\}$ (see [@yasmag 2.10]).
\[L:cntmoochiamx\] Let $M$ be an $R-$module with $\Ndim M=d>0$ and $x\in R$ such that $x M=M.$ Then $$\Ndim 0:_M x\leqslant
d-1.$$
Consider the ascending chain $$0\subseteq 0:_Mx\subseteq 0:_Mx^2 \subseteq \ldots.$$As $\Ndim M=d,$ there exists a positive integer $n$ such that $\Ndim (0:_Mx^{n+1}/0:_Mx^n) \leqslant d-1.$ Since $x M=M,$ the homomorphism $0:_Mx^{n+1}/0:_Mx^n \overset{x^n}\longrightarrow
0:_Mx$ is an isomorphism. Therefore $\Ndim 0:_M x\leqslant d-1.$
\[T:dlttdddpcp\] Let $M$ be a linearly compact $R-$module with $\Ndim M=d.$ Then $$H^I_i (M)=0$$ for all $i> d.$
Let $\mathcal{M}$ be a nuclear base of $M$. Then, by [@macdua 3.11], $M =
\underset{U\in\mathcal{M}}{\underleftarrow{\lim}} M/U.$ It follows from \[P:dddpghghn\] that $$H^I_i(M) \cong \underset{U\in\mathcal M}{\underleftarrow{\lim}}
H^I_i(M/U).$$ Note that $M/U$ is a discrete linearly compact $R-$module with $\Ndim M/U \leqslant \Ndim M.$ Thus we only need to prove the theorem for the case $M$ is a discrete linearly compact $R-$module. Let $L(M)$ be the sum of all artinian $R-$submodules of $M,$ by \[L:lmcpttnrrat\], $L(M)$ is atinian. From the proof of \[L:dddpbtdddpmdmx\], we have the isomorphisms $$H^I_i(M)\cong H^I_i(L(M))$$ for all $ i>0$. As $\Ndim L(M)\leqslant
\Ndim M=d,$ $H^I_i(L(M)) = 0$ for all $i>d$ by [@cuothe 4.8] and then the proof is complete.
\[R:rmdlttcpvdzg\] In [@cuothe 4.8, 4.10] we proved that if $M$ is an artinian module on a local noetherian ring $(R, \m),$ then $$\Ndim M= \max \big\{ i\mid H^{\m}_i(M)\not=0\big\},$$ where we use the convention that $\max(\emptyset)=-1$. Therefore it raises to the following natural question that whether the above equality holds true when $M$ is a semi-discrete linearly compact module? Unfortunately, the answer is negative in general. The following counter-example is due to H. Z" oschinger. Let $(R,\m)$ be a complete local noetherian domain of dimension $1$ and $K$ the field of fractions of $R.$ Consider $K$ as an $R-$module. Then $\Soc(K)=0$ and $\Coass(K)=\{0\},$ therefore $\Ndim K =1$ by [@zoslin 1.6 (a)]. Since $K/R$ is artinian, it follows by [@zoslin Theorem] that $K$ is a semi-discrete linearly compact $R-$module. As $xK=K$ for any non-zero element $
x\in \m$, $H^{\m}_0(K)=0$ by \[L:dktdptdcqdddp\]. Moreover, we obtain by \[L:dddpmdsb0tt\] that $H^{\m}_i(K)=0$ for all $i>0$ . Thus $$\Ndim K =1\not = -1= \max \big\{ i\mid H^{\m}_i(K)\not=0\big\}.$$
However, the following theorem gives an affirmative answer for the question when $\Ndim M \not= 1$.
\[T:dlttdddpcpnrr\] Let $(R,\m)$ be a local noetherian ring and $M$ a non zero semi-discrete linearly compact $R-$module. Then
(i) $\Ndim \Gamma_{\m}(M) = \max \big\{ i\mid
H^{\m}_i(M)\not=0\big\}$ if $ \Gamma_{\m}(M)\not= 0;$
\(ii) $\Ndim M= \max \big\{ i\mid
H^{\m}_i(M)\not=0\big\}$ if $\Ndim M\not= 1.$
\(i) Since $\Gamma_{\m}(M)$ is the artinian $R-$module, we obtain from [@cuothe 4.8, 4.10] that $$\Ndim
\Gamma_{\m}(M) = \max \big\{ i\mid
H^{\m}_i(\Gamma_{\m}(M))\not=0\big\}.$$ Thus (i) follows from \[C:dddpmbdddpgamamvdp\].
\(ii) First, note by virtue of [@zoslin 1.6 (a)] and \[R:cntrmarcpnrr\] (ii) that if $\Soc (M) = 0$ then $\Ndim M
\leqslant 1$. If $\Gamma_{\m}(M)=0$ then $\Soc (M)=0$ by \[L:lmbtcmdx\]. So we get from the hypothesis that $\Ndim M =0$. It follows that $M$ is a finitely generated $R-$module and $H^{\m}_0(M)\cong \widehat M \not = 0$, where $\widehat M$ is the $\m-$adic completion of $M$. Thus (ii) is proved in this case. Assume now that $\Gamma_{\m}(M)\not = 0$. By (i) we have only to show that $\Ndim M=\Ndim \Gamma_{\m}(M).$ Indeed, it is trivial for the case $\Ndim M=0$. Let $\Ndim M>1.$ From the short exact sequence $0 \longrightarrow \Gamma_{\m}(M) \longrightarrow M
\longrightarrow M/\Gamma_{\m}(M) \longrightarrow 0$ we get $$\Ndim M = \max\{\Ndim \Gamma_{\m}(M), \Ndim M/\Gamma_{\m}(M)\}.$$ Since $\Soc(M/\Gamma_{\m}(M))=0$, $\Ndim
(M/\Gamma_{\m}(M))\leqslant 1$. Thus $\Ndim M=\Ndim
\Gamma_{\m}(M)$ as required.
A sequence of elements $x_1 ,\ldots ,x_r$ in $R$ is said to be an [*$M-$coregular*]{} sequence (see [@ooimat 3.1]) if $0:_M
(x_1 , \ldots , x_r ) \not= 0$ and $0:_M (x_1 , \ldots , x_{i-1} )
\overset{x_i}\longrightarrow 0:_M (x_1 , \ldots , x_{i-1} ) $ is surjective for $i=1,\ldots , r.$ We denote by $\Width_I(M)$ the supremum of the lengths of all maximal $M-$coregular sequences in the ideal $I.$ Note by \[R:cntrmarcpnrr\] (i) and \[L:cntmoochiamx\] that $${\Width}_I(M)\leqslant \Ndim M<\infty$$ when $M$ is a semi-discrete $R-$module.
\[T:dtdrdddpcpnrr\] Let $M$ be a semi-discrete linearly compact $R-$module and $I$ an ideal of $R$ such that $0:_MI\not=0.$ Then all maximal $M-$coregular sequences in $I$ have the same length. Moreover $${\Width}_I(M)=\inf\{i/H^I_i(M)\not=0\}.$$
It is sufficient to prove that if $\{x_1,
x_2,\ldots,x_n\}$ is a maximal $M-$coregular sequence in $ I$, then $H^I_i (M)=0$ for all $i<n,$ and $H^I_n (M)\not=0.$ We argue by the induction on $n$. When $n=0,$ there does not exists an element $x$ in $I$ such that $xM=M.$ Then $H^I_0(M)\not=0$ by \[L:dktdptdcqdddp\].
Let $n>0.$ The short exact sequence $$0\longrightarrow 0:_Mx_1
\longrightarrow M\overset{x_1}\longrightarrow M\longrightarrow 0$$ gives rise to a long exact sequence $$\ldots\longrightarrow H^I_i(M)\overset{x_1}\longrightarrow H^I_i(M) \longrightarrow H^I_{i-1}(0:_M{x_1})\longrightarrow\ldots.$$ By the inductive hypothesis, $H^I_i(0:_M{x_1})=0 $ for all $i<n-1$ and $H^I_{n-1}(0:_M{x_1})\not=0.$ Therefore by virtue of \[L:dddpvpksvdn\] (ii), $H^I_i(M)=x_1H^I_i(M)=\underset{t>0}\bigcap x_1^tH^I_i(M)=0$ for all $i<n.$ Now, it follows from the exact sequence $$\ldots\longrightarrow H^I_n(M)\overset{x_1}\longrightarrow H^I_n(M) \longrightarrow H^I_{n-1}(0:_M{x_1})\longrightarrow 0$$ and $H^I_{n-1}(0:_M{x_1})\not=0$ that $H^I_n(M)\not=0$ as required.
\[R:rmdkgamanotoct\] We give here an example which shows that the condition $\Gamma _{\m} (M)\not= 0$ in Theorem \[T:dlttdddpcpnrr\] (i) is needful. Let $R$ be the ring and $K$ the $R-$module as in \[R:rmdlttcpvdzg\]. Set $M=N\oplus K$, where $N$ is a finitely generated $R-$module satisfying $\depth
_{\m} M \geq 1$. Then, it is easy to check that $\Gamma _{\m} (M)=
0$ and $H_i^{\m} (M)\cong H_i^{\m} (K) = 0$ for all $i\geq 1$ and $H_0^{\m} (M)\cong H_0^{\m} (N)\cong \widehat N\not= 0$. Therefore $$\Ndim \Gamma_{\m}(M) = -1 \not= 0=\max \big\{ i\mid
H^{\m}_i(M)\not=0\big\}.$$
We have seen in Remark \[R:rmdlttcpvdzg\] the existence of a non-zero semi-discrete module $K$ such that $H^{\m}_i (K)=0$ for all $i\geq 0$. Below, we give a characterization for this class of semi-discrete modules. This corollary also shows that we can not drop the condition $0:_M I\not= 0$ in the assumption of Theorem \[T:dtdrdddpcpnrr\].
\[C:hqddpttmi\] Let $(R,\m)$ be a local noetherian ring and $M$ a non-zero semi-discrete module. Then $H^{\m}_i(M)
=0$ for all $i\geq 0$ if and only if there exists an element $x\in \m$ such that $xM=M$ and $0:_Mx=0$.
Let $H^{\m}_i(M) =0$ for all $i\geq 0$. We obtain by \[L:dktdptdcqdddp\] that $xM=M$ for some $x\in \m$. On the other hand, it follows from the short exact sequence $0\longrightarrow
0:_Mx \longrightarrow M \overset{x}\longrightarrow M
\longrightarrow 0$ that $H^{\m}_i(0:_Mx)=0$ for all $i\geq 0.$ Since $0:_Mx$ is artinian by [@zoslin Corollary 1 (b0)], $0:_Mx=0$ by [@cuothe 4.10]. Conversely, suppose that $xM=M$ and $0:_Mx=0$, then for all $i\geq 0$ $$H^{\m}_i(M) = x
H^{\m}_i(M) = \underset{t>0}\bigcap x^tH^{\m}_i(M) =0$$ by \[L:dddpvpksvdn\] (ii).
Noetherian local homology modules {#S:tcntedddp}
==================================
First, the following criterion for a module to be noetherian is useful for the investigation of the noetherian property of local homology modules.
\[L:tcnote\] Let $J$ be a finitely generated ideal of a commutative ring $R$ such that $R$ is complete with respect to the $J-$adic topology and $M$ an $R-$module. If $M/JM$ is a noetherian $R-$module and $M$ is $J-$separated (i. e., $\underset{t>0}\bigcap
J^tM =0$), then $M$ is a noetherian $R-$module.
Set $$K = \underset{t\geq 0}\bigoplus J^tM/J^{t+1}M$$ the associated graded module over the graded ring$$Gr_{J} (R) = \underset{t\geq 0}\bigoplus\ J^t/J^{t+1} .$$ Let $x_1 , x_2 , \ldots , x_s$ be a system of generators of $J$ and $(R/J)[T_1 , \ldots ,T_s]$ the polynomial ring of variables $T_1 , T_2 , \ldots , T_s$. The natural epimorphism $$g: (R/J)[T_1 , \ldots ,T_s] \longrightarrow Gr_{J} (R)$$ leads $K$ to be an $(R/J)[T_1 , \ldots ,T_s]-$module. We write $K_t= J^tM/J^{t+1}M$ for all $t\geq 0,$ then $K_0 = M/J M$ is a noetherian $R/J-$module by the hypothesis. On the other hand, it is easy to check that $$K_{t+1} = \underset{i=1}{\overset{s}\sum} T_i K_t$$ for all $t\geq 0.$ Thus $K$ satisfies the conditions of [@kirart 1 (i)]. Then $K$ is a noetherian $(R/J )[T_1 , \ldots ,T_s]-$module and so $K$ is a noetherian $Gr_{J} (R)-$module. Moreover $M$ is $J-$separated by the hypothesis. Therefore $M$ is a noetherian $R-$module by [@atiint 10.25].
\[T:dltcnotemddddp\] Let $(R,\m)$ be a local noetherian ring and $M$ a semi-discrete linearly compact $R-$module. Then $H_i^{\frak{m}}(M)$ is a noetherian $\widehat{R}-$module for all $i\geq 0.$
We prove the theorem by induction on $i$. If $i=0,$ we have $H^{\frak{m}}_0 (M) \cong \Lambda_{\frak{m}} (M).$ As $M$ is a semi-discrete linearly compact $R-$module, $M/\m M$ is also a semi-discrete linearly compact $R/\m-$module. By virtue of [@macdua 5.2], $M/\m M$ is a finite dimensional vector $R/\m-$space. Then $\Lambda_{\m}(M)$ is a noetherian $\widehat{R}-$module by [@dieele 7.2.9]. Let $i>0.$ Combining \[L:dcddgiaomd\] with \[L:dktdptdcqdddp\] we may assume, by replacing $M$ with $\underset{t>0}\bigcap \m^tM,$ that there is an element $x\in \m$ such that $xM=M.$ Then the short exact sequence of modules $$0\longrightarrow 0:_Mx \longrightarrow M
\overset{x}\longrightarrow M \longrightarrow 0$$ gives rise to a long exact sequence of local homology modules $$\ldots\longrightarrow H^{\m}_i(M) \overset{x}\longrightarrow
H^{\m}_i(M) \overset{\delta}\longrightarrow
H^{\m}_{i-1}(0:_Mx)\longrightarrow \ldots.$$If $0:_Mx =0,$ then $H^{\m}_i(M)=x H^{\m}_i(M)=\underset{t>0}\bigcap x^tH^{\m}_i(M)=0$ for all $i\geq 0$ by \[L:dddpvpksvdn\] (ii). We now assume that $0:_Mx \not=0.$ By the inductive hypothesis, $H^{\m}_{i-1}(0:_Mx)$ is a noetherian $\widehat{R}-$module. Set $H=H^{\m}_i(M),$ we have $H/xH \cong \Im\delta \subseteq H^{\m}_{i-1}(0:_Mx).$ It follows that $H/xH$ is a noetherian $\widehat{R}-$module. Thus $H/\hat{\m}
H$ is also a noetherian $\widehat{R}-$module. Moreover, $\underset{t>0}\bigcap {\hat{\m}}^t H = \underset{t>0}\bigcap
{\m}^t H^{\m}_i(M) = 0.$ Therefore $H$ is a noetherian $\widehat{R}-$module by \[L:tcnote\].
\[T:dlmddddpnotetcd\] Let $(R,\m)$ be a local noetherian ring and $M$ a semi-discrete linearly compact $R-$module with $\Ndim
M=d$. Then $H^I_d(M)$ is a noetherian module on $\Lambda_I(R)$.
We argue by induction on $d$. If $d=0,$ $M$ is a finitely generated $R-$module, and so is $I-$separated. By \[T:dldtmditach\], $H^I_0(M)\cong \Lambda_I(M)\cong M,$ therefore $H^I_0(M)$ is a noetherian $\Lambda_I(R)-$module. Let $d>0.$ From \[L:dcddgiaomd\] we have $H^I_d(M)\cong
H^I_d(\underset{t>0}\cap I^tM).$ If $\Ndim (\underset{t>0}\cap
I^tM) < d,$ then $H^I_d(M) = 0$ by \[T:dlttdddpcp\] and then there is nothing to prove. If $\Ndim (\underset{t>0}\cap
I^tM) = d,$ by \[L:dktdptdcqdddp\] we may assume, by replacing $M$ with $\underset{t>0}\cap I^tM,$ that there is an element $x\in I$ such that $xM=M.$ Then, from the short exact sequence of modules $$0\longrightarrow 0:_Mx \longrightarrow M
\overset{x}\longrightarrow M \longrightarrow 0$$ we get an exact sequence of local homology modules$$H^I_d(M)
\overset{x}\longrightarrow H^I_d(M)
\overset{\delta}\longrightarrow H^I_{d-1}(0:_Mx).$$ Note by \[L:cntmoochiamx\] that $\Ndim(0:_Mx)\leqslant d-1.$ If $\Ndim(0:_Mx)<d-1,$ then $H^I_{d-1}(0:_Mx)=0$ by \[T:dlttdddpcp\] and therefore $$H^I_d(M)=xH^I_d(M)=\underset{t>0}\bigcap x^tH^I_d(M)=0$$ by \[L:dddpvpksvdn\] (ii). Assume that $\Ndim(0:_Mx)=d-1$. It follows by the inductive hypothesis that $H^I_{d-1}(0:_Mx)$ is a noetherian $\Lambda_I(R)-$module. On the other hand, we have $H^I_d(M)/xH^I_d(M) \cong \Im \delta\subseteq H^I_{d-1}(0:_Mx).$ Thus $H^I_d(M)/xH^I_d(M)$ is a noetherian $\Lambda_I(R)-$module. Therefore $H^I_d(M)/JH^I_d(M)$ is a noetherian $\Lambda_I(R)-$module, where $J=I\Lambda_I(R)$. Moreover, since $\underset{t>0}\bigcap J^tH^I_d(M) = \underset{t>0}\bigcap
I^tH^I_d(M) = 0$ and $\Lambda_I(R)$ is complete in $J-$adic topology, $H^I_d(M)$ is a noetherian $\Lambda_I(R)-$module by \[L:tcnote\] as required.
Macdonald duality {#S:dnmdn}
=================
Henceforth $(R,\m)$ will be a local noetherian ring with the maximal ideal $\m$. Suppose now that the topology on $R$ is the $\m-$adic topology.
Let $M$ be an $R-$module and $E(R/\m)$ the injective envelope of $R/\m.$ The module $D(M)=\Hom(M,E(R/\m))$ is called Matlis dual of $M.$ If $M$ is a Hausdorff linearly topology $R-$module, then [*Macdonald dual*]{} of $M$ is defined by $M^*=Hom(M,E(R/\m))$ the set of continuous homomorphisms of $R-$modules (see [@macdua §9]). In case $(R,\m)$ is local complete, the topology on $M^*$ is defined as in [@macdua 8.1]. Moreover, if $M$ is semi-discrete, then the topology of $M^*$ coincides with that induced on it as a submodule of $E(R/\m)^M,$ where $E(R/\m)^M=\underset{x\in M}\prod (E(R/\m))_x,$ $(E(R/\m))_x=E(R/\m)$ for all $x\in M$ (see [@macdua 8.6]).
\[L:dnmdnbmlmdnrr\] [(see [@macdua 5.8])]{} A $R-$module $M$ is semi-discrete if and only if $D(M)=M^*.$
\[L:dktddcltdas\] [(see [@macdua 5.7])]{} Let $M$ be a $R-$module and $u: M\longrightarrow A^*$ a homomorphism. Then the following statements are equivalent:
a\) $u$ is continuous, b) $\ker u$ is open, c) $\ker u$ is closed.
A $R-$module is $\m-$[*primary*]{} if each element of $M$ is annihilated by a power of $\m.$ A $R-$module $M$ is [*linearly discrete*]{} if every $\m-$primary quotient of $M$ is discrete. It should be noted that if $M$ is linearly discrete, then $M$ is semi-discrete. The direct limit of a direct system of linearly discrete $R-$modules is linearly discrete. If $f: M
\longrightarrow N$ is an epimorphism of Hausdorff linearly topologized $R-$modules in which $M$ is linearly discrete, then $f$ is continuous (see [@macdua 6.2, 6.7, 6.8]). Then I. G. Macdonald [@macdua] established the duality between linearly discrete and linearly compact modules as follows.
\[T:dnmdncplrrnrrlcp\] [( [@macdua 9.3, 9.12, 9.13])]{} Let $(R, \m)$ be a complete local noetherian ring.
\(i) If $M$ is linearly compact, then $M^*$ is linearly discrete (hence semi-discrete). If $M$ is semi-discrete, then $M^*$ is linearly compact.
\(ii) If $M$ is linearly compact or linearly discrete, then we have a topological isomorphism $\omega
:M\overset{\simeq}\longrightarrow M^{**}$.
The following duality theorem between local homology and local cohomology modules is the main result of this section.
\[T:dldndddpvddddp\] (i) Let $M$ be an $R-$module. Then for all $i\geq 0,$ $$H^I_i (D(M)) \cong D(H^i_I (M)).$$
\(ii) If $M$ is a linearly compact $R-$module, then for all $i\geq 0,$ $$H^I_i(M^*) \cong (H_I^i(M))^*.$$ Moreover, if $(R,\m)$ is a complete local noetherian ring, then $$H_I^i(M^*) \cong (H^I_i(M))^*.$$
\(iii) If $(R,\m)$ is a complete local noetherian ring and $M$ a semi-discrete linearly compact $R-$module, then we have topological isomorphisms of $R-$modules for all $i\geq 0,$ $$H_I^i(M^*) \cong (H^I_i(M))^*,$$ $$H^I_i(M^*) \cong (H_I^i(M))^*.$$
To prove Theorem \[T:dldndddpvddddp\] some auxiliary lemmas are necessary. First, we show that the Macdonald dual functor $(-)^*$ is exact on the category of linearly compact $R-$modules and continuous homomorphisms.
\[L:dnmdnkhop\] Let $$0 \lr M \overset{f}\lr N \overset{g}\lr P
\lr 0$$ be a short exact sequence of linearly compact $R-$modules, in which the homomorphisms $f, g$ are continuous. Then the induced sequence $$0 \lr P^* \overset{g^*}\lr N^* \overset{f^*}\lr M^* \lr 0$$ is exact.
By [@macdua 5.5] $f$ is an open mapping, so replace $M$ by $f(M)$ we may assume that $M$ is a close submodule of $N.$ Therefore, by [@macdua 5.9], for any continuous homomorphism $h: M\longrightarrow E(R/\m)$ there is a continuous homomorphism $\varphi: N\longrightarrow E(R/\m)$ which extends $h$. Thus $f^*$ is surjective. It is easy to see that $g^*$ is injective and $\Im g^*\subseteq\Ker f^*.$ So it remains to show that $\Ker f^*\subseteq\Im g^*.$ Let $\psi\in \Ker f^*,$ we have $\psi(\Ker g)=\psi(f(M))=0.$ Then $\psi$ induces a homomorphism $\phi: P\longrightarrow E(R/\m)$ such that $\phi \circ g=\psi.$ It follows $\ker\phi=g(\ker\psi).$ Since $\psi$ is continuous, $\ker\psi$ is open by \[L:dktddcltdas\]. Moreover, $g$ is open, so $\ker\phi$ is also open. Therefore $\phi$ is continuous by \[L:dktddcltdas\]. Thus $\psi\in\Im g^*.$ This finishes the proof.
Note that submodules and homomorphic images of a semi-discrete module are also semi-discrete. The following consequence shows that the converse is also true in the category of linearly compact $R-$modules.
\[C:dkdmdlcpnrr\] Let $$0 \lr M \overset{f}\lr N \overset{g}\lr P \lr 0$$ be a short exact sequence of linearly compact $R-$modules with continuous homomorphisms $f, g$. If $M$ and $P$ are semi-discrete, then $N$ is also semi-discrete.
It follows from \[L:dnmdnbmlmdnrr\] and the hypothesis that $P^*=D(P)$ and $M^*=D(M).$ We now have a commutative diagram $$\begin{matrix} 0 \lr& P^* &\overset{g^*}\lr& N^*&\overset{f^*}\lr& M^*& \lr 0\\
& \| && \downarrow j && \| \\
0 \lr & D(P)& \overset{D(g)}\lr& D(N)&\overset{D(f)}\lr& D(M)& \lr
0,\end{matrix}$$ in which $j$ is an inclusion and rows are exact by \[L:dnmdnkhop\] and [@rotani 3.16]. It follows that $N^*=D(N),$ thus $N$ is semi-discrete by \[L:dnmdnbmlmdnrr\] (i).
\[L:dnmdnctovaext\] Let $N$ be a finitely generated $R-$module and $M$ a linearly compact $R-$module. Then $$({\Tor}_i^R(N,M))^* \cong{\Ext}^i_R(N,M^*),$$ $${\Tor}_i^R(N,M^*) \cong ({\Ext}^i_R(N,M))^*$$for all $i\geq 0.$
Let $${\bold F_\bullet}: \cdots \longrightarrow F_i
\longrightarrow F_{i-1} \longrightarrow \cdots \longrightarrow F_1
\longrightarrow F_0 \longrightarrow N \longrightarrow 0$$ be a free resolution of $N,$ in which the free $R-$modules $F_i$ are finitely generated. Consider ${\bold F_\bullet}\bigotimes_R M$ as a complex of linearly compact $R-$modules with continuous differentials. Since the Macdonald dual functor $(-)^*$ is exact on the category of linearly compact $R-$modules and the continuous homomorphisms by \[L:dnmdnkhop\], it follows by [@norani 6.1 Theorem 1] that $$(H_i({\bold F_\bullet}\otimes_R M))^*\cong H^i(({\bold F_\bullet}\otimes_R M)^*).$$ On the other hand, by virtue of [@macdua 2.5] we have $$({\bold F_\bullet}\otimes_R M)^*\cong {\Hom}_R({\bold
F_\bullet}, M^*).$$ Therefore $$\begin{aligned}
({\Tor}_i^R(N,M))^*&\cong (H_i({\bold F_\bullet}\otimes_R M))^*\\
&\cong
H^i({\Hom}_R({\bold F_\bullet}, M^*))\\
&\cong {\Ext}^i_R(N,M^*).\end{aligned}$$The proof of the second isomorphism is similar.
If $M$ is a linearly topologized $R-$module, then the module $\Ext_R^i(R/I^t,M)$ is also a linearly topologized $R-$module by the topology defined as in \[L:extcptt\]. Since the local cohomology module $H_I^i(M)=\dlim \Ext^i_R(R/I^t , M)$ is a quotient module of $\underset{t}{\oplus} \Ext^i_R(R/I^t , M)$, it becomes a linearly topologized $R-$module with the quotient topology.
\[L:ddddpcmdrrtt\] Let $(R,\m)$ be a complete local noetherian ring. If $M$ is a semi-discrete linearly compact $R-$module, then $M$ is linearly discrete and therefore the local cohomology modules $H^i_I(M)$ are linearly discrete $R-$modules for all $i\geq
0.$
We first show that if $M$ is a semi-discrete linearly compact $R-$module, then $M$ is linearly discrete. Indeed, since $M$ is semi-discrete, $M^*$ is linearly compact and hence $M^{**}$ is linearly discrete by \[T:dnmdncplrrnrrlcp\] (i). On the other hand, since $M$ is a linearly compact $R-$module, we have by \[T:dnmdncplrrnrrlcp\] (ii) a topological isomorphism $M\cong M^{**}$. Therefore $M$ is linearly discrete. Now, by the same argument as in the proof of \[L:extcptt\] we can prove that $\{\Ext^i_R(R/I^t , M)\}_t$ is a direct system of semi-discrete linearly compact $R-$modules with the continuous homomorphisms, and therefore it is a direct system of linearly discrete modules. Thus, by [@macdua 6.7] $ H^i_I(M)=\dlim \Ext^i_R(R/I^t , M)$ are linearly discrete for all $i\geq 0$.
Now we are able to prove the duality theorem \[T:dldndddpvddddp\].
\(i) was proved in [@cuothe 3.3 (ii)].
\(ii) Note by [@macdua 2.6] that for a direct system $\{M_t\}$ of $R-$modules with the continuous homomorphisms we have an isomorphism $\inlim M_t^*\cong (\dlim
M_t)^*.$ Moreover, since $\big\{\Ext_R^i(R/I^t,M)\big\}_t$ forms a direct system of linearly compact $R-$modules with continuous homomorphisms by \[L:extcptt\], we get by \[L:dnmdnctovaext\] that
$$\begin{aligned}
H^I_i (M^*) &= \underset{t}{\underleftarrow{\lim}}{\Tor}^R_i (R/I^t , M^*)\\
&\cong \underset{t}{\underleftarrow{\lim}} ({\Ext}_R^i (R/I^t , M))^* \\
&\cong (\underset{t}{\underrightarrow{\lim}} {\Ext}_R^i (R/I^t ,
M))^* = (H^i_I (M))^*. \end{aligned}$$
To prove the second isomorphism note by [@macdua 9.14] that for an inverse system $\{M_t \}$ of linearly compact modules over complete local noetherian ring with continuous homomorphisms we have an isomorphism $(\inlim M_t)^*\cong \dlim M^*_t$, and that $\{\Tor^R_i(R/I^t,M)\}_t$ forms an inverse system of linearly compact $R-$modules with continuous homomorphisms by \[L:torcptt\]. It follows by \[L:dnmdnctovaext\] that $$\begin{aligned}
H^i_I(M^*) &= \dlim {\Ext}^i_R(R/I^t,M^*)\\
&\cong \dlim ({\Tor}^R_i(R/I^t,M))^*\\
&\cong (\inlim {\Tor}^R_i(R/I^t,M))^* = (H^I_i(M))^*. \end{aligned}$$
\(iii) Let us prove the first isomorphism. From (ii), it is the algebraic isomorphism. Thus, by [@macdua 6.8], we only need to show that both $H_I^i(M^*)$ and $(H_i^I(M))^*$ are linearly discrete. Indeed, it follows from \[P:dddpcpttlcptt\] and \[T:dnmdncplrrnrrlcp\] (i) that $(H^I_i(M))^*$ is linearly discrete. On the other hand, since $M$ is semi-discrete linearly compact, $M^*$ is linearly compact and linearly discrete. Therefore the local cohomology modules $H_I^i(M^*)$ are linearly discrete by \[L:ddddpcmdrrtt\], and the first topological isomorphism is proved. The second topological isomorphism follows from the first one and \[T:dnmdncplrrnrrlcp\] (ii).
\[C:hqdldddddp\] Let $(R,\m)$ be a complete local noetherian ring.
\(i) If $M$ is $R-$module, then for all $i\geq
0,$ $$H_I^i(M) \cong (H^I_i(M^*))^*,$$ $$H^I_i(M) \cong (H_I^i(M^*))^*.$$
\(ii) If $M$ is a semi-discrete linearly compact $R-$module, then we have topological isomorphisms of $R-$modules for all $i\geq 0,$ $$H_I^i(M) \cong (H^I_i(M^*))^*,$$ $$H^I_i(M) \cong (H_I^i(M^*))^*.$$
\(i) follows from \[T:dldndddpvddddp\] (ii), \[T:dnmdncplrrnrrlcp\] (ii).
\(ii) follows from \[T:dldndddpvddddp\] (iii) and \[T:dnmdncplrrnrrlcp\] (ii).
Local cohomology of semi-discrete\
linearly compact modules {#S:ddddpmdrr}
==================================
In this section $(R,\m)$ is a local noetherian ring with the $\m-$adic topology. We denote by $(\widehat R, \hat {\m})$ the $\m-$adic completion of $R$ with the maximal ideal $\hat{ \m}$ and $\widehat M$ the $\m-$adic completion of the module $M$. Recall that an artinian $R-$module $A$ has a natural structure as a module over $\widehat{R}$ as follows (see [@shaame 1.11]): Let $\hat a= (a_n) \in \widehat R$ and $x\in A$; since $\m^k x=0$ for some positive integer $k$, $a_nx$ is constant for all large $n$, and we define $\hat a x$ to be this constant value. Then we have the following generalization of this fact for $R-$modules.
\[L:mdcpnrrtvrm\] Let $M$ be a $R-$module. Then the following statements are true.
\(i) $M$ has a natural structure as a module over $\widehat{R}.$ Moreover, a subset $N$ of $M$ is a $\widehat{R}-$submodule if and only if $N$ is a closed $R-$submodule.
\(ii) Assume in addition that $M$ is a semi-discrete $R-$module. Then $M$ is also a semi-discrete $\widehat{R}-$module.
\(i) Assume that $\big\{ U_i\big\}_{i\in J}$ is a nuclear base of $M$ consisting of submodules. Then $M \cong
\underset{i\in J}{\underleftarrow{\lim}} M/U_i,$ in which $M/U_i$ is an artinian $R-$module for all $i\in J$ by [@macdua 3.11, 4.1, 5.5]. It should be noted by [@shaame 1.11] that an artinian module over a local noetherian ring $(R,\m)$ has a natural structure as an artinian module over $\widehat{R}$ so that a subset of $M$ is an $R-$submodule if and only if it is an $\widehat{R}-$submodules. Thus $\big\{ M/U_i \big\}_{i\in J}$ can be regard as an inverse system of artinian $\widehat{R}-$modules with $\widehat{R}-$homomorphisms. Therefore, pass to the inverse limits, $M$ has a natural structure as a module over $\widehat{R}.$
It is clear that a $\widehat R-$submodule of $M$ is a closed $R-$submodule. Now, if $N$ is a closed $R-$module of $M,$ then $N
\cong\underset{i\in J}{\underleftarrow{\lim}} N/(N\cap U_i).$ Since $$N/(N\cap U_i)\cong (N+U_i)/U_i\subseteq M/U_i,$$ $N/(N\cap U_i)$ can be considered as an artinian $R-$submodule of $M/U_i,$ so it is an artinian $\widehat{R}-$submodule. Moreover, the homomorphisms of the inverse system $\big\{ N/(N\cap U_i)
\big\}_{i\in J}$ are induced from the inverse system $\big\{ M/U_i
\big\}_{i\in J}.$ Therefore, by \[L:tcmdcptt4\] (iv) $N$ is an $\widehat{R}-$submodule of $M$.
\(ii) follows immediately from (i) by the fact that all submodules of a semi-discrete module are closed.
Remember that the [*(Krull) dimension*]{} $\dim_R M$ of a non-zero $R-$module $M$ is the supremum of lengths of chains of primes in the support of $M$ if this supremum exists, and $\infty$ otherwise. If $M$ is finitely generated, then $\dim M = \max\{\dim
R/\p\mid \p\in \Ass M\}$. For convenience, we set $\dim M= -1$ if $M=0$.
\[C:cnotevkrtrenrmu\] Let $M$ be a semi-discrete $R-$module. Then
\(i) $\Ndim_R M = \Ndim_{\widehat{R}} M;$
\(ii) $\dim_R M = \dim_{\widehat{R}} M.$
\(i) follows immediately from \[L:mdcpnrrtvrm\] (ii) and the definition of Noetherian dimension.
\(ii) In the special case $M$ is a finitely generated $R-$module, from \[T:dldtmditach\] we have $M \cong
\Lambda_{\m}(M) = \widehat{M}$, and therefore $\dim_R M =
\dim_{\widehat{R}} M$ by \[3, 6.1.3\]. For any semi-discrete linearly compact module $M$, there is by [@zoslin Theorem] a short exact sequence $0 \longrightarrow N \longrightarrow M
\longrightarrow A \longrightarrow 0$, where $N$ is finitely generated and $A$ is artinian. As $\dim_R A = \dim_{\widehat{R}} A
=0$ and $\dim_R N = \dim_{\widehat{R}} N,$ we get $\dim_R M =
\dim_{\widehat{R}} M.$
\[R:lytptcpnrrtdtn\] (i) Denote $\mathcal{C}$ the category of semi-discrete $R-$modules. It is well-known that the category $\mathcal{C}$ contains the category of artinian $R-$modules and also the category of finitely generated $R-$modules if $R$ is complete. However, there are many semi-discrete linearly compact $R-$modules which are neither artinian nor finitely generated. The first example for this conclusion is the module $K$ in Remark \[R:rmdlttcpvdzg\]. More general, let $R$ be complete ring, $A$ an artinian $R-$module with $\Ndim A>0$ and $N$ a finitely generated $R-$module with $\dim N>0$. Then $M=A\oplus N$ is semi-discrete . Further, let $Q$ be a quotient module of $M,$ then $Q$ is also a semi-discrete $R-$module.
\(ii) If $M\in$ $\mathcal{C}$, then by \[L:dnmdnbmlmdnrr\] the Matlis dual $D(M)$ and the Macdonald dual $M^*$ are the same. Moreover, $M$ is linearly discrete by \[L:ddddpcmdrrtt\] and can be regarded by \[L:mdcpnrrtvrm\] as an $\widehat R-$module, therefore the Macdonald dual functor $(-)^*$ is a functor from $\mathcal{C}$ to itself and we have by \[T:dnmdncplrrnrrlcp\] a topological isomorphism $\omega
:M\overset{\simeq}\longrightarrow M^{**}$. Thus, $(-)^{*}$ is an equivalent functor on the category $\mathcal{C}.$
\[L:ndimmsaobangdim\] Let $M$ be a semi-discrete linearly compact $R-$module. Then $${\Ndim}_R M^* = {\dim}_R M \ \ and \ \
{\Ndim}_R M = {\dim}_R M^*.$$
From \[L:mdcpnrrtvrm\] (ii) and \[C:cnotevkrtrenrmu\] we may assume that $(R,\m)$ is a complete ring. If $M$ is finitely generated $R-$module, $M^*$ is artinian. Keep in mind in our case that $M^*=D(M)$, then the equality $ \Ndim M^* =\dim M$ follows from the well-known facts of Matlis duality. If $M$ is artinian, then it is clear that $\Ndim M^* = \dim M=0.$ Suppose now that $M$ is semi-discrete . There is by [@zoslin Theorem] a short exact sequence $0\longrightarrow N \longrightarrow M \longrightarrow A
\longrightarrow 0$ in which $N$ is finitely generated and $A$ is artinian. Thus we get by Macdonald duality an exact sequence $0\longrightarrow A^* \longrightarrow M^* \longrightarrow N^*
\longrightarrow 0,$ where $N^*$ is artinian and $A^*$ is finitely generated. Then $$\begin{aligned}
\Ndim M^* &= \max\big\{
\Ndim N^*, \Ndim A^* \big\}\\
& = \max\big\{ \dim N, \dim A \big\} = \dim M .\end{aligned}$$ The second equality follows from \[R:lytptcpnrrtdtn\], (ii).
Now we are able to extend well-known results in Grothendieck’s local cohomology theory of finitely generated $R-$modules for semi-discrete modules.
\[T:mrttddddpg\] Let $M$ be a non zero semi-discrete linearly compact $R-$module. Then
\(i) $\dim_R M= \max \big\{ i\mid
H_{\m}^i(M)\not=0\big\}$ if $\dim_R M\not= 1;$
\(ii) $\dim_R (\widehat{M}) = \max \big\{ i\mid
H_{\m}^i(M)\not=0\big\}$ if $\widehat{M}\not= 0.$
\(i) Note by \[L:mdcpnrrtvrm\] (ii) and \[C:cnotevkrtrenrmu\] that $M$ is a semi-discrete $\widehat{R}-$module with $\dim_R M =
\dim_{\widehat{R}} M$. Moreover, the natural homomorphism $f:
R\longrightarrow \widehat{R}$ gives by [@broloc 4.2.1] an isomorphism $H^d_{\m}(M)\cong H^d_{\widehat{\m}}(M).$ Thus, we may assume without any loss of generality that $(R,\m)$ is a complete local noetherian ring. As $M$ is a semi-discrete linearly compact $R-$module, $M^*$ is also a semi-discrete linearly compact $R-$module by \[T:dnmdncplrrnrrlcp\] (i). Recall by [@macdua 5.6] that $M^* = 0$ if only if $ M=0$. Then, since $1\not= \dim M = \Ndim M^*$, it follows from \[L:ndimmsaobangdim\], \[T:dlttdddpcpnrr\] (ii) and \[T:dldndddpvddddp\] (ii) that $$\begin{aligned}
\dim M = \Ndim M^*
&= \max \big\{ i\mid
H^{\m}_i(M^*)\not=0\big\}\\
&= \max \big\{ i\mid
H_{\m}^i(M)^*\not=0\big\}\\
&= \max \big\{ i\mid H_{\m}^i(M)\not=0\big\}.\end{aligned}$$
\(ii) The continuous epimorphisms $M\lr M/{\m}^tM$ for all $t>0$ induce by \[L:lnkhmdcptt\] a continuous epimorphism $\pi: M\lr \widehat{M}.$ Moreover $\pi$ is the open homomorphism by [@macdua 5.5]. Thus $\widehat{M}$ is also a semi-discrete $R-$module. It follows from \[C:cnotevkrtrenrmu\] that $\dim_R\widehat{M}=\dim_{\widehat{R}}\widehat{M}.$ Hence, as in the proof of (i) we may assume without any loss of generality that $(R,\m)$ is a complete local noetherian ring. Note that $H^I_0(M)\cong \Lambda_I(M)$ and $H^0_I(M)\cong \Gamma_I(M),$ hence we have by \[T:dldndddpvddddp\] (ii) $$0\not=(\widehat M)^*=(H^{\m}_0(M))^* \cong \Gamma
_{\m}(M^*) .$$ Thus, by virtue of \[L:ndimmsaobangdim\], \[T:dlttdddpcpnrr\] (i) and \[T:dldndddpvddddp\] (ii) we get $$\begin{aligned}
\dim {\widehat M} = \Ndim (\widehat{M})^* = \Ndim
\Gamma_{\m}( M^*)
= &\max \big\{ i\mid
H^{\m}_i(M^*) \not=0 \big\}\\
= &\max \big\{ i\mid
(H_{\m}^i(M))^* \not=0 \big\}\\
=&\max \big\{ i\mid H_{\m}^i(M) \not=0 \big\}. \end{aligned}$$ The proof is complete.
\[R:rmcnt\] (i) The condition $\Ndim M\not=1$ in Theorem \[T:mrttddddpg\] (i) is necessary. Indeed, take the ring $R$ and the semi-discrete $R-$module $K$ as in Remark \[R:rmdlttcpvdzg\] and set $L=K^*$. It follows from \[T:dldndddpvddddp\] (iii) and \[R:rmdlttcpvdzg\] that $H_{\m}^i(L) \cong H^{\m}_i(K)^* =0$ for all $i\geq 0$. Hence $$\dim L = \Ndim K =1\not= -1 = \max \big\{ i\mid
H_{\m}^i(L)\not=0\big\}.$$
\(ii) The condition $\widehat M\not= 0$ in Theorem \[T:mrttddddpg\] (ii) can also not be dropped as the following example shows. Set $M=L\oplus A,$ where $A$ is an artinian $R-$module satisfying $\Width_{\m} A \geq 1$. Then there is an element $x\in \m$ such that $xM=M$, and therefore $\widehat M=0$. It is easy to see that $H_{\m}^i(M)=H_{\m}^i(L)=0$ for all $i\geq
1$ and $H_{\m}^0(M)\cong H_{\m}^0(A)=A$. Thus $$\dim \widehat M = -1 \not= 0= \max \big\{ i\mid
H_{\m}^i(M)\not=0\big\}.$$
To complete the unusual behaviour on the vanishing theorem of local cohomology for semi-discrete modules we give a characterization of semi-discrete modules, whose all local cohomology modules are vanished.
\[C:dtmodddptttb\] Let $M$ be a semi-discrete $R-$module. Then $H_{\m}^i(M)=0$ for all $i\geq 0$ if and only if there exists an element $x\in \m$ such that $xM=M$ and $0:_Mx=0$.
The conclusion follows from \[C:hqddpttmi\] by using \[T:dldndddpvddddp\] (ii) and \[R:lytptcpnrrtdtn\] (ii).
Recall that a sequence of elements $x_1 ,\ldots ,x_r$ in $R$ is said to be an [*$M-$regular*]{} sequence if $M/(x_1 , \ldots ,
x_r )M \not= 0$ and $M/(x_1 , \ldots , x_{i-1} )M
\overset{x_i}\longrightarrow M/(x_1 , \ldots , x_{i-1} )M$ is injective for $i=1,\ldots , r.$ Denote by $\depth_I(M)$ the supremum of the lengths of all maximal $M-$regular sequences in $I.$ Then we have
\[T:depthdddpsrcp\] Let $M$ be a $R-$module such that $M/IM\not=0.$ Then $${\depth}_I(M)=\inf\{i/H_I^i(M)\not=0\}.$$
Note by [@ooimat §3] that $x_1 ,\ldots ,x_r$ is an $M-$regular sequence if and only if it is a $D(M)-$coregular sequence. Since $M$ is , $D(M)=M^*$ and therefore ${\depth}_I(M)={\Width}_I(M^*).$ On the other hand, $M^*$ is a $R-$module by \[T:dnmdncplrrnrrlcp\](i) and $0:_{M^*}I \cong (M/IM)^*\not=0.$ Thus the conclusion follows by virtue of \[T:dtdrdddpcpnrr\] and \[T:dldndddpvddddp\] (ii).
Following is the artinianness of local cohomology modules.
\[T:ddddpcpnrrlartin\] Let $M$ be a semi-discrete linearly compact $R-$module with $\dim_R M = d.$ Then the following statements are true.
\(i) The local cohomology modules $H^i_{\m}(M)$ are artinian $R-$modules for all $i\geq 0;$
\(ii) The local cohomology module $H_I^d(M)$ is artinian.
Note first that if $A$ is an artinian $\widehat
R-$module, then $A$ is an artinian $R-$module. Therefore, from the independent of the base ring of local cohomology and \[L:mdcpnrrtvrm\] we may assume without loss of generality that $R$ is complete. Then, by applying the duality between local homology and local cohomology \[T:dldndddpvddddp\], the statement (i) follows from \[T:dltcnotemddddp\] and the statement (ii) from \[T:dlmddddpnotetcd\].
Finally, as an immediate consequence of Theorem \[T:ddddpcpnrrlartin\] we get the following well-known result.
\[C:ddddpmdhhsatin\] [(see [@broloc 7.1.3, 7.1.6])]{} Let $M$ be a finitely generated $R-$module with $\dim_R M = d.$ Then the local cohomology modules $H^i_{\m}(M)$ and $H^d_I(M)$ are artinian $R-$modules for all $i\geq 0.$
[**Acknowledgments.**]{} The authors have greatly enjoyed the perusal of the work by I. G. Macdonald [@macdua]. Some of the technical ideals exhibited in this paper are derived from this work. They would like to thank professor H. Zöschinger for showing the module $K$ in Remark \[R:rmdlttcpvdzg\]. The authors acknowledge support by the National Basis Research Program in Natural Science of Vietnam and the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
[9]{}
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|
---
abstract: 'We have studied the superconducting Si(111)-([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In surface using a $^3$He-based low-temperature scanning tunneling microscope (STM). Zero-bias conductance (ZBC) images taken over a large surface area reveal that vortices are trapped at atomic steps after magnetic fields are applied. The crossover behavior from Pearl to Josephson vortices is clearly identified from their elongated shapes along the steps and significant recovery of superconductivity within the cores. Our numerical calculations combined with experiments clarify that these characteristic features are determined by the relative strength of the interterrace Josephson coupling at the atomic step.'
author:
- Shunsuke Yoshizawa
- Howon Kim
- Takuto Kawakami
- Yuki Nagai
- Tomonobu Nakayama
- Xiao Hu
- Yukio Hasegawa
- Takashi Uchihashi
bibliography:
- 'MyEndNoteLibrary.bib'
title: 'Imaging Josephson Vortices on the Surface Superconductor Si(111)-([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In using a Scanning Tunneling Microscope'
---
The recent discovery of superconductivity in silicon surface reconstructions with metal adatoms was an unexpected surprise, because they are regarded as one of the thinnest two-dimensional (2D) materials ever possible [@Zhang_PbIn1ML; @Uchihashi_InR7R3Super; @Uchihashi_InR7R3Resistive; @Yamada_InR7R3Magneto; @Brun_PbSiDisorder]. This class of surface 2D materials has now become relevant for extensive superconductor researches in progress [@Oezer_PbHardSuper; @Qin_Pb2ML; @Wang_SingleFeSe; @Sekihara_1MLPb]. Notably, these new studies have been advanced by surface analytical techniques such as scanning tunneling microscopy (STM) [@Zhang_PbIn1ML; @Brun_PbSiDisorder; @Qin_Pb2ML; @Wang_SingleFeSe] and ultrahigh vacuum (UHV)-compatible transport measurement[@Uchihashi_InR7R3Super; @Uchihashi_InR7R3Resistive; @Yamada_InR7R3Magneto; @Tegenkamp_1DPb; @Yamazaki_R7R3].
One ubiquitous feature of these surface systems is the presence of atomic steps. Atomic steps are considered to strongly affect electron transport phenomena, because they potentially decouple neighboring surface terraces [@Crommie_StandingWave; @Hasegawa_StandingWave; @Uchihashi_1DIn; @Matsuda_StepR]. This could prevent superconducting currents from running over a long distance. The presence of supercurrents through atomic steps has indeed been demonstrated by direct electron transport measurements[@Uchihashi_InR7R3Super; @Uchihashi_InR7R3Resistive; @Yamada_InR7R3Magneto], and recent experiments indicated that atomic steps work as Josephson junctions [@Uchihashi_InR7R3Super; @Brun_PbSiDisorder]. Nevertheless, direct evidence of Josephson coupling has not been obtained yet, and possible local variation of its strength has remained an open issue. This problem is also closely related to Josephson junctions formed at the grain boundaries in thin films of high-$T_c$ cuprates, which are of technological importance [@Kogan_JJThinFilm; @Hilgenkamp_JJCuprate].
In this Letter, we report on compelling evidence of the Josephson coupling at atomic steps on the surface superconductor Si(111)-([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In \[referred to as ([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In\]. Zero-bias conductance (ZBC) images taken with a low-temperature (LT) STM reveal that vortices are present at atomic steps after magnetic fields are applied. The crossover behavior from Pearl to Josephson vortices is evident from their characteristic elongated shapes and significant recovery of superconductivity within their cores. This identification is strongly supported by our numerical calculations, which clarify their dependence on the interterrace Josephson coupling at the atomic step.
The experiment was performed using a UHV-LT-STM constructed at the Institute of Solid State Physics, University of Tokyo. The STM head was accommodated within a $^3$He-based cryostat combined with a solenoid superconducting magnet, where magnetic field was applied in the normal direction to the sample surface [@Nishio_PbIsland]. The temperature of the STM head $T_\mathrm{head}$ reaches below 0.5 K, which is sufficiently lower than the superconducting transition temperature $T_c\approx 3$ K of the ([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In surface [@Zhang_PbIn1ML; @Uchihashi_InR7R3Super; @Uchihashi_InR7R3Resistive; @Yamada_InR7R3Magneto]. Samples were prepared by thermal evaporation of In onto a clean Si(111) substrate followed by annealing in UHV [@Zhang_PbIn1ML; @Kraft_R7R3; @Rotenberg_R7R3; @Yamazaki_R7R3; @Uchihashi_InR7R3Super; @Uchihashi_InR7R3Resistive]. Subsequently, the surface ([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In structure was confirmed by reflection high energy electron diffraction (RHEED) and STM \[for representative data, see Figs. 1(a)(b)\]. The [[*dI/dV*]{}]{}spectra were recorded at a constant STM tip height in the ac lock-in detection mode by sweeping the sample bias voltage $V_\mathrm{s}$. ZBC images were taken at $V_\mathrm{s}=0$ mV in the same mode after the feedback was stabilized at $V_\mathrm{s}=20$ mV at each pixel point.
![(Color) (a) Representative RHEED pattern of a ([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In surface. Electron beam energy: 2.5 keV. (b) Representative STM image taken on a ([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In surface. Set point: 500 mV, 50 pA. (c) Zero-bias conductance (ZBC) image taken on a ([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In surface at $T_\mathrm{head} < 0.5$ K and at $B_\mathrm{ext}=0.04$ T. Set point: 20 mV, 200 pA. Bias modulation: 610 Hz, $200 \mu \mathrm{V}$. The bright round features show Pearl vortex cores. (d) Series of [[*dI/dV*]{}]{}spectra taken across the center of the left bright region. Set point: 20 mV, 600 pA. Bias modulation: 610 Hz, $50 \mu \mathrm{V}$. The curves are offset vertically for clarity. The locations for individual spectra are marked in the ZBC image in (c) in the same colors as used for spectral curves. The black curve is the result of fitting to the curve A using the Dynes formula. []{data-label="Fig1"}](fig1.eps){width="86mm"}
First, we characterized our samples by measuring vortices on a flat area. Figure 1(c) shows a ZBC image taken within a terrace of the ([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In surface under a magnetic field of $B_\mathrm{ext}=0.04$ T. The bright round regions (corresponding to high ZBC) indicate that vortices were created due to the penetration of magnetic field [@Hess_vortex1; @Tominaga_VortexSTM]. Namely, while ZBC is low in the superconducting region due to the presence of the energy gap $\Delta$, it recovers towards the normal-state value as $\Delta$ is suppressed within the vortex core [@Tinkham_Textbook]. To confirm this assignment, we obtained a series of site-dependent [[*dI/dV*]{}]{}spectra across the left bright feature \[Fig. 1(d)\]. At the location farthest from its center (marked as A), the [[*dI/dV*]{}]{}spectrum exhibited a characteristic superconducting energy gap structure with a dip around the zero bias and coherence peaks at $V_\mathrm{s} = \pm 0.60$ mV. Our fitting analysis based on the Dynes formula with s-wave gap function [@Dynes_QPLifetime] gives an energy gap $\Delta = 0.39 \mathrm{meV}$, quasi-particle lifetime broadening $\Gamma = 0.00 \mathrm{meV}$, and the sample temperature $T_\mathrm{sample} = 1.3$ K. [^1] (see the black line overlapped on Curve A). As the spectral site approached the center (marked as B), the zero-bias dip and the coherence peaks were both strongly suppressed, indicating breaking of superconductivity. We note that the vortices found here should be called Pearl vortices (PVs) because the present system consists of an atomically thin 2D superconductor [@PearlVortex; @Tafuri_PearlVortex]. [^2] For the following images, ZBC is normalized by the [[*dI/dV*]{}]{}value at a coherence peak at each pixel point to enhance the signal-to-noise ratio.
{width="160mm"}
Further experiments on wider surface regions allowed us to access more details of vortices in the present system. Figure 2(a) shows an STM topography image with an area of 500 nm $\times$ 1500 nm. The surface consists of flat terraces separated by steps with the single atomic height of 0.31 nm, which are indicated as $\alpha$, $\beta$, $\gamma$, and $\delta$ from top to bottom. ZBC images were taken on the same area under different magnetic fields of $B_\mathrm{ext}=0.08, 0.04, 0$ T in this order, as displayed in Figs. 2(b)-(d). The locations of the atomic steps are designated by thin solid lines. At $B_\mathrm{ext}=0.08$ T, PVs with bright round features formed a closely packed triangular lattice within each terrace. Reduction of magnetic field to $B_\mathrm{ext}=0.04$ T decreased the number of vortices on terraces as expected.
When the magnetic field was set to zero, vortices disappeared from the terraces, but slightly bright regions remained at some points along the steps \[Fig. 2(d)\]. Note that similar features were also present along the steps at finite fields \[Figs. 2(b)(c)\]. They are not simply regions where superconductivity is suppressed due to the presence of steps or disorder nearby. This is evident from the fact that the features change their positions under different magnetic fields, as seen from comparison of features A and A’. Similarly, comparison of regions C and C’ shows that ZBC increased at this location \[see Fig. 2(e) for the ZBC profiles\]. Furthermore, a sudden change in contrast is visible near feature B, indicating that it is mobile even under a constant field. The above observations clearly show that these bright features are vortices trapped at the atomic steps.
The vortices at steps are anomalous when compared to the PVs on terraces. Here we focus on vortices A’, B’, and C’ in Fig. 2(d). First, their shapes are elongated along the steps as seen from vortices A’ and B’; the full width at half maxima (FWHM) along and across the step are 162 and 80 nm for vortex A’, and 213 an 103 nm for vortex B’. [^3] Vortex C’ is largely spread along the step and appears to be disturbed by defects and/or temporal fluctuations. In contrast, PVs are isotropically round as seen from vortex D in Fig. 2(c), with a FWHW of $94\pm 5$ nm. Second, ZBC values measured at the centers are lower than those for PVs. This is quantitatively depicted in Fig. 2(e) as the ZBC profiles taken along the thick lines across vortices A’, B’, C’, and D. It means that the superconducting energy gap at the core recovers towards the zero-field value, while there is essentially no energy gap for a PV [@Hess_vortex1; @Tinkham_Textbook]. As explained below, these anomalies are the direct consequences of crossover to Josephson vortex (JV) and show that the atomic steps work as Josephson junctions. [^4]
![(Color) Numerically obtained spatial profile of the order parameter $\Psi(\bm{r})$ \[(a)(c)(e)\] and the zero energy density of state ${N}(E\!=\!0,\bm{r})$ \[(b)(d)(f)\]. The direction of an arrow in (a)(c)(e) denotes the phase $\phi(x,y)$ of the order parameter. The dashed lines indicate the place where the Josephson coupling was modeled as a reduced hopping strength $t_\mathrm{s}$. The length scale for $x$ and $y$ is the lattice constant $a$. Results in (a)(b), (c)(d), and (e)(f) are for hopping strength $t_\mathrm{s}/t=0.8,\ 0.4, \ 0.1$. We set the other parameters $\mu\!=\!-2.5t$ and $V\!=\!-3.0t$. []{data-label="Fig3"}](fig3.eps){width="80mm"}
Suppose that a vortex is created by penetration of magnetic field through a Josephson junction line and its surrounding region. Here the phase evolution due to supercurrent circulation around the core includes phase shifts $\Delta\phi$ at Josephson junctions. In the simplest case, $\Delta\phi$ is related to the supercurrent density $J_s$ through the following relation [@Tinkham_Textbook]: $$J_s = J_c \sin \Big[\Delta\phi-(2\pi/\Phi_0)\int \mathbf{A}(\bm{s}) \cdot d\mathbf{s} \Big],
\label{eq:Josephson_relation}$$ where $J_c$, $\Phi_0$, and $\int \mathbf{A}(\bm{s}) \cdot d\mathbf{s}$ denote the critical current density of the Josephson junction, magnetic flux quantum ($=h/2e$), and path integral of vector potential at the junction, respectively. This leads to two important properties regarding the vortex [@Blatter_VortexReview]. First, the circulation of supercurrent near the center is strongly deformed and the vortex core is elongated along the junction line by a factor of $(J_c/J_0)^{-1}$, where $J_0 (> J_c)$ is the critical current density in the superconducting regions. Second, the breaking of superconductivity around the core is weakened as $J_c/J_0$ decreases. The amplitude of the superconducting order parameter at the center $|\Psi_\mathrm{center}|$ is given by $$|\Psi_\mathrm{center}| \approx \left[1-(J_c/J_0)^2 \right] |\Psi_0|,
\label{eq:order_parameter_JV}$$ where $|\Psi_0|$ is the order parameter in the absence of magnetic field and supercurrent. The vortex should be called a JV when the supercurrent distribution near the junction line is nearly parallel and the suppression $\Delta |\Psi_\mathrm{center}|\equiv |\Psi_0|-|\Psi_\mathrm{center}| \approx (J_c/J_0)^2 |\Psi_0|$ is sufficiently smaller than $|\Psi_0|$. This terminology is consistent with the common usage of JVs in layered superconductors, which are created by magnetic field parallel to the layers [@Koshelev_JVReview; @Moll_TranstionAVtoJV]. [^5]
To compare the theoretical prediction with our experiment more directly, we numerically calculated the order parameter and the density of states (DOS) using the Bogoliubov-de Gennes (BdG) equation for a 2D tight-binding model: $$\begin{aligned}
~\label{eq:tight-bdg}
\sum_{j}\left(\begin{array}{cc}
\hat{K}_{i,j} &\hat{\Delta}_{i,j}\\
\hat{\Delta}^\ast_{j,i} & -\hat{K}^\ast_{i,j}
\end{array}\right)
\left(\begin{array}{c}
u_\gamma(\bm{r}_j)\\ v_\gamma(\bm{r}_j)
\end{array}\right)
=E_{\gamma}
\left(\begin{array}{c}
u_\gamma(\bm{r}_i)\\ v_\gamma(\bm{r}_i)
\end{array}\right).
$$ The single particle part is given by $\hat{K}_{i,j}=-t_{ij}\exp\left[i(\pi/\Phi_0)\int_{\bm{r}_i}^{\bm{r}_j}\bm{A}(\bm{s})\cdot d\bm{s}\right]-\mu\delta_{ij}$ with $t_{ij}$ the hopping strength. The Josephson junction was modeled as a straight line with one atomic spacing where the hopping strength $t_{\mathrm{s}}$ is reduced from a constant hopping strength $t$ elsewhere. Then the Josephson parameter $J_\mathrm{c}/J_0$ is represented by the ratio $t_\mathrm{s}/t$ according to Ambegaokar-Baratoff’s equation [@Ambegaokar-Baratoff]. Equation (\[eq:tight-bdg\]) was solved self-consistently [@Covaci2010; @Nagai2012; @NagaiPRB] to obtain the pair potential $\Delta(\bm r_{i})=\hat{\Delta}_{i,j}=\delta_{ij}V\sum_{\gamma}u_{\gamma}(\bm{r}_i)v_{\gamma}(\bm{r}_j)f(E_{\gamma})$ and DOS ${N}(E,\bm{r}_i)=\sum_{\gamma}|u_{\gamma}(\bm{r}_i)|^2\delta(E-E_{\gamma})$. [^6]
Figures 3(a)-(f) display the order parameter $\Psi(\bm r)=\Delta(\bm r)/V$ \[(a)(c)(e)\] and zero-energy DOS ${N}(E=0,\bm{r})$ \[(b)(d)(f)\] calculated for $t_\mathrm{s}/t=0.8, \ 0.4, \ 0.1$. For $\Psi(\bm r)$, its amplitude $|\Psi(\bm r)|$ and phase $\phi(\bm r)$ are shown in the upper and lower panels within each figure, respectively. The location of the Josephson coupling line (where $t_{ij}=t_\mathrm{s}$) is indicated by the dashed lines. While the suppression of $|\Psi(\bm r)|$ is strong and the spatial distribution of $\phi(\bm r)$ is almost cylindrically symmetric for $t_\mathrm{s}/t=0.8$, the former becomes weaker and the latter is elongated along the junction line as $t_\mathrm{s}/t$ is reduced to 0.4 and 0.1. Accordingly, the characteristics of ${N}(E=0,\bm{r})$ are changed; its magnitude around the center is decreased as $t_\mathrm{s}/t$ is reduced, while the spatial distribution becomes strongly elliptic. Considering that ZBC is proportional to DOS, this evolution directly corresponds to the observed changes for vortices A’, B’, and C’ in Fig. 2(d). Thus the coupling strength $J_c$ at steps $\alpha,\gamma,\delta$ decreases in this order. From the comparison of the experiment and the theory, $J_c/J_0$ is estimated to be $\sim 0.4$ for step $\gamma$ where vortex B’ is located. Step $\delta$ has a weak coupling $J_c/J_0 \ll 0.4$ and, according to the above definition, vortex C’ can be safely called a JV. We estimate $J_c=1.8 \ \mathrm{A/m}$ from the previous macroscopic transport measurement [@Uchihashi_InR7R3Super] and $J_0=19-62 \ \mathrm{A/m}$ from the present study, leading to $J_c/J_0=0.029-0.095$. [^7] This justifies our theoretical analysis because $J_c$ determined above should reflect the weakest interterrace coupling, being consistent with $J_c/J_0 \ll 0.4$ at step $\delta$.
The differences in $J_c/J_0$ clarified above may be attributed to the local atomic-scale structures along the steps. Figure 2(f) shows topographic images near steps $\alpha$, $\gamma$, $\delta$ where vortices A’, B’, C’ are located \[marked by the rectangles in Fig. 2(a)\]. Grooves are visible along step $\delta$, indicating that the superconducting indium layers did not grow up to the step edge. This should result in a weak electronic coupling between the upper and lower terraces [@Uchihashi_1DIn] and hence in a low $J_c/J_0$. In contrast, such a structure is nearly absent for step $\alpha$, which helps to establish a stronger interterrace coupling. Finally, we remark on possible JVs in Fig. 2(b) under a high magnetic field. All visible bright features in the image counts for number of vortices $N_\mathrm{vis}=26$, which is different from $N_\mathrm{theory}= B_\mathrm{ext}S/\Phi_0=29$ (imaging area $S=500\mathrm{nm} \times 1500\mathrm{nm}$, $B_\mathrm{ext}=0.08$ T, $\Phi_0= 2.07\times 10^{-15} \mathrm{Tm^2}$). The missing flux quanta are $N_\mathrm{theory}-N_\mathrm{vis} = 3$ and they should exist as JVs along step $\delta$.
In conclusion, we have observed the crossover from PV to JV at atomic steps on the ([[$\sqrt{7}\times\sqrt{3}$]{}]{})-In surface by taking ZBC images using a LT-STM. The present work provides compelling evidence and local information for Josephson coupling at atomic steps.
This work was financially supported by JSPS under KAKENHI Grants No. 25247053, No. 25286055, No. 25400385, No. 24340079 and by World Premier International Research Center (WPI) Initiative on Materials Nanoarchitectonics, MEXT, Japan. The calculations was performed using the supercomputing system PRIMERGY BX900 at the Japan Atomic Energy Agency.
[^1]: Energy gap $\Delta$ obtained here is smaller than $\Delta=0.57$ meV reported previously for this surface [@Zhang_PbIn1ML]. This may be due to the residual disorder found in the present sample.
[^2]: For a 2D superconductor with a thickness $d$, the characteristic length governing the magnetic field distribution is given by Pearl length $\Lambda=2\lambda^2/d$, where $\lambda$ is London penetration depth. The vortices interact with each other like $E_\mathrm{int} \propto \log r$ as long as $r<\Lambda$. The vortex is then called the Pearl vortex instead of the Abrikosov vortex in a three-dimensional (3D) superconductor, but their core structures are essentially the same. The magnetic flux size of a Pearl vortex is given by $\Lambda$, which is estimated to be as large as 4.4 mm here. Since the magnetic field distribution is considered to be uniform, it does not affect the structure of a vortex core (See Sec. 1 of Supplemental Material \[url\], which includes Ref. [@Raychaudhuri_InSuper]).
[^3]: See Sec. 2 of Supplemental Material \[url\].
[^4]: We stress that these observations were made possible through the STM measurement. Previous studies on JVs using scanning superconducting quantum interference devices on cuprates detected magnetic field distribution but did not access information on the vortex cores [@Hilgenkamp_JJCuprate].
[^5]: In a 3D system, the transition from Abrikosov to Josephson vortex occurs when the elongated core size exceeds the London penetration depth $\lambda$ \[A. Gurevich, Phys. Rev. B **46**, 3187 (1992)\]. In the present 2D system, vortices at junctions are quite different since the problem involves a nonlocal equation as opposed to the local sine-Gordon equation in the 3D case [@Kogan_JJThinFilm]. Hence this definition is not applicable here.
[^6]: See Sec. 3 of Supplemental Material \[url\], which includes Ref. [@Takigawa_NMRVortex].
[^7]: See Sec. 4 of Supplemental Material \[url\], which includes Ref. [@Raychaudhuri_InSuper].
|
---
abstract: |
We describe a linear isomorphism $\mathcal{K}$ from the enveloping algebra $\mathbf{U}(gl(n))$ to the algebra ${\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n))$ of polynomials in the entries of a “generic” square matrix of order $n$.
The isomorphism $\mathcal{K}$ maps any [*Capelli bitableau*]{} $[S|T]$ in $\mathbf{U}(gl(n))$ to the [*(determinantal) bitableau*]{} $(S|T)$ in ${\mathbb C}[M_{n,n}]$ and any [*Capelli \*-bitableau*]{} $[S|T]^*$ in $\mathbf{U}(gl(n))$ to the [*(permanental) \*-bitableau*]{} $(S|T)^*$ in ${\mathbb C}[M_{n,n}]$.
These results are far-reaching generalizations of the pioneering result of J.-L. Koszul [@Koszul-BR] on the Capelli determinant in $\mathbf{U}(gl(n))$ (see, e.g. [@Procesi-BR], [@Weyl-BR]).
We introduce [*column*]{} Capelli bitableaux and [*column*]{} Capelli \*-bitableaux in Section \[column exp\]; since they are mapped by the isomorphism $\mathcal{K}$ to [*monomials*]{} in ${\mathbb C}[M_{n,n}]$, this isomorphism can be regarded as a sharpened version of the PBW isomorphism for the enveloping algebra $\mathbf{U}(gl(n))$.
---
**On the action of the Koszul map**
**over the enveloping algebra**
**of the general linear Lie algebra**
A. Brini and A. Teolis
*$^\flat$ Dipartimento di Matematica, Università di Bologna*
*Piazza di Porta S. Donato, 5. 40126 Bologna. Italy.*
e-mail of corresponding author: [email protected]
**Keyword**: Enveloping algebras; Young tableaux; Lie superalgebras; Capelli determinants.
Introduction
============
The starting points of the present work are ([@Brini3-BR], [@Brini4-BR]):
- The linear operator $\mathcal{B} : {\mathbb C}[M_{n,n}] \rightarrow \mathbf{U}(gl(n))$ that maps any [*(determinantal) bitableau*]{} $(S|T)$ in ${\mathbb C}[M_{n,n}]$ to the [*Capelli bitableau*]{} $[S|T]$ in $\mathbf{U}(gl(n))$.
- The linear operator $\mathcal{B^*} : {\mathbb C}[M_{n,n}] \rightarrow \mathbf{U}(gl(n))$ that maps any [*(permanental) bitableau*]{} $(S|T)^*$ in ${\mathbb C}[M_{n,n}]$ to the [*Capelli \*-bitableau*]{} $[S|T]^*$ in $\mathbf{U}(gl(n))$.
The map $$\mathcal{K} : \mathbf{U}(gl(n)) \rightarrow {\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n))$$ introduced by Koszul in 1981 [@Koszul-BR] is proved to be the inverse of both $\mathcal{B}$ and $\mathcal{B}^*$. Then, $\mathcal{B} = \mathcal{B}^*$ and $\mathcal{B}$, $\mathcal{B}^*$, $\mathcal{K}$ are vector space isomorphisms: see Theorem \[BCKtheorem\] below, which can be regarded as a sharpened version of the PBW Theorem for the enveloping algebra $\mathbf{U}(gl(n))$.
The main objects in $\mathbf{U}(gl(n))$ - Capelli bitableaux, Capelli \*-bitableaux and right Young-Capelli bitableaux - are recalled in section \[sec 3\] by means of the *method of virtual variables*, which is in turn a superalgebraic extension of Capelli’s method of *variabili ausiliarie* [@Cap4-BR].
Since the [*standard*]{} bitableaux are a basis of ${\mathbb C}[M_{n,n}]$ ([@drs-BR], [@DKR-BR], [@DEP-BR], [@rota-BR]), then the [*standard*]{} Capelli bitableaux are a basis of $\mathbf{U}(gl(n))$. Since the [*costandard*]{} \*-bitableaux are a basis of ${\mathbb C}[M_{n,n}]$, then the [*costandard*]{} Capelli \*-bitableaux are a basis of $\mathbf{U}(gl(n))$.
In the polynomial algebra ${\mathbb C}[M_{n,n}]$, [*[column bitableaux]{}*]{} and [*[column \*-bitableaux]{}*]{} are, up to a sign, the [*[same monomials]{}*]{}. Their images in $\mathbf{U}(gl(n))$ - under the isomorphism $\mathcal{B}$ - are the [*[column Capelli bitableaux]{}*]{} and the [*[column Capelli \*-bitableaux]{}*]{}, respectively.
Although column Capelli bitableaux and column Capelli \*-bitableaux are far from being “monomials” in the enveloping algebra $\mathbf{U}(gl(n))$, their images under the Koszul isomorphism $\mathcal{K}$ are indeed *(commutative) monomials* in the polynomial algebra ${\mathbb C}[M_{n,n}]$. Therefore, column Capelli bitableaux and column Capelli \*-bitableaux play the same crucial role in $\mathbf{U}(gl(n))$ that monomials play in ${\mathbb C}[M_{n,n}]$.
Capelli bitableaux and Capelli \*-bitableaux expand - up to a global sign - into column Capelli bitableaux just in the same way as *determinantal* bitableaux and *permanental* \*-bitableaux expand into the corresponding monomials in ${\mathbb C}[M_{n,n}]$ (Laplace expansions).
The expressions of column Capelli bitableaux and column Capelli \*-bitableaux in $\mathbf{U}(gl(n))$ can be simply computed (Proposition \[column exp\] below).
Some topics of this paper were treated in a sketchy way in the present author’s notes [@Brini3-BR], [@Brini4-BR] (in the more general setting of superalgebras), in a rather cumbersome notation and almost without proofs. The emphasis is here on the role of *column* Capelli bitableaux and \*-bitableaux, and this leads to a new and transparent presentation.
The isomorphism $\mathcal{B}$ maps any *right symmetrized bitableau* $(S|\fbox{$T$}) \in {\mathbb C}[M_{n,n}]$ to the [*right Young-Capelli bitableau*]{} $[S|\fbox{$T$}]$ in $\mathbf{U}(gl(n))$. The basis of [*standard*]{} right Young-Capelli bitableaux acts in a quite remarkable way on the [*Gordan-Capelli basis*]{} of [*standard*]{} right symmetrized bitableaux (Remark \[action\], below). Furthermore, the elements of the *Schur-Sahi-Okounkov* basis of the center $\boldsymbol{\zeta}(n)$ of $\mathbf{U}(gl(n))$ admit an effective description as linear combinations of right Young-Capelli bitableaux (see, e.g. [@BriniTeolis-BR], [@Brini5-BR]).
Several examples/applications are provided throughout the paper. In particular, we show, in few lines, that the *row* Capelli bitableau $[12 \ldots n|12 \ldots n]$ equals the famous Capelli determinant in $\mathbf{U}(gl(n))$ [@Cap1-BR], [@Weyl-BR] (see Proposition \[CapDet\]).
Determinantal [*[Young bitableau]{}*]{} and permanental [*[Young \*-bitableau]{}*]{} in the polynomial algebra ${\mathbb C}[M_{n,n}]$ {#sec 2}
=====================================================================================================================================
Let $${\mathbb C}[M_{n,n}] = {\mathbb C}[(i|j)]_{i,j=1,\ldots,n}$$ be the polynomial algebra in the (commutative) “generic" entries $(i|j)$ of the matrix: $$M_{n,n} = \left[ (i|j) \right]_{i, j=1,\ldots,n}=
\left(
\begin{array}{ccc}
(1|1) & \ldots & (1|n) \\
(2|1) & \ldots & (2|n) \\
\vdots & & \vdots \\
(n|1) & \ldots & (n|n) \\
\end{array}
\right).$$
Given the *standard basis* $\big\{ e_{ij}; \ i, j = 1, 2, \ldots, n \big\}$ of the *general linear Lie algebra* $gl(n)$, the map $
e_{ij} \rightarrow (i|j)
$ induces an isomorphism $\mathbf{Sym}(gl(n)) \cong {\mathbb C}[M_{n,n}]$.
Let $\omega = i_1i_2 \cdots i_p$, $\varpi = j_1j_1 \cdots j_p$ be words on the alphabet $ \{1, 2, \ldots, n \}$.
Following [@rota-BR] and [@Brini1-BR], the [*[biproduct]{}*]{} of the two words $\omega$ and $\varpi$ $$\label{biproduct}
(\omega|\varpi) = (i_1i_2 \cdots i_p|j_1j_2 \cdots j_p)$$ is the is the *signed minor*: $$(\omega|\varpi) = (-1)^{{p} \choose {2}} \ det \Big( \ (i_r|j_s) \ \Big)_{r, s = 1, 2, \ldots, p} \in {\mathbb C}[M_{n,n}].$$
Let $S = (\omega_1, \omega_2, \ldots, \omega_p)$ and $T = (\varpi_1, \varpi_2, \ldots, \varpi_p)$ be Young tableaux on $\{1, 2, \ldots, n \}$ of the same shape $\lambda$.
Following again [@rota-BR] and [@Brini1-BR], the (determinantal) [*[Young bitableau]{}*]{} $$\label{bitableaux}
(S|T) =
\left(
\begin{array}{c}
\omega_1\\ \omega_2\\ \vdots\\ \omega_p
\end{array}
\right| \left.
\begin{array}{c}
\varpi_1\\ \varpi_2\\ \vdots\\ \varpi_p
\end{array}
\right)$$ is the *signed* product of the biproducts of the pairs of corresponding rows: $$\label{bitableau}
(S|T) =
\pm \ (\omega_1|\varpi_1)(\omega_2|\varpi_2) \cdots (\omega_p|\varpi_p),$$ where $$\label{crossing sign}
\pm = (-1)^{\ell(\omega_2)\ell(\varpi_1)+\ell(\omega_3)(\ell(\varpi_1)+\ell(\varpi_2))
+ \cdots +\ell(\omega_p)(\ell(\varpi_1)+\ell(\varpi_2)+\cdots+\ell(\varpi_{p-1}))},$$ and the symbol $\ell(w)$ denotes the length of the word $w$.
The [*[\*-biproduct]{}*]{} of the two words $\omega$ and $\varpi$ $$\label{biproduct}
(\omega|\varpi) = (i_1i_2 \cdots i_p|j_1j_2 \cdots j_p)$$ is the is the *permanent*: $$(\omega|\varpi)^* = \ per \Big( \ (i_r|j_s) \ \Big)_{r, s = 1, 2, \ldots, p} \in {\mathbb C}[M_{n,n}].$$
Let $S = (\omega_1, \omega_2, \ldots, \omega_p)$ and $T = (\varpi_1, \varpi_2, \ldots, \varpi_p)$ be Young tableaux on $\{1, 2, \ldots, n \}$ of the same shape $\lambda$.
Following again [@rota-BR] and [@Brini1-BR], the (permanental) [*[Young \*-bitableau]{}*]{} $$\label{bitableaux}
(S|T)^* =
\left(
\begin{array}{c}
\omega_1\\ \omega_2\\ \vdots\\ \omega_p
\end{array}
\right| \left.
\begin{array}{c}
\varpi_1\\ \varpi_2\\ \vdots\\ \varpi_p
\end{array}
\right)^*$$ is the product of the \*-biproducts of the pairs of corresponding rows: $$\label{bitableau}
(S|T)^* =
(\omega_1|\varpi_1)^*(\omega_2|\varpi_2)^* \cdots (\omega_p|\varpi_p)^*.$$
A *column* Young tableau of *depth* $h$ is a tableau of shape $(1^h)$. Then for a column Young bitableau, we have: $$\left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right)
=
(-1)^{h \choose 2}(i_1|j_1)(i_2|j_2) \cdots (i_h|j_h)$$ and for a column Young \*-bitableau, we have: $$\left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right)^*
=
(i_1|j_1)(i_2|j_2) \cdots (i_h|j_h).$$
The superalgebraic approach to the enveloping algebra $\mathbf{U}(gl(n))$ {#sec 3}
=========================================================================
We follow [@Brini1-BR], [@Brini2-BR], [@Brini3-BR], [@Brini4-BR].
Let $X = \{\alpha_1, \ldots, \alpha_{m_0} \} \cup \{\beta_1, \ldots, \beta_{m_1} \} \cup L$, $L =\{1, 2, \ldots, n \}$ ( $m_0, m_1$ “sufficiently large” ) denote the union of the alphabets of *virtual positive*, *virtual negative* and *proper negative* symbols, respectively. Let $\mathbf{U}(gl(m_0|m_1+n))$ denote the enveloping algebra of the *general linear Lie superalgebra* $gl(m_0|m_1+n)$, with basis $\{ e_{a,b}; \ a,b \in X \}$, $|e_{a,b}| = |a| + |b| \in {\mathbb Z}_2$. The *general linear Lie algebra* $gl(n))$ (with basis $\{ e_{a,b}; \ a,b \in L \}$, $|e_{a,b}| = |a| + |b| = 0 \in {\mathbb Z}_2$) is regarded as the subalgebra $gl(0|n)$ of $gl(m_0|m_1+n)$.
We recall ([@Brini3-BR], [@Brini4-BR]) that a product $$e_{a_mb_m} \cdots e_{a_1b_1} \in \mathbf{U}(gl(m_0|m_1+n))$$ is an [*irregular expression*]{} whenever there exists a right subword $$e_{a_i,b_i} \cdots e_{a_2,b_2} e_{a_1,b_1},$$ $i \leq m$ and a virtual symbol $\gamma \in A_0 \cup A_1$ such that $$\label{irrexpr-BR}
\# \{j; b_j = \gamma, j \leq i \} > \# \{j; a_j = \gamma, j < i \}.$$ The meaning of an irregular expression - in terms of the action of $\mathbf{U}(gl(m_0|m_1+n))$ - is that there exists a virtual symbol $\gamma$ and a right subsequence in which the symbol $\gamma$ is annihilated more times than it was already created.
Let $\mathbf{Irr}$ be the [*left ideal*]{} of $\mathbf{U}(gl(m_0|m_1+n))$ generated by the set of *irregular expressions* ([@Brini3-BR], [@Brini4-BR], see also [@Bri-BR]).
*([@Brini3-BR], [@BriUMI-BR])* The sum ${\mathbf{U}}(gl(0|n)) + \mathbf{Irr}$ is a direct sum of vector subspaces of $\mathbf{U}(gl(m_0|m_1+n)).$
The [*virtual algebra*]{} $Virt(m_0+m_1,n)$ is the subalgebra $$Virt(m_0+m_1,n) = \mathbf{U}(gl(0|n)) \oplus \mathbf{Irr} \subset {\mathbf{U}}(gl(m_0|m_1+n)).$$
The left ideal $\mathbf{Irr}$ of ${\mathbf{U}}(gl(m_0|m_1+n))$ is a two sided ideal of $Virt(m_0+m_1,n).$
The [*Capelli devirtualization epimorphism*]{} is the projection $$\mathfrak{p} : Virt(m_0+m_1,n) = \mathbf{U}(gl(0|n)) \oplus \mathbf{Irr} \twoheadrightarrow \mathbf{U}(gl(0|n)) = \mathbf{U}(gl(n))$$ with $Ker(\mathfrak{p}) = \mathbf{Irr}.$
In a formal way, *balanced monomials* are elements of the algebra ${\mathbf{U}}(gl(m_0|m_1+n))$ of the form:
- $e_{{i_1},\gamma_{p_1}} \cdots e_{{i_k},\gamma_{p_k}} \cdot
e_{\gamma_{p_1},{j_1}} \cdots e_{\gamma_{p_k},{j_k}},$
- $e_{{i_1},\theta_{q_1}} \cdots e_{{i_k},\theta_{q_k}} \cdot
e_{\theta_{q_1},\gamma_{p_1}} \cdots e_{\theta_{q_k},\gamma_{p_k}} \cdot
e_{\gamma_{p_1},{j_1}} \cdots e_{\gamma_{p_k},{j_k}},$
- and so on,
where $i_1, \ldots, i_k, j_1, \ldots, j_k \in L,$ i.e., the $i_1, \ldots, i_k, j_1, \ldots, j_k$ are $k$ proper (negative) symbols, and the $\gamma_{p_1}, \ldots, \gamma_{p_k}, \ldots, \theta_{q_1}, \ldots, \theta_{q_k}, \ldots$ are virtual symbols. In plain words, a balanced monomial is product of two or more factors where the rightmost one *annihilates* the $k$ proper symbols $ j_1, \ldots, j_k$ and *creates* some virtual symbols; the leftmost one *annihilates* all the virtual symbols and *creates* the $k$ proper symbols $ i_1, \ldots, i_k$; between these two factors, there might be further factors that annihilate and create virtual symbols only.
*([@Brini1-BR], [@Brini2-BR], [@Bri-BR], [@BriUMI-BR])* Every balanced monomial belongs to $Virt(m_0+m_1,n)$. Hence its image under the Capelli epimorphism $\mathfrak{p}$ belongs to $\mathbf{U}(gl(n)).$
Let $S$ and $T$ be the Young tableaux $$\label{tableaux}
S = \left(
\begin{array}{llllllllllllll}
i_{p_1} \ldots \ldots \ldots i_{p_{\lambda_1}} \\
i_{q_1} \ldots \ldots i_{q_{\lambda_2}} \\
\ldots \ldots \\
i_{r_1} \ldots i_{r_{\lambda_m}}
\end{array}
\right),
\quad
T = \left(
\begin{array}{llllllllllllll}
j_{s_1} \ldots \ldots \ldots j_{s_{\lambda_1}} \\
j_{t_1} \ldots \ldots j_{t_{\lambda_2}} \\
\ldots \ldots \\
j_{v_1} \ldots j_{v_{\lambda_m}}
\end{array}
\right).$$
To the pair $(S,T)$, we associate the [*[bitableau monomial]{}*]{}: $$\label{BitMon}
e_{S,T} =
e_{i_{p_1}, j_{s_1}}\cdots e_{i_{p_{\lambda_1}}, j_{s_{\lambda_1}}}
e_{i_{q_1}, j_{t_1}}\cdots e_{i_{q_{\lambda_2}}, j_{t_{\lambda_2}}}
\cdots \cdots
e_{i_{r_1}, j_{v_1}}\cdots e_{i_{r{\lambda_p}}, j_{v_{\lambda_p}}}$$ in ${\mathbf{U}}(gl(m_0|m_1+n)).$
Let $\beta_1, \ldots, \beta_{\lambda_1} \in A_1$, $\alpha_1, \ldots, \alpha_p \in A_0$ be sets of negative and positive *virtual symbols*, respectively. Set $$\label{Deruyts and Coderuyts}
D_{\lambda} = \left(
\begin{array}{llllllllllllll}
\beta_1 \ldots \ldots \ldots \beta_{\lambda_1} \\
\beta_1 \ldots \ldots \beta_{\lambda_2} \\
\ldots \ldots \\
\beta_1 \ldots \beta_{\lambda_p}
\end{array}
\right), \qquad
C_{\lambda} = \left(
\begin{array}{llllllllllllll}
\alpha_1 \ldots \ldots \ldots \alpha_1 \\
\alpha_2 \ldots \ldots \alpha_2 \\
\ldots \ldots \\
\alpha_p \ldots \alpha_p
\end{array}
\right)$$.
The tableaux $D_{\lambda}$ and $C_{\lambda}$ are called the [*[virtual Deruyts and Coderuyts tableaux]{}*]{} of shape $\lambda,$ respectively.
Given a pair of Young tableaux $S, T$ of the same shape $\lambda$ on the proper alphabet $L$, consider the elements $$\label{determinantal}
e_{S,C_{\lambda}} \ e_{C_{\lambda},T} \in {\mathbf{U}}(gl(m_0|m_1+n)),$$, $$\label{permanental}
e_{S,\widetilde{D_{\widetilde{\lambda}}}} \ e_{\widetilde{D_{\widetilde{\lambda}}},T} \in {\mathbf{U}}(gl(m_0|m_1+n)),$$ $$\label{rightYoungCapelli}
e_{S,C_{\lambda}} \ e_{C_{\lambda},D_{\lambda}} \ e_{D_{\lambda},T} \in {\mathbf{U}}(gl(m_0|m_1+n)).$$
Since elements (\[determinantal\]), (\[permanental\]) and (\[rightYoungCapelli\]) are balanced monomials in ${\mathbf{U}}(gl(m_0|m_1+n))$, they belong to the subalgebra $Virt(m_0+m_1,n)$.
We set $$\mathfrak{p} \Big( e_{S,C_{\lambda}} \ e_{C_{\lambda},T} \Big) = [S|T] \in {\mathbf{U}}(gl(n)),$$ and call the element $[S|T]$ a [*Capelli bitableau*]{} [@Brini3-BR], [@Brini4-BR].
We set $$\mathfrak{p} \Big( e_{S,\widetilde{D_{\widetilde{\lambda}}}} \ e_{\widetilde{D_{\widetilde{\lambda}}},T} \Big) = [S|T]^* \in {\mathbf{U}}(gl(n)),$$ and call the element $[S|T]$ a [*Capelli \*-bitableau*]{} [@Brini3-BR], [@Brini4-BR].
We set $$\mathfrak{p} \Big( e_{S,C_{\lambda}} \ e_{C_{\lambda},D_{\lambda}} \ e_{D_{\lambda},T} \Big)
= [S| \fbox{$T$}] \in {\mathbf{U}}(gl(n)).$$ and call the element $[S| \fbox{$T$}] $ a [*right Young-Capelli bitableau*]{} [@Brini2-BR].
The *bitableaux correspondence* maps $\mathcal{B}$ and $\mathcal{B}^*$ and the Koszul map $\mathcal{K}$ {#bbk maps}
=======================================================================================================
\[operator B\] The *bitableaux correspondence* map $$\label{map B}
\mathcal{B} : (S|T) \mapsto [S|T]$$ uniquely extends to a linear map $$\mathcal{B} : {\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n)) \rightarrow \mathbf{U}(gl(n)).$$
\[operator B\^\*\] The *\*-bitableaux correspondence* map $$\label{map B^*}
\mathcal{B}^* : (S|T)^* \mapsto [S|T]^*$$ uniquely extends to a linear map $$\mathcal{B}^* : {\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n)) \rightarrow \mathbf{U}(gl(n)).$$
The linear isomorphisms $\mathcal{B}$ and $\mathcal{B}^*$ were introduced in [@Brini4-BR], Theorem $1$ and Theorem $3$.
Eqs. (\[map B\]), (\[map B\^\*\]) indeed define [*linear*]{} operators since bitableaux in ${\mathbb C}[M_{n,n}]$ and Capelli bitableaux in $\mathbf{U}(gl(n))$ are ruled by the **same** *straightening laws* as well as \*-bitableaux and Capelli \*-bitableaux (see [@Brini3-BR], Proposition $7$).
Given $i, j = 1, 2, \ldots, n$, let $$\rho_{ij} : {\mathbb C}[M_{n,n}] \rightarrow {\mathbb C}[M_{n,n}]$$ be the linear operator $$\rho_{ij}(\mathbf{p}) = D_{ij}(\mathbf{p}) + (i|j) \cdot \mathbf{p},
\quad for \ every \ \mathbf{p} \in {\mathbb C}[M_{n,n}],$$ where $D_{ij}$ denotes the *polarization operator* defined by the following conditions:
- $D_{ij}$ is a derivation,
- $D_{ij} \big( (h|k) \big) = \delta_{jh} (i|k)$, for every $k = 1, 2, \ldots, n$.
\[main identity\] We have: $$[\rho_{ij},\rho_{hk}] = \rho_{ij} \rho_{hk} - \rho_{hk} \rho_{ij} =
\delta_{jh} \rho_{ik} - \delta_{ik} \rho_{hj}.$$
By the universal property of $\mathbf{U}(gl(n))$, Proposition \[main identity\] implies
The map $$e_{ij} \rightarrow \rho_{ij}, \quad e_{ij} \in gl(n)$$ defines an associative algebra morphism $$\tau : \mathbf{U}(gl(n)) \rightarrow End_{\mathbb C}[{\mathbb C}[M_{n,n}]].$$
Let $\varepsilon_1$ be the linear map *evalution at* $1$ $$\varepsilon_1 : End_{\mathbb C}[{\mathbb C}[M_{n,n}]] \rightarrow {\mathbb C}[M_{n,n}],$$ $$\varepsilon_1(\rho) = \rho(1) \in {\mathbb C}[M_{n,n}], \quad for \ every \ \rho \in End_{\mathbb C}[{\mathbb C}[M_{n,n}]].$$
The *Koszul map* is the (*linear*) composition map $$\mathcal{K} : \mathbf{U}(gl(n)) \rightarrow {\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n)) ,$$ $$\mathcal{K} = \varepsilon_1 \circ \tau.$$
We have:
1. $
\mathcal{K}(e_{i_1j_1}e_{i_2j_2} \cdots e_{i_hj_h}) = \rho_{i_1j_1}\rho_{i_2j_2} \cdots \rho_{i_hj_h}(1),
$ $e_{i_pj_p} \in gl(n)$, $p =1 2, \ldots ,h$.
2. $
\mathcal{K}(e_{ij}\mathbf{P}) = \rho_{ij}(\mathcal{K}(\mathbf{P})),
$ for every $\mathbf{P} \in \mathbf{U}(gl(n))$, $\ e_{ij} \in gl(n)$.
Expansion formulae for *column Capelli bitableaux* and *column Capelli \*-bitableaux* {#column exp}
=====================================================================================
Consider the *column Capelli bitableau* $$\left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]
=
\mathfrak{p} \Big(e_{i_1 \alpha_1} \cdots e_{i_h \alpha_h}e_{\alpha_1 j_1} \cdots e_{\alpha_h j_h} \Big) \in \mathbf{U}(gl(n)),$$ (where $\alpha_1, \ldots, \alpha_h$ are arbitrary *distict positive virtual* symbols) and the *column Capelli \*-bitableau* $$\left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]^*
=
\mathfrak{p} \Big(e_{i_1 \beta_1} \cdots e_{i_h \beta_h}e_{\beta_1 j_1} \cdots e_{\beta_h j_h} \Big) \in \mathbf{U}(gl(n))$$ (where $\beta_1, \ldots, \beta_h$ are arbitrary *distict negative virtual* symbols).
Remember that the *proper symbols* $i_1, i_2, \cdots, i_h,, \quad j_1, j_2, \cdots, j_h \in L = \{1, 2, \ldots, n \}$ are assumed to be *negative*.
\[generators\]
The families of column Capelli bitableaux and the column Capelli \*-bitableaux are systems of linear generators of $\mathbf{U}(gl(n))$.
Both *column Capelli bitabeaux* and *column Capelli \*-bitabeaux* are *row-commutative*, in simbols:
1. $$\left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]
=
\left[
\begin{array}{c}
i_{\sigma(1)}\\ i_{\sigma(2)} \\ \vdots \\ i_{\sigma(h)}
\end{array}
\right| \left.
\begin{array}{c}
j_{\sigma(1)}\\ j_{\sigma(2)} \\ \vdots \\ j_{\sigma(h)}
\end{array}
\right], \quad \sigma \in v\textbf{S}_h,$$
2. $$\left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]^*
=
\left[
\begin{array}{c}
i_{\sigma(1)}\\ i_{\sigma(2)} \\ \vdots \\ i_{\sigma(h)}
\end{array}
\right| \left.
\begin{array}{c}
j_{\sigma(1)}\\ j_{\sigma(2)} \\ \vdots \\ j_{\sigma(h)}
\end{array}
\right]^*, \quad \sigma \in v\textbf{S}_h,$$
We recall two basic expansion formulae, that describe the effect of picking out (on the left hand side) the first row of column Capelli bitableaux and column Capelli \*-bitableaux. These formulae play a crucial role in the theory of the Koszul map $\mathcal{K}$, and provide a simple way to compute the *actual* forms of column Capelli bitableaux and column Capelli \*-bitableaux as elements of $\mathbf{U}(gl(n))$ are *row-commutative*, in simbols:
\[column exp\] We have:
1. $$\begin{aligned}
& \left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right] =
\\
=&
\ (-1)^{h-1} e_{i_1j_1} \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right] +
(-1)^{h-2} \sum_{k=2}^h \ \delta_{i_kj_1} \ \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_1 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_k \\ \vdots \\ j_h
\end{array}
\right] \in \mathbf{U}(gl(n)).\end{aligned}$$
2. $$\begin{aligned}
& \left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]^* =
\\
=&
\ e_{i_1j_1} \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right]^* -
\sum_{k=2}^h \ \delta_{i_kj_1} \ \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_1 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_k \\ \vdots \\ j_h
\end{array}
\right]^* \in \mathbf{U}(gl(n)).\end{aligned}$$
For a proof, see e.g. [@Brini5-BR].
$$\begin{aligned}
\left[
\begin{array}{c}
1 \\ 2 \\ 3 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
2 \\ 3 \\ 4 \\ 3
\end{array}
\right] &=
- e_{12}\left[
\begin{array}{c}
2 \\ 3 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
3 \\ 4 \\ 3
\end{array}
\right]
+
\left[
\begin{array}{c}
1 \\ 3 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
3 \\ 4 \\ 3
\end{array}
\right]
+
\left[
\begin{array}{c}
2 \\ 3 \\ 1
\end{array}
\right| \left.
\begin{array}{c}
3 \\ 4 \\ 3
\end{array}
\right]
\\
&=
- e_{12}\left[
\begin{array}{c}
2 \\ 3 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
3 \\ 4 \\ 3
\end{array}
\right]
+
\ 2 \ \left[
\begin{array}{c}
1 \\ 3 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
3 \\ 4 \\ 3
\end{array}
\right]
\\
&=
- e_{12} \Big( e_{23} \left[
\begin{array}{c}
3 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
4 \\ 3
\end{array}
\right]
-
\left[
\begin{array}{c}
2 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
4 \\ 3
\end{array}
\right] \Big)
\\
& \phantom{=} \
+
2 \Big( e_{13}
\left[
\begin{array}{c}
3 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
4 \\ 3
\end{array}
\right]
-
\left[
\begin{array}{c}
1 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
4 \\ 3
\end{array}
\right] \Big)
\\
&=
e_{12}e_{23}e_{34}e_{23} -
e_{12}e_{24}e_{23}
- 2 e_{13}e_{34}e_{23}
+ 2 e_{14}e_{23} \in \mathbf{U}(gl(4)).\end{aligned}$$
Main results
============
\[K column\] $$\begin{aligned}
\mathcal{K} \big( \left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right] \big) =&
\left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right)
\\
=& \ (-1)^{h \choose 2} (i_1|j_1)(i_2|j_2) \dots (i_h|j_h)
\in {\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n)).\end{aligned}$$
$$\begin{aligned}
& \mathcal{K} \big( \left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right] \big) =
\\
=&
\ (-1)^{h-1} \mathcal{K} \big( e_{i_1j_1} \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right] \big) +
(-1)^{h-2} \mathcal{K} \big( \sum_{k=2}^h \ \delta_{i_kj_1} \ \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_1 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_k \\ \vdots \\ j_h
\end{array}
\right] \big)
\\
=&
\ (-1)^{h-1} \rho_{i_1j_1} \big(\mathcal{K} \big( \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right] \big) \big) +
(-1)^{h-2} \mathcal{K} \big( \sum_{k=2}^h \ \delta_{i_kj_1} \ \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_1 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_k \\ \vdots \\ j_h
\end{array}
\right] \big)
\\
=&
\ (-1)^{h-1} D_{i_1j_1} \big( \left(
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right) \big) + (-1)^{h-1} (i_1|j_1)\left(
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right)
\\
&
\qquad \qquad + (-1)^{h-2} \sum_{k=2}^h \ \delta_{i_kj_1} \ \left(
\begin{array}{c}
i_2 \\ \vdots \\ i_1 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_k \\ \vdots \\ j_h
\end{array}
\right)
\\
=&
\ (-1)^{h-1} (i_1|j_1)\left(
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right) =
\left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right).\end{aligned}$$
Consider the column Capelli bitableau $$\left[
\begin{array}{c}
1\\ 2 \\ 3
\end{array}
\right| \left.
\begin{array}{c}
2 \\ 1 \\ 1
\end{array}
\right] = e_{12} \left[
\begin{array}{c}
2 \\ 3
\end{array}
\right| \left.
\begin{array}{c}
1 \\ 1
\end{array}
\right] - \left[
\begin{array}{c}
1 \\ 3
\end{array}
\right| \left.
\begin{array}{c}
1 \\ 1
\end{array}
\right] = - e_{12}e_{21}e_{31} + e_{11}e_{31} \in \mathbf{U}(gl(n)).$$
We have $$\begin{aligned}
\mathcal{K} \big( \left[
\begin{array}{c}
1\\ 2 \\ 3
\end{array}
\right| \left.
\begin{array}{c}
2 \\ 1 \\ 1
\end{array}
\right] \big) &= \mathcal{K} \big( - e_{12}e_{21}e_{31} + e_{11}e_{31} \big)
\\
&=
\ \left(
\begin{array}{c}
1\\ 2 \\ 3
\end{array}
\right| \left.
\begin{array}{c}
2 \\ 1 \\ 1
\end{array}
\right)
\\
&= -(1|2)(2|1)(3|1) \in {\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n)).\end{aligned}$$
\[K \*-column\] $$\begin{aligned}
\mathcal{K} \big( \left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]^* \big) =&
\left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right)^*
\\
=& \ (i_1|j_1)(i_2|j_2) \dots (i_h|j_h)
\in {\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n)).\end{aligned}$$
$$\begin{aligned}
\mathcal{K} \big( \left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]^* \big) =&
\\
=&
\ \mathcal{K} \big( e_{i_1j_1} \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right]^* \big) -
\mathcal{K} \big( \sum_{k=2}^h \ \delta_{i_kj_1} \ \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_1 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_k \\ \vdots \\ j_h
\end{array}
\right]^* \big)\end{aligned}$$
$$\begin{aligned}
\phantom{ \quad \ \mathcal{K} \big( \left[
\begin{array}{c}
i_1
\end{array}
\right| \left.
\begin{array}{c}
j_1
\end{array}
\right]^* \big) =}&
\\
=&
\ \rho_{i_1j_1} \big( \mathcal{K} \big( \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right]^* \big) \big) -
\mathcal{K} \big( \sum_{k=2}^h \ \delta_{i_kj_1} \ \left[
\begin{array}{c}
i_2 \\ \vdots \\ i_1 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_k \\ \vdots \\ j_h
\end{array}
\right]^* \big)
\\
=&
\ D_{i_1j_1} \big( \left(
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right)^* \big) + (i_1|j_1)\left(
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right)^*
\\
&
\qquad \qquad - \sum_{k=2}^h \ \delta_{i_kj_1} \ \left(
\begin{array}{c}
i_2 \\ \vdots \\ i_1 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_k \\ \vdots \\ j_h
\end{array}
\right)^*
\\
=&
(i_1|j_1) \left(
\begin{array}{c}
i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_2 \\ \vdots \\ j_h
\end{array}
\right)^* =
\left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right)^*.\end{aligned}$$
Since $$\left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right)^* = (i_1|j_1)(i_2|j_2) \cdots (i_h|j_h) =
(-1)^{h \choose 2} \left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right),$$ we have
$$\left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]^* =
(-1)^{h \choose 2} \left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right] \in \mathbf{U}(gl(n)).$$
Since, Theorem \[operator B\] specializes to $$\mathcal{B} \big( \left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right) \big) = \left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]$$ and, Theorem \[operator B\^\*\] specializes to $$\mathcal{B^*}\big( \left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right)^* \big) = \left[
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right]^*,$$ from Remark \[generators\], we have
\[BCKtheorem\]
1. $\mathcal{B} = \mathcal{K}^{-1}$,
2. $\mathcal{B^*} = \mathcal{K}^{-1}$,
3. $
\mathcal{B^*} = \mathcal{B},
$
4. $\mathcal{B}, \ \mathcal{K}$ are linear isomorphisms.
Since the set of (determinantal) standard bitableaux $$\Big \{ (S|T); \ sh(S) =sh(T) = \lambda, \ \lambda_1 \leq n, \ S, T \ standard \Big \}$$ and the set of (permanental) costandard \*-bitableaux $$\Big \{ (U|V)^*; \ sh(U) =sh(V) = \mu, \ \widetilde{\mu_1} \leq n, \ U, V \ costandard \Big \}$$ are (linear) bases of ${\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n))$, then
The set of (determinantal) standard *Capelli bitableaux* $$\Big \{ [S|T]; \ sh(S) =sh(T) = \lambda, \ \lambda_1 \leq n, \ S, T \ standard \Big \}$$ and the set of (permanental) costandard *Capelli \*-bitableaux* $$\Big\{ [U|V]^*; \ sh(U) =sh(V) = \mu, \ \widetilde{\mu_1} \leq n, \ U, V \ costandard \Big\}$$ are (linear) bases of $\mathbf{U}(gl(n))$.
The *bitableaux correspondence isomorphism* $\mathcal{B}$ and the *Koszul isomorphism* $\mathcal{K}$ well-behave with respect to *right symmetrized bitableaux* $$(S|\fbox{$T$}) \in {\mathbb C}[M_{n,n}]$$ and [*right Young-Capelli bitableaux*]{} $$[S| \fbox{$T$} ] \in \mathbf{U}(gl(n)).$$
In plain words, any right Young-Capelli bitableaux $[S| \fbox{$T$} ]$ is the image - with respect to the linear operator $\mathcal{B}$ - of the right symmetrized bitableaux $(S|\fbox{$T$})$.
\[image symm\] We have: $$\begin{aligned}
&\mathcal{B} : (S|\fbox{$T$}) \mapsto [S| \fbox{$T$} ],
\\
&\mathcal{K} : [S| \fbox{$T$} ] \mapsto (S|\fbox{$T$}).\end{aligned}$$
Since the set $$\Big\{ (S|\fbox{$T$}); \ sh(S) = sh(T) = \lambda \vdash h, \ \lambda_1 \leq n, \ S, T \ standard \Big\}$$ is the *Gordan-Capelli basis* of ${\mathbb C}[M_{n,n}]$ (see, e.g. [@Bri-BR],[@Brini1-BR],[@Brini2-BR]), then
\[Y-C basis\] The set of right Young-Capelli bitableaux $$\Big\{ [S|\fbox{$T$}]; \ \ sh(S) = sh(T) = \lambda, \ \lambda_1 \leq n, \ S, T \ standard \Big\}$$ is a (linear) basis of $\mathbf{U}(gl(n))$.
\[action\] The basis elements $$\Big\{ \ [S|\fbox{$T$}]; \ S, T \ standard, \ sh(S) = sh(T) = \lambda \vdash k, \ \lambda_1 \leq n \Big\}$$ act in a quite remarkable way on Gordan-Capelli basis elements $$\Big\{ \ (U|\fbox{$V$}); \ U, V\ standard, \ sh(U) = sh(V) = \mu \vdash h, \ \mu_1 \leq n \Big\}.$$ Indeed, we have:
- If $h < k$, the action is zero.
- If $h = k$ and $\lambda \neq \mu$, the action is zero.
- If $h = k$ and $\lambda = \mu$, the action is nondegenerate triangular (with respect to a suitable linear order on standard tableaux of the same shape).
See *[@Brini2-BR]* and *[@Bri-BR]*, *Theorem* $10.1$.
Laplace expansions
==================
Laplace expansions in ${\mathbb C}[M_{n,n}]$ {#LaplExp}
--------------------------------------------
Recall that $$(i_1 i_2 \cdots i_h|j_1 j_2 \cdots j_h)
=
(-1)^{h \choose 2} \ det[ (i_s|j_t) ]_{s,t =1, 2, \ldots, h}
\in {\mathbb C}[M_{n,n}],$$ and, therefore, the biproduct $(i_1 i_2 \cdots i_h|j_1 j_2 \cdots j_h) \in {\mathbb C}[M_{n,n}]$ expands into column bitableaux as follows: $$\begin{aligned}
(i_1 i_2 \cdots i_h|j_1 j_2 \cdots j_h) &=
\sum_{\sigma \in \mathbf{S}_h} \ (-1)^{|\sigma|} \left(
\begin{array}{c}
i_{\sigma(1)}\\ i_{\sigma(2)} \\ \vdots \\ i_{\sigma(h)}
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right)
\\
&=
\sum_{\sigma \in \mathbf{S}_h} \ (-1)^{|\sigma|} \left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_{\sigma(1)}\\ j_{\sigma(2)} \\ \vdots \\ j_{\sigma(h)}
\end{array}
\right).\end{aligned}$$ Notice that, in the passage from monomials to column bitableaux, the sign $(-1)^{h \choose 2}$ disappears..
Recall that $$(i_1 i_2 \cdots i_h|j_1 j_2 \cdots j_h)^*
=
per[ (i_s|j_t) ]_{s,t =1, 2, \ldots, h}
\in {\mathbb C}[M_{n,n}],$$ and, therefore, the \*-biproduct $(i_1 i_2 \cdots i_h|j_1 j_2 \cdots j_h)^* \in {\mathbb C}[M_{n,n}]$ expands into column \*-bitableaux as follows: $$\begin{aligned}
(i_1 i_2 \cdots i_h|j_1 j_2 \cdots j_h)^* &=
\sum_{\sigma \in \mathbf{S}_h} \ \left(
\begin{array}{c}
i_{\sigma(1)}\\ i_{\sigma(2)} \\ \vdots \\ i_{\sigma(h)}
\end{array}
\right| \left.
\begin{array}{c}
j_1\\ j_2 \\ \vdots \\ j_h
\end{array}
\right)^*
\\
&=
\sum_{\sigma \in \mathbf{S}_h} \ \left(
\begin{array}{c}
i_1\\ i_2 \\ \vdots \\ i_h
\end{array}
\right| \left.
\begin{array}{c}
j_{\sigma(1)}\\ j_{\sigma(2)} \\ \vdots \\ j_{\sigma(h)}
\end{array}
\right)^*.\end{aligned}$$
The preceding arguments extend to bitableaux and to \*-bitableaux of any shape $\lambda, \ \lambda_1 \leq n.$ Given the Young tableaux of of eq.(\[tableaux\]) $$\label{tableaux}
S = \left(
\begin{array}{llllllllllllll}
i_{p_1} \ldots \ldots \ldots i_{p_{\lambda_1}} \\
i_{q_1} \ldots \ldots i_{q_{\lambda_2}} \\
\ldots \ldots \\
i_{r_1} \ldots i_{r_{\lambda_m}}
\end{array}
\right),
\quad
T = \left(
\begin{array}{llllllllllllll}
j_{s_1} \ldots \ldots \ldots j_{s_{\lambda_1}} \\
j_{t_1} \ldots \ldots j_{t_{\lambda_2}} \\
\ldots \ldots \\
j_{v_1} \ldots j_{v_{\lambda_m}}
\end{array}
\right).$$, a simple sign computation shows that $$(S|T) =
\sum_{\sigma_1, \ldots, \sigma_m } \ (-1)^{\sum_{k=1}^m \ |\sigma_k|} \
\left(
\begin{array}{c}
i_{p_{\sigma_1(1)}}\\ . \\ i_{p_{\sigma_1(\lambda_1)}} \\
\vdots \\
i_{r_{\sigma_m(1)}}\\ . \\ i_{r_{\sigma_m(\lambda_m)}}
\end{array}
\right| \left.
\begin{array}{c}
j_{s_1}\\ . \\ j_{s_{\lambda_1}} \\
\vdots \\
j_{v_1}\\ . \\ j_{v_{\lambda_m}}
\end{array}
\right)$$ $$\phantom{(S|T)} =
\sum_{\sigma_1, \ldots, \sigma_m } \ (-1)^{\sum_{k=1}^m \ |\sigma_k|} \
\left(
\begin{array}{c}
i_{p_1}\\ . \\ i_{p_{\lambda_1}} \\
\vdots \\
i_{r_1}\\ . \\ i_{r_{\lambda_m}}
\end{array}
\right| \left.
\begin{array}{c}
j_{s_{\sigma_1(1)}}\\ . \\ j_{s_{\sigma_1(\lambda_1)}} \\
\vdots \\
j_{v_{\sigma_m(1)}}\\ . \\ j_{v_{\sigma_m(\lambda_m)}}
\end{array}
\right)$$ where the multiple sums range over all permutations $\sigma_1 \in \mathbf{S}_{\lambda_1}, \ldots, \sigma_m \in \mathbf{S}_{\lambda_m}.$ Notice that only the signs of permutations remain.
Similarly, $$(S|T)^* =
\sum_{\sigma_1, \ldots, \sigma_m } \
\left(
\begin{array}{c}
i_{p_{\sigma_1(1)}}\\ . \\ i_{p_{\sigma_1(\lambda_1)}} \\
\vdots \\
i_{r_{\sigma_m(1)}}\\ . \\ i_{r_{\sigma_m(\lambda_m)}}
\end{array}
\right| \left.
\begin{array}{c}
j_{s_1}\\ . \\ j_{s_{\lambda_1}} \\
\vdots \\
j_{v_1}\\ . \\ j_{v_{\lambda_m}}
\end{array}
\right)^*$$ $$\phantom{(S|T)} =
\sum_{\sigma_1, \ldots, \sigma_m } \
\left(
\begin{array}{c}
i_{p_1}\\ . \\ i_{p_{\lambda_1}} \\
\vdots \\
i_{r_1}\\ . \\ i_{r_{\lambda_m}}
\end{array}
\right| \left.
\begin{array}{c}
j_{s_{\sigma_1(1)}}\\ . \\ j_{s_{\sigma_1(\lambda_1)}} \\
\vdots \\
j_{v_{\sigma_m(1)}}\\ . \\ j_{v_{\sigma_m(\lambda_m)}}
\end{array}
\right)^*.$$
Laplace expansions in $\mathbf{U}(gl(n))$
-----------------------------------------
Let $S$ and $T$ be the Young tableaux $$S = \left(
\begin{array}{llllllllllllll}
i_{p_1} \ldots \ldots \ldots i_{p_{\lambda_1}} \\
i_{q_1} \ldots \ldots i_{q_{\lambda_2}} \\
\ldots \ldots \\
i_{r_1} \ldots i_{r_{\lambda_m}}
\end{array}
\right),
\quad
T = \left(
\begin{array}{llllllllllllll}
j_{s_1} \ldots \ldots \ldots j_{s_{\lambda_1}} \\
j_{t_1} \ldots \ldots j_{t_{\lambda_2}} \\
\ldots \ldots \\
j_{v_1} \ldots j_{v_{\lambda_m}}
\end{array}
\right)$$ of eq.(\[tableaux\]).
By Theorem \[BCKtheorem\], the results of subsection \[LaplExp\] lead to the following *Laplace expansion* of Capelli bitableaux into column Capelli bitableaux and of Capelli \*-bitableaux into column Capelli \*-bitableaux.
$$\label{Exp1}
[S|T] =
\sum_{\sigma_1, \ldots, \sigma_m } \ (-1)^{\sum_{k=1}^m \ |\sigma_k|} \
\left[
\begin{array}{c}
i_{p_{\sigma_1(1)}}\\ . \\ i_{p_{\sigma_1(\lambda_1)}} \\
\vdots \\
i_{r_{\sigma_m(1)}}\\ . \\ i_{r_{\sigma_m(\lambda_m)}}
\end{array}
\right| \left.
\begin{array}{c}
j_{s_1}\\ . \\ j_{s_{\lambda_1}} \\
\vdots \\
j_{v_1}\\ . \\ j_{v_{\lambda_m}}
\end{array}
\right]$$
$$\label{exp2}
\phantom{[S|T]} =
\sum_{\sigma_1, \ldots, \sigma_m } \ (-1)^{\sum_{k=1}^m \ |\sigma_k|} \
\left[
\begin{array}{c}
i_{p_1}\\ . \\ i_{p_{\lambda_1}} \\
\vdots \\
i_{r_1}\\ . \\ i_{r_{\lambda_m}}
\end{array}
\right| \left.
\begin{array}{c}
j_{s_{\sigma_1(1)}}\\ . \\ j_{s_{\sigma_1(\lambda_1)}} \\
\vdots \\
j_{v_{\sigma_m(1)}}\\ . \\ j_{v_{\sigma_m(\lambda_m)}}
\end{array}
\right].$$
$$\label{exp3}
[S|T]^* =
\sum_{\sigma_1, \ldots, \sigma_m } \
\left[
\begin{array}{c}
i_{p_{\sigma_1(1)}}\\ . \\ i_{p_{\sigma_1(\lambda_1)}} \\
\vdots \\
i_{r_{\sigma_m(1)}}\\ . \\ i_{r_{\sigma_m(\lambda_m)}}
\end{array}
\right| \left.
\begin{array}{c}
j_{s_1}\\ . \\ j_{s_{\lambda_1}} \\
\vdots \\
j_{v_1}\\ . \\ j_{v_{\lambda_m}}
\end{array}
\right]^*$$
$$\label{exp4}
\phantom{[S|T]^*} =
\sum_{\sigma_1, \ldots, \sigma_m } \
\left[
\begin{array}{c}
i_{p_1}\\ . \\ i_{p_{\lambda_1}} \\
\vdots \\
i_{r_1}\\ . \\ i_{r_{\lambda_m}}
\end{array}
\right| \left.
\begin{array}{c}
j_{s_{\sigma_1(1)}}\\ . \\ j_{s_{\sigma_1(\lambda_1)}} \\
\vdots \\
j_{v_{\sigma_m(1)}}\\ . \\ j_{v_{\sigma_m(\lambda_m)}}
\end{array}
\right]^*.$$
By combining eqs. (\[exp1\]), (\[exp2\]), (\[exp3\]), (\[exp4\]) with the results of Proposition \[column exp\], one gets the explicit expansions as elements of $\mathbf{U}(gl(n))$.
We have
$$\begin{aligned}
[12|12] =&
\left[
\begin{array}{c}
1 \\ 2
\end{array}
\right| \left.
\begin{array}{c}
1 \\ 2
\end{array}
\right]
-
\left[
\begin{array}{c}
2 \\ 1
\end{array}
\right| \left.
\begin{array}{c}
1 \\ 2
\end{array}
\right]
\\
=&
-e_{11}e_{22} + e_{21}e_{12} - e_{22}
\\
=&
- \mathbf{cdet} \left(
\begin{array}{cc}
e_{11}+1 \ \ e_{12}
\\
e_{21} \ \ \ \ \ \ \ e_{22}
\end{array}
\right) \in \mathbf{U}(gl(n)).\end{aligned}$$
\[CapDet\] Consider the row Capelli bitableau $$[n \cdots 2 1 |12 \cdots n] \in \mathbf{U}(gl(n)).$$ We have:
1. $$\begin{aligned}
[n \cdots 2 1 |12 \cdots n] =
\mathbf{cdet}
\left(
\begin{array}{cccc}
e_{1 1}+(n-1) & e_{1 2} & \ldots & e_{1 n} \\
e_{2 1} & e_{2 2}+(n-2) & \ldots & e_{2 n}\\
\vdots & \vdots & \vdots & \\
e_{n 1} & e_{n 2} & \ldots & e_{n n}\\
\end{array}
\right),\end{aligned}$$
the *central* Capelli column determinant[^1] in $\mathbf{U}(gl(n))$.
2. $$\begin{aligned}
\mathcal{K} \left( [n \cdots 2 1 |12 \cdots n] \right) =&
\
\mathbf{det}
\left(
\begin{array}{ccc}
(1|1) & \ldots & (1|n) \\
(2|1) & \ldots & (2|n) \\
\vdots & & \vdots \\
(n|1) & \ldots & (n|n) \\
\end{array}
\right) \in {\mathbb C}[M_{n,n}].
\end{aligned}$$
We have
$$\begin{aligned}
[n \cdots 2 1 &|12 \cdots n] =
\sum_{\sigma \in \mathbf{S}_n} \ (-1)^{|\sigma|}
\left[
\begin{array}{c}
\sigma(n) \\ \sigma(n-1) \\ \vdots \\ \sigma(1)
\end{array}
\right| \left.
\begin{array}{c}
1 \\ 2 \\ \vdots \\ n
\end{array}
\right]
\\
=&
\sum_{\sigma \in \mathbf{S}_n} \ (-1)^{|\sigma|} \times
\\
& \ \Big( (-1)^{n-1} \ e_{\sigma(n) 1}
\left[
\begin{array}{c}
\sigma(n-1) \\ \sigma(n-2) \\ \vdots \\ \sigma(1)
\end{array}
\right| \left.
\begin{array}{c}
2 \\ 3 \\ \vdots \\ n
\end{array}
\right]
\\
& + (-1)^{n-2}
\sum_{k=1}^{n-1} \ \delta_{\sigma(k) 1}
\left[
\begin{array}{c}
\sigma(n-1) \\ \vdots \\ \sigma(n) \\ \vdots \\ \sigma(1)
\end{array}
\right| \left.
\begin{array}{c}
2 \\ \vdots \\ k \\ \vdots \\ n
\end{array}
\right] \Big)
\\
=&
\ (-1)^{n-1} \
\sum_{\sigma \in \mathbf{S}_n} \ (-1)^{|\sigma|}
\left(e_{\sigma(n) 1} + (n-1) \delta_{\sigma(n) 1} \right)
\left[
\begin{array}{c}
\sigma(n-1) \\ \sigma(n-2) \\ \vdots \\ \sigma(1)
\end{array}
\right| \left.
\begin{array}{c}
2 \\ 3 \\ \vdots \\ n
\end{array}
\right],\end{aligned}$$
by the expansion formula for *column Capelli bitableaux* (Proposition \[column exp\], item $1.$).
By iterating the same argument, $$\begin{aligned}
&[n \cdots 2 1 |12 \cdots n] =
\\
=&
\ (-1)^{n \choose 2} \times
\\
&
\sum_{\sigma \in \mathbf{S}_n} \ (-1)^{|\sigma|}
\left(e_{\sigma(n) 1} + (n-1) \delta_{\sigma(n) 1} \right)
\left(e_{\sigma(n-1) 2} + (n-2) \delta_{\sigma(n-1) 2} \right)
\cdots
e_{\sigma(1) n}
\\
=&
\sum_{\tau \in \mathbf{S}_n} \ (-1)^{|\tau|}
\left(e_{\tau(1) 1} + (n-1) \delta_{\tau(1) 1} \right)
\left(e_{\tau(2) 2} + (n-2) \delta_{\tau(2) 2} \right)
\cdots
e_{\tau(n) n}
\\
=&
\ \mathbf{cdet}
\left(
\begin{array}{cccc}
e_{1 1}+(n-1) & e_{1 2} & \ldots & e_{1 n} \\
e_{2 1} & e_{2 2}+(n-2) & \ldots & e_{2 n}\\
\vdots & \vdots & \vdots & \\
e_{n 1} & e_{n 2} & \ldots & e_{n n}\\
\end{array}
\right) \in \mathbf{U}(gl(n)),\end{aligned}$$ the *central* Capelli column determinant in $\mathbf{U}(gl(n))$.
Then $$\begin{aligned}
\mathcal{K} \left( [n \cdots 2 1 |12 \cdots n] \right) =&
\\
=& \
\mathcal{K} \Big(
\mathbf{cdet}
\left(
\begin{array}{cccc}
e_{1 1}+(n-1) & e_{1 2} & \ldots & e_{1 n} \\
e_{2 1} & e_{2 2}+(n-2) & \ldots & e_{2 n}\\
\vdots & \vdots & \vdots & \\
e_{n 1} & e_{n 2} & \ldots & e_{n n}\\
\end{array}
\right) \Big)
\\
=&
\ (n \cdots 2 1 |12 \cdots n)
\\
=&
\
\mathbf{det}
\left(
\begin{array}{ccc}
(1|1) & \ldots & (1|n) \\
(2|1) & \ldots & (2|n) \\
\vdots & & \vdots \\
(n|1) & \ldots & (n|n) \\
\end{array}
\right) \in {\mathbb C}[M_{n,n}].
\end{aligned}$$
The Capelli bitableau (of shape $\lambda = (2, 2)$: $$\label{excapbit}
\left[
\begin{array}{ccc}
1 & 2
\\
2 & 4
\end{array}
\right| \left.
\begin{array}{ccc}
2 & 3
\\
3 & 4
\end{array}
\right]
\in \mathbf{U}(gl(4))$$ equals (by eq. (\[Exp1\])) $$\left[
\begin{array}{ccc}
1 \\ 2
\\ 2 \\ 4
\end{array}
\right| \left.
\begin{array}{ccc}
2 \\ 3
\\ 3 \\ 4
\end{array}
\right]
-
\left[
\begin{array}{ccc}
1 \\ 2
\\ 2 \\ 4
\end{array}
\right| \left.
\begin{array}{ccc}
3 \\ 2
\\ 3 \\ 4
\end{array}
\right]
-
\left[
\begin{array}{ccc}
1 \\ 2
\\ 2 \\ 4
\end{array}
\right| \left.
\begin{array}{ccc}
2 \\ 3
\\ 4 \\ 3
\end{array}
\right]
+
\left[
\begin{array}{ccc}
1 \\ 2
\\ 2 \\ 4
\end{array}
\right| \left.
\begin{array}{ccc}
3 \\ 2
\\ 4 \\ 3
\end{array}
\right] \in \mathbf{U}(gl(4)),$$ where $$\label{exp1}
\left[
\begin{array}{ccc}
1 \\ 2
\\ 2 \\ 4
\end{array}
\right| \left.
\begin{array}{ccc}
2 \\ 3
\\ 3 \\ 4
\end{array}
\right] =
\ e_{12}e_{23}e_{23}e_{44} - 2e_{13}e_{23}e_{44},$$ $$\label{exp2}
-
\left[
\begin{array}{ccc}
1 \\ 2
\\ 2 \\ 4
\end{array}
\right| \left.
\begin{array}{ccc}
3 \\ 2
\\ 3 \\ 4
\end{array}
\right] =
\ - e_{13}e_{22}e_{23}e_{44} + e_{13}e_{23}e_{44},$$ $$\label{exp3}
-
\left[
\begin{array}{ccc}
1 \\ 2
\\ 2 \\ 4
\end{array}
\right| \left.
\begin{array}{ccc}
2 \\ 3
\\ 4 \\ 3
\end{array}
\right] =
\ - e_{12}e_{23}e_{24}e_{43} + e_{12}e_{23}e_{23} + e_{13}e_{24}e_{43}
- e_{13}e_{23} + e_{23}e_{14}e_{43} - e_{23}e_{13},$$ $$\label{exp3}
\left[
\begin{array}{ccc}
1 \\ 2
\\ 2 \\ 4
\end{array}
\right| \left.
\begin{array}{ccc}
3 \\ 2
\\ 4 \\ 3
\end{array}
\right] =
\ e_{13}e_{22}e_{24}e_{43} - e_{13}e_{22}e_{23} - e_{13}e_{24}e_{43}
- e_{13}e_{23}.$$
This example clarifies the difference between the PBW Theorem and the BCK Theorem.
The PBW Theorem establishes an isomorphism $\phi$ from the *graded object* $$Gr \left[ \mathbf{U}(gl(n)) \right] = \bigoplus_{h \in \mathbb{Z}^+} \ \frac {\mathbf{U}^{(h)}(gl(n))} {\mathbf{U}^{(h-1)}(gl(n))}$$ associated to the *filtered* algebra $\mathbf{U}(gl(n))$ to the symmetric algebra $\mathbf{Sym}(gl(n)) \cong {\mathbb C}[M_{n,n}] $. By eqs. (\[exp1\]), (\[exp2\]), (\[exp3\]), (\[exp4\]), the image under $\phi$ of the Capelli bitableau (\[excapbit\]) - *regarded as an element of the quotient space* $
\frac {\mathbf{U}^{(4)}(gl(n))} {\mathbf{U}^{(3)}(gl(n))}
$ - is immediately recognized to be the product determinants $$\label{proddet}
\mathbf{det}
\left(
\begin{array}{ccc}
(1|2) & (1|3)
\\
(2|2) & (2|3)
\end{array}
\right)
\times
\mathbf{det}
\left(
\begin{array}{ccc}
(2|3) & (2|4)
\\
(4|3) & (4|4)
\end{array}
\right) \in {\mathbb C}[M_{4,4}].$$
By the BCK Theorem, the product of determinants (\[proddet\]) is indeed the image under the Koszul isomorphism $\mathcal{K}$ of the Capelli bitableau (\[excapbit\]), *regarded as an element of $\mathbf{U}(gl(4))$*. Furthermore, the image under $\mathcal{K}$ of the Capelli \*-bitableau $$\left[
\begin{array}{ccc}
1 & 2
\\
2 & 4
\end{array}
\right| \left.
\begin{array}{ccc}
2 & 3
\\
3 & 4
\end{array}
\right]^*$$ equals the product permanents $$\mathbf{per}
\left(
\begin{array}{ccc}
(1|2) & (1|3)
\\
(2|2) & (2|3)
\end{array}
\right)
\times
\mathbf{per}
\left(
\begin{array}{ccc}
(2|3) & (2|4)
\\
(4|3) & (4|4)
\end{array}
\right) \in {\mathbb C}[M_{4,4}].$$
[99]{}
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[^1]: The symbol $\mathbf{cdet}$ denotes the column determinat of a matrix $A = [a_{ij}]$ with noncommutative entries: $\mathbf{cdet} (A) = \sum_{\sigma} \ (-1)^{|\sigma|} \ a_{\sigma(1), 1}a_{\sigma(2), 2} \cdots a_{\sigma(n), n}.$
|
IMSc/2000/06/24\
cond-mat/0006279
[**Tsallis Statistics: Averages and a Physical**]{}\
[**Interpretation of the Lagrange Multiplier $\beta$**]{}\
[S. Kalyana Rama]{}
Institute of Mathematical Sciences, C. I. T. Campus,
Taramani, CHENNAI 600 113, India.
email: [email protected]\
ABSTRACT
> Tsallis has proposed a generalisation of the standard entropy, which has since been applied to a variety of physical systems. In the canonical ensemble approach that is mostly used, average energy is given by an unnromalised, or normalised, $q$-expectation value. A Lagrange multiplier $\beta$ enforces the energy constraint whose physical interpretation, however, is lacking. Here, we use a microcanonical ensemble approach and find that consistency requires that only normalised $q$-expectation values are to be used. We then present a physical interpretation of $\beta$, relating it to a physical temperature. We derive this interpretation by a different method also.
PACS numbers: 05.30.-d, 05.70.Ce
[**1.**]{} Tsallis has proposed [@tsallis1] a one parameter generalisation of the standard Boltzmann-Gibbs entropy. It is given by $$\label{sq}
S_q = \frac{\sum p_i^q - 1}{1 - q}$$ where $p_i$ is the probability that the system is in a state labelled by $i$, the sum, here and in the following, runs over all the allowed states, and the parameter $q$ is a real number. We have set the Boltzmann constant $k$ equal to unity. In the limit $q \to 1$, $S_q = - \sum p_i ln p_i$, thus reducing to the standard Boltzmann-Gibbs entropy. The statistical mechanics that follows from the entropy $S_q$, referred to here as Tsallis statistics, is rich in applications and has been studied extensively. It retains the standard thermodynamical structure, leads to power-law distributions as opposed to the exponential ones that follow from the standard statistical mechanics, and has been applied to a variety of physical systems. See [@tsallis; @spl] for a thorough discussion, and [@brazil] for an exhaustive list of references.
Mostly, in these applications, a canonical ensemble approach is used [@tsallis1; @123]. The entropy $S_q$ is extremised subject to the constraint $\sum p_i = 1$, and an average energy constraint (equation (\[123\]) below), obtaining thus the probability distribution $p_i$. Various thermodynamical quantities are then calculated by standard methods[^1].
The averages can be taken to be given by unnormalised, or normalised, $q$-expectation values. Calculations are simpler with the former choice, whereas the later choice has all desireable properties. Also, various quantities calculated using these two choices, although not equal to each other, can be related by a set of formulae. Therefore, both choices have often been used. See [@123] and references therein for a detailed discussion. The average energy constraint is then enforced through a Lagrange multiplier, $\beta$. In the standard statistical mechanics, $\beta$ is the inverse of the temperature which can be physically measured. To the best of our knowledge, a similar physical interpretation of $\beta$ in Tsallis statistics is still lacking. (However, see the Note at the end of the paper.)
In the present letter, we use the microcanonical ensemble approach [@tsallis1] and calculate the temperature and specific heat using the entropy $S_q$. For classical ideal gas, various thermodynamical quantities have been calculated in [@prato; @abe] using the canonical ensemble approach. Upon comparing the specific heats obtained in the microcanonical and the canonical ensemble approaches, we find that these two approaches can be equivalent, and various quantities calculated in these two approaches can be equal to each other, only if the average is given by normalised $q$-expectation value, and not by unnormalised one.
We then present a physical interpretation of the temperature calculated in the microcanonical ensemble. Comparing then with the canonical ensemble results of [@abe], we obtain a physical interpretation of the Lagrange multiplier $\beta$. Interestingly, the same interpretation can be derived by a simple refinement of the Gibbsian argument of Plastino and Plastino [@plastino]. We present this derivation also.
The plan of the paper is as follows. We first give the relevant details [@123] and results [@prato; @abe] of the canonical ensemble approach. We then present our results, and close with a few comments.
[**2.**]{} In the canonical ensemble approach, the entropy $S_q$ is extremised subject to the constraint $\sum p_i = 1$, and an average energy constraint. One thus obtains the probability distribution $p_i$, and calculates the physical quantities by standard methods. We briefly present here the relevant aspects. We follow [@123], where more details can be found.
The average energy $U$ of the system can be taken to be given by unnormalised, or normalised, $q$-expectation value as follows: $$\label{123}
U_2 = \sum p_i^q \epsilon_i
\; , \; \; \; {\rm or} \; \; \;
U_3 = \frac{\sum p_i^q \epsilon_i}{\sum p_i^q} \; ,$$ where $\epsilon_i$ is the $i^{th}$ state energy. Here and in the following, the subscript $2$ ($3$) indicates that the averages are given by unnormalised (normalised) $q$-expectation values. Extremising $S_q$, with the average energy given by $U_2$ ($U_3$) in (\[123\]), gives $$\begin{aligned}
p_{i 2} & = & \frac{a_{i 2}}{Z_2}
\; , \; \; \;
a_{i 2} = \left( 1 - \beta_2 (1 - q) \epsilon_i
\right)^{\frac{1}{1 - q}} \label{pi2} \\
p_{i 3} & = & \frac{a_{i 3}}{Z_3}
\; , \; \; \;
a_{i 3} = \left( 1 - \frac{\beta_3 (1 - q)
(\epsilon_i - U_3)}{Y_3} \right)^{\frac{1}{1 - q}}
\; , \label{pi3} \end{aligned}$$ where $Z_{2(3)} = \sum a_{i 2(3)}$, and $\beta_2$ ($\beta_3$) is the Lagrange multiplier for $U_2$ ($U_3$). It also follows that $$Y_2 \equiv \sum p_{i 2}^q =
Z_2^{1 - q} + \beta_2 (1 - q) U_2 \; , \; \; \;
Y_3 \equiv \sum p_{i 3}^q = Y_3 = Z_3^{1 - q} \; .$$
[**3.**]{} Consider systems whose standard Boltzmann-Gibbs, namely $q = 1$, partition function is of the form $$\label{zbg}
Z_{BG} \propto l^a \beta^{- a}$$ where $a$ is a dimensionless parameter, $l$ is a characteristic length, and $\beta$ is the inverse temperature. For example, for a $d$-dimensional classical ideal gas with $N$ particles and volume $V$, $a = \frac{d N}{2}$ and $l \propto V^{\frac{1}{d}}$. For classical ideal gas, the above formalism has been applied, and various quantities such as energy, specific heat, etc. have been calculated in [@prato; @abe]. The average energies $U_2, U_3$ and the specific heats ${\cal C}_2, {\cal C}_3$ are given by [^2] $$\begin{aligned}
\beta_2 U_2 = \frac{a Y_2}{1 + (1 - q) a}
\; , & \; \; \; & {\cal C}_2 = a Y_2 \label{betac2} \\
\beta_3 U_3 = a Y_3 \; , & \; \; \; &
{\cal C}_3 = \frac{a Y_3}{1 - (1 - q) a} \; . \label{betac3} \end{aligned}$$ Note that the specific heats ${\cal C}_2$ and ${\cal C}_3$ are not equal to each other, and even have qualitatively different behaviour: ${\cal C}_2$ is always positive since $Y_2 \equiv
\sum p_{i 2}^q$ is always positive, whereas ${\cal C}_3$ becomes negative when $(1 - q) a > 1$. Explicit expressions for $Y_2$ and $Y_3$ can be found in [@prato; @abe], but are not needed here.
[**4.**]{} However, one can also use the microcanonical ensemble approach [@tsallis1]. For a given energy $E_{mc}$ of the system, the entropy is extremised subject only to the constraint $\sum p_{i (mc)} = 1$. The subscript $mc$, here and in the following, indicates that the microcanonical ensemble approach is used. Note that no averaging of the energy is involved in this approach and, hence, no choice is made.
The entropy $S_q$ is extremised in the case of equiprobability, [*i.e.*]{} when all the probabilities are equal, with the extremum being a maximum (minimum) if $q > 0$ ($q < 0$) [@tsallis1]. Thus, if $W$ is the number of allowed states then $$\label{sqmc}
p_{i (mc)} = \frac{1}{W}
\; \; \; {\rm and,} \; \; {\rm hence,} \; \; \;
S_q = \frac{W^{1 - q} - 1}{1 - q} \; .$$ Since the standard thermodynamical structure is preserved [@tsallis], the inverse temperature $\beta_{mc}$ and the specific heat ${\cal C}_{mc}$ can be calculated using the standard formulae: $$\begin{aligned}
\beta_{mc} & \equiv & \frac{\partial S_q}{\partial E_{mc}}
= \beta_* Y_{mc} \label{beta*} \\
{\cal C}_{mc} & \equiv & - \beta_{mc}^2 \left(
\frac{\partial^2 S_q}{\partial E_{mc}^2} \right)^{- 1}
= \frac{c_* Y_{mc}}{1 - (1 - q) c_*} \; , \label{c*}\end{aligned}$$ where we have defined $Y_{mc} = \sum p_{i (mc)}^q = W^{1 - q}$, $$\label{std}
\beta_* = \frac{\partial ln W}{\partial E_{mc}}
\; , \; \; \; {\rm and} \; \; \;
c_* = - \beta_*^2 \left(
\frac{\partial^2 ln W}{\partial E_{mc}^2} \right)^{- 1} \; .$$ For systems whose $q = 1$ partition function is given by (\[zbg\]), we have $$\label{sbg}
W \propto l^a E_{mc}^a \
\; \; \; {\rm and,} \; \; {\rm hence,} \; \; \;
\beta_* = \frac{a}{E_{mc}}
\; , \; \; \; {\rm and} \; \; \;
c_* = a \; .$$ Equations (\[beta\*\]) and (\[c\*\]) then give $$\label{betac}
\beta_{mc} E_{mc} = a Y_{mc} \; , \; \; \;
{\cal C}_{mc} = \frac{a Y_{mc}}{1 - (1 - q) a} \; .$$
[**5.**]{} For systems whose $q = 1$ partition function is given by (\[zbg\]), we now have expressions (\[betac2\]), (\[betac3\]), and (\[betac\]) for energy and specific heat, obtained using the canonical [@prato; @abe] and the microcanonical ensemble approach. The canonical ensemble approach involves an averaging of the energy, and the expressions (\[betac2\]) ((\[betac3\])) are for the case where the average is given by unnormalised (normalised) $q$-expectation value. However, in the microcanonical ensemble approach, no averaging of energy is involved and, hence, no choice is made.
Let us now compare the specific heats. The specific heats given by (\[betac3\]) and (\[betac\]) are identical, upto factors involving $Y_3$ and $Y_{mc}$, and differ distinctly from that given by (\[betac2\]). For example, the specific heat given by (\[betac2\]) is always positive, whereas the specific heats given by (\[betac3\]) and (\[betac\]) become negative when $(1 - q) a > 1$. Therefore, it follows that the microcanonical and the canonical ensemble approaches can be equivalent, and various quantities calculated in these two approaches can be equal to each other, only if the average is given by normalised $q$-expectation value, and not by unnormalised one.
[**6.**]{} From now on, we assume that the averages are given by normalised $q$-expectation values only. Therefore, energy and specific heat in the canonical ensemble approach are given by (\[betac3\]). We now present a physical interpretation of $\beta_*$ in (\[std\]) and, thus, also of $\beta_{mc}$ and $\beta_3$.
Consider a system obeying Tsallis statistics, with $q$ positive but otherwise arbitrary, with energy $E$ and number of allowed states $W(E)$, and enclosed within a container with which it can exchange energy only. Together, let them be isolated. Thus, if $E_c$ is the energy of the container then the total energy $E_{tot} = E + E_c$ is fixed. Let the container be choosen to obey the standard Boltzmann-Gibbs statistics, namely Tsallis statistics with $q = 1$. Therefore, if $W_c(E_c)$ is the number of allowed states of the container, then $$\label{betaphys}
\beta_{phys} = \frac{\partial ln W_c}{\partial E_c}$$ is the inverse of its temperature, which can be physically measured.
We would like to find the values $E$ of the system, and $E_c = E_{tot} - E$ of the container, at which the (system + container) is in equilibrium. But the analysis of such a composite system, where the constituent systems have different values of $q$, is highly nontrivial and is still an open problem[^3]. Nevertheless, it is reasonable to expect that [**(i)**]{} the entropy of the composite system is extremised in the case of equiprobability; [**(ii)**]{} the extremum is a maximum, at least in the case where the $q$’s of the constituent systems are all positive; and [**(iii)**]{} the maximum is a monotonically increasing function of the total number of allowed states of the composite system.
Although we are unable to justify these properties rigorously, they appear to be physically reasonable, and are satisfied by any single system obeying Tsallis statistics with $q > 0$ [@tsallis1], see equation (\[sqmc\]). Hence, we assume that [*any composite system, the $q$’s of whose constituent systems are all positive, also satisfies the properties [**(i)-(iii)**]{} given above.*]{} Since the entropy is extremised in equilibrium, our assumption then implies that [**(i’)**]{} when a composite system is in equilibrium, the energies of its constituents will be such as to maximise the total number of states of the composite system[^4]. Note that no assumption is made, or implied, about the explicit form of the entropy of the composite system by assuming the properties [**(i)-(iii)**]{} or [**(i’)**]{} for the composite system.
In the case of the (system + container) considered above, this assumption then implies that in equilibrium, the energy $E$ of the system, and the energy $E_c = E_{tot} - E$ of the container, will be such as to maximise the total number of states of the (system + container), given by $$W_{tot}(E_{tot}) = W(E) W_c(E_{tot} - E) \; .$$ Hence, with $E_{tot} = E + E_c$ fixed, we have that in equilibrium, $$\label{pathria}
\frac{\partial ln W(E)}{\partial E} =
\frac{\partial ln W_c(E_c)}{\partial E_c} \; .$$ It then follows from equations (\[beta\*\]), (\[std\]), and (\[betaphys\]) that $$\label{big0}
\beta_{phys} = \beta_* = \frac{\beta_{mc}}{Y_{mc}} \; ,$$ which relates $\beta_{mc}$ of the microcanonical ensemble approach to $\beta_{phys}$, the physical inverse temperature of the container. As clear from its derivation, the above relation is valid for any arbitrary system, whose $q = 1$ partition function is completely general.
Now, assuming the equivalence of the microcanonical and the canonical ensemble approach, we can set $E_{mc}(\beta_{mc}) =
U_3(\beta_3)$. For systems considered here, whose $q = 1$ partition function is given by (\[zbg\]), it then follows from equations (\[betac3\]), (\[betac\]), and (\[big0\]) that $$\label{big}
\frac{\beta_3}{Y_3} = \frac{\beta_{mc}}{Y_{mc}} =
\beta_{phys} \; ,$$ which relates $\beta_3$ to $\beta_{phys}$, the inverse temperature of the container, which can be physically measured. Equation (\[big\]) thus provides a physical interpretation of the Lagrange multiplier $\beta_3$ of the canonical ensemble approach.
[**7.**]{} The relation (\[big\]) between $\beta_{phys}$ and $\beta_3$ can also be derived by another method. Plastino and Plastino have derived the probability distribution of the form given in (\[pi2\]), for $q < 1$, by a Gibbsian argument [@plastino]. A simple refinement of their argument leads to the probability distribution of the form given in (\[pi3\]). Requiring it to be exactly identical with (\[pi3\]) then leads to (\[big\]).
The argument of [@plastino] is, briefly, the following. Consider a large, but finite, heat bath obeying the standard Boltzmann-Gibbs thermodynamics. Let $E_b$ be its energy and $\beta_{phys}$ its inverse temperature, which can be physically measured. Also, let the total number of states in the energy range $(E_b \pm \frac{\Delta}{2})$ be $\eta (E_b) \Delta$. Consider now a system weakly interacting with such a heat bath. Then, the probability $p_i (\epsilon_i)$ that the system is in a state $i$, with energy $\epsilon_i$, is given by $$p_i (\epsilon_i) \propto \eta (E_b - \epsilon_i) \Delta \; .$$ Assuming that $\eta (E) \propto E^{\alpha - 1}$, where $\alpha \gg 1$, one obtains $$\label{pipp}
p_i (\epsilon_i) \propto \left( 1 - \frac{\epsilon_i}{E_b}
\right)^{\alpha - 1} \; .$$ Also, $\beta_{phys} = \frac{\alpha - 1}{E_b}$. Let $q = \frac{\alpha - 2}{\alpha - 1}$. Then, $q < 1$, $\alpha - 1 = \frac{1}{1 - q}$, and $$p_i (\epsilon_i) \propto \left( 1 - \beta_{phys} (1 - q)
\epsilon_i \right)^{\frac{1}{1 - q}} \; ,$$ which is of the form given in (\[pi2\]).
By a simple refinement of the above argument, one can obtain the probability distribution of the form given in (\[pi3\]). The above expression for $\beta_{phys}$ assumes that the heat bath always has energy $E_b$, irrespective of the energy of the system. However, the actual energy of the heat bath is $E_b - \epsilon_i$ when the system is in state $i$. Therefore, if the average energy of the system is $U$ then the average energy of the heat bath is $E_b - U$. Hence, a more precise expression for $\beta_{phys}$ is given by $$\beta_{phys} = \frac{\alpha - 1}{E_b - U} \; .$$ Using this expression in (\[pipp\]), and with $q$ defined as above, one obtains $$\label{pi3pp}
p_i (\epsilon_i) \propto \left( 1 - \beta_{phys} (1 - q)
(\epsilon_i - U) \right)^{\frac{1}{1 - q}} \; ,$$ which is of the form given in (\[pi3\]). Requiring this distribution to be identical with (\[pi3\]) then gives $$\beta_{phys} = \frac{\beta_3}{Y_3} \; ,$$ which is the same relation as in (\[big\]) and relates $\beta_3$ of the canonical ensemble approach to $\beta_{phys}$, the physical inverse temperature of the bath. Assuming the validity of the argument of [@plastino] and our refinement of it, the above relation is valid for any arbitrary system, but with $q < 1$.
[**8.**]{} We close with a few comments. The normalised $q$-expectation values have, indeed, been found earlier to possess desireable properties and, hence, considered to be the appropriate ones. Here, we find that this result follows simply by requiring the equivalence between the microcanonical and the canonical ensemble approaches.
However, we have considered here only systems whose $q = 1$ partition function is given by (\[zbg\]). Hence, it is desireable to establish this result for any arbitrary system whose $q = 1$ partition function is completely general.
The relation (\[big0\]) between $\beta_{mc}$ and $\beta_{phys}$ is valid for any arbitrary system whose $q = 1$ partition function is completely general. The relation (\[big\]) between $\beta_3$ and $\beta_{phys}$ is derived, in the first method, only for systems whose $q = 1$ partition function is given by (\[zbg\]). Assuming the validity of the argument of [@plastino] and our refinement of it, the second method of derivation is valid for any arbitrary system, but with $q < 1$. Hence, a general relation between $\beta_3$ and $\beta_{phys}$, valid for any arbitrary system and for any value of $q$, is still lacking.
Also, Tsallis statistics is applied to a variety of diverse physical systems such as Levy flights, turbulence, etc. to name but a few [@tsallis; @spl]. It is not clear if each one of them can be modelled as a system within a container, or as a system weakly interacting with a large, but finite, heat bath - models which played a crucial role in the physical interpretation of $\beta_3$ presented here. On the other hand, however, one may instead assume that $\frac{\beta_3}{Y_3}$, or a suitable generalisation of it, is indeed a physical quantity as given in (\[big\]). Its study may then, perhaps, provide new insights into physical systems, to which Tsallis statistics is applied.
[**Note:**]{} While this work was being written, a paper by Abe et al [@recent] has appeared. In the prescription termed [*optimal Lagrange multipliers*]{} formalism [@olm] which they use, the combination $\frac{\beta_3}{Y_3}$ appears naturally. As shown in [@recent], certain key properties of the ideal gas then become identical in both the standard statistical mechanics and Tsallis statistics.
The referee has brought to our attention a paper by Abe [@aberef] where also the combination $\frac{\beta_3}{Y_3}$, termed [*a renormalised (inverse) temperature*]{}, appears naturally while establishing the zeroth law of thermodynamics using the classical ideal gas model.
[**Acknowledgement:**]{} We are grateful to G. Baskaran for introducing us to Tsallis statistics. We thank G. Baskaran, G. I. Menon, P. Ray, and B. Sathiapalan for many discussions. Also, we thank the referees for their comments, suggestions for improvement, and for bringing [@aberef] to our attention.
[999]{}
C. Tsallis, J. Stat. Phys. [**52**]{} (1988) 479. C. Tsallis, Braz. Jl. Phys. [**29**]{} 1 (1999). Braz. Jl. Phys. [**29**]{} (1999). This is a special issue on Nonextensive Statistical Mechanics and Thermodynamics, and can also be found at the URL http:$//$www.sbf.if.usp.br$/$WWW\_pages$/$Journals$/$ BJP$/$Vol29$/$Num1$/$index.htm An exhaustive, and regularly updated, list of references can be found at the URL http:$//$tsallis.cat.cbpf.br$/$biblio.htm C. Tsallis, R. S. Mendes, and A. R. Plastino, Physica [**A 261**]{} (1998) 534. A. R. Plastino, A. Plastino, and C. Tsallis, J. Phys. [**A:**]{} Math. Gen. [**27**]{} (1994) 5707; D. Prato, Phys. Lett. [**A 203**]{} (1995) 165. S. Abe, Phys. Lett. [**A 263**]{} (1999) 424; Erratum [**A 267**]{} (2000) 456. A. R. Plastino, and A. Plastino, Phys. Lett. [**A 193**]{} (1994) 140; Braz. Jl. Phys. [**29**]{} 50 (1999). See also J. D. Ramshaw, Phys. Lett. [**A 198**]{} (1995) 122. The probability distributions in (\[pi2\]) and (\[pi3\]) are related by $\beta_2 = \beta_3 \left( Y_3 + \beta_3 (1- q) U_3
\right)^{- 1}$. Then $p_{i 2} = p_{i 3}$ and, therefore, $Y_3 = Y_2$, $U_3 = \frac{U_2}{Y_2}$, and $\beta_3 =
\frac{\beta_2 Y_2^2}{Z_2^{1 - q}}$. Other quantities such as free energy, specific heat, etc. can also be correspondingly related [@123]. S. Abe, S. Martinez, F. Pennini, and A. Plastino, cond-mat/0006109. S. Martinez, F. Nicolas, F. Pennini, and A. Plastino,\
cond-mat/0006109. S. Abe, Physica [**A 269**]{} (1999) 403.
[^1]: Since the standard thermodynamical structure is preserved, such quantities can be calculated using the standard formulae [@tsallis]. However, they are mathematical constructs only, which may or may not have a physical meaning.
[^2]: As mentioned in section [**1**]{}, $(\beta_2, U_2, {\cal C}_2, \cdots)$ and $(\beta_3, U_3, {\cal C}_3, \cdots)$ can be related to each other by a set of formulae [@2to3].
[^3]: We thank the referee for emphasising this point.
[^4]: Alternatively, we may instead assume that [*the composite system satisfies the property [**(i’)**]{} only*]{}, which will suffice for our purposes here.
|
---
abstract: 'Although assigning $D_{s0}^+(2317)$ to the $I_3=0$ component $\hat F_I^+$ of iso-triplet four-quark mesons is favored by experiments, its neutral and doubly charged partners have not yet been observed. It is discussed why they were not observed in inclusive $e^+e^-\rightarrow c\bar c$ experiment and that they can be observed in $B$ decays.'
author:
- Kunihiko Terasaki
title: '$\bf{D_{s0}^+(2317)}$ as an iso-triplet four-quark meson[^1] '
---
The charm-strange scalar meson $D_{s0}^+(2317)$ has been observed in inclusive $e^+e^-$ annihilation . It is very narrow ($\Gamma < 3.8$ MeV [@BABAR-search]) and it decays dominantly into $D_s^+\pi^0$ while no signal of $D_s^{*+}\gamma$ decay has been observed. Therefore, the CLEO provided a severe constraint , $$R(D_{s0}^+(2317))
< 0.059, \label{eq:constraint}$$ where $R(S)= {\Gamma(S\rightarrow D_s^{*+}\gamma)}/
{\Gamma(S\rightarrow D_s^{+}\pi^0)}$ with $S=D_{s0}^+(2317)$. Similar resonances which are degenerate with it have been observed in $B$ decays: $B\rightarrow\bar D\tilde D_{s0}^+(2317)[D_s\pi^0,
D_s^{*+}\gamma]$ [@BELLE-BD], $B\rightarrow \bar D ({\rm or}\,\,\bar D^{*})
\tilde D_{s0}^+(2317)\break[D_s\pi^0]$ [@BABAR-B]. Here the new resonances have been denoted by $\tilde D_{s0}^+(2317)$\[observed channel(s)\] to distinguish them from the above $D_{s0}^+(2317)$, although they are usually identified to $D_{s0}^+(2317)$. It is because the resonance signals have been observed in the $D_s^{*+}\gamma$ channel in addition to the $D_{s}^+\pi^0$ [@BELLE-BD]. It is quite different from the previous $D_{s0}^+(2317)$.
As will be seen later, assigning $D_{s0}^+(2317)$ to the $I_3=0$ component $\hat F_I^+$ [@Terasaki-D_s] of iso-triplet scalar four-quark mesons, $\hat F_I\sim [cn][\bar s\bar n]_{I=1},\,(n=u,d)$, is favored by Eq. (\[eq:constraint\]). In this case, its narrow width might be wondered because $\hat F_I^+\rightarrow D_s^+\pi^0$ appears a [*fall-apart*]{} decay at a glance. However, its small rate can be realized by a small overlap between wavefunctions of the initial $|D_{s0}^+(2317)\rangle$ and final $\langle{D_s^+\pi^0}|$ states. Such a small overlap can be seen by decomposing a scalar four-quark state $|[qq][\bar q\bar q]\rangle$ with ${\bf 1_s}\times {\bf 1_s}$ of spin $SU(2)$ and ${\bf \bar 3_c}\times {\bf 3_c}$ of color $SU_c(3)$, which is the lowest lying four-quark state [@Jaffe], into a sum of $|\{q\bar q\}\{q\bar q\}\rangle$ states; $$\begin{aligned}
\hspace{-10mm}
|[qq]_{\bf \bar 3_c}^{\bf 1_s}
[\bar q\bar q]_{\bf 3_c}^{\bf 1_s}\rangle
&& =-\sqrt{1\over 4}\sqrt{1\over 3}
|\{q\bar q\}_{\bf 1_c}^{\bf 1_s}
\{q\bar q\}\}_{\bf 1_c}^{\bf 1_s}\rangle \nonumber\\
&&\hspace{4mm} + \sqrt{3\over 4}\sqrt{1\over 3}
|\{q\bar q\}_{\bf 1_c}^{\bf 3_s}
\{q\bar q\}\}_{\bf 1_c}^{\bf 3_s}\rangle + \cdots.
\label{eq:decomp}\end{aligned}$$ The color and spin wavefunction overlap between $|\hat F_I^+\rangle$ and $\langle{D_s^+\pi^0}|$ is given by the first term of the right-hand-side of Eq. (\[eq:decomp\]). To see its narrow width more explicitly, we estimate the rate for $\hat F_I^+\rightarrow D_s^+\pi^0$ by comparing it with $\hat\delta^{s+}\rightarrow \eta\pi^+$. Here we have assigned the observed $a_0(980)$, $f_0(980)$, $\kappa(800)$ and $f_0(600)$ [@PDG04] to scalar $[qq][\bar q\bar q]$ mesons, $\hat\delta^s$, $\hat\sigma^s$, $\hat\kappa$ and $\hat\sigma$ [@Jaffe]. However, the above overlap can be quite different from that of $\langle{\eta\pi^+}|$ and $|{\hat\delta^{s+}}\rangle$ at the scale of $m_{\hat\delta^s}\sim 1$ GeV, because a gluon exchange between $\{q\bar q\}$ pairs will reshuffle the above decomposition \[while such a reshuffling will be rare at the 2 GeV or a higher energy scale, because it is known that the $s$-quark at the 2 GeV scale is much more slim ($m_s\simeq 90$ MeV) [@Gupta] than the $s$-quark in the constituent quark model, i.e., the quark-gluon coupling at 2 GeV scale is much weaker than that at 1 GeV scale\]. With this in mind, we introduce a parameter $\beta_0$ describing the difference between overlaps of color and spin wavefunctions at the scale of $m_{\hat\delta^s}$ and at the scale of $m_{\hat F_I^+}$. In the limiting case that the full reshuffling around 1 GeV while no reshuffling at the scale of $m_{\hat F_I}$, we have $|\beta_0|^2={1}/{12}$ as seen in Eq. (\[eq:decomp\]). By using a hard pion approximation and the asymptotic $SU_f(4)$ symmetry, which have been reviewed comprehensively in Ref. [@suppl], we have $$\Gamma(\hat F_I^+\rightarrow D_s^+\pi^0)_{SU_f(4)}\simeq 5 - 10 \,\,
{\rm MeV}
\label{eq:width-SU_f(4)}$$ where the spatial wavefunction overlap is in the $SU_f(4)$ symmetry limit. Here, we have used $\Gamma(a_0(980)\rightarrow \eta\pi)_{\rm exp}= 50 - 100$ MeV and the $\eta$-$\eta'$ mixing angle $\theta_P\simeq -20^\circ$ [@PDG04] as the input data. Noting that the above $SU_f(4)$ symmetry overestimates by $20 - 30$ % in amplitude, we have $\Gamma(\hat F_I^+)\simeq \Gamma(\hat F_I^+\rightarrow D_s^+\pi^0)
\sim 3.5 - 7$ MeV [@HT-isospin]. It is sufficiently narrow.
Next, we study the radiative decay of $D_{s0}^+(2317)$ to see that its assignment to $\hat F_I^+$ is consistent with Eq. (\[eq:constraint\]). For later convenience, we study the typical three cases, $D_{s0}^+(2317)$ as (i) the iso-triplet $\hat F_I^+$, (ii) the iso-singlet $\hat F_0^+\sim [cn][\bar s\bar n]_{I=0}$ and (iii) the conventional scalar $D_{s0}^{*+}\sim \{c\bar s\}$, under the vector meson dominance (VMD) hypothesis. To test our approach, we study $D_s^{*+}\rightarrow D_s^+\gamma$ in the same way. Here, we take $VVP$ and $SVV$ couplings with spatial wavefunction overlap in the $SU_f(4)$ symmetry limit, where $V$, $P$ and $S$ denote a vector, a pseudoscalar and a scalar meson, respectively, and take the overlapping factor $|\beta_1|^2={1}/{4}$ between wavefunctions of a scalar four-quark and two vector-meson states, as seen in Eq. (\[eq:decomp\]). The results are listed in Table \[tab:1\], where the input data are taken from Ref. [@PDG04]. Comparing the rate for $\hat F_I^+\rightarrow D_s^{*+}\gamma$ in Table \[tab:1\] with Eq. (\[eq:width-SU\_f(4)\]), we obtain $R(\hat F_I^+) \sim (4.5 - 9)\times 10^{-3}$ [@HT-isospin], which is consistent with the constraint Eq. (\[eq:constraint\]). It implies that assigning $D_{s0}^+(2317)$ to $\hat F_I^+$ is favored by experiments.
[c c c c]{} Decay & Pole(s) & Input Data & Rate (keV)\
$D_s^{*+}\rightarrow D_s^+\gamma$ & $\phi,\,\psi$ & $\Gamma(\omega\rightarrow \pi^0\gamma)_{\rm exp}$ & [0.8]{}\
& [$\rho^0$]{} & $\Gamma(\phi\rightarrow a_0\gamma)_{\rm exp}$ & [45]{}\
$\hat F_0^+\rightarrow D_s^{*+}\gamma$ & [$\omega$]{} & $\Gamma(\phi\rightarrow a_0\gamma)_{\rm exp}$ & [4.7]{}\
& [$\phi,\,\psi$]{} & $\Gamma(\chi_{c0}\rightarrow \psi\gamma)_{\rm exp}$ & [35]{}\
In the cases of the above assignments (ii) and (iii), $D_{s0}^+(2317)\rightarrow D_s^+\pi^0$ is isospin non-conserving. The isospin non-conservation is assumed to be caused by the $\eta$-$\pi^0$ mixing as usual. The mixing parameter $\epsilon$ has been estimated [@Dalitz] as $\epsilon = 0.0105\pm 0.0013$, i.e., $\epsilon \sim O(\alpha)$ with the fine structure constant $\alpha$. It implies that isospin non-conserving decays are much weaker than the radiative ones. By using the hard pion approximation, the asymptotic $SU_f(4)$ symmetry and the above value of $\epsilon$, the rates for the isospin non-conserving decays can be obtained as listed in Table \[tab:2\].
[c l c]{} Decay & Input Data & Rate (keV)\
& [$\Gamma(\rho\rightarrow\pi\pi)_{\rm exp}$]{} & [0.05]{}\
& [$\Gamma(a_0\rightarrow \eta\pi)\simeq 70$ MeV]{} & [0.7]{}\
& [$\Gamma(K_0^{*0} \rightarrow K^+\pi^-)_{\rm exp}$]{} & [0.6]{}\
The results on the decays of $D_s^{*+}$ in Tables \[tab:1\] and \[tab:2\] lead to the ratio of decay rates $R(D_s^{*+})^{-1}\simeq 0.06$. This reproduces well the experimental value $R(D_s^{*+})^{-1}_{\rm BABAR}=0.062\pm 0.005\pm 0.006$. This means that the present approach is sufficiently reliable. The corresponding ratios of the decay rates in the cases (ii) and (iii) are also obtained as (ii) $R(\hat F_0^+)\simeq 7$ and (iii) $R(D_{s0}^{*+})\simeq 60$. They are much larger than the experimental upper bound. It should be noted that the isospin non-conserving decays are much weaker than the radiative decays, as expected intuitively above. The assignment of $D_{s0}^+(2317)$ to the iso-singlet $DK$ molecule [@BCL] leads to $R(\{DK\})\simeq 3$ which is much larger than the experimental upper bound in Eq. (\[eq:constraint\]). Hence, such an assignment should be rejected [@MS]. Thus, assigning $D_{s0}^+(2317)$ to an iso-singlet state ($\hat F_0^+$, $D_{s0}^{*+}$ or $DK$ molecule) is disfavored by experiments.
From the above considerations, it is natural to assign $D_{s0}^+(2317)$ to the iso-triplet $\hat F_I^+$. However, its neutral and doubly charged partners, $\hat F_I^{++}$ and $\hat F_I^0$, have not yet been observed [@BABAR-search]. With this in mind, we study the production of charm-strange scalar mesons ($\hat F_I^{++,+,0}$ and $\hat F_0^+$) by assigning $D_{s0}^+(2317)$ to $\hat F_I^{+}$, and discuss why experiments have observed $D_{s0}^+(2317)$ but not its neutral and doubly charged partners. To this aim, we consider their production through weak interactions as a possible mechanism, because the OZI-rule violating productions of multi-$q\bar q$-pairs and their recombinations into four-quark meson states are believed to be strongly suppressed at high energies. First, we recall that color mismatched decays which include rearrangements of colors in weak decay processes would be suppressed compared with color favored ones as long as non-factorizable contributions, which are actually small in $B$ decays and are expected to be much smaller at higher energies, are ignored. Next, we draw quark-line diagrams within the minimal $q\bar q$ pair creation, noting the OZI rule.
{width="75mm"} \[fig:via-c-cbar\]
Because there is no diagram yielding $\hat F_I^{++}$ production in this approximation, as seen in Fig. 1, it is easy to understand why no evidence of $\hat F_I^{++}$ was found in $e^+e^-\rightarrow c\bar c$ experiments. As will be seen in productions of $\tilde D_{s0}^+(2317)[D_s^+\pi^0]$ in the $B$ decays, their observed rates are comparable with color mismatched decays, so that rates for $\hat F_I^0$ and $\hat F_0^+$ productions in $e^+e^-\rightarrow c\bar c$ annihilation will be expected to be comparable with that through the color suppressed ones. Therefore, they would be much weaker (possibly by about two order of magnitude) than productions of $D_s^+\pi^-,\,D_s^{*+}\pi^-,\,D_s^{*+}\rho^- $, etc., and $D_s^+D_s^-,\,D_s^{*+}D_s^-,\,D_s^{*+}D_s^{*-}$, etc., created through the reaction depicted by Figs. 1(a) and (b). The $D_s^+\pi^-$ produced through Fig. 1(a) obscures the signal $\hat F_I^0\rightarrow D_s^+\pi^-$ events. In addition, the $D_s^{*+}$ and $\gamma$ from $D_s^{*-}\rightarrow D_s^-\gamma$ produced through the ordinary $e^+e^-\rightarrow c\bar c\rightarrow D_s^{*+}D_s^{*-}$ \[and through Fig. 1(b)\] obscure the signal of $\hat F_0^+\rightarrow D_s^{*+}\gamma$ events. Therefore, it is understood why the inclusive $e^+e^-$ annihilation experiment found no signal of scalar resonance in the $D_s^+\pi^-$ and $D_s^{*+}\gamma$ channels. In the case of $\hat F_I^+$, however, there do not exist large numbers of background events described by Figs. 1(a) and (b), because its main decay is $\hat F_I^+\rightarrow D_s^+\pi^0$. In fact, $D_{s0}^+(2317)$ has been observed in the $D_s^+\pi^0$ channel. This seems to imply that the production of four-quark mesons in hadronic weak decays plays an essential role [@production].
{width="70mm"} \[fig:Bu-8.eps\]
Because it is difficult to observe $\hat F_I^{++}$ and $\hat F_I^{0}$ in inclusive $e^+e^-\rightarrow c\bar c$ experiments, we study their productions in $B$ decays. First, we draw quark-line diagrams in the same way as the above. As expected in Figs. 2(a) and 3(b), $\tilde D_{s0}^+(2317)[D_s^+\pi^0]$, which can be identified to $\hat F_I^+$ because it decays dominantly into $D_s^+\pi^0$ as seen above, has been observed. The production of $\hat F_I^{++}$ is given by Fig. 2(b) which is of the same type as Fig. 2(a). In addition, the production of $\hat F_I^0$ is given by Fig. 3(a) which is again of the same type as Figs. 2(a) and (b), so that their production rates are not very different from each other; $B(B_u^+\rightarrow \hat F_I^{++}D^-)
\sim B(B_d^+\rightarrow \hat F_I^{0}\bar D^0)
\sim B(B_u^+\rightarrow \hat F_I^{+}\bar D^0)$, where $B(B_u^+\rightarrow \hat F_I^{+}\bar D^0)_{\rm exp}
\sim 10^{-3}$ [@BELLE-BD; @BABAR-B]. Besides, the BELLE observed indications of $\tilde D_{s0}^+(2317)[D_s^{*+}\gamma]$ which are conjectured to be signals of $\hat F_0^+\rightarrow D_s^{*+}\gamma$ because the production of $\hat F_0^+$ and $\hat F_I^+$ are depicted by the same diagrams and the $\hat F_I^+\rightarrow D_s^{*+}\gamma$ is much weaker than the $\hat F_I^+\rightarrow D_s^+\pi^0$ while the $\hat F_0^+\rightarrow D_s^{*+}\gamma$ is much stronger than the $\hat F_0^+\rightarrow D_s^{+}\pi^0$, as seen before.
{width="70mm"} \[fig:Bd-8.eps\]
As expected in Fig. 4(c), the BELLE [@BELLE-BK] observed $\bar B_d\rightarrow \tilde D_{s0}^+(2317)[D_s^+\pi^0]K^-$, and provided $B(\bar B_d\rightarrow \tilde D_{s0}^+(2317)K^-)
\cdot B(\tilde D_{s0}^+(2317)\rightarrow D_s^+\pi^0)
=(5.3^{+1.5}_{-1.3}\pm 0.7\pm 1.4)\times 10^{-5}$. If $\tilde D_{s0}^+(2317)[D_s^+\pi^0]$ is identified to $\hat F_I^+$ and $B(\tilde D_{s0}^+(2317)\rightarrow D_s^+\pi^0)\simeq 100\,\%$ is taken, $B(\bar B_d^0\rightarrow \hat F_I^+K^-)\sim 10^{-5}-10^{-4}$ would be obtained. Using it as the input data and noting that Figs. 4(c) and 5(c) are of the same type, we could estimate $B(B_u^-\rightarrow K^-\hat F_I^0)\sim 10^{-5}-10^{-4}$, if contributions from diagrams Figs. 5(a) and (b) cancel each other (because the phases of $\hat F_I^0$ in these diagrams have opposite signs arising from anti-symmetry property of its wavefunction).
{width="70mm"} \[fig:barBu-8.eps\]
{width="70mm"} \[fig:barBd-8.eps\]
In summary, we have seen that assigning $D_{s0}^+(2317)$ to an iso-triplet $\hat F_I^+$ is favored by experiments. In addition, we have discussed why inclusive $e^+e^-\rightarrow c\bar c$ experiments observed no evidence for its neutral and doubly charged partners $\hat F_I^0$ and $\hat F_I^{++}$. $\tilde D_{s0}^+(2317)[D_s^+\pi^0]$ which was observed in $B$ decays has been identified to $D_{s0}^+(2317)$. Indications of $\hat F_0^+$ also have been observed as $\tilde D_{s0}^+(2317)[D_s^{*+}\gamma]$ in $B$ decays. $\hat F_I^0$ and $\hat F_I^{++}$ will be observed in $B$ decays.\
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank the organizers of QNP06 for supporting local expense during the conference. This work is supported in part by the Grant-in-Aid for Science Research, Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 16540243).
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[^1]: Invited talk at [*QNP06*]{}, the 4th International Conference on Quarks and Nuclear Physics, June 5 – 10, 2006, Madrid, Spain
|
---
abstract: 'We propose a graphical notation by which certain spectral properties of complex systems can be rewritten concisely and interpreted topologically. Applying this notation to analyze the stability of a class of networks of coupled dynamical units, we reveal stability criteria on all scales. In particular, we show that in systems such as the Kuramoto model the Coates graph of the Jacobian matrix must contain a spanning tree of positive elements for the system to be locally stable.'
author:
- 'Anne–Ly Do,$^1$ Stefano Boccaletti,$^2$ and Thilo Gross $^3$'
title: Graphical notation reveals topological stability criteria for collective dynamics in complex networks
---
Discovering how the interactions between constituents of a system determine its macroscopic behavior is a central aim of physics. Ideally, we find properties in the system’s detailed organization that have direct system-level implications. Significant progress has been made in identifying such properties on a local level. For instance the implications of the degree distribution, the distribution of a network’s links among the nodes, is well understood [@Newman]. By contrast, identifying meso-scale properties that have a distinct effect on macroscopic behavior, despite recent advances [@Fortunato], remains a difficult challenge. A starting point is often to link the macro-behavior to spectral properties of certain matrices, such as the adjacency, the different graph Laplacians, or the Jacobian matrix of a dynamical system. Here again, the dependence on local properties, i.e., certain rows or diagonal elements, can be discovered with relative ease. By contrast, analytical insights on how meso-scale patterns affect the spectrum are difficult to obtain, as they typically require the evaluation of determinants leading to complex expressions.
In this Letter we propose a graphical notation, reminiscent of Feynman diagrams, that facilitates computing spectral implications of meso- and macro-scale structures. We illustrate the usage of the notation by investigating dynamical stability of stationary and phase-locked solutions in a class of symmetrically coupled dynamical systems, containing for instance the Kuramoto model [@Kuramoto; @Acebron]. Here, the proposed notation greatly reduces the complexity of the mathematical expressions and enables the derivation of topological stability criteria on all scales. In particular, we show that in the systems under consideration the graph defined by the off-diagonal elements of the Jacobian must contain a spanning tree of positive links to admit stable solutions.
Any given matrix ${\rm \bf J}$ can be interpreted as defining the connectivity of an abstract weighted network $\mathcal{G}$, the so-called Coates graph. In this network two nodes $i,j$ are connected if $J_{ij} \neq 0$. We can interpret a product of different matrix elements as defining a (not necessarily connected) network motif, e.g. $J_{ij}J_{jk}J_{ki}$ corresponds to a triangle path. The advantage of this graphical reading is that complex structures appearing in systems of equations can be described using network terminology.
![Examples for the graphical notation. Symbols denote the sum over all non-equivalent possibilities to build the depicted subgraph with the $q$ nodes $\in S$. Plotted are two example terms and their algebraic and topological equivalents.[]{data-label="Examples"}](Fig1.eps){width="48.00000%"}
It is useful to define a basis of symbols, $\times, |, \triangle, \Square, \pentagon, \ldots$, denoting sums over all non-equivalent cycles of length $n=1,2,3,4,5,\ldots$, respectively. We allow concatenation of these symbols by the ‘$\cdot$’ sign. For a given set $S$ of node indices, products of symbols denote the sum over all non-equivalent possibilities to realize the depicted motif with the nodes in $S$. For instance, if $S=\left\{i,j,k,l,m\right\}$ then $\times \cdot \square$ represents the sum over all products of elements that would be read as a cycle containing four nodes and a self-loop, e.g. $J_{ii}J_{jk}J_{kl}J_{lm}J_{mj}$ (cf. Fig. \[Examples\]). Finally, we define $\Phi_{S}$ as the sum over all products corresponding to all acyclic graphs that can be drawn by placing $\left|S\right|$ (undirected) links such that each link starts at a different node in $S$.
In the remainder of this Letter we illustrate the usage of the proposed notation by considering the synchronization of dynamical units. This subject was chosen as it is presently of broad interest in physics and appears in many fields including biology, ecology, and engineering [@Kurths; @Boccaletti; @Arenas]. The paradigmatic model proposed by Kuramoto [@Kuramoto] opened the field for detailed studies of the interplay between the structure of the interaction network and collective phenomena [@Mirollo; @Wu; @Pecora; @Kawamura]. These revealed the influence of various topological measures, such as the clustering coefficient, the diameter, and the degree or weight distribution, on the propensity to synchronize [@Chavez; @Nishikawa2006; @Lodato]. However, recent results [@Arenas; @Nishikawa; @Mori] indicate that beside global topological measures also details of the exact local configuration can crucially affect synchronization. This highlights synchronization of phase oscillators as a promising example in which it may be possible to understand the interplay between local, global, and mesoscale constraints on stability, that severely limit the operation of complex technical and institutional systems [@Motter; @Havlin].
To provide a specific example we focus on the Kuramoto model [@Acebron], describing a network of $N$ bidirectionally coupled phase oscillators $$\label{ODE}
\dot{x_i}=\omega_i+\sum_{j\neq i} A_{ij}\sin(x_j-x_i)\ , \quad \forall i\in 1\ldots N \ ,$$ where $x_i$ and $\omega_i$ are the phase and intrinsic frequency of oscillator $i$ and ${\rm \bf A}$ is the weight matrix of an undirected, network. The model can exhibit phase-locked states, which in a co-rotating frame correspond to steady states of . The local stability of such states is determined by the eigenvalues of the Jacobian matrix ${\rm \bf J} \in\mathbb{R}^{N\times N}$ defined by $J_{ik}=\partial\dot{x_{i}}/\partial x_{k}$. If all eigenvalues of ${\rm \bf J}$ are negative, then the state under consideration is locally stable.
For deriving topological stability criteria we apply Jacobi’s signature criterion (JSC), also called Sylvester criterion. The JSC states that a hermitian matrix ${\rm \bf J}$ with rank $r$ is negative definite iff all principal minors of order $q\leq r$ have the sign of $(-1)^q$ [@LiaoYu], i.e., iff $$\label{stability_cond}
{\rm sgn}\left(D_{\left|S\right|,S}\right)=(-1)^{\left|S\right|} \ \forall \ S\subset\left\{1,\ldots,N\right\}, \ \left|S\right|\leq r$$ where $D_{\left|S\right|,S}:=\det\left(J_{ik}\right)$, $i,k\in S$. Note that the whole set of equations constitutes a sufficient condition for stability, whereas each equation already constitutes a necessary condition.
Stability analysis by means of JSC is used in control theory [@LiaoYu] and has been applied to problems of different fields from fluid- and thermodynamics to offshore engineering and social networks [@Beckers; @Soldatova; @CaiWuChen; @DoRudolfGross]. An interesting property of the JSC is that it provides necessary conditions on all scales. However, application of the criterion were previously limited mostly to systems with few degrees of freedom because of the difficulties associated with computing large determinants.
In the following we rewrite the JSC conditions in the notation proposed above. By direct comparison we find $$\begin{aligned}
D_{1,S}&=&\times\\
D_{2,S}&=&\times \cdot \times -| \\
D_{3,S}&=&\times \cdot \times \cdot \times - \times \cdot | +2\triangle\\
D_{4,S}&=&\times \cdot \times\cdot \times\cdot \times -\times\cdot\times \cdot |+|\cdot|+2\times \cdot \triangle
- 2\Square\end{aligned}$$ and generally $$\label{formationrule}
D_{\left|S\right|,S}=\sum{\tiny \text{all combinations of symbols with $\sum n=\left|S\right|$}},$$ where symbols with $n>2$ appear with a factor of $2$ that reflects the two possible orientations in which the corresponding subgraphs can be paced out. Symbols with an even (odd) number of links carry a negative (positive) sign related to the sign of the respective index permutation in the Leibniz formula for determinants [@LA]. We note that for instance that $D_{6,S}$ has only 11 terms in the proposed graphical notation in contrast to 720 terms in conventional notation.
![Symbolic calculation of a determinant using the graphical notation. Consider the minor $D_{4,S}$ of the graph $\mathcal{G}$ sketched above. If $S$ is chosen to be the set of nodes plotted in grey, the terms of the fourth JSC, $D_{4,S}$, can be written as $\times\cdot\times\cdot\times\cdot\times=J_{11}J_{22}J_{33}J_{44}=(-1)^4(J_{12}+J_{13})(J_{21}+J_{23})(J_{31}+J_{32}+J_{34})(J_{43}+J_{45})=A+B+C+2D$, $-\times\cdot\times \cdot |=-(B+2C)$, $|\cdot|= C$, $2\times \cdot \triangle= -2D$ and $-2\Square= 0$. We can immediately read off that $D_{4,S}\equiv A = \Phi_{S}$, defined in the text. \[Subgraphs\]](Fig2.eps){width="47.00000%"}
The highly aggregated but somewhat awkward Eq. applies to all systems of symmetrically coupled one-dimensional units. A more convenient expression can be obtained by focusing specifically on systems having a zero row-sum (ZRS), such that $J_{ii}=-\sum_{j\neq i}J_{ij}$. In the Kuramoto model (and many other systems), the ZRS results from a force balance along the links. In the topological reading, it allows replacing all self-loops $J_{ii}$ by the negative sum over all links connected to respective node $i$. For the first term, $\times^{\left|S\right|}$, of a minor $D_{\left|S\right|, S}$ this leads to a sum over all graphs that can be drawn by placing $\left|S\right|$ (undirected) links such that every link starts from a distinct node in $S$. By elementary combinatorics it can be verified that all other terms of $D_{\left|S\right|, S}$ cancel exactly those subgraphs in $\times^{\left|S\right|}$ that contain cycles (cf. Fig. 2), leaving exactly $\Phi_{S}$ defined above. For systems with ZRS, we can thus express the minors as $$\label{det_n_2}
D_{\left|S\right|, S}=(-1)^{\left|S\right|}\Phi_{S}.$$ Thus the JSC stability conditions translate to $$\label{stability_cond2}
\Phi_{S}>0, \quad \forall \ S \ \text{with} \ \left|S\right|\leq r \ .$$ We remark that $\Phi_{S}$ with $\left|S\right|=N-1$ is the sum over all spanning trees of the network, such that Kirchhoff’s Theorem [@Schnakenberg; @Li] appears as the special case of Eq. .
The graphical stability conditions $\Phi_{S}>0$ conveniently conceals the complexity of the underlying determinants. Below we show that results can now be obtained by reasoning on the graphical level, i.e., without digging up the complex underlying expressions. Consider a network containing a tree-like branch that is only connected to the rest of the network by a single link. We choose $S$ as the set of nodes that are located in the branch and focus on the condition $\Phi_{S}>0$. The rules for constructing the network motifs in $\Phi_{S}$ imply they must contain one link starting from every node in $S$. By starting from the nodes of degree 1 (having only one link) and working downward one can see that there is only one motif contributing to $\Phi_{S}$, which consists of all links in the branch and the link connecting it to the rest of the network. The condition $\Phi_{S}>0$ therefore implies that the number of links in $\Phi_S$ associated with a negative elements must be even. Further, if this number were greater than zero, one could always find a part of the branch to which the same conditions apply, but which contains only one of the negative links, such that a stability condition on a smaller scale is violated. Thus, stability requires that all links appearing in such tree-like branches correspond to positive entries of ${\rm \bf J}$.
Using similar arguments as above the implications of different meso-scale motifs can be determined. The analysis reveals restrictions on (a) the number and position of potential negative links and (b) the absolute value of their weights. We find that restrictions of type (a) can be subsumed under one general condition: To admit stable solutions, a dynamical system with symmetric Jacobian and ZRS must possess a spanning tree made up entirely of positive elements.
To prove the statement above consider that in a network without a positive spanning tree it must be possible to partition the nodes into two nonempty sets $I_1$, $I_2$ such that $$\label{negativecut}
J_{ij}\leq 0 \quad \forall \ i,j \ | \ i\in I_1,\ j\in I_2 \ .$$ The idea is now to evaluate the stability conditions for different $S\subseteq I_1$ thereby exploiting that all links leading out of $I_1$ have negative weights. It is convenient to define $E^*$ as the set of links connecting $I_1$ and $I_2$, and $X=\left\{x_{1},\ldots,x_m\right\}$ as the subset of nodes $\in I_1$ incident to at least one link from $E^*$ (‘boundary nodes’). Further, we define $\sigma_i$ as the sum over all elements of $E^*$ incident to $x_i$, and, for any subset $Y$ of $X$, $\sigma_{Y}:=\prod_{m\in Y}\sigma_m$. Finally, for any subset $Y$ of $X$, we define $\tau_{Y}$ as the sum over all forests of $\mathcal{G}$ that (i) span $I_1$, and (ii) consists of $\left|Y\right|$ trees each of which contains exactly one element from $Y$. We can now write $$\label{expansion}
\Phi_{I_1\setminus C}=\sum_{B\subseteq X\setminus C}\sigma_B\tau_{B\cup C} \ ,$$ where $B$ and $C$ are disjoint subsets of $X$. We show that this is incompatible with the stability condition by the contradiction $$\begin{aligned}
\sum_{C\subseteq X}\underbrace{(-1)^{\left|C\right|}\sigma_{C}}_{\substack{>0\ \text{per} \\\text{construction}}}\underbrace{\Phi_{I_1\setminus C}}_{>0}&=\sum_{C\subseteq X}(-1)^{\left|C\right|}\sigma_{C}\sum_{B\subseteq X\setminus C}\sigma_B\tau_{B\cup C} \nonumber\\
&=\sum_{\substack{C\subseteq X\\B\subseteq X\setminus C}} (-1)^{\left|C\right|}\sigma_{B\cup C} \tau_{B\cup C}\nonumber\\
&=\sum_{\substack{A\subseteq X\\C\subseteq A}} (-1)^{\left|C\right|}\sigma_{A} \tau_{A}\nonumber\\
&=\sum_{A\subseteq X}\sigma_{A} \tau_{A}\underbrace{\sum_{C\subseteq A}(-1)^{\left|C\right|}}_{=0}=0 \ . \nonumber\end{aligned}$$ Therefore the existence of a spanning tree of positive elements is a necessary condition for stability.
Note that the (global) spanning tree criterion has distinct implications for meso-scale properties of $\mathcal{G}$. Any unbranched path can maximally contain one negative link, and the number of negative links in the network is limited by the number of independent cycles (Fig 3).
Let us now turn to stability conditions of type (b), which restrict the absolute value of potential negative links and result from smaller scale stability conditions: Consider for instance an unbranched segment of a cycle of $\mathcal{G}$ that consists of $d$ links $c_i$ with weights $w_i$, one of which, say $w_x$, is negative. Evaluating Eq. for a series of sets $S$ $\subset \left\{1,\ldots,d\right\}$, we find that stability requires $$\left|w_{x}\right|<\frac{\prod_{i\in I^*}w_i }{\sum \text{ \small all distinct products of $(d-2)$ factors $w_i$, $i\in I^*$}} ,$$ where $I^*= \left\{1,\ldots,d\right\}\setminus x$.
![Decomposition of a graph $\mathcal{G}$ in acyclic parts (black), unbranched segments of cycles (grey) and branching points (open black). Lines represent paths of $\mathcal{G}$ that may contain several nodes. Stability requires that (i) the acyclic parts only contain links with positive weights; (ii) any unbranched segment of a cycle contains at most one link with negative weight; (iii) the number of links with negative weight does not exceed the number of independent cycles (here, 5). \[Decomposition\]](Fig3.eps){width="37.00000%"}
Applied to the Kuramoto model, the criterion derived above implies that in any phase locked state a spanning tree must exist on which the phase difference between any two coupled oscillators obeys $|x_j-x_i|<\pi/2$. In networks where this condition is violated the criterion points to local interventions that enhance synchronizability.
We note that the results derived above are not contingent on the specific form of the Kuramoto model. They apply to all symmetrically coupled systems obeying the ZRS condition, which includes all systems of the form $$\label{generalization}
\dot{x_i}=C_i+\sum_{j\neq i} A_{ij}\cdot O_{ij}(x_j-x_i)\ , \quad \forall i\in 1\ldots N \ ,$$ where the $A_{ij}$ are the weights of a symmetric coupling matrix and $O_{ij}$ are odd functions. Besides systems of coupled oscillators, Eq. can for instance describe a range of variants of the Deffuant model of social opinion formation and ecological meta-population models. We emphasize that the derived results remain valid for heterogeneous networks containing different $A_{ij}$, $O_{ij}$, and $C_i$.
Although the ZRS simplifies the derivations above, similar calculations can be carried out for systems violating the ZRS condition, e.g. [@DoRudolfGross]. We applied the graphical notation to an adaptive extension of the Kuramoto model, where the ZRS is violated and the connection strength coevolves with the dynamics on the network according to ${\rm d/dt}(A_{ij})=\cos(x_j-x_i)-b\cdot A_{ij}$. Such systems have recently received much interest in physics because of their role in neuroscience [@ItoKaneko; @ZhouKurths; @Almendral]. Because of space constraints we postpone the detailed discussion of the model to a subsequent publication, but note that proceeding similarly as above leads to a stronger spanning tree condition for the adaptive system: Stability requires in this case the existence of a spanning tree where linked nodes obey $|x_j-x_i|<\pi/4$.
The proposed notation is also applicable to certain questions not pertaining to dynamics, for instance to the question of isospectrality of hermitian matrices [@Eskinetal; @Shirokov]. The key idea is that the characteristic polynomial $\chi$ of a hermitian matrix ${\rm\bf A}\in\mathbb{C}^{n\times n}$ can be expressed as $\chi(\lambda)=D_n({\rm\bf A}-\lambda {\rm\bf I})$. The structure of the graph $\mathcal{G}$ associated to ${\rm\bf A}-\lambda {\rm\bf I}$ reveals the symbols contributing to $\chi$. One can then determine which changes of off-diagonal entries leave all contributing symbols and thus the spectrum invariant.
In the present Letter, we proposed a graphical notation that facilitates the computation of spectral properties of complex systems. Applying this notation to systems of symmetrically-coupled one-dimensional dynamical units, we showed that any system obeying a simple force balance condition (ZRS) has to obey a global stability condition: Local dynamical stability of steady (e.g. phase-locked) states requires that the Coates graph of the Jacobian has a spanning tree of positive links. This criterion is complementary to results obtained by other methods (master stability function, ensemble simulations) and pertains to a large class of systems studied in physics, containing the Kuramoto model. Along with similar rules that can be derived analogously, the spanning-tree criterion has distinct meso-scale consequences, limiting for instance the number, position, and strength of negative elements in network motifs. Beyond the coupled oscillators, the proposed approach is applicable to questions ranging from adaptive networks in neuroscience to isospectrality problems in condensed matter.
We are grateful to Jeremias Epperlein and Stefan Siegmund for valuable discussions.
M.E.J. Newman, A.L. Barabási, D.J. Watts *The Structure and Dynamics of Networks* (Princeton University Press, Princeton, NJ, 2006). S. Fortunato, Phys. Rep. **486**, 75(2010). Y. Kuramoto, [*Lecture notes in physics vol. 39*]{} (Springer, New York, 1975). J.A. Acebron, L.L. Bonilla, C.J. Perez Vicente, F. Ritort, R. Spigler, Rev. Mod. Phys. **77**, 137 (2005). A. Pikovsky, M. Rosenblum, J. Kurths, [*Synchronization*]{} (Cambridge University Press, Cambridge, 2001). S. Boccaletti [*The synchronized dynamics of complex systems*]{} (Elsevier, Amsterdam, 2008). A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Phys. Rep. **469**, 93 (2008). R.E. Mirollo, S.H. Strogatz, Physica D **205**, 249 (2005). C.W. Wu, *Synchronization in complex networks of nonlinear dynamical systems* (World Scientific, Singapore, 2007). L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. **80**, 2109 (1998). Y. Kawamura, H. Nakao, K. Arai, H. Kori, Y. Kuramoto, Chaos **20**, 043110 (2010). M. Chavez, D.U. Hwang, A. Amann, H.G.E. Hentschel, S. Boccaletti, Phys. Rev. Lett. **94**, 218701 (2005). T. Nishikawa, A.E. Motter, Phys. Rev. E **73**, 065106 (2006). I. Lodato, S. Boccaletti, V. Latora, EPL **78**, 28001 (2007). T. Nishikawa, A.E. Motter, PNAS **107**, 10342 (2010). F. Mori, Phys. Rev. Lett. **104**, 108701 (2010). A.E. Motter, Phys. Rev. Lett. **93**, 098701 (2004). S.V. Buldyrev, R. Parshani, G. Paul, H.E. Stanley, S. Havlin, Nature **464**, 1025 (2010). X. Liao, P. Yu, [*Absolute Stability of Nonlinear Control Systems*]{} (Springer, Netherlands, 2008). J.M. Beckers, SIAM J. Numer. Anal. **29**(3), 701 (1992). E.D. Soldatova, J. Mol. Liq. **127**, 99 (2006). J. Cai, X. Wu, S. Chen, Math. Probl. Eng. **2007**, 86852 (2007). A.L. Do, L. Rudolf, T. Gross, New J. Phys. **12**, 063023 (2010). M.R. Adhikari, A. Adhikari, [*Textbook of Linear Algebra: Introduction to Modern Algebra*]{} (Allied Publishers, Mumbai, 2005). J. Schnakenberg, Rev. Mod. Phys. **48**(4), 571 (1976). M.Y. Li, Z. Shuai, J. Differ. Equations **248**, 1 (2010). J. Ito, K. Kaneko, Phys. Rev. Lett. **88**, 028701 (2001). C. Zhou, J. Kurths, Phys. Rev. Lett. **96**, 164102 (2006). J.A. Almendral, I. Leyva, D. Li, I. Sendiña-Nadal, S. Havlin, S. Boccaletti, Phys. Rev. E **82**, 016115 (2010). G. Eskin, J. Ralston, E. Trubowitz, Comm. Pure Appl. Math. **37**, 715 (1984). A.M. Shirokov, N.A. Smirnova, Y.F. Smirnov, Phys. Lett. B **434**, 237 (1998).
|
---
author:
- 'Marco Bertolini,'
- 'Ilarion V. Melnikov,'
- 'and M. Ronen Plesser'
bibliography:
- './bigref.bib'
title: Hybrid conformal field theories
---
Introduction {#s:intro}
============
Just what is a hybrid anyway? In constructing two-dimensional superconformal field theories (SCFTs) relevant for superstring vacua we are used to two sorts of massless fluctuating fields: those corresponding to a non-linear sigma model (NLSM), and those corresponding to a Landau-Ginzburg (LG) theory. The former define a classically conformally invariant system. Under favorable conditions, e.g. a Calabi-Yau target space and world-sheet supersymmetry, the background fields can be chosen to preserve superconformal invariance, and when the background is weakly coupled in a “large radius limit” (i.e. the background fields have small gradients), the theory reduces to a free-field limit. The latter have superpotential interactions that explicitly break scale invariance; however, under favorable conditions, e.g. a quasi-homogeneous superpotential, the IR limit of such a theory defines a non-trivial SCFT.
In each case, the utility of the description is two-fold: at a fundamental level, we can use the weakly coupled UV theory to define a SCFT; as a practical matter, the weakly coupled description, combined with non-renormalization theorems that follow from supersymmetry, allow us to identify and compute certain protected quantities such as chiral rings and massless spectra of the associated string vacua in terms of the UV degrees of freedom.
By now the reader has surely guessed what is meant by a hybrid [@Witten:1993yc; @Aspinwall:1993nu]: it is a two-dimensional theory that includes both types of massless fluctuating fields: ones that have classically conformally invariant NLSM self-interactions, as well as some that self-interact via a superpotential; of course an interesting hybrid also has interactions between the two types of degrees of freedom. A hybrid is a fibered theory, where the fiber is a LG theory with potential whose coefficients depend on the fields of the base NLSM. The potential is chosen so that its critical point set is the base target space. We then have two important questions: what are the criteria for a hybrid theory to flow to a SCFT? how do we generalize NLSM/LG techniques to compute physical quantities?
It is well-known that all of these descriptions— large radius limits of NLSMs, Landau-Ginzburg orbifolds (LGOs), and hybrid loci—arise as phases of (2,2), and more generally (0,2) gauged linear sigma models (GLSMs) [@Witten:1993yc]. The GLSM philosophy is that each phase should yield a limiting locus where at least protected quantities should be amenable to computation via the UV weakly-coupled field theory description. Such techniques are known for large radius NLSM and LGO phases but not for more general phases. In this work, we take a step in developing techniques for what we will call the “good hybrid” phases of a GLSM.[^1]
Although this does not cover a generic GLSM phase, and there are perhaps good reasons [@Aspinwall:2009qy] that we should not expect a simple description for a generic phase, it does increase the set of special points in the moduli space amenable to exact computations; this can lead to useful insights into stringy moduli space as in [@Aspinwall:2010ve; @Aspinwall:2011us; @Aspinwall:2011vp; @Blumenhagen:2011sq]. In addition, our definition of a good hybrid model, although inspired by the GLSM construction, will not explicitly invoke the GLSM. Thus, we are in principle providing a new class of UV theories that can lead to SCFTs without a known GLSM embedding.
In this note we will focus on hybrid theories with (2,2) world-sheet supersymmetry that are suitable for supersymmetric string compactification, i.e. ones with integral $\GUL\times\GUR$ R-symmetry charges; as in the case of LGO string vacua, this is achieved by taking an appropriate orbifold.
While such models offer a good point of departure, it is clear that a more general (0,2) hybrid framework will be both computationally useful and conceptually illuminating. We will describe (0,2) hybrids in a future work; for now we note that just like (2,2) LG models, the hybrids incorporate a class of Lagrangian deformations away from the (2,2) locus. These are obtained by smoothly deforming the (2,2) superpotential to a more general (0,2) form.
In what follows, we first give a broad geometric description of (2,2) hybrids, construct a Lagrangian for a good hybrid model and study its symmetries. With that basic structure in hand, we turn to a technique, valid in the large base volume limit and generalizing the well-known (2,2) and (0,2) LGO results of [@Kachru:1993pg; @Distler:1993mk], to compute the massless heterotic spectrum of a hybrid compactification. We then apply the techniques to a number of examples and conclude with a brief discussion of applications and further directions.
It is a pleasure to thank N. Addington, P.S. Aspinwall, A. Degeratu, S. Katz, D.R. Morrison, and E. Sharpe for useful discussions. IVM is grateful to University of Alberta and BIRS for hospitality while this work was being completed. MRP thanks the Aspen Center for Physics and NSF Grant 1066293 and the University of California at Santa Barbara for hospitality during work on this project, as well as the members of the Skywackers club for their hospitality at Larry Dennis flight park where some of this work was completed. MB thanks the Max-Planck-Institut für Gravitationsphysik (Golm) for hospitality during the completion of this work. MB and MRP are supported by NSF Grant PHY-1217109.
A geometric perspective {#s:geom}
=======================
The geometric setting for our theory is a (2,2) NLSM constructed with (2,2) chiral superfields. Consider a Kähler manifold $\bY_{\!\!0}$ equipped with a holomorphic function—the superpotential $W$—chosen so that its critical point set is a compact subset $B\subset \bY_0$. More precisely, $dW$, a holomorphic section of the cotangent bundle $T_{\bY_{\!\!0}}^\ast$, has the property that $dW^{-1}(0) = B \subset \bY_{\!\!0}$. We call this *the potential condition*. A LG model, with $\bY_{\!\!0} \simeq \C^n$ and $B$ being the origin, is a familiar example. A compact $\bY_{\!\!0}$ necessarily has a trivial superpotential, and the resulting theory is just a standard compact NLSM.
We say a geometry satisfying the potential condition has a *hybrid model* iff the local geometry for $B\subset \bY_{\!\!0}$ can be modeled by $\bY$ — the total space of a rank $n$ holomorphic vector bundle $X \to B$ over a compact smooth Kähler base $B$ of complex dimension $d$. The point of this definition is that the superpotential interactions will lead to a suppression of finite fluctuations of fields away from $B$, so that the low energy physics of the original NLSM will be well-approximated by the restriction to the hybrid model. Our main task will be to describe this low energy physics, and in what follows we will concentrate on the hybrid model geometry $\bY$. In many examples (e.g. the LG theories) $\bY \simeq \bY_{\!\! 0}$, but our results apply to the more general situation where $\bY$ is simply a local model. A simple example of a hybrid geometry, where $X = \cO(-2)$ over $B = \P^1$, is presented in appendix \[app:simplehybrid\].
In order to be reasonably confident that the low energy limit of a hybrid model is a (2,2) SCFT, we will need the geometry to satisfy several additional conditions intimately related to the existence of chiral symmetries and GSO projections. It will be easiest to discuss these after we introduce the explicit Lagrangian realization of this geometry. In our examples these features will already be present in the “UV” completion of the hybrid model, offered either by $\bY_{\!\!0}$ or some other high energy description such as a GLSM.[^2]
A final geometric comment, relevant for heterotic applications, concerns (0,2)-preserving deformations of these theories. (2,2) theories often admit a class of smooth (0,2) deformations, where the left-moving fermions couple to $\cE\to \bY$, a deformation of $T_{\bY}$, and the (0,2) superpotential is encoded by a holomorphic section $J \in \Gamma(\cE^\ast)$ with $J^{-1}(0) = B$. In the hybrid case there exist (0,2) deformations where $\cE = T_{\bY}$ but $d J \neq 0$; such a (0,2) superpotential cannot be integrated to a (2,2) superpotential $W$. Turning these on leads to a simple class of (0,2) hybrid models.
Action and symmetries {#s:action}
=====================
In this section we construct the (2,2) SUSY UV action for a hybrid model and analyze its symmetries. We begin with the necessary superspace formalism for a flat Euclidean world-sheet with coordinates $(z,\zb)$. Since we are interested in (0,2) deformations of (2,2) theories, it will be convenient for us to work with both (2,2) and (0,2) superspaces.[^3] Let’s start with the latter. Introducing Grassmann coordinates $\theta$ and $\thetab$, we obtain the supercharges $$\begin{aligned}
\cQ = -\frac{\p}{\p\theta} + \thetab \pbz,\qquad
\cQb = -\frac{\p}{\p\thetab} +\theta \pbz,\end{aligned}$$ where $\pb_{\!\zb} \equiv \p/\p \zb$. These form a representation of the (0,2) SUSY algebra: $\cQ^2 = \cQb^2 = 0$ and $\AC{\cQ}{\cQb} = -2\pbz$. The supercharges are graded by a $\GUR$ symmetry that assigns charge $\bqb = 1$ to $\theta$, and they anticommute with the supercovariant derivatives $$\begin{aligned}
\cD = \frac{\p}{\p\theta} + \thetab \pbz,\qquad
\cDb = \frac{\p}{\p\thetab} +\theta \pbz,\end{aligned}$$ that satisfy $\cD^2 =\cDb^2 = 0$ and $\AC{\cD}{\cDb} = 2\pbz$.
To build a (2,2) superspace we introduce additional Grassmann variables $\theta',\thetab'$ and form $\cQ'$, $\cQb'$, as well as $\cD'$ and $\cDb'$, by replacing $(\theta,\thetab,\pbz) \to (\theta',\thetab',\pz)$, where $\pz = \p /\p z$. These supercharges and derivatives are graded by $\GUL$ that assigns charge $\bq=1$ to $\theta'$.
Multiplets
----------
We are interested in Kähler hybrid models with target space $\bY$, and these can be constructed by using bosonic chiral (2,2) superfields and their conjugate anti-chiral multiplets[^4] denoted by $\cY^\alpha$ and $\cYb^{\alphab}$, with $\alpha,\alphab = 1,\ldots, \dim \bY$. These decompose into (0,2) chiral and anti-chiral multiplets as follows: $$\begin{aligned}
\label{eq:02sfields}
\cY^\alpha &= Y^\alpha + \sqrt{2} \theta' \cX^\alpha + \theta'\thetab' \pz Y^\alpha~, &&&
\cYb^{\alphab} & = \Yb^{\alphab} - \sqrt{2}\thetab' \cXb^{\alphab} -\theta'\thetab' \pz \Yb^{\alphab}~,\nonumber\\[2mm]
Y^\alpha & = y^\alpha + \sqrt{2}\theta \eta^\alpha +\theta\thetab \pbz y^{\alpha}~,&&&
\Yb^{\alphab} & = {\yb}^{\alphab} - \sqrt{2}\thetab \etab^{\alphab} -\theta\thetab \pbz \yb^{\alphab}~, \nonumber\\
\cX^\alpha & = \chi^\alpha + \sqrt{2}\theta H^\alpha +\theta\thetab \pbz \chi^{\alpha}~,&&&
\cXb^{\alphab} & = {\chib}^{\alphab} + \sqrt{2}\thetab \Hb^{\alphab} -\theta\thetab \pbz \chib^{\alphab}~.\end{aligned}$$ The $Y^\alpha$ are bosonic (0,2) chiral multiplets, while the $\cX^\alpha$ are chiral fermi multiplets, with lowest component a left-moving fermion $\chi^\alpha$; the $H^\alpha$ and their conjugates are auxiliary non-propagating fields.[^5]
Since $\bY$ is the total space of a vector bundle, it will occasionally be useful to split the $y^\alpha$ into base and fiber coordinates, which we will denote by $y^\alpha = (y^I, \phi^i)$, with $I = 1,\ldots, d$ and $i = 1,\ldots, n$. The $y^I$ are then coordinates on the base manifold $B$, while the $\phi^i$ parametrize the fiber directions.
The (2,2) hybrid action
-----------------------
The two-derivative (2,2) action is a sum of kinetic and potential terms, with $$\begin{aligned}
\label{eq:22action}
S_{\tkin} &= \frac{1}{4\pi} \int d^2 z ~\cD \cDb \cL_{\tkin},\qquad \cL_{\tkin} = \frac{1}{2} \cDb' \cD' \bK(\cY,\cYb),\nonumber\\
S_{\text{pot}} &= \frac{\sqrt{2}m}{4\pi} \int d^2z ~\cD \cW(Y,\cX) + \text{c.c.},\qquad
\cW =\frac{1}{\sqrt{2}} \cD' W(\cY)~.\end{aligned}$$ As is well-known, the kinetic term leads to a $\bY$ NLSM with a Kähler metric $g$. The superpotential $W$ is a holomorphic function on $\bY$ satisfying the potential condition, i.e. $dW(0)^{-1} = B$; $m$ is a parameter with dimensions of mass. If the metric $g$ is well-behaved, then the potential condition leads a suppression of field fluctuations away from $B \subset \bY$ via the bosonic potential $$\begin{aligned}
\label{eq:bospot}
S \supset \frac{|m|^2}{2\pi} \int d^2 z ~g^{\alpha\betab} \p_\alpha W \p_{\betab} \Wb~,\end{aligned}$$ and at low energies (as compared to $|m|$) the kinetic term can be taken to be quadratic in the fiber directions, i.e. the Kähler potential is $$\begin{aligned}
\label{eq:Kbf}
\bK= K(y^I,\yb^{\Ib}) + \phi h(y^I,\yb^{\Ib}) \phib + \ldots,\end{aligned}$$ where $K$ is a Kähler potential for a metric on $B$, $h$ is a Hermitian metric on $X \to B$, and $\ldots$ denotes neglected terms in the fiber coordinates. Using the base–fiber decomposition the metric $g_{\alpha\betab} = \p_\alpha \p_{\betab} \bK \equiv \bK_{\!\alpha\betab} $ then takes the form $$\begin{aligned}
g = (K_{I\Jb} - \phi \cF_{I\Jb} h \phib) dy^I d\yb^{\Jb} + D\phi h \Db\phib + \ldots,\end{aligned}$$ where $\cA = \p h h^{-1}$ is the Chern connection for the metric $h$, $\cF = \pb \cA$ is its (1,1) curvature, and $D\phi = d\phi +\phi \cA$ is the corresponding covariant derivative.
### Positivity of the metric and the case $\bY \simeq \bY_{\!\! 0}$ {#positivity-of-the-metric-and-the-case-by-simeq-by_-0 .unnumbered}
In many cases we need not worry about higher order corrections to $g$ in order to define a sensible theory. As in the simple case of LG models, this would be a situation where we need not consider the distinction between $\bY$ and $\bY_{\!\! 0}$ from above. Examining the form of $g$, we see that a necessary condition is that $\phi \cF_{I\Jb} h \phib$ is non-positive for all points in $\bY$.[^6] We say a bundle $X\to B$ is non-positive if it admits a Hermitian metric $h$ that satisfies this non-positivity condition.
Thus, to use (\[eq:Kbf\]) to define a UV-complete theory, we are led to a geometric question: what are the non-positive bundles over $B$? This is closely related to classical questions in algebraic geometry regarding positive and/or ample bundles, and using those classical results we can easily find sufficient conditions for non-positivity. Recall that a line bundle $L\to B$ is said to be positive if its (1,1) curvature form is positive; it is said to be negative if the dual bundle $L^\ast$ is positive [@Griffiths:1978pa; @Lazarsfeld:2004pa]. Taking $X = \oplus_i L_i$, a sum of negative and trivial line bundles, leads to many examples of non-positive bundles.
We should stress two points: first, even this set of examples leads to many previously unexplored SCFTs. Second, more generally, we do not need to assume that $\bY \simeq \bY_{\!\!0}$ or that $g$ has no higher-order terms in the fibers. The low energy limit of a UV theory with a hybrid model will be well-described by our action, and the potential condition will imply that the fiber corrections to the metric will not be important to the low energy physics. We will analyze one such example below, where $X$ is a sum of a positive and a negative bundle.
### (0,2) action {#action .unnumbered}
Since we are interested in heterotic applications as well as (0,2) deformations, it is useful to have the manifestly (0,2) supersymmetric action obtained by integrating over $\theta',\thetab'$ in (\[eq:22action\]). Absorbing the superpotential mass scale $m$ into $W$ the result is $$\begin{aligned}
\label{eq:02action}
\cL_{\tkin} &= \ff{1}{2} (\bK_\alpha \pz Y^\alpha- \bK_{\alphab} \pz \Yb^{\alphab}) + g_{\alpha\betab} \cX^\alpha \cXb^{\alphab}~, &
\cW & = \cX^\alpha W_\alpha~.\end{aligned}$$ where $\bK_{\!\alpha} \equiv \p \bK /\p Y^\alpha$, $W_\alpha \equiv \p W/\p Y^\alpha$, etc. It is a simple matter to obtain the classical equations of motion from the (0,2) action.[^7] The result is $$\begin{aligned}
\label{eq:02eom}
\cDb~\cXb_{\alpha} = \sqrt{2} W_\alpha,\qquad
\cDb\left[ g_{\alpha\betab} \p \Yb^{\betab} +g_{\alpha\betab,\gamma} \cXb^{\betab}\cX^\gamma \right] = \sqrt{2} \cX^\beta W_{\alpha\beta}~,\end{aligned}$$ where we defined the fermi superfield $\cXb_{\alpha} \equiv g_{\alpha\betab}(Y,\Yb) \cXb^{\betab}$.
### Component action {#component-action .unnumbered}
Finally, we can integrate over the remaining (0,2) superspace coordinates $\theta$ and $\thetab$ to obtain the component action. The auxiliary field $\Hb^{\alphab}$ is determined by the equations of motion (\[eq:02eom\]): $$\begin{aligned}
g_{\alpha\betab}\Hb^{\betab} = g_{\alpha\betab,\gammab}\etab^{\gammab}\chib^{\betab} + W_\alpha~,\end{aligned}$$ and using this as well as $\chib_\alpha \equiv g_{\alpha\betab} \chib^{\betab}$ we obtain $$\begin{aligned}
\label{eq:fullcomponent}
2\pi L & = g_{\alpha\betab} \left( \pbz y^\alpha\pz\yb^{\betab} + \etab^{\betab} \Dz \eta^\alpha\right) + \chib_{\alpha} \Dbz \chi^\alpha -\etab^{\betab} \eta^\alpha R_{\alpha\betab\gamma}^{~~~~\delta} \chib_{\delta} \chi^\gamma
- \chi^\alpha \eta^\beta D_\beta W_{\alpha} \nonumber\\
&\qquad+\chib^{\alphab}\etab^{\betab} D_{\betab}\Wb_{\alphab} + g^{\betab\alpha} W_\alpha \Wb_{\betab}~,\end{aligned}$$ where the covariant derivatives are defined with the Kähler connection $\Gamma^\alpha_{\beta\gamma} \equiv g_{\gamma\betab,\beta} g^{\betab \alpha}$, e.g. $$\begin{aligned}
\Db_{\zb} \chi^\alpha = \pbz \chi^\alpha + \pbz y^{\betab} \Gamma^\alpha_{\beta\gamma} \chi^\gamma~,\qquad
D_\alpha W_\beta = W_{\beta\alpha} - \Gamma^\gamma_{\alpha\beta} W_\gamma~,\end{aligned}$$ and the curvature is $R_{\alpha\betab\gamma}^{~~~~\delta} \equiv \Gamma^\delta_{\alpha\gamma,\betab}$. This is a complicated interacting theory, and in general it is not clear that one set of fields is preferred to another (say using $\chib_\alpha$ instead of $\chib^{\alphab}$); however, for the purpose of determining the massless spectrum, it turns out to be useful to introduce another field redefinition to keep track of the non-zero left-moving bosonic excitations: $$\begin{aligned}
\label{eq:rhodef}
\rho_\alpha \equiv g_{\alpha\alphab} \p \yb^{\alphab} + \Gamma^\delta_{\alpha\gamma}\chib_\delta \chi^\gamma~,\end{aligned}$$ in terms of which the left-moving kinetic terms take a strikingly simple form: $$\begin{aligned}
\label{eq:rhoaction}
2\pi L & = \rho_\alpha \pbz y^\alpha + \chib_\alpha \pbz \chi^\alpha + \eta^\alpha\left[ g_{\alpha\betab} \Dz \etab^{\betab} + \etab^{\betab} R_{\alpha\betab\gamma}^{~~~~\delta} \chib_\delta\chi^\gamma +\chi^\beta D_\alpha W_\beta
\right]~
\nonumber\\
&\qquad+\chib^{\alphab}\etab^{\betab} D_{\betab}\Wb_{\alphab} + g^{\betab\alpha} W_\alpha \Wb_{\betab}~.\end{aligned}$$ Unlike the other fields $\rho$ does not transform as a section of the pull-back of a bundle on $\bY$ under target space diffeomorphisms; this will have important consequences below.
Symmetries
----------
We now examine the symmetries of the hybrid Lagrangian.
### The $\bQb$ supercharge {#the-bqb-supercharge .unnumbered}
Our action respects (2,2) SUSY generated by the superspace operators $\cQ$ and $\cQb$, as well as their left-moving images. We define the action of the corresponding operators $\bQ$ and $\bQb$ by $$\begin{aligned}
\sqrt{2} \CO{\xi \bQ + \xib \bQb}{A} \equiv - \xi \cQ A -\xib \cQb A,\end{aligned}$$ where $\xi$ is an anti-commuting parameter and $A$ is any superfield. In order to avoid writing the graded commutator, we will use a condensed notation $\xib \bQb \cdot A \equiv \CO{\xib \bQb}{A}$. For our subsequent study of the right-moving Ramond ground states, we will be particularly interested in the action of $\bQb$. Using the superfields in (\[eq:02sfields\]), we obtain $$\begin{aligned}
\label{eq:Qbarfull}
\bQb \cdot y^\alpha & = 0, &
\bQb \cdot \chi^\alpha & = 0, &
\bQb \cdot \eta^\alpha &= \pbz y^\alpha~, &
\bQb \cdot H^\alpha & = \pbz \chi^\alpha~,
%
\nonumber\\
%
\bQb \cdot \yb^{\alphab} & =-\etab^{\alphab}~, &
\bQb \cdot \chib^{\alphab}& = \Hb^{\alphab}~, &
\bQb \cdot \etab^{\alphab} & = 0~,&
\bQb \cdot \Hb^{\alphab} & = 0~.\end{aligned}$$ The action of the remaining supercharges is easily obtained from this one by conjugation and/or switching left- and right-moving fermions. Eliminating the auxiliary fields by their equations of motion we obtain $$\begin{aligned}
\label{eq:QbarHeom}
\bQb \cdot y^\alpha & = 0~,&
\bQb \cdot \chi^\alpha &= 0~,&
\bQb \cdot \eta^\alpha &= \pbz y^\alpha~, \nonumber\\
\bQb \cdot \yb^{\alphab} & = -\etab^{\alphab}~,&
\bQb \cdot \chib_\alpha &= W_\alpha~, &
\bQb \cdot \etab^{\alphab} &= 0~.\end{aligned}$$ From (\[eq:02eom\]) it follows that up to the $\etab$ equations of motion we also have $\bQb\cdot \rho_\alpha = \chi^{\beta} W_{\beta\alpha}$. Hence we can decompose $\bQb$ as $\bQb = \bQb_0 + \bQb_W$, where the non-trivial action is $$\begin{aligned}
\label{eq:supersplit}
\bQb_0 \cdot \yb^{\alphab} &= -\etab^{\alphab}~,&
\bQb_0 \cdot \eta^\alpha &= \pbz y^\alpha~,&
\bQb_W \cdot \chib_\alpha & = W_\alpha~,&
\bQb_W \cdot \rho_\alpha & = \chi^\beta W_{\beta\alpha}~. \end{aligned}$$ These satisfy $\bQb_0^2 = \bQb_W^2 = \AC{\bQb_0}{\bQb_W} = 0$.[^8] $\bQb_0$ is the supercharge for the NLSM with $W=0$, while $\bQb_W$ incorporates the effect of a non-trivial potential.
### Chiral $\GU(1)$ symmetries {#chiral-gu1-symmetries .unnumbered}
The $\GUL\times\GUR$ symmetries play an important role in relating the UV hybrid model to the IR physics of the corresponding SCFT. In the classical NLSM with $W=0$ the presence of these symmetries is a consequence of the existence of an integrable, metric-compatible complex structure on $\bY$. In terms of components fields, the symmetries leave the bosonic fields invariant, while rotating the fermions as follows: $$\begin{aligned}
\GUL^0 ~:~ & \delta_{L}^0 \eta = 0,\qquad \delta^0_{L} \chi =- i \ep \chi~; &
\GUR^0 ~:~ & \delta_{R}^0 \eta = -i\ep \eta,\qquad \delta^0_{R} \chi = 0~,\end{aligned}$$ where $\ep$ is an infinitesimal real parameter. These naive symmetries are explicitly broken by the superpotential, but they can be improved if the geometry $(\bY,g)$ admits a holomorphic Killing vector $V$ satisfying $\cL_{V} W = W$.[^9] $V$ generates a non-chiral symmetry action $$\begin{aligned}
\delta_{V} Y^\alpha = i\ep V^\alpha(Y),\quad
\delta_{V} \Yb^{\alphab} = -i \ep \Vb^{\alphab}(\Yb);\qquad
\delta_{V} \cX^\alpha = i\ep V^\alpha_{,\beta} \cX^\beta~,
\delta_{V} \cXb^{\alphab} = -i \ep \Vb^{\alphab}_{,\betab} \cXb^{\betab}~,\end{aligned}$$ and it is easy to see that $\delta_{L,R}\equiv \delta_{L,R}^0 + \delta_V$ are symmetries of the classical action.
While $\GU(1)_{\text{diag}} \subset \GUL\times\GUR$ has a non-chiral action on the fermions and hence is non-anomalous, $\GUL$ is a chiral symmetry that will be anomaly free iff $c_1(T_{\bY}) = 0$, a condition satisfied when $\bY$ is a non-compact Calabi-Yau manifold, i.e. $\bY$ has a trivial canonical bundle $K_{\bY} \simeq \cO_{\bY}$. In what follows we assume $K_\bY$ is indeed trivial (this is stronger than $c_1(T_{\bY}) = 0$). When $X = \oplus_i L_i$, a sum of line bundles such that $\otimes_i L_i$ is negative, then since $K_{\bY} = K_B \otimes_i L_i^\ast$ the anti-canonical class of $B$ is very ample and $B$ is Fano.[^10]
In what follows we will denote the conserved charge for $\GUL$ ($\GUR$) by $J_0$ ($\Jb_0$) and its eigenvalues on various operators and states by $\bq$ ($\bqb$).
### R-symmetries for good hybrid models {#r-symmetries-for-good-hybrid-models .unnumbered}
We would like to identify the UV $\GUL\times\GUR$ symmetries described above with their counterparts in the conjectured IR SCFT. As usual, there is a small subtlety in doing this when $V$ is not unique. In practice this is easily achieved by picking a sufficiently generic superpotential and more generally, one could use $c$-extremization [@Benini:2012cz] to fix $\GUL\times\GUR$ up to the usual caveats of accidental IR symmetries.
More importantly, in order for the UV R-symmetry of the hybrid model to be a good guide to the IR physics, we need $V$ to be a vertical vector field, i.e. $\cL_{V} \pi^\ast(\omega) = 0$ for all forms $\omega \in \Omega^\bullet(B)$, and in particular the $\GUL\times\GUR$ symmetries fix $B$ point-wise. We denote a model where this is the case a *good hybrid*. As we show in Appendix \[app:VKill\] this implies $$\begin{aligned}
\label{eq:KVform}
V = \textstyle{\sum}_{i=1}^n q_i \phi^i\pp{\phi^i} + \text{c.c.}~\end{aligned}$$ for some real charges $q_i$. The $q_i$ have to be compatible with the transition functions defining $X \to B$, and since $\cL_V W = W$, and $W$ is polynomial in every patch, $q_i \in \Q_{\ge 0}$. In a LG theory, i.e $B$ a point, standard results show that if the potential condition is satisfied then without loss of generality $0 < q_i \le 1/2$ [@Kreuzer:1992bi; @Klemm:1992bx]. More generally, the potential condition requires that $W(y^I,\phi)$, thought of locally as a LG potential for the fiber fields $\phi$ depending on the “parameters” $y^I$, should be non-singular in a small neighborhood of any generic point in $B$. Hence, the range of allowed $q_i$ is the same for a hybrid theory as it is for LG models.
### The orbifold action {#the-orbifold-action .unnumbered}
Our main interest in the hybrid SCFTs is for applications to supersymmetric compactification of type II or heterotic string theories. For left-right symmetric theories this requires the existence of $\GUL\times\GUR$ symmetries with integral $\bq$, $\bqb$ charges of all (NS,NS) sector states [@Banks:1987cy]. Our hybrid theory, if it flows as expected to a $c=\cb =9$ SCFT in the IR will not satisfy this condition. Fortunately, the solution is the same as it is for Gepner models [@Gepner:1987vz] or LG orbifolds [@Vafa:1989xc; @Intriligator:1990ua]: we gauge the discrete symmetry $\Gamma$ generated by $\exp[2\pi i J_0]$, where $J_0$ denotes the conserved $\GUL$ charge; since all fields have $\bq-\bqb \in \Z$, the orbifold by $\Gamma$ is sufficient to obtain integral charges.
In the line bundle case with $q_i = n_i/d_i$ we then see that $\Gamma \simeq \Z_N$, with $N$ the least common multiple of $(d_1,\ldots,d_n)$. Since $\Gamma$ is embedded in a continuous non-anomalous symmetry we expect the resulting orbifold to be a well-defined quantum field theory, and the resulting orbifold SCFT will be suitable for a string compactification.
In addition to the introduction of twisted sectors and the projection, the orbifold has one important consequence for the physics of hybrid models: it allows us to consider more general “orbi-bundles,” where the fiber in $X\to B$ is of the form $\C^n/\Gamma$, and the transition functions are defined up to the orbifold action. For instance, we will examine a theory with $B = \P^3$ and $X = \cO(-5/2)\oplus\cO(-3/2)$, where the orbifold $\Gamma =\Z_2$ reflects both of the fiber coordinates.[^11]
The quantum theory and the hybrid limit
---------------------------------------
Having defined the classical hybrid model’s Lagrangian and discussed its symmetries, we now discuss the quantum theory. To orient ourselves in the issues involved, let’s recall the case of (2,2) LG models — the simplest examples of hybrids. These theories have a Lagrangian description at some renormalization scale $\mu$ as a free kinetic term for chiral multiplets, and a superpotential interaction with dimensionful couplings $m$. The theory is weakly coupled when $\mu \gg m$, and we can use the Lagrangian and (approximately) free fields to describe the theory. The low energy limit $\mu\to 0$ is then strongly coupled, and while $W$ is protected by SUSY non-renormalization theorems, the kinetic term receives a complicated but irrelevant set of corrections. There is by now overwhelming evidence that these do flow to the expected SCFTs, in accordance with the original proposals [@Martinec:1988zu; @Vafa:1988uu], and computations of RG-invariant quantities allow us to use the weakly coupled $\mu \gg m$ description to describe *exactly* the SCFT’s (c,c) chiral ring and more generally the $\bQb$-cohomology. Furthermore, the results extend to LGOs suitable for string compactification.[^12]
There is a small IR subtlety in using the weakly coupled LG description: the theory at $W=0$ is non-compact and has all the usual difficulties associated to non-compact bosons. This is of course not very subtle since the theory is free; however, more to the point, in using the weakly coupled description we still keep track of the R-charges and weights that follow from the superpotential and do not consider states supported away from the $W=0$ locus.
A more general hybrid theory has a similar structure, except that now there are two sorts of couplings: the superpotential couplings $m/\mu$, as well as the choice of Kähler class on the base $B$. Although the latter coupling is typically encoded in the kinetic D-term, it can also be expressed as a deformation of the twisted chiral superpotential. Hence, the Kähler class and superpotential couplings do not receive quantum corrections. Of course we do expect corrections to the D-terms, but these should be irrelevant just as they are in the LG case. Moreover, there is good evidence, based on GLSM constructions, that the hybrid models with a GLSM UV completion should flow to SCFTs with expected properties (i.e. correct central charges and R-symmetries), and we expect the same to hold for more general hybrid models. As in the LG case, the strict $W=0$ limit may be subtle, perhaps even more so, since it may require us to specify additional details about the geometry of $\bY$. However, we may use the same cure for these IR subtleties as we do in the LG case: use the R-charges and weights encoded by the superpotential and restrict attention to field configurations and states supported on $B$.
Assuming a hybrid model does flow to an expected SCFT, we would like to have techniques to evaluate RG-invariant quantities such as the $\bQb$-cohomology. It is here that there will be important conceptual and technical differences from the LG case due to the non-trivial base geometry $B$. For instance, we expect the $\bQb$-cohomology to depend on the choice of Kähler class on $B$. While there will not be a perturbative dependence, we do in general expect corrections from world-sheet instantons wrapping non-trivial cycles in $B$. These corrections are suppressed when $B$ is large, which leads us to define the hybrid analogue of the large radius limit of a NLSM: *the hybrid limit*, where the Kähler class of $B$ is taken to be arbitrarily deep in its Kähler cone. In what follows, we will study the $\bQb$-cohomology of a hybrid model in the hybrid limit.
Massless spectrum of heterotic hybrids {#s:spectra}
======================================
In this section we develop techniques to evaluate the massless spectrum for a compactification of the $\GE_8\times\GE_8$ heterotic string based on a $c=\cb=9$ (2,2) hybrid SCFT.[^13] We first review the standard prescription [@Gepner:1987vz; @Vafa:1989xc; @Kachru:1993pg] to obtain a modular invariant theory and identify world-sheet Ramond ground states with massless fermions in spacetime. We then discuss how to enumerate these ground states by studying the $\bQb$ cohomology in the hybrid limit.
Spacetime generalities
----------------------
In order to describe a heterotic string compactification, we complete our hybrid $c=\cb=9$ $N=(2,2)$ SCFT internal theory to a critical heterotic theory by adding ten left-moving fermions (with fermion number $F_\lambda$) that realize an $\so(10)$ level $1$ current algebra, a left-moving level $1$ hidden $\Le_8$ current algebra, and the free $c=4$, $\cb = 6$ theory of the uncompactified spacetime $\R^{1,3}$.
A modular invariant theory is obtained by performing left- and right- GSO projections. The left-moving GSO projection onto $e^{i\pi J_0} (-)^{F_\lambda} = 1$ is responsible for enhancing the linearly realized $\Lu(1)_{\text{L}}\oplus\so(10)$ gauge symmetry to the full $\Le_6$. The right-moving GSO projection has a similar action, combining $\Jb_0$ with the fermion number of the $\R^{1,3}$ theory. Its immediate spacetime consequence is $N=1$ spacetime supersymmetry, or equivalently, a relation, via spectral flow, between states in right-moving Neveu-Schwarz and Ramond sectors. Spacetime fermions arise in the (NS,R) and (R,R) sectors, and supersymmetry allows us to identify the full spectrum of supermultiplets in the spacetime theory from these states.
The spacetime theory obtained by this procedure will have a model-independent set of massless fermions: the gauginos of the hidden $\Le_8$, the gravitino, and the dilatino. In what follows we focus on the model-dependent massless spectrum. In particular, the hidden $\Le_8$ degrees of freedom are always restricted to their NS ground state and just make a contribution to the left-moving zero-point energy.
On-shell string states have vanishing left- and right-moving energies. For massless states there is no contribution to $\Lb_0$ from the $\R^{1,3}$ free fields; massless fermions are thus states in the (R,R) and (NS,R) sectors with vanishing left-moving and right-moving energies. In the (R,R) sector, massless states are associated to the ground states in the internal theory, related by spectral flow to (NS,NS) operators comprising the “chiral rings” [@Vafa:1988uu] of the theory. Massless states in the (NS,R) sector include states related to these by left-moving spectral flow as well as additional states. The main result of [@Kachru:1993pg] is a method for describing these states in LGO theories, which we here extend to hybrids. This relies on the familiar fact that since $$\begin{aligned}
\{\bQ,\bQb\} = 2\Lb_0~;\quad \bQ^2 = \bQb^2 = 0\end{aligned}$$ the kernel of $\Lb_0$ is isomorphic to the cohomology of $\bQb$.
The right-moving GSO projection is onto states with $\bqb\in \Z+\ff{1}{2}$; those with $\bqb=-1/2$ ($\bqb=1/2$) correspond to chiral (anti-chiral) multiplets, while states with $\bqb=\pm 3/2$ are gauginos in vector multiplets. The $\GUL$ charge $\bq$ determines the $\Le_6$ representation according to the decomposition $$\begin{aligned}
\label{eq:E6decomposition}
&\Le_6 \supset \so(10)\oplus \Lu(1)\nonumber\\
& \mathbf{78} = \mathbf{45}_0 \oplus \mathbf{16}_{-3/2} \oplus \mathbf{\overline{16}}_{3/2} \oplus \mathbf{1}_0 \nonumber\\
& \mathbf{27} = \mathbf{16}_{1/2} \oplus \mathbf{{10}}_{-1} \oplus \mathbf{1}_2 \nonumber\\
& \mathbf{\overline{27}} = \mathbf{\overline{16}}_{-1/2} \oplus \mathbf{{10}}_1 \oplus \mathbf{1}_{-2}~.\end{aligned}$$
As in the LG orbifold case [@Vafa:1989xc; @Kachru:1993pg], the GSO projection can be combined with the hybrid orbifold of $\Gamma=\Z_N$ to an orbifold by $\Z_2\ltimes\Z_N\cong \Z_{2N}$. Therefore we need to study the $2N$ sectors twisted by $[\exp(i\pi J_0)]^k, k=0,\dots,2N-1$.[^14] Spacetime CPT exchanges the $k$-th and the $(2N-k)$-th sectors, and CPT invariance means we can restrict our analysis to the $k=0,1,\dots,N$. sectors. The states arising in (R,R) ($k$ even) sectors give rise to $\Le_6$-charged matter. This is easy to see since in this case the ground states of the $\so(10)$ current algebra transform in $\rep{16}\oplus\brep{16}$. Massless $\Le_6$-singlets are of particular interest, and they can only arise from (NS,R) sectors, i.e. sectors with odd $k$.
Left-moving symmetries in cohomology {#ss:n2alg}
------------------------------------
The action of $\GUL$ commutes with $\bQb$, and following [@Witten:1993jg; @Silverstein:1994ih], we can find a representative for the corresponding conserved current in $\bQb$-cohomology, denoted by $H_{\bQb}$. Consider the operator $$\begin{aligned}
\cJ_L \equiv \cX^\beta(D_\beta V^\alpha -\delta^\alpha_\beta)\cXb_{\alpha}- V^\alpha g_{\alpha\betab} \pz\Yb^{\betab}~.\end{aligned}$$ Using (\[eq:02eom\]) and $\cL_{V} W = W$ it follows $\cDb \cJ_L = 0$. Observing that $\cQb$ and $\cDb$ are conjugate operators, $\cQb = -\exp\left[2\thetab\theta \pbz\right] \cDb \exp\left[2\theta\thetab \pbz\right]$, we conclude that $$\begin{aligned}
J_L \equiv \left.\cJ_L\right|_{\theta=0} = \chi^\beta(V^\alpha_{~,\beta} -\delta^\alpha_\beta)\chib_{\alpha}- V^\alpha\rho_\alpha\end{aligned}$$ is $\bQb$-closed and hence has a well-defined action on $H_{\bQb}$. Similarly, we can obtain the remaining generators of the left-moving $N=2$ algebra in $H_{\bQb}$. To find the energy-momentum generator $T$ we observe that $$\begin{aligned}
\cT_0 = -g_{\alpha\betab} \pz Y^\alpha \pz \Yb^{\betab} -\cX^\alpha \Dz \cXb_{\alpha} = -\pz Y^\alpha \left[g_{\alpha\betab}\pz\Yb^{\betab} - g_{\gamma\betab,\alpha} \cX^\gamma \cXb^{\betab}\right] - \cX^\alpha\pz\cXb_\alpha\end{aligned}$$ satisfies $\cDb \cT_0 = 0$, as does $$\begin{aligned}
\cT \equiv \cT_0 - \half\pz\cJ_L~.\end{aligned}$$ The lowest component of $\cT$ is $\bQb$-closed and given by $$\begin{aligned}
T &= -\p y^\alpha\rho_\alpha- \half\left(\chib_\alpha \pz \chi^\alpha+ \chi^\alpha \pz\chib_\alpha\right) -\half\pz\left[ \chi^\beta \chib_\alpha V^\alpha_{~,\beta} -V^\alpha \rho_\alpha\right]\ .\end{aligned}$$ The remaining generators of a left-moving $N=2$ algebra are obtained from the $\cDb$-closed fields $$\begin{aligned}
\cG^+ \equiv i\sqrt{2} \left[ \cXb_{\alpha} \pz Y^\alpha - \pz(\cXb_{\alpha} V^\alpha)\right],\qquad
\cG^- \equiv i\sqrt{2} \left[ \cX^\alpha g_{\alpha\betab} \pz\Yb^{\betab}\right]~,\end{aligned}$$ yielding the left-moving supercharges $G^{\pm}$ in $H_{\bQb}$: $$\begin{aligned}
G^+ = i\sqrt{2} \left[ \chib_{\alpha} \pz y^\alpha - \pz(\chib_{\alpha} V^\alpha)\right]~,\qquad
G^- \equiv i\sqrt{2} \chi^\alpha \rho_\alpha~.\end{aligned}$$
Reduction to a curved $bc-\beta\gamma$ system {#ss:bcbg}
---------------------------------------------
The action (\[eq:rhoaction\]) determines the OPEs for the left-moving degrees of freedom to be $$\begin{aligned}
\label{eq:OPEs}
y^\alpha(z) \rho_{\beta}(w) \sim \frac{1}{z-w}\delta^\alpha_\beta~,\qquad
\chi^\alpha(z)\chib_{\beta}(w) \sim \frac{1}{z-w}\delta^\alpha_\beta~.\end{aligned}$$ Using the normal ordering defined by these free-field OPEs we can define $T$, $J$, and $G^\pm$ in the quantum theory. This is particularly simple with our choice of fields and Killing vector $V$: the operators are quadratic in the fields, and it is easy to check that they indeed generate an $N=2$ algebra with central charge $$\begin{aligned}
\label{eq:centralcharge}
c = 3 d + 3\textstyle{\sum}_{i=1}^n (1-2q_i)~,\end{aligned}$$ which we recognize as the sum of the fiber LG central charge and the contribution from the base. The $\GUL$ charge $J_0$ and left-moving Hamiltonian $L_0$ are obtained in the standard fashion as $$\begin{aligned}
\label{eq:leftcharges}
J_0 = \oint \frac{dz}{2\pi i} J_L(z)~,\qquad L_0 = \oint \frac{dz}{2\pi i} z T(z)~,\end{aligned}$$ and the resulting charge and weight assignments for the fiber fields are given in table \[table:charges\] together with the $\GUR$ charge $\bqb$.
\[table:charges\]
$y^I$ $\rho_I$ $\chi^I$ $\chib_I$ $\phi^i$ $\rho_i$ $\chi^i$ $\chib_i$
-------- ------- ---------- ---------- ----------- ---------- ---------- ---------- -----------
$\bq$ $0$ $0$ $-1$ $1$ $q_i$ $-q_i$ $q_i-1$ $1-q_i$
$2h$ $0$ $2$ $1$ $1$ $q_i$ $2-q_i$ $1+q_i$ $1-q_i$
$\bqb$ $0$ $0$ $0$ $0$ $q_i$ $-q_i$ $q_i$ $-q_i$
: Weights and charges of the fields.
These currents are trivially annihilated by $\bQb_0$ and commute with $\bQb_W$, whose action is now realized as $$\begin{aligned}
\label{eq:bQbWop}
\bQb_W \equiv \oint \frac{dz}{2\pi i} ~ \left[ \chi^\alpha W_\alpha (y)\right] (z)~.\end{aligned}$$
It may seem a little bit puzzling that we have been able to reduce the entire problem to a free first order system. What, the reader may ask, encodes the target space geometry, for example? The answer, familiar from [@Nekrasov:2005wg; @Witten:2005px], is that the free field theory description only applies patch by patch in field-space. That is, we cover $\bY$ with open sets $U_{a}$ and local coordinates $x_a^\alpha$, and on each $U_{ab} = U_{a}\cap U_{b} \neq \emptyset$ $x_b=x_b(x_a)$, and we define the holomorphic transition functions $$\begin{aligned}
(T_{ba})^\alpha_{\beta} \equiv \frac{\p x^\alpha_b}{\p x^\beta_a},\qquad
(\cS_{ba})^\alpha_{\beta\gamma} \equiv (T_{ba}^{-1})^\alpha_\delta (T_{ba})^\delta_{\beta,\gamma}~.\end{aligned}$$ The left-moving fields then patch according to $$\begin{aligned}
\label{eq:patchfields}
y^\alpha_b &= x^\alpha_b(y_a)~,\quad
\chi^\alpha_b = (T_{ba})^{\alpha}_{\beta}\chi^\beta_a~,\quad
\chib_{b\alpha} = (T_{ba}^{-1})^\beta_\alpha \chib_{a\beta}~,\nonumber\\
\rho_{b\alpha} &= : (T_{ba}^{-1})^\beta_{\alpha}\left[ \rho_{b\beta} -\cS^\delta_{ba\beta\gamma} \chib_\delta\chi^\gamma\right]:~,\end{aligned}$$ where the transition functions are evaluated at $y_a$, e.g. $T_{ba} = T_{ba}(y_a)$. Note that the patching of $\rho$ requires a normal-ordering due to singularities in the $y-\rho$ and $\chib-\chi$ OPEs. Of course there are similar transformations for the right-moving fields $\yb$ and $\eta,\etab$. For instance, the $\etab^{\Ib}$ transform as sections of $y^\ast(\Tb_B)$.[^15]
These transition functions require a careful analysis when we expand about world-sheet instanton configurations, i.e. non-trivial holomorphic maps $\Sigma \to \bY$. This, together with non-trivial fermi zero modes in the background of an instanton will lead to world-sheet instanton corrections to $\bQb_0$.[^16] These corrections vanish in the hybrid limit where we expand about constant maps $\pz y = \pbz y = 0$, and the only non-trivial $\bQb_0$ action is on the anti-holomorphic zero modes $\bQb_0 \cdot \yb^{\alphab}_0 = -\etab^{\alphab}_0$. In fact, since the $\etab^{\ib}$ are $\bQb$-exact, as far as cohomology is concerned, we can safely ignore the $\etab^{\ib}$ as well as the anti-holomorphic bosonic fiber zero modes $\phib^{\ib}_0$. So, the only non-trivial $\bQb_0$ action is on the base anti-holomorphic zero modes: $\bQb_0 \cdot \yb^{\Ib}_0 = -\etab^{\Ib}_0$. In what follows we will drop the zero mode subscript on these right-moving fields with the understanding that $\yb$ and $\etab$ will denote the base antiholomorphic zero modes.
Massless states in the hybrid limit {#ss:massless}
-----------------------------------
Our task now is to work out, in each twisted sector, the set of GSO-even states that belong to $H_{\bQb}$ and carry left-moving energy $E= 0$. We construct the relevant states (i.e. the only ones with required energy and charges) in the Hilbert space as polynomials in the fermions and non-zero bosonic oscillator modes tensored with wavefunctions of the bosonic zero modes. In a generic twisted sector the bosonic zero modes correspond to the compact base $B$, while in less generic sectors there can be additional bosonic zero modes. However, since the non-compact bosonic modes will be lifted by the superpotential, in what follows all bosonic wavefunctions will be taken to be polynomial in the fiber fields.
The operators $T$ and $J_L$ can be used to grade the states according to their energy $E$ and left-moving charge $\bq$, and we can evaluate $\bQb$-cohomology on the states of fixed $E$ and $\bq$. An important simplification comes from working in the right-moving Ramond ground sector. A look at (\[eq:supersplit\]) shows that, as far as $\bQb$-cohomology is concerned, we can neglect any states containing oscillators in $\pbz y^\alpha$, as well as any non-zero mode of $\eta^\alpha$. We choose the Ramond ground state annihilated by the zero modes of $\eta^\alpha$, so our states will be constructed without $\eta^\alpha$ or right-moving bosonic oscillators. We will call the resulting space of states *the restricted Hilbert space* $\cH$. In general this will be infinite-dimensional even at fixed $E$ and $\bq$.
### Twisted modes and ground state quantum numbers {#twisted-modes-and-ground-state-quantum-numbers .unnumbered}
In this section we provide expressions for $E$, $\bq$ and $\bqb$ of the states in a fixed twisted sector. For simplicity, we work out the case $X = \oplus_i L_i$. The result extends immediately to orbi-bundles of the form $X = \oplus_i L_i^{x_i}$ for $x_i \in \Q$. It should be possible to treat the case of more general $X$ at the price of additional notation.
The first task is to describe the mode expansions of the fields and the quantum numbers of the ground states $|k\ra$. While we can restrict to right-moving (i.e. anti-holomorphic) zero modes, the left-moving oscillators need to be treated in detail. In each patch of the target space the moding of the left-moving fields in the $k$-th twisted sector is $$\begin{aligned}
\label{eq:modes}
y^\alpha(z)&=\sum_{r\in\Z-\nu_\alpha} y^\alpha_r z^{-r-h_\alpha} , \qquad \qquad \quad \chi^\alpha(z)=\sum_{r\in\Z-\nut_\alpha} \chi^\alpha_r z^{-r-\htld_\alpha}, \nonumber\\
\rho_\alpha(z)&=\sum_{r\in\Z+\nu_\alpha} {\rho_\alpha}_r z^{-r+h_\alpha-1} , \qquad \qquad \chib_\alpha(z)=\sum_{r\in\Z+\nut_\alpha} \chib_{\alpha r} z^{-r+\htld_\alpha-1},\end{aligned}$$ where $$\begin{aligned}
\label{eq:nus}
\nu_\alpha&=\frac{kq_\alpha}{2} \mod1~, &\nut_\alpha&=\frac{k(q_\alpha-1)}{2} \mod1~,&
\htld_{\alpha} -\frac{1}{2} &= h_\alpha = \frac{q_\alpha}{2}~.\end{aligned}$$ We choose $0\leq \nu_\alpha<1$ and $-1<\nut_\alpha\leq 0$ and recall that the oscillator vacuum $|k\ra$ is annihilated by all the positive modes. When $\chi,\chib$ have zero modes our conventions are that the ground state is annihilated by the $\chi_0$ modes.
The mode (anti)commutators follow from (\[eq:OPEs\]) and (\[eq:modes\]): $$\begin{aligned}
\CO{y^\alpha_r}{\rho_{\beta s}} = \delta^{\alpha}_{\beta} \delta_{r,-s}~,\qquad
\AC{\chi^\alpha_r}{\chib_{\beta s}} = \delta^{\alpha}_{\beta} \delta_{r,-s}~.\end{aligned}$$ Each oscillator carries the obvious $\bq,\bqb$ charges and contributes minus its mode number to the energy. By using this mode expansion to compute $1$-point functions of $T$ and $J_L$ in the oscillator vacuum $|k\ra$, we determine the quantum numbers of $|k\ra$. The left- and right-moving charges are given by $$\begin{aligned}
\bq_{|k\ra}&=\sum_\alpha \left[ (q_\alpha-1)(\nut_\alpha+\half)-q_\alpha(\nu_\alpha-\half) \right] , \nonumber\\
\bqb_{|k\ra}&=\sum_\alpha \left[ q_\alpha(\nut_\alpha+\half)+(q_\alpha-1)(-\nu_\alpha+\half) \right] ,\end{aligned}$$ and while the left-moving energy is $E_{|k\ra} =0$ for $k$ even, we have $$\begin{aligned}
E_{|k\ra} = -\frac{5}{8} + \half \sum_\alpha \left[ \nu_\alpha(1-\nu_\alpha)+\nut_\alpha(1+\nut_\alpha) \right] ,\end{aligned}$$ for $k$ odd. Note that this includes the usual $-c/24$ shift: $E=L_0-1$.
The oscillator vacuum $|k\ra$ we have constructed is not in general a state in the Hilbert space. To specify a state we need to prescribe a dependence on the bosonic zero modes so as to get a well-defined state, but from above we see that $|k\ra$ transforms as a section of a holomorphic line bundle $L_{|k\ra} $ over $B$. When $X=\oplus_iL_i$ we find (using $K_{\bY} = \cO_{\bY}$) $$\begin{aligned}
\label{eq:bundlegroundst}
L_{|k\ra}=\begin{cases}
\otimes_i L_i^{(\nut_i-\nu_i)} &\qquad \text{for } k \text{ even}, \\
\otimes_i L_i^{(\nut_i-\nu_i+\half)} & \qquad \text{for } k \text{ odd}~.
\end{cases}\end{aligned}$$ From (\[eq:nus\]) we see that if we set $\nu_I=0$ and $\nut_I = -k/2\mod 1$, then $\tau_\alpha = \nu_\alpha - \nut_\alpha$ is $$\begin{aligned}
\label{eq:tau}
\tau_\alpha = \begin{cases}
0 &\nu_\alpha = 0\\
1 &\nu_\alpha \ne 0
\end{cases}\quad\text{for even $k$}~;\quad
\tau_\alpha = \begin{cases}
1/2 & \nu_\alpha\le \half\\
3/2 & \nu_\alpha > \half
\end{cases}\quad\text{for odd $k$}~.\end{aligned}$$ This shows that $L_{|k\ra}$ is well-defined because $\tau_\alpha\in \Z$ for $k$ even and $\tau_\alpha\in \Z+\half$ for $k$ odd. A well-defined ground state can be of the form $$\begin{aligned}
|\Psi^k_0\ra = \Psi_0(y',\yb)_{\Ib_1\cdots \Ib_u}\etab^{\Ib_1}\cdots\etab^{\Ib_u}|k\ra\ ,\end{aligned}$$ where $y'$ denotes bosonic zero modes, the $\etab^{\Ib}$ are the right-moving superpartners of the base coordinates and $\Psi^k_u$ are (0,u) horizontal forms on $\bY$ valued in the holomorphic sheaf $L_{|k\ra}^\ast$. In sectors in which there are additional zero modes ($k=0$ is always an example of this) there are more general ground states, and in (R,R) sectors a subset of these ground states describes the massless spectrum.
This non-trivial vacuum structure is a generalization of familiar limiting cases of the hybrid construction. When $\bY = B$ a compact Calabi-Yau manifold, the Ramond ground state is a section of a trivial bundle (the square root of the trivial canonical bundle); in the LGO case each twisted sector has a unique ground state $|k\ra$.
### The double-grading and spectral sequence {#the-double-grading-and-spectral-sequence .unnumbered}
Our restricted Hilbert space $\cH$ at fixed $E$ and $\bq$ admits a grading by $\GUR$ charge, and $\bQb$ acts as a differential, $\bQb: \cH_{\bqb} \to \cH_{\bqb+1}$ that preserves the left-moving quantum numbers. A key observation, made in the LG case in [@Kachru:1993pg], that makes the cohomology problem tractable is that in fact $\cH$ admits a double-grading compatible with the split $\bQb = \bQb_0+ \bQb_W$ in (\[eq:supersplit\]). Let $U$ be an operator that assigns charge $+1$ to $\etab$, $-1$ to $\eta$, and leaves the other fields invariant. Although $U$ is not a symmetry of the theory when $W\neq 0$, we can still grade our restricted Hilbert space according to the eigenvalues $u$ of $U$ and $p\equiv \bq-u$, and since $\CO{U}{\bQb_0} = \bQb_0$ and $\CO{U}{\bQb_W} = 0$ we obtain a double-graded complex with $$\begin{aligned}
\bQb_0 : \cH^{p,u} \to \cH^{p,u+1},\qquad
\bQb_W :\cH^{p,u} \to \cH^{p+1,u}~\end{aligned}$$ acting, respectively, as anticommuting vertical and horizontal differentials. The cohomology of $\bQb$ is thus computed by a spectral sequence with first two stages $$\begin{aligned}
E_1^{p,u}=H^u_{\bQb_0}(\cH^{p,\bullet}),\qquad\text{and}\qquad E_2^{p,u}=H^p_{\bQb_W}H^u_{\bQb_0}(\cH^{\bullet,\bullet})~.\end{aligned}$$ In general, $E_{r+1}$ is obtained from $E_r$ as the cohomology of a differential $d_r$ acting as $$\begin{aligned}
d_r:E_r^{p,u}\to E_r^{p+r,u+1-r}\ .\end{aligned}$$ We have, for example, $d_0=\bQb_0$ and $d_1 = \bQb_W$. The differentials at higher stages are produced by a standard zig-zag construction [@Bott:1982df]. Since the range of $U$ is $0\le U\le d$ the differentials vanish for $r>\dim B$, and the sequence converges: $E_{\dim B+1}^{p,u}=E_\infty^{p,u}=H^{p,u}_{\bQb}(\cH^{\bullet,\bullet})$.
We now have almost all of the tools to describe the massless spectrum. In each twisted sector there is a geometric structure that organizes the states in the spectral sequence. On $\cH$ the $\bQb_0$ action is simply $$\begin{aligned}
\bQb_0 = -\etab^{\Ib} \frac{\p}{\p \yb^{\Ib}},\end{aligned}$$ so $\bQb_0$ cohomology amounts to restricting to horizontal[^17] Dolbeault cohomology groups, while $\bQb_W$ cohomology imposes further algebraic restrictions.
Since the geometry is typically non-compact the $\bQb_0$ cohomology groups are often infinite-dimensional. Fortunately we can obtain a well-defined counting problem because $\bQb_0$ respects the *fine grading* by a vector $\br = (r_1,\ldots, r_n)\in \Z^n$ that assigns grade $\br$ to a monomial $\prod_i \phi_i^{r_i}$.[^18] Restricting to a particular grade leads to finite-dimensional vector spaces that, as we show in appendix \[app:sheaf\], are readily computable in terms of sheaf cohomology over $B$. The fine grading is a refinement of the physically relevant grading by $\bq$ and $E$, and therefore it gives an effective method for evaluating the first stage in the spectral sequence $E_1^{p,u}$ at fixed twisted sector, $\bq$, and $E$.
The next step is to study the $\bQb_W$ cohomology, i.e. the second stage $E_2^{p,u}=H^p_{\bQb_W}\left(H^u_{\bQb_0}(\cH^{\bullet,\bullet})\right)$. Once the first two stages of the spectral sequence are determined, we are able to compute the cohomology of $\bQb$; higher derivatives are then determined by standard zig-zag arguments in terms of the two differentials $\bQb_0$ and $\bQb_W$.
The geometric structure depends on the twisted sector, and rather than presenting a universal framework at the price of opaque notation, we will next consider the relevant geometries in three separate situations:
1. The (R,R) sectors: $k \in 2\Z$. In this case since $E_{|k\ra} = 0$ we can restrict to zero modes for all the fields, which leads to a very transparent structure.
2. The untwisted (NS,R) sector: $k=1$. This and its CPT conjugate sector $k=2N-1$ are the only states with $E_{|k\ra} = -1$. In this case the geometry is simply $\bY$, and the spectrum involves an interplay between non-trivial base and fiber oscillators.
3. (NS,R) sectors with odd $k$ and $E_{|k\ra} >-1$. In this case the organizing geometry is a sub-bundle of $\bY \to B$, and while the choice of sub-bundle is $k$-dependent, the spectrum simplifies since base oscillators have $h=1$ and do not contribute to the massless states.
We consider these possibilities in turn in the next section.
Twisted sector geometry {#s:twisted}
=======================
To describe the geometric framework for the various twisted sectors we find it useful to distinguish base and fiber fields, with the latter differentiated according to the values of $\tau_\alpha$. More precisely, we split the coordinates $y^\alpha \to (y^{\alpha'},\phi^A)$, such that $\tau_{\alpha'} <1$ and $\tau_{A} \ge 1$. The $y^{\alpha'}$ decompose further into base and fiber directions: $y^{\alpha'} = (y^I,\phi^{i'})$, where $\tau_{i'} < 1$ (since $\nu_I =0$ for all the base fields $\tau_I< 1$ in all sectors). We decompose the bundle $X$ accordingly as $X = X_k \oplus \oplus_A L_A$ and define $$\begin{aligned}
\label{eq:bYk}
\bYk \equiv \text{tot} ( X_k \overset{\pi_k}{\longrightarrow} B).\end{aligned}$$ The utility of this is that the “light” fields, labeled by $\alpha'$, including the corresponding fermions, are organized by $\bYk$, while the remaining “heavy” fields, labeled by $A$, are organized by the pull-backs $\pi^\ast_k(L_A)$. The right-moving sector is considerably simpler: we restrict to zero modes, and as we described at the end of section \[ss:bcbg\], the only relevant ones are the zero modes $\yb^{\Ib}$ and their $\bQb_0$ superpartners $\etab^{\Ib}$. We now describe how this works in detail in various twisted sectors.
(R,R) sectors
-------------
In this case $E_{|k\ra} = 0$ as a consequence of the left-moving supersymmetry, and to describe the massless states we can restrict to zero modes for all the fields. A look back at the modes in (\[eq:modes\]) and (\[eq:nus\]) shows that the only fields with zero modes are the light fields. Among these the $\rho_{\alpha'}$ also have no zero modes, while the $\chi_{\alpha'}$ zero modes annihilate the vacuum state. Hence the most general state in the truncated Hilbert space is a linear combination of $$\begin{aligned}
|\Psi^s_u \ra = \Psi(y',\xb)^{\alpha'_1 \cdots \alpha'_s}_{\Ib_1 \cdots \Ib_{u}} \chib_{\alpha'_1} \chib_{\alpha'_2}\cdots \chib_{\alpha'_s} \etab^{\Ib_1} \cdots \etab^{\Ib_u} |k\ra~.\end{aligned}$$ The fermions $\chib_{\alpha'}$ and $\etab^{\Ib}$ transform respectively as sections of $T^\ast_{\bYk}$ and $\pi^\ast_k(\Tb_{B})$,[^19] while $|k\ra$ is a section of $L_{|k\ra} = \pi^\ast_k(\otimes_A L_A^\ast)$. Hence to be a well-defined state the wavefunction $\Psi^s_u$ must be a $(0,u)$ horizontal form valued in the holomorphic bundle $\cE^s = \wedge^s T_{\bYk} \otimes L_{|k\ra}^\ast$.
We can decompose the $\Psi$ according to their eigenvalues under the Lie derivative with respect to the restriction of the holomorphic Killing vector $V$ to $\bYk$ : $\cL_{V} \Psi = q_\Psi \Psi$.[^20] The resulting $|\Psi\ra$ has well-defined $\GUL\times\GUR$ charges: $$\begin{aligned}
\bq = \bq_{|k\ra} + q_{\Psi} + s~,\qquad
\bqb = \bqb_{|k\ra} + q_{\Psi} + u~.\end{aligned}$$ $\bQb_0$ acts by sending $\Psi^s_u \to -\pb \Psi^s_{u+1}$, and we can use the fine grading described in appendix \[app:sheaf\] to reduce $\bQb_0$ cohomology to computing the finite-dimensional vector spaces $H^\bullet_{\br}(\bYk,\cE^\bullet)$.
The result is still infinite-dimensional, since these cohomology groups will be non-zero for an infinite set of grades $\br$. This is a general feature of any sector with bosonic fiber zero modes. Fortunately, the action of $\bQb_W$, which takes the form $$\begin{aligned}
\bQb_W = W_{\alpha'} (y') \chi^{\alpha'}~,\end{aligned}$$ restricts the spectrum further. When $W$ is non-singular we expect a finite-dimensional result, and indeed, this is easy to prove for LG models.[^21] It would be useful to give a more general proof for hybrids. At any rate, we see from (\[eq:bQbWop\]) that the $\bQb_W$ action on our state is simply $$\begin{aligned}
\bQb_W : \Psi^s_u \mapsto (s W_{\alpha'_1} \Psi^{\alpha'_1 \alpha'_2 \cdots \alpha'_s})^{s-1}_u~.\end{aligned}$$ The spacetime interpretation of these states is either as $\Le_6$ gauginos ($\bq = \pm 3/2$) or the $\rep{16}_{\pm 1/2}$ components of $\rep{27}$s and $\brep{27}$s.
### $\bY = B$ {#by-b .unnumbered}
As a simple consistency check we can see that we correctly reproduce the expected spectrum from the unique $k=0$ (R,R) sector when $\bY = B$ a compact Calabi-Yau 3-fold. The non-vanshing $\bQb_0$-cohomology classes, given with multiplicities and $(\bq,\bqb)$ charges are $$\begin{aligned}
|0\ra_{-3/2,-3/2}^{\oplus 1} &&
|\Psi^3_0\ra_{3/2,-3/2}^{\oplus 1}&&
|\Psi^0_3\ra_{-3/2,3/2}^{\oplus 1}&&
|\Psi^3_3\ra_{3/2,3/2}^{\oplus 1}~, \nonumber\\[1.5mm]
|\Psi^1_1\ra_{-1/2,-1/2}^{\oplus h^1(T)}&&
|\Psi^2_2\ra_{1/2,1/2}^{\oplus h^1(T)}&&
|\Psi^2_1\ra_{1/2,-1/2}^{\oplus h^1(T^\ast)}&&
|\Psi^1_2\ra_{-1/2,1/2}^{\oplus h^1(T^\ast)}~.\end{aligned}$$ Comparing to (\[eq:E6decomposition\]), we see that the first line corresponds to the gauginos, while the second line corresponds to the $\brep{16}_{-1/2}$ and $\rep{16}_{1/2}$ components of $h^1(T)$ chiral $\brep{27}$ and $h^1(T^\ast)$ chiral $\rep{27}$ multiplets.
The $k=1$ sector {#ss:k1}
----------------
The $k=1$ sector is untwisted with respect to the LG orbifold action. It has the richest geometric structure and a number of universal features generalizing those observed for the LGO case [@Aspinwall:2010ve]. Since $\tau_\alpha = 1/2$ for all the fields, the geometry is simply $\bY_{\!\! 1} = \bY$, while the vacuum bundle $L_{|k\ra} = K_{\bY}$ is trivial. We also have $$\begin{aligned}
\bq_{|1\ra} = 0~,\qquad \bqb_{|1\ra} = -3/2~,\qquad E_{|1\ra} = -1~.\end{aligned}$$ Since $E_{|1\ra} = -1$ massless states may include non-zero modes of $\p y^I$ and $\rho_I$.
We now want to describe the operators that create zero-energy states from $|1\ra$. It turns out that hybrid theories for which some $q_i =1/2$ have additional zero-energy states that are not found in more generic theories. We will first describe the zero energy states present generically and then turn to the special states available due to fields with $q_i =1/2$.
### Generic $k=1$ operators {#generic-k1-operators .unnumbered}
Ignoring multiplets with $q_i = 1/2$, we list the operators that can carry weight $h\le1$:[^22] $$\begin{aligned}
\label{eq:k1ops}
\cO^{1,s} &= \Psi^{1s\alpha_1\cdots\alpha_s}(y)\chib_{\alpha_1}\cdots\chib_{\alpha_s}~,\qquad
\cO^2 =\Psi^2_{\alpha}(y) \chi^\alpha~,\qquad
\cO^3 = \Psi^3_{\alpha\beta}(y) \chi^\alpha\chi^\beta~,\qquad
\nonumber\\[1.5mm]
\cO^4 &= \Psi^4_{\alpha}(y) \p y^\alpha~,\qquad\qquad\qquad\quad~~
\cO^5 = ~:\Psi^{5\alpha}_{\beta}(y) \chib_\alpha \chi^\beta :~,\nonumber\\[1.5mm]
\cO^6 &=~ : \Psi^{6\alpha}(y) \rho_\alpha + \Psi^{6\alpha}_{~,\beta}(y) \chib_\alpha \chi^\beta :~.\end{aligned}$$ The index $s$ in $\cO^{1s}$ can take values $s=0,1,2,3$. In each case we only indicated the dependence on the left-moving fields; each $\Psi$ also depends on the $\yb$ and $\etab$ zero modes: $$\begin{aligned}
\label{eq:wavefuexp}
\Psi^t = \sum_{u=0}^d (\Psi^t_{u})_{\Ib_1 \cdots \Ib_u} \etab^{\Ib_1} \cdots \etab^{\Ib_u}~,\end{aligned}$$ and plugging in this expansion, we obtain a set of operators $\cO^t_u(z)$. We also used the normal ordering that follows from (\[eq:OPEs\]) to subtract off the $y\rho$ and $\chib\chi$ short-distance singularities. Since our free fields are only defined on open sets covering the target space $\bY$, just as in the $k$ even case the wavefunctions $\Psi^t_0$ have to transform as sections of appropriate holomorphic bundles $\cE^t$ over $\bY$. For instance, the fermi bilinear term appearing in $\cO^8$ is chosen to account for the unusual transition function of $\rho_\alpha$ in (\[eq:patchfields\]). That is, using (\[eq:patchfields\]), we find that for two patches $U_a$ and $U_b$ with $U_{ab} \neq \emptyset$ $\cO^6_{b} = \cO^6_{a}$ (i.e. $\cO^6$ is well-defined) iff $\Psi^6_0$ transforms as a section of $T_{\bY}$. Similarly, the remaining wavefunctions must transform in the expected way, e.g. $\Psi^{1s}_0$ as a section of $\wedge^s T_{\bY}$ and $\Psi^2_0$ as a section of $T^\ast_{\bY}$. The wavefunctions for $\Psi^t_{u>0}$ transform as (0,u) horizontal forms valued in $\cE^t$, and taking $\bQb_0$ cohomology means the $\Psi^t_u$ taken at a fine grade $\br$ define classes in $H^\bullet_{\br}(\bY,\cE^\bullet)$. As in the $k$ even case we need to consider all $\br$ that contain states with $h=1$ and non-trivial $\bQb_W$ classes. It is useful to introduce the following notation for the relevant holomorphic bundles $\cE^t$: $$\begin{aligned}
\label{eq:Bdef}
B_{s,t,q} \equiv \wedge^s T_{\bY} \otimes \wedge^t T^\ast_{\bY} \otimes \Sym^q (T_{\bY})~.\end{aligned}$$
If we grade the wavefunctions by the eigenvalue of the Lie derivative with respect to the symmetry vector $V$, i.e. $\cL_{V} \Psi^t_u = q \Psi$, then we obtain the following weights, charges and $\bQb_W$ action for these operators: $\bqb_{\cO}= q+ u$, and $$\begin{aligned}
\label{eq:eq:k1ops2}
\xymatrix@R=1.5mm@C=1.5mm{
\text{op.} & \cO^{1,s}_u &\cO^2_u &\cO^3_u &\cO^4_u &\cO^5_u &\cO^6_u\\
\bq_{\cO} &q+s &q-1 &q-2 &q &q &q \\
h_{\cO} &\frac{q+s}{2}
&\frac{q+1}{2}
&\frac{q+2}{2}
&\frac{q+2}{2}
&\frac{q+2}{2}
&\frac{q+2}{2}\\
\bQb_W\cdot
&sW_{\alpha_1}\Psi^{1s\alpha_1\cdots\alpha_s}\chib_{\alpha_2}\cdots\chib_{\alpha_s}
&0 &0 &0
&\Psi^{5\beta}_{u\gamma}W_\beta\chi^\gamma
&\chi^\alpha\p_\alpha(\Psi^{6\beta}W_\beta)
}\end{aligned}$$ Note that for $s>0$ the $\cO^{1,s}$ can carry negative eigenvalues under $\cL_V$, but it is not hard to show that they are bounded by $q >-s/2$. Using these operators we create states in the usual fashion: $|\cO^{t}_u\ra \equiv \lim_{z\to 0} \cO^{t}_u(z) |1\ra$. They carry energy $E=h_{\cO}-1$ and charges $\bqb = \bqb_{\cO}-3/2$ and $\bq = \bq_{\cO}$.
### Currents {#currents .unnumbered}
The $h_{\cO}=1$ $\bqb_{\cO}=0$ operators in $\bQb$ cohomology are conserved left-moving currents, and in a generic $k=1$ sector the corresponding states arise in the bottom row of the spectral sequence: $$\begin{aligned}
\label{eq:currents}
\xymatrix{|\cO^5_0\ra \oplus |\cO^6_0\ra \ar[r]^-{\bQb_W} & |\cO^2_0\ra}~, \end{aligned}$$ where $$\begin{aligned}
\Psi^5 &\in \oplus_{\br} H^0_{\br} (\bY,B_{1,1,0})~,&
\Psi^6 &\in\oplus_{\br} H^0_{\br}(\bY,B_{0,0,1})~,&
\Psi^2\in \oplus_{\br} H^0_{\br}(\bY,B_{1,0,0})~.\end{aligned}$$ Before taking cohomology, there are a number of states here, including, for example, holomorphic vector fields in $H^0(B,T_B)$ that lift to $\bY$ or various enhanced R-symmetries of the $W=0$ theory. Most of these states are lifted by the superpotential couplings. In fact, for a suitably generic $W$ the only current that survives is $J_L$, which corresponds to $\Psi^{5} = \iden$ and $\Psi^{6} = - V$; the resulting state is $\bQb_W$ closed as a result of $\cL_V W = W$. This gaugino corresponds to the linearly realized $\Lu(1)_L\subset \Le_6$. For less generic $W$ additional currents may appear, and of course they are accompanied by additional chiral $\bqb=-1/2$ states $|\cO^2_0\ra$ in the cokernel of $\bQb_W$. In spacetime each current corresponds to a gauge boson, and the appearance of extra currents reflects the spacetime Higgs mechanism.
### $\bY = B$ {#by-b-1 .unnumbered}
As in the $k=0$ case, we examine the case of trivial fiber and a CY target space. Taking $\bQb_0$ cohomology on the space of operators in (\[eq:k1ops\]), we find the following massless states with $\bqb < 0$ (for brevity we omit their conjugates with $\bqb>0$) $$\begin{aligned}
\cO^{1,0},\cO^5_0
& \to |1\ra_{0,-3/2}^{\oplus 1} \oplus \chib_\alpha \chi^\alpha|1\ra_{0,-3/2}^{\oplus 1}~
&&
\rep{45}_0\oplus\rep{1}_0
\nonumber\\[1mm]
%%%%%%%
\cO^{1,1},\cO^2
& \to |\cO^{1,1}_1\ra_{1,-1/2}^{\oplus h^1(T)}\oplus|\cO^2_1\ra_{-1,-1/2}^{\oplus h^1(T^\ast)}~
&&\rep{10}^{\oplus h^1(T)}_1\oplus\rep{10}_{-1}^{\oplus h^1(T^\ast)}
\nonumber\\[1mm]
%%%%%%%
\cO^{1,2},\cO^3
& \to |\cO^{1,2}_1\ra_{2,-1/2}^{\oplus h^1(\wedge^2T)} \oplus|\cO^3_1\ra_{-2,-1/2}^{\oplus h^1(\wedge^2T^\ast)}~
&&\rep{1}_{2}^{\oplus h^1(T^\ast)}\oplus\rep{1}_{-2}^{\oplus h^1(T)} \nonumber\\[1mm]
\cO^4,\cO^5_1,\cO^6 & \to |\cO^4_1\ra_{0,-1/2}^{\oplus h^1(T^\ast)} \oplus|\cO^5_1\ra_{0,-1/2}^{\oplus h^1(\End T)}
\oplus|\cO^6_1\ra_{0,-1/2}^{\oplus h^1(T)}~&&\rep{1}_0^{\oplus \{h^1(T)+h^1(T^\ast)+h^1(\End T)\}}\nonumber\\[1mm]\end{aligned}$$
It is not hard to extend this analysis to a more general (0,2) CY NLSM with $\su(n)$ bundle $\cV \neq T_B$. In particular, this offers certainly the most direct world-sheet perspective, in the spirit of [@Distler:1987ee], on marginal gauge-neutral deformations and agrees with spacetime [@Donagi:2009ra; @Anderson:2011ty] and world-sheet [@Melnikov:2011ez] results on marginal deformations in the large radius limit. This may be found in appendix \[app:02NLSM\].
### A hybrid example {#a-hybrid-example .unnumbered}
We will now illustrate how to set up the spectrum computation in a simple but non-trivial hybrid. We consider the “octic model”[^23] with $B = \P^1$ and $X = \cO(-2)\oplus\cO^{\oplus3}$. The quantum numbers of the ground states of the twisted sectors, as well as charges of the fiber fields are given in table \[tab:octicnums\].
[cc]{}
$k$ $E_{|k\ra}$ $\bq_{|k\ra}$ $\bqb_{|k\ra}$ $\ell_k$ $\nu_{i}$ $\nut_{i}$ $\nu_I$ $\nut_I$
----- -------------- --------------- ---------------- ---------- ------------- -------------- --------- -------------- -- -- --
$0$ $0$ $-\ff{3}{2}$ $-\ff{3}{2}$ $0$ $0$ $0$ $0$ $0$
$1$ $-1$ $0$ $-\ff{3}{2}$ $0$ $\ff{1}{8}$ $-\ff{3}{8}$ $0$ $-\ff{1}{2}$
$2$ $0$ $\ff{1}{2}$ $-\ff{3}{2}$ $-2$ $\ff{1}{4}$ $-\ff{3}{4}$ $0$ $0$
$3$ $-\ff{1}{2}$ $-1$ $-\half$ $0$ $\ff{3}{8}$ $-\ff{1}{8}$ $0$ $-\ff{1}{2}$
$4$ $0$ $-\half$ $-\half$ $-2$ $\ff{1}{2}$ $-\ff{1}{2}$ $0$ $0$
: Quantum numbers for the octic model.[]{data-label="tab:octicnums"}
&
$\phi^i$ $\rho_i$ $\chi^i$ $\chib_i$
-------- ------------- -------------- -------------- --------------
$\bq$ $\ff{1}{4}$ $-\ff{1}{4}$ $-\ff{3}{4}$ $\ff{3}{4}$
$\bqb$ $\ff{1}{4}$ $-\ff{1}{4}$ $\ff{1}{4}$ $-\ff{1}{4}$
: Quantum numbers for the octic model.[]{data-label="tab:octicnums"}
In this example as well as those that follow $\Pic B = H^{2}(B,\Z)$, and the vacuum bundle $L_{|k\ra}$ is determined by a class in $H^2(B,\Z)$. We label the class of the dual bundle $L_{|k\ra}^\ast$ by $\ell_k \in H^2(B,\Z)$. In this example $\ell_k$ is simply the degree of the line bundle over $\P^1$.
Let us consider as an example the states at $E=0$ and $\bq=2$ in the $k=1$ sector. We see from that these states belong to $\rep{1}_{2}$ of $\so(10)$. Energy and charge considerations show that the relevant operators from (\[eq:k1ops\]) are $\cO^{1,s}$, and the states fit in a double complex $$\begin{aligned}
\begin{xy}
\xymatrix@C=20mm@R=7mm{
\Psi_{[2]}^{\alpha\beta}\chib_\alpha\chib_\beta|1\ra & \Psi_{[5]}^\alpha\chib_\alpha |1\ra &0\\
\Psi^{\alpha\beta}_{[2]} \chib_\alpha\chib_\beta|1\ra
&
\Psi^{\alpha}_{[5]}\chib_\alpha |1\ra
&
{\begin{matrix}
\Psi_{[8]}|1\ra
\end{matrix}}
}
\save="x"!LD+<-3mm,0pt>;"x"!RD+<0pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<75mm,-3mm>;"x"!LU+<75mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-82mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-42mm,-4mm>*{-\ff{1}{2}}\restore
\save!RD+<-7mm,-4mm>*{\ff{1}{2}}\restore
\save!RD+<3mm,-4mm>*{p}\restore
\save!CL+<72mm,16mm>*{U}\restore
\end{xy}\end{aligned}$$ The wavefunctions satisfy $\cL_{V} \Psi^{\alpha\beta} = 0$ and $\cL_{V} \Psi^\alpha = \Psi^\alpha$; in practice this means that each $\Psi_{[d]}(y,\yb,\etab)$ is a quasi-homogeneous polynomial of degree $d$ in the fiber bosons $\phi^i$ if both indices are vertical, while it is of degree $d-1$ is one of the indices is horizontal. To limit clutter in the notation we suppressed the $\etab$s; their number is indicated by the $U$ grading. Recall that the horizontal grading is by $p=\bq-u$.
Taking $\bQb_0$ cohomology at the relevant $\bq,\bqb, E$ eigenvalues indicated by the subscripts, we obtain $$\label{octicTY}
\begin{xy}
\xymatrix@C=10mm@R=10mm{
\left[ H^1(\bY, B_{2,0,0}) \right]_{2,-1/2,0} & \left[ H^1(\bY, B_{1,0,0}) \right]_{2,1/2,0} &0\\
\left[ H^0(\bY, B_{2,0,0} ) \right]_{2,-3/2,0}
& \left[ H^0(\bY, B_{1,0,0}) \right]_{2,-1/2,0}
&
\left[ H^0(\bY, B_{0,0,0}) \right]_{2,1/2,0}
}
\save="x"!LD+<-3mm,0pt>;"x"!RD+<0pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<98mm,-3mm>;"x"!LU+<98mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-120mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-70mm,-4mm>*{-\ff{1}{2}}\restore
\save!RD+<-20mm,-4mm>*{\ff{1}{2}}\restore
\save!RD+<0mm,-4mm>*{p}\restore
\save!CL+<95mm,16mm>*{U}\restore
\end{xy}$$ To illustrate the counting, we concentrate on the dimension of $$\begin{aligned}
[H^0(\bY,B_{1,0,0})]_{2,-1/2,0}= [H^0(\bY,T_{\bY})]_{2,-1/2,0} = \bigoplus_{\sum_i r_i = 4} H^0_{\br}(\bY,T_{\bY})~.\end{aligned}$$ The computation is simple since $\bY \simeq \bY' \times \C^3$, where $\bY'$ is the total space of $\cO(-2) \to \P^1$. In this case, as we show in appendix (\[app:sheaf\]), the non-trivial graded cohomology groups are $$\begin{aligned}
H^0_{r_1}(\bY',\cO_{\bY'}) = \C^{2r_1+1},\qquad
H^0_{r_1} (\bY',T_{\bY'}) = \C^{4r_1+4}~.\end{aligned}$$ Decomposing $(T_{\bY})_{\br}$ according to (\[eq:TSES\]) we find two types of contributions to $H^0_{\br}(\bY,T_{\bY})$, those with $r_i \ge 0$, and those with $r_i=-1$ for $i=2,3,4$: $$\begin{aligned}
[H^0(\bY,T_{\bY})]_{2,-1/2,0} &= \bigoplus_{r_1=0}^4 \left[H^0_{r_1}(\bY',T_{\bY'}) \oplus H^0_{r_1} (\bY',\cO_{\bY'})^{\oplus 3}\right]\otimes \C^{\binom{6-r_1}{4-r_1}} \nonumber\\
&\qquad\oplus\left[ \bigoplus_{r_1=0}^5 H^0_{r_1}(\bY',\cO_{\bY'}) \otimes \C^{6-r_1} \right]^{\oplus 3}~ \nonumber\\
& = \C^{595}\oplus \C^{273} = \C^{868}.\end{aligned}$$ The factors of $\binom{6-r_1}{4-r_1}$ and $(6-r_1)$ arise from counting monomials, respectively, of degree $4-r_1$ in three variables and $5-r_1$ in two variables.
Computing the remaining cohomology groups in a similar fashion we obtain the $E_1$ stage of the spectral sequence $$\label{octicTYE1}
\begin{xy}
\xymatrix@C=10mm@R=10mm{
\C^{18}\ar[r]^{\bQb_W} & \C^{21} &0\\
\C^{126} \ar[r]^{\bQb_W}
& \C^{868} \ar[r]^{\bQb_W}
&
\C^{825}
}
\save="x"!LD+<-3mm,0pt>;"x"!RD+<10pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<37mm,-3mm>;"x"!LU+<37mm,5mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-45mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-26mm,-4mm>*{-\ff{1}{2}}\restore
\save!RD+<-4mm,-4mm>*{\ff{1}{2}}\restore
\save!RD+<3mm,-4mm>*{p}\restore
\save!CL+<33mm,14mm>*{U}\restore
\end{xy}$$ Finally, we turn to the computation of the $\bQb_W$ cohomology for these states and for simplicity consider the Fermat superpotential $$\begin{aligned}
W = S_{[8]}(\phi^1)^4+(\phi^2)^4+(\phi^3)^4+(\phi^4)^4~,\end{aligned}$$ where $S_{[8]}\in H^0(\P^1, \cO(8))$. From we see that for the states appearing at $p=-\ff{3}{2}$ $$\begin{aligned}
\label{eq:QWp32}
\bQb_W \left( \Psi^{\alpha\beta}_{[2]u} \chib_\alpha\chib_\beta\right) |1\ra = 2 \Psi^{\alpha\beta}_{[2]} W_\beta \chib_\alpha|1\ra~,\end{aligned}$$ and the derivatives of the superpotential that appear are ($a=2,3,4$) $$\begin{aligned}
W_a &= 4(\phi^a)^3~, &W_1&=4S_{[8]}(\phi^1)^3~, & W_I = \p_I S_{[8]} (\phi^1)^4~.\end{aligned}$$ The map has vanishing kernel, while the $\bQb_W$ action on the $p=-\ff{1}{2}$ states is $$\begin{aligned}
\label{eq:QWp12}
\bQb_W \left( \Psi^\alpha_{[5]} \chib_\alpha \right)|1\ra = \Psi^\alpha_{[5]} W_\alpha ~.\end{aligned}$$ Setting this to zero implies $\Psi^\alpha_{[5]} = \Phi^{\alpha\beta}_{[2]}W_{\beta}$ for some $ \Phi^{\alpha\beta}_{[2]}$ anti-symmetric in its indices. Hence the cohomology in the $(p,u)=(-\ff{1}{2},0)$ position is trivial, and the spectral sequence degenerates at $$\label{octicTYres}
\begin{xy}
\xymatrix@C=10mm@R=10mm{
0 & \C^3 &0\\
0
& 0
&
\C^{83}
}
\save="x"!LD+<-3mm,0pt>;"x"!RD+<10pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<25mm,-3mm>;"x"!LU+<25mm,5mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-38mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-22mm,-4mm>*{-\ff{1}{2}}\restore
\save!RD+<-4mm,-4mm>*{\ff{1}{2}}\restore
\save!RD+<3mm,-4mm>*{p}\restore
\save!CL+<22mm,14mm>*{U}\restore
\end{xy}$$ Here we count 86 anti-chiral states in the $\mathbf{1}_2$. These correspond to the $83$ polynomial and the $3$ non-polynomial deformations of complex structure of the octic hypersurface now determined from the hybrid’s point of view.
### Extra states in $k=1$ {#extra-states-in-k1 .unnumbered}
Multiplets with $q_i=\half$ can potentially give rise to additional massless states. In a LGO theory these genuinely correspond to massive multiplets that can be integrated out without affecting the IR physics. In general this is not so in the hybrid theory: if a $q_i = \half$ field is non-trivially fibered then its mass vanishes on the discriminant of $W$ in $B$, and the field cannot be integrated out globally over $B$. This leads to a rich structure entirely absent from LGO theories.
To describe the additional operators with $h= 1$ we sadly need a little more notation. Just in this section we use the indices $i', j'$, etc. to denote the multiplets with $q_{i'} = \half$; the $\alpha,\beta,\ldots$ continue to denote all the fields, while $I,J,\ldots$ denote the fields of the base geometry. Let $X_{\half} \equiv \oplus_{i'} L_{i'}$ and $\cA$ be a holomorphic (in fact diagonal) connection on the bundle $X_{\half}\to B$. The new operators are then $$\begin{aligned}
\label{eq:massiveops1}
\cO^{7} &= \Psi^{7i'j'k'm'}(y^I) \chib_{i'}\chib_{j'}\chib_{k'}\chib_{m'}~,\qquad
\cO^{8} = ~:\Psi^{8 i'j'}_{I}(y^I) \chib_{i'} \chib_{j'} \chi^I :~,\nonumber\\[1.5mm]
\cO^{9} &=~ :\Psi^{9 i'j'} (y^I) (\rho_{i'} +\chi^I \cA^{k'}_{Ii'} \chib_{k'}) \chib_{j'}:~.\end{aligned}$$ The wavefunctions are (0,u) forms valued in the following bundles: $$\begin{aligned}
\Psi^{7} &:~ \wedge^4 X_{\half}~,&
\Psi^{8} &:~ \wedge^2 X_{\half} \otimes T_{B}^\ast~,&
\Psi^{9} &:~ X_{\half}\otimes X_{\half}~.\end{aligned}$$ These operators have weight $h=1$ and charges $$\begin{aligned}
\label{eq:massiveops2}
\xymatrix@R=1.5mm@C=2.5mm{
~ &
\cO^7_u&
\cO^8_u&
\cO^9_u
\\
\bq&
2&
0&
0
\\
\bqb&
u-2&
u-1&
u-1
}\end{aligned}$$ The action of $\bQb_0$ on $\cO^7$ is simply to send $\Psi^7_u\to (-\pb \Psi^7)_{u+1}$. Since we used the holomorphic connection $\cA$ in $\cO^{9}$ to build a well-defined operator, the $\bQb_0$ action on $\cO^8_{u}+\cO^9_{u}$ is a bit more involved: $$\begin{aligned}
\label{eq:massiveobs1}
\bQb_0 \cdot (\cO^8_u+\cO^9_u) & = - (\pb \Psi^{9i'j'})_{\Ib_0\cdots\Ib_u} \etab^{\Ib_0}\cdots\etab^{\Ib_u} (\rho_{i'} +\chi^I \cA^{k'}_{Ii'} \chib_{k'}) \chib_{j'} \nonumber\\[1.5mm]
&\qquad+ \left[ \obs(\Psi^{9})^{k'j'}_{I} - \pb \Psi^{8k'j'}_I\right]_{\Ib_0\cdots\Ib_u}\etab^{\Ib_0}\cdots\etab^{\Ib_u} \chib_{k'}\chib_{j'}\chi^I~,\end{aligned}$$ where the linear map $$\begin{aligned}
\label{eq:massiveobs2}
\obs : \Omega^{0,u}(X_{\half}\otimes X_{\half}) \to \Omega^{0,u+1}(\wedge^2 X_{\half}\otimes T^\ast_B)\end{aligned}$$ is given by contracting $\Psi^9$ with the curvature $\cF = \pb \cA$ : $$\begin{aligned}
\obs(\Psi^{9})^{k'j'}_{I~\Ib_0\cdots\Ib_u} d\yb^{\Ib_0} \cdots d\yb^{\Ib_u} \equiv
\ff{1}{2}\left( \cF_{I\Ib_0~ i'}^{~~~k'}\Psi^{9i'j'}_{\Ib_1\cdots \Ib_u} - \cF_{I\Ib_0~ i'}^{~~~j'}\Psi^{9i'k'}_{\Ib_1\cdots \Ib_u} \right) d\yb^{\Ib_0} \cdots d\yb^{\Ib_u}~.\end{aligned}$$ It is easy to see that $\obs(\Psi^9)$ is $\pb$-closed when $\Psi^9$ is $\pb$-closed, so that $\cO^8_u+\cO^9_u$ is $\bQb_0$-closed iff $\pb \Psi^9 = 0$ and $\obs(\Psi^9)$ corresponds to the trivial class in $H^{u+1}(B, \wedge^2 X_{\half}\otimes T^\ast_B)$. We will meet examples of such possible “obstruction classes” below, but for now we simply note that $\obs$ vanishes in a number of important cases that often arise in particular examples. For instance, $\obs(\Psi^9_d)$ is clearly zero, and $\obs = 0$ for any $\Psi^{9} \in H^{\bullet}(B,L_{j'}\otimes L_{j'})$. A little less trivially, we can also show that $\obs$ vanishes for any $\Psi^{9} \in H^{\bullet}(B, \wedge^2 X_{\half}).$
The $\bQb_W$ action can also be determined;[^24] the results are: $$\begin{aligned}
\bQb_W \cdot \cO^7 &= 4 W_{i'} \Psi^{7i'j'k'm'} \chib_{j'}\chib_{k'}\chib_{m'}~,\qquad
\bQb_W \cdot \cO^8 = 2 W_{i'} \Psi^{8i'j'}_{I}\chib_{j'} \chi^I~,\nonumber\\
\bQb_W \cdot \cO^9 &= \Psi^{9i'j'}\left[(\rho_{i'} -\cA^{i'}_{I}\chib_{i'} \chi^I) W_{j'} +(\chi^\alpha W_{i'\alpha} -\chi^I\cA^{k'}_{Ii'} W_{k'} )\chib_{j'}\right]~.\end{aligned}$$
$k>1$ (NS,R) sectors
--------------------
Finally, we turn to (NS,R) sectors with $1<k<2N-1$. These sectors have, in general, two complications relative to the $k=1$ sector: in general $\bYk\neq \bY$, and $|k\ra$ may transform as a section of a nontrivial bundle over the base $B$.
### The vacuum {#the-vacuum .unnumbered}
Recalling the discussion above (\[eq:bYk\]), we split the coordinates as $y^\alpha \to (y^I,\phi^{i'},\phi^A)$. The quantum numbers of the vacuum are then write the vacuum energy as $$\begin{aligned}
E_{|k\ra}
&= -1 + \half\left[\sum_{i'} (\nu_{i'}-\frac{q_{i'}}{2}) + \sum_A (1-{\frac{q}{2}}-\nu_A)\right]~,\nonumber\\
\bq_{|k\ra}
&= \sum_{i'}(\frac{q_{i'}}{2}-\nu_{i'}) + \sum_A(1-\frac{q_A}{2}-\nu_A)~,\nonumber\\
\bqb_{|k\ra}
&= \sum_{i'}(\frac{q_{i'}}{2}-\half-\nu_{i'}) + \sum_A(\nu_A-\frac{q_A}{2}+\half) - \frac{d}{2}~,\end{aligned}$$ where $d$ is the dimension of the base $B$. Note that in the twisted sectors $1<k<2N-1$ we have $E_{|k\ra} > -1$. The vacuum bundle (\[eq:bundlegroundst\]) is given by $$\begin{aligned}
\label{eq:vactrans}
L_{|k\ra} = \otimes_A L_A^*\ .\end{aligned}$$
### Modes and transition functions {#modes-and-transition-functions .unnumbered}
Because we have $E_k>-1$ we can restrict attention to the subspace of the Hilbert space generated by the lowest modes of the left-moving fields. That is, we truncate (\[eq:modes\]) to $$\begin{aligned}
\label{eq:moding}
y^\alpha(z) &= z^{\nu_\alpha-q_\alpha/2}(y^\alpha + z^{-1}\rho^{\dagger\alpha})~, &
\rho_\alpha(z) &= z^{q_\alpha/2-\nu_\alpha}(\rho_\alpha + z^{-1}y^\dagger_\alpha)~,\nonumber\\
\chi^\alpha(z) &= z^{\nut_\alpha-q_\alpha/2-\half}(\chi^\alpha + z^{-1}\chib^{\dagger\alpha})~, &
\chib_\alpha(z) &= z^{q_\alpha/2+\half-\nut_\alpha}(\chib_\alpha + z^{-1}\chi^\dagger_\alpha)\ ,\end{aligned}$$ where in our restricted Hilbert space $\rho_I = 0$.
The transition functions for these oscillators follow by expanding (\[eq:patchfields\]). These show that $y^{\alpha'}$ are coordinates on $\bYk$, while $\chi^{\alpha'}$ ($\chib_{\alpha'}$) take values in $T_{\bYk}$ ($T^*_{\bYk}$). On the other hand, $\phi^A$ and $\lambda^A$ take values in $\hat Z_k = \pi_k^*(\oplus L_A)$ and $\rho_A$ and $\lambdab_A$ in $\hat Z_k^*$. As is the case for $k=1$, $\rho_{i'}$ is not a covariant operator due to the fermion bilinear term.
### Conserved charges {#conserved-charges .unnumbered}
Inserting (\[eq:moding\]) into our expressions for the conserved charges (\[eq:leftcharges\]) we find in our Hilbert space $$\begin{aligned}
L_0 &= \sum_\alpha \left[-\nu_\alpha\phi^\alpha\phi^\dagger_\alpha +
(1-\nu_\alpha)\rho_\alpha\rho^{\dagger\alpha} +
(1+\nut_\alpha)\chi^\alpha\chi^\dagger_\alpha -
\nut_\alpha \chib_\alpha\chib^{\dagger\alpha}\right]~, \nonumber\\
J_0 &= \sum_\alpha \left[(q_\alpha-1)(\chi^\alpha\chi^\dagger_\alpha - \chib_\alpha\chib^{\dagger\alpha}) -
q_\alpha\left(y^\alpha y^\dagger_\alpha +
\rho_\alpha\rho^{\dagger\alpha}\right)\right]~,\nonumber\\
\Jb_0 &= \sum_\alpha q_\alpha\left(-y^\alpha y^\dagger_\alpha - \rho_\alpha\rho^{\dagger\alpha} + \chi^\alpha\chi^\dagger_\alpha - \chib_\alpha\chib^{\dagger\alpha}\right)\ .\nonumber\end{aligned}$$
### States {#states .unnumbered}
We again list the operators that can carry weight $h< 1$, suppressing the right-moving $\etab^{\Ib}$ dependence. These can contain at most one operator with $h\ge\half$ and we organize them according to the nature of this operator as $$\begin{aligned}
\cO^{1,l,m} &= \Xi^{1lm}
{}^{\alpha',i'_2\cdots i'_l}_ {A_1\cdots A_m}
\chib_{\alpha'}\chib_{i'_2}\cdots\chib_{i'_l}\chi^{A_1}\cdots\chi^{A_m}\nonumber\\
\cO^{2,l,m} &= \Xi^{2lm}
{}^{i'_1\cdots i'_l}_ {\alpha',A_2\cdots A_m}
\chib_{i'_1}\cdots\chib_{i'_l}\chi^{\alpha'}\chi^{A_2}\cdots\chi^{A_m}\nonumber\\
\cO^{3,l,m} &= \Xi^{3lm}
{}^{B,i'_2\cdots i'_l}_ {A_1\cdots A_m}
\chib_B\chib_{i'_2}\cdots\chib_{i'_l}\chi^{A_1}\cdots\chi^{A_m}\\
\cO^{4,l,m} &= \Xi^{4lm}
{}^{i'_1\cdots i'_l}_ {B,A_1\cdots A_m}\phi^B
\chib_{i'_1}\cdots\chib_{i'_l}\chi^{A_1}\cdots\chi^{A_m}\nonumber\\
\cO^{5,l,m} &= \Xi^{5lm}
{}^{j',i'_1\cdots i'_l}_ {A_1\cdots A_m}
\left[\rho_{j'} - \cA_J{}^{k'}_{j'}\chi^J\chib_{k'}\right]
\chib_{i'_1}\cdots\chib_{i'_l}\chi^{A_1}\cdots\chi^{A_m}\nonumber\ .\end{aligned}$$ In constructing $\cO^5$ we have introduced a holomorphic (and diagonal) connection on $\oplus_{i'} L_{i'}$. Here the $\Xi^t$ include the dependence on $y^{\alpha'}$ and $\rho_A$, as well as on the right-moving zero modes of $\yb^{\Ib}$. We can make this more explicit by writing, for example, $$\begin{aligned}
\Xi^{1lm}{}^{\alpha',i'_2\cdots i'_l}_ {A_1\cdots A_m} =
\sum_{\vec t}
\Psi^{1lm}_{\vec t}(y){}^{\alpha',i'_2\cdots i'_l}_ {A_1\cdots A_m}
\prod_B \rho_B^{t_B+\sum_{a=1}^m \delta_{B,A_a}}\ ,\end{aligned}$$ in terms of a vector of integers $t_B\ge -1$ such that no negative powers of $\rho_B$ appear. $\cO^1$ will now create a well-defined state when acting on $|k\ra$ provided the wavefunction $\Psi^{1lm}_{\vec t}$ transforms as a section of a suitable bundle $\cE^{1lm}_{\vec
t}$ over $\bYk$ $$\begin{aligned}
\cE^{1lm}_{\vec t} =
T_{\bYk}\wedge\left(\wedge^{l-1}\pi_k^*(X_k)\right)
\otimes\left(\otimes_B(\pi_k^*L_B^{t_B+1})\right)\ .\end{aligned}$$ Note that this takes into account the transformation properties of the vacuum (\[eq:vactrans\]) and that the odd shift in the power of $\rho_B$ is now seen to be sensible. Incorporating the right-moving fermion zero modes, the wavefunction is in general a (0,u) horizontal form valued in this bundle. These can be fine graded as in \[app:sheaf\] by a vector of integers $\vec r = (r_{\alpha'})$.
Proceeding in an analogous way with the other operators we find that the wavefunctions take values in the following bundles, organized by $\vec t$ and the fine grading $\vec r$ $$\begin{aligned}
\cE^{1lm}_{\vec t,\vec r}(k) &=
\left[T_{\bY_k}\wedge\left(\wedge^{l-1}\pi_k^*(X_k)\right)
\otimes\left(\otimes_A(\pi_k^*L_A^{t_A+1})\right)\right]_{\vec
r}\nonumber\\
\cE^{2lm}_{\vec t,\vec r}(k) &=
\left[\left(\wedge^{l}\pi_k^*(X_k)\right)\otimes T^*_{\bY_k}
\otimes\left(\otimes_A(\pi_k^*L_A^{t_A+1})\right)\right]_{\vec r}\nonumber\\
\cE^{3lm}_{\vec t,\vec r}(k) &=
\oplus_B\left[\left(\wedge^{l-1}\pi_k^*(X_k)\right)
\otimes\left(\otimes_A(\pi_k^*L_A^{t_A+1})\right)\right]_{\vec r}\\
\cE^{4lm}_{\vec t,\vec r}(k) &=
\oplus_B\left[\left(\wedge^{l}\pi_k^*(X_k)\right)
\otimes\left(\otimes_A(\pi_k^*L_A^{t_A+1})\right)\right]_{\vec r}\nonumber\\
\cE^{5lm}_{\vec t,\vec r}(k) &=
\left[\pi_k^*(X_k)\otimes \left(\wedge^{l}\pi_k^*(X_k)\right)
\otimes\left(\otimes_A(\pi_k^*L_A^{t_A+1})\right)\right]_{\vec r}\nonumber\end{aligned}$$ We need to consider all $\vec t,\vec r$ that contain states $\cO|k\ra$ with $E=0$.
### $\bQb$ and cohomology {#bqb-and-cohomology .unnumbered}
On states of the form $\cO^1_u|k\ra,\ldots, \cO^4_u|k\ra$ $\bQb_0$ acts as $-\pb$ on horizontal (0,u) forms valued in holomorphic bundles over $\bYk$, and $\bQb_0$ cohomology is the horizontal Dolbeault cohomology. The action on states of the form $\cO^5|k\ra$ has an added term of the sort already familiar from (\[eq:massiveobs1\],\[eq:massiveobs2\]) for the “massive” states in the $k=1$ sector: $$\begin{aligned}
\label{eq:QbPtwo}
\bQb_0 \cO_u^5|k\ra = -\etab^{\Kb}\left[\pb_{\Kb} \cO_u^5{}^{j'}+
\cF_{\Kb J}{}^{j'}_{k'}\chi^J\chib_{j'} (\Xi^{5k'}_{u})^{i'_1\cdots i'_l}_{A_1\cdots A_m}\chib_{i'_1}\cdots\chib_{i'_l}\chi^{A_1}\cdots\chi^{A_m} \right]|k\ra\ ,\end{aligned}$$ where $\cF$ is the curvature of $\cA$. For $\pb$-closed $\Psi^5$, the additional “obstruction” term is $\pb$-closed and gives a linear map $$\begin{aligned}
\obs : \Omega^{0,u}(\cE^{5l,m}) \to \Omega^{0,u+1}(\cE^{4(l+1),m}\otimes\pi^\ast_k T_B^\ast)~.\end{aligned}$$ If $\obs(\Psi^5)$ is exact, then we can construct a $\bQb_0$-closed state just as we saw in the $k=1$ case. We have not encountered a nontrivial obstruction term in any of the examples we considered, and in \[subsec:cpt\] we argue that this will be the case in any well-defined model.
The action of $\bQb_W$ is given by the mode expansion of $$\begin{aligned}
\bQb_W = \oint~\frac{dz}{2\pi i} \chi^\alpha W_\alpha = \chi^\alpha\oint ~\frac{dz}{2\pi i}
z^{\nut_\alpha-q_\alpha/2-1/2} W_\alpha + \chib^{\dagger\alpha}\oint~ \frac{dz}{2\pi i}
z^{\nut_\alpha-q_\alpha/2-3/2} W_\alpha \ ,\end{aligned}$$ where we write $$\begin{aligned}
W_\alpha =
W_\alpha\left(z^{\nu_\beta-q_\beta/2}(\phi^\beta +
z^{-1}\rho^{\dagger\beta})\right)\ .\end{aligned}$$ We can use the homogeneity relation $W_\alpha(\lambda^q\phi^\beta) = \lambda^{1-q_\alpha}
W_\alpha(\phi^\beta)$ and simplify this to $$\begin{aligned}
\label{eq:Wmodes}
\bQb_W = \chi^\alpha\oint \frac{dz}{2\pi i}
z^{\nut_\alpha} W_\alpha\left(z^{\nu_\beta}(\phi^\beta +
z^{-1}\rho^{\dagger\beta})\right) +
\chib^{\dagger\alpha}\oint \frac{dz}{2\pi i}
z^{\nut_\alpha-1} W_\alpha\left(z^{\nu_\beta}(\phi^\beta +
z^{-1}\rho^{\dagger\beta})\right) \ .\end{aligned}$$
Comments on CPT {#subsec:cpt}
----------------
The spectrum we obtain should be invariant under CPT. This means that for any massless state with charge $(\bq,\bqb)$ in the $k$ sector we should find a massless state with charge $(-\bq,-\bqb)$ in the $2N-k$ sector. In this section we will discuss how this works for sectors with odd $k$. To avoid additional notational elaborations we will make the simplifying assumption that $\nut < 0$ for all fields.[^25] As we will now argue, CPT invariance essentially reduces to Serre duality for Dolbeault cohomology on $B$, as well as a natural dual action of $\bQb_W$.
### A pairing on the Hilbert spaces {#a-pairing-on-the-hilbert-spaces .unnumbered}
The two-point function in the CFT is a natural pairing between the conjugate sectors respecting charge conservation and pairing states with the same energy, and given the quantum orbifold symmetry we expect that the Hilbert spaces of states in the $|k\ra$ and $|2N-k\ra$ sectors are dual to each other in this way.
From the expressions above it is clear that the vacua satisfy $$\begin{aligned}
E_{|2N-k\ra} = E_{|k\ra};\qquad (\bq_{|2N-k\ra},\bqb_{|2N-k\ra}) = (-\bq_{|k\ra},d-\bqb_{|k\ra})\end{aligned}$$ while the moding in the conjugate sectors is related by $$\begin{aligned}
\label{eq:conjnus}
\nu_\alpha\leftrightarrow 1-\nu_\alpha;\qquad \nut_\alpha\leftrightarrow
-1-\nut_\alpha\ .\end{aligned}$$ This implies that the fields $\phi^{i'}$ for which $\tau=1/2$ in the $k$ sector have $\tau=3/2$ in the conjugate $2N-k$ sector, and vice versa, so that we have $$\begin{aligned}
\bYk &= {\rm tot} (\oplus_{i'} L_{i'}{\rightarrow} B)\qquad &L_{|k\ra} &=
\otimes_A L_A^*\nonumber\\
\bY_{\!\!2N-k} &={\rm tot} (\oplus L_A{\rightarrow} B)\qquad &L_{|2N-k\ra} &=
\otimes_{i'} L_{i'}^*~.\end{aligned}$$ In particular $L_{|k\ra}\otimes L_{|2N-k\ra} = K^\ast_B$. For any state with weight $h$ and charge $(\bq,\bqb)$ in the $k$ sector, we can find a state with the same weight and charge $(-\bq,d-\bqb)$ in the $2N-k$ sector by exchanging the oscillator excitations according to $$\begin{aligned}
y^\alpha\leftrightarrow\rho_\alpha\qquad\chi^\alpha\leftrightarrow\chib_\alpha\ .\end{aligned}$$ This is enough to show that at the level of left-moving oscillators the two-point function leads to a pairing between the state spaces defined above, which respects $\bq$ and violates $\bqb$ by $d$. If we denote $\cH^{tlm}_{\vec t,\vec r}(k) = \Gamma(\cE^{tlm}_{\vec t,\vec r}(k))$, then the pairing takes the form $$\begin{aligned}
\label{eq:pairs}
\cH^{1\oplus 2\oplus 3}{}^{lm}_{\vec t,\vec r}(k)\times
\cH^{1\oplus 2\oplus 3}{}^{ml}_{\vec r,\vec t}(2N-k)
&\to \C\nonumber\\
\cH^{4lm}_{\vec t,\vec r}(k)\times\cH^{5ml}_{\vec r,\vec t}(2N-k) &\to
\C\ ,\end{aligned}$$
### $\bQb_0$ and Serre duality {#bqb_0-and-serre-duality .unnumbered}
The pairing descends to $\bQb_0$ cohomology, and in a reasonable physical theory this must be nondegenerate. This will be the case if $$\begin{aligned}
H^\bullet_{\bQb_0}\left(\cH^{lm}_k(\vec t,\vec r)\right) =
\left[H^{d-\bullet}_{\bQb_0}\left(\cH^{ml}_{2N-k}(\vec r,\vec t)\right)\right]^*\ .\end{aligned}$$ For the first line in (\[eq:pairs\]), in which $\bQb_0$ acts as $-\pb$, this is in fact equivalent to Serre duality. For simplicity let’s see first how this works in $\cH^{111}_{\vec r,\vec t}$. The fine grading on $H^\bullet(T_{\bY_k})$ can be obtained from the long exact sequence (LES) following from the short exact sequence (SES) (\[eq:Tsections\]) $$\begin{aligned}
\xymatrix{0 \ar[r]& \oplus_i (\pi_k^\ast L_i)_{\br+\bx_i} \ar[r] &
(T_{\bYk})_{\br} \ar[r] &(\pi_k^\ast T_B)_{\br} \ar[r] & 0}\ ,\end{aligned}$$ which we here encounter twisted by a vector bundle (so still exact) as $$\begin{aligned}
\label{eq:sestwst}
\xymatrix{0 \ar[r]& \oplus_i (\pi_k^\ast
L_i)_{\br+\bx_i}\otimes\Vh_{\vec t} \ar[r] &
(T_{\bYk})_{\br}\otimes\Vh_{\vec t} \ar[r] &(\pi_k^\ast T_B)_{\br} \otimes\Vh_{\vec t}\ar[r] & 0}\ ,\end{aligned}$$ where $$\begin{aligned}
\Vh_{\vec t} =
\pi_k^*\left[\oplus_B\left(\otimes_A(L_A^{t_A+1})\right)\right]\ .\end{aligned}$$ The bundles on either end of the SES are pulled back from $B$, and we can use (\[eq:sheafpullback\]) to compute their cohomology. Thus $$\begin{aligned}
H^\bullet_{\vec r}\left(\bYk,\oplus_i (\pi_k^\ast
L_i)_{\br+\bx_i}\otimes\Vh_{\vec t}\right)
=
H^\bullet\left(B,\oplus_{i,B}\left(
\otimes_A(L_A^{t_A+1})
\otimes\left(\otimes_j (L_j^*)^{r_j}
\right)\right)\right)\ ,\end{aligned}$$ while $$\begin{aligned}
H^\bullet_{\vec r}\left(\bYk,(\pi_k^\ast\, T_B)_{\br} \otimes\Vh_{\vec
t}\right) =
H^\bullet\left(B,T_B\otimes\left(\oplus_B\left(\otimes_A(L_A^{t_A+1})
\otimes \left(\otimes_j (L_j^*)^{r_j}\right)\right) \right)\right)\ .\end{aligned}$$ Recalling that $K_B = \otimes_\alpha L_\alpha$, these are Serre dual, respectively, to $$\begin{aligned}
&H^{d-\bullet}\left(B,\oplus_{i,B}\left(
\otimes_A(L_A^*)^{t_A}
\otimes\left(\otimes_j (L_j)^{r_j+1}
\right)\right)\right)\nonumber\\
&= H^{d-\bullet}\left(\bY_{\!\!2N-k},\oplus_{i,B}\left(\pi_{2N-k}^*(L_A^*)_{\vec
t - \vec y_A}\otimes\left(\otimes_j(\hat L_j^{r_j+1})\right)\right)\right)\end{aligned}$$ and $$\begin{aligned}
&H^{d-\bullet}\left(B,T^*_B\otimes\left(\oplus_B\left(\otimes_A(L_A^*)^{t_A}
\otimes \left(\otimes_j (L_j^{r_j+1})\right)\right)
\right)\right)\nonumber\\
&= H^{d-\bullet}\left(\bY_{\!\!2N-k},\left(\pi^*_{2N-k}\,T^*_B\right)_{\vec t}\otimes\left(
\oplus_j(\pi_{2N-k}^*(L_j)^{r_j+1})\right)\right)\ .\end{aligned}$$ Inserting this result into the dual LES we find $$\begin{aligned}
H^\bullet( (T_{\bYk})_{\br}\otimes\Vh_{\vec t}) =
\left[H^{d-\bullet}( (T_{\bY_{\!\!2N-k}})_{\vec t}\otimes\Vh_{\vec r})\right]^*\end{aligned}$$ with a suitable natural definition for $\Vh_{\vec r}$.
Higher powers of the tangent/cotangent bundles are fine graded by recursively using the same SES and the dual, so recursively applying this argument we find that Serre duality implies CPT in the sense above whenever we can use $\bQb_0 = -\pb$. This argument will fail if nontrivial obstruction classes arise in (\[eq:QbPtwo\]), because no such obstruction can arise for the dual states in $\cH^4$. We conclude that in reasonable physical theories there will be no nontrivial obstructions in the twisted sectors.
### $\bQb_W$ and CPT {#bqb_w-and-cpt .unnumbered}
Given that the cohomology of $\bQb_0$ produces a spectrum consistent with CPT, we can also show that the action of $\bQb_W$ is consistent with this. Consider a monomial in $W_\alpha$ that contributes to $\bQb_W$ in the $k$ sector a term $$\begin{aligned}
\chi^\alpha\prod_\beta\left[(\phi^\beta)^{m_\beta}
(\rho^{\dagger\beta})^{n_\beta}\right]\ .\end{aligned}$$ This means that $$\begin{aligned}
\sum_\beta\left[\nu_{\beta,k}(m_\beta+n_\beta) - n_\beta\right] =
- \nut_{\alpha,k} -1\ .\end{aligned}$$ Using (\[eq:conjnus\]) we see that this implies $$\begin{aligned}
\sum_\beta\left[\nu_{\beta,2N-k}(m_\beta + n_\beta) - m_\beta\right]
= \sum_\beta\left[-\nu_{\beta,k}(m_\beta + n_\beta)+n_\beta \right] =
\nut_{\alpha,k}+1 = -\nut_{\alpha,2N-k}\ ,\end{aligned}$$ which means that the same monomial contributes a term $$\begin{aligned}
\chib^{\dagger\alpha}\prod_\beta\left[(\phi^\beta)^{n_\beta}
(\rho^{\dagger\beta})^{m_\beta}\right]\end{aligned}$$ to $\bQb_W$ in the $2N-k$ sector. This acts in precisely the appropriately dual way on the states as mapped above, showing that CPT is maintained as a symmetry after taking $\bQb_W$ cohomology.
Examples {#s:Examples}
========
In this section we will apply the techniques developed in the previous sections to a number of hybrid examples. In each case we will focus on characterizing first order deformations that preserve (0,2) superconformal invariance and the $\Le_8\oplus \Le_6$ spacetime gauge symmetry.
The infinitesimal deformations which preserve (2,2) symmetry parametrize the tangent space of the (2,2) moduli space. They are not obstructed and in a large radius limit are identified with complex structure and complexified Kähler moduli of the CY. There is a well-known correspondence between the (2,2) moduli and the $\Le_6$-charged matter, and we will borrow the large radius notation by denoting the number of chiral $\mathbf{27}$’s and $\mathbf{\overline{27}}$’s in the hybrid computation by $h^{1,1}$ and $h^{2,1}$ respectively.
More interesting are the deformations which only preserve (0,2) superconformal invariance. The computation of the number of massless gauge singlets associated to these deformations, which we indicate as $\mathcal{M}$, is the main goal of this section. These singlets arise in (NS,R), i.e. the odd $k$ sectors. In the following we will compute $\cM$ in three examples that illustrate a number of technical and conceptual points.
1. For the first example we choose the simplest possible base, i.e. $B=\P^1$. This is a good warm-up for more difficult cases and is of interest in its own right since the model can be found as a phase of a GLSM without a large radius limit in its Kähler moduli space. In fact, it can be shown [@Aspinwall:1994cf] that $h^{1,1}=1$, and the only other phase is a LGO.
2. In the second example we describe a model in the broader orbi-bundle set-up with $B=\P^3$. It will be clear that most of our discussion above was restricted to the case in which $X$ is a sum of line bundles solely for ease of exposition. This example also give us a chance to compute a higher order differential (it will turn out to be zero).
3. In the last example we consider the case in which one of the line bundles defining $X$ is positive, and $B=\mathbb{F}_0$ is not a projective space.
While our construction does not depend on a GLSM embedding, all of these models do arise as phases of a GLSM. That gives us the possibility to compare the hybrid spectrum with the spectrum known in other phases. What we discover is that while in the hybrid limit extra singlets appear at a particular complex structure or Kähler form, there is no evidence of world-sheet instanton corrections to masses of $\Le_6$ singlets.
A hybrid with no large radius
-----------------------------
We begin with the model $X=\cO(-2)\oplus\cO^{\oplus4}$ and $B=\P^1$ with superpotential $$\begin{aligned}
\label{genWnoLR}
W = \sum_{p=0}^2 F_{[2p]}(\phi^1)^p. \end{aligned}$$ Some notational clarifications are in order: it is convenient to distinguish between the trivial and non-trivial fiber indices, so let $a,b=2,\dots,5$; moreover, let $F_{[d]}$ be a generic polynomial of degree $4-d$ in the $\phi^a$’s, whose coefficients belong to $H^0(\P^1,\cO(d))$. The left- and right-moving charges for the fields and the quantum numbers of the twisted ground states are summarized in table \[table:noLR\].
The orbifold action $\Gamma=\Z_8$ introduces $7$ twisted sectors; because of CPT invariance to compute the number of massless $\Le_6$-singlets it is sufficient to study the $k=1$ and $k=3$ sectors.
### $k=1$ sector {#k1-sector .unnumbered}
[cc]{}
$k$ $E_{|k\ra}$ $\bq_{|k\ra}$ $\bqb_{|k\ra}$ $\ell_k$ $\nu_a,\nu_1$ $\nut_a,\nut_1$
----- -------------- --------------- ---------------- ---------- ----------------------- ------------------------- -- -- --
$0$ $0$ $-\ff{3}{2}$ $-\ff{3}{2}$ $0$ $0,0$ $0,0$
$1$ $-1$ $0$ $-\ff{3}{2}$ $0$ $\ff{1}{8},\ff{1}{4}$ $-\ff{3}{8},-\ff{1}{4}$
$2$ $0$ $\half$ $-\ff{3}{2}$ $-2$ $\ff{1}{4},\ff{1}{2}$ $-\ff{3}{4},-\ff{1}{2}$
$3$ $-\ff{1}{2}$ $-1$ $-\half$ $-2$ $\ff{3}{8},\ff{3}{4}$ $-\ff{1}{8},-\ff{3}{4}$
$4$ $0$ $-\half$ $-\half$ $0$ $\ff{1}{2},0$ $-\half,0$
: Quantum numbers for the $X=\cO(-2)\oplus\cO^{\oplus4}\rightarrow \P^1$ model.[]{data-label="table:noLR"}
&
$\phi^i,\phi^1$ $\rho_i,\rho_1$ $\chi^i,\chi^1$ $\chib_i,\chib_1$
-------- ------------------- --------------------- ------------------------- ---------------------
$\bq$ $\ff{1}{4},\half$ $-\ff{1}{4},-\half$ $-\ff{3}{4},-\ff{1}{2}$ $\ff{3}{4},\half$
$\bqb$ $\ff{1}{4},\half$ $-\ff{1}{4},-\half$ $\ff{1}{4},\ff{1}{2}$ $-\ff{1}{4},-\half$
: Quantum numbers for the $X=\cO(-2)\oplus\cO^{\oplus4}\rightarrow \P^1$ model.[]{data-label="table:noLR"}
The $E_1$ stage of the spectral sequence is obtained by taking $H_{\bQb_0}(\cH)$ as described in section \[ss:massless\] and we reproduce here the result, where the subscripts denote the dimension of the respective cohomology groups $$\label{eq:spectrseqE1noLRk1}
\begin{matrix}\vspace{20mm}\\E_1^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=15mm@R=5mm{
H^1\left(\bY,B_{1,0,0}\right)_{3}
\ar[r]^{\bQb_W}&
{\begin{matrix}
H^1\left(\bY,B_{0,0,1} \right)_{10}
\\\oplus\\
H^1\left(\bY, B_{1,1,0} \right)_{63}
\end{matrix}} \ar[r]^{\bQb_W} &
H^1\left(\bY, B_{0,1,0} \right)_{35} \\
&
{\begin{matrix}
H^0\left(\bY,B_{0,0,1}\right)_{20}
\\\oplus\\
H^0\left(\bY,B_{1,1,0} \right)_{17}
\end{matrix}} \ar[r]^{\bQb_W}
&
{\begin{matrix}
H^0\left(\bY, B_{0,1,0}\right)_{176}
\end{matrix}}
}
\save="x"!LD+<-6mm,0pt>;"x"!RD+<40pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<120mm,-3mm>;"x"!LU+<120mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-62mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-105mm,-4mm>*{-\ff{5}{2}}\restore
\save!RD+<-18mm,-4mm>*{-\half}\restore
\save!RD+<13mm,-3mm>*{p}\restore
\save!CL+<117mm,21mm>*{U}\restore
\end{xy}$$
The lowest row of the sequence provides an example of the universal structure of currents we indicated above in (\[eq:currents\]), and for generic $W$ the kernel is one-dimensional, corresponding to the $\GUL$ symmetry. By choosing a particular form of the superpotential (\[genWnoLR\]) we can increase $\ker \bQb_W$, and the additional vectors correspond to an enhanced symmetry at the special locus in the moduli space.
In order to compute the cohomology of the top row of let us list all the states contributing at $E_1^{p,1}$: $$\label{eq:k1cohomnoLR}
\begin{xy}
\xymatrix@C=20mm@R=10mm{
V \rho_1 \chib_1|1\ra_{3}
\ar[r]^-{\bQb_W} &
{ \begin{matrix}
H_{[2]} \chib_1 \chi^I |1\ra_{30} \\\oplus\\
G_{[1]} \chib_b \chi^I |1\ra_{16} \\\oplus\\
G_{[1]} \chib_1 \chi^b |1\ra_{16} \\\oplus\\
\Phi_I \phi^1 \chib_1 \chi^I |1\ra_{1} \\\oplus\\
G_{[2]} \rho_1|1\ra_{10} \\\oplus\\
\Psi_I \pz y^I |1\ra_1
\end{matrix} }
\ar[r]^-{\bQb_W} &
G_{[4]} \chi^I |1\ra_{35}
}
\end{xy}$$ where $G_{[d]}$ and $H_{[d]}$ are generic polynomials of degree $d$ in the $\phi^a$’s with coefficients in $H^1(\P^1,\cO(-2))$ and $H^1(\P^1,\cO(-4))$, respectively, while $\Psi_I, \Phi_I \in H^1(\P^1,\cO(-2)) $. First, consider the map on the left. We have the state $V \rho_1 \chib_1|1\ra$ where $V \in H^1(\bY, T_{\bY}\otimes T_{\bY})\simeq H^1\left(\P^1, \cO(-4) \right) $. Under $\bQb_W$ it maps to $$\begin{aligned}
\label{eq:massiveP1noLR}
\bQb_W V \rho_1 \chib_1|1\ra &= V \left( \p_1 W \rho_1+ \p_{11}W \chi^1 \chib_1 + \p_{1I}W \chi^I \chib_1 \right) |1\ra + V\left( \p_{1a}W \chi^a\chib_1 \right)|1\ra ~.\end{aligned}$$ Since $\p_1W, \p_{1a}W \in \Gamma(\P^1,\cO(2))$, it follows that $V\p_1W, V\p_{1a}W \in H^1 \left( \P^1, \cO(-2)\right)$. To compute the dimension of the cokernel of this map we first note that if we restrict the superpotential to its Fermat form, namely $W= \sum_{i=2}^5 (\phi^i)^4 + S_{[4]}(\phi^1)^2$, we have $$\begin{aligned}
\bQb_W V \rho_1 \chib_1|1\ra &= 2V \left( \phi^1 S_{[4]} \rho_1+ S_{[4]} \chi^1 \chib_1 + \phi^1 \p_IS_{[4]}\chi^I \chib_1 \right) |1\ra ~.\end{aligned}$$ Since $V \phi^1 S_{[4]} \in H^1 \left( \P^1, \cO(-2)\right)$ and $h^1(\P^1, \cO(-2))=1$ the kernel at Fermat is 2-dimensional.
Adding to $W$ a term of the form $S_{[2]}\phi^2\phi^3\phi^1+ T_{[2]}\phi^4\phi^5\phi^1$, where $S_{[2]}, T_{[2]}\in \Gamma(B,\cO(2))$, we find that reads $$\begin{aligned}
\bQb_W V \rho_1 \chib_1|1\ra &= \underbrace{V \left( \p_1W \rho_1+ \p_{11}W \chi^1 \chib_1 + \p_{1I}W \chi^I \chib_1 \right) |1\ra}_{\mbox{Fermat}} \nonumber\\
& \quad + V S_{[2]}\left( \phi^2 \chi^3 + \phi^3\chi^2\right) \chib_1 |1\ra + V T_{[2]}\left( \phi^4 \chi^5 + \phi^5\chi^4\right) \chib_1 |1\ra ~,\end{aligned}$$ and the map is injective for $W$ generic enough. Now, for the map on the right in we have $$\begin{aligned}
\bQb_W \left( \Psi_{ab}\rho_1 +\Psi_{ab,I} \chi^I\chib_1 \right)\phi^a\phi^b |1\ra &= \p_\alpha\left( \Psi_{ab} \p_1 W\right) \chi^\alpha \phi^a\phi^b |1\ra \nonumber\\
\bQb_W \Sigma_{abI}\chi^I\chib_1 \phi^a\phi^b|1\ra & = -\Sigma_{abI}\chi^I \p_1 W \phi^a\phi^b |1\ra \nonumber\\
\bQb_W V_a^b \phi^a \chi^I \chib_b |1\ra &= -V_a^b \phi^a \p_b W \chi^I |1\ra \nonumber\\
\bQb_W \Phi_I \phi^1 \chi^I \chib_1 |1\ra &= -\Phi_I \phi^1 \p_1 W \chi^I |1\ra \end{aligned}$$ The cokernel of this map is thus any object of the form $\Psi_{abcd} \phi^a\phi^b\phi^c\phi^d \chi^I |1\ra$ for $\Psi_{abcd}\in H^1(\P^1,\cO(-2))$, which cannot be written as $\p_1 W \phi^a\phi^b \chi^I |1\ra$ or $\phi^a\p_b W \chi^I |1\ra$. We find a $9$-dimensional space. Thus, the $E_2$ stage of the spectral sequence is $$\begin{matrix}\vspace{10mm}\\E_2^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=8mm@R=8mm{\;\;\;
0 &
\C^{45}
&
\C^{9} \\ \;\;\;
&
\C^1 &
\C^{139}
}
\save="x"!LD+<3mm,0pt>;"x"!RD+<40pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<45mm,-3mm>;"x"!LU+<45mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-36mm,-4mm>*{-\ff{5}{2}}\restore
\save!RD+<-23mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-6mm,-4mm>*{-\ff{1}{2}}\restore
\save!RD+<12mm,-4mm>*{p}\restore
\save!CL+<42mm,12mm>*{U}\restore
\end{xy}$$ and obviously all higher differentials vanish. Hence the spectral sequence degenerates already at this stage, $E_\infty = E_2$. Thus, in this sector we count $45+139=184$ chiral and $9$ anti-chiral $\Le_6$-singlets.
### $k=3$ sector {#k3-sector .unnumbered}
The $k=3$ ground state has a non-trivial vacuum bundle $L_{|3\ra} = \cO(2)$ and, as discussed in section \[s:twisted\], we must distinguish between light and heavy fields. In particular we have $A=1$, $i' = 2,\dots,5$, $\alpha' = (I,i')$, while the geometry is determined by $\bY_{\!\!3}$, the total space of $\cO^{\oplus4}\xrightarrow{\pi_3}\P^1$. The expansion of $\bQb_W$ in this sector takes the form $$\begin{aligned}
\label{QbWk3noLR}
\bQb_W = \chib^A{}^\dag \p_A W + \chi^A \p_{Ai}W \rho^i{}^\dag + \chib^{\alpha'}{}^\dag \p_{\alpha' i} W \rho^i{}^\dag + \chi^{\alpha'} \p_{\alpha' ij}W \rho^i{}^\dag\rho^j{}^\dag~.\end{aligned}$$ The $E_1$ stage of the spectral sequence is given by $$\label{E1k3noLR}
\begin{matrix}\vspace{10mm}\\E_1^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=12mm@R=10mm{
\;\;\; H^1(\bY_{\!\!3}, \pi_3^\ast\left( L_6^\ast \right)\otimes \wedge^2 T_{\bY_{\!\!3}} )_{18} \ar[r]^-{\bQb_W} &
H^1(\bY_{\!\!3}, T_{\bY_{\!\!3}} )_{16} \\
\;\;\; 0 &
{\begin{matrix}
H^0(\bY_{\!\!3}, \wedge^2 T_{\bY_3} \otimes T^\ast_{\bY_3})_{6} \\\oplus\\
H^0(\bY_{\!\!3},T_{\bY_3} )_1
\end{matrix}}
}
\save="x"!LD+<1mm,0pt>;"x"!RD+<33pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<107mm,-3mm>;"x"!LU+<107mm,8mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-76mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-20mm,-4mm>*{-\half}\restore
\save!RD+<11mm,-3mm>*{p}\restore
\save!CL+<104mm,23mm>*{U}\restore
\end{xy}$$ Now, the only non-trivial map is at $U=1$, where $$\begin{aligned}
\label{eq:bqbWk3noLR}
\bQb_W V^{ab}\rho_1\chib_a\chib_b |3\ra = 2V^{ab} \rho_1\p_{1a}W \chib_b |3\ra \neq 0 ~.\end{aligned}$$ The RHS never vanishes, giving a $6$-dimensional image. Hence, the spectral sequence degenerates at the $E_2$ term $$\begin{matrix}\vspace{8mm}\\E_2^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=8mm@R=8mm{\;\;\;
\C^{12}
&
\C^{10} \\
0 &
\C^{7}
}
\save="x"!LD+<3mm,0pt>;"x"!RD+<30pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<32mm,-3mm>;"x"!LU+<32mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-23mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-6mm,-4mm>*{-\ff{1}{2}}\restore
\save!RD+<10mm,-3mm>*{p}\restore
\save!CL+<29mm,12mm>*{U}\restore
\end{xy}$$ Hence we count 19 chiral and 10 anti-chiral states for a total of $222$ $\Le_6$ singlets. By similar methods we compute $h^{2,1}=61$ and $h^{1,1}=1$, yielding $\cM=160$.
The orbi-bundle
---------------
Now we present an example in which $X$ is not a sum of line bundles, but a more general orbi-bundle. Let us take $B=\P^3$ and $X=\cO(-5/2)\oplus\cO(-3/2)$ along with the quasi-homogeneous superpotential $$\begin{aligned}
W= S_5 (\phi^1)^2 + S_4\phi^1\phi^2+ S_3 (\phi^2)^2 , \end{aligned}$$ where $S_d\in H^0(B,\cO(d))$. The ground state quantum numbers and charges of the fields are given in table \[table:orbinums\], and to find the singlets we need only consider the first twisted sector.
The first stage of the spectral sequence is $$\begin{matrix}\vspace{20mm}\\E_1^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=10mm@R=5mm{
{\begin{matrix}
H^3\left(\bY,B_{1,0,1} \right)_6
\\\oplus \\
H^3\left(\bY,B_{2,1,0} \right)_{15}
\end{matrix}}
\\
0 & 0 \\
0 &
H^1\left(\bY,B_{1,1,0} \right)_{2}
& 0 \\
0 &
{\begin{matrix}
H^0\left(\bY,B_{0,0,1} \right)_{21}
\\\oplus\\
H^0\left(\bY,B_{1,1,0} \right)_{6}
\end{matrix}} \ar[r]^{\bQb_W} &
{\begin{matrix}
H^0\left(\bY, B_{0,1,0}\right)_{295}
\end{matrix}}
}
\save="x"!LD+<-6mm,0pt>;"x"!RD+<35pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<110mm,-3mm>;"x"!LU+<110mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-95mm,-4mm>*{-\ff{5}{2}}\restore
\save!RD+<-55mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-16mm,-4mm>*{-\half}\restore
\save!RD+<12mm,-3mm>*{p}\restore
\save!CL+<107mm,34mm>*{U}\restore
\end{xy}$$ The bottom row is the only place where we can have cokernel, and for generic superpotential we find $\dim \ker \bQb_W=1$.
[cc]{}
$k$ $E_{|k\ra}$ $\bq_{|k\ra}$ $\bqb_{|k\ra}$ $\ell_k$ $\nu_i$ $\nut_i$
----- ------------- --------------- ---------------- ---------- ------------- -------------- -- -- --
$0$ $0$ $-\ff{3}{2}$ $-\ff{3}{2}$ $0$ $0$ $0$
$1$ $-1$ $0$ $-\ff{3}{2}$ $0$ $\ff{1}{4}$ $-\ff{1}{4}$
$2$ $0$ $\half$ $-\ff{3}{2}$ $-4$ $\ff{1}{2}$ $-\ff{1}{2}$
: Quantum numbers for the $X=\cO(-5/2)\oplus\cO(-3/2)\rightarrow \P^3$ model.[]{data-label="table:orbinums"}
&
$\phi^i$ $\rho_i$ $\chi^i$ $\chib_i$
-------- ------------- -------------- -------------- --------------
$\bq$ $\ff{1}{2}$ $-\ff{1}{2}$ $-\ff{1}{2}$ $\ff{1}{2}$
$\bqb$ $\ff{1}{2}$ $-\ff{1}{2}$ $\ff{1}{2}$ $-\ff{1}{2}$
: Quantum numbers for the $X=\cO(-5/2)\oplus\cO(-3/2)\rightarrow \P^3$ model.[]{data-label="table:orbinums"}
Thus, the $E_2$ stage of the spectral sequence is $$\begin{matrix}\vspace{10mm}\\E_2^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=5mm@R=5mm{
&
\C^{21}
\\
& 0 & 0 \\
& 0 & \C^2 & 0 \\
& 0 &
\C & \C^{269} & 0
}
\save="x"!LD+<5mm,0pt>;"x"!RD+<15pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<43mm,-3mm>;"x"!LU+<43mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-41mm,-4mm>*{-\ff{5}{2}}\restore
\save!RD+<-28mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-15mm,-4mm>*{-\ff{1}{2}}\restore
\save!RD+<-2mm,-4mm>*{\ff{1}{2}}\restore
\save!RD+<4mm,-4mm>*{p}\restore
\save!CL+<40mm,20mm>*{U}\restore
\end{xy}$$ All higher differentials vanish, and the spectral sequence degenerates at the $E_2$ term. We then count 271 chiral and 21 antichiral states corresponding to massless $\Le_6$ singlets. We also computed by similar methods the number of charged singlets, $h^{2,1} = 90$ and $h^{1,1}=2$, corresponding to the (2,2) moduli, which we can subtract from the total number of neutral singlets to find $\cM=200$.
### A higher differential ? {#a-higher-differential .unnumbered}
It is worth noting that the spectral sequence for computing the number of $\rep{1}_2\subset\mathbf{27}$ states degenerates only at the $E_4$ term, giving us an example of a possible higher differential. At zero energy and $\bq=2$ we have $$\begin{matrix}\vspace{10mm}\\E_1^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=5mm@R=6mm{
0 \\
{H^2(\bY,B^1_{3,0,0})}_{1} & 0 \\
0 & 0
& 0 \\
0 & {H^0(\bY,B_{2,0,0})}_{120}
\ar[r]^-{\bQb_W} &
{H^0(\bY,B_{1,0,0})}_{905}
\ar[r]^-{\bQb_W} &
{H^0(\bY,B_{0,0,0})}_{875}
}
\save="x"!LD+<-2mm,0pt>;"x"!RD+<15pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<101mm,-3mm>;"x"!LU+<101mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-118mm,-4mm>*{-\ff{5}{2}}\restore
\save!RD+<-85mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-50mm,-4mm>*{-\half}\restore
\save!RD+<-15mm,-4mm>*{\half}\restore
\save!RD+<5mm,-3mm>*{p}\restore
\save!CL+<98mm,22mm>*{U}\restore
\end{xy}$$ Trivially $d_2=0$, thus $E_3=E_2$, but there is one more map we have to compute, in fact $$\begin{matrix}\vspace{10mm}\\E_3^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=7mm@R=7mm{
0 \\
\C \ar[rrrdd]^-{d_3}& 0\\
0 & 0 & 0 \\
0 & 0 & 0 & \C^{90}
}
\save="x"!LD+<00mm,0pt>;"x"!RD+<15pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<31mm,-3mm>;"x"!LU+<31mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-41mm,-4mm>*{-\ff{5}{2}}\restore
\save!RD+<-29mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-18mm,-4mm>*{-\ff{1}{2}}\restore
\save!RD+<-5mm,-4mm>*{\ff{1}{2}}\restore
\save!RD+<4mm,-4mm>*{p}\restore
\save!CL+<28mm,22mm>*{U}\restore
\end{xy}$$ Let us recall that an element $b\in \cH$ represents a cohomology class in $E_3$ if there exist $c_1,c_2 \in \cH$ such that $$\begin{aligned}
\bQb_0 b &=0~, & \bQb_W b &= \bQb_0 c_1 ~ , &\bQb_W c_1 &= \bQb_0 c_2~ ,\end{aligned}$$ and $d_3$ on the cohomology class $[b]_3$ is given by $$\begin{aligned}
\label{d3coho}
d_3[b]_3 = [\bQb_W c_2]_3 ~.\end{aligned}$$ Thus, we just chase down the state $\etab^{\Jb}\etab^{\Kb}V^I_{\Jb\Kb}\chib_1\chib_2\chib_I |1\ra \in E_3^{-5/2,3}$ as prescribed in (\[d3coho\]) $$\begin{aligned}
\xymatrix@C=5mm{0 & 0\\
\etab^{\Jb}\etab^{\Kb}V^I_{\Jb\Kb}\chib_1\chib_2\chib_I |1\ra \ar[r]^-{\bQb_W} \ar[u]^-{\bQb_0}&
\etab^{\Jb}\etab^{\Kb}V^I_{\Jb\Kb}\epsilon^{\alpha\beta\gamma} \p_\alpha W \chib_\beta\chib_\gamma |1\ra \ar[u]^-{\bQb_0}\\
&\etab^{\Jb}S^I_{\Jb}\epsilon^{\alpha\beta\gamma} \p_\alpha W \chib_\beta\chib_\gamma |1\ra
\ar[u]^{\bQb_0} \ar[r]^-{\bQb_W}&
\etab^{\Jb}S^I_{\Jb}\epsilon^{\alpha\beta\gamma} \p_\alpha W \p_{[\beta} W\chib_{\gamma]} |1\ra \\
& & R^I \epsilon^{\alpha\beta\gamma} \p_\alpha W \p_{[\beta} W\chib_{\gamma]} |1\ra
\ar[u]^{\bQb_0} \ar[r]^-{\bQb_W}&
0 ~ .
}\end{aligned}$$ The coefficients satisfy $$\begin{aligned}
V^I_{\Jb\Kb} &= -(\pb S)^I_{\Jb\Kb}~, &S^I_{\Jb} &= - (\pb R)^I_{\Jb}~.\end{aligned}$$ We just showed that $d_3$, while in principle allowed, vanishes, and the spectral sequence degenerates at the $E_4=E_2$ term. In this sector we count $h^{2,1}=90$ and $h^{1,1}=1$ and the “missing" Kähler modulus is to be found in the $k=3$ sector, as expected.
A positive line bundle
----------------------
For our last example we consider $X=\cO(-3,-3)\oplus\cO(1,1)$ and $B=\mathbb{F}_0$. The novelty here is that we allow a positive line bundle over a non-projective base.
A non degenerate superpotential is given by $$\begin{aligned}
W= (\phi^1)^4 S_{[12,12]} + (\phi^1)^3\phi^2 S_{[8,8]} + (\phi^1\phi^2)^2 S_{[4,4]} + \phi^1(\phi^2)^3 S_{[0,0]}~,\end{aligned}$$ where $S_{[m,n]} \in \Gamma(\mathbb{F}_0,\cO(m,n))$ and the quantum numbers for this theory are listed in table \[table:F0nums\]. Studying the (R,R) sectors we find $h^{1,1}=3$ and $h^{2,1}=243$, and to count the remaining $\Le_6$ singlets we need to consider the $k=1$ and $k=3$ sectors.
### $k=1$ sector {#k1-sector-1 .unnumbered}
In the first twisted sector the spectral sequence at $\bq=0$ is $$\begin{matrix}\vspace{20mm}\\E_1^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=15mm@R=5mm{
{\begin{matrix}
{H^2(\bY,B_{1,1,0})}_{39} \\\oplus\\
{H^2(\bY,B_{0,0,1})}_{9}
\end{matrix}}
\ar[r]^-{\bQb_W} &
{H^2(\bY,B_{0,1,0})}_{39} \\
{H^1(\bY,B_{1,1,0})}_{10}
\ar[r]^-{\bQb_W} &
{H^1(\bY,B_{0,1,0})}_2 \\
{\begin{matrix}
{H^0(\bY,B_{1,1,0})}_{63} \\\oplus\\
{H^0(\bY,B_{0,0,1})}_{27}
\end{matrix}}
\ar[r]^-{\bQb_W}
&
{H^0(\bY,B_{0,1,0})}_{825}
}
\save="x"!LD+<-2mm,0pt>;"x"!RD+<50pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<75mm,-3mm>;"x"!LU+<75mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<15mm,-3mm>*{p}\restore
\save!CL+<72mm,28mm>*{U}\restore
\save!RD+<-59mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-18mm,-4mm>*{-\half}\restore
\end{xy}$$ It is not hard to verify that both the maps $\bQb_W\big|_{U=1}$ and $\bQb_W\big|_{U=2}$ are surjective for sufficiently generic $W$, and as we already saw in the discussion about the general $k=1$ sector, there is only one state at $\bqb=-\ff{3}{2}, u=0$ in the kernel of $\bQb_W$. The spectral sequence degenerates at the $E_2$ term $$\begin{matrix}\vspace{10mm}\\E_2^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=10mm@R=5mm{
\mathbb{C}^9 \\
\mathbb{C}^8 & 0 \\
\mathbb{C} & \mathbb{C}^{736} &
}
\save="x"!LD+<-6mm,0pt>;"x"!RD+<0pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<30mm,-3mm>;"x"!LU+<30mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-1mm,-4mm>*{p}\restore
\save!CL+<27mm,15mm>*{U}\restore
\save!RD+<-37mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-19mm,-4mm>*{-\half}\restore
\end{xy}$$ Thus we count $744$ chiral and $9$ anti-chiral massless $\Le_6$-singlets and one vector.
[cc]{}
$k$ $E_{|k\ra}$ $\bq_{|k\ra}$ $\bqb_{|k\ra}$ $\ell_k$ $\nu_i$ $\nut_i$
----- -------------- --------------- ---------------- ----------- ------------- -------------- -- -- --
$0$ $0$ $-\ff{3}{2}$ $-\ff{3}{2}$ $0$ $0$ $0$
$1$ $-1$ $0$ $-\ff{3}{2}$ $0$ $\ff{1}{8}$ $-\ff{3}{8}$
$2$ $0$ $\half$ $-\ff{3}{2}$ $(-2,-2)$ $\ff{1}{4}$ $-\ff{3}{4}$
$3$ $-\ff{3}{4}$ $-\half$ $-1$ $0$ $\ff{3}{8}$ $-\ff{1}{8}$
$4$ $0$ $-1$ $-1$ $(-2,-2)$ $\ff{1}{2}$ $-\ff{1}{2}$
: Quantum numbers for the $X=\cO(-3,-3)\oplus\cO(1,1)\rightarrow \mathbb{F}_0$ model.[]{data-label="table:F0nums"}
&
$\phi^i$ $\rho_i$ $\chi^i$ $\chib_i$
-------- ------------- -------------- -------------- --------------
$\bq$ $\ff{1}{4}$ $-\ff{1}{4}$ $-\ff{3}{4}$ $\ff{3}{4}$
$\bqb$ $\ff{1}{4}$ $-\ff{1}{4}$ $\ff{1}{4}$ $-\ff{1}{4}$
: Quantum numbers for the $X=\cO(-3,-3)\oplus\cO(1,1)\rightarrow \mathbb{F}_0$ model.[]{data-label="table:F0nums"}
### $k=3$ sector {#k3-sector-1 .unnumbered}
In the $k=3$ sector all the fields are “light”, $L_{|3\ra}$ is trivial, and the geometry is again encoded in the full $\bY_{\!\!3} = \bY$. The spectral sequence starts then as $$\begin{matrix}\vspace{15mm}\\E_1^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=15mm@R=5mm{
{\begin{matrix}
{H^2(\bY,\Sym^2 T_{\bY})}_{27} \\\oplus\\
{H^2(\bY,\wedge^2 T_{\bY}\otimes T^\ast_{\bY})}_6
\end{matrix}} \ar[r]^-{\bQb_W}
& {H^0(\bY,\cO)}_1 \\
0 & 0 \\
{H^0(\bY,\Sym^2 T_{\bY})}_9
\ar[r]^-{\bQb_W}
&
{H^0(\bY,\cO)}_{58}
}
\save="x"!LD+<-2mm,0pt>;"x"!RD+<35pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<76mm,-3mm>;"x"!LU+<76mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-56mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-12mm,-4mm>*{-\ff{1}{2}}\restore
\save!RD+<11mm,-4mm>*{p}\restore
\save!CL+<73mm,21mm>*{U}\restore
\end{xy}$$ It can be shown that the map $\bQb_W\big|_{U=0}$ is injective while the map $\bQb_W\big|_{U=2}$ is surjective. Therefore the second stage of the spectral sequence is $$\begin{matrix}\vspace{10mm}\\E_2^{p,u}:\end{matrix}
\begin{xy}
\xymatrix@C=10mm@R=5mm{
\mathbb{C}^{32} & 0 \\
0 & 0 \\
0& \mathbb{C}^{49} &
}
\save="x"!LD+<-2mm,0pt>;"x"!RD+<0pt,0pt>**\dir{-}?>*\dir{>}\restore
\save="x"!LD+<30mm,-3mm>;"x"!LU+<30mm,2mm>**\dir{-}?>*\dir{>}\restore
\save!RD+<-1mm,-4mm>*{p}\restore
\save!CL+<27mm,15mm>*{U}\restore
\save!RD+<-36mm,-4mm>*{-\ff{3}{2}}\restore
\save!RD+<-17mm,-4mm>*{-\half}\restore
\end{xy}$$ The spectral sequence degenerates at the $E_2$ term, and we find 49 chiral and 32 anti-chiral singlets.
Summarizing, we count 834 massless chiral $\Le_6$-singlets and once we subtract the moduli we obtain $\cM=588$.
Discussion {#s:discuss}
==========
We have described a class of perturbative vacua for heterotic string compactifications and a limit in which their properties are computable. We have illustrated these computations in models with (2,2) world-sheet supersymmetry, although the methods clearly extend to more general (0,2) theories.
Our class of (2,2) models fits in with a number of other constructions. To describe this we proceed in increasing dimension $d$ of the base $B$ and assume this is Fano. For $d=1$ this means $B=\P^1$ and the $c=6$ LGO theory on the fiber determines a one-parameter family of K3 compactifications. Models with no large radius limit in the Kähler moduli space, such as the first example in section \[s:Examples\], are obtained when the monodromies of the family are not simultaneously geometrical in any duality frame. It seems likely that any such model would be obtained as a limit in some GLSM, but we have not shown this.
For $d=2$ the base is a del Pezzo surface and the $c=3$ LGO theory on the fiber can be interpreted as determining in Weierstrass form an elliptic fibration over $B$. This can be smooth if the discriminant is nonsingular in $B$, in which case the model will have a large-radius limit. It is not clear how to construct a GLSM embedding for a hybrid with a non-toric base.
For $d=3$ there are many possible choices for $B$, but the $c=0$ LGO theory is quadratic, and hence appears to be trivial. Since the fiber fields are massive at generic points on the base, one might think the low-energy theory would be a NLSM with target space $B$, but this cannot be correct, as this would not be conformally invariant. This naïve discussion omits the orbifold action. Since at low energy there are no excitations in the fiber direction, one can try [@Addington:2012zv] to describe the resulting model as a NLSM with target space a double cover of $B$ branched over the singular locus of $W(y,\phi)$ [*considered as a function of $\phi$ only*]{}. This leads to a geometric interpretation of the limiting point we called the hybrid limit. It is not directly related to a symplectic quotient construction and, if the model has a large-radius limit, it is not birational to the target space at this limit. The relationship between the two descriptions is unclear. It would be interesting to study, among other things, the behavior of the D-brane spectrum and moduli in a type-IIA compactification near such a hybrid limit.
The models we have studied have been “good” hybrids, in which the R-symmetry does not act on the base. Limiting points of GLSMs often produce hybrids for which this does not hold. The hybrid limit for “good” hybrids is expected to lie at infinite distance in the moduli space of SCFTs; it should be possible to determine the approximate moduli space metric in the hybrid limit. We expect that the approximation should improve as the hybrid limit is approached and the distance to the hybrid limit deep in the Kähler cone of $B$ will diverge. It would be interesting to verify this in detail. In [@Aspinwall:2009qy] “pseudo” hybrids were defined as hybrid limits lying at finite distance; the behavior of the D-brane spectrum near these limits was found to be quite different from that expected near a “good” hybrid. It seems natural to conjecture that “good” hybrids and “true” (not “pseudo”) hybrid limits coincide.
Although we focused on models with (2,2) world-sheet supersymmetry, the methods extend naturally to a much larger class of models with (0,2) supersymmetry. This larger class presents an array of interesting questions. As a first foray in this direction, the massless $\Le_6$ singlets in (NS,R) sectors belong to (anti-) chiral multiplets containing massless scalars. Expectation values for these represent marginal deformations of the world-sheet SCFT preserving (0,2) supersymmetry. We do not at present have effective techniques to determine which of these are exactly marginal, and the structure of the moduli space of (0,2) SCFTs is still largely unknown.
In general one expects [@Dine:1986zy; @Distler:1987ee] that away from the hybrid limit the (0,2) models we construct will be destabilized by world-sheet instantons wrapping cycles in $B$. In some classes of models this expectation has been thwarted, and the anticipated corrections are absent [@Silverstein:1995re; @Basu:2003bq; @Beasley:2003fx]. Even in cases in which no known argument precludes such corrections they have been found less generally than one might expect [@Aspinwall:2010ve; @Aspinwall:2011us]. It would be very interesting to investigate this issue in the context of hybrid models, in which the structure of the relevant instantons – associated to rational curves in $B$ rather than in a Calabi–Yau threefold, may provide a simpler context for their study.
More generally, we can construct (0,2) hybrid models that are not deformations of (2,2) models by taking the left-moving fermions to be sections of a holomorphic bundle $\cE\to \bY$ and a (0,2) superpotential given by a section $J \in \Gamma(\cE^\ast)$ with $J^{-1}(0) = B$. It is to be expected that most such models will not have a limit in which they are described by a (0,2) NLSM or one in which they reduce to a (0,2) LGO theory, so that these will determine a large class of new perturbative vacua of the heterotic string. These models will be considered in a forthcoming work.
Hybrid geometry : an example {#app:simplehybrid}
============================
Let $B = \P^1$ and take $\bY$ to be the total space of $X = \cO(-2) \to \P^1$. We cover $\bY$ by two patches ${\cU}_{u}$ and $\cU_{v}$, with local coordinates $(u,\phi_u)$ and $(v,\phi_v)$, respectively : $$\begin{aligned}
u = v^{-1}, \qquad \phi_u = v^2\phi_v \qquad\text{on} \quad {\cU}_{u}\cap{\cU}_v = \C^\ast.\end{aligned}$$ The projection $\pi :\bY \to B$ is simply $(u,\phi_u) \to u$ and $(v,\phi_v) \to v$ in the two patches. The transition function for $\sigma = \sigma^1_u \p_{u} + \sigma^2_u \p_{{\phi_u}}$, a section of $T_{\bY}$, is $$\begin{aligned}
\begin{pmatrix}
\sigma^1_u & \sigma^2_{u}
\end{pmatrix} =
\begin{pmatrix}
\sigma^1_v & \sigma^2_{v}
\end{pmatrix}
\begin{pmatrix}
-v^{-2} & 2v\phi_v \\
0 & v^2
\end{pmatrix}
~.\end{aligned}$$ $T_{\bY}$ belongs to a family of rank $2$ holomorphic bundles $\cV_\ep\to {\bY}$ with transition function $$\begin{aligned}
\label{eq:extrans}
\begin{pmatrix}
\sigma^1_u & \sigma^2_{u}
\end{pmatrix} =
\begin{pmatrix}
\sigma^1_v & \sigma^2_{v}
\end{pmatrix} M_{\ep}~, \qquad
M_{\ep} \equiv
\begin{pmatrix}
-v^{-2} & 2\ep v\phi_v \\
0 & v^2
\end{pmatrix}
~.\end{aligned}$$ When $\ep = 0$ the bundle splits: $\cV_{\ep=0} = \pi^\ast \cO(2)\oplus\pi^\ast \cO(-2)$; more generally $\cV_{\ep}$ is an irreducible rank $2$ bundle over $\bY$.
An example of a quasi-homogeneous superpotential depending on a parameter $\alpha$ is $$\begin{aligned}
W_u = (\alpha+u^8) \phi_u^4~,\qquad W_v = (\alpha v^8+1) \phi_v^4~.\end{aligned}$$ Clearly $W_u = W_v$ on the overlap. Computing the gradient in the two patches, we obtain $$\begin{aligned}
dW_u = 8 u^7 \phi_u^4 du + 4 (\alpha+u^8) \phi_u^3 d\phi_u~,\qquad
dW_v =8 \alpha v^7 \phi_v^4 dv + 4 (\alpha v^8+1) \phi_v^3 d\phi_v~.\end{aligned}$$ It is then easy to see that for $\alpha \neq 0$ we have $dW^{-1}(0) = B$. A more general superpotential respecting the same quasi-homogeneity is $$\begin{aligned}
W_u = S_u(u) \phi_u^4~,\qquad W_v = S_v(v) \phi_v^4~,\end{aligned}$$ where $S_{u,v}$ is the restriction of $\Sigma \in H^0(B,\cO(8))$ to $\cU_{u,v}$. The potential condition is satisfied for generic choices of $\Sigma$.
We can see how the fibration affects the naive chiral ring $R_p$ of the LG fiber theory over a point $p\in B$: $\dim R_p$ jumps in complex co-dimension $1$ but stays finite if the potential condition is satisfied. In our example $R_u = \{1,\phi_u,\phi_u^2\}$ for $u^8+\alpha \neq 0$, while at the $8$ special points $R = \{1,\phi_u,\phi_u^2,\phi_u^3\}$. If $\alpha = 0$ then the potential condition is violated, and $\dim R_0 = \infty$.
### A (0,2) deformation {#a-02-deformation .unnumbered}
Taking the left-moving bundle to be $\cE = \cV_{\ep}$, we obtain a class of (0,2) theories. The most general (0,2) superpotential that respects the same quasi-homogeneity as $dW$, $J \in \Gamma(\cE^\ast)$, takes the form $$\begin{aligned}
J_u =
\begin{pmatrix}
T_u(u)\phi_u^4 \\ 4S_u(u) \phi_u^3
\end{pmatrix}, \qquad
J_v =\begin{pmatrix}
T_v(v)\phi_v^4 \\ 4S_v(v) \phi_v^3
\end{pmatrix}~, \qquad\end{aligned}$$ where $S$ and $T$ are holomorphic functions constrained by $J_u = M_\ep^{-1} J_v$ when $u\neq v$. $S_{u,v}$ are restrictions of $\Sigma$ as above, while $T_{u,v}$ are given by $$\begin{aligned}
T_u(u) = -\left.\Sigmat\right|_{u} + 8\ep u^{-1} \left(S_u(u) - S_u(0) \right)~, \qquad
T_v(v) = \left.\Sigmat\right|_{v} + 8\ep v^7 S_u(0)~,\end{aligned}$$ where $\Sigmat\in H^0(B,\cO(6))$. The potential condition is satisfied for generic $\Sigma$ and $\Sigmat$. Setting $\ep=1$ and $T_u = \p_u S_u$, we recover the (2,2) potential from above. On the other hand, taking $\ep =0$, we see that $T$ is just given by restriction of holomorphic sections of $\cO(6)$.
We can compare the number of holomorphic deformation parameters in the (2,2) or (0,2) superpotentials. $W$ depends on $9$ holomorphic parameters specifying section $\Sigma$. The more generic (0,2) superpotential $J$, on the other hand, depends on $16$ parameters, independent of $\ep$; as a check, we see that demanding that $J$ is integrable to $W$ reduces the parameters to $9$.
### Metrics for $\bY$ and $\cE$ {#metrics-for-by-and-ce .unnumbered}
It is well known that $\bY$ admits an ALE Kähler Ricci-flat metric with Kähler potential [^26] $$\begin{aligned}
K_{\text{CY}} = \sqrt{1+L} + \frac{1}{2} \log\frac{ \sqrt{1+L}-1}{\sqrt{1+L}+1}~, \qquad
L \equiv 4 \phi\phib(1+u\ub)^2~.\end{aligned}$$ This is obviously well-defined with respect to the patching. To leading order in the fiber coordinates, we find that up to irrelevant constants $$\begin{aligned}
K_{\text{CY}} = K + O(|\phi|^4), \qquad K = \log(1+u\ub) + (1+u\ub)^2 \phi\phib~.\end{aligned}$$ $K$ leads to a complete non-Ricci-flat metric on $X$: $$\begin{aligned}
g_X =\begin{pmatrix} g_{u\ub} & g_{u\phib} \\ g_{\phi\ub} & g_{\phi\phib} \end{pmatrix}
= \begin{pmatrix} (1+u\ub)^{-2} + 2 (1+2u\ub)\phi\phib & 2\ub\phi(1+u\ub)\\
2 u\phib(1+u\ub) & (1+u\ub)^2 \end{pmatrix}.\end{aligned}$$ To $O(|\phi|^4)$ this agrees with the Kähler metric obtained by symplectic reduction from $\C^3$.
We can also endow $\cE$ with a Hermitian metric. In our example with $\cE = \cV_\ep$, a convenient choice is $$\begin{aligned}
(\sigma,\tau) \equiv \sigma \cG \overline{\tau} ,\qquad
\cG = \begin{pmatrix} (1+u\ub)^{-2} + 2\ep\epb (1+2u\ub)\phi\phib & 2\ep\ub\phi(1+u\ub)\\
2\epb u\phib(1+u\ub) & (1+u\ub)^2 \end{pmatrix}.\end{aligned}$$ Setting $\ep =1$, we obtain a Hermitian, in fact Kähler, metric on $T_{\bY}$. Setting $\phi =0$ we obtain the bundles restricted to $B$. As we might expect, $T_{\bY}|_{B} = \cV_{\ep}|_{B} = \cO(2) \oplus \cO(-2)$.
The explicit Ricci-flat metric on $\bY$ is fairly complicated, and generalizations to other spaces are typically not available. Fortunately, we do not need the explicit form of the metric for our analysis: by construction the superpotential restricts low energy field configurations to $B$, and the details of the metric on $\bY$ away from the base become irrelevant to the IR physics.
Vertical Killing vectors {#app:VKill}
========================
In this appendix we examine holomorphic vertical Killing vectors on $\bY$ and prove that with our assumptions they act homogeneously on the fiber directions.
Let $V= V^\alpha \pp{y^\alpha} + \text{c.c.}$ be an holomorphic vector field on $\bY$, i.e. $V^\alpha_{,\betab} = 0$. Then the Killing equation for a Kähler metric $g_{\alpha\betab}$ takes the form $$\begin{aligned}
\p_\gamma (g_{\alpha\betab} V^\alpha) + \p_{\betab} (g_{\gamma\alphab}\Vb^{\alphab}) = 0~.\end{aligned}$$ Using the base/fiber decomposition $y^\alpha = (u^I,\phi^i)$, the hybrid metric has components $$\begin{aligned}
g_{I\Jb} = G_{I\Jb} + \phi h_{I\Jb}\phib,\qquad
g_{i\Jb} = h_{i\mb \Jb} \phib^{\mb},\qquad
g_{I\jb} = \phi^m h_{m\jb I},\qquad
g_{i\jb} = h_{i\jb}~.\end{aligned}$$ Since $V$ is vertical, we have $V = V^i \pp{\phi^i} + \text{c.c.}$, and a moment’s thought shows that $V^i(u,\phi)$ transforms as a section of $\pi^\ast(X)$. In this case the Killing equation reduces to $$\begin{aligned}
\p_\gamma( g_{i\betab} V^i) + \p_{\betab} (g_{\gamma\ib} \Vb^{\ib} ) = 0~,\end{aligned}$$ and decomposing it further along base/fiber directions leads to two non-trivial conditions. First, from $\betab,\gamma = \jb,k$ we obtain $$\begin{aligned}
\p_k V^i + h^{\jb i} \p_{\jb} \Vb^{\ib}_{\jb} h_{k\ib} = 0~.\end{aligned}$$ Since $h$ is $\phi$-independent and $\p_m \Vb^{\ib}_{\jb} = 0$, we conclude that $$\begin{aligned}
V^i = A^i_k(u) \phi^k + B^i(u),\qquad \Ab^{\ib}_{\kb} = (A^i_k)^\ast = -h^{\ib i} A^{k}_{i} h_{k\kb}~.\end{aligned}$$ The latter restriction on $A \in H^0(B, X\otimes X^\ast)$, combined with its holomorphy leads to $D_J A = 0$. The remaining non-trivial conditions are obtained by taking $\betab,\gamma = J, \kb$ in the Killing equation, and they lead to $D_J B = 0$ for $B \in H^0(B, X)$.
So, we have learned that vertical automorphic Killing vectors are characterized by covariantly constant sections $A \in \Gamma(X\otimes X^\ast)$ and $B \in \Gamma (X)$, with the additional restriction $$\begin{aligned}
(A^i_k)^\ast = -h^{\ib i} A^{k}_{i} h_{k\kb}~.\end{aligned}$$ In fact, we can always shift away the global section $B$ by a redefinition of the $\phi^i$; moreover, for a generic choice of metric $h$ the only solution for $A$ is a diagonal anti-Hermitian $u$-independent matrix; demanding $\cL_V W = W$ will fix the eigenvalues (up to an overall $i$) to be the charges $q_i$.
A little sheaf cohomology {#app:sheaf}
=========================
In this section we present some useful results for reducing sheaf cohomology on $\bY$ to computations on the base $B$ in the case that $X =\oplus_i L_i$. In order to compute $\bQb_0$ cohomology we need an effective method to evaluate $$\begin{aligned}
\label{eq:sheafwant}
H^\bullet_{\br}(\bY, \pi^\ast(\cE)\otimes \wedge^s T_{\bY} \otimes \wedge^t T^\ast_{\bY}),\end{aligned}$$ where $\cE$ is some bundle (or more generally sheaf) on $B$, and $\br$ is the restriction to fine grade $\br$. Recall that the grading $\br\in \Z^{n}$ assigns to every monomial $\prod_i\phi_i^{r_i}$ grade $\br = (r_1,\ldots,r_n)$; in particular $\phi_i$ has grade $\bx_i$ with $(\bx_i)_j = \delta_{ij}$. Since $\bY$ is non-compact the grade restriction is necessary to obtain a well-posed counting problem. For instance, the structure sheaf $\cO_{\bY}$ clearly has infinite-dimensional cohomology group $H^0(\bY,\cO_{\bY})$.
### Graded cohomology of a pulled-back sheaf {#graded-cohomology-of-a-pulled-back-sheaf .unnumbered}
Suppose $s=t=0$ in (\[eq:sheafwant\]). As we now show, $$\begin{aligned}
\label{eq:sheafpullback}
H^\bullet_{\br}(\bY,\pi^\ast(\cE)) \simeq H^\bullet(B, \cE\otimes \LL_{\br}),\end{aligned}$$ where $\LL_{\br} \to B$ is the line bundle $\LL_{\br} \equiv \otimes_i (L_i^\ast)^{r_i}$.
The proof follows from the basic geometry. First, to describe the line bundles $L_i\to B$, we work with a cover $\cU = \{ U_a\}$ for $B$ with local coordinates $u_a^I$ in each patch, so that on overlaps $U_{ab}\neq \emptyset$ sections of $L_i$ satisfy $$\begin{aligned}
\lambda^i_b (u_b) = \lambda^i_a(u_a) g^i_{ab}(u_a)~,\end{aligned}$$ where the $g^i_{ab}$ are the transition functions defining the bundle $L_i$. The sections $\sigma_a$ of a sheaf $\cE \to B$ satisfy $$\begin{aligned}
\sigma_b (u_b) = \sigma_a(u_a) G_{ab}(u_a)~,\end{aligned}$$ where the $G_{ab}$ are the transition functions for $\cE$, and sections of $\pi^\ast(\cE) \to \bY$ patch as $$\begin{aligned}
\label{eq:overY}
\sigma_b (u_b,\phi_b) = \sigma_a(u_a,\phi_a) G_{ab}(u_a)~,\end{aligned}$$ with $\phi^i_b = \phi^i_a g^i_{ab}(u_a)~$. Since the transition functions for $\pi^\ast(\cE)$ are identical to the transition functions for $\cE$ over $B$, at fixed grade (\[eq:overY\]) takes the form $$\begin{aligned}
\prod_i (\phi_b^i)^{r_i} \xi_b(u_b) =
\prod_i (\phi_a^i)^{r_i} \xi_a(u_a) G_{ab}(u_a) \iff
\xi_b(u_b) = \xi_a(u_a) G_{ab}(u_a) \prod_i \left[ g^i_{ab}(u_a)\right]^{-r_i}~.\end{aligned}$$ Hence the space of sections of $\pi^\ast(\cE)_{\br}$ over $\bY$ is isomorphic to the space of sections of $\cE\otimes \LL_{\br}$ over $B$. The grading is compatible with Čech cohomology (i.e. with defining chains for higher intersections $U_{a_1\cdots a_k}$ and taking cohomology of the Čech differential), and (\[eq:sheafpullback\]) holds.
### The tangent bundle {#the-tangent-bundle .unnumbered}
Having reduced the graded cohomology of a pull-back sheaf to a cohomology problem on the base, we now turn to the tangent bundle. This is of course not in general the pull-back of a sheaf from $B$, as we explictly saw in appendix \[app:simplehybrid\]. However, $T_{\bY}$ fits into a short exact sequence $$\begin{aligned}
\xymatrix{0 \ar[r]& \pi^\ast(X) \ar[r] & T_{\bY} \ar[r] &\pi^\ast(T_B) \ar[r] & 0}~.\end{aligned}$$ This is easy to see explicitly. In an open neighborhood $U_a$ a vector field $\Sigma$ takes the form $$\begin{aligned}
\Sigma_a = V_{a} \pp{u_a} + \nu_a \pp{\phi_a},\end{aligned}$$ and on overlaps $U_{ab}$ $$\begin{aligned}
V_b = V_a \ff{\p u_b}{\p u_a},\qquad \nu_b = \nu_a g_{ab} + \phi_a \cL_V g_{ab}~,\end{aligned}$$ where $g_{ab}$ are the transition functions for $X$. Hence, we see that a section $\nu$ of $X$ lifts to a section of $T_{\bY}$ with $V=0$, while a section of $T_{\bY}$ at $\phi = 0$ yields a section of $T_B$.
This short exact sequence can be decomposed with respect to the fine grading. Working again in the case $X = \oplus_i L_i$, the transition functions for sections of $T_{\bY}$ can be written explicitly as $$\begin{aligned}
(\sigma^0_b,\sigma^1_b,\ldots,\sigma^n_b) =
(\sigma^0_a,\sigma^1_a,\ldots,\sigma^n_a)
\begin{pmatrix}
\frac{\p u_b}{\p u_a} & \phi^1_a \p g^1_{ab} & \phi^2_a \p g^2_{ab} & \cdots & \phi^n_a \p g^n_{ab}\\
0 & g^1_{ab} & 0 & \cdots & 0 \\
0 & 0 & g^2_{ab} & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & g_{ab}^n
\end{pmatrix}~.\end{aligned}$$ Hence, sections of $T_{\bY}$ also admit a fine grading, which we define $$\begin{aligned}
\label{eq:Tsections}
(\Sigma)_{\br} \equiv (\sigma^0_{\br},\sigma^1_{\br+\bx_1},\sigma^2_{\br+\bx_2},\ldots,\sigma^n_{\br+\bx_n})~.\end{aligned}$$ This means the short exact sequence for $T_{\bY}$ can be decomposed according to $\br$ as $$\begin{aligned}
\label{eq:TSES}
\xymatrix{0 \ar[r]& \oplus_i (\pi^\ast L_i)_{\br+\bx_i} \ar[r] & (T_{\bY})_{\br} \ar[r] &(\pi^\ast T_B)_{\br} \ar[r] & 0}~.\end{aligned}$$ Using the induced long exact sequence on cohomology, together with (\[eq:sheafpullback\]), we can evaluate $H^{\bullet}_{\br}(\bY,T_{\bY})$. Taking appropriate products one can generalize this result to compute all desired cohomology groups in (\[eq:sheafwant\]).
We should mention one small subtlety in grading the sections of $T_{\bY}$: from (\[eq:Tsections\]) we see that there can be non-trivial contributions for $r_i =-1$. More precisely, $(T_{\bY})_{\br} = 0$ whenever any $r_i <-1$ or $r_i=r_j=-1$, and if a single $r_i =-1$ we have $$\begin{aligned}
(T_{\bY})_{\br} = (\pi^\ast L_i)_{\br+\bx_i}~,\end{aligned}$$ in which case $H^{\bullet}_{\br} (\bY,T_{\bY}) = H^{\bullet} (B, \LL_{\br})$.
### Application to $X = \cO(-2)$ and $B= \P^1$ {#application-to-x-co-2-and-b-p1 .unnumbered}
In this case the grading is one-dimensional $\br = (r)$, the grading bundle is $\LL_s = (\cO(-2)^\ast)^{s} = \cO(2s)$, and for any $r\ge 0$ the structure sheaf cohomology is $$\begin{aligned}
H^0_r (\bY,\cO_{\bY}) = H^0(B,\cO(2r)) \simeq \C^{2r+1}~,\quad
H^q_r(\bY,\cO_{\bY}) = 0,\qquad\text{for}~q>0~.\end{aligned}$$ For the tangent sheaf the short exact sequence $$\begin{aligned}
\xymatrix{0 \ar[r]& (\pi^\ast \cO(-2))_{r+1} \ar[r] & (T_{\bY})_{r} \ar[r] &(\pi^\ast \cO(2) )_{r} \ar[r] & 0}\end{aligned}$$ leads to the long exact sequence in cohomology $$\begin{aligned}
\xy {\ar(0.05,-10)*{};(0.05,-11)*{}};
\xymatrix{
0 \ar[r] & H^0(B,\cO(2r)) \ar[r] & H^0_{r} (\bY,T_{\bY}) \ar[r] & H^0(B,\cO(2r+2))
\ar@{-} `d[l]`[llld] \\
H^1(B,\cO(2r)) \ar[r] & H^1_{r} (\bY,T_{\bY}) \ar[r] & H^1(B,\cO(2r+2)) \ar[r] & 0
}
\endxy\end{aligned}$$ At grade $0$ we obtain $$\begin{aligned}
\xymatrix{
0 \ar[r] & \C \ar[r] & H^0_{0} (\bY,T_{\bY}) \ar[r] & \C^3
\\
0 \ar[r] & H^1_{0} (\bY,T_{\bY}) \ar[r] & 0 \ar[r] & 0
}\end{aligned}$$ Hence, $H^0_{0} (\bY, T_{\bY}) \simeq \C^4$, and $H^1_0(\bY,T_{\bY}) = 0$. More generally, for any non-negative grade $$\begin{aligned}
\label{eq:octictangent}
H^0_r (\bY,T_{\bY}) = H^0_r(B,\cO(2r))\oplus H^0_r(B,\cO(2r+2)) \simeq \C^{4r+4},\qquad
H^1_r(\bY,T_{\bY}) = 0~.\end{aligned}$$
### A note on horizontal representatives {#a-note-on-horizontal-representatives .unnumbered}
In order to evaluate $\bQb_0$ cohomology we needed to study the finely graded Dolbeault cohomology of horizontal forms on $\bY$ valued in a holomorphic sheaf $\cF$. One might wonder what is the relationship between these horizontal forms and more general Dolbeault classes in $H^{(0,u)}_{\pb} (\bY,\cF)$. In fact, every such class has a horizontal representative, which is why our results on finely graded cohomology describe horizontal Dolbeault cohomolgy as well. This is rather intuitive, since the fiber space is simply $\C^n$ (or $\C^n/\Gamma$ for orbi-bundles), but for completeness we give a sketch of the proof.[^27]
The statement is trivial at $u=0$, so we consider $u=1$. Let $\tau \in \ker \pb \cap \Omega^{(0,1)}(\bY,\cO_{\bY})$. In any patch $U_a$ we have $$\begin{aligned}
\tau_a = \omega_{a\Ib} d\ub^{\Ib}_{a} + \sigma_{a\ib} d\phib^{\ib}_{a}~.\end{aligned}$$ We define $\eta_a(u_a,\ub_{a},\phi_{a},\phib_\alpha)$ via the line integral $$\begin{aligned}
\eta_\alpha = \int_0^{\phib_a} ~d\zb^{\ib} \sigma_{a\ib}(u_a,\ub_a,\phi_a,\zb)~.\end{aligned}$$ Since $\pb \tau = 0$ implies $\sigma_{a \ib,\jb} =\sigma_{a\jb,\ib}$, the line integral does not depend on the choice of contour from $0$ to $\phib$; moreover, a change of variables $\zb^{\ib} = \gb^{\ib}_{ba} \yb^{\ib}$ in the integral shows that $\eta_a = \eta_b$ on any $U_{ab}\neq \emptyset$, so that $\eta$ patches to a function on $\bY$. Therefore $\tau' = \tau-\pb \eta$ is a (0,1) horizontal form, and a moment’s thought shows that $\pb \tau'=0$ implies that it has a holomorphic dependence on the fiber coordinates.
One can generalize the argument to $u>1$ and more general holomorphic sheaf $\cF \to \bY$. The analogous construction yields $\eta$, a section of $\Omega^{(0,u-1)}(\bY,\cF)$, such that $\tau'= \tau-\pb \eta$ is a horizontal representative of $[\tau] \in H^{(0,u)}_{\pb}(\bY,\cF)$.
Massless spectrum of a (0,2) CY NLSM {#app:02NLSM}
====================================
In this appendix we apply the first-order techniques developed in section \[ss:bcbg\] to marginal deformations of (0,2) NLSMs with CY target space $B$ and a left-moving $\SU(n)$ bundle $\cV$. We assume $\ch_2(\cV) = \ch_2(T_B)$ and $\cV$ is a stable bundle. This ensures that the NLSM is conformally invariant to all orders in $\alpha'$ perturbation theory [@Witten:1985bz; @Dine:1986zy]. Our techniques allow us to determine the massless spectrum to all orders in $\alpha'$. The results for the (R,R) sector and for the gauge-charged matter are exactly the same as those obtained by a Born-Oppenheimer approach in [@Distler:1987ee]. However, the massless gauge-neutral chiral matter has to our best knowledge not been studied directly in the NLSM. The first-order formulation of $\bQb_0$ cohomology turns out to be perfectly suited to this task and should be thought of as a first step in systematically including any non-perturbative world-sheet effects.
In parallel with the analysis of the $k=1$ sector in section \[ss:k1\], we first list the operators that can give rise to massless singlets. We need to slightly alter our notation in comparison to the $T_B = \cV$ analysis of section \[ss:k1\]; just in this appendix we use $I,J,\ldots$ for tangent/cotangent indices, while the $\alpha,\beta$ indices will refer to sections of the left-moving bundle $\cV$ and its dual $\cV^\ast$. We will continute to denote the bosonic coordinates by $y,\yb$. Thus, $\chi^\alpha$ ($\chib_\alpha$) transforms as a section of the pullback of $\cV$ ($\cV^\ast$). In particular, the $\chi$ kinetic term is $$\begin{aligned}
2\pi L \supset \chib_\alpha \Dbz \chi^\alpha = \chib_\alpha ( \pbz \chi^\alpha + \pbz y^I \cA^\alpha_{ I\beta} \chi^\beta)~, \end{aligned}$$ where $\cA$ is a HYM connection on $\cV$ with traceless curvature $\cF= \pb \cA$.
Using the connection, we can easily describe the full set of operators that can give rise to gauge-netural massless states in the (NS,R) sector (we ignore the universal gravitino and dilatino states and drop the normal ordering): $$\begin{aligned}
\cO^4(z) = \Psi^4_I \p y^I,\qquad
\cO^{5+6}(z) = \Psi^{5\alpha}_{\beta} \chib_\alpha \chi^\beta + \Psi^{6 I} (\rho_I - \cA^\alpha_{I \beta} \chib_\alpha \chi^\beta)~.\end{aligned}$$ As in our discussion of states in the $k=1$ sector we suppressed the expansion of each of these in $\etab$; taking that into account the wavefunctions correspond to the following bundles: $$\begin{aligned}
\Psi^4 \in \Gamma(\oplus_u \Omega^{(0,u)}(T^\ast_B))~,\qquad
\Psi^5 \in \Gamma(\oplus_u \Omega^{(0,u)}(\End \cV))~,\qquad
\Psi^6 \in \Gamma(\oplus_u \Omega^{(0,u)}(T_B))~.\end{aligned}$$ These states are $\bQb_0$ closed iff $\Psi^4$, $\Psi^5$ and $\Psi^6$ are $\pb$-closed and $$\begin{aligned}
\obs(\Psi^6) + \pb \Psi^5 = 0~,\end{aligned}$$ where $\obs(\Psi^6_u)$ is a (0,u+1) $\pb$-closed form valued in (traceless) endomorphisms of $\cV$ $$\begin{aligned}
\obs(\Psi^6)^\alpha_{\beta \Jb_0 \cdots \Jb_u} \equiv \Psi^{6I}_{[\Jb_1 \cdots\Jb_u} \cF^\alpha_{\Jb_0] I\beta}~.\end{aligned}$$ Taking cohomology, $[\obs(\Psi^6_u)] \in H^{u+1}(B,\End \cV)$. As explained in [@Anderson:2011ty], at $u=1$ this is the Atiyah class [@Atiyah:1957xx]—an obstruction to extending infinitesimal complex structure deformations of the base $B$ to infinitesimal complex structure deformations of the holomorphic bundle $\cV\to B$. Thus, our states fit into the complex $$\begin{aligned}
\xymatrix@R=1mm@C=1cm{
\cO^4_0 \ar[r]^-{\bQb_0} &
\cO^4_1 \ar[r]^-{\bQb_0} &\cO^4_2 \ar[r]^-{\bQb_0} &\cO^4_3\\
{\begin{matrix} \cO^5_0 \\ \oplus \\ \cO^6_0 \end{matrix}} \ar[r]^-{\bQb_0}&
{\begin{matrix} \cO^5_1 \\ \oplus\\
\cO^6_1
\end{matrix}} \ar[r]^-{\bQb_0} &
{\begin{matrix}
\cO^5_2 \\ \oplus\\
\cO^6_2
\end{matrix}} \ar[r]^-{\bQb_0} &
{\begin{matrix} \cO^5_3 \\ \oplus \\ \cO^6_3 \end{matrix}}
}\end{aligned}$$ Taking $\bQb_0$ cohomology we find $$\begin{aligned}
\xymatrix@R=1mm{
0 & H^1(T^\ast) \ar[r]^{0} & H^2(T^\ast) & 0 \\
{\begin{matrix} H^0(\End\cV) \\ \oplus \\H^0(T) \end{matrix}} \ar[r]^{\obs_0} &
{\begin{matrix} H^1(\End\cV) \\ \oplus \\H^1(T) \end{matrix}} \ar[r]^{\obs_1} &
{\begin{matrix} H^2(\End\cV) \\ \oplus \\H^2(T) \end{matrix}} \ar[r]^{\obs_2} &
{\begin{matrix} H^3(\End\cV) \\ \oplus \\H^3(T) \end{matrix}} .
}\end{aligned}$$ For traceless $\End \cV$ on the CY 3-fold $B$ $$\begin{aligned}
H^0(B,\End\cV) = H^3(B,\End\cV)=0~,\qquad H^2(\End\cV) \simeq \overline{H^1(B,\End\cV)}~,\end{aligned}$$ so that the complex reduces to $$\begin{aligned}
\xymatrix{
0 & H^1(T^\ast) \ar[r]^{0}& H^2(T^\ast) & 0 \\
0 &
{\begin{matrix} H^1(\End\cV) \\ \oplus \\H^1(T) \end{matrix}} \ar[r]^{\obs_1} &
{\begin{matrix} H^2(\End\cV) \\ \oplus \\H^2(T) \end{matrix}} &
0
}\end{aligned}$$ The only Atiyah obstructions arise in $H^1(B,T) \to H^2(B,\End\cV)$, and hence there are $$\begin{aligned}
h^1(T^\ast) + h^1(T) + h^1(\End\cV) - \dim\ker \obs_1\end{aligned}$$ massless gauge-neutral singlets.
The patient reader who has made it to this last appendix may perhaps be aware that in a (0,2) NLSM with a tree-level $H$-flux there are additional obstructions similar to the $H^1(B,T) \to H^2(B,\End\cV)$ map just discussed [@Melnikov:2011ez]. The $B$-field coupling will alter the $\eta$ equations of motion and lead to $H$-flux appearing in $\bQb_0 \cdot \rho$, and we expect that including this contribution should reproduce the result of [@Melnikov:2011ez]. It would be useful to check that in detail.
[^1]: Along the way we obtain a simple and direct description of the massless spectrum for the large radius limit of a (0,2) NLSM — an application to CY NLSMs with non-standard embedding may be found in appendix \[app:02NLSM\] .
[^2]: It would be interesting to find hybrid examples where these features emerge accidentally.
[^3]: Our superspace conventions are those of [@Melnikov:2011ez]; more details may be found in [@Distler:1995mi] or [@West:1990tg].
[^4]: Recall that a chiral superfield $A$ satisfies the constraints $\cDb A = \cDb' A = 0$; more general (2,2) multiplets (twisted chiral and semi-chiral) are reviewed in, for instance, [@Lindstrom:2005zr].
[^5]: A comment on Euclidean conventions: the charge conjugation operator $\cC$, inherited from Minkowski signature, conjugates the complex bosons and acts as $\cC(\chi) = \chib$ and $\cC(\chib) =-\chi$ for every fermion $\chi$.
[^6]: Suppose there is a point $p\in B$ and $\phi_0 \in \pi^{-1}(p)$ such that the Hermitian form $\phi_0 \cF_{I\Jb} h \phib_0$ has a positive eigenvalue. Then taking $\phi = t \phi_0$, for sufficiently large $t$ the metric $g$ will cease to be positive.
[^7]: If $A$ and $B$ are (0,2) superfields, then $\cD\cDb(A B) |_{\theta,\thetab =0} = 0 \quad \forall B \implies A = 0$; any chiral (anti-chiral) superfield, say $\delta \cX$ ($\delta \cXb$), can be expressed as $\cDb P$ ($\cD \Pb$) for some superfield $P$.
[^8]: If we keep the terms in $\bQb \cdot \rho$ proportional to $\etab$ equations of motion and decompose that into a $W$-independent and $W$-dependent contributions, we find that the decomposition $\bQb = \bQb_0 + \bQb_W$ into a pair of nilpotent anti-commuting operators holds without use of equations of motion; for us the result of (\[eq:supersplit\]) will be sufficient.
[^9]: Holomorphic Killing vectors satisfy $V^\alpha_{,\betab} = 0$ and $\cL_V g = 0$. They are a familiar topic in supersymmetry—see, e.g., Appendix D of [@Wess:1992cp]. Note that on a compact Kähler manifold a Killing vector field is holomorphic, but this can fail on a non-compact manifold. Killing vectors on Kähler manifolds are further discussed in [@Ballmann:2006le; @Moroianu:2007fk].
[^10]: A variety is Fano iff its anti-canonical class is ample; Fano varieties are quite special: for instance $H^i(B,\cO) =0$ for $i>0$, $\Pic(B)\simeq H^2(B,\Z) $; in addition, they are classified in dimension $d \le 3$ and admit powerful criteria for evaluating positivity of bundles [@Lazarsfeld:2004pa].
[^11]: A GLSM embedding of this hybrid model is given in section 2.5 of [@Addington:2013gpa] .
[^12]: These typically have non-trivial (a,c) rings encoded in the twisted sectors, and that ring structure is not easy to access directly via the LG orbifold description.
[^13]: The $\SO(32)$ case can be handled in an entirely analogous fashion.
[^14]: That is, schematically, in the $k$-th twisted sector fields satisfy $\phi(ze^{2\pi i},\zb e^{-2\pi i}) = [\exp(i\pi J_0)]^k\phi(z,\zb)$. We will make these periodicities more precise shortly.
[^15]: As we are working on a flat world-sheet throughout this paper, we do not keep track of the world-sheet spinor properties of the fermionic degrees of freedom.
[^16]: Since $\bQb_W$ is associated to a chiral superpotential, we do not expect it to be corrected by world-sheet instantons.
[^17]: We mean in the sense of the fiber–base geometry of $\bY$.
[^18]: This grading has a simple physical interpretation: the $W=0$ theory has $n$ $\GU(1)$ symmetries that rotate the fiber fields separately.
[^19]: The pull-back to the world-sheet is irrelevant since in the hybrid limit we consider constant maps.
[^20]: The Lie derivative has a well-defined action even when $L_{|k\ra}$ is non-trivial because $V$ is a vertical vector, while the transition functions for $L_{|k\ra}$ only depend on $B$.
[^21]: The result follows from the finite-dimensionality of the Koszul cohomology groups associated to the ideal $\la W_1,\cdots,W_n\ra \in \C[\phi_1,\ldots,\phi_n]$ for a non-singular superpotential [@Kawai:1994qy; @Melnikov:2009nh].
[^22]: Working with fields, as opposed to modes, avoids complications in patching the non-trivial bosonic oscillators on the base. These complications do not arise in sectors with $E_{|k\ra} >-1$.
[^23]: The name comes from the large radius phase of this much-studied example. Let $X_0$ be an octic hypersurface in the two-parameter toric resolution of the weighted projective space $\P^4_{\{2,2,2,1,1\}}$. The hybrid model $\cO(-2)\oplus\cO^{\oplus3}\rightarrow\P^1$ arises as one of the phases of the corresponding GLSM [[@Morrison:1994fr]]{}.
[^24]: A little care is required in using point-splitting and the free OPE in computing the action on $\cO^9$.
[^25]: When this is not the case there are, as in the (R,R) sectors, $\chi$ and $\chib$ zero-modes. It should be possible to extend the CPT discussion to these situations as well.
[^26]: Constructions of such metrics for line bundles over $\P^{n-1}$, which generalize the classic work of Eguchi and Hanson [@Eguchi:1980jx], go back to [@Calabi:1979rf; @Freedman:1981rf]; [@Higashijima:2002jt] gives an elegant generalization for line bundles over symmetric spaces. These are also the only explicitly known ALE metrics with $\SU(n)$ $n\ge 3$ holonomy [@Joyce:2000cm].
[^27]: This essentially follows the standard proof [@Bott:1982df] that $H^{k}_{dR}(\R^n,\R) =0$ for $k>0$.
|
---
abstract: 'We devise a simple modification that essentially doubles the efficiency of the BB84 quantum key distribution scheme proposed by Bennett and Brassard. We also prove the security of our modified scheme against the most general eavesdropping attack that is allowed by the laws of physics. The first major ingredient of our scheme is the assignment of significantly different probabilities to the different polarization bases during both transmission and reception, thus reducing the fraction of discarded data. A second major ingredient of our scheme is a refined analysis of accepted data: We divide the accepted data into various subsets according to the basis employed and estimate an error rate for each subset [*separately*]{}. We then show that such a refined data analysis guarantees the security of our scheme against the most general eavesdropping strategy, thus generalizing Shor and Preskill’s proof of security of BB84 to our new scheme. Up till now, most proposed proofs of security of single-particle type quantum key distribution schemes have relied heavily upon the fact that the bases are chosen uniformly, randomly and independently. Our proof removes this symmetry requirement.'
author:
- |
Hoi-Kwong Lo[^1]\
Department of Electrical and Computer Engineering;\
and Department of Physics\
University of Toronto\
10 King’s College Road, Toronto, ON Canada M5S 3G4\
H. F. Chau[^2]\
Department of Physics, University of Hong Kong,\
Pokfulam Road, Hong Kong\
and\
M. Ardehali
title: Efficient Quantum Key Distribution Scheme And Proof of Its Unconditional Security
---
Keywords: Quantum Cryptography, Quantum Key Distribution\
Introduction {#S:Intro}
============
Since an encryption scheme is only as secure as its key, key distribution is a big problem in conventional cryptography. Public-key based key distribution schemes such as the Diffie-Hellman scheme [@DH] solve the key distribution problem by making computational assumptions such as that the discrete logarithm problem is hard. However, unexpected future advances in algorithms and hardware (e.g., the construction of a quantum computer [@Shor94; @Shor95]) may render many public-key based schemes insecure. Worse still, this would lead to a [*retroactive*]{} total security break with disastrous consequences. This is because an eavesdropper may save a message transmitted in the year 2003 and wait for the invention of a new algorithm/hardware to decrypt the message decades later. A big problem in conventional public-key cryptography is that there is, in principle, nothing to prevent an eavesdropper with infinite computing power from passively monitoring the key distribution channel and thus successfully decoding any subsequent communication.
Recently, there has been much interest in using quantum mechanics in cryptography. (The subject of quantum cryptography was started by S. Wiesner [@Wiesner] in a paper that was written in about 1970 but remained unpublished until 1983. For reviews on the subject, see [@sciam; @pt; @book].) The aim of quantum cryptography has always been to solve problems that are impossible from the perspective of conventional cryptography. This paper deals with quantum key distribution [@BB84; @Mor; @Ekert] whose goal is to detect eavesdropping using the laws of physics.[^3] In quantum mechanics, measurement is not just a passive, external process, but an integral part of the formalism. Indeed, thanks to the quantum no-cloning theorem [@Dieks82; @WZ82], passive monitoring of unknown transmitted signals is strictly forbidden in quantum mechanics. Moreover, an eavesdropper who is listening to a channel in an attempt to learn information about quantum states will almost always introduce disturbance in the transmitted quantum signals [@BBM]. Such disturbance can be detected with high probability by the legitimate users. Alice and Bob will use the transmitted signals as a key for subsequent communications only when the security of quantum signals is established (from the low value of error rate).
Although various QKD schemes have been proposed, the best-known one is still perhaps the first QKD scheme proposed by Bennett and Brassard and published in 1984 [@BB84]. Their scheme, which is commonly known as the BB84 scheme, will be briefly discussed in Section \[sec:bb84\]. Here it suffices to note two of its characteristics. First, in BB84 each of the two users, Alice and Bob, chooses for each photon between two polarization bases randomly (that is, the choice of basis is a random variable), *uniformly* (that is, with equal probability) and independently. For this reason, half of the times they are using different basis, in which case the data are rejected immediately. Consequently, the efficiency of BB84 is at most 50%. Second, a simple-minded error analysis is performed in BB84. That is to say, all the accepted data (those that are encoded and decoded in the same basis) are lumped together and a *single* error rate is computed.
In contrast, in our new scheme Alice and Bob choose between the two bases randomly, independently but *not* uniformly. In other words, the two bases are chosen with *substantially different* probabilities. As Alice and Bob are now much more likely to be using the same basis, the fraction of discarded data is greatly reduced, thus achieving a significant gain in efficiency. In fact, we are going to show in this paper that the efficiency of our scheme can be made asymptotically close to unity. (The so-called orthogonal quantum cryptographic schemes have also been proposed. They use only a single basis of communication and, according to Goldenberg, it is possible to use them to achieve efficiencies greater than $50\%$ [@gold; @koashi]. Since they are conceptually rather different from what we are proposing, we will not discuss them here.)
Is the new scheme secure? If a simple-minded error analysis like the one that lumps all accepted data together were employed, an eavesdropper could easily break a scheme by eavesdropping mainly along the predominant basis. To ensure the security of our scheme, it is crucial to employ a refined data analysis. That is to say, the accepted data are further divided into two subsets according to the actual basis used by Alice and Bob and the error rate of each subset is computed *separately*. We will argue in this paper that such a refined error analysis is sufficient in ensuring the security of our improved scheme, against the *most general* type of eavesdropping attack allowed by the laws of quantum physics. This is done by using the technique of Shor and Preskill’s proof [@shorpre] of security of BB84 — a proof that built on the earlier work of Lo and Chau [@qkd] and of Mayers [@mayersqkd].
Our scheme is worth studying for several reasons. First, unlike the entanglement-based QKD scheme proposed by Lo and Chau in Ref. [@qkd], the implementation of our new scheme does *not* require a quantum computer. It only involves the preparation and measurement of single photons as in standard BB84. Second, none of the existing schemes based on non-orthogonal quantum cryptography has an efficiency more than $50\%$. (We shall say a few word on the so-called orthogonal quantum cryptography in Section \[sec:conclusion\].) By showing in this paper that the efficiency of our new scheme can be made asymptotically close to 100%, we know that QKD can be made arbitrarily efficient. Our idea is rather general and can be applied to improve the efficiency of some other existing single particle based QKD schemes such as the six-state scheme[@bruss; @six]). Note that the efficiency of quantum cryptography is of practical importance because it may play an important role in deciding the feasibility of practical quantum cryptographic systems in any future application. Third, our scheme is one of the few QKD schemes whose security have been rigorously proven. Finally, all previous proofs of security seem to rely heavily on the fact that the two bases are chosen randomly and uniformly. Our proof shows that such a requirement is redundant. Another advantage of our security proof is that it does not depend on asymptotic argument and hence can be applied readily to realistic situation involving only a relatively small amount of quantum signal transmission.
The organization of our paper is as follows. The basic features and the requirements of unconditional security will be reviewed in Section \[sec:basic\]. In Section \[sec:bb84\], we will review the BB84 scheme and Shor-Preskill proof for completeness. Readers who are already familiar with the BB84 scheme and Shor-Preskill proof may browse through Section \[sec:basic\] and skip Section \[sec:bb84\]. An overview of our proof of security of an efficient QKD scheme will be given in Section \[sec:overview\], which is followed by Section \[s:proof\] which ties up some loose ends. Finally, we give some concluding remarks in Section \[sec:conclusion\].
Basic Features and Requirements of a Quantum Key Distribution Scheme {#sec:basic}
====================================================================
basic procedure {#subsec:procedure}
---------------
The aim of a QKD scheme is to allow two cooperative participants (commonly known as Alice and Bob) to establish a common secret key in the presence of noise and eavesdropper (commonly known as Eve) by exploiting the laws of quantum physics. More precisely, it is commonly assumed that Alice and Bob share a small amount of initial authentication information. The goal is then to expand such a small amount of authentication information into a long secure key. In almost all QKD schemes proposed so far, Alice and Bob are assumed to have access to a classical public unjammable channel as well as a quantum noisy insecure channel. That is to say, we assume that everyone, including the eavesdropper Eve, can listen to the conversations but cannot change the message that send through the public classical channel. In practice, an authenticated classical channel should suffice. On the other hand, the transmission of quantum signal can be done through free air [@exp; @Buttler; @free] or optical fibers [@hughes00; @MZG95; @townsend98] in practice. The present state-of-the-art quantum channel for QKD can transmit signals up to a rate of $4\times 10^5$ qubits per second over a distance of about 10 km with an error rate of a few percent [@Buttler; @hughes00; @townsend98].[^4] The quantum channel is assumed to be insecure. That is to say that the eavesdropper is free to manipulate the signal transmitted through the quantum channel as long as such manipulation is allowed by the known laws of physics.
Using the above two channels, procedures in all secure QKD schemes we know of to date can be divided into the following three stages:
1. *Signal Preparation And Transmission Stage*: Alice and Bob separately prepare a number of classical and quantum signals. They may keep some of them private and transmit the rest to the other party using the secure classical and insecure quantum channels. They may iterate the signal preparation and transmission process a few times.
2. *Signal Quality Check Stage*: Alice and Bob then (use their private information retained in the signal preparation and transmission stage, the secure classical channel and their own quantum measurement apparatus to) test the fidelity of their exchanged quantum signals that have just been transmitted through the insecure and noisy quantum channel. Since a quantum measurement is an irreversible process some quantum signals are consumed in this signal quality check stage. The aim of their test is to estimate the noise and hence the upper bound for the eavesdropping level of the channel from the sample of quantum signals they have measured. In other words, the process is conceptually the same as a typical quantity control test in a production line — to test the quality of products by means of destructive random sampling tests. Alice and Bob abort and start all over again in case they believe from the result of their tests that the fidelity of the remaining quantum signal is not high enough. Alice and Bob proceed to the final stage only if they believe from the result of their tests that the fidelity of the remaining quantum signal is high.
3. *Signal Error Correction and Privacy Amplification Stage*: Alice and Bob need to correct errors in their remaining signals. Moreover, they would like to remove any residual information Eve might still have on the signals. In other words, Alice and Bob would like to distill from the remaining untested quantum signals a smaller set of almost perfect signals without being eavesdropped or corrupted by noise. We call this process privacy amplification. Finally, Alice and Bob make use of these distilled signals to generate their secret shared key.
security requirement
--------------------
A QKD scheme is said to be secure if, for any eavesdropping strategy by Eve, either a) it is highly unlikely that the state will pass Alice and Bob’s quality check stage or b) with a high probability that Alice and Bob will share the same key, which is essentially random and, furthermore, Eve has a negligible amount of information on their shared key.[^5]
Bennett and Brassard’s Scheme (BB84) {#sec:bb84}
====================================
Basic idea of the BB84 scheme {#subsec:basicidea}
-----------------------------
We now briefly review the basic ingredients of the BB84 scheme and the ideas behind its security. Readers who are already familiar with BB84 and the Shor-Preskill proof may choose to skip this section to go directly to our biased scheme in Section 4. In BB84 [@BB84], Alice prepares and transmits to Bob a batch of photons each of which is independently in one of the four possible polarizations: horizontal ($0^\circ$), vertical ($90^\circ$), $45^\circ$ and $135^\circ$. For each photon, Bob randomly picks one of the two (rectilinear or diagonal) bases to perform a measurement. While the measurement outcomes are kept secret by Bob, Alice and Bob publicly compare their bases. They keep only the polarization data that are transmitted and received in the same basis. Notice that, in the absence of noises and eavesdropping interference, those polarization data should agree. This completes the signal preparation and transmission stage of the BB84 scheme. We remark that the laws of quantum physics strictly forbid Eve to distinguish between the four possibilities with certainty. This is because the two polarization bases, namely rectilinear and diagonal, are complementary observables and quantum mechanics forbids the simultaneous determination of the eigenvalues of complementary observables.[^6] More importantly, any eavesdropping attack will lead to a disagreement in the polarization data between Alice and Bob, which can be detected by them through public classical discussion. More concretely, to test for tampering in the signal quality check stage, Alice and Bob choose a random subset of the transmitted photons and publicly compare their polarization data. If the quantum bit error rate (that is, the fraction of polarization data that disagree) is unreasonably large, they throw away all polarization data and start all over again. On the other hand, if the quantum bit error rate is acceptably small, they should then move on to the signal error correction and privacy amplification stage by performing public classical discussion to correct remaining errors.
Proving security of a QKD scheme turned out to be a very tricky business. The problem is that, in principle, Eve may have a quantum computer. Therefore, she could employ a highly sophisticated eavesdropping attack by entangling all the quantum signals transmitted by Alice. Moreover, she could wait to hear the subsequent classical discussion between Alice and Bob during both the signal quality check and the error correction and privacy amplification stages before making any measurement on her system.[^7] One class of proofs by Mayers [@mayersqkd] and subsequently others [@benor; @biham] proved the security of the standard BB84 directly. Those proofs are relatively complex. Another approach by Lo and Chau [@loqkd; @qkd] dealt with schemes that are based on quantum error-correcting codes. It has the advantage of being conceptually simpler, but requires a quantum computer to implement. These two classes of proofs have been linked up by the recent seminal work of Shor and Preskill [@shorpre], who provided a simple proof of security of the BB84 scheme. They showed that an eavesdropper is no better off with standard BB84 than a QKD scheme based on a specific class of quantum error-correcting codes. So long as from Eve’s view, Alice and Bob [*could have*]{} performed the key generation by using their quantum computers, one can bound Eve’s information on the key. It does not matter that Alice and Bob did not really use quantum computers.
entanglement purification
-------------------------
To recapitulate Shor and Preskill’s proof, we shall first introduce a QKD scheme based on entanglement purification and prove its security. Our discussion in the next few subsections essentially combines those of Shor and Preskill[@shorpre] and Gottesman and Preskill[@squeeze].[^8]
Entanglement purification was first proposed by Bennett, DiVincenzo, Smolin and Wootters (BDSW) [@BDSW]. Its application to QKD was first proposed by Deutsch [*et al.*]{} [@deutsch]. A convincing proof of security based on entanglement purification was presented by Lo and Chau [@qkd]. Finally, Shor and Preskill[@shorpre] noted its connection to BB84.
Suppose two distant observers, Alice and Bob, share $n$ impure EPR pairs. That is to say, some noisy version of the state $$| \Phi^{(n)} \rangle = | \Phi^+ \rangle^{\otimes n}$$ where $ | \Phi^+ \rangle =
{ 1 \over \sqrt 2} ( | 00 \rangle + | 11 \rangle ) $. They may wish to distill out a smaller number, say $k$, pairs of perfect EPR pairs, by applying only classical communications and local operations. This process is called entanglement purification [@BDSW]. Suppose they succeed in generating $k$ perfect EPR pairs. By measuring the resulting EPR pairs along a common axis, Alice and Bob can obtain a secure $k$-bit key.
Of course, a quality check stage must be added in QKD to guarantee the likely success of the entanglement purification procedure (for any eavesdropping attack that will pass the quality check stage with a non-negligible probability). A simple quality check procedure is for Alice and Bob to take a random sample of the pairs and measure each of them randomly along either $X$ or $Z$ axis and compute the bit error rate (i.e., the fraction in which the answer differs from what is expected from an EPR pair). Suppose they find the bit error rates for the $X$ and $Z$ bases of the sample to be $p_X$ and $p_Z$ respectively. For a sufficiently large sample size, the properties of the sample provide good approximations to those of the population. Therefore, provided that the entanglement purification protocol that they employ can tolerate slightly more than $p_X$ and $p_Z$ errors in the two bases, we would expect that their QKD scheme is secure. This point will be proven in subsequent discussions in subsection \[subsec:quality\].
Let us introduce some notations.
[*Definition: Pauli operators.*]{} We define a Pauli operator acting on $n$ qubits to be a tensor product of individual qubit operators that are of the form $I = \pmatrix{1 & 0 \cr 0 & 1}$, $X = \pmatrix{0 & 1 \cr 1 & 0}$, $Y = \pmatrix{0 & -i \cr i & \ 0}$ and $Z = \pmatrix{1 & \ 0 \cr 0 & -1}$.
For example, ${\cal P} = X \otimes I \otimes Y \otimes Z$ is a Pauli operator.
We shall consider entanglement purification protocols that can be conveniently described by [*stabilizers*]{}[@gottesman96; @gottesmanthesis]. A stabilizer is an Abelian group whose generators, $M_i$’s, are Pauli operators.
Consider a fixed but arbitrary $[[n,k,d]]$ stabilizer-based quantum error-correcting code (QECC). The notation $[[n,k,d]]$ means that it encodes $k$ logical qubits into $n$ physical qubits with a minimum distance $d$. As noted in [@BDSW], the encoding and decoding procedure of Alice and Bob can be equivalently described by a set of Pauli operators, $M_i$, with both Alice and Bob measuring the same operator $M_i$. To generate the final key from the encoded qubits, Alice and Bob eventually apply a set of operators, say $\bar{Z}_{a,A}$ and $\bar{Z}_{a,B}$ respectively, for $a= 1, 2,
\cdots, k$. In Shor and Preskill’s proof, all Alice’s (Bob’s respectively) operators commute with each other.
If the $n$ EPR pairs were perfect, Alice and Bob would obtain identical outcomes for their measurements, $ M_{i,A}$ and $M_{i,B}$. Moreover, because of the commutability of the operators, those measurements would not disturb the encoded operations, $\bar{Z}_{a,A} \otimes \bar{Z}_{a,B}$, each of which will give $+1$ as its eigenvalue for the state of $n$ perfect EPR pairs. This is because measurements $\bar{Z}_{a,A}$ and $\bar{Z}_{a,B}$ produce the same $+1$ or $-1$ eigenvalues.
What about $n$ noisy EPR pairs? Suppose Alice and Bob broadcast their measurement outcomes for $ M_{i,A}$ and $M_{i,B}$ respectively. The product of their measurement outcomes of $ M_{i,A}$ and $M_{i,B}$ gives the error syndrome of the state, which is now noisy. Since the original QECC can correct up to $t \equiv \lfloor { d -1 \over 2}
\rfloor$ errors, intuitively, provided that the number of bit-flip and phase error errors are each less than $t$, Alice and Bob will successfully correct the state to obtain the $k$ encoded EPR pairs. Now, they can measure the encoded operations $\bar{Z}_{a,A} \otimes \bar{Z}_{a,B}$ to obtain a secure $k$-bit key.
Reduction to Pauli strategy {#subsec:quality}
---------------------------
[*Definition: Correlated Pauli strategy.*]{} Recall that a Pauli operator acting on $n$ qubits is defined to be a tensor product of individual qubit operators that are of the form $I$, $X$, $Y$ and $Z$. We define a correlated Pauli strategy, $({\cal P}_i , q_i)$, to be one in which Eve applies only Pauli operators. That is to say that Eve applies a Pauli operator ${\cal P}_i$ with a probability $q_i$.
The argument in the last subsection is precise only for a specific class of eavesdropping strategies, namely the class of correlated Pauli strategies. In this case, the numbers of bit-flip and phase errors are, indeed, well-defined. What about a general eavesdropping attack? In general, Alice and Bob’s system is entangled with Eve’s system. Does it still make any sense to say that Alice and Bob’s system has no more than $t$ bit-flip errors and no more than $t$ phase errors? Surprisingly, it does. Instead of having to consider all possible eavesdropping strategies by Eve, it turns out that it is sufficient to consider the Pauli strategy defined above. In other words, one can assume that Eve has applied some Pauli operators, i.e., tensor products of single-qubit identities and Pauli matrices, on the transmitted signals with some [*classical*]{} probability distribution. More precisely, it can be shown that the fidelity of the recovered $k$ EPR pairs is at least as big as the probability that i) $t$ or fewer bit-flip errors and ii) $t$ or fewer phase errors [*would have*]{} been found if a Bell-measurement had been performed on the $n$ pairs.
Mathematically, the insight can be stated as the following theorem:
[**Theorem 1 (from [@squeeze; @shorpre; @qkd])**]{}: Suppose Alice and Bob share a bipartite state of $n$ pairs of qubits and they execute a stabilizer-based entanglement purification procedure that can be described by the measurement operators, $M_i$, with both Alice and Bob measuring the same $M_i$. Suppose further that the procedure leads to a $[[n,k,d]]$ QECC which corrects $t \equiv \lfloor { d - 1 \over 2} \rfloor $ bit-flip errors and also $t$ phase errors. Then, the fidelity of the recovered state, after error correction, as $k$ EPR pairs $$F \equiv \langle \bar{\Phi}^{(k)} | \rho_R | \bar{\Phi}^{(k)}
\rangle \geq
{\rm Tr} \left( \Pi_S \rho \right) .
\label{e:reduction}$$ Here, $\bar{\Phi}^{(k)}$ is the encoded state of $k$ EPR pairs, $\rho_R $ is the density matrix of the recovered state after quantum error correction, $\rho$ is the density matrix of the $n$ EPR pairs before error correction and $\Pi_S$ represents the projection operator into the Hilbert space, called ${\cal H}_{\rm good}$, which is spanned by Bell pairs states that differ from $n$ EPR pairs in no more than $t$ bit-flip errors and also no more than $t$ phase errors.
[**Proof of Theorem 1**]{}:
One can regard $\rho$ as the reduced density matrix of some pure state $ | \Psi \rangle_{SE}$ which describes the state of the system, $S$ and an ancilla (the environment, $E$, outside Alice and Bob’s control). Now, in the recovery procedure, Alice and Bob couple some auxiliary reservoir, $R$, prepared in some arbitrary initial state, $| 0 \rangle_R$, to the system. Initially, let us decompose the pure state $| \Psi \rangle_{SE} \otimes | 0 \rangle_R $ into a “good” component and a “bad” component, where the good component is defined as: $$| \Psi_{good} \rangle =
( \Pi_S \otimes I_{ER} ) | \Psi \rangle_{SE} \otimes | 0 \rangle_R$$ and the bad component is given by: $$| \Psi_{bad} \rangle =
( ( I_S - \Pi_S ) \otimes I_{ER} )
| \Psi \rangle_{SE} \otimes | 0 \rangle_R .$$
Now, the recovery procedure will map the two components, $| \Psi_{good} \rangle $ and $ | \Psi_{bad} \rangle $, unitarily into $| \Psi'_{good} \rangle $ and $ | \Psi'_{bad} \rangle $. Since the recovery procedure works perfectly in the subspace, ${\cal H}_{\rm good}$, we have $$| \Psi'_{good} \rangle = | \bar{\Phi}^{(k)} \rangle_S
\otimes | junk \rangle_{ER}.$$
Let us consider the norm of the good component: $$\begin{aligned}
\langle \Psi'_{good} | \Psi'_{good} \rangle & =&
\langle \Psi_{good} | \Psi_{good} \rangle \nonumber \\
& =& {\rm Tr} \left( \Pi_S \rho \right) . \end{aligned}$$
Now, the fidelity of the final state as an $k$-EPR pairs is given by: $$\begin{aligned}
F &=& ~_{SER}\langle \Psi' | \left( | \bar{\Phi}^{(k)} \rangle_S
~_S \langle \bar{\Phi}^{(k)} | \right) \otimes I_{ER} | \Psi'
\rangle_{SER}
\label{e:fidelity1} \\
~ &=& ~_{SER}\langle \Psi'_{good} | \left( | \bar{\Phi}^{(k)}
\rangle_S
~_S \langle \bar{\Phi}^{(k)} | \right) \otimes I_{ER} |
\Psi'_{good} \rangle_{SER}
\nonumber \\
~ & & + ~_{SER}\langle \Psi'_{bad} | \left( | \bar{\Phi}^{(k)}
\rangle_S
~_S \langle \bar{\Phi}^{(k)} | \right) \otimes I_{ER} |
\Psi'_{bad} \rangle_{SER}
\nonumber \\
~ & & + ~_{SER}\langle \Psi'_{good} | \left( | \bar{\Phi}^{(k)}
\rangle_S
~_S \langle \bar{\Phi}^{(k)} | \right) \otimes I_{ER} |
\Psi'_{bad} \rangle_{SER}
\nonumber \\
~ & & + ~_{SER}\langle \Psi'_{bad} | \left( | \bar{\Phi}^{(k)}
\rangle_S
~_S \langle \bar{\Phi}^{(k)} | \right) \otimes I_{ER} |
\Psi'_{good} \rangle_{SER}
\label{e:fidelity2} \\
~ &=& {\rm Tr} \left( \Pi_S \rho \right) \nonumber \\
~ & & + ~_{SER}\langle \Psi'_{bad} | \left( | \bar{\Phi}^{(k)}
\rangle_S
~_S \langle \bar{\Phi}^{(k)} | \right) \otimes I_{ER} |
\Psi'_{bad} \rangle_{SER}
\nonumber \\
~ & & + ~_{SER}\langle \Psi'_{good} | \Psi'_{bad} \rangle_{SER}
\nonumber \\
~ & & + ~_{SER}\langle \Psi'_{bad} | \Psi'_{good} \rangle_{SER}
\label{e:fidelity3} \\
~ &=& {\rm Tr} \left( \Pi_S \rho \right) \nonumber \\
~ & & + ~_{SER}\langle \Psi'_{bad} | \left( | \bar{\Phi}^{(k)}
\rangle_S
~_S \langle \bar{\Phi}^{(k)} | \right) \otimes I_{ER} |
\Psi'_{bad} \rangle_{SER}
\label{e:fidelity4} \\
~ &\geq & {\rm Tr} \left( \Pi_S \rho \right)
\label{e:fidelity5} \end{aligned}$$ where the orthogonality of the states, $ | \Psi'_{good}
\rangle_{SER}$ and $ | \Psi'_{bad} \rangle_{SER} $, is used in Eq. (\[e:fidelity4\]). Q.E.D.
quality check procedure
-----------------------
In the last subsection, we showed that, provided that a Bell measurement, if had been performed, would have shown that the numbers of bit-flip errors and phase errors are both no more than $t$, Alice and Bob will succeed in generating a secure key. In reality, there is no way for two distant observers, Alice and Bob, to verify such a condition directly. Fortunately, Alice and Bob can perform some quality check procedure by randomly sampling their pairs. We have the following Proposition:
[**Proposition 1 ([@qkd], particularly, its supplementary notes VI)**]{}: Suppose Alice prepares $N$ EPR pairs and sends a half of each pair to Bob via a noisy channel (perhaps controlled by Eve). Alice and Bob may randomly select $m$ of those pairs and perform a random measurement along either the $X$ or the $Z$ axis. Suppose, for the moment, that they compute the bit error rates of the tested sample in the two bases separately, thus obtaining $p_X^{sample}$ and $p_Z^{sample}$. Then, these two error rates are good estimates of those of the population (and therefore, also the remaining untested pairs). In particular, one can apply [*classical*]{} random sampling theory to estimate confidence levels for the error rates in the two bases for the population (and thus the untested pairs).
[*Proof of Proposition 1*]{}: Let us summarize the overall strategy of the proof. One imagines applying the mathematical operation of Bell measurements on the $N$ imperfect EPR pairs before the error correction procedure, but [*after*]{} Eve’s eavesdropping. Consider the resulting state. It could have been obtained by a different eavesdropping strategy on the part of Eve, which applies Pauli operators to the N-EPR-pair state with some probability distribution. Finally, it suffices to consider only this limited class of eavesdropping strategies.
Let us consider the state of the $N$ EPR pairs after Eve’s eavesdropping attack. For each of the $m$ tested pair along the $Z$-basis, consider the projection operators, $ P^{i, z}_{||}$ and $ P^{i, z}_{anti-||}$ for the two [*coarse-grained*]{} outcomes (parallel and anti-parallel) of the measurement performed on the $i$-th pair. Specifically, $$\begin{aligned}
P^{i,z}_{||} & = & |00\rangle_i \,\langle 00|_i + |11\rangle_i
\,\langle 11|_i
\nonumber \\
& = & |\Phi^+\rangle_i \,\langle \Phi^+|_i + |\Phi^-\rangle_i
\,\langle
\Phi^-|_i , \end{aligned}$$ $$\begin{aligned}
P^{i,z}_{anti-||} & = & |01\rangle_i \,\langle 01|_i + |10\rangle_i
\,\langle
10|_i \nonumber \\
& = & |\Psi^+\rangle_i \,\langle \Psi^+|_i + |\Psi^-\rangle_i
\,\langle
\Psi^-|_i , \end{aligned}$$ where $|\Phi^\pm\rangle = \frac{1}{\sqrt{2}} (|00\rangle \pm
|11\rangle)$ and $|\Psi^\pm\rangle = \frac{1}{\sqrt{2}} (|01\rangle \pm |10\rangle)$.
Similarly, for each of the $m$ test pair along the $X$-axis, consider the projection operators, $ P^{k, x}_{||}$ and $ P^{k, x}_{anti-||}$, for the two [*coarse-grained*]{} outcomes (parallel and anti-parallel) of the measurement performed on the $k$-th tested pair. Namely, $$\begin{aligned}
P^{k,x}_{||} &
= & \frac{1}{4} ( |0\rangle_k + |1\rangle_k ) \otimes (|0\rangle_k
+
|1\rangle_k )
(\langle 0|_k + \langle 1|_k) \otimes (\langle 0|_k + \langle 1|_k) \nonumber \\
& & +\frac{1}{4} ( |0\rangle_k - |1\rangle_k ) \otimes (|0\rangle_k
-
|1\rangle_k )
(\langle 0|_k - \langle 1|_k) \otimes (\langle 0|_k - \langle 1|_k) \nonumber \\
& = & |\Phi^+\rangle_k \,\langle\Phi^+|_k
+ |\Psi^+\rangle_k \,\langle\Psi^+|_k , \end{aligned}$$ $$\begin{aligned}
P^{k,x}_{anti-||} &
= & \frac{1}{4} ( |0\rangle_k + |1\rangle_k ) \otimes (|0\rangle_k
-
|1\rangle_k )
(\langle 0|_k + \langle 1|_k) \otimes (\langle 0|_k - \langle 1|_k) \nonumber \\
& & + \frac{1}{4} ( |0\rangle_k + |1\rangle_k ) \otimes
(|0\rangle_k -
|1\rangle_k )
(\langle 0|_k + \langle 1|_k) \otimes (\langle 0|_k - \langle 1|_k) \nonumber \\
& = & |\Phi^-\rangle_k \,\langle\Phi^-|_k
+ |\Psi^-\rangle_k \,\langle\Psi^-|_k . \end{aligned}$$
The above four equations clearly show that using local operations and classical communications only (LOCCs), Alice and Bob can effectively perform a coarse-grained Bell’s measurement with these four projection operators.
Now, consider the operator, $M_B$, which represents a complete measurement along $N$-Bell basis. Since $M_B$, $ P^{i, x}_{||}$, $ P^{i, x}_{anti-||}$, $ P^{k, z}_{||}$ and $ P^{k, z}_{anti-||}$ all refer to a single basis (namely, the $N$-Bell basis), they clearly commute with each other. Therefore, they can be simultaneously diagonalized. Thus, a pre-measurement $M_B$ by say Eve will in no way change the outcome for $ P^{i, x}_{||}$, $ P^{i, x}_{anti-||}$, $ P^{k, z}_{||}$ and $ P^{k, z}_{anti-||}$. Therefore, we may as well consider the case when such a pre-measurement is performed. By doing so, we have reduced the most general eavesdropping strategy to a restricted class that involves only Pauli operators. Consequently, the problem of estimation of the error rates of the two bases is classical. Q.E.D.
We emphasize that the key insight of Proposition 1 is the “commuting observables” idea: Consider the set of Bell measurements, $X \otimes X$ and $Z \otimes Z$, on all pairs of qubits. All such Bell measurements commute with each other. Therefore, without any loss of generality, we can assign classical probabilities to their simultaneous eigenstates and perform classical statistical analysis. This greatly simplifies the analysis.
More concretely, provided that total number of the EPR pairs goes to infinity, the classical de Finetti’s theorem applies to the random test sample of $m$ pairs. Moreover, for a sufficiently large $N$, it is common in classical statistical theory to assume a normal distribution and use it to estimate the mean of the population and establish confidence levels. Therefore, with a high confidence level, for the remaining untested pairs, the error rates $p_X^{untested} < p_X^{sample} +
{\varepsilon}$ and $p_Z^{untested} < p_Z^{sample} + {\varepsilon}$.
The next question is: how do the two error rates (for the $X$ and $Z$ bases) relate to the bit-flip and phase errors in the underlying quantum error correcting code? Suppose, as in our discussion so far, Alice and Bob generate their final key by measuring along the $Z$-axis only. In this case, it should not be hard to see that the bit-flip error has an error rate $p_Z^{untested}$ and the phase error has an error rate $p_X^{untested}$.
However, in BB84, it is common practice to allow Alice and Bob to generate the key by measuring each pair along either the $X$ or $Z$-axis with uniform probabilities. Mathematically, as discussed in [@shorpre; @six], this is equivalent to Alice’s applying either i) a Hadamard transform or ii) an identity operator to the qubit before sending it to Bob. Therefore, in this case, it should not be too hard to see that the bit-flip error is given by the averaged error rate $( p_X^{untested} + p_Z^{untested})/2$ of the two bases. Similarly, the phase error rate is given by the same expression. For this reason, it is, in fact, unnecessary in Shor and Preskill’s proof for Alice and Bob to compute the two error rates separately. In other words, a simple-minded error analysis in which they lump all polarization data (from both rectilinear or diagonal bases) together and compute a single sample bit error rate, call it $e^{sample}$ is sufficient for the quality check stage.
Now, suppose a QECC $[[n,k,d]]$ is chosen such that the maximal tolerable error rate, $e^{max} = {t \over n} \equiv {\lfloor { (d - 1) /2} \rfloor \over
n}
> e^{sample} + {\varepsilon}$. Then, for any eavesdropping strategy that will pass the quality check stage with a non-negligible probability, it is most likely that the remaining untested $n$ EPR pairs will have less then $t$ bit-flip errors and also less than $t$ phase errors. Therefore, the error correction will most likely succeed and Alice and Bob will share a $k$-EPR-pair state with high fidelity.
The following theorem shows that once Alice and Bob share a high fidelity $k$-EPR-pair state, then they can generate a key such that the eavesdropper’s mutual information is very small.
[**Theorem 2**]{} ([@qkd]): Suppose two distant observers, Alice and Bob, share a high fidelity $k$-EPR-pair state, $\rho$, such that $ \langle \Phi^{(k)} | \rho | \Phi^{(k)} \rangle > 1 - \delta$ where $\delta \ll 1$ and they generate a key by measuring the state along say the $Z$-axis, then the eavesdropper’s mutual information on the key is bounded by $$S (\rho) <
- ( 1 - \delta) \log_2 ( 1 - \delta) - \delta \log_2 { \delta \over ( 2^{2k} - 1)} = \delta \times \left( { 1 \over \log_e 2} + 2k +
\log_2 ( 1/ \delta) \right) + O (\delta^2).$$
[**Proof**]{}: Let us recapitulate the proof presented in Section II of supplementary material of [@qkd]. The proof consists of two Lemmas. Lemma A says that high fidelity implies low entropy. Lemma B says that the entropy is a bound to the eavesdropper’s mutual information with Alice and Bob.
More concretely, Lemma A says the following: If $ \langle \Phi^{(k)}| \rho | \Phi^{(k)} \rangle >
1 - \delta $ where $\delta \ll 1$, then the von Neumann entropy satisfies $S (\rho) < - ( 1 - \delta) \log_2 ( 1 - \delta) -
\delta \log_2 { \delta \over ( 2^{2k} -1 )} $. Proof of Lemma A: If $ \langle \Phi^{(k)}| \rho | \Phi^{(k)} \rangle >
1 - \delta $, then the largest eigenvalue of the density matrix $\rho$ must be larger than $ 1 - \delta$. Therefore, the entropy of $\rho$ is, bounded above by that of a density matrix, $\rho_0 =
diag ( 1 - \delta, { \delta \over ( 2^{2k} -1 )},
{ \delta \over ( 2^{2k} -1 )}, \cdots, { \delta \over ( 2^{2k} -1 )}
)$, which has an entropy $- ( 1 - \delta) \log_2 ( 1 - \delta) -
\delta \log_2 { \delta \over ( 2^{2k} -1 )} $.
Lemma B, which is a corollary of Holevo’s theorem [@holevo], says the following: Given any pure state $\phi_{A'B'}$ of a system consisting of two subsystems, $A'$ and $B'$, and any generalized measurements $X$ and $Y$ on $A'$ and $B'$ respectively, the entropy of each subsystem $S( \rho_{A'})$ (where $\rho_{A'}$ is the reduced density matrix, $Tr_{B'} | \phi_{A'B'} \rangle \langle \phi_{A'B'} | $) is an upper bound to the amount of mutual information between $X'$ and $Y'$.
Now, Suppose Alice and Bob share a bipartite state $\rho_{AB}$ of fidelity $ 1- \delta$ to $k$ EPR pairs. By applying Lemma A, one shows that the entropy of $\rho_{AB}$ is bounded by $S (\rho) < - ( 1 - \delta) \log_2 ( 1 - \delta) -
\delta \log_2 { \delta \over ( 2^{2k} -1 )} $.
Let us now introduce Eve to the picture and consider the system consisting of the subsystem, $A'$, of Eve and the subsystem, $B'$, of combined Alice-Bob. (i.e., $B' = AB$.) Let us consider the most favorable situation for Eve where she has perfect control over the environment. In this case, the overall (Alice-Bob-Eve) system wavefunction can be described by a pure state, $\phi_{A'B'}$ where Eve controls $A'$ and the combined Alice-Bob controls $B'$. By Lemma B, Eve’s mutual information with Alice-Bob’s system is bounded by $( 1 - \delta) \log_2 ( 1 - \delta)
- \delta \log_2 { \delta \over ( 2^{2k} -1 )} $. Q.E.D.
[*Remark 1*]{}: It is not too hard to see that Alice and Bob will most likely share a common key that is essentially random in the above procedure.
[*Remark 2*]{}: Suppose we limit the eavesdropper’s information, $I_{eve}$, to be less than ${\varepsilon}$, Theorem 2 shows that, as the length, $k$ of the final key increases, the allowed infidelity, $\delta$, of the state must decrease at least as $ O (1/k)$.
reduction to BB84
-----------------
Shor and Preskill considered a special class of quantum error correcting codes, namely Calderbank-Shor-Steane (CSS) codes. They showed that a QKD that employs an entanglement purification protocol (EPP) based on a CSS code can be reduced to BB84. Let us follow their arguments in two steps.
### from entanglement purification protocol to quantum error-correcting code protocol
From the work of BDSW [@BDSW], it is well known that any entanglement purification protocol with only one-way classical communications can be converted into a quantum error-correcting code. Shor and Preskill applied this result to an EPP-based QKD scheme. Let us recapitulate the procedure of an EPP-based QKD scheme. Alice creates $N$ EPR pairs and sends half of each pair to Bob. She then measures the check bits and compares them with Bob. If the error rate is not too high, Alice then measures $M_{i,A}$ and publicly announces the outcomes to Bob, who measures $M_{i,B}$. This allows Alice and Bob to correct errors and distill out $k$ perfect EPR pairs. Alice and Bob then measure $\bar{Z}_{a,A}$ and $\bar{Z}_{a,B}$, the encoded $Z$ operators, to generate the key.
Note that, by locality, it does not matter whether Alice measures the check bits before or after she transmits halves of EPR pairs to Bob. Similarly, it does not matter whether Alice measures her syndrome (i.e., the stabilizer elements, $M_{i,A}$) before or after the transmission. Now, if she measures her check bits before the transmission, it is equivalent to choosing a random BB84 state, $ | 0 \rangle, | 1 \rangle, | + \rangle = { 1 \over \sqrt{2}}
( | 0 \rangle + | 1 \rangle ), | - \rangle =
{ 1 \over \sqrt{2}
} ( | 0 \rangle - | 1 \rangle )$. If Alice measures her syndromes before the transmission, it is equivalent to encoding halves of $k$ EPR pairs in an $[[n,k,d]]$ QECC, ${\cal C}_{s_A}$, and sending them to Bob, where ${\cal C}_{s_A}$ is the corresponding quantum code for the syndrome, $s_A$, she found.
Finally, suppose Alice measures her halves of the encoded $k$ EPR pairs before the transmission, it is equivalent to Alice preparing one of the $2^k$ mutually orthogonal codeword states in the quantum code, ${\cal C}_{s_A}$, to represent a $k$-bit key and sending the state to Bob. In summary, the above discussion reduces a QKD protocol based on EPP to a QKD protocol based on a class of $[[n,k,d]]$ QECC, ${\cal C}_{s_A}$’s.
### from error-correcting protocol to BB84
So far, we have not specified which class of QECCs to employ. Notice that, for a general QECC, the QECC protocol still requires quantum computers to implement (for example, the operators $M_{i,A}$). Here comes a key insight of Shor and Preskill: If one employs Calderbank-Shor-Steane (CSS) codes [@CS; @steane], then the scheme can be further reduced to standard BB84, which can be implemented [*without*]{} a quantum computer. CSS codes have the nice property that the bit-flip and phase error correction procedures are totally decoupled from each other. In other words, the error syndrome is of the form of a pair $(s_b, s_p)$ where, $s_b$ and $s_p$ are respectively the bit-flip and phase error syndrome. Without quantum computers, there is no way for Alice and Bob to compute the phase error syndrome, $s_p$. However, this is not really a problem because phase errors do not change the value of the final key, which is all that Alice and Bob are interested in. For this reason, Alice and Bob can basically drop the phase-error correction procedure.
Let us first introduce the CSS code. Consider two classical binary codes, $C_1$ and $C_2$, such that, $$\{ 0 \} \subset C_2 \subset C_1 \subset F^n_2,
\label{e:CSScodes1}$$ where $F^n_2$ is the binary vector space of the $n$ bits and that both $C_1$ and $C_2^{\perp}$, the dual of $C_2$, have a minimal distance, $d= 2t +1$, for some integer, $t$. The basis vectors of a CSS code, $\cal C$, are: $$v \to | \psi(v) \rangle=
{ 1 \over | C_2|^{1/2}} \sum_{w \in C_2} | v + w \rangle,
\label{e:CSScodes2}$$ where $v \in C_1$. Note that, whenever $v_1 - v_2 \in C_2$, they are mapped to the same state. In fact, the basis vectors are in one-one correspondence with the cosets of $C_2$ in $C_1$. The dimension of a CSS code is $2^k$ where $k = {dim (C_1) - dim
(C_2)}$. In standard QECC convention, the CSS code is denoted as an $[[n,k,d]]$ QECC.
One can also construct a whole class of CSS codes, ${\cal C}_{z,x}$, from $\cal C$, where the basis vectors of ${\cal C}_{z,x}$ are of the form $$v \to | \psi(v)_{z,x} \rangle=
{ 1 \over | C_2|^{1/2}} \sum_{w \in C_2} ( -1)^{x \cdot w}
| v + w + z \rangle,
\label{e:cssshifted}$$ where $v \in C_1$.[^9]
Let us introduce some notation. Recall the definition of Pauli matrices. The operator $\sigma_x$ corresponds to a bit-flip error, $\sigma_z$ a phase error and $\sigma_y$ a combination of both bit-flip and phase errors. It is convenient to denote the Pauli operator acting on the $k$-th qubit by $\sigma_{a(k)}$ where, $a \in \{x, y, z\}$. Given a binary vector $s \in F^n_2$, let $$\sigma^{[s]}_a
= \sigma^{s_1}_{a(1)} \otimes \sigma^{s_2}_{a(2)} \otimes \cdots
\sigma^{s_n}_{a(n)} .$$ By definition, the eigenvalues of $\sigma^{[s]}_a$ are $+1$ and $-1$.
Let $H_1$ be the parity check matrix for the code $C_1$ and $H_2$ be the parity check matrix for $C_2^{\perp}$. For each row, $r \in H_1$, consider an operator, $\sigma^{[r]}_z$. Applying to a quantum state, their simultaneous eigenvalues give the bit-flip error syndrome. For each row, $s \in H_2$, consider an operator, $\sigma^{[s]}_x$. Applying to a quantum state, their simultaneous eigenvalues give the phase error syndrome. For instance, when applied to the state, $\psi (v)$ in Eq. (\[e:cssshifted\]), we find the bit-flip error syndrome, $s_b$, and the phase error syndrome, $s_p$ to be: $$\begin{array}{cc}
s_b = H_1 (z),
&
s_p = H_2 (x).
\end{array}$$
Let us look at the QECC-based QKD scheme as a whole. Alice is supposed to pick a random vector $v \in C_1$, random $x_A$ and $z_A$ and encode it as $| \psi(v)_{z_A,x_A} \rangle$. After Bob’s acknowledgement of his receipt of the state, Alice then announces the values of $x_A$ and $z_A$ to Bob. Bob measures the state and obtains his own syndrome, the values of $x_B$ and $z_B$. The relative syndrome, the values of $x_A \times x_B$ and $z_A \times
z_B$, is the actual error syndrome of the channel. Bob then corrects the errors and measures along the $z$-axis to obtain a string $v + w
+ z_A$ for some $w \in C_2$. He then subtracts $x_A$ to obtain $v+ w$. Finally, Bob applies the generator matrix[^10], $G_2$, of the dual code $C^{\perp}_2$ (i.e., the parity check matrix of the code $C_2$) to generate the key, $$G_2 (v + w) = G_2 (v) + G_2 (w) = G_2 (v).
\label{e:keygeneration}$$ Notice that the key is in one-one correspondence with the coset $C_2$ in $C_1$ because of the mapping $G_2 (v) \to v + C_2$.[^11]
Here comes the key point: Since Bob measures along the $z$-axis to generate the key, the phase errors really do not change the value of the key. Therefore, it is not necessary for Alice to announce the phase error syndrome, $x_A$, to Bob. Therefore, without affecting the security of the scheme, Alice is allowed to prepare a state $\psi (v)_{z_A,x_A}$ and then discard, rather than broadcast the value of $x_A$. Equivalently, she is allowed to prepare an [*averaged*]{} state $\psi (v)_{z_A, x_A}$ over all values of $x_A$. The averaging operation destroys the phase coherence and, from Eq. (\[e:cssshifted\]), leads to a classical mixture of $| v + w + z_A \rangle$ in the $z$-basis.
As a whole, the error correction/privacy amplification procedure for the resulting BB84 QKD scheme goes as follows: Alice sends $ | u \rangle$ to Bob through a quantum channel. Bob obtains $ u + e$ due to channel errors. Alice later broadcasts $u + v$, for a random $v \in C_1$. Bob subtracts it from his received string to obtain $ v + e$. He corrects the errors using the code $C_1$ to obtain a codeword, $v \in C_1$. He then applies the matrix, $G_2$, to generate the final key $G_2 (v)$, which is in one-one correspondence with a coset of $C_2$ in $C_1$.
[*Remark 3*]{}: Upon reduction from CSS code to BB84, the original bit-flip error correction procedure of $C_1$ becomes a classical error correction procedure. On the other hand, the phase error correction procedure becomes a privacy amplification procedure. (And, it is achieved by extracting the coset of $C_2$ in $C_1$ by using the generator matrix, $G_2$, of the dual code $C^{\perp}_2$.)
[*Remark 4*]{}: Note that the crux of this reduction is to demonstrate that Eve’s view in the original EPP picture can be made to be exactly the same as in BB84. Therefore, the fact that Alice and Bob [*could have*]{} executed their QKD with quantum computers is sufficient to guarantee the security of QKD. They do not actually need quantum computers in the actual execution. Another way to saying what is going on is that Alice and Bob are allowed to [*throw away*]{} the phase error syndrome information without weakening security. By throwing such phase error syndrome away, the scheme becomes implementable with only classical computers, and, therefore, does not require quantum computers.
Acceptable error rate
---------------------
If one only aims to decode noise patterns up to half of the minimal distance $d$ (as in much of conventional coding theory), then, given that above quantum code uses $C_1$ and $C^{\perp}_2$ that have large minimal distances, it achieves the quantum Gilbert-Varshamov bound for CSS codes[@CS; @steane]. As the length of the code, $n$ goes to infinity, the number of encoded qubits goes to $ [ 1 - 2H (2 e ) ]n $, where $e$ is the measured bit error rate in the quantum transmission. Here, the factor of $2$ in front of $H$ arises because one has to deal with both phase and bit-flip errors in a quantum code. In the classical analog, the factor of $2$ in front of $H$ does not appear. (The factor of $2$ inside $H$ ensures that the distance between any two codewords is at least twice of the tolerable error rate.)
However, in fact, the same CSS code can decode, with vanishing probability of error, up to [*twice*]{} of the above error rate. That is to say, it can achieve the quantum Shannon bound for [*non-degenerate*]{} codes. Asymptotically, the number of encoded qubits goes to $ [ 1 - 2 H ( e ) ] n$. The maximal tolerable error rate would be about $11 \%$.
The reason for the improvement is that the code only needs to correct the [*likely*]{} errors, rather than all possible errors at such a noise level. We remark that this is highly reminiscent of a result in classical coding theory which states that Gallager codes, which are based on very low density parity check matrices, can achieve the Shannon bound in classical coding theory[@MacKay]. In the classical case, the intuition is that in a very high-dimensional binary space, while two spheres of radius $r$ whose centers are a distance $d$ apart have a non-zero volume of intersection for any $r$ greater than $d/2$, the [*fractional*]{} overlap is [*vanishingly small*]{} provided that $r < d$.
To achieve the Shannon bound in the quantum code case, it is necessary to ensure that the errors are randomly distributed among the $n$ qubits. As noted by Shor and Preskill, this can be done by, for example, permuting the $n$ qubits randomly.
[*Remark 5*]{}: In the original Mayers’ proof, the maximal tolerable error rate is about $7\%$. As noted by Shor and Preskill, Mayers’ proof has a hidden CSS code structure. Mayers considered some (efficiently decodable) classical codes, $C_1$, and a random subcode, $C_2$, of $C_1$. It turns out that, the dual, $C^{\perp}_2$, of a random subcode of $C_1$ is highly likely to be a good code. However, Mayers’ proof considered the correction of [*all*]{} phase errors, rather than [*likely*]{} phase errors within the error rate. For this reason, as the length, $n$, of the codeword goes to infinity, the number of encoded qubits asymptotically approaches $ [ 1 - H( e ) - H ( 2 e ) ] n$, the first $H$ comes from error correction and the second comes from privacy amplification. Thus, key generation is possible only up to $7\%$. Shor and Preskill extended Mayers’ proof by noting that it is necessary to correct only [*likely*]{} phase errors, but not all phase errors within the error rate. They also randomize the errors by adding the permutation step mentioned in the above paragraphs.
Shor and Preskill’s protocol of BB84 {#subsec:shorpreskill}
------------------------------------
In the last few subsections, we have already discussed the main steps of Shor and Preskill’s proof. For completeness, we will list here all the steps of Shor and Preskill’s protocol of BB84 scheme.
\(1) Alice sends a sequence of say $(4 + \delta_1 )n $, where $\delta_1$ is a small positive number, photons each in one of the four polarizations (horizontal, vertical, 45 degrees and 135 degrees) chosen randomly and independently.
\(2) For each photon, Bob chooses the type of measurement randomly: along either the rectilinear or diagonal bases.
\(3) Bob records his measurement bases and the results of the measurements.
\(4) Subsequently, Bob announces his bases (but [*not*]{} the results) through the public unjammable channel that he shares with Alice.
[*Remark 6*]{}: Notice that it is crucial that Bob announces his basis only after his measurement. This ensures that during the transmission of the signals through the quantum channel the eavesdropper Eve does not know which basis to eavesdrop along. Otherwise, Eve can avoid detection simply by measuring along the same basis used by Bob.
\(5) Alice tells Bob which of his measurements have been done in the correct bases.
\(6) Alice and Bob divide up their polarization data into two classes depending on whether they have used the same basis or not.
[*Remark 7*]{}: Notice that on average, Bob should have performed the wrong type of measurements on half of the photons. Here, by a wrong type of measurement we mean that Bob has used a basis different from that of Alice. For those photons, he gets random outcomes. Therefore, he throws away those polarization data. We emphasize that this immediately implies that half of the data are thrown away and the efficiency of BB84 is bounded by 50%.
With high probability, at least $\approx 2n$ photons are left. (If not, they abort.) Assuming that no eavesdropping has occurred, all the photons that are measured by Bob in the correct bases should give the same polarizations as prepared by Alice. Besides, Bob can determine those polarizations by his own detectors without any communications from Alice. Therefore, those polarization data are a candidate for their raw key. However, before they proceed any further, it is crucial that they test for tampering. For instance, they can use the following simplified method for estimating the error rate. (Going through BB84 would give us essentially the same result, namely that all accepted data are lumped together to compute a [*single*]{} error rate.)
\(7) Alice and Bob randomly pick a subset of photons from those that are measured in the correct bases and publicly compare their polarization data for preparation and measurement. For instance, they can use $\approx n$ photons for such testing. For those results, they estimate the error rate for the transmission. Of course, since the polarization data of photons in this subset have been announced, Alice and Bob must sacrifice those data to avoid information leakage to Eve.
We assume that Alice and Bob have some idea on the channel characteristics. If the average error rate $\bar{e}$ turns out to be unreasonably large (i.e., $\bar{e} \geq e_{\rm max}$ where $e_{\rm max}$ is the maximal tolerable error rate), then either substantial eavesdropping has occurred or the channel is somehow unusually noisy. In both cases, all the data are discarded and Alice and Bob may re-start the whole procedure again. Notice that, even then there is no loss in security because the compromised key is never used to encipher sensitive data. Indeed, Alice and Bob will derive a key from the data only when the security of the polarization data is first established.
On the other hand, if the error rate turns out to be reasonably small (i.e., $\bar{e} < e_{\rm max}$), they go to the next step.
\(8) Reconciliation and privacy amplification: Alice and Bob can independently convert the polarizations of the remaining $n$ photons into a [*raw*]{} key by, for example, regarding a horizontal or 45-degree photon as denoting a ‘0’ and a vertical or 135-degree photon a ‘1’.
Alice and Bob pick a CSS code based on two classical binary codes, $C_1$ and $C_2$, as in Eqs. (\[e:CSScodes1\]) and (\[e:CSScodes2\]), such that both $C_1$ and $C_2^{\perp}$, the dual of $C_2$, correct up to $t$ errors where $t$ is chosen such that the following procedure of error correction and privacy amplification will succeed with a high probability.
(8.1) Let $v$ be Alice’s string of the remaining $n$ unchecked bits.
Alice picks a random codeword $u \in C_1$ and publicly announces $u + v$.
(8.2) Let $v + \Delta $ be Bob’s string of the remaining $n$ unchecked bits. (It differs from Alice’s string due to the presence of errors $\Delta $.) Bob subtracts Alice’s announced string $u + v$ from his own string to obtain $u + \Delta$, which is a corrupted version of $u$. Using the error correcting property of $C_1$, Bob recovers a codeword, $u$, in $C_1$.
(8.3) Alice and Bob use the coset of $u + C_2$ as their key.
[*Remark 8*]{}: As noted before, there is a minor subtlety [@shorpre]. To tolerate a higher channel error rate of up to about $11\%$, Alice should apply a random permutation to the qubits before their transmission to Bob. Bob should then apply the inverse permutation before decoding.
[*Remark 9*]{}: Depending on the desired security level, the number of test photons in Step (7) can be made to be much smaller than $n$. If one takes the limit that the probability that Eve can break the system is fixed but arbitrary, then the number of test photons can be made to be of order $\log n$ only. On the other hand, if the probability that Eve can break the system is chosen to be exponentially small in $n$, then it is necessary to test order $n$ photons.
Overview of efficient BB84 {#sec:overview}
==========================
In this section, we will give an overview of the efficient BB84 scheme and provide a sketch of a simple proof of its security.
bias
----
The first major new ingredient of our efficient BB84 scheme is to put a bias in the probabilities of choosing between the two bases.
Recall the fraction of rejected data of BB84 is likely to be at least $50\%$. This is because in BB84 Alice and Bob choose between the two bases randomly and independently. Consequently, on average Bob performs a wrong type of measurement half of the time and, therefore, half of the photons are thrown away immediately. The efficiency will be increased if Alice prepares and Bob measures their photons with a biased choice of basis. Specifically, they first agree on a fixed number $0< p \leq 1/2$. Alice prepares (Bob measures) each photon randomly, independently in the rectilinear and diagonal basis with probabilities $p$ and $1-p$ respectively. Clearly, the scheme is insecure when $p=0$. Nonetheless, we shall show that in the limit of large number of photon transfer, this biased scheme is secure in the limit of $p\rightarrow 0^+$. Hence, the efficiency of this biased scheme is asymptotically doubled when compared to BB84.
Notice also that the bias in the probabilities might be produced passively by an apparatus, for example, an unbalanced beamsplitter in Bob’s side. Such a passive implementation based on a beamsplitter eliminates the need for fast switching between different polarization bases and is, thus, useful in experiments. This may not be obvious to the readers why a beamsplitter can create a probabilistic implementation. If one uses a beamsplitter, rather than a fast switch, one gets a superposition of states and not a mixture. However, provided that the subsequent measurement operators annihilate any state transmitting in one of the two paths, the probabilities of the outcomes will be the same for either a mixture or a superposition. More concretely, suppose one can model the problem by decomposing the Hilbert space into two subspaces ${\cal H} = {\cal H}_1 \oplus {\cal H}_2 $ where ${\cal H}_1$ is the Hilbert subspace corresponding to the first path and ${\cal H}_2$ the second respectively. Consider the two sets of measurement operators, $\{P_i\}$’s and $\{Q_j \}$’s respectively, where $P_i | \psi \rangle = 0$ for all $ | \psi \rangle \in {\cal H}_2$ and $Q_j | \psi \rangle = 0$ for all $ | \psi \rangle \in {\cal H}_1$. Let us write $ | u \rangle = | u_1 \rangle + | u_2 \rangle $ where $ | u_1 \rangle \in {\cal H}_1$ and $ | u_2 \rangle \in {\cal H}_2$.
Now, the probability of the outcome corresponding to the measurement $P_i$ is given by $$| \langle u | P_i | u \rangle| =
| \langle u_1 | P_i | u_1\rangle |$$ and the probability of the outcome corresponding to the measurement $Q_j$ is given by $$| \langle u | Q_j | u \rangle | =
| \langle u_2 | Q_j | u_2 \rangle | .$$ Those probabilities are exactly the same as those given by a mixture of $ |u_1 \rangle$ and $| u_2 \rangle$.
Refined Error Analysis
----------------------
In the original BB84 scheme, all the accepted data (those for which Alice and Bob measure along the same basis) are lumped together to compute a [*single*]{} error rate. In this subsection, we introduce the second major ingredient of our scheme — a refined error analysis. The idea is for Alice and Bob to divide up the accepted data into two subsets according to the actual basis (rectilinear or diagonal) used. After that, a random subset of photons is drawn from each of the two sets. They then publicly compare their polarization data and from there estimate the error rate for each basis [*separately*]{}. They decide that the run is acceptable if and only if both error rates are sufficiently small.
The requirement of having estimated error rates separately in both bases to be small is more stringent that the original one. In fact, if a naive data analysis, where only a single error rate is computed by Alice and Bob, had been employed, our new scheme would have been insecure. To understand this point, consider the following example of a so-called biased eavesdropping strategy by Eve.
For each photon, Eve 1) with a probability $p_1$ measures its polarization along the rectilinear basis and resends the result of her measurement to Bob; 2) with a probability $p_2$ measures its polarization along the diagonal basis and resends the result of her measurement to Bob; and 3) with a probability $1-p_1 -p_2$, does nothing. We remark that, by varying the values of $p_1$ and $p_2$, Eve has a whole class of eavesdropping strategies. Let us call any of the strategies in this class a biased eavesdropping attack.
Consider the error rate $e_1$ for the case when both Alice and Bob use the rectilinear basis. For the biased eavesdropping strategy under current consideration, errors occur only if Eve uses the diagonal basis. This happens with a [*conditional*]{} probability $p_2$. In this case, the polarization of the photon is randomized, thus giving an error rate $e_1 = p_2/2$. Similarly, errors for the diagonal basis occur only if Eve is measuring along the rectilinear basis. This happens with a conditional probability $p_1$ and when it happens, the photon polarization is randomized. Hence, the error rate for the diagonal basis $e_2 = p_1/2$. Therefore, Alice and Bob will find, for the biased eavesdropping attack, that the average error rate $$\bar{e} = { p^2 e_1 + ( 1 - p)^2 e_2 \over
p^2 + ( 1 - p)^2 }
= { p^2 p_2 + ( 1 -p )^2 p_1 \over
2 [ p^2 + ( 1 - p)^2] } .$$
Suppose Eve always eavesdrops solely along the diagonal basis (i.e., $p_1 =0$ and $p_2 = 1$), then $$\bar{e} = { p^2 \over
2 [ p^2 + ( 1 - p)^2] } \to 0$$ as $p$ tends to $0$. Hence, with the original error estimation method in BB84, Alice and Bob will fail to detect eavesdropping by Eve. Yet, Eve will have much information about Alice and Bob’s raw key as she is always eavesdropping along the dominant (diagonal) basis. Hence, a naive error analysis fails miserably.
In contrast, the refined error analysis can make our scheme secure against such a biased eavesdropping attack. Recall that in a refined error analysis, the two error rates are computed [*separately*]{}. The key observation is that these two error rates $e_1= p_2/2$ and $e_2= p_1/2$ depend only on Eve’s eavesdropping strategy, but [*not*]{} on the value of ${\varepsilon}$. This is so because they are [*conditional*]{} probabilities. Consequently, in the case that Eve is always eavesdropping along the dominant (i.e., diagonal) basis, Alice and Bob will find an error rate of $e_1= p_2/2 = 1/2$ for the rectilinear basis. Since $1/2$ is substantially larger than $e_{max}$, Alice and Bob will successfully catch Eve.
Procedure of efficient QKD
--------------------------
We now give the complete procedure of an efficient QKD scheme. Its security will be discussed in Subsection \[ss:proofeff\] and more details of a proof of its security will be given in Section \[s:proof\].
[**Protocol E: Protocol for efficient QKD**]{}
\(1) Alice and Bob pick a number $0 < p \leq 1/2$ whose value is made public. Let N be a large integer. Alice sends a sequence of N photons to Bob. For each photon Alice chooses between the two bases, rectilinear and diagonal, with probabilities $p$ and $1 -p $ respectively. The value of $p$ is chosen so that $N (p^2 - \delta') = m_1 = \Omega ( \log N) $, where $\delta'$ is some small positive number and $m_1$ is the number of test photons in the rectilinear basis in Step (7).
\(2) Bob measures the polarization of each received photon independently along the rectilinear and diagonal bases with probabilities $p$ and $1-p$ respectively.
\(3) Bob records his measurement bases and the results of the measurements.
\(4) Bob announces his bases (but [*not*]{} the results) through the public unjammable channel that he shares with Alice.
\(5) Alice tells Bob which of his measurements have been done in the correct bases.
\(6) Recall that each of Alice and Bob uses one of the two bases — rectilinear and diagonal. Alice and Bob divide up their polarization data into four cases according to the actual bases used. They then throw away the two cases when they have used different bases. The remaining two cases are kept for further analysis.
\(7) From the subset where they both use the rectilinear basis, Alice and Bob randomly pick a fixed number say $m_1$ photons and publicly compare their polarizations. (Since $N (p^2 - \delta') = m_1$, for a large $N$, it is highly likely that at least $m_1$ photons are transmitted and received in the rectilinear basis. If not, they abort.) The number of mismatches $r_1$ tells them the estimated error rate $e_1 = r_1/m_1$. Similarly, from the subset where they both use the diagonal basis, Alice and Bob randomly pick a fixed number say $m_2$ photons and publicly compare their polarizations. The number of mismatches $r_2$ gives the estimated error rate $e_2 = r_2/ m_2$.
Provided that the test samples $m_1$ and $m_2$ are sufficiently large, the estimated error rates $e_1$ and $e_2$ should be rather accurate. As will be given in Subsection \[ss:summary\], $m_1$ and $m_2$ should be at least of order $\Omega ( \log k)$, where $k$ is the length of the final key. Now they demand that $e_1, e_2 < e_{\rm max} - \delta_e $ where $e_{\rm max}$ is a prescribed maximal tolerable error rate and $\delta_e$ is some small positive parameter. If these two independent constraints are satisfied, they proceed to step (8). Otherwise, they throw away the polarization data and re-start the whole procedure from step (1).
\(8) Reconciliation and privacy amplification: For simplicity, in what follows, we will take $m_1 =m_2 =N (p^2 -
\delta') $. Alice and Bob randomly pick $n= N [ ( 1- p)^2 - p^2 -\delta' ]$ photons from those untested photons that are transmitted and received in the diagonal basis. Alice and Bob then independently convert the polarizations of those $n$ photons into a [*raw*]{} key by, for example, regarding a 45-degree photon as denoting a ‘0’ and a 135-degree photon a ‘1’.
[*Remark 10*]{}: Note that the raw key is generated by measuring along a single basis, namely the diagonal basis. This greatly simplifies the analysis without compromising efficiency or security.
Alice and Bob pick a CSS code based on two classical binary codes, $C_1$ and $C_2$, as in Eqs. (\[e:CSScodes1\]) and (\[e:CSScodes2\]), such that both $C_1$ and $C_2^{\perp}$, the dual of $C_2$, correct up to $t$ errors where $t$ is chosen such that the following procedure of error correction and privacy amplification will succeed with a high probability.
(8.1) Let $v$ be Alice’s string of the remaining $n$ unchecked bits.
Alice picks a random codeword $u \in C_1$ and publicly announces $u + v$.
(8.2) Let $v + \Delta $ be Bob’s string of the remaining $n$ unchecked bits. (It differs from Alice’s string due to the presence of errors $\Delta $.) Bob subtracts Alice’s announced string $u + v$ from his own string to obtain $u + \Delta$, which is a corrupted version of $u$. Using the error correcting property of $C_1$, Bob recovers a codeword, $u$, in $C_1$.
(8.3) Alice and Bob use the coset of $u + C_2$ as their key.
[*Remark 11*]{}: As noted before, there is a minor subtlety [@shorpre]. To tolerate a higher channel error rate of up to about $11\%$, Alice should apply a random permutation to the qubits before their transmission to Bob. Bob should then apply the inverse permutation before decoding.
Outline proof of Security of efficient QKD scheme {#ss:proofeff}
-------------------------------------------------
In this subsection, we will give the general strategy of proving the unconditional security of efficient QKD scheme and discuss some subtleties. Some loose ends will be tightened in Section \[s:proof\].
First of all, we would like to derive the relationship between the error rates in the two bases ($X$ and $Z$) in biased BB84 and the bit-flip and phase error rates in the underlying entanglement purification protocol (EPP). Actually, this depends on how the key is generated. If the key is generated only from polarization data in say the $Z$-basis, then clearly, the bit-flip error rate is simply the $Z$-basis bit error rate and the phase error rate is simply the $X$-basis bit error rate. On the other hand, if the key is generated only from polarization data in say the $X$-basis, then the bit-flip error rate is simply the $X$-basis bit error rate and the phase error rate is simply the $Z$-basis bit error rate.
More generally, if a key is generated by making a fraction, $q$, of the measurements along the $Z$-basis and a fraction, $1-q$, along the $X$-basis, then the bit-flip and phase error rates are given by weighted averages of the bit error rates of the two bases: $$\begin{aligned}
e^{bit-flip} &=& q e_1 + ( 1- q) e_2 \nonumber \\ \cr
e^{phase} &=& q e_1 + ( 1-q)e_2 ,
\label{eq:bitflip} \end{aligned}$$ where $e_1$ and $e_2$ are the bit error rates of the $Z$ and the $X$ bases respectively.
Now, in a refined data analysis, Alice and Bob separate data from the two bases into two sets and compute the error rates in the two sets individually. This gives them individual estimates on the bit error rates, $e_1$ and $e_2$, of the $Z$ and $X$ bases respectively. They demand that both error rates must be sufficiently small, say, $$0 \leq e_1 , e_2 < e_{max} - \delta_e .
\label{eq:channel}$$
From Eqs. (\[eq:bitflip\]), we see that, provided that the bit error rates of the $X$ and $Z$ bases are sufficiently small (such that Eqs. (\[eq:channel\]) are satisfied), we have $$0 \leq e^{bit-flip}, e^{phase} < 11\% ,
\label{eq:signal}$$ which says that both bit-flip and phase-flip signal error rates of the underlying EPP are small enough to allow CSS code to correct. Therefore, Shor and Preskill’s argument carries over directly to establish the security of our efficient QKD scheme, if Alice and Bob apply a refined data analysis. This completes our sketch of the proof of security.
We remark that the error correction and privacy amplification procedure that we use are exactly the same as in Shor-Preskill’s proof. The point is the following: Once the error rate for both the bit-flip and phase errors are shown to be correctable by a quantum (CSS) code, the procedure for error correction and privacy amplification in their proof can be carried over directly to our new scheme.
practical issues {#ss:practical}
----------------
Several complications deserve attention. First, Alice and Bob only have estimators of $e_1$ and $e_2$, the bit error rates of the two bases, from their random sample. They need to establish confidence levels on the actual bit error rates of the population (or more precisely, those of the [*untested*]{} signals) from those estimators. Second, Alice and Bob are interested in the bit-flip and phase error rates of the EPP, rather than the bit error rates of the two-bases. Some conversion of the confidence levels has to be done. Given that the two bases are weighted differently, such a conversion looks non-trivial. Third, Alice and Bob have to deal with finite sample and population sizes whereas many statistics textbooks takes the limit of infinite population size. Indeed, it is commonplace in statistics textbooks to take the limit of infinite population size and, therefore, assume a normal distribution. Furthermore, in practice, Alice and Bob are interested in bounds, not approximations (which might over-estimate or under-estimate) which many statistics textbooks are contented with.
Another issue: it is useful to specify the constraints on the bias parameter, $q$, and the size of the test samples, $m_1$ and $m_2$. Indeed, in order to demonstrate the security of an efficient scheme for QKD, it is important to show that the size of the test sample can be a very small fraction of the total number of transmitted photons.
We shall present some basic constraints here. As will be shown in Section \[s:proof\], these basic constraints turn out to the most important ones. We see from Remark 2 that, if one limits the eavesdropper’s information, $I_{eve}$, to less than a small fixed amount, then, as the length, $k$, of the key increases, the allowed infidelity in Theorem 2, $\delta$, of the state must decrease at least as $O(1/k)$. Suppose $m_1$ and $m_2$ signals are tested for the two different bases respectively, it is quite clear that $\delta$ is at least $e^{ O(m_i)}$. This leads to a constraint that $m_i $ is at least $\Omega(\log k)$.[^12] Suppose $N$ photons are transmitted and Alice sends photons along the rectilinear and diagonal bases with probabilities, $p$ and $1-p$ respectively. Then, the average number of particles available for testing along the rectilinear basis is only $N p^2$. Imposing that $m_i$ is no more than order $N p^2$, we obtain $ N p^2 = \Omega (\log k)$.
Details of Proof of security of efficient QKD {#s:proof}
=============================================
We will now tighten some of the loose ends in the proof of unconditional security of our efficient QKD protocol, Protocol E.
Using only one basis to generate the raw key
--------------------------------------------
Recall that, in a refined data analysis, Alice and Bob separate data from the two bases into two sets and compute the error rates in the two sets individually. This gives them individual estimates on the bit error rates, $e_1$ and $e_2$, of the $Z$ and $X$ bases respectively. Alice and Bob demand that both error rates must be sufficiently small, say, $$0 \leq e_1 , e_2 < e_{max} - \delta_e ,
\label{eq:channel2}$$ where $\delta_e$ is some small positive parameter. From the work of Shor-Preskill, $e_{max}$ is about 11%.
We would like to derive the relationship between the error rates in the two bases ($X$ and $Z$) in biased BB84 and the bit-flip and phase error rates in the underlying entanglement purification protocol (EPP). Actually, this depends on how the key is generated. In our protocol E, the raw key is generated only from polarization data in the $X$-basis (diagonal basis), the bit-flip error rate is simply the $X$-basis bit error rate and the phase error rate is simply the $Z$-basis (rectilinear basis) bit error rate. Therefore, no non-trivial conversion between the error rates of the two bases and the bit-flip and phase error rates needs to be performed. This greatly simplifies our analysis without compromising the efficiency nor security of the scheme.
Therefore, we have: $$0 \leq e^{phase}_{sample} , e^{bit-flip}_{sample} <
e_{max} - \delta_e ,
\label{eq:channel3}$$ where $\delta_e$ is some small positive parameter and $e_{max}$ is about 11%.
Using classical random sampling theory to establish confidence levels
---------------------------------------------------------------------
A main point of Shor-Preskill’s proof is that the bit-flip and phase error rates of the random sample provide good estimates of the population bit-flip and phase error rates. Indeed, our refined data analysis, as presented in [@patent] and earlier version of the current paper, has been employed by Gottesman and Preskill [@squeeze] in their recapitulation of Shor and Preskill’s proof. Gottesman and Preskill assumed that Alice and Bob generate the key by always measuring along the $Z$-axis. We remark that the problem of establishing confidence levels of the population from the data provided by a random sample is strictly a problem in [*classical*]{} random sampling theory because the relevant operators all commute with each other. See subsection \[subsec:quality\] for details.
It should be apparent that Gottesman-Preskill’s reformulation of Shor-Preskill’s proof and its accompanying analysis of classical statistics carry over to our efficient QKD scheme, provided that we employ the prescribed refined data analysis.
Let us now give more details of the argument that the sample (bit-flip and phase) error rates provide good estimates of the population (bit-flip and phase) error rates. It is simpler to take the limit of $N$ goes to infinity. In this case, the classical de Finetti’s representation theorem applies [@caves]. The de Finetti’s theorem states that the number, $r_1$, of phase errors in the test sample of $m_1$ photons is given by: $$p (r_1, m_1) = { m_1 \choose r_1 } \int_0^1 z^{r_1} ( 1-z)^{m_1 -
r_1}
P^1_{ \infty } (z) dz$$ for some ‘probability of probabilities’ (i.e., a non-negative function, $ P^1_{\infty}$). Physically, it means that one can imagine that each photon is generated by some unknown independent, identical distribution that is chosen with a probability, $P^1_{\infty} (z) $.
Similarly, for the bit-flip errors, its number, $r_2$, in the test sample of $m_2$ photons is given by: $$p (r_2, m_2) = { m_2 \choose r_2} \int_0^1 z^{r_2} ( 1-z)^{m_2 -
r_2}
P^2_{\infty} (z) dz$$ for some ‘probability of probabilities’, $P^2_{\infty} (z) dz $.
We are interested in the case of a finite population size, $N$. Fortunately, a similar expression still exists[@lee; @renes; @jaynes] and it can be written in terms of hypergeometric functions: $$p (r_2, m_2) = \sum_{ n=r_2}^{ N - m_2 + r_2 }
[ C(m_2, r_2) C(N-m_2, n -r_2)/ C(N,n) ] P (n, N)$$ where $C(a, b)$ is the number of ways of choosing $b$ objects from $a$ objects and $P (n, M)$ is the ‘probability of probabilities’.
An upper bound, which will be sufficient for our purposes, can be found in the following Lemma.
[**Lemma 1**]{}. Suppose one is given a population of $n_{\rm total}$ balls out of which $p n_{\rm total}$ of them are white and the rest are black. One then picks $n_{\rm test}$ balls randomly and uniformly from this population *without replacement*. Then, the probability of getting at most $\left\lfloor \lambda n_{\rm test} \right\rfloor$ white balls, $Pr_{\rm wr} (X<\left\lfloor \lambda n_{\rm test} \right\rfloor)$, satisfies the inequality $$\begin{aligned}
Pr_{\rm wr} (X\leq \left\lfloor \lambda n_{\rm test}
\right\rfloor) <
2^{-n_{\rm test} \{ A(\lambda,p) - n_{\rm test} / [(n_{\rm total}
- n_{\rm test})\ln 2] \} }
\label{E:Pr_inequality} \end{aligned}$$ provided that $n_{\rm test} > 1$ and $0 \leq \lambda < p$, where $$A(\lambda,p) = -H(\lambda) - \lambda \log_2 p - (1-\lambda)
\log_2 (1-p)
\label{E:A_Def}$$ with $H(\lambda) \equiv -\lambda \log_2 \lambda - (1-\lambda)
\log_2
(1-\lambda)$ being the well-known binary entropy function.
Furthermore, $A(\lambda,p) \geq 0$ whenever $0\leq \lambda \leq p
< 1$ and the equality holds if and only if $\lambda = p$.
[*Proof*]{}: We denote the probability of getting exactly $j$ white balls by $Pr_{\rm wr}
(X=j)$. Clearly, $$\begin{aligned}
& & Pr_{\rm wr} (X=j) \nonumber \\
& = & \frac{ \left( \!\!\!\begin{array}{c} n_{\rm test} \\ j
\end{array}
\!\!\!\right) \!(p n_{\rm total}\!-\!j\!+\!1)_j \ ([1-p]
n_{\rm total}\!-\!n_{\rm test}\!+\!j\!+\!1)_{n_{\rm test}-j}}{
(n_{\rm total}\!-\!n_{\rm test}\!+\!1)_{n_{\rm test}}} ~,
\label{E:Pr_wr_def} \end{aligned}$$ where $(x)_j \equiv x (x+1) (x+2) \cdots (x+j-1)$. Eq. (\[E:Pr\_wr\_def\]) is called the hypergeometric distribution whose properties have been studied in great detail. In particular,
Sródka showed that [@Probability_Bound] $$\begin{aligned}
Pr_{\rm wr} (X=j) & < & \left( \!\!\!\begin{array}{c} n_{\rm
test} \\ j
\end{array} \!\!\!\right) p^j (1-p)^{n_{\rm test}-j} \left( 1 -
\frac{n_{\rm test}}{n_{\rm total}} \right)^{-n_{\rm test}} \times \nonumber \\
& & ~~~~\left[ 1 + \frac{6 n_{\rm test}^2 +
6 n_{\rm test} - 1}{12n_{\rm total}} \right]^{-1} \nonumber \\
& < & \left( \!\!\!\begin{array}{c} n_{\rm test} \\ j \end{array}
\!\!\!\right) p^j (1-p)^{n_{\rm test}-j} \left( 1 -
\frac{n_{\rm test}}{n_{\rm total}} \right)^{-n_{\rm test}}
\label{E:Pr_wr_bound} \end{aligned}$$ whenever $n_{\rm test} > 1$.
Consequently, $$\begin{aligned}
& & Pr_{\rm wr} (X\leq \left\lfloor \lambda n_{\rm test}
\right\rfloor)
\label{e:lemma1a} \\
& < & \left( 1 - \frac{n_{\rm test}}{n_{\rm total}}
\right)^{-n_{\rm test}}
\,\sum_{j=0}^{\left\lfloor \lambda n_{\rm test} \right\rfloor}
\left(
\!\!\!\begin{array}{c} n_{\rm test} \\ j \end{array}
\!\!\!\right) p^j
(1-p)^{n_{\rm test}-j}
\label{e:lemma1b} \\
& < & \left( 1 - \frac{n_{\rm test}}{n_{\rm total}}
\right)^{-n_{\rm test}}
\,2^{n_{\rm test} [H(\lambda) + \lambda \log_2 p + (1-\lambda)
\log_2 (1-p)]}
\label{e:lemma1c} \\
& < &
2^{-n_{\rm test} \{ -H(\lambda) - \lambda \log_2 p - (1-\lambda)
\log_2
(1-p) - n_{\rm test} / [(n_{\rm total} - n_{\rm test}) \ln 2]\}}
\label{e:lemma1d} \end{aligned}$$ whenever $0\leq \lambda < p$. Note that we have used the inequality in [@Coding] to obtain Eq. (\[e:lemma1c\]) and the inequality $- { x \over 1 -x} \leq \ln ( 1-x) \leq -x \leq 0$ to obtain Eq. (\[e:lemma1d\]) respectively. Hence, Eq. (\[E:Pr\_inequality\]) holds.
Finally we want to show that $A(\lambda,p) \geq 0$ whenever $0\leq
\lambda \leq p <
1$; and the equality holds if and only if $\lambda = p$. This fact follows directly from the observations that $A(\lambda,\lambda) = 0$, $\partial A /
\partial p \geq 0$ whenever $0\leq \lambda \leq p < 1$ and the equality holds if and only if $\lambda = p$. Q.E.D.
Note that Lemma 1 gives a precise bound, not just an approximation. The upshot of Lemma 1 is that the probability that the sample mean deviates from the population mean by any arbitrary but fixed non-zero amount can be shown to be [*exponentially*]{} small in $n_{test}$, as discussed in subsection \[ss:practical\]. In effect, Lemma 1 gives the conditional probability, $\varepsilon_1$, that the signal quality check stage is passed, given that more than $t \equiv
\left\lfloor (d-1)/2 \right\rfloor$ out of the $n$ pairs of shared entangled particles between Alice and Bob are in error. We will choose $n_{test} = m_1 =m_2$ in our Protocol E.
Bounding fidelity
-----------------
Given any eavesdropping strategy that will pass the verification test with a probability, $\varepsilon_2$, it is important to obtain a bound on the fidelity of the recovered state as $k$ EPR pairs, after quantum error correction and quantum privacy amplification. We have the following Theorem.
[**Theorem 3. (Adapted from [@qkd])**]{} Suppose Alice and Bob perform a stabilizer-based EPP-based QKD and, for the verification test, randomly sample along at least two of the three bases, $X$ and $Y$ and $Z$ and compute their error rates. Suppose further that the CSS code used in the signal privacy amplification stage acts on $n$ imperfect pairs of qubits to distill out $k$ pairs of qubits. Given any fixed but arbitrary eavesdropping strategy by Eve, define the following probabilities: $$p = P( {\rm EPP~succeeds}) ,$$ $$\varepsilon_1 = P({\rm verification~passed}~|
{\rm EPP~fails}),$$ and $$\bar{\varepsilon}_1 = P({\rm verification~failed}~|
{\rm EPP~succeeds}),$$ (In statistics language, $\varepsilon_1$ and $\bar{\varepsilon}_1$ are the type I and II errors respectively.) Then, for any Eve’s cheating strategy whose probability of passing the verification test is greater than $\varepsilon_2$, the fidelity of the remaining untested shared entangled state immediately after the quantum privacy amplification is greater than $1 - \varepsilon_1 / \varepsilon_2 $.
[*Proof*]{}: From Theorem 1 and Proposition 1, one can, indeed, apply classical arguments to the problem by assigning classical probabilities to the $N$-Bell-basis states. Given any fixed but arbitrary eavesdropping strategy, the fidelity of the remaining untested entangled state is given by: $$\begin{aligned}
F &\geq& { P( {\rm verification~passed~and~EPP~succeeds}) \over
P ( {\rm verification~passed}) } \nonumber \\
~&=& { P( {\rm EPP~succeeds} ) P ({\rm
verification~passed}~|{\rm EPP~succeeds}) \over
P( {\rm EPP~succeeds} ) P ({\rm
verification~passed}~|{\rm EPP~succeeds}) + P( {\rm EPP~fails} ) P
({\rm
verification~passed}~|{\rm EPP~fails})} \nonumber \\
~&=& { P( {\rm EPP~succeeds} ) P ({\rm
verification~passed}~|{\rm EPP~succeeds}) \over
P( {\rm EPP~succeeds} ) P ({\rm
verification~passed}~|{\rm EPP~succeeds}) + P( {\rm EPP~fails} ) P
({\rm
verification~passed}~|{\rm EPP~fails}) } \nonumber \\
~&=& { p ( 1 - \bar{\varepsilon}_1 ) \over
p ( 1 - \bar{\varepsilon}_1 ) + ( 1- p) \varepsilon_1 }
\nonumber \\
~&\geq & 1 - \frac{\varepsilon_1}{p (1-\bar{\varepsilon}_1) +
( 1- p) \varepsilon_1 } ~,
\label{E:F1} \end{aligned}$$
Now, for any Eve’s cheating strategy whose probability of passing the verification test is greater than $\varepsilon_2$, we have $ p (1-\bar{\varepsilon}_1) +
( 1- p) \varepsilon_1 > \varepsilon_2$ and, hence, from Eq. (\[E:F1\]), $$F > 1 - \frac{\varepsilon_1}{\varepsilon_2} ~. \label{E:F2}$$ This completes the proof of Theorem 3. Q.E.D.
Summary of the proof {#ss:summary}
--------------------
We will now put all the pieces together and show that a rigorous proof of security is possible with the number of test particles, $m_1 =m_2 =n_{test}$, scaling logarithmically with the length $k$ of the final key. Consequently, the bias in an efficient BB84 scheme can be chosen such that $N ( p^2 - \delta') = n_{test}$ for a small $\delta$. In other words, $p = O ( \sqrt{(\log k) / N })$, which goes to zero as $N$ goes to infinity.
Given a signal quality check that involves only $n_{test}$ photons, from Lemma 1, we see that the conditional probability, $ \varepsilon_1$, that the signal quality check stage is passed, given that more than $t \equiv
\left\lfloor (d-1)/2 \right\rfloor$ out of the $n$ pairs of shared entangled particles between Alice and Bob are in error is exponentially small in $n_{test}$. i.e., $$\varepsilon_1 = O (2^{- n_{test} \alpha}),
\label{e:epsilon1}$$ for some positive constant $\alpha$.
Let Alice and Bob pick a security parameter, $$\varepsilon_2 = 2^{-u},
\label{e:defineu}$$ and consider only eavesdropping strategies that will pass the signal quality check with a probability at least $ \varepsilon_2$. We require that $$\varepsilon = { \varepsilon_1 \over \varepsilon_2}
\ll 1.
\label{e:approximation}$$
Recall from Theorem 3 that for any eavesdropping strategy that will pass the signal quality check test with a probability at least $\varepsilon_2$, has its fidelity bounded by $ 1 - \varepsilon$. i.e., $$F \geq 1 - \varepsilon.$$
Now, from Theorem 2, the eavesdropper’s mutual information with the final key is bounded by $$I_{eve}^{Bound} = \varepsilon ( 2 k + \log_2( 1/ \varepsilon ) + { 1 \over
\log_e 2}) .
\label{e:bound}$$
Consider a fixed but arbitrary value of $I_{eve}^{Bound}$, the constraint on the eavesdropper’s mutual information on the final key: i.e., $$I_{eve}^{Bound} = 2^{- s},
\label{e:securitys}$$ where $s$ is a positive security parameter. In the large $k$ limit, Eq. (\[e:bound\]) implies that $$\varepsilon = O ( 2^{-s}/k).
\label{e:orderk}$$ Substituting Eq. (\[e:approximation\]) into Eq. (\[e:orderk\]), we see that $${ k \varepsilon_1 \over 2^{-s} \varepsilon_2} = O (1).
\label{e:order1}$$ Substituting Eqs. (\[e:epsilon1\]) and (\[e:defineu\]) into Eq. (\[e:order1\]), we find that $${ k 2^{- n_{test} \alpha} \over 2^{-(u + s) } } = O (1).
\label{e:order2}$$
Now, for fixed but arbitrary values of the security parameters, $s$ and $u$, we see that, in fact, the number of test photons, $n_{test}$, is required to scale only as $ O (\log k)$, i.e., the logarithm of the final key length. Consequently, the only constraint on the bias $p$ is that there are enough photons for performing the verification test. This gives rise to the requirement that $N ( p^2 - \delta') = n_{test} = O (\log k)$, i.e., $$p = O ( \sqrt{(\log k) / N }).$$
This completes our proof of security of Protocol E, an efficient QKD scheme. We remark that the error correction and privacy amplification procedure in Protocol E are exactly the same as in Shor-Preskill’s proof.
As a side remark, if one insists that the eavesdropper’s information is exponentially small in $N$, then one can take $s = c N$, for some positive constant, $c$. From Eq. (\[e:order2\]), this will require $n_{test}$ to be proportional to $N$. A number of earlier papers make such an assumption. However, in this paper, we note that this requirement can be relaxed. For instance, it is consistent to pick $s = c N^{a'}$ where $ 0 \leq a' \leq 1$. In this more general case, we have from Eq. (\[e:order2\]) that asymptotically $\alpha n_{test} \sim c N^{a'}$. Consequently, $$\begin{aligned}
\alpha N p^2 &\geq& \alpha n_{test} \sim c N^{a'} \nonumber \\
p^2 & = &\Omega ( { c N^{a' -1} \over \alpha } ) .
\label{e:a'} \end{aligned}$$ From Eq. (\[e:a’\]), it is clear that for all values of $a' \in [0,1]$, the probability $p$ can be chosen to be arbitrarily small, but non-zero. This completes our analysis for the security of an efficient QKD scheme where each of Alice and Bob picks the two polarization bases with probabilities $p$ and $1-p$.
Concluding Remarks {#sec:conclusion}
==================
In this paper, we presented a new quantum key distribution scheme and proved its unconditional security against the most general attacks allowed by quantum mechanics.
In BB84, each of Alice and Bob chooses between the two bases (rectilinear and diagonal) with equal probability. Consequently, Bob’s measurement basis differs from that of Alice’s half of the time. For this reason, half of the polarization data are useless and are thus thrown away immediately. We have presented a simple modification that can essentially double the efficiency of BB84. There are two important ingredients in this modification. The first ingredient is for each of Alice and Bob to assign significantly different probabilities (say ${\varepsilon}$ and $1 - {\varepsilon}$ respectively where ${\varepsilon}$ is small but non-zero) to the two polarization bases (rectilinear and diagonal respectively). Consequently, they are much more likely to use the same basis. This decisively enhances efficiency.
However, an eavesdropper may try to break such a scheme by eavesdropping mainly along the predominant basis. To make the scheme secure against such a biased eavesdropping attack, it is crucial to have the second ingredient — a refined error analysis — in place. The idea is the following. Instead of lumping all the accepted polarization data into one set and computing a [*single*]{} error rate (as in BB84), we divide up the data into various subsets according to the actual polarization bases used by Alice and Bob. In particular, the [*two*]{} error rates for the cases 1) when both Alice and Bob use the rectilinear basis and 2) when both Alice and Bob use the diagonal basis, are computed separately. It is only when both error rates are small that they accept the security of the transmission.
We then prove the security of efficient QKD scheme, not only against the specific attack mentioned above, but also against the most general attacks allowed by the laws of quantum mechanics. In other words, our new scheme is [*unconditionally*]{} secure. Moreover, just like the standard BB84 scheme, our protocol can be implemented without a quantum computer. The maximal tolerable bit error rate is 11%, the same as in Shor and Preskill’s proof. If we allow Eve to get a fixed but arbitrarily small amount of information on the final key, then the number of test particles, $n_{test}$, is required only to scale logarithmically with the length $k$ of the final key. Consequently, the bias in an efficient BB84 scheme can be chosen such that $N ( p^2 - \delta') = n_{test}$ for a small $\delta$ and where $N$ is the total number of photons transmitted. In other words, $p = O ( \sqrt{(\log k) / N
})$, which goes to zero as $N$ goes to infinity. More generally, suppose we pick the security parameter to be $s$ (for an eavesdropper’s information $I_{eve} \leq 2^{-s}$) such that $s = c N^{a'}$ where $ 0 \leq a' \leq 1$. We find that this can be achieved by testing $n_{test}$ random photons where $\alpha n_{test} \sim c N^{a'}$. Furthermore, each of Alice and Bob may pick the two polarization bases with probabilities $p$ and $1-p$ such that $p^2 = \Omega ( { c N^{a' -1} \over \alpha }) $. Therefore, $p$ can, indeed, be made arbitrarily small but non-zero.
This is the first time that a single-particle quantum key distribution scheme has been proven to be secure without relying on a symmetry argument — that the two bases are chosen randomly and uniformly. Our proof is a generalization of Shor and Preskill’s proof [@shorpre] of security of BB84, a proof that in turn built on earlier proofs by Lo and Chau [@qkd] and also by Mayers [@mayersqkd].
We remark that our idea of efficient schemes of quantum key distribution applies also to other schemes such as Biham, Huttner and Mor’s scheme [@Mor] which is based on quantum memories. Our idea also applies the six-state scheme [@bruss], which has been shown rigorously to tolerate a higher error rate of up to 12.7% [@six].
As a side remark, Alice and Bob may use different biases in their choices of probabilities. In other words, our idea still works if Alice chooses between the two bases with probabilities ${\varepsilon}$ and $1- {\varepsilon}$ and Bob chooses with probabilities ${\varepsilon}'$ and $1- {\varepsilon}'$ where ${\varepsilon}\not= {\varepsilon}'$.
We thank Gilles Brassard for many helpful discussions and suggestions. Enlightening discussions with Daniel Gottesman, Debbie Leung, Norbert Lütkenhaus, John Preskill and Peter Shor on various proofs of the unconditional security of QKD schemes are also gratefully acknowledged. We also thank Peter Shor for suggesting to us the possibility that Shor and Preskill’s proof may be generalized to prove the unconditional security of efficient quantum key distribution scheme. We gratefully acknowledge enlightening discussions with Chris Fuchs, Stephen Lee, and Joe Renes about statistical analysis. We thank P. L. H. Yu for his useful discussions on hypergeometric function and pointing out some references to us. We also thank anonymous referees for their many comments, which are very useful for improving the presentation of the current paper. HFC is supported by the RGC grant HKU 7143/99P of the Hong Kong SAR Government. Parts of this work were done while M. Ardehali was with NEC Japan and while H.-K. Lo was at the Institute for Advanced Study, Princeton, NJ, Hewlett-Packard Laboratory, Bristol, UK and MagiQ Technologies, Inc., New York.
[*Notes Added*]{}: An entanglement-based scheme with an efficiency greater than $50\%$ has also been discussed in a recent preprint by two of us (H.-K. Lo and H.F. Chau) [@qkd]. Recent proofs of the unconditional security of various QKD schemes have been provided by H. Inamori [@hitoshi1; @hitoshi2], H. Aschauer and H. J. Briegel [@hans] and by D. Gottesman and J. Preskill [@squeeze]. Recently, it has been shown [@two] by D. Gottesman and one of us (H.-K. Lo) that two-way classical communications can be used to increase substantially the maximal tolerable bit error rate in BB84 and the six-state scheme. The result presented in the current paper can be combined with [@two] to obtain, for example, an efficient BB84 scheme that can tolerate a substantially higher bit error rate (say, 18.9 percent) than in Shor-Preskill’s proof. It has been shown in a recent preprint [@imperfect] that even imperfect devices can provide perfect security in QKD within the entanglement purification approach employed in the present paper. Finally, a proof of the unconditional security of another well-known QKD scheme, B92 scheme published by Bennett in 1992 [@b92], has recently been presented [@proofb92].
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[^1]: email: [email protected]
[^2]: email: [email protected]
[^3]: Another class of applications of quantum cryptography has also been proposed [@BBCS; @BCJL]. Those applications are mainly based on quantum bit commitment and quantum one-out-of-two oblivious transfer. However, it is now known [@Mayers2; @LoChau1; @LoChau2; @Lo] that unconditionally secure quantum bit commitment and unconditionally secure quantum one-out-of-two oblivious transfer are both impossible. Furthermore, other quantum cryptographic schemes such as a general two-party secure computation have also been shown to be insecure [@Lo; @LoChau2]. For a review, see [@special].
[^4]: In experimental implementations, coherent states with a Poisson distribution in the number of photons are often employed. To achieve unconditional security, it is important that the operational parameters are chosen such that the fraction of multi-photon signals is sufficiently small. This may substantially reduce the key generation rate[@ilm]. In the current paper, we restrict our attention to perfect single photon signals as assumed in standard BB84 and various security proofs.
[^5]: Naively, one might think that the security requirement should simply be: conditional on passing the quality check stage, Eve has a negligible amount of information on the key. However, such a strong security requirement is, in fact, impossible to achieve [@mayersqkd; @qkd]. The point is that a determined eavesdropper can always replace all the quantum signals from Alice by some specific state prepared by herself. Such a strategy will most likely fail in the quality check. But, if it is lucky enough to pass, then Eve will have perfect information on the key shared by Alice and Bob.
[^6]: Mathematically, observables in quantum mechanics are represented by Hermitian matrices. Complementary observables are represented by non-commuting matrices and, therefore, cannot be simultaneously diagonalized. Consequently, their simultaneous eigenvectors generally do not exist.
[^7]: As demonstrated by the well-known Einstein-Podolsky-Rosen paradox, classical intuitions generally do not apply to quantum mechanics. This is a reason why proving security of QKD is hard.
[^8]: There are some subtle differences between the original Shor and Preskill’s proof and the one elaborated by Gottesman and Preskill. First, in the original Shor and Preskill’s proof, Alice and Bob apply a simple-minded error rate estimation procedure in which they lump all polarization data of their test sample together into a single set and compute a single bit error rate. In contrast, in Gottesman and Preskill’s elaboration, Alice and Bob separate the polarization data according to the bases in which they are transmitted and received. The two bit error rates for the rectilinear and diagonal bases are computed separately. In essence, they are employing the refined data analysis idea, which was first presented in a preliminary version of this manuscript [@prelim]. Second, in Gottesman and Preskill’s discussion, the final key is generated by measuring along a [*single*]{} basis, namely the $Z$-basis. (Because of this prescription, they call the error rates of the two bases simply bit-flip and phase errors. To avoid any potential confusion, we will not use their terminology here.) In contrast, in Shor and Preskill’s original proof, the final key is generated from polarization data obtained in both bases.
[^9]: Note that our notation is different from both Refs. [@shorpre] and [@squeeze] in that we have interchanged $x$ and $z$ in Eq. (\[e:cssshifted\]) as well as in the definition of ${\cal C}_{z,x}$. In our notation, $z$ denotes the bit-flip error syndrome and $x$ denotes the phase error syndrome.
[^10]: Gottesman and Preskill’s paper stated that the parity check matrix, $H_2$, of the dual code $C^{\perp}_2$ should be used. But, it should really be the generator matrix.
[^11]: This is a well-known result in classical coding theory.
[^12]: Notice that this constraint is [*weaker*]{} than the usual constraint of $m_i = \Omega (N)$ imposed by various other proofs[@mayersqkd; @biham]. In the next section, we will see that it is, indeed, unnecessary to impose $m_i = \Omega (N)$.
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[H]{}
Introduction
============
A proper definition of M-theory as a non perturbative framework for superstring theories is still an open problem. It is strongly shaded, among other problems, by a widely incomplete knowledge of the very structure of the world-volume theory of the M5-branes.
A conjectural formulation of the M5-brane theory has been given in [@open] in terms of a six-dimensional string theory, called Little String Theory (LST), which should describe the world-volume theory of the M5-brane. This arises naturally once we understand the M-theory 5-brane not only as the magnetic dual of the membrane but also as Dirichlet surface for membranes. This picture is natural from the point of view of the equivalence of M-theory compactification on a circle $S^1$ and type IIA string theory. In fact, under $S^1$ compactification, the M5-brane generates both the NS-5branes which are magnetic duals to fundamental strings and the D4-branes on which fundamental strings can end. At the same time the fundamental strings are generated by membranes wrapping the circle.
Despite the simplicity and naturalness of the above picture, LST shows several peculiar features which make its actual implementation via the usual tools of superstring theory, to say the least, tricky.
A proper analysis [@bb] of the boundary degrees of freedom of the membranes ending on the 5-brane gives as a result that the low energy spectrum of LST is given by the $(0,2)$ self-dual tensor multiplet. Its bosonic content is given by a 2-form with self-dual curvature which couples minimally to the strings and 5 bosons in the vector representation of the local $SO(5)$ group describing the transverse dispalcements of the 5-brane in the 11 dimensional target space of M-theory (i.e. these 5 bosons are a section of the normal bundle of the 5-brane world-volume embedded in 11 dimensions).
A first consequence of this analysis is that LST should not be understood in the usual perturbative terms. It is, because of self-duality, a symmetric dyon with equal electric and magnetic charge and, applying the Dirac quantization rule $eg'+ge'=2\pi n$, we find $e^2=\pi n$. This means that the elementary charge $e_0=\sqrt{\pi}$ is not an adjustable parameter and therefore, assuming that the charge is a non constant function of the string coupling, it is natural to expect that LST does not admit a usual [*perturbative*]{} world-sheet formulation. Another unusual feature is the LST target space dimension which is 6 and this means that LST does not fulfill the criticality bound as a string theory. Moreover a further possibly problematic aspect is given by its unusual non-gravitational low energy spectrum.
Notwithstanding the open questions concerning a proper [*off-shell*]{} formulation of LST, it is possible to study some of its [*on-shell*]{} properties by indirect methods and to check the validity of the above model.
The specific issue we are going to treat here is a counting of BPS multiplets of states in LST. These are encoded in the generalized supersymmetric Witten index of the theory. We will propose an on-shell framework to calculate these objects within a six dimensional perspective on a world-volume six manifold of the product form $T^2\times M_4$. The particular form of the six-manifold allows the use of a duality map between M-theory on $T^2$ and type IIB theory in the limit of vanishing $T^2$-volume to check the validity of our results. Under this duality map, M5-branes wrapped on the above product manifold correspond to D3-branes wrapped on $M_4$. This map therefore reverts our problem to a calculation of a supersymmetric path integral in a corresponding suitable four-dimensional gauge theory whose coupling is identifyed with the $T^2$ modular parameter.
The result of our calculation will be to explicitly recognize the presence of tensionless string states in correspondence with the intersection between different 5-branes in a given bound state and to give a precise formula to count their multiplicities.
This talk is mainly based on [@io; @io2].
The M5-branes index on $T^2\times M_4$
======================================
To properly embed our problem in M-theory, we consider M-theory on $W=Y_6\times T^2\times R^3$, where $Y_6$ is a Calabi-Yau threefold of general holonomy. Let $M_4$ be a supersymmetric simply connected four-cycle in $Y_6$ which we take to be smooth. Under these assumptions $M_4$ is automatically equipped with a Kaehler form $\omega$ induced from $Y_6$ and is simply connected. We consider then $N$ M5-branes wrapped around $C=T^2\times M_4$.
It can be shown that in this specific geometrical set-up [@io2] the potential anomalies which tend to ruin gauge invariance of the world-volume theory are absent and that it is then meaningful to define a supersymmetric index for the above 5-branes bound states by extending the approach in [@vafa']. As it is well known, the supersymmetric index is independent on smooth continue parameters and as a consequence we have that this counting of supersymmetry preserving states have to be independent on the $T^2$ volume.
In [@io2] the calculation of the supersymmetric index was performed in the limiting cases of small and of large volumes of the $T^2$ and shown to agree while in [@io] a six dimensional framework was proposed which reproduces the above calculations as a one loop string supersymmetric path-integral. We will review the last construction.
Let us start with the single M5-brane case. The bosonic spectrum of the low energy world-volume theory of this 5-brane is given by a 2-form $V$ with self-dual curvature and five real bosons taking values in the normal bundle $N_C$ induced by the structure of the embedding as $T_W|_C=T_C\oplus N_C$. Passing to the holomorphic part and to the determinants and using the Calabi-Yau nature of $Y_6$, it follows that the five transverse bosons are respectively, three non-compact real scalars $\phi_i$ and one complex section $\Phi$ of $K_{M_4}=\Lambda^{-2} T^{(1,0)}_{M_4}$, which is the canonical line bundle of $M_4$. A (partially) twisted chiral $(0,2)$ supersymmetry of the type considered in [@baulieuwest] completes the spectrum. It is given by a doublet of complex anti-commuting fields which are $(2,0)-$forms in six dimensions and a doublet of complex anti-commuting fields which are scalars in six dimensions.
The calculation of the supersymmetric index for this part of the spectrum can be done with path-integral techniques. It is classically exact because of boson/fermion exact cancellation and it results in a zero-modes amplitude for the self-dual tensor field. This can be analyzed following the results of [@mans]. It consists of a $\theta$-function of the lattice of the self-dual harmonic three forms. The $\theta$-function is not completely specified in [@mans] because of the possible inequivalent choices of its characteristics [$\left[\matrix{\alpha\cr\beta}\right]$]{}. It is $$\theta\left[\matrix{\alpha\cr\beta}\right](Z^0|0)
=\sum_k e^{i\pi \left((k+\alpha)Z^0(k+\alpha)+2(k+\alpha)\beta\right)},$$ where $Z^0$ is a period matrix of the relevant six-manifold cohomology that we specify shortly. Let $\left\{E^{(6)},\tilde E^{(6)}\right\}$ be a symplectic basis of harmonic 3-forms on the six-manifold at hand such that, in matrix notation, $$\int E^{(6)}E^{(6)}=0,\quad
\int \tilde E^{(6)}E^{(6)}=1,\quad
\int \tilde E^{(6)}\tilde E^{(6)}=0.$$ We can expand $\tilde E^{(6)}=X^0E^{(6)}+Y^0{}^*E^{(6)}$, where ${}^*$ is the Hodge operator. Then $Z^0$ is defined as $Z^0=X^0+iY^0$. In general, under modular transformations (i.e. global diffeomeorphisms of the 5-brane world-volume) these $\theta$-functions transform among each others. If we ask for a single candidate closed under modular transformations, we see [@gust] that the natural choice is given by [$\left[\matrix{\alpha\cr\beta}\right]=\left[\matrix{0 \cr 0}\right]$]{}.
In our case the world volume is in the product form $T^2\times M_4$ and [@io] we have $\frac{1}{2}b_3\left(T^2\times M_4\right)=b_2\left(M_4\right)$ and $Z^0=-\tau^{(1)}Q+i\tau^{(2)}1$, where $Q$ is the intersection matrix on $M_4$, $1$ is the unity matrix and $\tau=\tau^{(1)}+i\tau^{(2)}$ is the modulus of the $T^2$. The relevant $\theta$-function is then (Z\^0)(Z\^0|0) = \_k e\^[ik Z\^0 k]{}= \_[m]{}q\^[(m,\*m-m)]{}[|q]{}\^[(m,\*m+m)]{} \_(q,|q) \[uno\]where $q=e^{2i\pi\tau}$ and $\Lambda$ is the lattice of integer period elements in $H^2(M_4,{\bf R})$.
To count the full BPS spectrum of the theory a second sector is still lacking. In fact, the 5-brane theory is completed in the UV by the little string theory which has BPS saturates strings which eventually have to be kept into account in the calculation of the complete supersymmetric index. Even if a full off-shell model for this six-dimensional string theory is not available at the moment, it is possible to follow an on-shell simple calculation scheme for their contribution to the supersymmetric index.
As the susy index is given by a trace on the string Hilbert space, it corresponds to a one-loop string path integral. Moreover, as it is usual in these index calculations, the semiclassical approximation is exact. Now, since $M_4$ is simply connected, the only contributions to this path integral can arise from string world sheets wrapping the $T^2$ target itself. Therefore the configuration space of $n$ of these string world-sheets [^1] will be given by the symmetric product $(M_4)^n/S_n$ whose points parametrize the transverse positions. Now, since the supersymmetric index calculated the Euler characteristics of the configuration space, we claim that the full contribution from these string BPS configurations is given by q\^[-\_[M\_4]{}/24]{} \_n q\^n (M\_4\^n/S\_n)= q\^[-\_[M\_4]{}/24]{} \_[n>0]{}=(q)\^[-\_[M\_4]{}]{} \[due\]where we fixed a global multiplicative factor $q^{-\chi_{M_4}/24}$ because of modularity requirements and we used well known results from [@vw]. In [@io] it is shown that this result can be obtained also via a natural generalization of the construction done in [@bps5] for toroidal and K3 compactifications to the generic simply connected Kahler case.
The complete supersymmetric index for a single five-brane on $T^2\times M_4$ is then given multiplying the factor (\[uno\]) and the factor (\[due\]) and reads $${\cal I}_1^{T^2\times M_4}=\frac{\theta_\Lambda}{\eta^\chi}\, .$$
We now pass to the analysis of the multi 5-brane case. We will start assuming that the BPS multi five-brane bound states are classified by holomorphic branched coverings of the relevant world volume manifold and by a choice of spin structure on it. Notice that this will result in recostructing exactly the relevant supersymmetric index as it can be calculated from the zero $T^2$-volume limit. In this limit [@io; @io2] we find in fact the four dimensional gauge theory corresponding to a dual multi D3-brane bound state in type IIB and the calculation is under full control In [@io] also an independent duality argument is given which supports this picture when the six manifold admits a regular $S^1$ fibration. Suppose therefore that we are dealing with a BPS bound state of $N$ 5-branes wrapped on a six manifold ${\cal C}$. Then we have to consider the space of rank N holomorphic coverings of ${\cal C}$ in the M-theory target space. Generically this space of coverings will be disconnected and reducible into components consisting of connected irreducible coverings of rank less or equal to $N$. Each of these irreducible holomorphic coverings corresponds to an irreducible bound state. Then the general structure formula for the supersymmetric index of an irreducible bound state of $N$ 5-branes is given by the lifting to the spectral cover of the index formula. In formulas, if ${\cal C}_N$ is a generic holomorphic covering of ${\cal C}$, then ${\cal C}_N=\cup_j {\cal C}_j^{irr}$, where $\sum j=N$ and ${\cal C}_j^{irr}$ is irreducible. The relative contribution to the index is ${\cal I}_N^{\cal C}=\prod_j {\cal I}_1^{{\cal C}_j^{irr}}$. This formula has to be understood still schematically since we didn’t specified the sum over the spin structures. In the case ${\cal C}=T^2\times M_4$, a generic irreducible holomorphic covering is of the form $\tilde T^2\times \Sigma_r$, where $\tilde T^2$ is a torus with modulus $\tilde\tau=(a\tau+b)/d$ which covers $a\cdot d=n_r$ times the original $T^2$ and $d>b\geq 0$ and $\Sigma_r$ is a rank $r$ holomorphic branched covering of $M_4$. The above multiplicities are of course constrained by $n_r\cdot r=j$ which is the rank of the six dimensional irreducible covering ${\cal C}_j^{irr}$ above. The full supersymmetric index is then given by summing all along the whole set of possible covering structures. Let us perform the sum with respect to the $T^2$ coverings first. To do it by preserving modularity, we multiply each factor by a rescaling term $d^{-w-\bar w}$, where $(w,\bar w)$ are the modular weights of ${\cal I}|_{n_r=1}$, and then we sum over all the triples $(a,b,d)$ fulfilling the above condition. This operation coincides exactly with the definition of the Hecke modular operator ${\cal H}_{n_r}$ and our partial sum now reads $
{\cal I}_{(n_r,r)}^{T^2\times M_4}={\cal H}_{n_r}
{\cal I}_{(1,r)}^{T^2\times M_4}
$. By including the sum over the appropriate spin structures and explicitating the dependence over the given irreducible rank $r$ covering $\Sigma_r$, we finally get \_[n\_r,r]{}= [H]{}\_[n\_r]{} \_ \[eg\]where $\varepsilon$ is a label for the square-roots of the canonical line bundle (spin structures) on $M_4$ with respect to a given one as ${\cal O}_\varepsilon\otimes K^{1/2}$ with ${\cal O}_\varepsilon^2=1$, $x=[{\cal O}_\varepsilon^{\otimes a+1}]$ shifts correspondingly the lattice of integer periods $\Lambda^{\Sigma_a}$ on $H^2(\Sigma_a,R)$.
As it was already mentioned in the introduction, the multi 5-brane interacting model we have in mind consists in a picture where the 5-branes, beside being magnetic duals of the membranes, are also Dirichlet hypersurfaces for them, the boundaries of the membranes being strings in the 5-brane world-volume. In this picture one expects that tensionless BPS string states appear when two or more 5-branes intersect each other. In our geometric picture this is the branching locus of the covering six manifold. Indeed the supersymmetric index (\[eg\]) reveals clearly the apparing of these extra BPS string states. In our particular case, the $T^2$ holomorphic self-covering is unbranched and the little strings contribution is explicitly exposed in the Dedekind $\eta$-function. In fact it enters in the form $\left(\frac{1}{\eta}\right)^{\chi_{\Sigma_r}}$, where $\Sigma_r$ is a rank $r$ holomorphic covering of $M_4$. The explicit dependence on the branching locus or the covering $B_r$ appears once we apply the Hurwitz formula $\chi_{\Sigma_r}=r\cdot\chi_{M_4}-\chi_{B_r}$ which holds for the covering structure. We find therefore that \^[-\_[\_r]{}]{}= \^r \^[\_[B\_r]{}]{} \[tre\]We read this formula for multiplicities of massless BPS string states as a superposition of the independent BPS states pertaining to each brane copy which is corrected by a further term corresponding to the interacting theory.
Conclusions and Open Problems
=============================
In this talk we have reviewed a six dimensional framework for the evaluation of the supersymmetric index of M-theory five-branes on $T^2\times M_4$ based on a [*on shell*]{} model for BPS states in LST. The appearing of extra massless BPS states in coincidence with 5-branes intersection is clearly encoded as a by-product in our formulas.
As we explained in the previous section, the geometric model encoding the structure of the BPS bound states of 5-branes we refer to is based on the idea that the relevant informations about them are encoded in the (geometric) moduli space of branched holomrphic coverings of the six manifold on which the system lives. Although we have different arguments to justify this model, a proof internal to the six dimensional interacting theory is still lacking (together with the [*off-shell*]{} model itself) and it sounds – see how these structures appear in D-brane physics [@DDD; @bbtt] – that the direct solution of this specific point will be available only once the 5-brane (non-local) analogous geometrical structure to the D-branes gauge bundle structure will be given.
Moreover, the above formula for the supersymmetric index (lets consider for simplicity a single 5-brane) should generalize to $$\Theta(Z^0_{\cal C}) / \N(Z^0_{\cal C})$$ for a 5-brane on a generic supersymmetric six cycle ${\cal C}$ where ${\N(Z^0_{\cal C})}^{-1}$ has to be a modular function of the period matrix $Z^0_{\cal C}$ of the six manifold which reduces to $\eta(\tau)^{-\chi_{M_4}}$ in the case ${\cal C}=T^2\times M_4$ and which calculates the BPS multiplicities of LS states in the general case.
Another interesting point would be to generalize the above treatment to amplitudes of BPS (surface) operators of the type considered in [@surf].
The interested reader is referred to [@io3] for further developments in the subjects treated here.
[**Acknowledgments**]{} I would like to warmly thank M. Bertolini for interesting discussions, the Departments of Physics of Padua and Turin Universities where these results have been presented and the organizers of the Corfu’ RTN meeting. Work supported by the European Commission RTN programme HPRN-CT-2000-00131 as subcontractor of Leuven University.
[77]{}
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[^1]: We are assuming here that only one type of BPS strings has to be counted. This is supported also by the analysis of the short representation of $(0,2)$ supersymmetry in six dimensions given in [@agmh].
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---
abstract: 'The impossibility of superluminal communication is a fundamental principle of physics. Here we show that this principle underpins the performance of several fundamental tasks in quantum information processing and quantum metrology. In particular, we derive tight no-signaling bounds for probabilistic cloning and super-replication that coincide with the corresponding optimal achievable fidelities and rates known. In the context of quantum metrology, we derive the Heisenberg limit from the no-signaling principle for certain scenarios including reference frame alignment and maximum likelihood state estimation. We elaborate on the equivalence of assymptotic phase-covariant cloning and phase estimation for different figures of merit.'
author:
- 'P. Sekatski, M. Skotiniotis, W. Dür'
bibliography:
- 'nosignaling.bib'
title: 'No-signaling bounds for quantum cloning and metrology'
---
Introduction
============
Nothing can travel faster than the speed of light. This is one of the pillars of modern physics and an explicit element of Einstein’s theory of relativity. Any violation of this principle would lead to problems with local causality giving rise to logical contradictions. This principle not only applies to matter, but also to information, rendering superluminal communication impossible. Whilst not explicitly contained in the postulates of quantum mechanics all attempts to construct or observe violations of this principle have failed, leading us to believe that this is indeed a basic ingredient of our description of nature. In fact, modifications of quantum mechanics, e.g., by allowing non-linear dynamics, would lead to signaling and a violation of this fundamental principle [@Gisin:89; @*Gisin:90; @Polchinski:91; @Simon:01]. It is therefore natural to assume that no-signaling holds and try to deduce what follows under such an assumption.
Indeed, the no-signaling principle has been used to derive bounds and limitations on several physical processes and tasks. These include the observation that a perfect quantum copying machine would allow for superluminal communication [@Herbert:82; @Ghirardi:79; @Simon:99], limitations on universal quantum $1 \to 2$ cloning [@Gisin:98; @Ghosh:99] and $1 \to M$ cloning [@Gedik:13], a security proof for quantum communication [@Barrett:05], optimal state discrimination [@Bae:11], and bounds on the success probability of port-based teleportation [@Garcia:13]. However, no-signaling alone is not restrictive enough as it allows for stronger non-local correlations than possible within quantum mechanics [@Popescu:94], and several attempts have been made to further supplement the no-signaling principle in order to retrieve quantum mechanical correlations [@Brassard:06; @*Linden:07; @*; @Skrzypczyk:09; @Pawlowski:09; @Navascues:10; @Fritz:13].
Here we derive limitations on optimal quantum strategies from fundamental principles. In particular we show
- Tight no-signaling bound on probabilistic phase-covariant quantum cloning.
- Asymptotically tight no-signaling bound on unitary super-replication.
- A derivation of the Heisenberg limit for metrology from the no-signaling condition.
- Equivalence between asymptotic quantum cloning and phase estimation.
- Quantum protocols that achieve the bounds placed by no-signaling.
We assume the Hilbert space structure of pure states and show how the no-signaling principle directly leads to tight bounds on different fundamental tasks in quantum information processing and quantum metrology. We start by showing how the impossibility of faster-than-light communication between Alice and Bob can be used to provide upper bounds on Bob’s ability to perform certain tasks, even if Bob has access to supra-quantum resources. Not only does the no-signalling principle allow us to prove ultimate limits on these fundamentally important tasks, it also allows us to demonstrate the optimality of known protocols and shed light on the recently discovered possibility of probabilistic super-replication of states [@Ch:13] and operations [@DSS:14; @Ch:15].
We derive a no-signaling bound on the global fidelity of $N\to M$ probabilistic phase-covariant cloning [@Ch:13]. Our derivation is constructive and we provide the optimal deterministic quantum protocol that achieves the bound [@Ch:13]. In similar fashion, we derive a no-signaling bound on the replication of unitary operations [@DSS:14], which is tight in the large $M$ limit. Furthermore, we derive the Heisenberg limit of quantum metrology solely from the no-signaling principle, more specifically for phase reference alignment [@Buzek:99; @Berry:00; @Ch:04]. We find a tight no-signaling bound on the maximal likelihood and a bound with the correct scaling on the fidelity of reference frame alignment for phase both for the uniform prior as well as for a non-uniform prior probability distribution.
We show that the no-signaling condition can be used to establish bounds on the performance of quantum information tasks for which no bounds are known, or for which the brute force optimization of the tasks is hard. This demonstrates an alternative approach to establish the possibilities and limitations of quantum information processing, which is based on fundamental principles rather than actual protocols. We emphasize that this approach is not limited to the specific tasks discussed here, but is generally applicable.
We also discuss the correspondence between asymptotic phase-covariant quantum cloning and state estimation for different figures of merit, solving the open problem of whether asymptotic cloning, quantified by the global fidelity, is equivalent to state estimation [@Yang:13]. Finally we supplement our approach by a general argument, extending that of [@Simon:01], showing that optimal quantum protocols are at the edge of no-signaling.
No-signaling {#sec:no-signaling}
============
In this section we describe the operational setting underpinning all three tasks we consider (cloning, replication of unitaries, and metrology), as well as the no-signalling condition. All three tasks we consider can be described in the following operationally generic setting. A party, Bob, possess an $N$-qubit state, \_B = \_[**v**]{} a\_[**v**]{} \_B =\_[n=0]{}\^N p\_n \_[\_B]{}, where ${\bf v}$ runs over all $N$-bit stringsand $\ket{\tilde n}_B$ is a superposition over all states with Hamming weight $|{\bf v}|=n$.. Bob then receives, via a remote preparation scenario to be described shortly, the action of a unitary operator $U_\theta^{\otimes N}$ such that $$\ket{\Phi^N_\theta}=U_\theta^{\otimes N}\ket{\Phi^N},
\label{state}$$ where $U_\theta=e^{i\theta H}$ with $H$ an arbitrary Hamiltonian acting on $2$-level systems (qubits) with *spectral radius* $\sigma(H)$, $\theta$ uniformly chosen from $(0,2\pi\sigma(H)]$.
Bob has to process $U_\theta^{\otimes N}$ for some quantum information task in an optimal way. In particular, we do not demand that Bob’s processing be described by linear maps, nor do we demand that the mapping from valid quantum states to probability distributions be given by the Born rule. All that we require of Bob’s processing outcomes is that they should be valid inputs for someone whose processing power is limited by quantum theory. We choose such a setting because our goal is rather pragmatic—we wish to derive upper bounds on *quantum* information tasks. Hence, throughout this work we shall assume that all of Bob’s *static* resources, i.e., pure states of physical systems, ensembles of pure states, and probability distributions, are described within the framework of quantum theory, but Bob’s *dynamical* resources, i.e., processing maps, are not. In fact imposing no-signaling condition for quantum static resources is equivalent to imposing quantum mechanics (see Sec.\[sec: Discussion\]), but when a direct optimization over quantum strategies is unfeasible the no-signaling argument can help to show that a known strategy is optimal.
What Bob has to output varies depending on which task he performs. For example, if the required task is the cloning of the state $\ket{\psi(\theta)}$, then Bob has to output an $M$-qubit state, $\rho^M_\theta$, that is a close approximation to $\ket{\psi(\theta)}^{\otimes M}$. If the required task is the replication of the unitary operator $U_\theta$ then Bob has to output a quantum channel acting on the Hilbert space of $M$ qubits that is a close approximation to $U_\theta^{\otimes M}$. Finally, if the required task is to estimate the parameter $\theta$, then Bob must output a probability distribution corresponding to his updated knowledge about parameter $\theta$. We denote the outcome of Bob’s processing, be it a quantum state, channel, or probability distribution, by $\P(\theta)$.
To incorporate the no-signaling condition we consider that Bob holds one part of a suitably chosen entangled state \_[AB]{} = \_[n=0]{}\^N c\_n \_A \_B which he shares with Alice, where Alice keeps the $(N+1)$-level system spanned by $\{\ket{n}_A\}$ and Bob holds the $2^N$-dimensional system spanned by $\{\ket{\tilde n}_B\}$. The state $\ket{\Psi}_{AB}$ can always be chosen such that Bob receives the action of $U^{\otimes N}_\theta$ on an arbitrary input state. This is achieved by Alice first performing $(U^{\otimes N}_\theta\otimes {\mbox{$1 \hspace{-1.0mm} {\bf l}$}}) \ket{\Psi}_{AB}$, followed by a measurement in the *Fourier basis* $\{ \ket{ k} \propto \sum_n e^{i n \frac{2\pi k }{N+1}} \ket{n}_A\}$ with $k = 0, \ldots, N$ (see Fig. \[fig1\]). If Alice obtains outcome $k$ then Bob’s state becomes $$\ket{\Phi^N_{\theta+\frac{2\pi\,k}{N+1}}}= U_{\theta+\frac{2\pi\, k}{N+1}}^{\otimes N}\ket{\Phi^N}.$$ As all outcomes, $\ket{k}$, are equally likely Bob ends up with a random state from the ensemble $\{\ket{\Phi^N_{\theta+\frac{2\pi\,k}{N+1}}}, \, k\in(0,\ldots, N)\}$.
The no-signaling condition now requires that Bob, who does not know which unitary $U_\theta, \, \theta\in(0,2\pi]$ was chosen by Alice, can not learn $\theta$ from the above ensemble no matter what processing power, quantum or otherwise, Bob has at his disposal. If this were not the case then Alice and Bob, who are spatially separated, can use the above construction to perform faster-than-light communication. Denoting the outcome of Bob’s processing by $\mathcal{P}(\theta | k)$ the no-signaling condition requires that the mixture \[ns general\] () =\_[k=0]{}\^[N]{} (| k) is independent of $\theta$ chosen by Alice.
Note that the no-signaling bound derived above is based on a particular way to embed a quantum information processing task into a communication scenario. The bound turns out to be tight in the present context but is not in general. We will come back to this point from a more general perspective in the Sec. \[sec: Discussion\].
![Generic setting for faster-than-light communication. Alice and Bob share an entangled state $\ket{\Psi}_{AB}$. By applying $U_\theta^{\otimes N}$ followed by a suitable measurement Alice can prepare any ensemble $\{ p_k \prjct{\Phi_{\theta|k}} \}$, which Bob processes into $\mathcal{O}(\theta)=\sum_k p_k \mathcal{P}(\theta|k)$. The no-signaling condition imposes that $\mathcal{O(\theta)}$ is independent of $\theta$ chosen by Alice.[]{data-label="fig1"}](Fig1.pdf){width="8cm"}
Probabilistic phase covariant cloning {#sec:PCC}
=====================================
We first apply the no-signaling condition to the case of phase-covariant quantum cloning (PCC). The latter task involves cloning an unknown state from the set $\{\ket{\psi(\theta)}=U(\theta)\ket\psi)\}$ [@Bruss:00; @*D'Ariano:03; @Ch:13]. We focus on PCC of equatorial states, $\ket{\psi(\theta)}=1/\sqrt{2}(\ket{0}+e^{i\theta}\ket{1})$, which play a crucial role in proving the security of quantum key distribution [@Barrett:05]. Specifically, we provide a bound for the optimal PCC of $N$ qubits into $M>N$ qubits and show that this bound is achievable by a deterministic quantum mechanical strategy, if one drops the restriction of separable $N$-qubit input states. The latter strategy involves the use of a suitable $N$-partite entangled input state on which $U_\theta^{\otimes N}$ is applied. We then show how our deterministic strategy is equivalent to the probabilistic PCC of [@Ch:13], by introducing a suitable filter operation that maps $\ket{\psi(\theta)}^{\otimes N}$ to the suitable $N$-partite entangled state.
A *deterministic*, phase-covariant quantum cloning machine is some transformation, $\mathcal{C}$, whose input is $N$ copies of an unknown equatorial qubit state $\ket{\psi(\theta)}$, that outputs an $M$-qubit state $\rho^M(\theta) =\mathcal{C}\left((\prjct{\psi(\theta)})^{\otimes N}\right)$. Optimal deterministic cloning machines, be it state-dependent [@Bruss:00; @*D'Ariano:03; @Wo:82] or state-independent [@Buzek:96; @Gisin:97; @Bruss:98; @Werner:98], have been constructed and tight bounds, for the case of $1\to 2$ cloning, based on the no-signaling condition have been derived [@Gisin:98; @Ghosh:99]. A probabilistic cloning machine is more powerful in that it allows for a much higher number of copies at the cost of succeeding only some of the time. Indeed, probabilistic PCC, when successful, can output up to $N^2$ faithful copies of $\ket{\psi(\theta)}$. However, the probability of success is exponentially small [@Ch:13].
If the input state to the probabilistic PCC machine is remotely prepared by Alice, as explained in Sec. \[sec:no-signaling\], then the no-signaling condition on the output of Bob’s probabilistic PCC procedure has to be independent of $\theta$, i.e., \[ns\_for\_cloning\] \^M = \^M() =\_[k0]{}\^N \_[+ ]{}\^M. Following [@Ch:13], we quantify the success of the cloning procedure by the *worst case global cloning fidelity* \[mf\] F\_[wc]{}\^[C\_[NM]{}]{} = \_F\_C(\^M\_, ()\^[M]{}), where $F_C(\rho_\theta^M, (\prjct{\psi(\theta)})^{\otimes M})=\mathrm{Tr}\left(\rho_\theta^M (\prjct{\psi(\theta)})^{\otimes M}\right)$ is the *global* fidelity between the output of the cloner and $M$ perfect copies of the input state.
Recalling that the Uhlmann fidelity, $F_U(\rho,\sigma)= \tr \sqrt{\sqrt{\sigma} \rho \sqrt{\sigma}}$ it follows that $F_C(\rho_\theta^M, (\prjct{\psi(\theta)})^{\otimes M}) =F_U(\rho_\theta^M, (\prjct{\psi(\theta)})^{\otimes M})^2$. Moreover, as the worst case fidelity is always smaller or equal than the mean fidelity the following bound holds $$\begin{aligned}
\nonumber
F_{wc}^{\mathcal C_{N\to M}}\leq \left( \int \frac{d\theta}{2\pi} F_U(\rho^M_\theta, (\prjct{\psi(\theta)})^{\otimes M}) \right)^2. \end{aligned}$$ Thus an upper bound for the worst case global cloning fidelity can be obtained by obtaining an upper bound on the mean Uhlmann fidelity.
In order to upper bound the mean Uhlmann fidelity we first rewrite the latter as $$\begin{aligned}
\label{Fidelity2}
&\int \frac{d\theta}{2\pi} F_U(\rho^M_\theta, (\prjct{\psi(\theta)})^{\otimes M})=
\int_0^{2\pi}\sum_{k=0}^N \frac{d\theta}{2\pi(N+1)} \nonumber \times\\ \nonumber
& F_U(\rho^M_{\theta+\frac{2\pi k}{N+1}},
(U_{\frac{2\pi k}{N+1}}\prjct{\psi(\theta)}U^{\dagger}_{\frac{2\pi k}{N+1}})^{\otimes M})\\
&\leq \int_0^{2\pi} \frac{d\theta}{2\pi} F_U\left(\rho^M, \sum_{k=0}^N\frac{(U_{\frac{2\pi k}{N+1}}\prjct{\psi(\theta)}
U^{\dagger}_{\frac{2\pi k}{N+1}})^{\otimes M}}{N+1}\right),\end{aligned}$$ where we have used the joint concavity of the Uhlmann fidelity, $F_U\left(\sum_i p_i\,\rho_i, \sum_i p_i\sigma_i\right)\geq
\sum_i p_i F_U(\rho_i,\sigma_i)$ in the last line of Eq. . As $\ket{\psi(\theta)}=U_\theta\ket{+}$, and $[U_{\frac{2\pi k}{N+1}},U_\theta]=0$, unitary invariance of the fidelity, $F(\rho ,U\sigma U^\dagger)=F(U^\dag \rho U,\sigma)$, allows us to shift the action of $U(\theta)^{\otimes M}$ onto $\rho^M$ and the integrand of Eq. reads \_0\^[2]{} F\_U(U\_\^\^M U\_, \_[k=0]{}\^N). Finally using the concavity of the Uhlmann fidelity, $F\left(\sum_i p_i\,\rho_i, \sigma\right)\geq \sum_i p_i F(\rho_i, \sigma)$), to move the integral over $\theta$ inside the argument for the Uhlmann fidelity, and defining the maps $\mathcal{G}_{\mathbb{Z}_{N+1}}[\cdot]\equiv\frac{1}{N+1}\sum_{k=0}^NU^{\otimes M}_{\frac{2\pi k}{N+1}}\,
(\cdot)\, U^{\dagger\,\otimes M}_{\frac{2\pi k}{N+1}}$ and $\mathcal{G}^{\dagger}_{\mathbb{U}(1)}[\cdot]\equiv \frac{1}{2\pi}
\int_0^{2\pi}\mathrm{d}\theta\, U^{\dagger\,\otimes M}_\theta \,(\cdot)\, U^{\otimes M}_\theta$, we obtain the desired upper bound for the worst case global cloning fidelity \[Fidelity4\] F\_[wc]{}\^[C\_[NM]{}]{} F\_U(\^\_[(1)]{}\[\^M\],\_[\_[N+1]{}]{} \[()\^[M]{}\])\^2.
We now proceed to give an explicit expression for the upper bound of Eq. . The maps $\mathcal{G}$ impose a *block-diagonal* structure on any density matrix on which they act, making it it easy to find $\rho_M$ that maximizes Eq. . As the state $(\prjct{+})^{\otimes M}$ is symmetric under permutations it suffices to maximize over all symmetric $\rho^M$. For any permutation symmetric $\rho^M$, $\mathcal{G}^{\dagger}_{\mathbb{U}(1)}[\rho^M]$ is diagonal and can be written as \[block1\] \^\_[(1)]{}\[\^M\] = \_[n=0]{}\^M p\_n , where $\{\ket{n,M}\}_{n=0}^{M}$ is an orthonormal basis spanning the symmetric subspace of $M$ qubits with $n$ qubits in state $\ket{1}$ and $M-n$ qubits in state $\ket{0}$. Correspondingly, we may write \[block2\] \_[\_[N+1]{}]{}\[()\^[M]{}\]=\_[=0]{}\^N, where $\ket{\phi^{(\lambda)}}=\sum_{n |\, n\mod (N+1) =\lambda}\sqrt{\frac{\binom{M}{n}}{2^M}}\ket{n,M}$ are unnormalized pure symmetric states with the sum running over all $n$ that have a reminder $\lambda$ after division by $N+1$. The states $\ket{\phi^{(\lambda)}}$ have non-zero overlap with the symmetric states $\ket{n,M}$ whenever $n \mod (N+1) =\lambda$.
Because of the block-diagonal structure we rewrite the mean Uhlmann fidelity as $$\begin{aligned}
F_U(\mathcal{G}^\dagger_{\mathbb{U}(1)}[\rho^M],\mathcal{G}_{\mathbb{Z}_{N+1}}[(\prjct{+})^{\otimes M}])= \nonumber\\
\sum_{\lambda=0}^N \sqrt{\bra{\phi^{(\lambda)}} \mathcal{G}_{\mathbb{U}(1)}[\rho^M] \ket{\phi^{(\lambda)}}}.\end{aligned}$$ Denoting by $p_\lambda = \sum_{\{n| n \mod (N+1) =\lambda\}} p_n$ the probability of projecting $\mathcal{G}^\dagger_{\mathbb{U}(1)}[\rho^M]$ on the sector with a given $\lambda$ we can maximize the Uhlmann fidelity by optimizing each sector $\lambda$ independently. This is achieved by finding the $n$ in each sector $\lambda$ such that the overlap ${\langle \phi^{(\lambda)} |n,M\rangle}$ is maximized. The maximum Uhlmann fidelity then reads $$\begin{aligned}
\max_{p_n} F_U(\bigoplus_{n=0}^M p_n \prjct{n,M},\bigoplus_{\lambda=0}^N\prjct{\phi^{(\lambda)}} ) =\nonumber\\
\max_{p_\lambda} \sqrt{p_\lambda \max_n |{\langle \phi^{(\lambda)} |n,M\rangle}|^2}= \sqrt{\sum_\lambda \max_n |
{\langle \phi^{(\lambda)} |n,M\rangle}|^2}.\end{aligned}$$
The probability $|{\langle \phi^{(\lambda)} |n,M\rangle}|^2 = \frac{1}{2^M}\binom{M}{n}$ is given by the binomial distribution if $n\mod (N+1) = \lambda$ and is zero otherwise. Thus, it is always optimal to choose $n\mod (N+1)$ closest to $\frac{M}{2}$. Doing so for all $\lambda$ we find that the maximal fidelity is given by the square root of the sum of the $N+1$ largest terms of the binomial distribution $\frac{1}{2^M}\binom{M}{n}$. Hence, the upper bound for the worst case global cloning fidelity reads \[fidelity6\] F\_[wc]{}\^[C\_[NM]{}]{} \_[=0]{}\^N , where $\lfloor\cdot\rfloor$ denotes the floor function. Finally, noting that the binomial distribution, $\frac{1}{2^M}\binom{M}{n}$ can be approximated by a Gaussian $\mathcal{N}(\mu=M/2, \sigma=\sqrt{M}/2)$, the upper bound in Eq. can be approximated, for $M\gg N$, by $$F_{wc}^{\mathcal C_{N\to M}}\leq \mathrm{erf}\left(\frac{N+1}{\sqrt{2 M}}\right).
\label{asymptoticfidelity}$$
We note that as long as $M=\mathcal{O}(N^2)$ the cloning fidelity approaches unity in the limit $N\to\infty$. Indeed, one can make an even stronger claim. Any replication procedure that respects the no-signaling condition and produces a number of replicas $M=\mathcal{O}(N^{2+\alpha})$ does so with a cloning fidelity that tends to zero exponentially fast.
We now show how one can achieve the no-signaling bound of Eq. using a deterministic quantum mechanical strategy. Instead of $N$-copies of $\ket{\psi(\theta)}$, suppose Bob prepares the entangled state \[deterministic\] \_[=0]{}\^N . Bob now applies the cloning map $\mathcal{C}: \ket{N,\lambda} \mapsto \ket{M,\lfloor \frac{M-N}{2}\rfloor + \lambda}$, that maps totally symmetric $N$-qubit states to totally symmetric $M$-qubit states. This strategy achieves the bound of Eq. as the latter is valid for all input states. We pause to note that the above result does not contradict the well know limits for deterministic cloning, as in the latter Bob is forced to input $N$-copies of a qubit state.
The bound of Eq. is the ultimate bound that can be achieved even by a probabilistic strategy. Indeed, the best probabilistic quantum mechanical PCC attains precisely the no-signaling bound of Eq. [@Ch:13]. In fact there is an easy way to see how the probabilistic strategy of [@Ch:13] and the deterministic strategy described above are related. Starting from $N$ copies of the state $\ket{\psi(\theta)}$, the probabilistic PCC of [@Ch:13] has Bob first apply the probabilistic filter that projects onto the state $\ket{\Phi^N}$ of Eq. and succeeds with probability $p_{yes} = |{\langle \Phi^N |+\rangle}^{\otimes N}|^2$. As such a filter commutes with the unitary $U_\theta^{\otimes N}$ it can be seen as part of the overall state preparation. The advantage, then, of probabilistic PCC can be simply understood as a passage from the standard quantum limit in quantum metrology, achieved for separable input states, to the Heisenberg limit achieved by entangled input states. Notice that no probabilistic advantage exists for the case of $1\to M$ cloning. For the latter, the fidelity of Eq. takes the simple form $F^{\mathcal C_{1\to M}}
= \frac{1}{2^{M-1}}\binom{M}{\frac{M-1}{2}}$ for $M$ odd and $F^{\mathcal C_{1\to M}} = \frac{1}{2^{M}}\binom{M+1}
{\frac{M}{2}+1}$ for $M$ even and is known to be achievable by a deterministic strategy [@D'Ariano:03].
![Equivalence between probabilistic PCC and deterministic PCC using arbitrary states. The filter in the probabilistic cloning can be viewed as part of a probabilistic preparation of a general state from a separable N-qubit state. Allowing for arbitrary input states makes the preparation process deterministic.[]{data-label="fig2"}](Fig2.pdf){width="6cm"}
Replication of unitaries
========================
We now consider the task where Bob has to output an approximation, $V_\theta$, of $U^{\otimes M}_\theta$ having received only $N$ uses of the black box implementing the unitary transformation $U_\theta$ [@Ch:08; @DSS:14]. The figure of merit that one uses is the global Jamiołkowski fidelity (process fidelity) [@Gilchrist:05; @*Dur:05], \[processfidelity\] F(U\^[M]{}\_, V\_) = \_[V\_]{} , averaged over all $\theta$, where $\ket{\psi_{U^{\otimes M}_\theta}} = ({\mbox{$1 \hspace{-1.0mm} {\bf l}$}}\otimes U_\theta^{\otimes M}) \ket{\Phi^+}$ and $\rho_{V_\theta} = {\mbox{$1 \hspace{-1.0mm} {\bf l}$}}\otimes V_\theta (\prjct{\Phi^+})$, with $\ket{\Phi^+}=1/\sqrt{2^M}\sum_{n}\ket{n}\ket{n}$, where $n$ are the $M$-qubit bit strings, are the corresponding Choi-Jamiołkowski states [@Jamiolkowski:72; @*Choi:72] for $U^{\otimes M}_\theta$ and $V_\theta$ respectively. It was shown in [@DSS:14] that when $M<N^2$ Bob can approximate $U_\theta^{\otimes M}$ almost perfectly, i.e., with process fidelity approaching unity in the large $N$ limit. We now show that the protocol in [@DSS:14] saturates the no-signaling bound.
In order to apply the no-signaling condition for the case of unitary replication in an easy way we consider the following communication scenario. Alice prepares the Choi-Jamiołkowski state corresponding to $U^{\otimes N}_\theta$, $\ket{\psi_{U^{\otimes N}_\theta}}$, at Bob’s side, which he can then use to probabilistically implement $U^{\otimes N}_\theta$ on an arbitrary input state [@Dur:01]. Consequently, the protocol for which we shall derive a no-signaling bound is inherently probabilistic. We note that a bound for a probabilistic protocol is automatically a bound for a deterministic protocol as well, as the former are less restrictive than the latter.
The no-signaling constraint for unitary replication takes the form \_N\^M = \_[k=0]{}\^[N]{} V\_[+ ]{} (), and is independent of $\theta$. As the worst case process fidelity (Eq. ) is identical to the worst case global cloning fidelity used for PCC (Eq. ) the no-signaling bound for probabilistic replication of unitaries reads \[unitaryrepbound\] F\_[wc]{}(U\^[M]{}\_, V\_) \_[=0]{}\^N . This bound is achieved, in the limit of large $M$, by the deterministic strategy in [@DSS:14], for which the fidelity is independent of $\theta$. This implies that probabilistic processes offer no advantage in this case. Thus, the optimal deterministic replication of unitary operations allowed by quantum mechanics is at the edge of no-signaling.
Quantum Metrology
=================
We now apply the no-signaling condition to provide bounds for quantum metrology. The latter task involves the use of $N$ systems, known as the probes, prepared in a suitable state $\ket{\psi}\in\mathcal{H}^{\otimes
N}$, and subjected to a dynamical evolution described by a completely positive map, $\mathcal{E}_\theta$, that imprints the value of $\theta$ onto their state, i.e., $\rho_\theta=\mathcal{E}_\theta(\prjct{\psi})$. Information about the value of $\theta$ is retrieved by a suitable measurement of the $N$ probes. The goal in quantum metrology is to choose the initial state $\ket{\psi}$ and final measurement such that the value of $\theta$ can be inferred as precisely as possible.
If the $N$ quantum probes are prepared in a separable quantum state, i.e., $\ket{\psi}=\ket{\phi}^{\otimes N}$, then the mean square error with which $\theta$ can be estimated, optimizing over all allowable measurements, scales inversely proportional with $N$ [@GLM04]. This limit is known as the *standard quantum limit*. If, however, the $N$ probes are prepared in a suitably entangled state, then the mean square error with which $\theta$ can be estimated scales inversely proportional with $N^2$ [@GLM04]. This limit is known as the *Heisenberg limit*. By allowing for a probabilistic strategy, the Heisenberg limit in precision can be obtained even with separable states [@Fiurasek:06; @Gendra:13a; @*Gendra:13b]. Recently, it was shown that both the standard and Heisenberg limits are related with the maximum replication rates corresponding to a deterministic and probabilistic PCC strategies respectively [@Ch:13; @Gendra:14].
We now show how the no-signaling condition implies that the ultimate bound in precision for metrology is the Heisenberg limit, even if supra-quantum processing is allowed. We shall consider two particular examples of Bayesian quantum metrology; phase alignment, where the relevant parameter to be estimated is the phase of a local oscillator, $\theta\in (0,2\pi]$, which is initially completely unknown [@BRS:07] (Sec. \[sec:metrology\]), and phase diffusion where our prior knowledge of the parameter, initially described by a delta function around some value $\theta_0$, diffuses over time [@Demkowicz:11] (Sec. \[sec:diffusion\]). We stress that whereas analytical bounds for phase alignment are known, for phase diffusion bounds are known only for small number of probes [@Demkowicz:11]. This is due to the fact that the optimal strategy is difficult to compute, even numerically. Nevertheless, our no-signaling constraint allows us to place an upper bound on the optimal fidelity of estimation for asymptotically many probe systems. We emphasize that a similar strategy can be applied to a variety of quantum information processing tasks, where limitations of the processes can be gauged by fundamental principles.
In Sec. \[Assymptotic Cloning\] we establish the relationship between optimal quantum cloning protocols and measure and prepare strategies. In particular, we show that a measure and prepare strategy that maximizes the alignment fidelity is asymptotically equivalent to a quantum cloning machine that maximizes the [*per copy*]{} fidelity, whereas a measure and prepare strategy that optimizes the maximum likelihood of estimation is asymptotically equivalent to a quantum cloning machine that maximizes the global fidelity. This latter result answers the open question of Yang [*et al.*]{} [@Yang:13] as to whether asymptotic cloning, where the quality of the cloned state is quantified by the global fidelity, is equivalent to state estimation.
Metrology with uniform prior {#sec:metrology}
----------------------------
Consider the problem of phase alignment, i.e., estimating a completely unknown phase, $\theta$. We will utilize two different ways of quantifying the precision of estimation of $\theta$: the *maximum likelihood* of a correct guess, $\mu=p(\theta|\theta)$ [@Ch:04], and the *fidelity* of alignment, given by the payoff function $f=\cos^2\left(\frac{\theta-\theta'}{2}\right)$ [@Berry:00]. For the case of phase alignment the no-signaling condition (Eq. ) takes the form $$\begin{aligned}
\label{ns alignment}
p(\theta'|\theta)=\frac{1}{N+1}\sum_{k=0}^N p\left(\theta'| \theta + \frac{2\pi k}{N+1}\right)\end{aligned}$$ and is independent of $\theta$ (the same holds for a measurement with discrete outcomes). Note that we make no assumptions on how Bob obtains the probability distribution of Eq. . In particular we do not restrict Bob’s processing to be quantum mechanical. We only require that the inputs and outputs to Bob’s processing apparatus be valid quantum states and probability distributions respectively.
### Maximal likelihood {#sec: maximal likelihood}
For the case where the precision is quantified by the maximum likelihood the no-signaling bound (Eq. ) gives $p(\theta|\theta) \leq (N+1)p(\theta)$. If the estimate $\theta'$ is unbiased, all outcomes are equally likely and the no-signaling bound takes the simple form $p(\theta|\theta) \leq N+1$. The bound is known to be achievable using the state [@Ch:04] \[ml state\] =\_n .
### Alignement fidelity {#sec: alignment fidelity}
For the case where the precision is quantified by the fidelity of alignment, for each choice of $\theta,\, \theta'$ the fidelity must be properly weighted by the *joint probability distribution*, $p(\theta',\theta)=p(\theta'|\theta)p(\theta)$. The average fidelity of alignment is thus \[meanfid\] |[f]{}= ’\^2() p(’|) The probability distribution that both maximizes the average fidelity and is compatible with no-signaling is $$\begin{aligned}
\label{step probability}
p(\theta'|\theta)=
\left\{\begin{array}{cl}
\frac{N+1}{2\pi} & \text{if}\quad |\theta' -\theta| \leq \frac{\pi}{N+1}\\
0 & \text{otherwise}
\end{array}\right.\end{aligned}$$ as we now show.
Our aim is to distribute the probability distribution of Eq. amongst $N+1$ terms subject to the constraint that $\int\mathrm{d}\theta' p(\theta')=1$ such that the average fidelity of Eq. is maximized. Without loss of generality assume that $\theta\in \left(0,\frac{2\pi}{N+1}\right)$. If this is not the case we can always relabel the measurement outcomes $k\in(0,\ldots,N)$ such that $\theta$ lies in $\left(0,\frac{2\pi}{N+1}\right)$. As $\cos^2\left(\frac{\theta'-\theta}{2}\right)$ is largest when $\theta-\theta'=0$ the average fidelity is optimized by setting $p(\theta'|\theta+\frac{2\pi k}{N+1})=0$ for $k\neq 0$. As this is true for all randomly chosen $\theta$, and using the constraint $\int\mathrm{d}\theta'\, p(\theta')=1$, it follows that $p(\theta'|\theta)=\frac{N+1}{2\pi}$ for $|\theta'-\theta|\leq
\frac{\pi}{N+1}$ and zero everywhere else.
We now derive the maximum average fidelity (Eq. ) compatible with no-signaling. As the conditional probability distribution $p(\theta'|\theta)$ of Eq. depends only on the difference $\theta'-\theta$ we may write the average fidelity as |f=1-\_[-]{}\^\_[-]{}\^’ p(’-)\^2() \[fid1\] where we have used the identity $\cos^2(x)=1-\sin^2(x)$. As the integrand in Eq. depends only on the difference $\theta'-\theta$ we may define $\phi=\theta'-\theta$ and $\mathrm{d}\phi=\mathrm{d}\theta'$ so that |f=1-\_[-]{}\^ p()\^2(). \[fid2\] Substituting the no-signaling probability distribution of Eq. in place of $p(\phi)$ in Eq. one obtains |f=1-\_\^ \^2(). \[fid3\] In the limit of large $N$ the limits of integration in Eq. become narrower and we can use the small angle approximation to write $\sin(\phi/2)\approx \phi/2$. Substituting the latter into Eq. and evaluating the integral one obtains the average fidelity $\bar f\approx1-\frac{\pi^2}{12N^2}$. The maximum average fidelity achievable by a quantum mechanical strategy is $\bar{f}\approx 1- \frac{\pi^2}{ 4 N^2}$ [@Berry:00] achieved by the input state \[Berry Wisemann state\] \_n () . This fidelity is strictly smaller than the bound achieved by no-signaling. Nevertheless, the no-signaling bound gives rise to the right scaling with respect to $N$.
Correspondance between asymptotic cloning and phase estimation {#Assymptotic Cloning}
--------------------------------------------------------------
Every estimation strategy can be used in a measure and prepare cloning protocol (henceforth referred to as m&p), where Bob first estimates the $N$-copy input state and, based on his estimate, prepares an $M$-qubit state. There are two free choices in every m&p protocol: (a) what is the optimal estimation strategy, i.e., which figure of merit to chose, and, (b) which output state to prepare, i.e., do we prepare M-copies of $\ket{\psi(\theta)}$ or some suitable $M$-partite entangled state. Similarly, there are two figures used in the literature thus far, to quantify the quality of a quantum cloning machine: (a) the *global fidelity* [@Ch:13] defined by the overlap of the M-qubit output of the cloning machine, $\rho^M$, with the the ideal $M$-copy state $\ket{\psi}^{\otimes M}$ (this is the figure of merit that we considered in the previous section) and (b) the *per copy fidelity* [@Gisin:98; @Ghosh:99; @Bruss:00; @Buzek:96; @Gisin:97; @Bruss:98; @Werner:98] which is the average of the overlap of the reduced single qubit output state $\rho^M|_n= \tr_{\textrm{all} \setminus n} \rho^M$ with a perfect single-qubit state $\ket{\psi}$.
In this subsection we discuss how the optimal m&p strategies compare with optimal cloning when the number of copies $M$ goes to infinity. In particular, we will show that the optimal m&p strategy based on the *alignment fidelity* is equivalent to an asymptotic quantum cloning machine which optimizes the *per copy fidelity* of the clones, whereas the optimal m&p strategy based on the maximum likelihood is equivalent to an asymptotic quantum cloning machine which optimizes the *global fidelity*. Whereas the equivalence between asymptotic cloning, quantified by the [*per copy*]{} fidelity, and a corresponing m&p protocol was known for both deterministic [@Bae:06] as well as probabilistic cloning [@Gendra:14] the same question concerning asymptotic cloning, quantified by the *global* fidelity, has been an open problem [@Yang:13].
### Per copy cloning fidelity and alignement fidelity
We begin by discussing the equivalence between a m&p strategy based on the alignment fidelity and an asymptotically optimal cloning machine that maximizes the *per copy* fidelity. As the equivalence between a deterministic m&p strategy and the optimal [*per copy*]{} cloning fidelity is known [@Bae:06] we focus on the more general case of a probabilistic m&p strategy. The latter directly translates into the alignment fidelity of the estimation strategy as the optimal output state simply consists of preparing copies of $\ket{\psi(\theta')}$, where $\theta'$ is Bob’s estimate of $\theta$. Consequently, the best m&p average per copy cloning fidelity, given by $\int \,p(\theta) p(\theta'|\theta) \tr \prjct{\psi(\theta')} \prjct{\psi(\theta)} d\theta\,d\theta'$, equals the alignment fidelity $ \int \,p(\theta)p(\theta'|\theta) \cos^2(\frac{\theta-\theta'}{2}) d\theta\,d\theta'$. Hence, the optimal phase alignment protocol [@Berry:00], achieved for the input state Eq., directly translates into the optimal probabilistic m&p cloning with per copy fidelity $\bar f=1-\frac{\pi^2}{4 N^2}$. Again this probabilistic strategy provides a drastic improvement over the optimal deterministic cloning strategy, where the average per copy fidelity is $f= 1-\frac{1}{N}$ in the large $N$ limit [@D'Ariano:03].
### Global cloning fidelity and maximal likelihood
Let us now turn to the global fidelity. The naive m&p strategy consists in preparing $M$-copies $\ket{\psi(\theta')}^{\otimes M}$ all pointing in the estimated direction $\theta'$. In this case the output state is $$\label{m&pstate}
\rho^M =\frac{1}{2\pi}\int\,d\theta\,d\theta' p(\theta'|\theta) \prjct{\psi(\theta)}^{\otimes M} d\theta.$$ The cloning fidelity is now given by $$\label{m&p fidelity1}
F^{N\to M}_{m\&p}=\left|\frac{1}{2\pi}\int\,d\theta\,d\theta'\,p(\theta'|\theta)|{\langle \psi(\theta') |\psi(\theta)\rangle}|^{2M}\right|$$ Now in the limit $M\to\infty$ the overlap $|{\langle \psi(\theta') |\psi(\theta)\rangle}|^{2M} \to \frac{2\sqrt{\pi}}{\sqrt{M}}\delta(\theta-\theta')$ where the constant of proportionality is obtained by integrating over the entire range of either $\theta$ or $\theta'$. Inserting this expression back into Eq. yields the global fidelity for this m&p protocol of $$\label{m&p fidelity2}
F^{N\to M}_{m\&p}=\sqrt{\frac{1}{\pi M}}p(\theta|\theta).$$
The global fidelity of this m&p protocol is directly proportional to the maximum likelihood for phase estimation. However, we note that when one substitutes the optimal maximal likelihood $p(\theta|\theta)=N+1$, achieved by the input state in Eq. , Eq. is smaller than the global fidelity (Eq. ), which we proved to be the optimal fidelity achievable by the no-signaling condition, by a factor of $\sqrt{2}$. A similar discrepancy was already noted in [@Yang:13] for deterministic cloning, where the authors also showed for two simple cases (with small $N$) how to build m&p strategies that attain the optimal asymptotic global fidelity. This was done by allowing Bob to output more general states.
In the following we derive the optimal probabilistic m&p strategy that attains the asymptomatic global fidelity of the probabilistic phase-covariant cloner (Eq. ) for an arbitrary number of input copies $N$. We consider a m&p protocol based on maximum likelihood estimation (we shall discuss its optimality in the end of the section) but allow Bob to output a general state $U_\theta^{\otimes M}\ket{\Psi^M}$.
Without loss of generality let us assume that $\theta=0$. The strategy discussed above would let Bob output the state $$\label{m&p state2}
\rho_0^{m\&p} =\int p(\theta|0) U_\theta^M \prjct{\Psi^M} U_\theta^{M \dag} d\theta,$$ where the probability distribution $p(\theta| 0)=\tr \prjct{\Phi^N_{m.l.}} E(\theta)=\frac{1}{(N+1)2\pi} \sum_{n,\bar n=0}^N
e^{i \theta(n-\bar n)}$. Here the optimal POVM elements are known to be covariant [@Ch:04] and are given by $E(\theta)=\sum_{n,m=0}^N e^{i\theta(n-m)}{\mathinner{|{n}\rangle}\!\!\mathinner{\langle{m}|}}$. The corresponding m&p cloning fidelity is $F_{m\&p}^{N\to M} = \tr \rho_0^{m\&p} \prjct{\psi(0)}^{\otimes M}$. Using the cyclic property of the trace to shift the action of the unitaries $U_\theta^M$ onto $\prjct{\psi(0)}^{\otimes M}$ and carrying out the integration the global cloning fidelity reads \[m&p fidelity3\] F\_[m&p]{}\^[NM]{} = \_M\^N, where \_M\^N = \_[m,|m=0]{}\^M \_N( m-|m) with the coherence decay term $\Delta_N(m-\bar m)$ given by $$\begin{aligned}
\nonumber\label{Delta terms}
&\Delta_N(m-\bar m) = \int e^{i \theta (m-\bar m)} \bra{\Phi_{m.l}^N}E(\theta)\ket{\Phi_{m.l}^N } \frac{d\theta}{2\pi} =\\ &\int \sum_{n,\bar n=0}^N e^{i\theta(n+m - \bar n -\bar m)} \frac{d\theta}{2\pi(N+1)}
=\max\{1-\frac{|m-\bar m|}{N+1},0\}.\end{aligned}$$
Having established the form of the m&p fidelity (Eq. ) we can now proceed to optimize this expression and obtain the corresponding optimal state $\ket{\Psi^M}$. First, we note that the optimal state should have maximal support over those values of $m$ that lie in the interval $(\frac{M}{2}\pm\sqrt{\frac{M}{4}})$ since as for this range of values the binomial coefficients in $O_M^N$ are large. Second, $\ket{\Psi^M}$ should be roughly constant in the range $[m,m+N+1]$ such that that all the coherence terms, ${\mathinner{|{m}\rangle}\!\!\mathinner{\langle{\bar m}|}}$, add up constructively, i.e., $\sum_{\bar m-m} \Delta_N(m-\bar m)=N+1$. In the limit $M\to \infty$ both of these requirements can be satisfied simultaneously. In particular, choosing $|{\langle m |\Psi^M\rangle}|^2=\frac{1}{\sqrt{2\pi} \sigma}e^{\frac{-(m-M/2)}{2\sigma^2}}$ leads to a m&p fidelity of \[m&p asymptotic\] F\_[m&p]{}\^[NM]{} = (1+ O()\^2 + O()\^2). We note that Eq. corresponds to the asymptotic expansion of the optimal cloning fidelity of Eq. . Choosing $\sigma=M^{\frac{1}{2}-\epsilon}$, for $0\leq\epsilon<\frac{1}{2}$ yields the optimal m&p fidelity that is equivalent to the asymptotically optimal cloning machine whose performance is quantified by the global fidelity.
Note that the entire argument above is applicable even if one considers a different estimation strategy, i.e., a different figure of merit. Indeed, the only thing that changes if one changes the estimation strategy (going to a general input state $\ket{\Phi^N}$) is the coherence decay $\Delta_N(m-\bar m)$ in Eq. . However its contribution of the coherence terms to the cloning fidelity $\sum_{m} \Delta_N(m) = p_{\ket{\Phi^N}}(0|0)$ is given by the maximal likelihood, which establishes a correspondence between the asymptotic global fidelity of the m&p cloner and the maximal likelihood of the estimation (remark also that the optimal input state Eq. and Eq. match for $M\to \infty$). Of course the same correspondence holds for deterministic cloning, for which the maximal likelyhood for $N
$-copies state $\ket{\psi(\theta)}^{\otimes N}$ is simply obtained as $\bra{\psi(\theta)}^{\otimes N} E(\theta) \ket{\psi(\theta)}
^{\otimes N}= \frac{1}{2^N}(\sum_{j=0}^N \sqrt{\binom{N}{j}})^2$ and the optimal global fidelity is known to be $\frac{1}{2^{N
+M}} \binom{M}{M/2}(\sum_{j=0}^N \sqrt{\binom{N}{j}})^2 $ [@D'Ariano:03].
To summarize (see table), for phase-covariant cloning the m&p strategy based on maximal likelihood estimation (Sec. \[sec: maximal likelihood\]) is optimal with respect to the *global* cloning fidelity whereas the m&p strategy based on the alignment fidelity of estimation (Sec. \[sec: alignment fidelity\]) is optimal with respect to the per copy fidelity of cloning. Both m&p strategies attain the optimal cloning for any fixed $N$ and $M \to \infty$, and this is true both for optimal deterministic as well as probabilistic cloning. We believe that same correspondence should hold for universal cloning, however this is beyond the scope of this paper. The correspondence between the different m&p strategies and optimal cloning machines is summarized in the following table
------------------------- ------- ---------------------------------
Estimation scenario for Optimal asymptotic cloning
M&P cloning (probabilistic or deterministic
Maximal likelihood $\to$ Global fidelity
Alignment fidelity $\to$ Per copy fidelity
------------------------- ------- ---------------------------------
Metrology with general prior {#sec:diffusion}
----------------------------
Let us now consider a more general metrological scenario where Bob has some prior knowledge, $p(\theta)$, of the parameter $\theta$. Following [@Demkowicz:11] we consider the prior, $p(\theta; t) =\frac{1}{2\pi}\big(1 +2 \sum_{n=1}^\infty\cos(n\theta)e^{-n^2 t}\big)$, that arises from a diffusive evolution of $p(\theta)=\delta(\theta)$. The mean fidelity (Eq. ) now reads \[mean fidelity prior\] |f\_t= 1-’ p(’|) p(; t) \^2() An efficient algorithm optimizing $\bar f_t$ for moderate $N$ was derived in [@Demkowicz:11]. However, the optimization becomes intractable, even numerically, when $N$ increases. Indeed, optimizing $\bar f_t$ for large $N$ is in general a hard task. Nevertheless the no-signaling constraint allows us to derive an upper bound for $\bar f_t$ for large enough $N$ as we now show.
Our goal is to minimize the integrand of Eq. under the no-signaling constraint of Eq. . For a fixed value of $t$ the product g(;’, t)=p(; t) \^2() in Eq. obtains its minimum value when $\theta-\theta'=0$. In addition $g(\theta;\theta',t)$ is monotonically increasing so long as the derivative of $g(\theta;\theta',t)$ around $\theta'=\theta$ is greater than zero. This is true so long as \^2() < ()\^2 := \^2(), where $m\equiv\min_\theta p(\theta; t)$ and $M =\max_\theta | \partial_\theta p(\theta; t)|$. Outside the interval $[\theta' -\Delta, \theta'+\Delta] $ the function $g(\theta;\theta',t)$ is larger than $m \sin^2(\Delta/2)$. Therefore, $g(\theta;\theta',t)$ attains its global minimum in the finite interval satisfying the condition \^2() <m \^2()=.
Now consider the narrowest probability distribution compatible with no-signaling given by $\frac{N+1 }{2 \pi}p(\theta')$, where $p(\theta')$ is the probability distribution given in Eq. , for $|\hat \theta_\ell -\theta|<\frac{\pi}{N+1}$ and zero elsewhere. For large enough $N$ this probability distribution is contained entirely in the interval $[\theta' -\Delta, \theta'+\Delta] $ where $g(\theta;\theta', t)$ attains its minimum and therefore minimizes the integrant of Eq. . Plugging this probability into Eq. (\[mean fidelity prior\]) and using the condition $\int \mathrm{d}\theta' p(\theta') =1$ leads to |f\_t 1 - \_4(0, e\^[-t]{}), where $\vartheta_4(0, e^{-t}) = 1 + 2\sum_{n=1}^\infty (-1)^n e^{-n^2 t}= m$ is the Jacobi theta function ranging from $0$, when $p(\theta;0)=\delta(\theta)$, to $1$, when $p(\theta;\infty)= 1/2\pi$. Again we discover that the ultimate bound in precision scales inversely proportional to $N^2$.
![The lower bound on the asymptotically achievable error $\frac{1}{2}\vartheta_4(0,e^{-t})= \frac{6 N^2}{\pi^2}(1-\bar f_t)$ (bottom curve) and the error of the prior $\int p(\theta; t) \sin^2(\theta/2) d\theta$ (top curve) as functions of $t$.[]{data-label="fig4"}](Fig4.pdf){width="6cm"}
Discussion {#sec: Discussion}
==========
Tightness of bounds
-------------------
We have shown that the no-signaling condition can set upper bounds on several important quantum information tasks, such as cloning, unitary replication, and metrology. In the case of PCC and unitary replication we have shown that the no-signaling bound coincides with the optimal quantum mechanical strategy implying that quantum mechanical strategies for PCC cloning and unitary replication are at the edge of no-signaling. However, for the case of metrology, and in particular for the average fidelity of estimation, we see that there is a gap between the no-signaling bound and the optimal quantum strategy. Could this gap be an indication of the existence of a supra-quantum strategy, compatible with no-signaling, that outperforms the best quantum mechanical strategy? The answer is no, as we now explain.
In deriving the no-signaling constraint of Eq. we only considered one particular way for Alice and Bob to attempt for faster-than-light communication; using a suitably entangled state $\ket{\Psi}_{AB}$. This, in turn led to the sharp probability distribution of Eq. . However, one can construct a communication scenario where the probability distribution of Eq. can lead to signaling as we now show.
Let us first consider the qubit case ($N=1$). Let Alice and Bob share the entangled state $\ket{\Psi}_{AB} = \cos(\varepsilon)\ket{00} +\sin(\varepsilon)\ket{11}$. Alice can chose to measure her system in either the computational basis $\{\ket{0},\ket{1}\}$ or the x-basis $\{\ket{\pm}=\frac{\ket{0}\pm \ket{1}}{\sqrt{2} }\}$ steering Bob’s state into the ensembles $\mathcal{E}^{(1)}=\{ \cos^2(\varepsilon) \prjct{0}, \sin^2(\varepsilon)
\prjct{1}\}$ and $\mathcal{E}^{(2)}=\{\frac{1}{2} \prjct{\varepsilon}, \frac{1}{2} \prjct{-\varepsilon}\}$ respectively, where $\ket{\pm \varepsilon} =\cos(\varepsilon)\ket{0} \pm \sin(\varepsilon)\ket{1}$, as shown in Fig. \[fig3\].
This construction obviously holds if all the states are rotated by the same angle. In particular, we can always set this angle such that the probability distribution in Eq. yields $p(\theta'| \ket{0})= p(\theta'| \ket{-
\varepsilon})=0$ and $p(\theta'| \ket{1})= p(\theta'| \ket{\varepsilon})=\frac{1}{\pi}$. In this case the two ensembles give a different probability to observe the outcome $\theta'$, $p(\theta'| \mathcal{E}^{(1)}) =\frac{\sin^2(\varepsilon)}
{\pi}$ and $p(\theta'| \mathcal{E}^{(2)}) =\frac{1}{2 \pi}$. Hence, Bob can distinguish the two ensembles with non-zero probability and infer Alice’s choice of measurement instantaneously.
Let us now consider the general case. Any probability distribution $p(\theta)$ defines a continuous ensemble $\mathcal{E}^{(p)} =\{ p(\theta) \prjct{\Phi_\theta^N}\}$, where $\ket{\Phi_\theta^N}=\sum_{n=0}^N \psi_n e^{i \theta n}\ket{n}$ are the $N$-qubit states of Eq. . Without loss of generality we consider $p(\theta)$ such that \[FT\] p() =(1+ 2 \_[k=1]{}\^p\_k (k )). The density matrix for the ensemble $\mathcal{E}^{(p)}$ is given by $\rho = \sum_{n,m =0}^N \psi_n \psi_m^* {\mathinner{|{n}\rangle}\!\!\mathinner{\langle{m}|}} p_{|n-m|}$, in such a way that it only depends on the first $N$ coefficients, $p_k$, of the Fourier series in Eq. . For any two distributions $p_1(\theta)$ and $p_2(\theta)$ that are identical in the first $N$ components of the Fourier series the ensembles $\mathcal{E}^{(p_1)}$ and $\mathcal{E}^{(p_2)}$ give rise to the same density matrix $\rho$, and therefore can not be distinguished by Bob.
In particular, the ensemble given by the probability distributions $p_1(\theta)= \frac{1}{2\pi}$ and $p_2(\theta) = \frac{1}{2\pi}(1 + \cos(M \theta))$, for $M>N$, correspond to the same density matrix. However, with the outcome probability distribution of Eq. Bob can distinguish the two probability distributions with non-zero probability as $p(\theta'| \mathcal{E}^{(p_2)}) - p(\theta'| \mathcal{E}^{(p_1)}) = \frac{N+1}{2\pi }\int_{-\frac{\pi}{N+1}}^{\frac{\pi}{N+1}}(p_2(\theta)-p_1(\theta))d\theta = \mathrm{sinc}(\frac{M\pi}{N+1})\neq 0$. Therefore, the probability distribution of Eq. leads to signaling when Alice can chose to prepare $\mathcal{E}^{(p_1)}$ or $\mathcal{E}^{(p_2)}$.
More generally the above argument implies that any outcome probability $p(\theta'| \theta)$ compatible with no-signaling has to satisfy $\int p(\theta'|\theta)\cos(M\theta)d\theta =0$ for $M>N$, i.e., the Fourier components $p_k$ of $p(\theta'| \theta)$ are necessarily zero for $k>N$. Therefore, for finite $N$, probability distributions with sharp edges such as the one in Eq. are ruled out.
![The two ensemble decompositions of $\rho_\varepsilon$ for a qubit, leading to faster-than-light communication for the probability distribution Eq. (represented by the red semi-circle).[]{data-label="fig3"}](Fig3.pdf){width="2.5cm"}
A tighter no-signaling bound can be obtained if we optimize over all possible no-signaling scenarios, i.e., over all possible bi-partite entangled states $\ket{\Psi}_{AB}$. In fact any ensemble $\{p_k, \rho_k\}$ corresponding to a density matrix, $\rho_B = \sum_k p_k \rho_k$, at Bob’s side can be remotely prepared by Alice, if they initially share a suitable entangled state $\ket{\Psi}_{AB}$ (that only depends on $\rho_B$) and Alice does an appropriate measurement [@Gisin:89; @Hughston:93].
Quantum Mechanics at the edge of no-signaling
---------------------------------------------
The above argument shows that the probability distribution of Eq. is only valid for one possible no-signaling scenario, and that in order to obtain a tighter bound we should consider all possible states shared between Alice and Bob and all possible measurements at Alice’s side that steer Bob’s partial state into different ensembles of pure states that correspond to the same density matrix. Would such an optimization close the gap between our no-signaling bound and the optimal quantum strategy?
Following [@Simon:01], we now show that such an optimization is not even necessary, as the only processing compatible with no-signaling is given by the Born rule, i.e., the probability of some measurement outcome $\ell$ for the input state $\rho$ is given by $P_\ell = \tr \rho E_\ell$ for some positive operator $E_\ell$.
Indeed, any ensemble leading to the same density matrix for Bob can be remotely prepared by Alice [@Gisin:89; @Hughston:93]. This together with the no-signaling condition implies the linearity of Bob’s processing, $\mathcal{P}$. The latter states that for any two ensembles $\{p_k, \, \rho_k \}$ and $\{q_k, \, \sigma_k \}$, corresponding to the same $\rho_B$, the ensembles after the processing $\{p_k, \, \mathcal{P}(\rho_k) \}$ and $\{q_k, \, \mathcal{P}(\sigma_k) \}$ can not be distinguished with a non-zero probability. Adding the assumption that probabilities are attributed to quantum states via the Born rule (as it is done in [@Simon:01]) the condition above implies equality on the density matrices corresponding to the processed ensembles: \[linearity1\] \_k p\_k (\_k) = \_k q\_k (\_k) (\_B).
This shows that any dynamical evolution of quantum states that respects no-signaling is necessarily described by a completely positive (CP) map [@Simon:01]. The results of [@Simon:01] are concerned with situations where the outputs of Bob’s processing are quantum mechanical states. In this case [@Simon:01] implies that the optimal quantum mechanical strategies are at the edge of no-signaling.
However, in the case of quantum metrology the outputs are probability distributions. We remark than in [@Simon:01] the validity of the Born rule was assumed and used to derive the possibility for remote state preparation of any ensemble and to get the linearity constraint of Eq. from the indistinguishably of processed ensembles. In this case supra-quantum metrology is ruled out. However, if we make no assumptions on how probabilities are assigned to measurement outcomes of quantum states but only take remote state preparation as an experimental fact, the no-signaling constraint implies the Born rule already. In this case no-signaling again implies the indistinguishably of two ensembles $\{p_k, \, \rho_k \}$ and $\{q_k, \, \sigma_k \}$ corresponding to the same density matrix $\rho_B$ which in turn implies the linearity of the probability assignment rule. The probability $P_\ell$ to observe some outcome $\ell$ has to satisfy \[linearity2\] \_k p\_k P\_(\_k) = \_k q\_k P\_(\_k) P\_( \_B), i.e., outcome probabilities *only depend on the density matrix* but not on a particular ensemble. Note that one can easily construct probability assignment rules for pure states that do not satisfy Eq. (see the example from the previous section) so it is not something one has to impose a priori. However, as we just saw assuming no-signaling together with the practical possibility for steering enforces linearity.
It is well known that the only probability assignment rule compatible with linearity is the Born rule—$P_\ell(\rho_B)=\tr \rho_B E_\ell$ for some positive operator $E_\ell$ [@Holevo:82]. For systems of dimension $d>2$ this can also be seen as a consequence of Gleason’s theorem (it suffices to consider all ensembles of pure states forming an orthonormal basis). Moreover, similar result for the equivalence of CP dynamics and the Born rule being enforced by linearity are known to hold in a more general context of probabilistic theories with purification [@Ch:10]. In summary, we have shown that also in the case of quantum metrology the optimal quantum mechanical strategies are at the edge of no-signaling.
Probabilistic vs deterministic bounds
-------------------------------------
Notice that all the no-signaling bounds derived here are concerned with probabilistic strategies. This is most transparent in our derivation of a no-signaling bound for unitary replication. In some cases, the optimal deterministic strategy coincides with the optimal probabilistic one as is the case with replication of unitaries. In other cases the optimal probabilistic strategy can be made deterministic if one drops some restrictions on the input alphabet, as is the case for the cloning of states when one allows for entangled input states $U_\theta^{\otimes N} \ket{\Phi^N}$ instead of separable input states $\ket{\psi(\theta)}^{\otimes N}$. How does one decide if a deterministic bound still holds when one allows for probabilistic tasks? If this is not the case can one achieve the probabilistic performance deterministically by allowing for more general input states?
It is known that, in general, any probabilistic strategy can be decomposed into a filter, $F$, acting on the input state and mapping it to an output state in the same Hilbert space, followed by a deterministic transformation [@Ch:13]. Moreover, one is usually interested in probabilistic strategies where all states from the input alphabet $\{ \ket{\Phi_i}\}$ have the same probability of success $p_s = \tr F \prjct{\Phi_i} F^\dag$. In this case what the probabilistic advantage has to offer is the possibility to modify the alphabet to any other alphabet reachable by a filter $\{\ket{\Phi_i^F}=\frac{1}{\sqrt{p_s}} F \ket{\Phi^F_i} \}$ [^1]. So the question about the strength of the deterministic bound is actually whether the input alphabet $\{ \ket{\Phi_i}\}$ is the best among alphabets $\{\ket{\Phi_i^F}\}$ for the particular task.
If this is not the case, then the probabilistic strategy can always be made deterministic by starting with the optimal alphabet. As we saw in Sec. \[sec:PCC\] for the case of PCC the $N$-copy input states are not optimal leaving room for probabilistic improvement, whereas in the case of universal cloning no probabilistic advantage exists as the symmetry group of the input alphabet, $SU(2)$, forces the filter to be the identity. In fact, there is a substantial improvement in cloning fidelity if one allows entangled $N$-qubit states as inputs into the cloning machine, but such states are not reachable by any filter [@Ch:13]. The entangled states that yield such a substantial improvement are exactly those states that maximize the average fidelity of alignment for a Cartesian frame of reference [@Ch:04a].
Conclusion
==========
In this paper we derived no-signaling bounds for various quantum information processing tasks. These include phase covariant cloning of states and unitary operations, as well as quantum metrology. In the latter case we showed the validity of the Heisenberg limit purely from the no-signaling condition. In general, following [@Simon:01], we have shown that the optimal probabilistic quantum mechanical strategy is at the edge of no-signaling also for the case of metrology. Furthermore, we have found that for some tasks, such as PCC of states and unitaries, the optimal probabilistic and deterministic strategies coincide. These results show a direct connection between the no-signalling principle and the ultimate limits on quantum cloning and metrology. This connection provides a new insight into the [*physical*]{} origin of these limits, in contrast to the previously known limits based on optimization, using e.g. semidefinte programs.
On the one hand it is clear that a bound for probabilistic strategies is also a bound for deterministic ones. However, it might be possible to derive tighter no-signaling bounds for deterministic strategies. It is an interesting open question how to incorporate the requirement that the protocol be deterministic in a no-signaling scenario.
On the other hand, there are several tasks for which the optimal quantum strategy is not known. In such cases the techniques and methods we provide here can be particularly useful in deriving limitations to these tasks based on no-signaling. We have demonstrated one such example for Bayesian metrology for arbitrary prior, however the methods we introduce are applicable in a broader context. This provides an alternative approach to study the possibilities and limitations of quantum information processing.
Acknowledgements
================
We thank Giulio Chiribella, Nicolas Gisin, and Christoph Simon for valuable remarks and comments. This work was supported by the Austrian Science Fund (FWF): P24273-N16 and the Swiss National Science Foundation grant P2GEP2\_151964.
[^1]: If the input alphabet is generated by the action of a unitary on a fixed seed state $\ket{\Psi_i} = U_i \ket{\Psi}$, then the filter commutes with the unitary and modifies the seed state to $\ket{\Psi^F} =\frac{1}{\sqrt{p_s}} F \ket{\Psi}$
|
---
abstract: 'We study the behaviour of a superconductor in a weak static gravitational field for temperatures slightly greater than its transition temperature (fluctuation regime). Making use of the time-dependent Ginzburg–Landau equations, we find a possible short time alteration of the static gravitational field in the vicinity of the superconductor, providing also a qualitative behaviour in the weak field condition. Finally, we compare the behaviour of various superconducting materials, investigating which parameters could enhance the gravitational field alteration.'
author:
- '<span style="font-variant:small-caps;">Giovanni Alberto Ummarino</span>'
- '<span style="font-variant:small-caps;">Antonio Gallerati</span>'
bibliography:
- 'bibliografia2019ott.bib'
title: ' **Exploiting weak field gravity-Maxwell symmetry in superconductive fluctuations regime** '
---
Introduction {#sec:Intro}
============
It is since 1966, with the paper of DeWitt [@DeWitt:1966yi], that there is great interest in the interplay between the theory of gravitation and superconductivity [@Kiefer:2004hv]. In the following years were produced a lot of theoretical papers about this topic [@papini1967detection; @Papini:1970cw; @rothen1968application; @rystephanick1973london; @hirakawa1975superconductors; @minasyan1976londons; @anandan1977gravitational; @anandan1979intgra; @anandan1984relthe; @anandan1994relgra; @ross1983london; @Felch:1985pre; @dinariev1987relativistic; @peng1991electrodynamics; @peng1990new; @peng1991interaction; @li1991effects; @li1992gravitational; @torr1993gravitoelectric; @de1992torsion], until Podkletnov and Nieminem declared to have observed a gravitational shielding in a disk of YBaCuO (YBCO) [@podkletnov1992possibility], an high-$\Tc$ superconductor (HTCS). Of course, after the publication of this paper, other groups tried to repeat the experiment obtaining controversial results [@li1997static; @de1995alternative; @unnikrishnan1996does; @tajmar2009measuring; @tajmar2011evaluation; @podkletnov2003investigation; @poher2017enhanced], so that the question is still open.
Many researchers tried to give a theoretical explanation [@ciubotariu1996absence; @agop1996gravitational; @agop2000some; @agop2000local; @ivanov1997gravitational; @ahmedov1999general; @ahmedov2005electromagnetic; @Tajmar:2002gm; @deMatos:2006tn; @Tajmar:2004ww; @tajmar2008electrodynamics; @ning2004gravitational; @chiao2006interface; @de2007gravitoelectromagnetism; @de2008electromagnetic; @de2008gravitational; @de2010physical; @de2012modified; @inan2017interaction; @inan2017new; @atanasov2017geometric; @Sbitnev2019] of the experimental results of Podkletnov and Nieminem in subsequent years, although, in our opinion, the clearest work was made by Modanese in 1996 [@modanese1996theoretical; @modanese1996role], interpreting the experimental results in the frame of a quantum field formulation. However, the complexity of the formalism makes it difficult to extract quantitative predictions.
In a previous work [@Ummarino:2017bvz], we determined the possible alteration of a static gravitational field in a superconductor making use of the time-dependent Ginzburg–Landau equations [@Cyrot_1973; @zagrodzinski2003time; @alstrom2011magnetic], providing also an analytic solution in the weak field condition [@PhysRevD.39.2825; @ruggiero2002gravitomagnetic]. Now, we develop quantitative calculations in a range of temperatures very close but higher than the critical temperature, in the regime of fluctuations [@larkin2002fluctuation].
Weak field approximation {#sec: Weak field}
========================
Let us consider a nearly flat spacetime configuration (weak gravitational field) where the metric $\gmetr$ can be expanded as: $$\gmetr~\simeq~\emetr+\hmetr\;,
\label{eq:gmetr}$$ with $\hmetr$ small perturbation of the flat Minkowski metric[^1]. The inverse metric in the linear approximation is given by $$\invgmetr~\simeq~\invemetr-\invhmetr\;,$$ and the Christoffel symbols, to linear order in $\hmetr$ are written as $$\label{eq:Gam}
\Gam[\lambda][\mu][\nu]~\simeq~\frac12\,\invemetr[\lambda][\rho]\,
\left(\dm[\mu]\hmetr[\nu][\rho]+\dm[\nu]\hmetr[\rho][\mu]-\dm[\rho]\hmetr[\mu][\nu]\right)\;.$$ The Riemann tensor is defined as $$\Ruddd\= 2\,\dm[{[}\lambda]\Gam[\sigma][\nu{]}][\mu]
\+2\,\Gam[\sigma][\rho][{[}\lambda]\,\Gam[\rho][\nu{]}][\mu]\;,$$ while the Ricci tensor is given by the contraction $\Ricci=\Ruddd[\sigma][\mu][\sigma][\nu]$. To linear order in $\hmetr$, the latter reads [@Ummarino:2017bvz] $$\Ricci~\simeq~\dmup[\rho]\dm[{(}\mu]\hmetr[\nu{)}][\rho]-\frac12\,\dd^2\hmetr-\frac12\,\dm\dm[\nu]h\;,
\label{eq:Ricci}$$ with $h=\hud[\sigma][\sigma]$. The Einstein equations [@Wald:1984rg; @misner1973gravitation] are written as $$\GEinst\=\Ricci-\dfrac12\,\gmetr\,R
\=8\pi\GN\;T_{\mu\nu}\;,$$ and the l.h.s. in first-order approximation reads $$\begin{split}
\GEinst&~\simeq~\dmup[\rho]\dm[{(}\mu]\hmetr[\nu{)}][\rho]-\frac12\,\dd^2\hmetr-\frac12\,\dm\dm[\nu]h
-\frac12\,\emetr\left(\dmup[\rho]\dmup[\sigma]\hmetr[\rho][\sigma]-\dd^2h\right)\;.
\end{split}$$ Introducing the symmetric tensor $$\bhmetr\=\hmetr-\frac12\,\emetr\,h\;,$$ the above expression simplifies in [@Ummarino:2017bvz] $$\GEinst~\simeq~
\dmup[\rho]\dm[{(}\mu]\bhmetr[\nu{)}][\rho]-\frac12\,\dd^2\bhmetr
-\frac12\,\emetr\,\dmup[\rho]\dmup[\sigma]\bhmetr[\rho][\sigma]
\=\dmup[\rho]\left(
\dm[{[}\nu]\bhmetr[\rho{]}][\mu]+\dmup[\sigma]\emetr[\mu][{[}\rho]\,\bhmetr[\nu{]}][\sigma]
\right)\;.$$ If we now define the tensor $$\label{eq:Gscr}
\Gscr~\equiv~
\dm[{[}\nu]\bhmetr[\rho{]}][\mu]+\dmup[\sigma]\emetr[\mu][{[}\rho]\,\bhmetr[\nu{]}][\sigma]\;,$$ the Einstein equations can be rewritten in the compact form: $$\label{eq:Einst}
\;\GEinst\=\dmup[\rho]\Gscr\=8\pi\GN\;T_{\mu\nu}\;.$$ We can impose a gauge fixing using the *harmonic coordinate condition* [@Wald:1984rg]: $$\Box x^\mu=0
\;\qLrq\;
\dm\left(\sqrt{-g}\,\invgmetr\right)=0
\;\qLrq\;
\invgmetr\,\Gam\,=\,0\;,
\label{eq:gaugefix}$$ also called *De Donder gauge*[^2]. If we now use eqs. and in the last of previous , we find, in first-order approximation $$0~\simeq~ \dm\invhmetr-\frac12\,\dmup[\nu]h\;,$$ that is, we have the relations $$\label{eq:gaugecond0}
\dm\invhmetr\simeq\frac12\,\dmup[\nu]h
\;\quad\Leftrightarrow\;\quad
\dmup\hmetr\simeq\frac12\,\dm[\nu]h\;,$$ that, in turns, imply the *Lorenz gauge condition*: $$\dmup\bhmetr~\simeq~0\;.$$ The above result simplifies expression for $\Gscr$, which takes the form $$\Gscr~\simeq~\dm[{[}\nu]\bhmetr[\rho{]}][\mu]\;.
\label{eq:Gscr0}$$
Gravito-Maxwell equations
-------------------------
Now, let us define the fields [@Ummarino:2017bvz]
[6]{}[12]{}\[eq:fields0\] $$\begin{aligned}
\Eg&~\equiv~E_i~=\,-\,\frac12\,\Gscr[0][0][i]~=\,-\,\frac12\,\dm[{[}0]\bhmetr[i{]}][0]\;,\\[\jot]
\Ag&~\equiv~A_i\=\frac14\,\bhmetr[0][i]\;,\\[\jot]
\Bg&~\equiv~B_i \=\frac14\,{\varepsilon_i}^{jk}\,\Gscr[0][j][k]\;,\end{aligned}$$
where, using , we have $$\Gscr[0][i][j]\=\dm[{[}i]\bhmetr[j{]}][0]
\=\frac12\left(\dm[i]\bhmetr[j][0]-\dm[j]\bhmetr[i][0]\right)\=4\,\dm[{[}i]A_{j{]}}\;.$$ First, we find $$\Bg\=\frac14\,{\varepsilon_i}^{jk}\,4\,\dm[{[}j]A_{k{]}}
\={\varepsilon_i}^{jk}\,\dm[j]A_k=\nabla\times\Ag
\;\qLq\;
\nabla\cdot\Bg\=0\;.$$ Then, one also has $$\nabla\cdot\Eg\=\dmup[i]E_i~=\,-\dmup[i]\frac{\Gscr[0][0][i]}{2}
~=\,-8\pi\GN\;\frac{T_{00}}{2}
\=4\pi\GN\;\rhog\;,
$$ using eq. and having defined $\rhog\equiv-T_{00}$. If we take the curl of $\Eg$, we obtain $$\nabla\times\Eg\={\varepsilon_i}^{jk}\,\dm[j]E_k
~=\,-{\varepsilon_i}^{jk}\,\dm[j]\frac{\Gscr[0][0][k]}{2}
~=\,-\frac14\,4\;\dm[0]\,{\varepsilon_i}^{jk}\,\dm[j]A_k
~=\,-\dm[0]B_i~=\,-\frac{\dd\Bg}{\dd t}\;,$$ while, for the curl of $\Bg$, $$\label{eq:gravMaxwell4}
\begin{split}
\nabla\times\Bg&\={\varepsilon_i}^{jk}\,\dm[j]B_k
\=\frac14\,{\varepsilon_i}^{jk}\,
{\varepsilon_k}^{\ell m}\,\dm[j]\Gscr[0][\ell][m]
\=\frac12\left(\dmup\Gscr[0][i][\mu]-\dm[0]\Gscr[0][0][i]\right)\=
\\[2\jot]
&\=\frac12\left(8\pi\GN\;T_{0i}-\dm[0]\Gscr[0][0][i]\right)
\=4\pi\GN\;j_i+\frac{\dd E_i}{\dd t}
\=4\pi\GN\;\jg\+\frac{\dd\Eg}{\dd t}\;,
\end{split}$$ using again eq. and having defined $\jg \equiv j_i \equiv T_{0i}$.
Following the above prescriptions, we obtained for the fields the set of equations: $$\label{eq:gravMaxwell}
\begin{split}
\nabla\cdot\Eg&\=4\pi\GN\,\frac{m^2}{e^2}\;\rhog\= \frac{\rhog}{\epsg}\;;\\[2\jot]
\nabla\cdot\Bg&\=0 \;;\\[2\jot]
\nabla\times\Eg&~=-\dfrac{\dd\Bg}{\dd t} \;;\\[2\jot]
\nabla\times\Bg&\=4\pi\GN\,\frac{m^2}{c^2\,e^2}\,\jg
\+\frac{1}{c^2}\,\frac{\dd\Eg}{\dd t}
\= \mug\;\jg\+\frac{1}{c^2}\,\frac{\dd\Eg}{\dd t}\;,\qquad
\end{split}$$ having restored physical units [@Ummarino:2017bvz]. This equations are formally equivalent to Maxwell equations, with $\Eg$ and $\Bg$ gravitoelectric and gravitomagnetic field respectively, having defined the vacuum gravitational permittivity and the vacuum gravitational permeability as: $$\epsg=\frac{1}{4\pi\GN}\,\frac{e^2}{m^2}\;,\qqquad
\mug=4\pi\GN\,\frac{m^2}{c^2\,e^2}\;.$$ For example, on the Earth surface, $\Eg$ is simply the Newtonian gravitational acceleration and the $\Bg$ field is related to angular momentum interactions [@agop2000local; @agop2000some; @braginsky1977laboratory; @huei1983calculation; @peng1990new].
Generalized Maxwell equations
-----------------------------
Now we introduce the generalized electric/magnetic field, scalar and vector potentials, containing both electromagnetic and gravitational terms: $$\E=\Ee+\frac{m}{e}\,\Eg\,;\quad\;
\B=\Be+\frac{m}{e}\,\Bg\,;\quad\;
\phi=\phi_\textrm{e}+\frac{m}{e}\,\phig\,;\quad\;
\A=\Ae+\frac{m}{e}\,\Ag\,,
\label{eq:genfields}$$ where $m$ and $e$ are the mass and electronic charge, respectively, the subscripts identifying the electromagnetic and gravitational contributions.
The generalized Maxwell equations for the fields then become [@Ummarino:2017bvz; @Behera:2017voq]: $$\label{eq:genMaxwell}
\begin{split}
\nabla\cdot\E&\=\left(\frac1\epsg+\frac{1}{\epsz}\right)\,\rho \;;\\[2\jot]
\nabla\cdot\B&\=0 \;;\\[2\jot]
\nabla\times\E&~=-\dfrac{\dd\B}{\dd t} \;;\\[2\jot]
\nabla\times\B&\=\left(\mug+\muz\right)\,\jj
\+\frac{1}{c^2}\,\dfrac{\dd\E}{\dd t} \;,
\end{split}$$ with $$\rhog\=\frac{e}{m}\,\rho\;,\qqquad
\jg\=\frac{e}{m}\;\jj\;,$$ where $\epsz$ and $\muz$ are the electric permittivity and magnetic permeability in the vacuum, and $\rho$ and $\jj$ are the electric charge density and electric current density respectively.
We have shown how to define a new set generalized Maxwell equations for generalized electric $\E$ and magnetic $\B$ fields, in the limit of weak gravitational fields. In the following sections we will use this results to study the behaviour of a superconductor in the fluctuation regime, i.e. very close to its critical temperature $\Tc$.
The model
=========
The behaviour of a superconductor in the vicinity of its critical temperature has been extensively studied. This particular region of temperature is characterized by thermodynamic fluctuations of the order parameter, giving rise to a gradual increase of the resistivity of the material from zero to its normal state value, for temperatures $T>\Tc$. This happen because, above the critical temperature $\Tc$, the order parameter fluctuations create superfluid regions in which electrons are accelerated. For temperatures larger than $\Tc$, the average size of these regions is much greater than the mean free path, though it decreases with the rise in temperature of the sample.
The described regime can be studied by using the time-dependent Ginzburg-Landau equations [@Cyrot_1973]. Of course, we have to be sufficiently far from the critical point for this description to be valid (essentially we are dealing with a mean field theory). Moreover, here we suppose we deal with sufficiently dirty materials, in order to observe the effects of the fluctuations over a sizable range of temperature, i.e. the electronic mean free path $\ell$ in the normal material has to be less than 10 .
The time-dependent Ginzburg-Landau equations can be written, for temperatures larger than $\Tc$, with just the linear term, in the gauge-invariant form [@hurault1969nonlinear; @schmid1969diamagnetic]: $$\Gamma\left(\hbar\,\dt[]-2\,i\,e\,\Phi\right)\psi\=
\frac{1}{2m}\left(\hbar\,\dt[]-2\,i\,e\,\A\right)\psi
+\alpha\,\psi\;,$$ where $\psi(\x,t)$ is the order parameter, $\A(\x,t)$ is the potential vector and $\Phi(\x,t)$ is the electric potential. Moreover, once defined $\epsilon(T)=\sqrt{\frac{T-\Tc}{\Tc}}$, we also have $$\alpha=-\frac{\hbar^{2}}{2\,m\,\xi^{2}}\;,\qquad\;
\xi=\xi(T)=\frac{\xi_0}{\sqrt{\epsilon(T)}}\;,\qquad\;
\Gamma=\frac{\abs{\alpha}}{\epsilon(T)}\,\frac{\pi}{8\,\kB\,\Tc}\;,$$ where $\xi_0=\xi(0)$ is the BCS coherence lenght. If we put $$\psi(\x,t)\=f(\x,t)\,\exp\big(i\,g(\x,t)\big)\;,$$ we obtain two equations for the functions $f(\x,t)$ and $g(\x,t)$:
$$\begin{aligned}
\Gamma\,\hbar\,\dt[f]&\=
\alpha\,f+\frac{\hbar^{2}}{2m}\,\Delta f-\frac{1}{2}m\,\vs^{2}\,f\;,
\label{eq:dtf}
\\[2.5\jot]
\Gamma\,\hbar\,f\,\dt[g]&\=
2\,e\,\Gamma\,\Phi\,f-\frac{\hbar^{2}}{2m}\,f\,
\Delta g-2\,\hbar\;\vvs\cdot\nabla f\;,
\label{eq:dtg}\end{aligned}$$
where $$\vvs=\frac{1}{m}\left(\hbar\,\nabla g+2\,\frac{e}{c}\,\A\right)
\label{eq:vs}$$ is the superfluid speed and where the associated current density is $$\js~=\,-2\,\frac{e}{m}\,|\psi|^{2}\left(h\,\nabla g+2\,\frac{e}{c}\,\A\right)
~=\,-2\,e\,f^{2}\,\vvs\;.$$ Now, we consider a fluctuation of the wave vector for the function $f$. Let $f_k$ be the value of $f$ for a fluctuation of the wave vector $\kk$. The above equations can be recast in a more convenient form:
$$\begin{aligned}
\Gamma\,\hbar\,\dt[f_{k}]&\=
\alpha\,f_{k}-\frac{\hbar^{2}}{2m}\,k^{2}\,f_{k}-\frac{1}{2}\,m\,\vs^2\;,
\label{eq:dtfk}
\\[1.5\jot]
\dt[\vvs]&~=\,-2\,\frac{e}{m}\,\E\;
\label{eq:dtvs}\end{aligned}$$
where the last expression is obtained by using eq. and $\nabla\Phi=-\E-\frac{1}{c}\dt[\A]$ and taking the gradient of eq. . By integrating from zero to $t$, we obtain $$\Gamma\,\hbar\,\dt[f_{k}]\=
\left(\alpha-\frac{\hbar^{2}}{2m}\,k^{2}-2\,\frac{e^2}{m}\,E^{2}\,t^{2}\right)f_{k}\;,$$ so that $f_{k}$ is given by $$f_{k}(t)\=f_{k}(0)\,\exp\left(\frac{\left(\alpha-\frac{\hbar^2}{2m}k^{2}\right)t-\frac{2}{3}\,\frac{e^2}{m}\,E^{2}\,t^3}{\Gamma\,\hbar}\right)\;,$$ with $f_{k}^{2}(0)=\frac{\kB\,T}{2\left(|\alpha|+\frac{\hbar^2}{2m}k^{2}\right)}$ as it was calculated in [@de2018superconductivity]. Then, the current $\jsk(t)$ can be written as $$\jsk(t)\=4\,\frac{e^{2}}{m}\,\E\;t\;f_{k}^{2}(0)\,\exp\left(2\,\frac{\left(\alpha-\frac{\hbar^2}{2m}k^{2}\right)t-\frac{2}{3}\,\frac{e^2}{m}\,E^{2}\,t^3}{\Gamma\,\hbar}\right)\;,$$ At this point we sum over $\kk$. The simpler situation is a three-dimensional sample whose dimensions are greater than the correlation length $\xi$, so that we obtain $$\langle\,\js(t)\rangle\=
\frac{1}{8\pi^{3}}\,\int_{0}^{+\infty}
\jsk(k,t)\,4\pi\,k^{2}\,dk\;. $$ The potential vector $\A(x,y,z,t)$ can be calculated from: $$\A(x,y,z,t)=\frac{1}{4\pi}\int\frac{\js(t)\;dx'\,dy'\,dz'}{\sqrt{(x-x')^{2}+(y-y')^{2}+(z-z')^{2}}}\;,$$ when the time variations of external fields are small. The generalized electric field $\E(x,y,z,t)$ of eq. , in the case under consideration, can be written as $$\E(x,y,z,t)~=\,-\frac{1}{c}\dt[\A(x,y,z,t)]+\frac{m}{e}\,\g~=\,-\frac{1}{c}\dt[\js(t)]\;\mathcal{C}(x,y,z)+\frac{m}{e}\,\g\;,$$ where we have considered the static weak (Earth-surface) gravitational field contribution $\g$, and where $\mathcal{C}(x,y,z)$ is a geometrical factor that depends on the shape of the superconductor and on the space point where we calculate the gravitational fluctuations caused by the presence of the superconductor itself. Of course, when $\E=\frac{m}{e}\,\g$ we are in the weak field regime and we can neglect the term proportional to $t^{3}$ in the exponential. In the latter case, for the realisation of an experiment, one needs a weak magnetic field (we are around $\Tc$) in order to have the superconductor in the normal state, and turn off the magnetic field at the time $t=0$.
Results
=======
In Figures \[fig:APdisk\] and \[fig:YBdisk\] we show the variation of the gravitational field as a function of time, measured on the axis of a superconductive disk with bases parallel to the ground, at a fixed distance $d$ from the base surface, respectively for low-$\Tc$ (Al and Pb) and high-$\Tc$ superconductors (YBCO and BSCCO). The effect is calculated in the range of temperature where superconductive fluctuations are present. The system is initially at a temperature very close to $\Tc$, and we put it in the normal state by using a weak static magnetic field (near $\Tc$ the upper critical field tends to zero). At the time $t=0$, we remove the magnetic field so that the system goes in the superconductive state.
It is interesting to note that, in a very short initial time interval, the gravitational field is reduced w.r.t. its unperturbed value. After that, it increases up to a maximum value at the time $t=\tau_{0}$ and then decreases to the standard external value. In our previous paper, in the regime under $\Tc$, we found a weak shielding of the external gravitational field [@Ummarino:2017bvz], with the corresponding solution for a simple case. The value $\Delta$ is the maximum variation of the external gravitational field: in principle, field variation is measurable (especially in high-$\Tc$ superconductors), while the problem lies in the very short time intervals in which the effect manifests itself.
In Fig. \[fig:YBdeltag\] it is shown the field variation effect as a function of distance from the disk surface, measured along the axis of the disk at the fixed time $t=\tau_0$ that maximizes the effect. In Table \[tab:param\] we summarize the experimental data for the superconductive materials under consideration.
It is instructive to study the values of the parameters that maximize the effect in intensity and time interval. After simple but long calculations, it is possible to demonstrate that $\tau_{0}\propto (T-\Tc)^{-1}$, so it is fundamental to be very close to the critical temperature in order to increase the time range in which the effect takes place. The maximum value of the correction for the external field is obtained for $t=\tau_{0}$ and is proportional to $\xi^{-1}(T)$: this means that the effect is larger in high-$\Tc$ superconductors, having the latter small coherence length. Clearly this behaviour makes the experimental detection difficult, since if we are close to $\Tc$ we find an increase for the value of $\tau_{0}$ together with a decrease for the alteration of gravitational field, owing to the coherence length divergence at $T=\Tc$.
[@ T[p]{}[0.125]{}| M[p]{}[0.11]{} M[p]{}[0.095]{} M[p]{}[0.095]{} M[p]{}[0.11]{} M[p]{}[0.13]{} M[p]{}[0.13]{} ]{} & () & T() & \_[0]{}() & (T)() & \_[0]{}() & (/\^[2]{})\
Al & 1.175 & 1.176 & 15500 & 531313 & 7.4510\^[-10]{} & 5.3710\^[-10]{}\
Pb & 7.220 & 7.221 & 870 & 73924 & 7.4510\^[-10]{} & 2.3710\^[-8]{}\
YBCO & 89.0 & 89.1 & 30 & 895 & 7.5010\^[-12]{} & 2.4110\^[-5]{}\
BSCCO & 111.0 & 111.1 & 10 & 333 & 7.5010\^[-12]{} & 8.0810\^[-5]{}\
![The variation of gravitational field as a function of time in the vicinity of a superconductive sample of Al (green solid line) and one of Pb (orange dot-dashed line). The field is measured along the axis of the disk, with bases parallel to the ground, at a fixed distance $d=0.5\,\cm$ above the disk surface. The radius of the disk is $R=10\,\cm$ and the thickness is $h=1\,\cm$.[]{data-label="fig:APdisk"}](leg_AL-PBdisk.pdf){width="\textwidth"}
![The variation of gravitational field as a function of time in the vicinity of a superconductive disk of YBCO (blue solid line) and BSCCO (purple dot-dashed line). The field is measured along the axis of the disk, with bases parallel to the ground, at a fixed distance $d=0.5\,\cm$ above the disk surface. The radius of the disk is $R=10\,\cm$ and the thickness is $h=1\,\cm$.[]{data-label="fig:YBdisk"}](leg_YBCO-BSCCOdisk.pdf){width="\textwidth"}
![The variation of gravitational field as a function of distance in the vicinity of a superconductive sample of YBCO (grey solid line) and one of BSCCO (light blue dot-dashed line). The field is measured along the axis of the disk, with bases parallel to the ground, at the fixed time $t=\tau_0=7.50\cdot10^{-12}\,\s$ that maximizes the variation. The radius of the disk is $R=10\,\cm$ and the thickness is $h=1\,\cm$.[]{data-label="fig:YBdeltag"}](leg_YBCO-BSCCOdeltag.pdf){width="\textwidth"}
Conclusions
===========
We have calculated the possible alteration of a static gravitational field in the vicinity of a superconductor in the regime of fluctuations. We have also shown that the effect should be weak (though perceptible), but it occurs in very short time intervals, making direct measurements difficult to obtain. Probably some ingredient for a complete depiction of the gravity-superfluid interaction has to be included, as long as it exists, for a more detailed characterization of the phenomenon.
Clearly, the goal is to obtain non-negligible experimental evidences (gravitational field perturbations) in workable time scales, trying to optimize contrasting effects by choosing appropriate temperature and sample coherence length. At present, the best option is to choose a HTCS (since very short coherence length increases the intensity of perturbation) and put the system at a temperature very close to $\Tc$ (increase of time range where the effect occurs). It is also possible that the simultaneous presence of an electromagnetic field with particular characteristics, together with a suitable setting for the geometry of the experiment, could increase the magnitude of the effects under consideration.
Acknoledgments {#acknoledgments .unnumbered}
==============
This work was supported by the MEPhI Academic Excellence Project (contract No. 02.a03.21.0005) for the contribution of prof. G. A. Ummarino. We also thank Fondazione CRT that partially supported this work for dott. A. Gallerati.
[^1]: we work in the mostly plus convention, [$\emetr=\mathrm{diag}(-1,+1,+1,+1)$]{}
[^2]: the requirement of a coordinate condition plays the role of a gauge fixing, uniquely determining the physical configuration and removing indeterminacy; in harmonic coordinates, the metric satisfies a manifestly Lorenz-covariant condition, so that the De Donder gauge becomes a natural choice. Moreover, if one considers the weak-field expansion of the EH action in De Donder gauge, the action itself (as well as the graviton propagator) takes a particularly simple form.
|
**Quantum Hall Fluid of Vortices in a Two Dimensional**
**Array of Josephson Junctions**
\
\
Ady Stern
Physics Department, Harvard University, Cambridge, Ma. 02138
\
\
**Abstract**
A two dimensional array of Josephson junctions in a magnetic field is considered. It is shown that the dynamics of the vortices in the array resembles that of electrons on a two–dimensional lattice put in a magnetic field perpendicular to the lattice. Under appropriate conditions, this resemblance results in the formation of a quantum Hall fluid of vortices. The bosonic nature of vortices and their long range logarithmic interaction make some of the properties of the vortices’ quantum Hall fluid different from those of the electronic one. Some of these differences are studied in detail. Finally, it is shown that a quantum Hall fluid of vortices manifests itself in a quantized Hall electronic transport in the array.
\
\
[**1. Introduction**]{}
This paper discusses a quantized Hall effect (QHE) state of vortices in a two dimensional (2D) array of Josephson junctions. Motivated by the analogy between Magnus force acting on a vortex moving in a two–dimensional ideal fluid and Lorenz force acting on a charge in a magnetic field, we study the transport of vortices in a Josephson junction array. In particular, we focus on the case in which the charging energy of the array is minimized when the number of Cooper pairs on each element of the array is not an integer. We find that for a certain range of parameters the vortices are expected to form a quantum Hall fluid, and the resistivity of the array is expected to show a QHE behavior. While some of the properties of the quantum Hall fluid formed by the vortices are similar to those of the well known Laughlin fluid, formed by electrons in QHE systems, we find that the logarithmic interaction between the vortices leads to interesting modifications of other properties.
The paper is organized in the following way: in Section (2) we review the classical and quantum mechanical analogies between the 2D dynamics of charged particles under the effect of a magnetic field and that of vortices in a 2D fluid. These analogies, arising from the analogy between Magnus and Lorenz forces, motivate the introduction of the system we analyze – a Josephson junction array in a magnetic field, and the study of a quantized Hall effect in that system. Section (2) is concluded with a precise formulation of the problem to be studied. In Section (3) we analyze the transport of vortices in this array. We show that the dynamics of the vortices can be mapped on that of charged particles under the effect of a magnetic field, a lattice–induced periodic potential and a mutual interaction. The mutual interaction is composed of a logarithmic “static” part as well as a velocity–dependent short ranged part. In Section (4) we analyze the quantized Hall effect associated with the transport of the vortices, and its observable consequences. In particular, we study the unique features of the QHE for logarithmically interacting particles. Conclusions are presented in Section (5).
The possibility of Quantum Hall phenomena in Josephson junction arrays was recently discussed in two other works, one of Odintsov and Nazarov [@Odintsov], and the other of Choi[@Choi]. The regime of parameters we consider is different from the ones considered by these authors. We comment briefly on this difference and its implications in section (2).
[**2. Transport of vortices in a Josephson junction array – introduction and motivation**]{}
The classical dynamics of vortices in two–dimensional ideal fluids is well known to resemble that of charged particles under the effect of a strong magnetic field [@Lamb]. An electron in a magnetic field is subject to Lorenz force, while a vortex in an ideal fluid is subject to Magnus force. Both forces are proportional and perpendicular to the velocity. Two electrons in a strong magnetic field encircle each other, and so do two vortices in an ideal fluid. The dynamics of both are well approximated by an Hamiltonian that includes only a potential energy $V(x,y)$, where $x,y$ are the planar coordinates, and for which $x$ and $y$ are canonically conjugate. This approximation is known as the guiding center approximation for the electronic problem, and as Eulerian dynamics for the vortex problem. This close resemblance naturally raises the possibility of analogies between transport phenomena of electrons in a magnetic field and those of vortices in ideal fluids.
A vortex in a fluid can be viewed as an excitation in which each fluid particle is given an angular momentum $l$ relative to the vortex center [@Feynman]. Consequently, the velocity field ${\vec v({\vec r})}$ of the fluid satisfies $\int_\Gamma {\vec v}\cdot d{\vec l}={{2\pi l}\over m}$, where $m$ is the mass of a fluid particle, and $\Gamma$ is a curve that encloses the vortex center. When the vortex center moves with a velocity $\vec u$, and the fluid is at rest far away from the vortex center, the vortex center is subject to a Magnus force, given by $F_{Magnus}=2\pi l n {\vec u}\times{\hat z}$, where $n$ is the number density of the fluid far away from the vortex core. Being both proportional and perpendicular to the velocity of the vortex center, Magnus force obviously resembles the Lorenz force acting on an electron moving in the $x-y$ plane under the effect of a magnetic field $B\hat z$. This Lorenz force is given by $F_{Lorenz}={{eB}\over c}{\vec u}\times {\hat z}$, where $\vec u$ is the velocity of the electron. Thus, the role played by the product ${e\over c}B$ in the latter case is played by the product $2\pi l n$ in the former. While the fluid density plays a role analogous to that of a magnetic field, a fluid current plays a role analogous to that of an electric field. To see that, note that in a frame of reference in which the electron is at rest, the force it is subject to looks as if it arises from an electric field, given by ${B\over c}{\vec u}\times{\hat z}$. Similarly, in a frame of reference in which the vortex center is at rest, the force it is subject to seems to arise from the motion of the fluid. Since the fluid current density is $\vec J=n\vec u$, in the vortex rest frame the Magnus force is $F_{Magnus}=2\pi l
{\vec J}\times{\hat z}$, and ${\vec J}\times{\hat z}$ plays a role analogous to that of an electric field. Thus, while the fluid density affects the vortex dynamics in the same way a magnetic field affects electronic dynamics, the fluid current plays the role of an electric field. Maxwell’s equation ${\vec\nabla}\times{\vec E}
+{{\partial B}\over{\partial t}}=0$ is then analogous to the continuity equation in the fluid ${\vec\nabla}\cdot{\vec J}+
{{\partial n}\over{\partial t}}=0$. [@Thouless][@Orlando]
Quantum mechanics introduces two new ingredients to the analogies discussed above. The first is the quantum of angular momentum $l$, given by $\hbar$ (or alternatively, the quantum of vorticity, $h\over m$) [@Onsager]. The second is the quantization of the magnetic flux, the integral of the magnetic field over area. This quantization is most clearly seen through the Aharonov–Bohm effect [@Aharonov]: the Aharonov–Bohm phase shift accumulated by an electron traversing a closed path in a magnetic field is $2\pi$ times the number of flux quanta it encircles. Combining these two ingredients together, one should expect a quantization associated with the integral of the number density over area, i.e., with the number of particles. This quantization should manifest itself in the phase accumulated by a vortex carrying a single quantum of angular momentum, $\hbar$, when it traverses a closed path in a fluid. Indeed, as shown first by Arovas, Schrieffer and Wilczek [@Arovas], such a vortex does accumulate a geometric (Berry) phase [@Berry], and this phase is $2\pi$ times the number of fluid particles it encircles. [*The analog of a flux quatum is then a single fluid particle*]{}[@MPAFisher]. Note that the analogy between vortex dynamics in a fluid and electron dynamics in a magnetic field does not depend on the fluid being charged, and is valid for neutral fluids as well.
Quantum transport of 2D electrons in a magnetic field crucially depends on the electronic filling factor, the ratio between the density of conduction electrons and the density of flux quanta. For a very low filling factor, $(\ll 1)$, electrons are expected to form a Wigner lattice. At higher filling factors, the quantized Hall effect takes place [@Prange]. Similarly, we expect transport of vortices in a fluid to depend on a vortex “filling factor”, the ratio between the density of vortices and the density of fluid particles. However, in continious two dimensional fluids this ratio is usually much smaller than one, and the vortices indeed form an Abrikosov lattice.
How can the vortices “filling factor” be made larger? In this work we make the vortex filling factor larger by considering a lattice structure. As is well known, all properties of electrons on a lattice are invariant to the addition of a magnetic flux quantum to a lattice plaquette. Similarly, when we consider vortex transport on a lattice-structured fluid, we find all properties to be invariant to the addition of a single fluid particle to a lattice site. It is this periodicity that allows us to make the effective filling factor much larger than the ratio between the density of vortices and the density of fluid particles.
Based on the foregoing general considerations, we study in this paper a Josephson junction array in a magnetic field. Josephson junction arrays were extensively studied in recent years [@Mooij][@Eckern]. The array we consider is composed of identical small super–conducting dots coupled by a nearest–neighbors Josephson coupling $E_J$, and by a capacitance matrix $\hat C$. For definitness, we consider a square lattice of the superconducting dots. Generalizations to other lattices are straight forward. A perpendicular magnetic field induces vortices in the configuration of the superconducting phase. We denote the average number of vortices per lattice plaquette by ${\overline n}_v$. Each of the dots carries a dynamical number of Cooper pairs, denoted by $n_i$ (for the $i$’th dot), as well as positive background charges. The charging energy of the array is minimized for a certain set of values of $n_i$, which we denote by $n_{x,i}$. (In our notation, charge is always expressed in units of $2e$, i.e., $n_i,n_{x,i}$ are dimensionless.) While the $n_i$’s are operators with integer eigenvalues, $n_{x,i}$ are real parameters, that are closely related to the chemical potential of the dots. In this work we consider the case in which for all sites $n_{x,i}=n_x$. The Hamiltonian describing the array is [@Eckern], $$H={{(2e)^2}\over{2}}{\sum_{ ij}} (n_i-n_x){\hat C}^{-1}_{ij}(n_j-n_x)+
E_J{\sum_{\langle ij\rangle}}
(1-\cos(\phi_i-\phi_j-\int_i^j {\vec A}\cdot {\vec dl}))
\label{ham}$$ where ${\sum_{\langle ij\rangle}}$ denotes a sum over nearest neighbors, $n_i$ is the number of Cooper pairs on the $i'$th dot, $\phi_i$ is the phase of the superconducting order parameter on the $i$’th dot, $\vec A$ is the externally put vector potential and the integral is taken between the sites $i$ and $j$. A factor of ${2e}\over c$ is understood to be absorbed in $\vec A$. The matrix ${\hat C}^{-1}$ is the inverse of the capacitance matrix $\hat C$. Generally, the matrix $\hat C$ includes elements coupling a dot to its nearest neighbors, to the substrate and to neighbors further away. The matrix elements of both $\hat C$ and ${\hat C}^{-1}$ are a function of the distance between the sites $i$ and $j$. For short distances the electrostatic energy is determined by nearest neighbors capacitance only, and all other capacitances can be ignored. The $ij$ matrix element of $C^{-1}$ is then ${{2\pi}\over{C_{nn}}}\log
{|r_i-r_j|}$, where $C_{nn}$ is the nearest neighbors capacitance [@Eckern]. For large distances, the electrostatic interaction depends also on capacitance to the substrate and capacitance to neighbors further away. The inverse capacitance matrix then decays with the distance. Throughout most of our discussion we assume that the size of the array is small enough such that the charging energy is determined by nearest neighbors capacitance only. Then, the charging energy involves one energy scale, $E_C\equiv{{e^2}\over {2C_{nn}}}$. The effect of other capacitances is briefly discussed in section (3).
Since our main interest in this study is focused on transport phenomena of vortices, we constrain ourselves to arrays in which $E_J\widetilde{>}
E_C$. In that regime of parameters vortices are mobile enough not to be trapped within plaquettes, but their rest energy is large enough such that quantum fluctuations of vortex–antivortex pair production can be neglected. Arrays in which $E_J\widetilde{>}
E_C$ were studied experimentally by van der Zant [*et.al.*]{} [@Zant], and were found to show a magnetic field tuned transition from an almost super conducting state to an almost insulating state. At weak magnetic fields the density of vortices is low, and their ground state is the Abrikosov lattice. The array is then super–conducting. The transition to the insulating state, at a critical value of the magnetic field, is interpreted as caused by a transition of the vortices from a lattice phase to a correlated super–fluid–like phase [@MPAFisher][@Zant].
As mentioned in section (1), QHE phenomena in Josephson junction arrays were discussed in two recent preprints. The first, by Odintsov and Nazarov [@Odintsov], focuses on the regime $E_C\gg E_J$, and discusses a quantum Hall fluid of Cooper–pairs. The second, by Choi [@Choi], focuses on the regime $E_J\gg E_C$, and discusses a quantum Hall fluid of vortices. The quantum fluid we discuss in this paper has some similarity to the one discussed by Choi. However, the difference in the regime discussed, as well as our detailed study of the effect of the logarithmic vortex–vortex interaction, make some of our conclusions different from those of Choi.
Due to the lattice structure of the array, the spectrum and eigenstates of the Hamiltonian (\[ham\]) are manifestly periodic with respect to $n_x$, with the period of one Cooper–pair ($n_x=1$). This periodicity is similar to the periodicity of the spectrum of electrons on a lattice with respect to the addition of one flux quantum per plaquette. Thus, although the ratio between the density of vortices and the density of charges in the system is very small, the physically meaningfull ratio is the ratio of ${\overline n}_v$ to $(n_x-[n_x])$ (where $[n_x]$ is the largest integer smaller than $n_x$), and this ratio is not necessarily small. Following this observation, [*we limit ourselves from now on to the case $0\le n_x<1$*]{}.
Having described in detail the Josephson junction array to be considered, we conclude this section by formulating precisely the question to be studied, namely, [*how do the physical properties of the array depend on the ratio between the vortex density ${\overline n}_v$ and the charge density $n_x$?*]{} We start our examination of that question by deriving an effective action for the vortices in the array.
[**3. The effective action for the vortices**]{}
The Hamiltonian (\[ham\]) describes the Josephson junctions array in terms of the sets of variables $\{n_i\},\{\phi_i\}$. In this section we derive an equivalent description of the array in terms of the vortex density $\rho^{vor}$, the vortex current ${\vec J}^{vor}$ and gauge fields the vortices interact with. Our goals in attempting to derive this description are three–fold. The first goal is to verify the validity of our assertion that vortices are subject to Magnus force, and that $n_x$ plays a role analogous to that of a magnetic field in electronic dynamics. The second goal is to study the mutual interactions between vortices. The third goal is to estimate the mass of the vortices. The first two goals are relatively easy to achieve. Estimating the mass of the vortex, however, turns out to be a harder task, which we are able to handle only approximately.
The effective action for vortices in a Josephson junction array was first discussed by Eckern and Schmid, who considered the Hamiltonian (\[ham\]), with $n_x=0$. More generally, the effective action for singularities in the phase configuration of a complex field was discussed in various other contexts in physics. A particularly convenient method for the derivation of such an action is the “duality transformation”, developed and used by Jose [*et.al.*]{} [@Jose], Berezhinskii [@Berezhinskii], Peskin [@Peskin], Fisher and Lee [@Fisher] and others. This method was applied to analyze the motion of vortices in Josephson junction arrays (again, for the case $n_x=0$) by Fazio, Geigenmuller and Schon [@Fazio].
In our derivation of the effective action, we follow Fazio, Geigenmuller and Schon[@Fazio] by applying the duality transformation to obtain an effective action for vortices on a lattice. The action resulting from the duality transformation (Eq. (\[action\]) below) describes the vortices as bosons on a lattice interacting with an externally put vector potential, as well as with a dynamical vector potential. The externally put vector potential, which we denote by $\vec{\cal K}^{ext}$, satisfies ${\vec \nabla}\times
{\vec{\cal K}}^{ext}=2\pi\hbar n_x$. The interaction with the dynamical vector potential mediates a vortex–vortex interaction, composed of two parts. The first part is the familiar logarithmic interaction. Its strength is proportional to the Josephson energy $E_J$. The second part, induced mostly by the charging energy of the array, is a short ranged velocity–velocity interaction. The latter makes the vortices massive, since it includes a self interaction term, quadratic in the vortex velocity. However, the mass defined by this interaction is a “bare mass”, that does not take into account the periodic potential exerted on the vortices by the lattice. Generally speaking, the periodic potential changes the bare mass into an effective band mass. In an attempt to estimate the band mass we write the continuum limit of the vortices action. In the continuum language, vortices are massive particles interacting with an external vector potential, with a periodic lattice potential and with one another. Our analysis of this rather complicated dynamics follows the way the dynamics of electrons on a lattice is analyzed. We start by neglecting vortex–vortex interactions. We are then faced with a single particle problem, in which a massive vortex interacts with a static vector potential ${\vec{\cal K}}^{ext}$, and with a periodic lattice potential. This problem is identical to the problem of an electron under the effect of a uniform magnetic field and a lattice periodic potential, whose solution is well known. When $n_x\ll 1$ the effect of the periodic potential can be accounted for by changing the “bare mass” to an effective mass. We limit ourselves to this case, and estimate the resulting effective mass. Then, we incorporate the vortex–vortex interactions back into the action.
Before turning into the details of the derivation sketched in the last paragraph, we pause to define a notation. We denote 3–vectors by bold–faced letters, and their two spatial components by vector arrows. The electromagnetic potential is then ${\bf A}=(A_0,A_x,A_y)=(A_0,{\vec A})$. We number array sites by a subscript $i$. The bond connecting a site $i$ to its neighbor on the right side is denoted by the subscript $i,x$. Similarly, the bond connecting the $i$’th site to the site above it is denoted by the subscript $i,y$. The difference operator $\vec\Delta$, a discretized version of $\vec\nabla$, is defined accordingly. When operating on a scalar $\phi$, for example, the $x$–component of $\vec\Delta_i$ is $\phi_j-\phi_i$ where $j$ is the neighbor to the right side of $i$.
Our derivation of the effective action starts by considering the partition function $${\cal Z}={\rm tr}e^{-\beta H}=\int D\{n_i\}\int
D\{\phi_i\}e^{-{1\over\hbar}S(\{n_i(t)\},\{\phi_i(t)\})}
\label{parfunc}$$ where the action $S(\{n_i(t)\},\{\phi_i(t)\})\equiv
\int_0^\beta dt \left[ i\sum_i
\hbar n_i(t){\dot\phi}_i(t)-H(\{n_i(t)\},\{\phi_i(t)\})\right ]$ and the Hamiltonian is given by (\[ham\]). The path integral is to include all paths satisfying $n_i(\beta)=n_i(0)$ and $\phi_i(0)=\phi_i(\beta)$. The variables $n_i$ are integers and therefore the path integral has to be performed stepwize [@Swanson]. We limit ourselves to zero temperature, i.e., $\beta=\infty$.
The first step of the derivation follows closely previous works [@Fazio], and is therefore given in Appendix A. Using the Villain approximation and the duality transformation method, the path integral over the charge and phase degrees of freedom, $n_i$ and $\phi_i$, is transformed to a path integral over a 3–component integer field $\bf J^{vor}_i$ describing the vortex charge and density, and a 3–component real gauge field $\bf {\cal K}_i$, to which $\bf J^{vor}_i$ is coupled. This gauge field describes the charge degrees of freedom, to which it is related through its derivatives. The field strengths associated with this gauge field, ${1\over{2\pi\hbar}}\epsilon^{\mu\nu\sigma}
\partial_\mu{\cal K}_{i,\nu}$ are the Cooper–pairs current and density on the $i$’th site. In terms of $\bf J^{vor}$ and $\bf{\cal K}$, and in a gauge in which ${\vec\Delta}\cdot{\vec{\cal K}}=0$ the action is given by, $$\begin{array}{ll}
S^{vor}=
\sum_i\left\{ i(\rho^{vor}_i-{{\overline n}_v}){\cal K}_{i,0} +\
\ i{\vec J}^{vor}_i
\cdot({\vec{\cal K}}_i+{\vec{\cal K}}^{ext}_i)+{1\over{8\pi^2
E_J}}[(\Delta_t{\vec{\cal K}}_i)^2
+({\vec\Delta}{\cal K}_{0i})^2]\right\}
\\ \\
+{{e^2}\over {2\pi^2\hbar^2}}{\sum_{ ij}}
({\vec\Delta}\times{\vec{\cal K}}_i)
{\hat C}^{-1}_{ij}
({\vec\Delta}\times{\vec{\cal K}}_j)
\end{array}
\label{action}$$ where ${\vec\Delta}\times{\vec{\cal K}}^{ext}=2\pi\hbar n_x$. This action describes the vortices as bosons on a lattice, interacting with an externally put gauge field $\vec{\cal K}^{ext}$ whose spatial curl is a constant, given by $2\pi\hbar n_x$, and with a dynamical gauge field $\bf{\cal K}$. As expected from the similarity between Magnus and Lorenz forces, a moving vortex is affected by the Cooper–pairs on the dots in the same manner a charged particle is affected by a magnetic field. Moreover, the Josephson currents between the dots affect the vortices in the same way an electric field affects charged particles. The last three terms of the action include the self energy of the field $\bf{\cal K}$. They are simply understood once the relation between $\bf{\cal K}$ and the Cooper–pairs currents and densities is taken into account. The first two are the kinetic energy of the Josephson currents (the transverse part of that current is ${{e}\over{\pi\hbar}}{\vec\Delta}{\cal K}_0$, and ${{e}\over{\pi\hbar}}\dot{\vec{\cal K}}$ is the longitudinal part). The last term is the charging energy (the net charge on the $i$’th dot is ${1\over{2\pi\hbar}}{\vec\Delta}\times{\vec{\cal K}}_i$). The transverse part of the current satisfies a two dimensional Gauss law ${\vec\Delta}^2{\cal K}_0=4\pi^2E_J\rho^{vor}$ and mediates a logarithmic interaction between the vortices. The excitation spectrum of $\vec{\cal K}$ is the spectrum of longitudinal oscillations of the Cooper–pairs, i.e., the plasma spectrum of the array.
Our next step is a formulation of the continuum limit of the action (\[action\]). When doing that, two points should be handled carefully. The first is the periodic potential exerted by the lattice on the vortex. This potential was studied in detail by Lobb, Abraham and Tinkham[@Lobb]. Since Currents do not flow uniformly within the array, the energy cost associated with a creation of a vortex depends on the position of its center within a plaquette (i.e., on the precise distribution of the currents circulating its core). This energy cost is periodic with respect to a lattice spacing of the array, and is independent of the sign of the vorticity. The origin of this potential can be visualized using the analogy with 2D electrostatics. In that analogy, a vortex is analogous to a charge in a two–dimensional world. A vortex on a lattice is then analogous to a charge in a two dimensional world [*in which the dielectric constant varies periodically with position*]{}. The electrostatic energy of such a charge varies periodically with position, too, and is independent of the sign of the charge. This energy cost can then be interpreted as a periodic potential exerted by the lattice on the vortices. The characteristic energy scale for that potential is $E_J$. Its amplitude and functional form were studied in Ref. [@Lobb]. The amplitude was found to be $0.2E_J$ and $0.05 E_J$ for square and triangular lattices, respectively.
A convenient way to incorporate the periodic dependence of the vortex potential energy on the position of the vortex core within a plaquette is by replacing the Josephson energy $E_J$ in the action (\[action\]) by a periodically space dependent function $\epsilon_J({\vec r})$, that is non–zero only along lattice bonds. The period of $\epsilon_J({\vec r})$ is obviously the lattice spacing. The energy cost involved with the Josephson currents then becomes $\int d{\vec r}
{1\over{8\pi^2
\epsilon_J({\vec r})}}[({{\vec \nabla}}{\cal K}_{0}({\vec r},t))^2+
({\dot{\vec{\cal K}}}({\vec r},t))^2]$, and that energy cost confines the currents to the lattice bonds. The effect of the spatial dependence of $\epsilon_J({\vec r})$ on the interactions mediated by ${\cal K}_0,
{\vec{\cal K}}$ is discussed below.
The second point to be handled carefully when transforming to a a continuum action is the short distance cut–off on the capacitance matrix. The model we employ does not attempt to describe statics and dynamics of Cooper–pairs within superconducting dots. Thus, its continuum version should not allow for excitations of $\vec{\cal K}$ at wavelengths shorter than the lattice spaing. This constraint is taken into account by introducing a high wave–vector cut–off to the capacitance matrix, as it was done in Ref. [@Eckern].
Taking into account the two points discussed above, the continuum limit of the action (\[action\]) is, $$\begin{array}{ll}
S&=\int dt\int d{\vec r}\Bigg\{i[\rho^{vor}({\vec r},t)-{\overline n}_v]
{\cal K}_0({\vec r},t)+i{\vec J}({\vec r},t)\cdot [{\vec{\cal K}}
({\vec r},t)
+{\vec{\cal K}}^{ext}({\vec r})]\nonumber \\ \nonumber\\
&\hspace{2.5in}+{1\over{8\pi^2\epsilon_J({\vec r})}}[({\vec \nabla}
{\cal K}_0({\vec r},t))^2+
({\dot{\vec{\cal K}}}({\vec r},t))^2]
\Bigg\} \nonumber\\ \nonumber\\
&+{{e^2}\over{2\pi^2\hbar^2}}
\int dt \int d{\vec r}\int d{\vec r}'
\left[{\vec \nabla}\times{\vec{\cal K}}({\vec r},t)\right]
{\hat C}^{-1}({\vec r}-{\vec r}')
\left[{\vec \nabla}\times{\vec{\cal K}}({\vec r}',t)\right]
\label{conact}
\end{array}$$
When the fields ${\cal K}_0, {\vec{\cal K}}$ are integrated out they mediate mutual interactions between vortices and self interactions of a vortex with itself. The spatial dependence of $\epsilon_J({\vec r})$ does not significantly affect mutual interactions between vortices whose distance is much larger than one lattice spacing. It does, however, potentially affect the self interaction.
Consider the action associated with a single vortex. As discussed above, due to the spatial dependence of $\epsilon_J({\vec r})$, the interaction of the vortex with ${\cal K}_0$ yields a periodically space dependent potential energy [@Lobb]. The coupling to $\vec{\cal K}$ results in a kinetic energy. To see that, note that the vortex current corresponding to a moving vortex whose center is at ${\vec r}_0(t)$ is ${\vec J}^{vor}={\dot{\vec r}_0(t)}\delta\left({\vec r}-
{\vec r}_0(t)\right)$. Substituting this expression in the action (\[conact\]), we find the part of the action that depends on the vortex velocity, $\dot{\vec r}_0$, and the gauge field it interacts with, $\vec{\cal K}$, to be $$\begin{array}{ll}
& i\int dt {\dot{\vec r}_0(t)}\cdot{\vec{\cal K}}({\vec r}_0,t) \\ \\ +
&\int dt\Bigg\{\int d{\vec r}
{1\over{8\pi^2\epsilon_J({\vec r})}}({\dot{\vec{\cal K}}}({\vec r},t))^2
+{{e^2}\over{2\pi^2\hbar^2}}
\int d{\vec r}\int d{\vec r}'\left[{\vec \nabla}\times{\vec{\cal K}}
({\vec r},t)\right]
{\hat C}^{-1}({\vec r}-{\vec r}')
\left[{\vec \nabla}\times{\vec{\cal K}}({\vec r}',t)\right]\Bigg\}
\end{array}
\label{svdyn}$$ The gauge field $\vec{\cal K}$ can be integrated out, with the resulting effective action for the vortex velocity $\dot{\vec r}_0$ being non–local in time. However, as pointed out by Eckern and Schmid [@Eckern] and by Fazio [*et.al.*]{} [@Fazio], the time non–locality can be neglected as long as the characteristic frequencies involved in ${\dot{\vec r}_0(t)}$ are smaller than the Josephson plasma frequency ${1\over\hbar}\sqrt{8E_JE_C}$. This neglect is possible due to the gap in the excitation spectrum of $\vec{\cal K}$, a gap that makes the time non–locality short ranged [@Notea]. Having neglected the time non–locality, we find the effective action for the vortex velocity $\dot{\vec r}_0$ to be, $$\int dt{1\over 2}m_{bare}{\dot{\vec r}_0(t)}^2
\label{baremass}$$ where $m_{bare}$, the vortex bare mass, is defined by $m_{bare}\equiv{{\pi^2\hbar^2}\over{4E_C}}$. [@Eckern] Thus, the interaction of the vortex with $\vec{\cal K}$ results in a kinetic term.
Having integrated out both ${\cal K}_0$ and $\vec{\cal K}$ we have turned the single vortex action into an action of a charged particle interacting with a periodic potential and a magnetic field $2\pi\hbar n_x$. For $n_x\ll 1$ the effect of the periodic potential is to change the bare mass into an effective band mass. Since the lowest band is the relevant one for bosons, the band mass is always larger than the bare one [@Kittel]. The effective band mass for $n_x=0$ was studied both theoretically and numerically by Geigenmuller [@Geigenmuller] and by Fazio [*et.al.*]{}[@Fazio] (see also references therein). While a qualitative estimate of the mass is easy to arrive at, a quantitative determination depends on the precise details of the periodic potential, and is therefore hard to obtain. Qualitatively, the tight binding limit, in which $E_J\gg E_C$, is distinguished from the weak periodic potential limit, in which the opposite condition applies. In the former, the effective band mass is $$m_{band}\sim \hbar^2
\sqrt{{\alpha_1 }\over{E_JE_C}}e^{\sqrt{\alpha_2 {E_J\over E_C}
}}$$ where $\alpha_{1,2}$ are numbers of order unity [@Fazio][@Geigenmuller]. In the latter, $$m_{band}\sim m_{bare}\left(1+({E_J\over E_C})^2\right )$$ The regime of parameters we are interested in lies between the two limits. It is therefore reasonable to assume that the band mass is larger than, but of the order of, the bare one.
We now turn to discuss the many vortices configuration. As we have argued above, the discreteness of the array does not significantly affect the mutual interaction between vortices. Therefore, for the study of this interaction we may replace $\epsilon_J({\vec r})$ by $E_J$. Then, the action (\[conact\]) can be written in momentum space as $$\begin{array}{ll}
S=
\int dt\int {{d{\vec q}}\over{(2\pi)^2}}\left\{i[\rho^{vor}_{-{\vec q}}
-{\overline n}_v\delta({\vec q})]{\cal K}_{0{\vec q}}
+i{\vec J}^{vor}_{\vec q}\cdot({\vec{\cal K}}_{-{\vec q}}
+{\vec{\cal K}}^{ext}_{-{\vec q}})
+{1\over{8\pi^2E_J}}[|{\vec q}{\cal K}_{0{\vec q}}|^2+
|{\dot{\vec{\cal K}}}_{\vec q}|^2]
\right. \\ \\
+\left.{{e^2}\over {2\pi^2\hbar^2}}
|{\vec q}\times{\vec{\cal K}}_{\vec q}|^2{\hat C}^{-1}({\vec q})\right\}
\end{array}
\label{conactt}$$ where $\rho^{vor}_{\vec q},
J^{vor}_{\vec q},{\vec{\cal K}}_{\vec q}, {\vec{\cal K}}^{ext}_{\vec q},
{\hat C}^{-1}({\vec q})$ are the Fourier transforms of the corresponding quantities. The momentum space representation used in Eq. (\[conactt\]) is convenient for the integration of the field $\bf{\cal K}$. The integration out of the time component, ${\cal K}_0$, yields a density–density interaction between the vortices, of the form $
{1\over 2}\int d{\vec q}{{E_J}\over{q^2}}|\rho^{vor}_{{\vec q}}
-{\overline n}_v\delta({\vec q})|^2$, where $q\equiv|{\vec q}|$. When transformed back to real space, this interaction is $${\pi E_J}\int d{\vec r}\int d{\vec r}'
(\rho^{vor}({\vec r})
-{\overline n}_v)\log{|{\vec r}-{\vec r}'|}(\rho^{vor}({\vec r}')-
{\overline n}_v)
\label{logint}$$ Similarly, the integration of $\vec{\cal K}$ yields a current–current interaction between the vortices. In the gauge we use, $\vec{\cal K}_{\vec q}$ has only transverse components. It therefore mediates an interaction only between the transverse component of the vortex currents. Integrating out $\vec{\cal K}$, and neglecting again the slight non–locality in time, we find that the current–current interaction is given, in momentum space, by $${\hbar^2\over{8 e^2}}\int {{d{\vec q}}\over{ q^2{\hat C}^{-1}(q)}}
|J^{vor}_{\perp{\vec q}}|^2
={\hbar^2\over{16 E_C}}
\int {{d{\vec q}}} |J^{vor}_{\perp{\vec q}}|^2
\label{curcur}$$ where $J^{vor}_{\perp{\vec q}}\equiv
{{{\vec q}}\over q}\times{\vec J}^{vor}_{{\vec q}}$ is the transverse component of ${\vec J}^{vor}_{{\vec q}}$, and the integral over ${\vec q}$ is cut–off at $q=2\pi$. The current–current interaction is described in real space as, $${\hbar^2\over{16 E_C}}
\int d{\vec r}\int d{\vec r}'\int {{d{\vec q}}} [{\vec J}^{vor}
({\vec r})\times{\vec q}]
[{\vec J}^{vor}({\vec r}')\times{\vec q}] {1\over
{q^2}}e^{i{\vec q}\cdot ({\vec r}-{\vec r}')}
\label{velvel}$$ As pointed out in Ref. [@Eckern], this current–current interaction includes both a self interaction term, that assigns a mass $m_{bare}$ to each vortex, and a velocity–velocity interaction between different vortices. The former was discussed in the context of a single vortex. The latter is short ranged. For large separations, it is inversly proportional to the square of the distance between the interacting vortices.
Eqs. (\[logint\]) and (\[velvel\]) both neglected the effect of the lattice structure of the array on the vortex–vortex interactions. Similar to the common practice in the analysis of electrons on a lattice, we assume that the sole effect of the lattice is to modify the single vortex mass from the bare mass $m_{bare}$ to the effective band mass $m_{band}$. The current–current interaction in Eq. (\[velvel\]) includes a self interaction that assigns a mass $m_{bare}$ to each vortex. To account for the modification of the mass by the lattice, we add another kinetic term to the action, of the form ${M^*}\int d{\vec r}
{{{\vec J}^{vor}({\vec r})^2}\over
{2\rho^{vor}({\vec r})}}$, where ${M^*}\equiv m_{band}-m_{bare}$. Altogether, then, the effective vortices action becomes, $$\begin{array}{ll}
S^{vor}({\bf J^{vor}})=\int dt\Bigg\{&\int d{\vec r}\Big[
{M^*}{{{\vec J}^{vor}({\vec r})^2}\over
{2\rho^{vor}({\vec r})}}+i{\vec J}^{vor}({\vec r})\cdot
{\vec{\cal K}}^{ext}({\vec r})\Big]\\ \\
&+
{\hbar^2\over{16 E_C}}
\int d{\vec r}\int d{\vec r}'\Big[\int {{d{\vec q}}}
[{\vec J}^{vor}({\vec r})\times{\vec q}]
[{\vec J}^{vor}({\vec r}')\times{\vec q}] {1\over
{|{\vec q}|^2}}e^{i{\vec q}\cdot ({\vec r}-{\vec r}')}\\ \\
&+{{\pi E_J}}
(\rho^{vor}({\vec r})
-{\overline n}_v)\log{|{\vec r}-{\vec r}'|}(\rho^{vor}
({\vec r}')-{\overline n}_v)\Big]
\Bigg\}
\end{array}
\label{voracti}$$
Eq. (\[voracti\]) is a concise description of the dynamics of the vortices, since the only dinamical fields it includes are those of the vortices. It describes the vortices as interacting particles of mass $m_{band}$ and an average density ${\bar n}_v$, under the effect of a “magnetic field” $2\pi\hbar n_x$. The vortices “filling factor” is then indeed ${\bar n}_v\over n_x$.
The current–current interaction term in the action (\[voracti\]) is somewhat inconvenient for calculations. Thus, in our analysis of the quantum Hall fluid of vortices in the next section we choose to reintroduce $\vec{\cal K}$, and consider the vortices as particles of mass ${M^*}$ interacting with a dynamical vector potential $\vec{\cal K}$, as well as with ${\vec{\cal K}}^{ext}$ and with one another.
We conclude this section with a few remarks regarding the dependence of its results on the form of the capacitance matrix. The capacitance matrix determines the bare mass of the vortex (see Eqs. (\[svdyn\]) and (\[baremass\])) and the form of the vortex current–current interaction (see Eq. (\[velvel\])). So far we considered a capacitance matrix that includes only nearest neighbors coupling. The inverse capacitance matrix describes then a two dimensional Coulomb interaction between Cooper–pairs on the superconducting dots. In Fourier space, it is proportional to $1\over q^2$. Inclusion of capacitances to the ground and/or capacitance between dots that are not nearest neighbors result in a screening of that interaction. Then, at small $q$, ${\hat C}^{-1}(q)\propto q^{-\alpha}$, with $0\le\alpha<2$. This screening has two consequences. First, the kinetic energy cost involved in a vortex motion, i.e., its bare mass, is affected. Second, the excitation spectrum of $\vec{\cal K}$ becomes gapless. We now examine these consequences. Consider a vortex moving in a constant velocity ${\vec v}$. As seen from Eq. (\[svdyn\]), a moving vortex acts like a source for the vector potential $\vec{\cal K}$. The (imaginary time) wave equation for $\vec{\cal K}$ can be derived from Eq. (\[svdyn\]). In Fourier space its solution is, $${\cal K}_{\perp{\vec q},\omega}=iv_\perp
{{ \delta(\omega-{\vec q}\cdot{\vec v}})
\over{{\omega^2\over {4\pi^2 E_J}}+
{{e^2}\over{\pi^2\hbar^2}}q^2{\hat C}^{-1}(q)}}
\label{wave}$$ where $v_\perp\equiv{{{\vec q}\times\vec v}\over q}$ is the transverse part of the velocity vector. The longitudinal component of $\vec{\cal K}$ vanishes in the gauge we use. The kinetic energy cost associated with the motion of a vortex is the energy cost of the fields ${\dot{\vec{\cal K}}}$ and ${\vec \nabla}\times{\vec{\cal K}}$ it creates. It is composed of two parts. The first, $\int {d{\vec r}}{1\over{8\pi^2E_J}}
{\dot{\vec{\cal K}}}^2$, is the kinetic energy cost of the longitudinal currents created by the motion of the vortex. The second, ${{e^2}\over{2\pi^2\hbar^2}}\int d{\vec r}\int d{\vec r}'
\left[{\vec \nabla}\times{\vec{\cal K}}({\vec r})\right ]
{\hat C}^{-1}({\vec r}-{\vec r}')\left
[{\vec \nabla}\times{\vec{\cal K}}({\vec r}')\right]$ is the cost in charging energy. A moving vortex induces voltage drops between the superconducting dots, and those result in a charging energy cost, determined by the capacitance matrix. The first energy cost is proportional to $v^4$, while the second is proportional to $v^2$. Thus, the bare mass is determined by the charging energy. Transforming Eq. (\[wave\]) to real space, and substituting into the expression for the charging energy, we observe that the charging energy is finite as long as $\alpha>0$, and diverges logarithmically with the system size for $\alpha=0$. The gapless excitations of $\vec{\cal K}$, characteristic of $\alpha<2$, play a role when vortices accelarate or decelarate. Then, the coupling of the vortices to these excitations (the “spin waves” [@Fazio]) becomes a weak mechanism for dissipation of a vortex kinetic energy [@Eckern]. For the present context we note that the effect of a weak dissipative mechanism on the quantized Hall effect was studied by Hanna and Lee [@Hanna]. While some properties of the effect are affected by such a mechanism, its main features are not.
[**4. The quantum Hall fluid of vortices**]{} 0.5cm[**4.1 General discussion**]{}
In the previous section we established the mapping of the vortex dynamics in a Josephson junction array on the problem of interacting charged particles in a magnetic field. We have also identified the ratio ${\overline n}_v\over n_x$ as the vortices filling factor. In this section we examine the formation of QHE fluid state of vortices at appropriate filling factors. While so far we have emphasized the similarities between the dynamics of the vortices and that of electrons in a magnetic field, in this section we must study the differences between the two. We begin by a general discussion of two of the differences. Then, we turn in the next subsection to a detailed calculation, using the Chern Simon Landau Ginzburg approach to the QHE, developed by Zhang, Hansson, Kivelson and Lee [@Zhang].
The first difference is in the statistics: while vortices are bosons, electrons are fermions. This difference changes the values of the “magic” filling factors, and eliminates the possibility of a QHE in the absence of interactions. The filling factors at which bosons form quantum Hall fluids are $p\over q$, where $p,q$ are integers, and one of them is even [@Read]. Fermi liquids of the type discussed by Halperin, Lee and Read [@Halperinb] form at filling factors $1\over(2n+1)$, where $n$ is an integer.
The second difference is in the interaction: the logarithmic interaction between vortices is of longer range than the Coulomb interaction between electrons. This difference leads to a modification of the quantized Hall conductance, a modification of the charge of Laughlin’s quasiparticles, and, perhaps most interestingly, to a modification of one of the diagonal elements of the linear response function. While for a short range interaction these diagonal elements vanish in the long wavelength low frequency limit $({\vec q},\omega\rightarrow 0)$, reflecting the lack of longitudinal dc conductance in the QHE state, we find that the logarithmic interaction makes one of the diagonal elements non–zero. In fact, rather than describing insulator–type zero longitudinal response, as expected from a QHE system, this element describes a longitudinal response of the type usually associated with a superconductor. These consequences of the logarithmic interaction are all derived in detail in the next subsection, where we also study the difference between the linear response function and the conductivity. In this subsection we preceed the derivation by a discussion of a thought experiment that makes the role of the logarithmic interaction physically transparent. The thought experiment we consider was extensively used in the study of the Quantized Hall Effect, e.g., by Laughlin [@Laughlin] and Halperin [@Halperin], and was proved very useful in understanding various aspects of the effect.
Consider a “conventional” [*electronic*]{} quantum Hall system, in which a disk shaped two dimensional electron gas (2DEG) is put in a strong magnetic field, and a thin solenoid threads the disk at its center. The flux through the solenoid is time dependent, and is denoted by $\Phi(t)$. If the electrons on the disk are in a QHE state, the current density at any point is perpendicular to the [*total*]{} electric field at that point. The time dependence of the flux induces an electric field in the azymuthal direction, given by ${{e}\over{2\pi rc}}\dot\Phi(t)$, where $c$ is the speed of light, and $r$ is the distance from the center. Due to the finite Hall conductance, this electric field induces a radial current, and, consequently, a charge accumulation at the center of the disk. The charge accumulated at the center during the interval $0<t<t_0$ is given by ${{e\sigma_{xy}}\over{\Phi_0}}(\Phi(t_0)-\Phi(0))$ (where $\sigma_{xy}$ is the dimensionless Hall conductivity and $\Phi_0\equiv {{hc}\over e}$ is the flux quantum). This charge accumulation, in turn, creates a radial electric field. Now, if the electrons interact via a Coulomb interaction, the radial electric field is proportional to $1\over r^2$, i.e., it decays faster than the azymuthal one. Then, far away from the center the electric field is predominantly azimuthal, and the currents are predominantly radial. However, if the electrons interact via a logarithmic interaction, both the radial and azymuthal components of the electric field are inversly proportional to $r$, and thus their ratio is independent of $r$. The current then has both radial and azimuthal components, and their ratio is independent of $r$, too. Moreover, the azymuthal component of the current is proportional to the flux at the center, and not to its time derivative.
The two components of the current and the charge accumulated in the center can be calculated using classical equations of motion, since in the absence of impurities, the classical and quantum mechanical calculations coincide. Consider, therefore, the hydrodynamical equation of motion of a fluid of electrons in a magnetic field, whose electronic density and velocity fields are denoted by $\rho({\vec r})$ and ${\vec v}({\vec r})$, respectively. Assuming a uniform positive background charge density $\overline\rho$ on the disk, this equation of motion is $$m\rho({\vec r}){\dot{\vec v}}({\vec r})=
-\rho({\vec r}){\vec v}({\vec r})\times{\vec B}-
\int d{\vec r}'{\vec \nabla} V_{e-e}({\vec r}-{\vec r}')(\rho({\vec r}')-
{\overline\rho})\rho({\vec r})+
{{\dot\Phi}\over{2\pi r}}\rho({\vec r})
\label{conte}$$ where ${\dot{\vec v}}({\vec r})$ is the complete time derivative of the velocity field, $V_{e-e}$ is the electron–electron interaction potential, and we use a system of units where $e=c=1$. The initial conditions corresponding to the scenario discussed in the previous paragraph are $\rho({\vec r})={\overline\rho}$, ${\vec v}({\vec r})=0$ and $\Phi(t=0)=0$. Due to the circular symmetry of both Eq. (\[conte\]) and its initial and boundary conditions, the current and density remain circularly symmetric when the flux is turned on, and the electron–electron interaction term can be written as $V_{e-e}'(r)Q(r)\rho(r)$ where $Q(r)\equiv\int_0^r dr'
2\pi r'(\rho(r')-{\overline\rho})$ is the net charge within a distance $r$ from the origin, and $V_{e-e}'(r)\equiv{{\partial V_{e-e}(r)}
\over{\partial r}}$. The conservation of charge constraint implies ${\dot Q}(r)=-2\pi r \rho(r)v_r(r)$, where $v_r$ is the radial component of $\vec v$. The azymuthal component of Eq. (\[conte\]) can therefore be written as, $$2\pi r\rho(r){\dot v}_\phi={{\rho(r)}\over m}{\dot\Phi}-\omega_c {\dot Q}
\label{hydro}$$ with $\omega_c\equiv{B\over m}$. For values of $r$ far away from the center but not close to the edge the density $\rho(r)$ remains approximately equal to $\bar\rho$ all along the process, and thus $Q(r)$ is $r$–independent. Within that approximation, and for such values of $r$, the azymuthal equation can be integrated and substituted in the radial one. The latter then becomes, $${\bar\rho}{\dot v}_r={{\omega_c^2Q}
\over {2\pi r}}-{{\bar\rho}
\over m}{{\omega_c\Phi}\over{2\pi r}}-{1\over m}V_{e-e}'(r)
Q(r){\bar\rho}+{1\over r}{\bar\rho}{\vec v}_\phi(r)^2
\label{radial}$$ where the last term is the centrifugal force. Suppose now that the flux $\Phi$ is turned adiabatically on from zero to $\Phi(t_0)$ in the interval $0<t<t_0$. For times $t\gg t_0$ the velocity field is purely azimuthal and ${\dot v}_r=0$. Then, if the potential gradient ${\vec \nabla} V_{e-e}$ decays faster than $1\over r$, so does also the azymuthal velocity $v_\phi$, and $$Q=\nu{{\Phi(t_0)}\over{\Phi_0}}
\label{coulomb}$$ where $\nu={{\overline\rho}\over B}\Phi_0$ is the filling factor. If $\Phi(t_0)=\Phi_0$ then the charge accumulated near the origin is the charge of Laughlin’s quasiparticle, namely $-e\nu$.
However, in the case of a logarithmic interaction, $V_{e-e}(r)=-V_0\log{r}$, $$\begin{array}{ll}
v_\phi={\Phi\over{2\pi r m}}
{{V_0\nu}\over{\hbar\omega_c+V_0\nu}}\\ \\
Q ={\Phi\over\Phi_0}{\nu\over {1+{{ V_0\nu}\over{\hbar\omega_c}} } }
\end{array}
\label{loglog}$$ The azymuthal current is then indeed inversly proportional to the distance from the origin, and proportional to the flux $\Phi$. In fact, the current is related to the vector potential created by the solenoid, ${\vec A}^{sol}$, via a London–type equation $${\vec \nabla}\times{\vec J}={{\bar\rho}\over m}
{{V_0\nu}\over{\hbar\omega_c+V_0\nu}}
{\vec \nabla}\times{\vec A}^{sol}
\label{london}$$ Thus, the longitudinal response of the electrons on the disk to the vector potential created by the solenoid resembles the longitudinal response of a two dimensional superconducting disk in a similar situation.
Two conclusions can be drawn from the above thought experiment. First, for logarithmically interacting particles, the charge of the Laughlin quasiparticle does not equal the filling factor, but depends on the interaction. And second, the transverse part of the linear response function, relating a transverse current to an externally applied transverse vector potential, resembles that of a superconductor. Since this response function is proportional to the transverse current–current correlation function, the latter should be expected to resemble a supeconductor, too. Both conclusions are substantiated in the next subsection, and are applied to the study of the quantum Hall fluid of vortices.
[**4.2 A study of the vortices QHE state by the Chern Simon Landau Ginzburg approach**]{}
In this subsection we use the Chern Simons Landau Ginzburg approach to further analyze the properties of the quantized Hall fluid of vortices formed at appropriate values of ${{{\overline n}_v}\over n_x}$. We start by writing a Landau–Ginzburg action that describes the dynamics of the vortices. We then perform a Chern–Simons singular gauge transformation that attaches an even number of fictitious Cooper pairs to each vortex. The resulting action, in which the order parameter describes transformed “composite” bosons, is convenient for a saddle point analysis. We find the uniform density saddle point that describes a superfluid of composite bosons. By expanding the action to quadratic order around that saddle point we calculate the response function of the vortices to an external probing field. This response function, denoted by ${\hat\Sigma}$, is the ratio of the vortex density and current $\bf J^{vor}$ to an infinitesimal probing field $\bf{\cal K}^p$ applied externally to the system. The matrix ${\hat\Sigma}$ is calculated by adding an external probing field $\bf{\cal K}^p$ to the Lagrangian, and integrating out all the other fields to obtain an effective Lagrangian $L^{eff}$ in terms of $\bf{\cal K}^p$ only. The three components of $\bf J^{vor}$ are then given by $J^{vor}_\alpha=-{{\partial
L^{eff}}\over {\partial {\cal K}^p_\alpha}}$, and the elements of ${\hat\Sigma}$ are [@Zhang] $$\Sigma_{\alpha\beta}={{\partial{\bf
J^{vor}_\alpha}}\over{\partial{\bf{\cal K}^p_\beta}}}
\Bigg|_{{\bf{\cal K}^p_\beta}=0}=
- {{\partial\ }\over{\partial
{\bf{\cal K}^p_\alpha}}}{{\partial\ }\over
{\partial{\bf{\cal K}^p_\beta}}}L^{eff}
({\bf{\cal K}^p})\Bigg|_{{\bf{\cal K}^p}=0}
\label{sigmat}$$ The physical meaning of ${\hat\Sigma}$, as well as the important distinction between the response to $\bf{\cal K}^p$ and the response to the [*total*]{} field $\bf{\cal K}^p+{\cal K}$, are discussed after the calculation is presented.
The Landau–Ginzburg action that describes the properties of the vortices as they were found in section (3) is, $$\begin{array}{ll}
S_{LG}({\tilde\psi},{\bf{\cal K}})
=
\int dt \Bigg\{&\int d{\vec r} \Big[\hbar{\tilde\psi^*}
\partial_t{\tilde\psi}+
{1\over {2{M^*}}}
|(i\hbar{\vec \nabla}-{\cal K}^{ext}-{\cal K}){\tilde\psi}|^2+
{1\over{8\pi^2E_J}}{\dot{\vec{\cal K}}}^2\Big]\\ \\
&+\int d{\vec r}\int d{\vec r}'
\Big[\pi E_J(|{\tilde\psi}({\vec r})|^2-{\overline n}_v)
\log|{\vec r}-{\vec r}'|(|{\tilde\psi}({\vec r}')|^2-{\overline n}_v)\\ \\
&+{{e^2}\over {2\pi^2\hbar^2}}
[{\vec \nabla}\times{\vec{\cal K}}({\vec r})]{\hat C}^{-1}({\vec r}-{\vec r}')
[{\vec \nabla}\times{\vec{\cal K}}({\vec r}')]\Big]\Bigg\}
\end{array}
\label{lg}$$ The fields ${\tilde\psi},{\bf {\cal K}^{ext}},
{\bf{\cal K}}$ all depend on ${\vec r}$ and on $t$. For the brevity of the expressions we omit this dependence whenever this omission does not lead to confusion. The field $\tilde\psi$, the order parameter for the vortices, satisfies bosonic commutation relations.
Restricting our derivation to the “fundamental” fractions $1\over\eta$, where $\eta$ is an even number, our first step in analysing the action (\[lg\]) is the Chern Simon singular gauge transformation, in which the field ${\tilde\psi}({\vec r},t)$ is transformed to $$\psi({\vec r},t)=e^{i\eta\int d{\vec r}'{\rm arg}({\vec r}-{\vec r}')
|{\tilde\psi}({\vec r}',t)|^2}\ \ {\tilde\psi}({\vec r},t)$$ where ${\rm arg}({\vec r}-{\vec r}')$ is the angle the vector ${\vec r}-{\vec r}'$ forms with the $x$ axis. Since $\eta$ is an even integer the field $\psi$ has the same statistics as $\tilde\psi$, i.e., bosonic. Note that $|{\tilde\psi}({\vec r},t)|=|\psi({\vec r},t)|$. The singular gauge transformation shifts the phase of the field. Denoting the phase of $\psi({\tilde\psi})$ by $\theta
({\tilde\theta})$, the Chern–Simons transformation amounts to ${\vec \nabla}\theta({\vec r},t)={\vec \nabla}
{\tilde\theta}({\vec r},t)-{1\over\hbar}{\vec a}({\vec r},t)$ where ${\vec a}({\vec r},t)\equiv \hbar\eta\int d{\vec r}'
{{{\hat z}\times({\vec r}-{\vec r}')}\over
{|{\vec r}-{\vec r}'|^2}}|{\tilde\psi}({\vec r}',t)|^2$. The Chern–Simons field ${\vec a}$ has a gauge freedom, which we fix below. Writing $\psi({\vec r},t)\equiv
\sqrt{n_v({\vec r},t)}e^{i\theta({\vec r},t)}$, the above Landau–Ginzburg functional becomes $$\begin{array}{lll}
S_{LG}(n_v,\theta,{\bf{\cal K}},{\bf a})
=
\int dt \Bigg\{\int d{\vec r} \Big[in_v(\hbar\partial_t\theta -a_0)+
{n_v\over {2{M^*}}}(\hbar{\vec \nabla}\theta-{\cal K}^{ext}-
{\cal K}+{\vec a})^2+ \\ \\
{\hbar^2\over {2{M^*}}}({\vec \nabla}\sqrt{n_v})^2+
{1\over {8\pi^2E_J}}{\dot{\vec{\cal K}}}^2+{i\over{4\pi\eta\hbar}}
\epsilon^{\mu\nu\sigma}a_\mu\partial_\nu a_\sigma\Big]\\
\\
+\int d{\vec r}\int d{\vec r}'\Big[\pi E_J(n_v({\vec r})-
{\overline n}_v)\log|{\vec r}-{\vec r}'|
(n_v({\vec r}')-{\overline n}_v)\\ \\
+{{e^2}\over {2\pi^2\hbar^2}}[{\vec \nabla}\times
{\vec{\cal K}}({\vec r})]{\hat C}^{-1}({\vec r}-{\vec r}')
[{\vec \nabla}\times{\vec{\cal K}}({\vec r}')]\Big]\Bigg\}
\end{array}
\label{lgcs}$$ While the Euler–Lagrange equations of motion for the original field $\tilde\psi$ required its phase to be multiply valued (for a minimization of the kinetic energy), the equations of motion for the transformed field $\psi$ allow for a solution in which the magntiude and the phase of the field are constant. It is straight forward to see that the action (\[lgcs\]) is minimized when, $$n_v({\vec r},t)={\overline n}_v
\label{nv}$$ $${\vec \nabla}\theta({\vec r},t)=0
\label{theta}$$ $${\vec a}({\vec r},t)={\vec{\cal K}}^{ext}({\vec r})
\label{va}$$ $${\cal K}({\vec r},t)=0
\label{sa}$$ This minimum of the action describes a state in which the vortex density is constant on the average and the Chern–Simons field ${\vec a}$ cancels ${\cal K}^{ext}$ on the average. We now expand the action around these minimum values. Around the saddle point the phase $\theta$ is singly valued, and thus we can choose a gauge in which $\theta=0$ identically, and the field $\psi$ is real. Writing, then, $n_v={\overline n}_v+\delta n_v$, $\psi=\sqrt{{\overline n}_v}+{{\delta n_v}\over{2\sqrt{{\overline n}_v}}}$ and ${\vec a}={\vec{\cal K}}^{ext}+{\delta{\vec a}}$, we find that the quadratic deviations from the extermum point (\[nv\])–(\[sa\]) are described by the following action, $$\begin{array}{ll}
S_{LG}(\delta n_v,{\bf{\cal K}},{\bf a})\approx\int dt\int d{\vec r}
[ia_0\delta{n_v}+{{\bar n}_v\over {2{M^*}}}({\vec{\cal K}}+{\delta{\vec a}})^2+
{\hbar^2\over {8{M^*}}}{{({\vec \nabla}\delta{n_v})^2}\over{{\overline n}_v}}+
{1\over{8\pi^2 E_J}}{\dot{\vec{\cal K}}}^2
+{i\over{4\pi\eta\hbar}}
\epsilon^{\mu\nu\sigma}a_\mu\partial_\nu a_\sigma] \\
\\
+\int dt\int d{\vec r}\int d{\vec r}'\Big[\pi E_J\delta n_v({\vec r})
\log|{\vec r}-{\vec r}'|
\delta n_v({\vec r}')
+{{e^2}\over {2\pi^2\hbar^2}}({\vec \nabla}\times{\vec{\cal K}}({\vec r}))
{\hat C}^{-1}({\vec r}-{\vec r}')
({\vec \nabla}\times{\vec{\cal K}}({\vec r}'))\Big]
\end{array}
\label{quad}$$
For the calculation of the conductivity matrix and the correlations functions, we imagine coupling the vortices to an infinitesimal 3–vector probing field $\bf{\cal K}^p$. Naturally, linear response functions are more conveniently described in Fourier space. Thus, we write the action (\[quad\]) in the presence of the probing field $\bf{\cal K}^p$, in Fourier space, as $$\begin{array}{ll}
S_{LG}(\psi,{\bf{\cal K}},{\bf a},{\bf {\cal K}^p})\approx\int {{d\omega}
\over{2\pi}}\int
{{d{\vec q}}\over{(2\pi)^2}}
&\Bigg[i(a_0({\bf q})+{\cal K}^p_0({\bf q}))
\delta{n_v}(-{\bf q})+{{\bar n}_v\over {2{M^*}}}|{\vec{\cal K}^p}({\bf q})+
{\vec{\cal K}}({\bf q})+{\delta{\vec a}}({\bf q})|^2\\ \\ & +
{\hbar^2\over {8{M^*}}}{{|{\vec q}\delta{n_v}({\bf q})|^2}
\over{{\overline n}_v}} +
{1\over{8\pi^2E_J}}|\omega{\vec{\cal K}}({\bf q})|^2
+{i\over{4\pi\hbar\eta}}
\epsilon^{\mu\nu\sigma}a_\mu({\bf q})q_\nu a_\sigma({\bf q})\\
\\ &
+{{2\pi^2 E_J}\over{{\vec q}^2}}|\delta n_v({\bf q})|^2
+{{e^2}\over{2\pi^2\hbar^2}}|{\vec q}\times{\vec{\cal K}}({\bf q})|^2
{\hat C}^{-1}({\vec q})\Bigg]
\end{array}
\label{quadq}$$ where ${\bf q}_0\equiv\omega$. We choose the Coulomb gauge for $\bf{\cal K}^p$, i.e., ${\vec q}\cdot{\vec{\cal K}^p}({\vec q})
=0$. Thus, the probing field becomes a 2–component vector. Since the three components of $\bf J^{vor}$ are constrained by the conservation of vorticity, $\bf J^{vor}$ is effectively a two–component vetor, too, and $\Sigma_{\alpha\beta}$ is a $2\times 2$ matrix. The indices $\alpha$ and $\beta$ take the values $0$ (for the time component) and $\perp$ (for the component perpendicular to ${\vec q}$). The integration of the fields $\delta n_v,{\vec {\cal K}},{\bf a}$ is easily carried out, since (\[quadq\]) is quadratic in all fields. The resulting effective Lagrangian is, $$L^{eff}({\cal K}^p)=
%\Bigg\{
%{
\frac{
{{({\cal K}^p_\perp)^2}\over 2}\Big [{ {{{M^*}\omega^2}\over
{2\pi\hbar\eta\bar n_v}}+
{{2\pi E_J}\over{\hbar\eta}}+{{\hbar q^4}\over{8\pi{M^*}{\bar n_v}\eta}}
}\Big ]
+{{q^2({\cal K}^p_0)^2}\over 2 D}-
{i q{\cal K}^p_0{\cal K}^p_\perp}
}
%\over
{ {{{M^*}\omega^2}\over{{D}\bar
n_v}}+{4\pi^2 E_J\over {D}}+2\pi\hbar\eta
+{{\hbar^2 q^4}\over{4{M^*}{\bar n_v}D}}}
%}
%\Bigg\}
\label{eleff}$$ where ${D}\equiv 2\pi\hbar\eta
\left[{{M^*}\over{\bar n_v}}+{1\over{
{e^2\over{\pi^2\hbar^2}}
q^2{\hat C}^{-1}(q)+{{\omega^2}\over {4\pi^2E_J}} }}\right]^{-1}$. In the limit of $q,\omega\rightarrow 0$ and for ${\hat C}(q)=C_{nn}q^2$ $D=2\pi\hbar\eta
\left[{{M^*}\over{\bar n_v}}+2m_{bare}\right]^{-1}$.
Eq. (\[sigmat\]) expresses the matrix elements of ${\hat\Sigma}$ in terms of second derivatives of this effective Lagrangian with respect to ${\cal K}_0^p$ and ${\cal K}_\perp^p$. Each of the four components of the matrix $\Sigma_{\alpha\beta}$ warrants a short discussion. First, the transverse component of the vortex current and the transverse component of the gauge field $\bf{\cal K}^p$ are related, in the limit ${\vec q},\omega\rightarrow 0$, by $$J^{vor}_\perp=-{ {2\pi E_JD}\over{ 4\pi^2\hbar\eta E_J+
2\pi\hbar^2\eta^2 D}}{\cal K}^p_\perp
\label{perp}$$ This London–type of relation was anticipated by Eq. (\[london\]). It is characteristic of superconductors, and is very different from the insulating behaviour characteristic of the diagonal components of the response functions in QHE systems. This difference results from the static vortex–vortex interaction being of a long range. Defering the discussion of the effect of Eq. (\[perp\]) on the longitudinal conductivity to a later stage, we now point out its effect on the vortex current–current correlation function. By the fluctuation–dissipation theorem, $$\begin{array}{l}
\langle{ J^{vor}_\perp
J^{vor}_\perp}\rangle_{{\vec q},\omega}=
{\rm Im}\Sigma_{\perp\perp}({\vec q},\omega)
\\ \\
\int d\omega'{\rm P}({1\over{\omega-\omega'}})\langle{
J^{vor}_\perp J^{vor}_\perp}\rangle_{{\vec q},\omega'}
={\rm Re}\Sigma_{\perp\perp}({\vec q},\omega)
\end{array}
\label{fdkk}$$ where $\rm P$ denotes the principal part of the integral, and the second line is an application of Kramers–Kronig relations [@Forster]. For an insulator, ${\rm Re}\Sigma({\vec q},\omega)\propto\omega^2$ when $\omega\rightarrow 0$. This is also the case for QHE systems with short range interactions [@Zhangb]. For a superconductor ${\rm Re}{\hat\Sigma}({\vec q},\omega)$ approaches a constant in the $\omega\rightarrow 0$ limit. As we now see, so is also the case for a QHE system in which the interactions are logarithmic. In the particular problem we study, this constant is $-{ {2\pi E_JD}\over{4\pi^2\hbar\eta E_J+2\pi\hbar^2\eta^2 D}}$.
Second, we note that the compressibility of the vortex fluid vanishes in the limit ${\vec q},\omega\rightarrow 0$, as is manifested by the absence of low frequency poles in the density–density correlation function. Like its electronic analog, the quantum Hall fluid of vortices is incompressible.
Third, the Hall component of the linear response function is given, in the limit ${\vec q},\omega\rightarrow 0$, by, $$\Sigma_{0,\perp}={{-iq}\over{{{4\pi^2 E_J}\over D}+2\pi\hbar\eta}}
\label{hall}$$ If the vortex–vortex interaction was of shorter range, the long wavelength limit of $\Sigma_{0,\perp}$ would satisfy $\Sigma_{0,\perp}={{iq}\over{2\pi\hbar\eta}}$, seemingly demonstrating the quantization of the Hall conductivity [@Zhang]. We further comment on the difference between the two expressions below.
The qualitative effect the logarithmic interaction has on the transverse and Hall components of the linear response function raises the following question: does the conductivity of the system we study have the properties of the conductivity matrix of a QHE system, namely, zero longitudinal conductivity and quantized Hall conductivity? To answer this question, we clarify the relation between the response function $\Sigma_{\alpha\beta}$ and the vortex conductivity matrix. A similar relation was discussed, in the context of the QHE, by Halperin [@Halperin], Laughlin [@Laughlinb], Halperin Lee and Read [@Halperinb] and Simon and Halperin [@Simon]. The transport of vortices in the array is probed by externally applied (number) density and current of Cooper–pairs, given, respectively, by ${1\over{2\pi\hbar}}{\vec \nabla}\times{\cal K}^p$, and $-{1\over{2\pi\hbar}}{\vec \nabla}{\cal K}_0^p-{1\over{2\pi\hbar}}
{\dot{\vec{\cal K}^p}}$. The matrix $\hat\Sigma$ is defined such that ${\bf J^{vor}}={\hat\Sigma}{\bf{\cal K}^p}$. However, the vortices themselves contribute to the Cooper–pair density and current, with the most trivial contribution being the circulation of current around each vortex center. The [*total*]{} Cooper–pair density and current are therefore given by the derivatives of a total gauge field, composed of the probing field $\bf{\cal K}^p$ and the field induced by the vortex density and current, denoted by $\bf{\cal K}^{ind}$. The latter is proportional to the vortex density and current $\bf J^{vor}$. Thus, we can define a matrix $\hat V$ such that $${\bf{\cal K}^{ind}}\equiv{\hat V} {\bf J^{vor}}={\hat V}{\hat\Sigma}
{\bf {\cal K}^p}
\label{vmat}$$ Consequently, the total field is ${\bf {\cal K}^{tot}}=
(1+{\hat V}{\hat\Sigma}) {\bf {\cal K}^p}$, and $${\bf J^{vor}}={\hat\Sigma}(1+{\hat V}{\hat\Sigma})^{-1}{\bf{\cal K}^{tot}}
\label{aaa}$$ Thus, the matrix ${\hat\Sigma}(1+{\hat V}{\hat\Sigma})^{-1}$ relates ${\bf {\cal K}^{tot}}$ to the vector $\left (\begin{array}{c}\rho^{vor}
\\ J_\perp^{vor}
\end{array}\right )$. The vortex conductivity matrix $\sigma^{vor}$, relating the vortex current $\left(\begin{array}{c}J_{||}^{vor}\\ J_\perp^{vor}
\end{array}\right )$ to the [*total driving force vector*]{} $\left(\begin{array}{c}-i{{\vec q}\over{2\pi\hbar}}{\cal K}_0^{tot} \\ -
i{\omega\over{2\pi\hbar}}{\vec{\cal K}}^{tot}\end{array}\right )$ is then, $$\sigma^{vor}=\left(\begin{array}{ll}
-{\omega\over q}& 0 \\
0 & 0
\end{array}\right)
{\hat\Sigma}(1+{\hat V}{\hat\Sigma})^{-1}
\left(\begin{array}{ll}
i{{2\pi\hbar}\over q} & 0\\
0& i{{2\pi\hbar}\over\omega}
\end{array}\right)
\label{condu}$$ where the leftmost matrix converts $\rho^{vor}$ to $J^{vor}_{||}$.
Eq. (\[condu\]) defines the vortex conductivity matrix in terms of the matrices ${\hat\Sigma}$ and $\hat V$. The matrix ${\hat\Sigma}$ is defined by Eqs. (\[sigmat\]) and (\[eleff\]). The matrix $\hat V$, relating $\bf J^{vor}$ to ${\cal K}^{ind}$, is specified by the action (\[quadq\]) to be $${\hat V}=\left ( \begin{array}{cc}
{{4\pi^2 E_J}\over q^2} & 0 \\ \\
0 & 1\over{ {\omega^2\over {4\pi^2E_J}}+{e^2\over{\pi^2\hbar^2}}
q^2{\hat C}^{-1}(q)}
\end{array}
\right )
\label{vvv}$$ The upper left element describes the field ${\cal K}_0$ created by a vortex density $\rho^{vor}$. The gradient of that field is the transverse current circulating around the vortex center. The bottom right element describes the field ${\cal K}_\perp$ created by a transverse vortex current $J^{vor}_\perp$, and is obtained from Eq. (\[quadq\]) by taking its derivative with respect to ${\cal K}_\perp({\bf q})$.
Substituting the matrices $\hat\Sigma$ and $\hat V$ to Eq. (\[condu\]), we find, to leading order in $q,\omega$, $${\hat\sigma}^{vor}=\left( \begin{array}{ll}
i{{\omega{M^*}}\over{\eta^2 n_v}} & 1\over\eta \\ \\
-{1\over\eta} & -i{{\omega{M^*}}\over{\eta^2 n_v}}
\end{array}\right )
\label{almost}$$ In the limit ${\vec q},\omega\rightarrow 0$ the diagonal terms vanish, and the vortex current satisfies $$\vec J^{vor}=-{i\over{\eta 2\pi\hbar}}{\hat z}\times({\vec q}{\cal K}_0+
\omega{\vec{\cal K}})
\label{sofsof}$$ Eqs. (\[almost\]) and (\[sofsof\]) describe a quantized Hall effect: the current is purely perpendicular to the total “driving force”, and the Hall conductivity is quantized. Contrasting Eqs. (\[perp\]) and (\[hall\]) with Eq. (\[sofsof\]) we can finally summarize the effect of the logarithmic interaction on the linear response of the system: the dc conductivity, which is the $q,\omega\rightarrow 0$ response to the [*total*]{} driving force, has the usual form of the quantum Hall conductivity, and is unaffected by the interaction. The correlation functions, on the other hand, determined by the response to the [*externally applied*]{} driving force, are affected by the interactions even in the $q,\omega\rightarrow 0$ limit, with the most notable effect being on the transverse current–current correlation function.
We conclude this section by relating the vortices conductivity, calculated above, to the electric conductivity and resistivity, which are the quantities typically measured in experiments. The electric conductivity is the matrix relating voltage drops (or, in the continuum limit, electric fields) between superconducting dots to the electric Josephson current flowing in the array. The electrostatic potential at a point ${\vec r}$ is given by ${{e}\over{\pi\hbar}}
\int d{\vec r}' {\hat C}^{-1}({\vec r}-{\vec r}')
{\vec \nabla}\times{\vec{\cal K}}({\vec r}')$. Thus, in Fourier space the $\vec q$ component of the electrostatic potential is $i{{e}\over{\pi\hbar}}
q{\hat C}^{-1}(-{\vec q}){\cal K}_\perp({\vec q})$ and the [*longitudinal*]{} electric field is $-{{e}\over{\pi\hbar}}
q^2{\hat C}^{-1}(-{\vec q}){\cal K}_\perp({\vec q})$. Now, by deriving an equation of motion for ${\cal K}_\perp$ from the action (\[conactt\]), we see that a $dc$ [*transverse*]{} vortex current ${\vec J}^{vor}_\perp$ creates a field ${\vec{\cal K}}_\perp$ given by $${\vec J}^{vor}_{\perp{\vec q}}={{e^2}\over{\pi^2\hbar^2}}
q^2{\hat C}^{-1}({\vec q}){\cal K}_\perp({\vec q})$$ i.e., a transverse vortex current ${\vec J}^{vor}_{\perp{\vec q}}$ induces a longitudinal electric field ${{\pi\hbar}\over{e}}{\vec J}^{vor}_{\perp{\vec q}}$. A similar argument regarding the relation of the longitudinal vortex current to the transverse electric field leads to the conclusion that a vortex current ${\vec J}^{vor}_{{\vec q}}$ creates an electric field ${{\pi\hbar}\over{e}}{\hat z}
\times{\vec J}^{vor}_{\perp{\vec q}}$. The Josephson charge current, on the other hand, is $-i{e\over{\pi\hbar}}{\hat z}\times
({\vec q}{\cal K}_0+{\omega{{\cal K}_\perp}})$, i.e., it is proportional to the “driving force” acting on the vortices. Thus, the matrix relating the Josephson current to the electric field is proportional to the matrix relating the driving force acting on the vortices to the vortex current, or, explicitly, $$\rho^{el}={2\pi\hbar\over{(2e)^2}}\sigma^{vor}$$ where $\rho^{el}$ is the electric [*resistivity*]{} matrix of the array [@Choi]. This result can be simply concluded from Eq. (\[sofsof\]). The right hand side of that equation is the Josephson current, divided by $2e$. The left hand side is proportional and perpendicular to the electric field. The electric field is then proportional and perpendicular to the Josephson current, with the proportionality constant being ${{2\pi\hbar}
\over{(2e)^2\eta}}$. The quantum Hall fluid of vortices manifests itself in electronic properties of the array – the longitudinal electric resistivity vanishes, and the Hall electric resistivity is quantized.
[**5. Conclusions**]{}
In the previous sections we presented a study of the transport of vortices in an array of Josephson junctions described by the Hamiltonian (\[ham\]). In particular, we focused on a quantum Hall fluid formed by the vortices at appropriate values of ${\bar n}_v\over n_x$. In this section we summarize the results of this study, and comment on a few open questions.
Our study was motivated by the analogy between Magnus force acting on vortices and Lorenz force acting on charges in a magnetic field. In this analogy, fluid density plays a role analogous to a magnetic field, and fluid current density plays a role analogous to an electric field. Quantum mechanics extends the analogy further: a fluid particle is found to play a role analogous to that of a flux quantum. This analogy motivates the search for a quantized Hall effect for the vortices. The vortices’ filling factor is identified with the ratio of the vortex density to the fluid density. This ratio is very small for superconducting films, and it is this smallness that motivates the study of the Josephson junction array. Due to the periodicty of the spectrum of the Hamiltonian (\[ham\]) with respect to the parameter $n_x$, the effective filling factor becomes ${\bar n_v}\over {n_x(mod 1)}$, which can be made of order unity.
The dynamics of the vortices in a Josephson junction array was studied in section (3). It is found to be that of massive interacting charged particles under the effect of a magnetic field and a periodic potential. The magnetic field is $2\pi\hbar n_x$. The effect of the periodic potential is taken into account in an effective mass approximation, changing the mass from a bare mass to an effective band mass. The effective mass is exponentially large for $E_J\gg E_C$, and of the order of ${{\pi^2\hbar^2}\over{4E_C}}$ for $E_J\widetilde{>}E_C$. Being interested in a phenomenon resulting from a motion of vortices, we obviously consider the latter regime. The mutual interaction between vortices consists of a velocity independent logarithmic interaction, whose strength is proportional to $E_J$, and a short ranged velocity–velocity interaction.
In view of the mapping of the dynamics of the vortices on that of massive interacting charged particles in a magnetic field, the existence of a quantum Hall fluid phase is to be expected. In Section (4) we examine some of the properties of that phase, but we leave unexplored some other important propoerties. Most notable among the latter are the regime of vortices filling factors at which the quantum Hall fluid is the lowest energy state, and the energy gap for excitations above that fluid.
Our study of the quantum Hall fluid is performed by means of the Chern Simon Landau Ginzburg approach to the quantum Hall effect. When ${{\bar n}_v\over n_x}={1\over \eta}$ with $\eta$ being an even integer, the vortices Landau–Ginzburg action is found to have a saddle point corresponding to a quantum Hall fluid. By hierarchical construction such saddle points can be found for ${{\bar n}_v\over n_x}={p\over q}$, with $p,q$ being one even and one odd integer. The properties of the vortices quantum Hall fluid are studied within a quadratic expansion of the action around the corresponding saddle point. We find that the vortices conductivity matrix shows a typical QHE behavior, i.e., zero diagonal elements and quantized non–diagonal elements. However, we find the ${\vec q},\omega\rightarrow 0$ limit of the current–current correlation functions in the ground state to be different from those of a typical quantum Hall state, due to the long range logarithmic interaction. In particular, the transverse current–current correlation function is predicted to bahave like that of a superconductor, rather than an insulator. For large arrays (larger than an effective London penetration length) the vortex–vortex interaction is screened. Then, both the conductivity matrix and the correlation functions are expected to behave, in the $dc$ limit, like those of a typical quantum Hall state.
A necessary condition for the quantum Hall fluid to be the lowest energy state is presumably that the ground state at $n_x=0$ and ${\bar n}_v\ne 0$ (infinite filling factor) is a superfluid of vortices, i.e., an insulator. The observation, by van der Zant [*et. al.*]{} [@Zant], of a magnetic field tuned transition points at the regime of parameters in which this condition is satisfied, namely, $E_J\approx E_C$ and $0.3> {\bar n}_v>0.15$. In this regime of parameters we expect the quantum Hall fluid to be the ground state at large filling factors, and the Abrikosov lattice to be the ground state at small filling factors. This expectation is based on the phase diagram of a two dimensional electron gas. For the latter, if the ground state at zero magnetic field is a Fermi liquid, then the ground state at large filling factors $(\widetilde{>}0.2)$ is the quantum Hall fluid and the ground state at low filling factors is the Wigner lattice.
[**Appendix A – The Villain approximation and the duality transformation**]{}
The starting point of this appendix is the expression of the partition function as a path integral over the phase and number sets of variables $\{\phi_i\},\{n_i\}$, Eq. (\[parfunc\]). Using the Villain approximation and the duality transformation we transform that path integral to a path integral over an integer 3–component vector field $\bf J^{vor}$, describing the vortex density and current, and a real 3–component vector gauge field, $\bf{\cal K}$. The action in terms of $\bf J^{vor}$ and $\bf {\cal K}$, to be derived below, is given by Eq. (\[action\]). The following derivation follows the method of Fazio [*et.al.*]{}[@Fazio]
In the Villain approximation the imaginary time integral is done in discrete steps, where the size of each step, denoted by $\tau_0$, is of the order of the inverse Josephson plasma frequency $\omega_J\equiv \hbar^{-1}\sqrt{8E_JE_C}$. Each term in the Josephson energy part of the path integral is approximated by a Villain form (we put $\hbar=1$ throughout the appendix, and restore its value in the final formula), $$\begin{array}{ll}
e^{-\tau_0 E_J (1-\cos(\phi_{ij}-A_{ij}))}
\approx\sum_{v_{ij}=-\infty}^{\infty}&e^{-{1\over 2}\tau_0 E_J(\phi_{ij}-A_{ij}
+2\pi v_{ij})^2}\\
&=\sum_{v_{ij}=-\infty}^{\infty}\sqrt{\tau_0\over{2\pi E_J}}
\int dp_{ij}e^{-{{p_{ij}^2\tau_0}\over{2E_J}}+
i{ p}_{ij}\tau_0 (\phi_{ij}-\vec A_{ij}+2\pi v_{ij})}
\end{array}
\label{villain}$$ This approximation is valid for $E_J\tau_0\widetilde{>}1$, and gets better as $E_J\tau_0$ gets larger. However, it retains the most important feature of the Josephson energy, the periodicity with respect to $\phi$, for all values of $E_J\tau_0$. Alogether, then, the approximation we discuss holds for $E_J>E_C$. The significance of the field $\vec v_i$ can be understood by noting that ${\vec\Delta}\times{\vec v}_i$ describes the density of vortices [@Polyakov]. As for the real variable $ p_{ij}$, as shown below, it describes the Josephson current along the bond $ij$. Since the Josephson energy includes a sum over all lattice bonds, the Villain approximation introduces a variable $p_{ij}$ to each lattice bond. Thus, we can regard $p$ as a [*vector*]{} defined for each lattice $site$, such that $p_{ix}$ corresponds to the bond $i,x$ and $p_{iy}$ corresponds to the bond $i,y$. Similarly, the difference $\phi_i-\phi_j$, the integral $\int_i^j{\vec A}\cdot dl$ and the variables $v_{ij}$ can be regarded as vectors ${\vec\Delta}\phi_i$, ${\vec A}_i$ and $\vec v_i$.
Next we apply the Poisson resummation formula to the $n_i$ dependent part of the action. By doing that we make the $n_i$ variables real numbers rather than integers and add a time component to the integer–valued vector field ${\vec v}_i$. The partition function then becomes, $$\begin{array}{lll}
Z=\sum_{\{{\bf{ v}_i(t)}\}}
&\int D\{n_i(t)\}\int D\{{\vec p}_i(t)\}\int \{D\phi_i(t)\}&\\ \\
&\exp\Bigg\{\int_0^\beta dt\Big[
\sum_iin_i({\dot\phi}_i+ 2\pi
v_{0i})&-{{(2e)^2}\over 2}{\sum_{ ij}} (n_i-n_x){\hat C}^{-1}_{ij}(n_j-n_x)
\\ \\ & &-\sum_i{{{\vec p}_i^2}\over{2 E_J}}+
i{\vec p}_i\cdot({\vec\Delta}\phi_i-{\vec A}_i+2\pi{\vec v}_i)\Big]\Bigg\}
\end{array}
\label{poisson}$$ where the path integral should be performed stepwize [@Swanson]. For the brevity of this expression we omitted the explicit time dependence of $n_i,\phi_i,{\vec p}_i, {\vec v}_i$ in the stepwize integrated action. This form allows us to understand the physical significance of $\vec p$. The only $\vec A$–dependent term in the action is $-i{\vec p}_i\cdot
{\vec A}_i$. The derivative of the Lagrangian with respect to $\vec A$ is the current. Thus, $\vec p_i$ is the Josephson current flowing through the site $i$.
The path integral over the phase variables $\phi_i(t)$ can now be performed. The phase $\phi_i$ at the site $i$ is coupled to the charge $n_i$ (via the term $in_i{\dot\phi_i}$) and to the vector $\vec p$ at the site $i$ and its nearest neighbors. The integration over $\phi_i$ yields conservation of charge constraint on the integration over $n_i,{\vec p}_i$, in the form $$\Delta_tn_i+{\vec\Delta}\cdot{\vec p}_i=0
\label{dcoc}$$ where the definition $\Delta_tn_i\equiv
{1\over\tau_0}[n_i(t+\tau_0)-n_i(t)]$ makes the difference operator $\bf\Delta$ a 3–vector.
The constraint (\[dcoc\]) is nothing but a discretized form of a two dimensional conservation of charge equation. Like the latter, it can be solved by defining a 3–vector field $\bf{\cal K}$ that satisfies, $$\epsilon^{\alpha\beta\mu}\Delta_\alpha{{\cal K}}_{i,\beta}=2\pi{\bf p}_{i,\mu}
\label{curl}$$ where ${\bf p}_i
\equiv(n_i, p_{i,x}, p_{i,y})$. The three components of $\bf{\cal K}_i$ are real, like those of ${\bf p}_i$. The definition (\[curl\]) of $ {\cal K}_i$ is not unique and it becomes unique only when a gauge is fixed. The partition function is, of course, independent of that gauge. The constrained path integral over ${\bf p}_i$ is replaced now by path integrals over $\bf{\cal K}_i$, constrained by the gauge condition. Here we choose to work in the Coulomb gauge, in which ${\vec\Delta}\cdot{\vec{\cal K}}=0$. In that gauge the partition function becomes, $$\begin{array}{ll}
Z=\sum_{\bf v_i(t)} \int D\{{\bf{\cal K}_i}\}
&\exp{\int_0^\beta dt} \left[-{{e^2}\over {2\pi^2}}
{\sum_{ ij}} ({\vec\Delta}\times{\vec{\cal K}}_i)
{\hat C}^{-1}_{ij}
({\vec\Delta}\times{\vec{\cal K}}_j)\right.
\\ \\ & \left. -\sum_i{1\over{8\pi^2 E_J}}(\Delta_t{\vec{\cal K}}_i^2
+{\vec\Delta}{\cal K}_{0i}^2)+i\sum_i[{\bf\Delta}\times({\bf{\cal K}}+
{\bf{\cal K}}^{ext})]_i\cdot({\bf v_i}-{1\over{2\pi}}{\bf A}_i)
\right]
\end{array}
\label{vilsav}$$ where ${\vec{\cal K}}^{ext}$ is defined by ${\vec\Delta}
\times{\vec{\cal K}}^{ext}=2\pi
n_x$. We are now one step away from having an effective action for the vortices. The remaining step is an integration by parts of the last term in the action in (\[vilsav\]). After performing that integration, the following action is obtained: $$\begin{array}{ll}
S^{vor}=\int_0^\beta dt
\sum_i\Big\{ &i(\rho^{vor}_i-{\bar n_v}){\cal K}_{0i} +i{\vec J}^{vor}_i
\cdot({\vec{\cal K}}_i+{\vec{\cal K}}^{ext}) \\ \\
&+ {{e^2}\over{2\pi^2\hbar^2}}\sum_j ({\vec\Delta}\times{\vec{\cal K}}_i)
{\hat C}^{-1}_{ij}
({\vec\Delta}\times{\vec{\cal K}}_j)+{1\over{8\pi^2 E_J}}
((\Delta_t{\vec{\cal K}}_i)^2
+({\vec\Delta}{\cal K}_{0i})^2)\Big\}
\end{array}
\label{actiona}$$ where the vortex 3–vector current ${\bf J^{vor}}$ is defined as ${\bf J^{vor}}={\bf\Delta}\times{\bf v}$, the average density of vortices is given by ${\bar n}_v={B\over\Phi_0}$ and the value of $\hbar$ has been restored. Equation (\[actiona\]) is the starting point of the discussion in section (3).
[**Acknowledgements**]{}
I am indebted to B.I. Halperin, S. Simon and D.H. Lee for instructive discussions, and to M.Y. Choi for sending me his preprint prior to publication. I am grateful to the Harvard Society of Fellows for financial support. Part of this work was done in the Aspen center of physics, to which I am grateful for hospitality. Part of this work was supported by NSF Grant No. DMR-91-15491.\
\
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abstract: 'We show that in a two-component Bose gas with Rashba spin-orbit coupling (SOC) two atoms can form bound states (Rashbons) with any intra-species scattering length. At zero center-of-mass momentum there are two degenerate Rashbons due to time-reversal symmetry, but the degeneracy is lifted at finite in-plane momentum with two different effective masses. A stable Rashbon condensation can be created in a dilute Bose gas with attractive intra-species and repulsive inter-species interactions. The critical temperature of Rashbon condensation is about six times smaller than the BEC transition temperature of an ideal Bose gas. Due to Rashba SOC, excitations in the Rashbon condensation phase are anisotropic in momentum space.'
author:
- 'Rong Li, Lan Yin'
title: 'Pair condensation in a dilute Bose gas with Rashba spin-orbit coupling'
---
. In recent several years one major progress in ultracold atom physics was the realization of spin-orbit coupling (SOC) in Bose-Einstein condensation [@Lin11] and ultracold Fermi gases [@PWang; @Law]. In contrast to the intrinsic SOC of electrons in atoms, SOC in neutral atoms refers to the coupling between spin and center-of-mass momentum of atoms. Bose gases with SOC have displayed very rich phase diagrams. In experiments where the SOC is an equal-weight combination of Rashba and Dresselhaus SOCs, several phase transitions, including magnetic to spin-mixed states and normal to magnetics states, were observed in Bose gases [@Lin11; @Zhang13; @Zhang12], consistent with theoretical studies [@Li]. In a uniform Bose gas with Rashba SOC, competition between plane-wave and spin-stripe phases were predicted [@CWang; @Ho; @Yu13]. New phases, such as half-vortex and skyrmion-lattice phases, were predicted in trapped systems [@Wu; @Hu; @Sinha; @Ozawa]. In this paper, we are going to show that a stable pairing state can appear in a two-component Bose gas with Rashba SOC.
In contrast to the well-observed BCS-BEC crossover in Fermi gases [@Regal], pairing state of Bose gas is an exotic but never observed phenomena. Although Feshbach molecules of Bose atoms have been created in experiments [@Donley; @Xu; @Thompson], rapid particle-loss rate due to strong inelastic collision near the resonance severely limits the molecule lifetime, making it impossible to reach a condensed state. Despite experimental difficulties, the pairing state of a Bose gase has been explored theoretically [@Rad; @Romans; @Yin], but it was found unstable even with a weak attractive interaction away from the resonance [@Jeon; @Basu; @Yu10].
The successful creation of SOC offers a new opportunity to realize the pairing state in a Bose gas. As in the fermion case, Rashba SOC changes the atom density of states (DOS) which has strong effects on pairing and produces Rashbon [@Vya1], the two-body bound state with negative scattering length. In the following, we first study the two-body bound states of bosons with Rashba SOC. We find that at zero center-of-mass momentum bound states (Rashbons) can exist with arbitrary intra-species scattering length, while SOC has virtually no effect on the bound state created by the inter-species interaction. Next, we study the possibility of Rashbon condensation in a Bose gas with Rashba SOC. We find that Rashbon condensation can be stabilized by a repulsive inter-species interaction. The Rashbon condensation may be realized in a dilute Bose gas with weak intra-species attraction and inter-species repulsion. One signature of this phase is the anisotropic excitation spectrum. The Rashbon transition temperature is about six times smaller than the ideal BEC temperature.
[*Model*]{}. We consider a two-component Bose gas with Rashba SOC, described by the Hamiltonian $\mathcal{\hat{H}}=\mathcal{\hat{H}}_0+\mathcal{\hat{H}}_{\text{int}}$, where $$\mathcal{\hat{H}}_0=\sum_{{\bf k},\sigma}\epsilon_{{\bf k}}c^\dag_{{\bf
k}\sigma}c_{{\bf k}\sigma}+\sum_{\bf k}[S({\bf k}_{\perp})c^\dag_{{\bf k}\downarrow}c_{{\bf
k}\uparrow}+h.c.],$$ $c_{{\bf k}\sigma}$ represents the annihilation operator of a boson with wave-vector ${\bf k}$ and spin $\sigma$, $S({\bf
k}_{\perp})=\hbar^2 \kappa(k_x+i k_y)/m$, $\kappa$ is the strength of Rashba SOC, ${\bf k}_{\perp}$ is the projection of ${\bf
k}$ in the x-y plane, and $\epsilon_{\bf k}=\hbar^2 k^2/2m$. The single-atom Hamiltonian $\mathcal{\hat{H}}_0$ can be easily diagonalized, yielding helical excitations with energies given by $\xi_{{\bf k}\pm}=\epsilon_{\bf k}\pm\hbar^2
\kappa k_\perp/m$. The s-wave interaction between atoms is given by $$\mathcal{\hat{H}}_{\text{int}}=\frac{1}{2V}\sum_{{\bf k},{\bf
k}^\prime,{\bf q};\sigma,\sigma'}g_{\sigma\sigma'}c^\dag_{{\bf
k}^\prime\sigma}c^\dag_{{\bf q}-{\bf
k}^\prime\sigma'}c_{{\bf q}-{\bf
k}\sigma'}c_{{\bf k}\sigma},$$ where $V$ is the volume. The inter-species coupling constants satisfy $g_{\uparrow\downarrow}=g_{\downarrow\uparrow}=4\pi \hbar^2 a'/m$, where $a'$ is the inter-species scattering length. In the following, we consider only the symmetric case with two identical intra-species coupling constants, $g_{\uparrow\uparrow}=g_{\downarrow\downarrow}=4\pi \hbar^2 a/{m}$, where $a$ is the intra-species scattering length. In this symmetric case, the system is invariant under time-reversal transformation $({\bf k},\sigma)\rightarrow(-{\bf k},-\sigma)$.
[*Two-body bound states*]{}. We first study two-body bound states described by the wave function $$|\Psi\rangle_{{\bf q}}=\frac{1}{2}\sum_{{\bf k},\sigma, \sigma'}
\psi_{\sigma \sigma'}({\bf k},{\bf q}-{\bf k})c^\dag_{
{\bf k}\sigma}c^\dag_{{\bf q}-{\bf k}\sigma'}|0\rangle,$$ where $\psi_{\sigma \sigma'}({\bf k},{\bf k}')=\psi_{\sigma'
\sigma}({\bf k}',{\bf k})$ is a coefficient. By solving the eigenvalue problem, $H |\Psi\rangle_{\bf q} =E_{\bf q}
|\Psi\rangle_{\bf q} $, we can obtain wavefunction and eigenenergy of bound states. At ${\bf q}=0$, the eigenequation can be further written as $$\label{eigeneq}
M_{\bf k} \psi_{\bf k}'={1 \over V}G\sum_{\bf p}\psi_{\bf p}',$$ where $\psi_{\bf k}'$ is a four component-vector given by $$\psi_{\bf k}'=[\psi_{\uparrow\uparrow}({\bf k},-{\bf k}),
\psi_{\downarrow\downarrow}({\bf k},-{\bf k}),\psi_{\uparrow\downarrow}({\bf k},-{\bf k}),\\
\psi_{\uparrow\downarrow}({\bf -k},{\bf k})],$$ $M_{\bf k}$ is the matrix of eigenenergy minus kinetic energy and SOC, $$M_{\bf k}=\left[\begin{array}{cccc}
\mathcal{E}_{\bf k} & 0 & S^*({\bf k}_{\perp})& -S^*({\bf k}_{\perp}) \\
0&\mathcal{E}_{\bf k}& -S({\bf k}_{\perp})& S ({\bf k}_{\perp}) \\
S({\bf k}_{\perp}) & -S^*({\bf k}_{\perp})& \mathcal{E}_{\bf k}& 0 \\
-S({\bf k}_{\perp})& S^*({\bf k}_{\perp}) & 0 & \mathcal{E}_{\bf k}
\end{array}\right],$$ $\mathcal{E}_{\bf k}=E_0-2\epsilon_{\bf k}$, and G is the matrix of coupling constants, $$G=\left[\begin{array}{cccc}
g_{\uparrow\uparrow} & 0 & 0& 0 \\
0 & g_{\uparrow\uparrow} & 0 & 0 \\
0 & 0 & g_{\uparrow\downarrow}/2 & g_{\uparrow\downarrow}/2\\
0 & 0 & g_{\uparrow\downarrow}/2 & g_{\uparrow\downarrow}/2
\end{array}\right].$$ Define the vector $Q=G \sum_{\bf k}\psi_{\bf k}'/V$, from Eq. (\[eigeneq\]) we can obtain an equation for $Q$, $$\label{eigeneqp}
Q={1 \over V}G\sum_{\bf k}M^{-1}_{\bf k} Q.$$ Using the symmetry $S({\bf k}_{\perp})=-S(-{\bf k}_{\perp})$, we find that $$\sum_{\bf k}M^{-1}_{\bf k}=\sum_{\bf k}\det|M^{-1}_{\bf k}|\left[\begin{array}{cccc}
A_{\bf k} & 0 & 0& 0 \\
0 & A_{\bf k} & 0 & 0 \\
0 & 0 & A_{\bf k} & B_{\bf k}\\
0 & 0 & B_{\bf k} & A_{\bf k}
\end{array}\right],$$ where $A_{\bf k}=\mathcal{E}_{\bf k}^3-2\mathcal{E}_{\bf k}|S({\bf k}_{\perp})|^2$, $B_{\bf k}=-2\mathcal{E}_{\bf k}|S({\bf k}_{\perp})|^2$, and $\det|M_{\bf k}|=\mathcal{E}_{\bf k}^2[\mathcal{E}_{\bf k}^2-4|S({\bf k}_{\perp})|^2]$. Eq. (\[eigeneqp\]) has three different solutions, two intra-species bound states with $Q_3=Q_4=0$ and one inter-species bound state with $Q_1=Q_2=0$ and $Q_3=Q_4$. Due to the symmetry $\psi_{\downarrow\uparrow}({\bf k},-{\bf k})=\psi_{\uparrow\downarrow}(-{\bf k},{\bf k})$, the solutions always satisfy $Q_3=Q_4$ which is also guaranteed by the $G$-matrix elements $G_{34}=G_{43}=G_{33}=G_{44}$ in Eq. (\[eigeneqp\]). The $G$-matrix can also be chosen as a diagonal matrix with $G_{33}=G_{44}=g_{\uparrow\downarrow}$, but then the unphysical solution with $Q_3 \neq Q_4$ has to be taken out by hand.
For the two degenerate bound states at ${\bf q}=0$ created by the intra-species interaction, their eigenenergy $E_0$ is determined from the equation $1/g_{\uparrow\uparrow}=\sum_{\bf k}\det|M^{-1}_{\bf k}|A_{\bf k}/V$ which yields $$\frac{m}{4\pi \hbar^2 a}=\frac{1}{2V}\sum_{\bf k}[\frac{1}{\epsilon_{\bf
k}}-\frac{1}{2\epsilon_{\bf k}-E_0}-\frac{1}{4\xi_{{\bf k}+}-2E_0}
-\frac{1}{4\xi_{{\bf k}-}-2E_0}],
\label{EEE}$$ where the first r.h.s. term is due to $T$-matrix correction. Eq. (\[EEE\]) shows that these bound states are Rashbons which can exist with any intra-species interaction, whereas in a simple Bose gas without SOC two-body bound states only exists in the repulsive regime. The binding energy defined by $E_b=-E_0-
2\epsilon_{\kappa}$ is presented in Fig. \[fig1\](a) where $\epsilon_{\kappa}=\hbar^2 \kappa^2/2m$. When the intra-species interaction is tuned from attraction to repulsion, the binding energy monotonically increases with $1/(\kappa a)$. We find that in the limit of $\kappa a\rightarrow 0^-$, the binding energy has the asymptotic form $E_b\rightarrow 8\epsilon_\kappa\exp\{4[1/(\kappa a)-1]\}$; at resonance $1/a=0$, $E_b=0.132\epsilon_\kappa$, much smaller than that in the fermion case [@Yu11]; when $\kappa a\rightarrow 0^+$, $E_b\rightarrow\hbar^2/(m a^2)$, recovering the result of a dilute Bose gas without SOC.
The degeneracy of Rashbons at ${\bf q}=0$ is protected by time-reversal symmetry. One Rashbon wavefunctions is given by $$\begin{aligned}
\psi_{\uparrow \uparrow }({\bf k},-{\bf k})&=& \frac{\mathcal{N}}{\mathcal{E} _{{\bf k}}}\frac{\mathcal{E} _{{\bf k}}^2-2|S({\bf k}_{\perp})|^2}{\mathcal{E} _{{\bf k}}^2-4|S({\bf k}_{\perp})|^2}, \nonumber \\
\psi_{\downarrow \downarrow }({\bf k},-{\bf k})&=&-\frac{2 \mathcal{N}}{\mathcal{E} _{{\bf k}}}\frac{ S^2({\bf k}_{\perp})}{\mathcal{E} _{{\bf k}}^2-4|S({\bf k}_{\perp})|^2}, \nonumber \\
\psi_{\uparrow \downarrow }({\bf k},-{\bf k})&=&-\frac{\mathcal{N}S({\bf k}_{\perp})}{\mathcal{E} _{{\bf k}}^2-4|S({\bf k}_{\perp})|^2},
\label{intra1}\end{aligned}$$ where $\mathcal{N}$ is a normalization constant. The other Rashbon wavefunction can be obtained by time-reversal transformation $\psi'_{\sigma \sigma'}({\bf k},-{\bf k})=\psi^*_{-\sigma -\sigma'}(-{\bf k},{\bf k})$.
![Binding energy and effective masses of Rashbons. (a) Rashbon binding energy versus $1/(\kappa a)$ at ${\bf q}_\perp=0$. At resonance, it is given by $E_b=0.132 \epsilon_\kappa$. (b) In the limit ${\bf q}_\perp \rightarrow 0$, two Rashbon effective masses can be obtained from Rashbon binding energies.[]{data-label="fig1"}](Fig1.eps){width="0.99\columnwidth"}
Rashbon appearance in the attractive regime is due to the increase in atom DOS at low energies by SOC. The density of state of the lower helicity excitation $\xi_{{\bf k}-}$ is a constant at energy minimum $\xi_{{\bf k}-}=-\epsilon_\kappa$ for $k_\perp=\kappa$ and $k_z=0$, which leads to an infrared divergence at zero binding energy on r.s.h. of Eq. (\[EEE\]) and consequently Rashbon appearance in the attractive regime. Rashbons in the weakly attractive regime may be helpful for experimental observation. Since the system is far away from resonance, the particle loss rate due to inelastic collision may be suppressed.
At ${\bf q}_\perp\neq0$, the bound-state eigenergy problem cannot be reduced to a simple equation. We numerically solve for bound state energies, and find that the two Rashbons have two different effective masses, as shown in Fig. \[fig1\] (b). The lift of Rashbon degeneracy is not surprising, because two Rashbons are no longer connected by time-reversal symmetry at finite ${\bf q}_\perp$ and the Rashbon degeneracy is no longer protected by time-reversal symmetry. The two Rashbon effective masses behave differently with the intra-species scattering length $a$. The bigger effective mass $m^*_+$ reaches maximum at resonance, while the smaller effective mass $m^*_-$ decreases monotonically with $1/(\kappa a)$. In the limit $a \rightarrow 0^-$, we obtain $m^*_+=8m$ and $m^*_-=8m/3$; at resonance, $m^*_+= 9.29m$ and $m^*_-=2.36m$; in the limit $a \rightarrow 0^+$, both effective masses recover the results without SOC, $m^*_\pm\rightarrow 2m$.
When the in-plane momentum $\hbar q_\perp$ exceeds a critical value $\hbar q_c$, the Rashbon dissociates into excited atoms. We find that the critical wavevector $q_c$ is different for different Rashbons, approximately satisfying the condition for Rashbon dissociation in the effective-mass approximation, $E_0+\hbar^2q_{c\pm}^2/(2m^*_\pm)\approx-2\epsilon_\kappa$. The critical momenta vanish in the limit of weakly attractive interaction $a \rightarrow 0^-$, and diverge in the opposite limit $a \rightarrow 0^+$.
For the bound state created by the inter-species interaction, its eigenenergy at ${\bf q} = 0$ is given by $1/g_{\uparrow\downarrow}=\sum_{\bf k}\det|M^{-1}_{\bf k}|(A_{\bf k}+B_{\bf k})/V$, yielding $$\frac{m}{4\pi \hbar^2 a'}=\frac{1}{V}\sum_{\bf k} (\frac{1}{2\epsilon_{\bf
k}}+\frac{1}{E_0-2\epsilon_{\bf k}})
\label{inter1}$$ which is the same as that without SOC and gives the same result $E_b=\hbar^2/(m {a'}^2)$, whereas in the fermion case the inter-species bound state is strongly affected by SOC [@Yu11]. The wave function of this bound state is also the same as that without SOC, $$\begin{aligned}
\psi_{\uparrow\uparrow}({\bf k},-{\bf k})&=&\psi_{\downarrow\downarrow}({\bf k},-{\bf k})=0, \nonumber \\
\psi_{\uparrow\downarrow}({\bf k},-{\bf k})&=&\frac{\mathcal{N'}}{\mathcal{E} _{\bf k}},
\label{inter2}\end{aligned}$$ where $\mathcal{N'}$ is a normalization factor.
The qualitative difference between Rashbon and the inter-species bound state can be explained in terms of symmetries of their wavefunctions as given in Eq. (\[intra1\]) and(\[inter2\]). The bound state created by the inter-species interaction consists of s-wave pairs of atoms with different helicities, whereas in Rashbon two atoms are either with the same helicity or p-wave symmetrized with different helicities. In consequence, the binding energy of the bound state created by the inter-species interaction depends on DOS of the pair energy of different helicities $\xi_{{\bf k}+}+\xi_{{\bf k}-}=2\epsilon_{ k}$ which is independent of SOC. Thus SOC has no effect on the binding energy of the bound state created by the inter-species interaction. In contrast, the Rashbon binding energy depends on not only DOS of pair energy of different helicities, but also DOS of pair energy of the same helicity which is half of the atom DOS with the same helicity. The atom DOS is a constant at the lowest energy $-\epsilon_\kappa$ producing an infrared divergence at zero binding energy on r.s.h. of Eq. (\[EEE\]). The change of atom DOS by SOC is responsible for the Rashbon existence in the attractive regime. For comparison, in the fermionic case [@Vya1; @Yu11], there is no s-wave intra-species interaction due to Fermi-Dirac statistics, and the inter-species bound state consists of p-wave pairs of atoms with the same helicity. The bound state is a Rashbon because of the DOS effect due to SOC.
[*Rashbon condensation*]{}. Rashbons are composite bosons obeying Bose-Einstein statistics. We consider the possibility of Bose-Einstein condensation of Rashbons in a Bose gas with Rashba SOC. The Rashbon condensation can be described by pairing order parameters $\Delta_{\uparrow\uparrow}=g_{\uparrow\uparrow}\sum_{\bf k}\langle c_{-{\bf
k}\uparrow}c_{{\bf k}\uparrow}\rangle/V$ and $\Delta_{\downarrow\downarrow}=g_{\downarrow\downarrow}\sum_{\bf k}\langle c_{-{\bf
k}\downarrow}c_{{\bf k}\downarrow}\rangle/V$. In general, if inter-species bound states condense, another pairing order parameter $\Delta_{\uparrow\downarrow}=g_{\uparrow\downarrow}\sum_{\bf k}\langle c_{-{\bf
k}\uparrow}c_{{\bf k}\downarrow}\rangle/V$ needs to be introduced. Rashbon condensation is not directly coupled to the condensation of inter-species bound states. In the dilute limit with weakly attractive intra-species interaction and repulsive inter-species interaction, the Rashbon binding energy is much smaller than the binding energy of the inter-species bound state. In the following, we consider the system with only Rashbon condensation and focus on the spin-balanced case, $g_{\uparrow\uparrow}=g_{\downarrow\downarrow}$ and $|\Delta_{\uparrow\uparrow}|=|\Delta_{\downarrow\downarrow}|=\Delta$. In general there may be a phase difference between $\Delta_{\uparrow\uparrow}$ and $\Delta_{\downarrow\downarrow}$. Without losing generality we define $\Delta_{\uparrow\uparrow}=e^{i\theta}\Delta$, $\Delta_{\downarrow\downarrow}=e^{-i\theta}\Delta$ and $\Delta>0$. The mean-field Hamiltonian of the Rashbon condensation phase is given by $$\frac{H_{MF}}{V}=\frac{1}{2V}\sum_{\bf k}\{ B^+_{\bf k} H_{\bf k}B_{\bf
k}-2\xi_{\bf k}\}-\frac{\Delta^2}{g_{\uparrow\uparrow} }-(2g_{\uparrow\uparrow}+g_{\uparrow\downarrow})n^2,$$ where $B^+_{\bf k}$ is the field operator with four components $B^+_{\bf k}=[c^\dag_{{\bf k}\uparrow},c_{{\bf
-k}\uparrow},c^\dag_{{\bf k}\downarrow},c_{{\bf -k}\downarrow}]$, $n$ is the density of each spin component, the matrix $H_{\bf k}$ is given by $$\begin{aligned}
\label{MFH}
H_{\bf k}&=&\left[\begin{array}{cccc}
\xi_{\bf k} & \Delta_{\uparrow\uparrow} & S^*({\bf k}_{\perp})& 0 \\
\Delta^*_{\uparrow\uparrow}& \xi_{\bf k} & 0 & -S({\bf k}_{\perp}) \\
S({\bf k}_{\perp}) & 0 & \xi_{\bf k}& \Delta_{\downarrow\downarrow} \\
0 & -S^*({\bf k}_{\perp}) & \Delta^*_{\downarrow\downarrow} & \xi_{\bf k}
\end{array}\right],\end{aligned}$$ $\xi_{\bf k}=\epsilon_{\bf k}-\mu+2g_{\uparrow\uparrow}n+g_{\uparrow\downarrow}n$, and $\mu$ is chemical potential.
![Pairing order parameter $\Delta$ versus $1/(\kappa a)$ for different densities in the dilute limit $\kappa \gg n^{1/3}$. For fixed $\kappa$, the order parameter $\Delta$ increases monotonously with $n$ and $1/a$. At resonance, for $ \kappa / n^{1/3}=60$, the order parameter $\Delta=0.0075\epsilon_\kappa$ is much smaller than the binding energy $E_b=0.132\epsilon_\kappa$.[]{data-label="OP"}](Fig2.eps){width="0.95\columnwidth"}
The mean-field Hamiltonian Eq. (\[MFH\]) can be diagonalized by generalized Bogoliubov transformation. The single-particle excitations form two branches with excitation energies given by $$\label{Ek}
\varepsilon_{{\bf k}\pm}=\left[\xi^2_{\bf k}+|S({\bf k}_{\perp})|^2-\Delta^2\pm 2
|S({\bf k}_{\perp})|\sqrt{\xi^2_{\bf k}-\Delta^2\cos^2\varphi_{\bf k}}\right]^\frac{1}{2},$$ where $\varphi_{\bf k}=\phi_{\bf k}+\theta$ and $\phi_{\bf k}=\arg(k_x+i k_y)$. The pairing order parameters and density can be obtained self-consistently, yielding the following equations at zero temperature $$\begin{aligned}
\frac{1}{ g_{\uparrow\uparrow}}&=&\frac{1}{4V}\sum_{\bf k}[\frac{2}{\epsilon_{\bf k}}-\frac{1
}{\varepsilon_{{\bf k}+}}-\frac{ 1}{\varepsilon_{{\bf k}-}}-\frac{|S({\bf k}_{\perp})|\cos^2\varphi_{\bf k}
}{\sqrt{\xi^2_{\bf k}-\Delta^2\cos^2\varphi_{\bf
k}}}(\frac{1 }{\varepsilon_{{\bf k}+}}-\frac{1}{ \varepsilon_{{\bf k}-}})], \label{Gap}\\
n&=&\frac{1}{4V}\sum _{\bf k} [\frac{\xi _{\bf k} }{\varepsilon_{{\bf k}-}}(1-\frac{|S({\bf k}_{\perp})|}{\sqrt{\xi^2_{\bf k} -\Delta^2\cos^2\varphi_{\bf k} }})\nonumber +\frac{\xi _{\bf k} }{\varepsilon_{{\bf k}+}}(1+\frac{|S({\bf k}_{\perp})|}{\sqrt{\xi^2_{\bf k} -\Delta^2\cos^2\varphi_{\bf k} }})-2].
\label{Num}\end{aligned}$$ We numerically solve Eq. (\[Gap\]) and find that the mean-field solution always exist in the dilute limit $n\rightarrow 0$, as shown in Fig. \[OP\].
![Anisotropy of lower quasi-particle excitation energy $\varepsilon_{{\bf k}-}$ at $\mu'/\epsilon_\kappa=-1.2$, $\Delta /\epsilon_\kappa=0.18$ and $k_z=0$, where $\mu'=\mu-2g_{\uparrow\uparrow}n-g_{\uparrow\downarrow}n$. (a) $\varepsilon_{{\bf k}-}$ along x-axis (solid line) and y-axis (dash line) are plotted as functions of $k_\perp/\kappa$ for $\theta=0$. The anisotropy is stronger at low energies and weaker at higher energies. (b) $\varepsilon_{{\bf k}-}$ at $k_\perp=\kappa$ versus $\phi_{\bf k}$ for different $\theta$.[]{data-label="Three"}](Fig3.eps){width="0.99\columnwidth"}
In Rashbon condensation, quasi-particle excitation energies given in Eq. (\[Ek\]) are anisotropic, dependent on the angle $\varphi_{\bf k}=\phi_{\bf k}+\theta$. This anisotropy is stronger at low energies when $k_\perp$ is near $\kappa$, as shown in Fig. \[Three\](a). At higher energies, the anisotropy becomes weaker and eventually disappears. This anisotropic effect is caused by the coupling between pairing order parameters $\Delta _{\uparrow\uparrow}$ and $\Delta_{\downarrow\downarrow}$ due to SOC. For a spin-up atom with wave-vector ${\bf k}$, SOC can flip its spin down with a phase $\phi_{\bf k}$. This phase becomes $\phi_{\bf k}+\pi$ for the spin-up atom with opposite wave-vector $-{\bf k}$. These two spin-flips can turn an atom pair from total spin-up to total spin-down states with phase $2\phi_{\bf k}+\pi$. If $2\phi_{\bf k}+\pi+2\theta=2l\pi$ where $l$ is an integer, spin-flips are encouraged and the quasi-particle energy $\varepsilon_{{\bf k}-}$ is at minimum. If $2\phi_{\bf k}+2\theta=2l\pi$, spin-flips are discouraged and the quasi-particle energy is at maximum. As shown in Fig. \[Three\](b), the quasi-particle energy $\varepsilon_{{\bf k}-}$ shows a periodic behavior as a function of $\phi_{\bf k}$ with period $\pi$.
In the following, we focus on Rashbon condensation in the dilute limit with attractive intra-species interaction, $\kappa \gg n^{1/3}$ and $(-a)^{-1} \gg n^{1/3}$. Since in the dilute limit the distance between Rashbons is the largest length scale, the structure of Rashbons is not affected by the weak interaction between Rashbons, which is very similar to the BEC limit of BEC-BCS crossover in Fermi gases. In this limit, Eq. (\[Gap\]) can be solved analytically, and we find that the order parameter $\Delta $ is much smaller than Rashbon binding energy, $$\Delta \approx 4\sqrt{2\pi}(\frac{n}{\kappa ^{3}})^\frac{1}{2}(\epsilon_\kappa E_b)^\frac{1}{2}\ll E_b.$$ The attractive intra-species interaction tends to make the system unstable. If the Rashbon condensation is stable, the positive compressibility condition $\partial \mu/\partial n>0$ must be satisfied. We find that in the dilute limit this stability condition is given by $\kappa(a'+2a)>3/2$. Therefore a repulsive inter-species interaction with $\kappa>3/(2a'+4a)\gg n^{1/3}$ is required to stabilize the Rashbon condensation in a dilute Bose gas with Rashba SOC.
In the Rashbon condensation phase, in addition to single-particle excitations, there are also pair excitations. At the transition temperature $T_c$ of Rashbon condensation, pair excitations are quadratically dispersed. In the dilute limit with attractive intra-species interaction, they have effective masses approximately as same as those of Rashbons in vacuum. Since in this limit the Rashbon binding energy is much bigger than $k_B T_c$, single-particle excitations can be neglected at $T_c$, and only excited Rashbons contribute to the density at $T_c$, $$\label{Den}
n=\frac{1}{V}\sum_{{\bf q},s}{'}\frac{1}{e^{\beta (E_{{\bf
q}s}-E_0)}-1},$$ where $s=\pm$, $E_{{\bf q}\pm} \approx E_0+\hbar^2q^2_z/(4m)+\hbar^2q^2_\perp/(2m^*_\pm)$ are Rashbon energies in the effective mass approximation, and $\sum{'}$ denotes the summation over ${\bf q}$ for $|q_\perp| \leq q_{c\pm}$. From Eq. (\[Den\]), we obtain the transition temperature $$\label{Tc}
T_c=[\sqrt{2}(m^*_++m^*_-)/m]^{-\frac{2}{3}}T_a\approx 0.164 T_a,$$ where $T_a=2\pi \zeta^{-\frac{2}{3}}(\frac{3}{2})\hbar^2n^{2/3}/(k_B m)$ is the critical temperature of an ideal Bose gas and $\zeta(x)$ is the Riemann zeta function. Eq. (\[Tc\]) shows that the transition temperature of Rashbon condensation $T_c$ in the dilute limit is about six times smaller than the BEC transition temperature of an ideal Bose gas.
In current experiments in $^{87}$Rb, the strength of Rashba SOC is limited by the wavelength of the Raman laser $\lambda=804.1$nm, $\kappa\leq7.8\times10^6$ m$^{-1}$ [@Lin11]. With background intra-species scattering length $a_{bg}=100a_0$ and density of the order of $10^{13}$ cm$^{-3}$ [@Long13], the dilute region of Rashbon condensation is hardly reachable. With the new proposal to generate Rashba SOC [@And13; @Ken13], if $\kappa$ can be enhanced to $2\times10^8$ m$^{-1}$ and scattering lengths can be tuned to $a=-95a_0$ and $a'>330a_0$, Rashbon condensation may be observed around 29nK with $n=10^{13}$ cm$^{-3}$ in $^{87}$Rb.
[*Discussion and conclusion.*]{} We have shown that Rashbon condensation can be mechanically stable in a dilute Bose gas with Rashba SOC and weakly attractive intra-species interaction. In this dilute region, we expect that the particle loss rate is suppressed because of its density dependence. As shown in experiments on $^{85}$Rb in the dilute region [@Donley; @Thompson], the loss rate of Feshbach molecules is much smaller than the molecule binding energy. Now with the help of Rashba SOC, the Rashbon binding energy is exponentially small, and the lifetime of dilute Rashbon condensation is expected to be long enough for experimental observations.
There are a lot of interesting questions to be answered about Rashbon condensation. Collective excitations in this phase are worth to explore. Another important question is whether or not at a higher density there is a quantum phase transition between Rashbon condensation and mixture of atom and Rashbon condensates. We plan to address these issues in future studies.
In summary, we find that two Bose atoms with Rashba SOC can form a Rashbon with any intra-species interaction. In contrast, the bound state created by the inter-species interaction is not affected by SOC. At zero center-of-mass momentum there are two degenerate Rashbons with the degeneracy protected by time-reversal symmetry. The degeneracy is lifted at finite in-plane momentum with two different effective masses. We explore the possibility of Rashbon condensation in a dilute Bose gas with Rashba SOC and attractive intra-species interaction. We find that Rashbon condensation can be stabilized by a repulsive inter-species interaction. In Rashbon condensation, the single-particle excitation energy is anisotropic, due to coupling between pairing order parameters by SOC. The transition temperature of Rashbon condensation is about six times smaller than that of BEC in an ideal Bose gas.
[**Acknowledgement**]{}. We would like to thank Z. Q. Yu, W. Zhang, and T.-L. Ho for helpful discussions. This work is supported by NSFC under Grant No 11274022.
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|
---
abstract: 'We present a fast solver for the 3D high-frequency Helmholtz equation in heterogeneous, constant density, acoustic media. The solver is based on the method of polarized traces, coupled with distributed linear algebra libraries and pipelining to obtain an empirical online runtime $ \cO(\max(1,R/n) N \log N)$ where $N = n^3$ is the total number of degrees of freedom and $R$ is the number of right-hand sides. Such a favorable scaling is a prerequisite for large-scale implementations of full waveform inversion (FWI) in frequency domain.'
address: |
Massachusetts Institute of Technology,\
Department of Mathematics and Earth Resources Laboratory,\
77 Massachusetts Ave,\
Cambridge, MA 02139.\
Université Catholique de Louvain,\
1, Place de l’Université,\
B-1348 Louvain-la-Neuve, Belgium\
Total E. & P. Research & Technology USA\
1201 Louisiana St.,\
Houston, TX 77002.\
Computational Research Division,\
Lawrence Berkeley National Laboratory,\
1 Cyclotron road,\
Berkeley, CA 94720.\
University of California Irvine,\
Department of Mathematics,\
540 Rowland Hall,\
Irvine, CA 92963.
author:
- 'Leonardo Zepeda-Núñez, Adrien Scheuer, Russell J. Hewett, and Laurent Demanet'
bibliography:
- 'GRF\_integral\_formulations.bib'
date: December 2017
title: The method of polarized traces for the 3D Helmholtz equation
---
Introduction {#section:introduction}
============
Efficient modeling of time-harmonic wave scattering in heterogeneous acoustic or elastic media remains a difficult problem in numerical analysis, yet it has broad application in seismic inversion techniques, as shown by [@Chen:Inverse_scattering_via_Heisenberg's_uncertainty_principle; @Pratt:Seismic_waveform_inversion_in_the_frequency_domain;_Part_1_Theory_and_verification_in_a_physical_scale_model; @Virieux_Operto:An_overview_of_full-waveform_inversion_in_exploration_geophysics]. In the constant density acoustic approximation, time-harmonic wave propagation is modeled by the Helmholtz equation, $$\triangle u({\mathbf{x}}) + \omega^2 m ({\mathbf{x}}) u({\mathbf{x}}) = f_s({\mathbf{x}}), \qquad \mbox{in } \Omega \label{eq:Helmholtz}$$ with absorbing boundary conditions, and where $\Omega $ is a $3$D rectangular domain, $\triangle$ is the 3D Laplacian, ${\mathbf{x}}= (x,y,z)$, $m = 1/c^2({\mathbf{x}})$ is the squared slowness for velocity $c({\mathbf{x}})$, $u$ is the wavefield, and $f_s$ are the sources, indexed by $s = 1,..., R$. There is no essential obstruction to extending the techniques presented in this paper to the case of heterogeneous density, or to the elastic, viscoelastic or poroelastic time-harmonic equations.
Throughout this paper we assume that Eq. \[eq:Helmholtz\] is in the high-frequency regime, i.e., when $\omega \sim n$, where $n$ is the number of unknowns in each dimension. Alternatively, in the situation where the domain $\Omega$ grows and the frequency is fixed, the problem can be rescaled in such a way that $\omega$ grows in proportion to the domain size, which can also considered to be a “high-frequency” regime. For this paper, we focus on the former scenario.
Given the importance of solving Eq. \[eq:Helmholtz\] in geophysical contexts, there has been a renewed interest in developing efficient algorithms to solve the ill-conditioned linear system resulting from its discretization. Recent progress toward an efficient solver, i.e., a solver with linear complexity, has generally focused on three strategies:
- *Fast direct solvers*, such as the ones introduced by [@Xia:multifrontal; @Wang:H_multifrontal; @Gillman_Barnett_Martinsson:A_spectrally_accurate_solution_technique_for_frequency_domain_scattering_problems_with_variable_media; @Amestoy_Weisbecker:compressed_MUMPS], which couple multifrontal techniques (e.g., [@GeorgeNested_dissection; @Duff_Reid:The_Multifrontal_Solution_of_Indefinite_Sparse_Symmetric_Linear]) with compressed linear algebra (e.g., [@Bebendorf:2008]) to obtain efficient direct solvers with small memory footprint. However, they suffer the same sub-optimal asymptotic complexity as standard multifrontal methods (e.g., [@Demmel_Li:superlu; @Amestoy_Duff:MUMPS; @Davis:UMFPACK]) in the high-frequency regime.
- *Classical preconditioners*, such as [incomplete factorization preconditioners]{} (e.g., [@Grote_Schenk:algebraic_multilever_preconditioner_Helmholtz_equation]) and [multigrid-based preconditioners]{} (e.g., [@Brandt_Livshits:multi_ray_multigrid_standing_wave_equations; @Erlangga:shifted_laplacian; @Sheikh_Lahaye_Vuik:On_the_convergence_of_shifted_Laplace_preconditioner_combined_with_multilevel_deflation; @Calandra_Grattonn:an_improved_two_grid_preconditioner_for_the_solution_of_3d_Helmholtz]), which are relatively simple to implement but suffer from super-linear asymptotic complexity and may need significant tuning to achieve effective run-times.
- *Sweeping-like preconditioners* (e.g., [@GanderNataf:LU_incomplete; @EngquistYing:Sweeping_H; @EngquistYing:Sweeping_PML; @Chen_Xiang:a_source_transfer_ddm_for_helmholtz_equations_in_unbounded_domain; @Cheng_Xiang:A_Source_Transfer_Domain_Decomposition_Method_For_Helmholtz_Equations_in_Unbounded_Domain_Part_II_Extensions; @CStolk_rapidily_converging_domain_decomposition; @GeuzaineVion:double_sweep; @Liu_Ying:Recursive_sweeping_preconditioner_for_the_3d_helmholtz_equation; @ZepedaDemanet:the_method_of_polarized_traces]), which are a relatively recent domain decomposition based approach that has been shown to achieve linear or nearly-linear asymptotic complexity.
The method in this paper belongs to the third category. Sweeping preconditioners and their generalizations, i.e., domain decomposition techniques coupled with high-quality transmission/absorption conditions, offer the right mix of ideas to attain linear or near-linear complexity in 2D and 3D, provided that the medium does not have large resonant cavities [@ZepedaDemanet:the_method_of_polarized_traces]. These methods rely on the sparsity of the linear system to decompose the domain in layers, in which classical sparse direct methods are used to compute the interactions within the layer. Interactions across layers are computed by sequentially sweeping through the sub-domains in an iterative fashion.
For current applications, empirical runtimes are a more practical measure of an algorithm’s performance than asymptotic complexity. This requirement has led to a recent effort to reduce the runtimes of preconditioners with optimal asymptotic complexity by leveraging parallelism. For example, @Poulson_Engquist:a_parallel_sweeping_preconditioner_for_heteregeneous_3d_helmholtz introduce a new local solver, i.e., a solver for the subproblem defined on each layer, which carefully handles communication patterns between layers to obtain impressive timings. While most sweeping algorithms require visiting each subdomain in sequential fashion, [@Stolk:An_improved_sweeping_domain_decomposition_preconditioner_for_the_Helmholtz_equation] introduced a modified sweeping pattern, which changes the data dependencies during the sweeps to improve parallelism. Finally, @ZepedaDemanet:the_method_of_polarized_traces introduced the method of polarized traces, which reduces the solver’s run-time by leveraging parallelism and fast summation methods. This paper builds on top of the general framework of the method of polarized traces, which we elaborate on in the sequel.
To date, most studies focus on minimizing the parallel runtime or complexity of a single solve with a single right-hand side. However, in the scope of seismic inversion, where there can be many thousands of right-hand sides, it is important to consider the overall runtime or complexity of solving *all* right-hand sides. In this context, linear complexity is $\cO(RN)$, where $N$ is the total number of degrees of freedom (we assume that $N = n^3$ and let $n$ be the number of degrees of freedom in a single dimension of a 3D volume) and $R$ is the number of right-hand sides.
In this paper, we present a solver for the 3D high-frequency Helmholtz equation with a [*sublinear*]{} online parallel runtime, given by $$\cO( \alpha_{\text{pml}}^2 \max(1,R/L) N \log{N}),$$ where $N = n^3$ is the total number of unknowns, $L \sim n$ is the number of subdomains in a layered domain decomposition, and $\alpha_{\text{pml}}$ is the number of points needed to implement a high-quality absorbing boundary condition between layers. We achieve this complexity by comprehensive parallelization of all aspects of the algorithm, including exploiting parallelism in local solves and by pipelining the right-hand sides. Thus, as long as $R \sim n^2$ (3D), there is a mild $R/L \sim n$ factor impacting the asymptotic complexity. The solver in this paper is based on the method of polarized traces [@ZepedaDemanet:the_method_of_polarized_traces], a layered domain decomposition method which exploits:
- local solvers, using efficient sparse direct solvers at each subdomain,
- high-quality transmission conditions between subdomains, implemented via perfectly-matched layers (PML; [@Berenger:PML; @Johnson:PML]), and
- an efficient preconditioner based on polarizing conditions imposed via incomplete Green’s integrals.
These concepts combine to yield a global iterative method that converges in a small number of iterations. The method has two stages: an offline stage, that can be precomputed independently of the right-hand sides, and an online stage, that is computed for each right-hand side or by batch processing.
One advantage of the method of polarized traces is that only the degrees of freedom at the interfaces between layers are needed for the bulk of the computation, because the volume problem is reduced to an equivalent surface integral equation (SIE) at the interfaces between layers. Due to this efficiency, the algorithm requires a smaller memory footprint, which helps make feasible pipelining the right-hand sides. Pipelining for domain decomposition methods has been previously considered [@Stolk:An_improved_sweeping_domain_decomposition_preconditioner_for_the_Helmholtz_equation], albeit without complexity claims and without a fully tuned communication strategy between subdomains. Moreover, we are unaware of any complexity claims within the context of inversion algorithms, in particular, with focus on full waveform inversion [@Taratola:Inversion_of_seismic_reflection_data_in_the_acoustic_approximation], where there are many right-hand sides that are strongly frequency dependant.
Complexity claims
-----------------
Suppose that $L$, the number of layers in the domain decomposition, scales as $L \sim n$, i.e., each layer has a constant thickness in number of grid points[^1]. Each layer is further extended by $\alpha_{\text{pml}}$ grid points in order to implement the PML. It has been documented in [@Poulson_Engquist:a_parallel_sweeping_preconditioner_for_heteregeneous_3d_helmholtz] that $\alpha_{\text{pml}}$ needs to grow with problem frequency, $\alpha_{\text{pml}} \sim \log\omega$, in order to obtain a number of Krylov solver iterations to convergence that scales as $\log \omega$. Additionally, as above, for the 3D problem it is typical that the number of sources $R \sim n^2$, because as frequency and resolution increase, the number of sources in both the in-line and cross-line direction must also increase [@Brossier:2D_and_3D_frequency-domain_elastic_wave_modeling_in_complex_media_with_a_parallel_iterative_solver].
Finally, given that we solve the 3D problem in a high-performance computing (HPC) environment, we assume that the number of computing nodes in the HPC cluster is $\cO(n^3 \log(n)/M)$, with $L \sim n$ layers, $\cO(n^2\log(n)/M)$ nodes inside each layer, and $M$ is the memory of a single node. The assumptions on node growth come from the fact that computing nodes have finite memory, and thus more nodes are needed to solve larger problems. As numerical examples will show, using more nodes per layer reduces the runtime per Krylov iteration by enhancing the parallelism of the solves at each subdomain, provided that a carefully designed communication pattern is used, to keep the communication overhead low.
We summarize the asymptotic runtimes in Table \[table:complexity\]. Like other related methods, the offline stage is linear in $N$ and is independent of the number of right-hand sides. The online runtime is sub-linear, in the sense that linear complexity would be $\cO(RN)$.
Stage Polarized traces
--------- -------------------------------------------------------
offline $\cO\left(\alpha_{\text{pml}}^3 N \right)$
online $\cO( \alpha_{\text{pml}}^2 \max(1,R/L) N \log{N}) $
: Runtime of both stages of the algorithm. Note that typically $ \alpha_{\text{pml}} \sim \log \omega$.[]{data-label="table:complexity"}
Related work {#section:related_work}
------------
Modern linear algebra techniques, in particular nested dissection methods [@GeorgeNested_dissection] coupled with $\cH$-matrices [@Hackbusch:Hierarchical_matrices] have been applied to the Helmholtz problem, yielding, for example: the hierarchical Poincaré-Steklov solver [@Gillman_Barnett_Martinsson:A_spectrally_accurate_solution_technique_for_frequency_domain_scattering_problems_with_variable_media], solvers using hierarchical semi-separable (HSS) matrices [@Wang:H_multifrontal; @Wang_de_Hoop:Massively_parallel_structured_multifrontal_solver_for_time-harmonic_elastic_waves_in_3D_anisotropic_media; @Wang_Li_Sia_Situ_Hoop:Efficient_Scalable_Algorithms_for_Solving_Dense_Linear_Systems_with_Hierarchically_Semiseparable_Structures], or block low-rank (BLR) matrices [@Amestoy_Weisbecker:compressed_MUMPS; @Amestoy:Fast_3D_frequency_domain_full_waveform_inversion_with_a_parallel_block_low-rank_multifrontal_direct_solver_Application_to_OBC_data_from_the_North_Sea].
Multigrid methods, once thought to be inefficient for the Helmholtz problem, have been successfully applied to the Helmholtz problem by @Calandra_Grattonn:an_improved_two_grid_preconditioner_for_the_solution_of_3d_Helmholtz and @Stolk. Although these algorithms do not result in a lower computational complexity, their empirical run-times are impressive due to their highly parallelizable nature. Due to the possibility for efficient parallelization, there has been a renewed interest on multilevel preconditioners such as the one in [@Hu_Zhang:Substructuring_Preconditioners_for_the_Systems_Arising_from_Plane_Wave_Discretization_of_Helmholtz_Equations].
Within the geophysical community, the analytic incomplete LU (AILU) method was explored by @Plessix_Mulder:Separation_of_variable_preconditioner_for_iterativa_Helmholtz_solver and applied in the context of 3D seismic imaging, resulting in some large computations [@Plessix:A_Helmholtz_iterative_solver_for_3D_seismic_imaging_problems]. A variant of Kazmarc preconditioners [@Gordon:A_robust_and_efficient_parallel_solver_for_linear_systems] have been studied and applied to time-harmonic wave equations by @Brossier:2D_and_3D_frequency-domain_elastic_wave_modeling_in_complex_media_with_a_parallel_iterative_solver. Although these solvers have, in general, relatively low memory consumption they tend to require many iterations to converge, thus hindering practical run-times.
Domain decomposition methods for solving PDEs date back to [@Schwarz:Uber_einen_Grenzubergang_durch_alternierendes_Verfahren], in which the Laplace equation is solved iteratively (for a more recent treatise, see @Lions:on_the_Schwarz_alternating_method_I). The application of domain decomposition to the Helmholtz problem was first proposed by @Despres:domain_decomposition_hemholtz. [@Cessenat_Despres:application_of_an_ultra_weak_variational_formulation_of_elliptic_pdes_to_the_2_d_helmholtz_problem] further refined this approach with the development of the ultra-weak variational formulation (UWVF) for the Helmholtz equation, in which the basis functions in each element, or sub-domain, are solutions to the local homogeneous equation. The UWVF approach motivated a series of related methods, such as the partition of unity method of @babuska_melenk:partition_of_unity_method, the least squared method of @Monk_Wang:A_least-squares_method_for_the_Helmholtz_equation:, the discontinuous enrichment method by @Farhat:The_discontinuous_enrichment_method, and Trefftz methods by @Gittelson_Hipmair_Perugia:Trefftz, and @Perugia:trefft, among many others. A recent and thorough review of Trefftz and related methods can be found in ([@Hiptmair_Moiola_Perugia:A_Survey_of_Trefftz_Methods_for_the_Helmholtz_Equation]).
The results in @Lions:on_the_Schwarz_alternating_method_I and @Despres:domain_decomposition_hemholtz have inspired the development of various domain decomposition algorithms, which are now classified as Schwarz algorithms[^2]. However, the convergence rate of such algorithms is strongly dependent on the boundary conditions prescribed at the interfaces between subdomains. [@Gander_Nataf:Optimized_Schwarz_Methods_without_Overlap_for_the_Helmholtz_Equation] introduces an optimal, non-local boundary condition for domain interfaces, which is then approximated by an optimized Robin boundary condition. This last work lead to the introduction of the framework of optimized Schwarz methods in [@Gander:Optimized_Schwarz_Methods] to described optimized boundary conditions that provides high convergence. The design of better interface approximations has been studied in [@Gander_Kwok:optimal_interface_conditiones_for_an_arbitrary_decomposition_into_subdomains; @Geuzaine:A_quasi-optimal_nonoverlapping_domain_decomposition_algorithm_for_the_Helmholtz_equation; @Gander_Zhang:Domain_Decomposition_Methods_for_the_Helmholtz_Equation:_A_Numerical_Investigation; @Gander:Optimized_Schwarz_Methods_with_Overlap_for_the_Helmholtz_Equation; @Gander:Optimized_Schwarz_Method_with_Two_Sided_Transmission_Conditions_in_an_Unsymmetric_Domain_Decomposition] among many others.
[@Engquist_Zhao:Absorbing_boundary_conditions_for_domain_decomposition] introduced absorbing boundary conditions for domain decomposition schemes for elliptic problems and the first application of such techniques to the Helmholtz problem traces back to the AILU factorization ([@GanderNataf:ailu_for_hemholtz_problems_a_new_preconditioner_based_on_an_analytic_factorization]). The sweeping preconditioner, introduced in [@EngquistYing:Sweeping_H; @EngquistYing:Sweeping_PML], was the first algorithm to show that those ideas could yield algorithms with quasi-linear complexity. There exists two variants of the sweeping preconditioner which involved using either $\cH$-matrices [@EngquistYing:Sweeping_H] or multi-frontal solvers [@EngquistYing:Sweeping_PML] to solve the local problem in each thin layer. These schemes are extended to different discretizations and physics by [@Tsuji_engquist_Ying:A_sweeping_preconditioner_for_time-harmonic_Maxwells_equations_with_finite_elements; @Tsuji_Poulson:sweeping_preconditioners_for_elastic_wave_propagation; @Tsuji_Ying:A_sweeping_preconditioner_for_Yees_finite_difference_approximation_of_time-harmonic_Maxwells_equations]. Since the introduction of the sweeping preconditioner, several related algorithms with similar claims have been proposed, such as the source transfer preconditioner ([@Chen_Xiang:a_source_transfer_ddm_for_helmholtz_equations_in_unbounded_domain; @Cheng_Xiang:A_Source_Transfer_Domain_Decomposition_Method_For_Helmholtz_Equations_in_Unbounded_Domain_Part_II_Extensions]), the rapidly converging domain decomposition ([@CStolk_rapidily_converging_domain_decomposition]) and its extensions ([@Stolk:An_improved_sweeping_domain_decomposition_preconditioner_for_the_Helmholtz_equation]), the double sweep preconditioner ([@GeuzaineVion:double_sweep]) and the method of polarized traces ([@ZepedaDemanet:the_method_of_polarized_traces]).
Organization
------------
The remainder of this paper is organized as follows: we provide the numerical formulation of the Helmholtz equation, present the reduction to a surface integral equation, and introduce the method of polarized traces for solving the SIE. Next, we elaborate on the parallelization and communication patterns and examine the empirical complexities and runtimes. Finally, we provide results from several experiments to support our claims.
Formulation {#section:formulation}
===========
For this study, we discretize Eq. \[eq:Helmholtz\] using the standard second order finite difference method on a regular mesh of $\Omega$, with a grid of size $n_x \times n_y \times n_z$ and a grid spacing $h$. Note, the method of polarized traces is not restricted to second-order finite differences, but using higher-order finite difference schemes makes the numerical implementation slightly more complicated[^3]. Absorbing boundary conditions are imposed via perfectly matched layers (PMLs) as described by @Berenger:PML and @Johnson:PML.
We describe our PML implementation in detail because the quality and structure of the PML implementation strongly impact the convergence properties of the method. Following [@Berenger:PML; @Johnson:PML], the PML’s are implemented via a complex coordinate stretching. First, we define an extended domain ${\widehat}\Omega$ such that $\Omega \subset {\widehat}\Omega$ and we extend the Helmholtz operator from Eq. \[eq:Helmholtz\] to that domain as follows: $$\label{eq:extended_operator_PML}
\cH = {\widehat}\triangle + m\omega^2 \qquad \text{ in } {\widehat}\Omega,$$ where $m$ is an extension of the squared slowness to ${\widehat}\Omega$ and the extended Laplacian ${\widehat}\triangle$ is constructed by replacing the partial derivatives in the standard Laplacian $\triangle = \partial_{xx} + \partial_{yy} + \partial_{zz}$ with coordinate-stretched partial derivatives defined on ${\widehat}\Omega$: $$\partial_x \rightarrow \beta_x({\mathbf{x}}) \partial_x, \,\, \partial_y \rightarrow \beta_y({\mathbf{x}}) \partial_y, \,\, \partial_z \rightarrow \beta_z({\mathbf{x}}) \partial_z.$$ The complex dilation function $\beta_x({\mathbf{x}})$ (and similarly $\beta_y({\mathbf{x}})$ and $\beta_z({\mathbf{x}})$) is defined as $$\label{appendix:eq:def_alpha}
\beta_x({\mathbf{x}}) = \frac{1}{1+ i \frac{\sigma_x({\mathbf{x}})}{ \omega } },$$ where the PML profile function $\sigma_x({\mathbf{x}})$ (and similarly $\sigma_y({\mathbf{x}})$ and $\sigma_z({\mathbf{x}})$) is, $$\sigma_x({\mathbf{x}}) = \left \{\begin{array}{rl}
\frac{C}{\delta_{\text{pml}}} \left (\frac{x } {\delta_{\text{pml}}} \right)^2, & \text{if } x \in (-\delta_{\text{pml}}, 0 ),\\
0 , & \text{if } x \in [ 0, L_x ], \\
\frac{C}{\delta_{\text{pml}}} \left (\frac{x - L_x }{\delta_{\text{pml}}}\right)^2, & \text{if } x \in ( L_x , L_x + \delta_{\text{pml}} ),\\
\end{array} \right .$$ where $L_x$ is the length of $\Omega$ in the $x$ dimension and $\delta_{\text{pml}}$ is the length of the extension. In general, $\delta_{\text{pml}}$ grows slowly with the frequency, i.e., $\delta_{\text{pml}} \propto \cO(\log{\omega})$, in order to obtain enough absorption as the frequency increases. The constant $C$ is chosen to provide enough absorption. In practice, $\delta_{\text{pml}}$ and $C$ can be seen as parameters to be tuned for accuracy versus efficiency.
The extended Helmholtz operator provides the definition of the global continuous problem, $$\label{eq:Helmholtz_pml}
\cH u = f_s, \qquad \text{in } {\widehat}\Omega,$$ which is then discretized using finite differences to obtain the discrete global problem, $$\mathbf{H} {\mathbf{u}}= {\mathbf{f}}_s. \label{eq:discrete_Helmholtz}$$
In the method of polarized traces, $\Omega$ is decomposed into a set of $L$ layers, $\{\Omega^{\ell}\}_{\ell=1}^{L}$. Without loss of generality, we assume that the decomposition is in the $z$ dimension. Each subdomain $\Omega^{\ell}$ is extended to include an absorbing region, as above, yielding the extended subdomain ${\widehat}\Omega^{\ell}$. For boundaries of ${\widehat}\Omega^{\ell}$ shared with ${\widehat}\Omega$, the absorbing layer is considered to be inherited from the global problem. For the intra-layer boundaries of $\Omega^{\ell}$, i.e., those due to the partitioning of $\Omega$, the extension to the additional absorbing layers in ${\widehat}\Omega^{\ell}$ are necessary to prevent reflections at layer interfaces which are detrimental to convergence.
Then, the local Helmholtz problem is $$\cH^{\ell} v^{\ell} := \triangle^{\ell} v^{\ell} ({\mathbf{x}}) + m^{\ell} \omega^2 v^{\ell} ({\mathbf{x}})= f_s^{\ell}({\mathbf{x}}) \qquad \text{in } {\widehat}\Omega^{\ell}, \label{eq:local_Helmholtz}$$ where $m^{\ell}$ and $f^{\ell}$ are the local restrictions of the model parameters and source functions to ${\widehat}\Omega^{\ell}$, generated by extending $m \chi_{\Omega^{\ell}}$ and $f \chi_{\Omega^{\ell}}$ to ${\widehat}\Omega^{\ell}$, where $\chi_{\Omega^{\ell}}$ is the characteristic function of $\Omega^{\ell}$. The local Laplacian $\triangle^{\ell}$ is defined using the same coordinate stretching approach as above, except on ${\widehat}\Omega^{\ell}$. As before, $\delta_{\text{pml}}$, and thus $\alpha_{\text{pml}}$, must scale as $\log{\omega}$ to obtain the convergence rate claimed in this paper.
We discretize the local problem in Eq. \[eq:local\_Helmholtz\] resulting in the discrete local Helmholtz system $$\mathbf{H}^{\ell} {\mathbf{v}}^{\ell} = {\mathbf{f}}_s^{\ell}. \label{eq:discrete_local_Helmholtz}$$ For the finite difference implementation in this paper, we assume a structured, equispaced Cartesian mesh with mesh points ${\mathbf{x}}_{i,j,k} = (x_i,y_j, z_k) = (ih,jh, kh)$. Assuming the same ordering @ZepedaDemanet:the_method_of_polarized_traces, we write the global solution in terms of the depth index, $$\label{eq:global_ordering}
{\mathbf{u}}= ({\mathbf{u}}_1, {\mathbf{u}}_2, ..., {\mathbf{u}}_{n_z}),$$ where ${\mathbf{u}}_k$ is a plane sampled at constant depth $z_k$, or in MATLAB notation, $$\label{eq:trace_ordering}
{\mathbf{u}}_k = (u_{:,:,k}).$$
Let ${\mathbf{u}}^{\ell}$ be the local restriction of ${\mathbf{u}}$ to $\Omega^{\ell}$, i.e., ${\mathbf{u}}^{\ell} = \chi_{\Omega^{\ell}}{\mathbf{u}}$. Following the above notation, ${\mathbf{u}}^{\ell}_k$ is the local solution trace in the plane at local depth $z_k^{\ell}$. For notational convenience, we renumber the local depth indices so that ${\mathbf{u}}^{\ell}_1$ and ${\mathbf{u}}^{\ell}_{n^{\ell}}$ are the top and bottom planes of the bulk domain. Points due to the PML are not considered[^4]. Finally, let $$\begin{aligned}
\label{eq:traces_definition}
\underline{{\mathbf{u}}} = \left ({\mathbf{u}}^{1}_{n^1} , {\mathbf{u}}^{2}_{1}, {\mathbf{u}}^{2}_{n^2}, ...,{\mathbf{u}}^{L-1}_{1}, {\mathbf{u}}^{L-1}_{n^{L-1}}, {\mathbf{u}}^{L}_{1} \right)^{t},\end{aligned}$$ be the vector of interface traces for all $L$ layers.
To map solution vectors at fixed depth planes back to the discretized whole volume of $\Omega_{\ell}$, we define the Dirac delta at a fixed depth, $$(\delta(z-z_p) {\mathbf{v}}_q)_{i,j,k} = \left \{ \begin{array}{cl}
0 & \text{if } k \neq p, \\
\frac{({\mathbf{v}}_q)_{i,j}}{h^3} & \text{if } k = p.
\end{array}
\right .$$ This definition of the numerical Dirac delta is specific to a classical finite difference discretization. If the discretization changes, it is still possible to define a numerical Dirac delta using the approach developed in [@ZepedaDemanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation].
Accuracy
--------
Solving the Helmholtz equation in the high-frequency regime is notoriously because:
- it is difficult to efficiently discretize the PDE, and
- the resulting linear system is difficult to solve in a scalable and efficient fashion.
In this paper, we focus on the second issue. However, for completeness, we provide a brief overview of difficulties associated to the discretization.
From the Shannon-Nyquist sampling theorem, an oscillatory function at frequency $\omega$ requires $\cO(\omega^d)$ degrees of freedom to be accurately represented, without aliasing. For example, to accurately represent the solution of Eq. \[eq:Helmholtz\], only $\cO(\omega^3)$ degrees of freedom are required. Obviously, accuracy is still limited by the error in the discretization of the differential operator. Even if the medium is very smooth, standards methods based on finite differences and finite elements are subject to pollution error, i.e., the ratio between the error of the numerical approximation and the best approximation cannot be bounded independently of $\omega$ [@Babuska:A_Generalized_Finite_Element_Method_for_solving_the_Helmholtz_equation_in_two_dimensions_with_minimal_pollution; @Ihlenburg_Babuska:Finite_element_solution_of_the_Helmholtz_equation_with_high_wave_number_Part_I:_The_h_version_of_the_FEM; @Babuska:Stable_Generalized_Finite_Element_Method].
The direct consequence of pollution error is that the approximation error, i.e., the error between the analytical and the numerical solution to the linear system, increases with the frequency, even if $n \sim \omega$. Thus, to obtain a bounded approximation error independent of the frequency, it is required to oversample the wavefield, relative to the Shannon-Nyquist criterion, i.e., $n$ needs to grow faster than $\omega$. Unfortunately, oversampling provides discretizations with a suboptimal number of degrees of freedom with respect to the frequency. To alleviate pollution error, several new approaches have been proposed, which can be broadly classified into two groups:
1. methods using standard polynomial bases with modified variational formulations [@GOLDSTEIN:The_weak_element_method_applied_to_Helmholtz_type_equations; @Melenk_Sauter:Wavenumber_Explicit_Convergence_Analysis_for_Galerkin_Discretizations_of_the_Helmholtz_Equation; @Melenk_Parsania_Sauter_13:General_DG_Methods_for_Highly_Indefinite_Helmholtz_Problems; @Moiola_Spence:Is_the_Helmholtz_Equation_Really_Sign_Indefinite; @Graham_Lohndorf_Melenk_Spence:When_is_the_error_in_the_h-BEM_for_solving_the_Helmholtz_equation_bounded_independently_of_k];
2. methods based on well known variational formulations but using non-standard basis, such as plane waves [@Hiptmair_Moiola_Perugia:A_Survey_of_Trefftz_Methods_for_the_Helmholtz_Equation; @Perugia_Pietra_Russo:PW_VEM], polynomials modulated by plane waves [@Betcke_Phillips:Approximation_by_dominant_wave_directions_in_plane_wave_methods; @DBLP:journals/jcphy/NguyenPRC15], or other specially adapted functions.
Even though the methods mentioned above have been successful in reducing pollution error, the resulting linear systems cannot, in general, be solved in quasi-linear time or better because the matrices either have a high degree of interconnectivity or are extremely ill-conditioned. However, some new fast algorithms have recently been proposed for solving the Helmholtz equation without pollution error with quasi-linear complexity for media that are homogeneous up to smooth and compactly supported heterogeneities (@ZepedaZhao:Fast_Lippmann_Schwinger_solver [@Fang_Qian_Zepeda_Zhao:Learning_Dominant_Wave_Directions_For_Plane_Wave_Methods_For_High_Frequency_Helmholtz_Equations]; and references therein). In the case of highly heterogeneous media the accuracy of finite elements has not been extensively studied, although methods of efficient discretizations for highly heterogeneous media, coupled with fast algorithms are emerging [@Taus_Demanet_Zepeda:HDG_Helmholtz].
In this paper, we assume that waves will propagate in very general and highly heterogeneous media, thus, we do not have a theoretical framework to assess the accuracy. Instead, we use numerical experiments to check the accuracy of the solution. Our numerical experiments show, for the cases considered in this paper, using 10 points per wavelength results in roughly 1 digit of accuracy at the highest frequency considered.
Reduction to a surface integral equation
----------------------------------------
The global solution is related to the local layer solutions by coupling the subdomains using the Green’s representation formula (GRF) within each layer. The resulting surface integral equation (SIE), posed at the interface between layers, effectively reduces the problem from the global domain $\Omega$ to the interfaces between layers. The resulting SIE has the form $$\underline{\mathbf{M}} \underline{{\mathbf{u}}} = \underline{{\mathbf{f}}}, \label{eq:SIE}$$ where $\underline{\mathbf{M}}$ is formed by interface-to-interface Green’s functions, $\underline{{\mathbf{u}}}$ is defined in Eq. \[eq:traces\_definition\], and $\underline{{\mathbf{f}}}$ is the right-hand-side, formed as in Line 8 of Alg. \[alg:SIE\_solver\].
The matrix $\underline{\mathbf{M}}$ is a block banded matrix (Fig. \[fig:M\_polarized\], left) of size $2(L-1)n^2 \times 2(L-1)n^2$. Theorem 1 of @ZepedaDemanet:the_method_of_polarized_traces gives that the solution of Eq. \[eq:SIE\] is exactly the restriction of the solution of Eq. \[eq:discrete\_Helmholtz\] to the interfaces between layers.
Following [@ZepedaDemanet:the_method_of_polarized_traces], if the traces of the exact solution are known, then it is possible to apply the GRF to locally reconstruct exactly the global solution within each layer. Equivalently, the reconstruction can be performed by modifying the local source with a measure supported on the layer interfaces and solving the local system with the local solver, as seen in lines 11-12 of Alg. \[alg:SIE\_solver\], where a high-level sketch of the algorithm to solve the 3D high-frequency Helmholtz equation is given.
To efficiently solve the 3D problem, it is critical that the matrix $\underline{\mathbf{M}}$ is never explicitly formed. Instead, a matrix-free approach [@ZepedaDemanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation] is used to apply the blocks of $\underline{\mathbf{M}}$ via applications of the local solver, using equivalent sources supported at the interfaces between layers, as shown in Alg. \[alg:applyM\]. Moreover, as seen in Alg. \[alg:applyM\], the application of $\underline{\mathbf{M}}$ is easily implemented in parallel, with a small communication overhead. The only non-embarrassingly parallel stage of Alg. \[alg:SIE\_solver\] is the solution of Eq. \[eq:SIE\], which is inherently sequential.
Given that $\underline{\mathbf{M}}$ is never explicitly formed, an iterative method is the natural choice for solving Eq. \[eq:SIE\]. In practice, the condition number of $\underline{\mathbf{M}}$ is very large and it has a wide spectrum is the complex plane, which implies that a large number of iterations are required to achieve convergence. To alleviate this problem, we apply the method of polarized traces, as an efficient preconditioner for Eq. \[eq:SIE\], which we describe below.
Offline computation \[alg:offline\_computation\]
$\mathbf{H}^{\ell} = \triangle^{\ell} + m^{\ell} \omega^2 $ $\mathbf{L}^{\ell} \mathbf{U}^{\ell} = \mathbf{H}^{\ell} $
Online computation using the SIE reduction \[alg:SIE\_solver\]
$ \mathbf{f}^{\ell} = \mathbf{f}\chi_{\Omega^{\ell}} $ ${\mathbf{v}}^{\ell} = (\mathbf{H}^{\ell})^{-1} \mathbf{f}^{\ell} $ $ \underline{\mathbf{f}} = \left ( {\mathbf{v}}^1_{n^1}, {\mathbf{v}}^2_1 ,{\mathbf{v}}^2_{n^2} ,\hdots ,{\mathbf{v}}^L_1 \right )^{t} $ $\underline{{\mathbf{u}}} = \left( \underline{\mathbf{M}} \right )^{-1} \underline{{\mathbf{f}}} $ $ \begin{array}{ll} \mathbf{g}^{\ell} = & {\mathbf{f}}^{\ell} + \delta(z_{1}-z){\mathbf{u}}^{\ell-1}_{n^{\ell-1}} - \delta(z_{0}-z){\mathbf{u}}^{\ell}_{1} \\
& - \delta(z_{n^{\ell}+1}-z){\mathbf{u}}^{\ell}_{n^{\ell}} + \delta(z_{n^{\ell}}-z){\mathbf{u}}^{\ell+1}_{1}
\end{array}$ $ {\mathbf{u}}^{\ell} = (\mathbf{H}^{\ell})^{-1} \mathbf{g}^{\ell}$ ${\mathbf{u}}= ({\mathbf{u}}^1 , {\mathbf{u}}^2, \hdots, {\mathbf{u}}^{L-1}, {\mathbf{u}}^{L})^t $
Application of the boundary integral matrix $\underline{\mathbf{M}}$ \[alg:applyM\]
$ {\widetilde}{{\mathbf{f}}}^{1} = - \delta(z_{n^{1}+1}-z){\mathbf{v}}^{1}_{n^{\ell}} + \delta(z_{n^{1}}-z){\mathbf{v}}^{2}_{n^{1}} $ $ \mathbf{w}^{1} = (\mathbf{H}^{1})^{-1} {\widetilde}{{\mathbf{f}}}^{1}$ ${\mathbf{u}}^{\ell}_{n^{\ell}} = \mathbf{w}^{\ell}_{n^{\ell}} - {\mathbf{v}}^{\ell}_{n^{\ell}} $ $ \begin{array}{ll} {\widetilde}{{\mathbf{f}}}^{\ell} = & \delta(z_{1}-z){\mathbf{v}}^{\ell-1}_{n^{\ell-1}} - \delta(z_{0}-z){\mathbf{v}}^{\ell}_{1} \\
& - \delta(z_{n^{\ell}+1}-z){\mathbf{v}}^{\ell}_{n^{\ell}} + \delta(z_{n^{\ell}}-z){\mathbf{v}}^{\ell+1}_{1}
\end{array}$ $ \mathbf{w}^{\ell} = (\mathbf{H}^{\ell})^{-1} {\widetilde}{{\mathbf{f}}}^{\ell} $ ${\mathbf{u}}^{\ell}_{1} = \mathbf{w}^{\ell}_{1} - {\mathbf{v}}^{\ell}_{1}; \qquad {\mathbf{u}}^{\ell}_{n^{\ell}} = \mathbf{w}^{\ell}_{n^{\ell}} - {\mathbf{v}}^{\ell}_{n^{\ell}} $ $ {\widetilde}{{\mathbf{f}}}^{L} = \delta(z_{1}-z){\mathbf{v}}^{L-1}_{n^{L-1}} - \delta(z_{0}-z){\mathbf{v}}^{L}_{1}$ $ \mathbf{w}^{L} = (\mathbf{H}^{L})^{-1} {\widetilde}{{\mathbf{f}}}^{L}$ ${\mathbf{u}}^{L}_{1} = \mathbf{w}^{L}_{1} - {\mathbf{v}}^{L}_{1}$
Method of Polarized traces
==========================
Reducing the Helmholtz problem to a SIE allows us to efficiently parallelize most of the computation required to solve Eq. \[eq:discrete\_Helmholtz\]. The only remaining sequential bottleneck is the solution of Eq. \[eq:SIE\]. Given the size and the distributed nature of $\underline{\mathbf{M}}$, iterative methods, such as GMRES ([@Saad_Schultz:GMRES]) or Bi-CGSTAB ([@van_der_Vorst:BiCGSTAB]), are the logical approach for solving Eq. \[eq:SIE\]. However, numerical experiments indicate that the condition number of $\underline{\mathbf{M}}$ scales as $\cO(h^{-2})$, or as $\cO(\omega^2)$ in the high-frequency regime [@ZepedaDemanet:the_method_of_polarized_traces]. The number of iterations required for schemes like GMRES to converge is proportional to the condition number of the system, yielding poor scalability for solving the SIE at high frequencies. To alleviate this problem, we use the method of polarized traces to convert the SIE to an equivalent problem, which is easily preconditioned. This preconditioned system only requires $\cO(\log \omega)$ GMRES iterations[^5], i.e., it is comparatively independent of the frequency. Here, we provide a high-level review of the method of polarized traces and its implementation and we direct the reader to @ZepedaDemanet:the_method_of_polarized_traces for a detailed exposition.
Preconditioner
--------------
As seen in the previous discussion, Eq. \[eq:SIE\] is the result of decomposing the domain into a set of layers and reducing the Helmholtz problem to an equivalent SIE on the interfaces between the subdomains. To precondition the SIE with the method of polarized traces, the solution at the interfaces is decomposed in up- and down-going components such that $$\underline{{\mathbf{u}}} = \underline{{\mathbf{u}}}^{\uparrow} + \underline{{\mathbf{u}}}^{\downarrow},$$ which defines the polarized wavefield $$\underline{\underline{{\mathbf{u}}}} = \left (\begin{array}{c}
\underline{{\mathbf{u}}}^{\downarrow} \\
\underline{{\mathbf{u}}}^{\uparrow}
\end{array}
\right ).$$ By introducing the polarized wavefield, we have deliberately doubled the unknowns and produced an underdetermined system. To close the system, we impose annihilation, or polarizing, conditions (see Section 3 of [@ZepedaDemanet:the_method_of_polarized_traces]) that are encoded in matrix form as $$\underline{\mathbf{A}}^{\uparrow} \underline{{\mathbf{u}}}^{\uparrow}=0, \,\,\, \text{ and } \,\,\, \underline{\mathbf{A}}^{\downarrow} \underline{{\mathbf{u}}}^{\downarrow}=0. \label{eq:annihilation_conditions}$$
Requiring that the solution satisfies both Eq. \[eq:SIE\] and the annihilation conditions yields another equivalent formulation, $$\underline{\underline{\mathbf{M}}} \, \underline{\underline{{\mathbf{u}}}} = \underline{\underline{{\mathbf{f}}}}_s, \label{eq:polarized_SIE}$$ where $$\underline{\underline{\mathbf{M}}} = \left [ \begin{array}{cc}
\underline{\mathbf{M}} & \underline{\mathbf{M}} \\
\underline{\mathbf{A}}^{\downarrow} & \underline{\mathbf{A}}^{\uparrow}
\end{array}
\right ], \,\, \text{and} \,\,
\underline{\underline{{\mathbf{f}}}}_s = \left ( \begin{array}{c}
\underline{\mathbf{f}}_s \\
0
\end{array}
\right ).$$ Following a series of basic algebraic operations and permutations (see [@ZepedaDemanet:the_method_of_polarized_traces] for the full details), we obtain an equivalent formulation of the polarized SIE matrix in Eq. \[eq:polarized\_SIE\], given by $$\underline{\underline{\mathbf{M}}} = \left [ \begin{array}{cc}
\underline{\mathbf{D}}^{\downarrow} & \underline{\mathbf{U}} \\
\underline{\mathbf{L}} & \underline{\mathbf{D}}^{\uparrow}
\end{array}
\right ].$$ There exist straightforward, parallel algorithms for applying the block matrices $\underline{\mathbf{D}}^{\downarrow}$, $\underline{\mathbf{D}}^{\uparrow}$, $\underline{\mathbf{L}}$, and $\underline{\mathbf{U}}$. By construction $\underline{\mathbf{D}}^{\downarrow}$ and $\underline{\mathbf{D}}^{\uparrow}$ can be easily inverted using block forward and backward substitution because they are block triangular with identity blocks on their diagonals. The blocks that appear in the sparsity pattern of $\underline{\underline{\mathbf{M}}}$ (Fig. \[fig:M\_polarized\]; right) are a direct manifestation of interactions between the layer interfaces.
While the resulting block linear system can be solved using standard matrix-splitting iterations, such as block Jacobi iteration or block Gauss-Seidel iteration [@Saad:iterative_methods_for_sparse_linear_systems], it is natural to continue to use GMRES to solve the system due to the parallel nature of applying the constituent blocks of $\underline{\underline{\mathbf{M}}}$. However, the structure of $\underline{\underline{\mathbf{M}}}$ is convenient for using a single iteration of Gauss-Seidel as a preconditioner, $$\mathbf{P}\underline{\underline{\mathbf{M}}} \, \underline{\underline{{\mathbf{u}}}} = \mathbf{P}\underline{\underline{{\mathbf{f}}}}_s, \label{eq:precond_polarized_SIE}$$ where the preconditioning matrix is $$\label{eq:preconditioner_GS}
\mathbf{P} = \left(
\begin{array}{ll}
\mathbf{\underline D}^\downarrow & \mathbf{O} \\
\mathbf{\underline L} & \mathbf{\underline D}^\uparrow \\
\end{array}
\right)^{-1}.$$ In the subsequent sections, we will elaborate on the physical and numerical meanings of the constituent blocks of $\underline{\underline{\mathbf{M}}}$ and $\mathbf{P}$.
### Polarization
The main novelty of the method of polarized traces is due to the polarization conditions, which are encoded in the matrices $\mathbf{\underline{A}}^{\uparrow}$ and $\mathbf{\underline{A}}^{\downarrow}$. The polarizing conditions provide a streamlined way to define an iterative solver using standard matrix splitting techniques, and thus an efficient preconditioner for Krylov methods, such as GMRES.
The polarization conditions are constructed by projecting the solution on two orthogonal sets, physically given by waves traveling upwards and downwards. Similar constructs are well-known to the geophysics community, as methods that decompose wavefields into distinct down- and up-going components are the backbone of several imaging techniques (see [@Zhang:The_Theory_of_True_Amplitude_One-Way_Wave_Equation_Migrations] and references therein). Commonly, the decomposition is obtained using discretizations of pseudo-differential operators, which can be interpreted as separating the wavefield into a set of wave-atoms traveling in the different directions which are then propagated accordingly. Methods for decomposing and locally extrapolating directionally decomposed wavefields are well documented ([@Wu:Wide-angle_elastic_wave_one-way_propagation_in_heterogeneous_media_and_an_elastic_wave_complex_screen_method; @Collino_Joly:Splitting_of_Operators_Alternate_Directions_and_Paraxial_Approximations_for_the_Three_Dimensional_Wave_Equation; @Ristow:3D_implicit_finite-difference_migration_by_multiway_splitting; @deHoop:generalization_of_the_phase_screen_approximation_for_the_scattering_of_acoustic_waves]).
In our case, we rewrite the decomposition condition as an integral relation between the Neumann and Dirichlet data of the wavefield, which ultimately leads to the annihilation conditions in Eq. \[eq:annihilation\_conditions\]. The pair composed of the Neumann and Dirichlet traces should lie within the null space of an integral operator defined on an interface, which allows the decomposition of the total wavefield into the up- and down-going components, with each having a clear physical interpretation. In particular, an up-going wavefield is a wavefield generated by a source located beneath the interface and it satisfies a radiation condition at positive infinity and a down-going wavefield is a wavefield generated by a source located above the interface and it satisfies a radiation condition at negative infinity. As detailed in [@ZepedaDemanet:the_method_of_polarized_traces], defining the decomposition in this manner allows us to extrapolate each component in a stable manner using an incomplete Green’s integral.
The extrapolation of up-going components is performed algorithmically by the inversion of $\underline{\mathbf{D}}^{\uparrow}$ and in the same fashion the extrapolation of down-going components is performed by the inversion of $\underline{\mathbf{D}}^{\downarrow}$. Moreover, the application of the operator $\underline{\mathbf{L}}$ isolates the up-going reflections due to down-going waves interacting with the material in each subdomain, and similarly for the operator $\underline{\mathbf{U}}$.
The application of the preconditioner to a decomposed wavefield, $$\mathbf{P} \left( \begin{array}{c} \underline{{\mathbf{v}}}^{\downarrow} \\
\underline{{\mathbf{v}}}^{\uparrow}
\end{array}
\right) = \left(
\begin{array}{c}(\underline{\mathbf{D}}^{\downarrow})^{-1} \underline{{\mathbf{v}}}^{\downarrow} \\
(\underline{\mathbf{D}}^{\uparrow})^{-1} \mathbf{r}
\end{array}
\right ),$$ for $\mathbf{r} = \left ( {\mathbf{v}}^{\uparrow} -\underline{\mathbf{L}} (\underline{\mathbf{D}}^{\downarrow})^{-1} \underline{{\mathbf{v}}}^{\downarrow} \right)$, can be physically interpreted as follows:
1. $(\underline{\mathbf{D}}^{\downarrow})^{-1} \underline{{\mathbf{v}}}^{\downarrow}$: extrapolate the down-going components by propagating them downwards,
2. $\mathbf{r} = \left ( {\mathbf{v}}^{\uparrow} -\underline{\mathbf{L}} (\underline{\mathbf{D}}^{\downarrow})^{-1} \underline{{\mathbf{v}}}^{\downarrow} \right)$: compute the local reflection of the extrapolated field and add them to the up-going components,
3. $(\underline{\mathbf{D}}^{\uparrow})^{-1} \mathbf{r}$: extrapolate the up-going components by propagating them upwards.
Algorithms
----------
As with the application of $\underline{\mathbf{M}}$ in Alg. \[alg:applyM\], we construct matrix-free methods for solving $ ( \underline{\mathbf{D}}^{\downarrow})^{-1}$ and $ (\underline{\mathbf{D}}^{\uparrow})^{-1}$ (Algs. \[alg:downwardsSweep\] and \[alg:upwardsSweep\]), where local solves are applied in an inherently sequential fashion. To complete the preconditioner, a matrix-free (and embarrassingly parallel) algorithm for applying $\underline{\mathbf{L}}$ is given in Alg \[alg:upwardsReflections\]. Similar algorithms for applying $\underline{\mathbf{U}}$, $\underline{\mathbf{D}}^{\uparrow}$, and $\underline{\mathbf{D}}^{\downarrow}$, as well as the complete matrix-free algorithm for applying $\underline{\underline{\mathbf{M}}}$, are provided in the Appendix.
In solving the systems for $ ( \underline{\mathbf{D}}^{\downarrow})^{-1}$ and $ (\underline{\mathbf{D}}^{\uparrow})^{-1}$, each application of the local solver is local to each layer, which means that some communication is required to transfer solution updates from one layer to the next. The sequential nature of the method for solving these systems implies that only one set of processors, those assigned to the current layer, are working at any given stage of the algorithm. This is illustrated in Fig. \[fig:nodes\_load\], where each block represents a local solve and the execution path moves from left to right. As explained in [@ZepedaDemanet:the_method_of_polarized_traces], it is possible to apply $\underline{\mathbf{D}^{\downarrow}}$ and $\underline{\mathbf{L}}$ simultaneously, thus decreasing the number of local solves per layer.
Downward sweep, application of $ ( \mathbf{\underline{D}}^{\downarrow} )^{-1}$ \[alg:downwardsSweep\]
${\mathbf{u}}^{\downarrow,1}_{n^1} = -{\mathbf{v}}^{\downarrow,1}_{n^1}$ ${\mathbf{u}}^{\downarrow,1}_{n^1+1} = -{\mathbf{v}}^{\downarrow,1}_{n^1+1} $ $ {\widetilde}{{\mathbf{f}}}^{\ell} = \delta(z_{1}-z){\mathbf{u}}^{\downarrow,\ell-1}_{n^{\ell-1}} - \delta(z_{0}-z){\mathbf{u}}^{\downarrow,\ell-1}_{n^{\ell-1}+1} $ $ \mathbf{w}^{\ell} = (\mathbf{H}^{\ell})^{-1} {\widetilde}{{\mathbf{f}}}^{\ell} $ ${\mathbf{u}}^{\downarrow,\ell}_{n^{\ell}} = \mathbf{w}_{n^{\ell}} - {\mathbf{v}}^{\downarrow,\ell}_{n^{\ell}}$ ${\mathbf{u}}^{\downarrow,\ell}_{n^{\ell}+1} = \mathbf{w}_{n^{\ell}+1}-{\mathbf{v}}^{\downarrow,\ell}_{n^{\ell}+1} $ $\underline{{\mathbf{u}}}^{\downarrow} = \left ({\mathbf{u}}^{\downarrow,1}_{n^1} , {\mathbf{u}}^{\downarrow,1}_{n^1+1}, {\mathbf{u}}^{\downarrow,2}_{n^2}, ..., {\mathbf{u}}^{\downarrow,L-1}_{n^{L-1}}, {\mathbf{u}}^{\downarrow,L-1}_{n^{L-1}+1} \right)^{t} $
Upward sweep, application of $ ( \mathbf{\underline{D}}^{\uparrow} )^{-1}$ \[alg:upwardsSweep\]
${\mathbf{u}}^{\uparrow,L}_{0} = -{\mathbf{v}}^{\uparrow,L}_{0}$ ${\mathbf{u}}^{\uparrow,L}_{1} = -{\mathbf{v}}^{\uparrow,L}_{1} $ $ {\widetilde}{{\mathbf{f}}}^{\ell} = - \delta(z_{n^{\ell}+1}-z){\mathbf{u}}^{\uparrow,\ell+1}_{0} + \delta(z_{n^{\ell}}-z){\mathbf{u}}^{\uparrow,\ell+1}_{1} $ $ \mathbf{w}^{\ell} = (\mathbf{H}^{\ell})^{-1} {\widetilde}{{\mathbf{f}}}^{\ell} $ ${\mathbf{u}}^{\uparrow,\ell}_{1} = \mathbf{w}^{\ell}_{1} - {\mathbf{v}}^{\uparrow,\ell}_{1}$ ${\mathbf{u}}^{\uparrow,\ell}_{0} = \mathbf{w}^{\ell}_{0}-{\mathbf{v}}^{\uparrow,\ell}_{0} $ $\underline{{\mathbf{u}}}^{\uparrow} = \left ({\mathbf{u}}^{\uparrow,2}_{0} , {\mathbf{u}}^{\uparrow,2}_{1}, {\mathbf{u}}^{\uparrow,3}_{0}, ..., {\mathbf{u}}^{\uparrow,L}_{0}, {\mathbf{u}}^{\uparrow,L}_{1} \right)^{t} $
Upward reflections, application of $ \mathbf{\underline{L}}$ \[alg:upwardsReflections\]
$ \begin{array}{ll} {\mathbf{f}}^{\ell} = & \delta(z_{1}-z){\mathbf{v}}^{\downarrow,\ell}_{0} - \delta(z_{0}-z){\mathbf{v}}^{\downarrow,\ell}_{1} \\
& - \delta(z_{n^{\ell}+1}-z){\mathbf{v}}^{\downarrow,\ell+1}_{0} + \delta(z_{n^{\ell}}-z){\mathbf{v}}^{\downarrow,\ell+1}_{1}
\end{array}$ $ \mathbf{w}^{\ell} = (\mathbf{H}^{\ell})^{-1} {\mathbf{f}}^{\ell} $ ${\mathbf{u}}^{\uparrow,\ell}_{1} = \mathbf{w}^{\ell}_{1} - {\mathbf{v}}^{\downarrow,\ell}_{1}$ ${\mathbf{u}}^{\uparrow,\ell}_{0} = \mathbf{w}^{\ell}_{0} $ $ {\mathbf{f}}^{L} = \delta(z_{1}-z){\mathbf{v}}^{\uparrow,L}_{0} - \delta(z_{0}-z){\mathbf{v}}^{\uparrow,L}_{1} $ $ \mathbf{w}^{L} = (\mathbf{H}^{L})^{-1} {\mathbf{f}}^{L} $ ${\mathbf{u}}^{\uparrow,L}_{1} = \mathbf{w}^{L}_{1} - {\mathbf{v}}^{\downarrow,L}_{1}$ ${\mathbf{u}}^{\uparrow,L}_{0} = \mathbf{w}^{L}_{0} $ $\underline{{\mathbf{u}}}^{\uparrow} = \left ({\mathbf{u}}^{\uparrow,2}_{0} , {\mathbf{u}}^{\uparrow,2}_{1}, {\mathbf{u}}^{\uparrow,3}_{0}, ..., {\mathbf{u}}^{\uparrow,L}_{0}, {\mathbf{u}}^{\uparrow,L}_{1} \right)^{t} $
### Physical intuition
We deliberately present the preconditioner in a purely algebraic fashion, as it is instructive for implementing the method. However, there is a physical interpretation of the steps in the preconditioner, which we describe below.
As alluded to previously, the application of the preconditioner, and in particular the block back-substitution in Algs. \[alg:downwards\] and \[alg:upwards\], can be seen as a sequence of depth extrapolation steps. Indeed, lines $4$ and $5$ in Alg. \[alg:downwardsSweep\] are the discrete counterpart of the incomplete Green’s integral defined by $$u^{\downarrow}({\mathbf{x}}) = \int_{\Gamma_{\ell-1, \ell}} \left (G^{\ell}({\mathbf{x}},{\mathbf{y}}) \partial_z u^{\downarrow}({\mathbf{y}}) - \partial_z G^{\ell}({\mathbf{x}},{\mathbf{y}}) u^{\downarrow}({\mathbf{y}}) \right) dS_y,$$ which is equivalent to the Rayleigh integral used to extrapolate a wavefield measured in surface towards the interior of the Earth by [@Berkhout:Seismic_Migration_imaging_of_acoustic_energy_by_wavefield_extrapolation]. Likewise, Lines 4 and 5 of Alg. \[alg:upwardsSweep\] are the discrete counterpart to an up-going discrete Green’s integral.
The quality of the extrapolation depends directly on the quality of the approximation of the local Green’s function $G^{\ell}$ with respect to the global Green’s function. In the reductive case, if the local Green’s function is precisely the global Green’s function, the method will converge in two iterations, see [@Gander:Optimized_Schwarz_Methods]. However, this is equivalent to solving the global problem, which is prohibitively expensive. Instead, we compute a local approximation of the Green’s function such that the Green’s representation formula is valid within the layer only, not globally. As expected with domain decomposition methods, incorrect local approximations introduce numerical artifacts, which are typically due to truncating the domain in a manner that is inconsistent with the underlying physics. In the method of polarized traces, these issues are mitigated with judicious use of high-order absorbing boundary conditions in the form of PML’s. As a physical consequence, the local Green’s function can only see local features within a particular layer. Far-field interactions, reflections induced by material changes in the other layers, will not be observed by the local Green’s function and must be handled iteratively, by sequentially sweeping through the domains.
An important consequence of the Green’s integral representation is that it completely eliminates the difficulties that most domain decomposition methods have with seamlessly connecting sub-domains together. Rather than assigning data dependent boundary conditions, the coupling is performed using potentials defined on the physical interfaces, and the absorbing boundary conditions in an extended domain effectively dampen spurious reflections. The transmission conditions given by the discrete Green’s representation formula are algebraically exact, thus there is no need for tuning parameters.
Parallelization Strategies
==========================
The computational effort needed to solve industrial scale 3D problems requires aggressive parallelization and optimization of the algorithm. To obtain a scalable implementation, the algorithm and code must be designed to balance the utilization and occupancy of three key resources: CPU, memory, and communication network. In this section, we describe our parallel implementation of the method of polarized traces, with a focus on maximizing the utilization of these resources.
In the previous section, we formally introduced a matrix-free approach for preconditioning the SIE system on layer interfaces. However, this approach still relies on local solves that are implemented using a direct solver. It is possible to use specially designed iterative local solvers by nesting the method of polarizes traces within each layer [@ZepedaDemanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation] or a recursive version of the sweeping factorization [@Liu_Ying:Recursive_sweeping_preconditioner_for_the_3d_helmholtz_equation]. However, such approach would require a complicated code with a very carefully implemented communication pattern. For simplicity, and to broaden the portability of the framework, we use a hybrid approach, where the local solves use off-the-shelf numerical linear algebra libraries and the polarization is matrix-free. We will address the parallelism on two fronts: parallelism by layer and parallelism within the layers.
Pipelining
----------
First, we address parallelism due to the layer decomposition. Primarily, the parallelism across layers is due to the SIE and the preconditioner used to help solve it. There are five trivially parallel (by layer) applications of the local solver: four due to $\underline{\underline{\mathbf{M}}}$ and one due to the appearance of $\underline{\mathbf{L}}$ in the preconditioner. However, in the preconditioner application there are sequential bottlenecks due to the applications of $(\mathbf{\underline{D}}^{\uparrow})^{-1}$ and $(\mathbf{\underline{D}}^{\downarrow})^{-1}$ via block back-substitution. Despite the trivial parallel nature of the other local solver applications, applying the preconditioner using Algs. \[alg:downwardsSweep\] and \[alg:upwardsSweep\] permits work to be done on only one layer at a time, thus forcing the majority of the computer to remain idle. This is illustrated in the top half of Fig. \[fig:nodes\_load\], where each blue box represents a local solve and algorithm execution moves from left to right. Supposing that each local solve costs $\gamma(n)$ time, then following Fig. \[fig:nodes\_load\], each GMRES iteration can be performed in $5 \gamma(n) + 2L \gamma(n)$, ignoring communication costs.
![Sketch of the load of each node in the GMRES iteration.[]{data-label="fig:nodes_load"}](load_nodes.pdf){width="8.5cm"}
To alleviate the sequential bottleneck, we leverage the fact that seismic problems have thousands of right-hand sides and introduce pipelining. Pipelining allows us to process multiple right-hand sides simultaneously, each at different levels of progress through the sweeps, which helps balance the computational load on the layers, reducing the idle time and increasing the computational efficiency. The pipelining principle is demonstrated in the bottom half of Fig \[fig:nodes\_load\], where the boxes represent a local solve and the blue, green, and orange colors indicate distinct right-hand sides. Pipelining allows the layers to perform work for different right-hand sides simultaneously. Indeed, as long as there are at least $2L$ right-hand sides, the pipeline can be completely full and all available compute resources will be occupied. For the pipelined algorithm, again disregarding communication costs, the runtime of a GMRES iteration is $5R\gamma(n) + 2(L + R)\gamma(n)$. Recalling that $L \sim n$, $R\sim n$, and $\gamma(n) = \cO(\alpha^2_{\text{pml}} n^2 \log n) $, the cost ratio for solving $R$ right-hand sides compared to one right-hand side is constant, $$\frac{5R\gamma(n) + 2(L + R)\gamma(n)}{5 \gamma(n) + 2L \gamma(n)} = \cO(1).$$
One of the advantages of the method of polarized traces is that the memory requirement to store the intermediate representation of solution is lower than other methods, because it requires solutions for the degrees of freedom involved in the SIE only. Thus, for each right-hand side, only $N/q$ data need to be stored, where $q$ is the thickness of the interface. This reduction in storage, when combined with the reduced storage due to the relatively small number of GMRES iterations required for convergence, yields a smaller memory footprint for the outer GMRES iteration than methods requiring to update the full volume. It is possible to further reduce the memory footprint by using Bi-CGSTAB instead of GMRES, keeping the computational cost almost constant [@ZepedaDemanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation].
Local Solves: Parallel Multi-frontal Methods
--------------------------------------------
The method polarized traces is a highly modular framework for solving the Helmholtz equation: in practice, one can use any existing algorithm or package for solving linear equations to perform the local solves, including the method of polarized traces itself [@ZepedaDemanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation]. To obtain good parallel performance, we use a high-performance distributed linear algebra library to solve the local problems within each layer. Due to the sparsity pattern of the linear system at each layer, a typical recipe for the local solves is to re-order the degrees of freedom to increase stability and reduce numerical fill-in, perform a multi-frontal factorization, and solve the resulting factorized system with forward- and backward-substitution, often called triangular solves. There exists a myriad of techniques to parallelize the multi-frontal factorization and the triangular solves (see [@davis:A_survey_of_direct_methods_for_sparse_linear_systems] for a recent and extensive review of different techniques for solving sparse systems). A popular approach is to use supernodal elimination trees (see [@Ashcraft:Progress_in_Sparse_Matrix_Methods_for_Large_Linear_Systems_On_Vector_Supercomputers]) defined through nested dissection, which results in highly scalable factorizations [@Gupta:Highly_Scalable_Parallel_Algorithms_for_Sparse_Matrix_Factorization], albeit with less efficient triangular solves [@Gupta_Kumar:A_high_performance_two_dimensional_scalable_parallel_algorithm_for_solving_sparse_triangular_systems]. To avoid the poor scalability of dense triangular solves due to this approach, @Raghavan:Efficient_Parallel_Sparse_Triangular_Solution_Using_Selective_Inversion introduced a scheme called [*selective inversion*]{}, which is applied by @Poulson_Engquist:a_parallel_sweeping_preconditioner_for_heteregeneous_3d_helmholtz specifically for the Helmholtz problem. Although very efficient, using the techniques mentioned before would require lengthy and complex code, we use instead off-the-shelf libraries, which can be effortlessly changed if necessary.
For the results in this paper, we use STRUMPACK [@Rouet_Li_Ghysels:A_distributed-memory_package_for_dense_Hierarchically_Semi-Separable_matrix_computations_using_randomization] to perform the local solves. STRUMPACK is a state-of-the-art distributed sparse linear solver library that relies on supernodal factorizations,[^6] a 2D block-cyclic distribution of the matrix, and a static mapping technique to assign task to MPI processes based on [*proportional mapping*]{} [@Pothen_Sun:A_Mapping_Algorithm_for_Parallel_Sparse_Cholesky_Factorization] to achieve good parallel performance. The implementation is competitive with other distributed linear algebra solvers with liberal licenses, such as SuperLU-DIST (cf. [@li_demmel03:SuperLU_DIST]) and MUMPS (cf. [@Amestoy_Duff:MUMPS]), while providing the user more freedom to arrange the distribution of the matrix and right-hand sides, within a distributed memory enviroment, in a manner that is optimal for specific applications.
A strong advantage of STRUMPACK is that the factorization and solve processes can be accelerated using compressed linear algebra, in particular HSS compression with nested bases, using randomized sampling techniques. In general, using compressed formulations allows a reduced memory footprint and in some cases faster algorithms. For the high-frequency regime, it is known that solvers based on compression techniques do not provide a lower asymptotic complexity, due to the fact that the ranks of the off-diagonal blocks are frequency dependent [@Engquist_Zhao:approximate_separability_of_green_function_for_high_frequency_Helmholtz_equations]. However, these solvers still tend to provide smaller memory footprints for the cases we consider in this paper, albeit, with much bigger runtimes. Therefore, we deliberately do not considered the performance of the adaptive compression in this paper and leave such treatment for future work.
The STRUMPACK solver addresses parallelism within layers in two ways: classical distribution of tasks with MPI and synchronous processing within each task with OpenMP, which we exploit for the results presented in our numerical results. STRUMPACK’s hybrid parallelism model allows us to maximize the utilization of computational resources. We have designed the distribution of MPI tasks to exploit highly asynchronous communication patterns, thus reducing the communication time subtantially.
Finally, in most of the experiments shown in the sequel, we process at most one right-hand side per layer. It is possible to solve more than one right-hand side per layer, and take advantage of BLAS3 routines. Moreover, a quick inspection to the algorithms within the preconditioner, shows that the right-hand sides are sparse. Indeed, the sources are supported on the interfaces, and the solutions are only needed at the boundaries. In principle, it is possible to take advantage of the sparsity of the solution and the right-hand-side to reduce the constants by removing some branches from the elimination tree. These techniques would certainly reduce the constants but they would have little impact on the asymptotic scaling; hence, they were not explored in this study.
Communication
-------------
As is common in massively parallel applications, there is a bottleneck due to the communication between parallel tasks, which is strongly dependent on the distribution of the tasks on the cluster. For this implementation, we assume a very simple topology for the distribution of unknowns. We distribute the parallel tasks following the layer structure. For each slab of unknowns, we assign $\cO(n^2)$ tasks, and using MPI directives enforce that the tasks are contiguous within physical computing nodes. Each slab is divided into $\cO(n^2)$ cubes, as illustrated in Fig. \[fig:DDM\_cubes\], and the parallel tasks associated with that slab are divided evenly and contiguously amongst the cubes. Each cube contains a contiguous block of the solution, as shown in Fig. \[fig:DDM\_cubes\], and the associated entries of the local matrix.
Within the slab, the topology is designed so that each cube only communicates with its neighboring cubes in the same slab, generally inside the local solver. Across slabs, cubes only communicate with the cube in the same position on the adjacent slabs immediately above and below it, as illustrated in Fig. \[fig:communication\]. Under this particular topology, we can distinguish two main communications bottlenecks:
- the communication between parallel tasks within the distributed linear algebra solver, and
- the communication of the boundary data between slabs, during the application of the preconditioner.
As a consequence of using third-party distributed linear algebra solvers, we have little control over the communication pattern, particularly because STRUMPACK uses MC64 [@Duff_Koster:On_Algorithms_For_Permuting_Large_Entries_to_the_Diagonal_of_a_Sparse_Matrix], for enhancing stability and ParMetis [@Karypis:A_Parallel_Algorithm_for_Multilevel_Graph_Partitioning_and_Sparse_Matrix_Ordering] to optimally reorder the matrix in order to reduce fill-in during the factorization. To have the desired distribution of the degrees of freedom among the cubes, we reorder the matrix with a $Z$ ordering, so that smallest division corresponds exactly to the degrees of freedom within a cube. Then, the matrix is then assembled and passed to the linear solver in a distributed fashion.
Communication between slabs is a product of the application of the preconditioner in Eq. \[eq:preconditioner\_GS\], in which back- and forward-substitution are used to apply $(\mathbf{\underline{D}}^{\uparrow})^{-1}$ and $(\mathbf{\underline{D}}^{\downarrow})^{-1}$. Algs. \[alg:downwards\] and \[alg:upwards\]) require a local solve in each slab, followed by communication of the trace information to the next slab, in which another local solve is performed. This operation is repeated until all the slabs are visited within the sweep.
By dividing the slabs into cubes, the communication of the trace information between slabs is very efficient. Given that each cube communicates with the cube directly above and below, it is possible to perform asynchronous point-to-point communication between the cubes of two adjacent slabs as shown in Fig. \[fig:communication\]. This allows the communication to be performed in nearly constant time, up to saturation of the network. Moreover, the trace information is already distributed for distributed assembly of the right-hand side within the subsequent slab. As stated before, it is possible to use topologies better suited for the multi-frontal solver, such as the one due to @Poulson_Engquist:a_parallel_sweeping_preconditioner_for_heteregeneous_3d_helmholtz. However, such an implementation requires a very precise understanding of the re-ordering mechanism within the solver, which we do not have for black-box solvers.
Complexity of Polarized Traces
==============================
The run-time complexity of the polarized traces preconditioner is driven by the costs of computation and communication. Achieving optimal performance requires delicately balancing the parallel distribution of the problem depending on the characteristics of the target HPC system. In this section we develop models for computation and communication costs, which guide problem parameter selection in HPC environments.
Computation
-----------
As before, each layer has $\cO((n_z + \alpha_{\text{pml}})\times n^2)$ grid points, i.e., they are $n_z$ grid points thick with $\alpha_{\text{pml}}$ additional points due to the PML. We have that $n_z = \cO(1)$ because $L \sim n$, which implies that we are solving a quasi-2D problem. The additional cost is due only to the points used to implement the absorbing boundary conditions. When applying 2D nested dissection to the quasi-2D problem we have $\cO(\alpha_{\text{pml}} n)$ degrees of freedom in the biggest front, thus leading to a complexity of $\cO(\alpha_{\text{pml}}^3n^3)$ for the factorization of the systems local to each layer and $\cO(\alpha_{\text{pml}}^2n^2 \log{n})$ for the application of the triangular solve [@Duff_Reid:The_Multifrontal_Solution_of_Indefinite_Sparse_Symmetric_Linear]. Sequentially, the complexity of Alg. \[alg:offline\_computation\] is $\cO(\alpha_{\text{pml}}^3 N^{4/3})$, but given that the loop in line $2$ of Alg. \[alg:offline\_computation\] is embarrassingly parallelizable, Alg. \[alg:offline\_computation\] can be performed in $\cO(\alpha_{\text{pml}}^3 N)$ time[^7]. Due to the sequential nature of Algs. \[alg:downwardsSweep\] and \[alg:upwardsSweep\], applying the preconditioner requires $2L$ local solves per iteration, applied sequentially. Consequently, the total complexity for the application of the preconditioner is $\cO(\alpha_{\text{pml}}^2 N \log{N})$. For $\alpha_{\text{pml}} \sim \log{n} $, at most $\cO(\log{n})$ iterations are empirically needed to converge, thus the complexity of the solver is linear (up to poly-logarithmic factors), provided that $L \sim n$ and that the number of iterations for convergence grows slowly.
It is possible to relax the restriction that $L \sim n$, instead allowing $L \sim n^{b}$ where $b < 1 $. However, in this regime, maintaining the overall linear complexity requires that we exchange the multi-frontal solver for an iterative solver [@Liu_Ying:Recursive_sweeping_preconditioner_for_the_3d_helmholtz_equation; @ZepedaDemanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation]. The main disadvantage of this approach is that it reduces the possible parallelism due to using multi-frontal solvers, which makes an efficient implementation of the pipelining difficult and makes the communication patterns more complicated.
Pipelining
----------
We introduced pipelining of $R$ right-hand sides to alleviate the sequential nature of applying the preconditioner. To understand the run-time impact of pipelining multiple right-hand sides, we consider its impact on the complexity of applying $\underline{\underline{\mathbf{M}}}$ and applying the preconditioner. Given that the run-time cost of each local solve is $\cO(\alpha_{\text{pml}}^2n^2 \log n)$ and recalling that applying $\underline{\underline{\mathbf{M}}}$ is embarrassingly parallel, the cost of applying $\underline{\underline{\mathbf{M}}}$ to $R$ right-hand sides is $\cO(\alpha_{\text{pml}}^2 R n^2 \log{n})$. As long as $R \sim L \sim n $, applying the preconditioner costs $\cO(L \alpha_{\text{pml}}^2 n^2 \log {n})$. However, when $R \gtrsim L$, the additional right-hand-sides are treated sequentially, resulting in a cost of $\cO(\alpha_{\text{pml}}^2 R n^2 \log {n})$. Using the fact that $L \sim n$ and that $N = n^3$, we obtain the advertised runtime of $\cO( \alpha_{\text{pml}}^2 \max(1,R/L) N \log {N} ))$.
Communication
-------------
We treat the distributed linear algebra solver as a black box, and therefore do not analyze the costs of communication within the local solve. Thus, we only consider the cost of communication due to the global solve, that is the costs of communicating between subdomains across layers. We assume that that each subdomain has fast access to its corresponding patch of both the $R$ wavespeeds and the $R$ sources. Moreover, we assume that each subdomain assembles and stores its portion of the global solution[^8].
The offline stage of the algorithm, assembly of the local matrices and the local factorizations, is embarrassingly parallel under the assumptions described above. The online part, has three stages:
- the preparation of the right-hand side (Lines 2-8 in Alg. \[alg:SIE\_solver\]);
- the solve of the SIE (Lines 9 in Alg. \[alg:SIE\_solver\]); and
- the assembly of the global solution (Lines 10-14 in Alg. \[alg:SIE\_solver\]).
Of these, the first and third stages require no communication under the above assumptions.
Solving the SIE has two main phases: the application of $\underline{\underline{\mathbf{M}}}$ and the application of the preconditioner. Applying $\underline{\underline{\mathbf{M}}}$, as shown in Alg. \[alg:applyM\], is an embarassingly parallel operation with zero communication. On the other hand, applying the preconditioner, which is fully sequential, requires communication of $\cO(n^2)$ unknowns from one layer to the next. In our implementation, these unknowns are distributed evenly between $\cO(n^2)$ MPI tasks. Using a point-to-point communication strategy, each of the MPI tasks assigned to a layer only communicates with one corresponding MPI task in the adjacent (above and below) layers, as illustrated in Fig. \[fig:communication\]. Thus, by exploiting asynchronous communication, the communication between layers can be performed in $\cO(1)$ time up to saturation of the bandwidth, which is asymptotically negligible with respect to solving the local linear systems. This communication must be performed $\cO(L)$ times during each sweeping operation in Algs. \[alg:downwardsSweep\] and \[alg:upwardsSweep\]. Consequently, total communication cost of applying the preconditioner is $\cO(n)$, up to saturation of the bandwidth.
Numerical Experiments
=====================
In this section, we present the results of several numerical experiments used to verify the complexity described above. In particular, we demonstrate the performance of the 3D preconditioner in various heterogeneous media for a single source and then illustrate the impact of pipelining on parallel performance. Our polarized traces implementation is written in C and compiled with the 2015 Intel compiler suite. The current implementation is uses IEEE double precision floating point. To perform the local solves, we use STRUMPACK v1.1.0 with Intel MKL support for fast linear algebra operations. The preconditioner is parallelized with MPI and STRUMPACK is parallelized with both MPI and OpenMP. The experiments were performed on Total’s “Laure” SGI ICE-X cluster, where each computing node contains two 8-core Intel Sandy Bridge processors, 64GB of RAM, and are connected with an Infiniband interconnect.
Homogeneous media
-----------------
First, we demonstrate the effectiveness of the preconditioner by solving the Helmholtz problem in homogeneous media. With no reflectors in the medium, the convergence of the algorithm is only dependent on the frequency and the quality of the absorbing boundary condition at the layer interfaces. In this experiment, as well as the subsequent experiments, we hold $\alpha_{\text{pml}}$ constant, with sufficient points to minimize artificial reverberations while simultaneously keeping the number of iterations low. For higher frequencies, to preserve the low iteration count we would need to scale $\alpha_{\text{pml}}$ as $\cO(\log{n})$.
In this experiment, we test 4 problem sizes, $n=50,$ $100,$ $200,$ and $400$, which corresponds to frequencies of $8$, $16$, $32$, and $64$ Hz. Source frequency is scaled with problem size to stay in the high-frequency regime and sources are assumed to be point sources. The number of layers is also scaled with the problem, $L=5,$ $10,$ $20,$ and $40$. Due to memory limitations on the computing node, in some cases the nodes were saturated before all cores could be assigned to an MPI task. For these cases, we allow the remaining cores to be used for vectorized processing with OpenMP. The outer GMRES iteration is run until the residual is reduced to $10^{-7}$, which is excessive in a production, single-precision environment; however, it shows the favorable behavior of the solver under more challenging conditions. Lower tolerances can produce misleading results when frequencies are not high enough, thus only revealing a pre-asymptotic behavior. For each configuration, we report the wall-clock times for initialization, matrix assembly, matrix factorization, and total online time for $R=1$ and $R=L$ with pipelining. Additionally, we track the number of GMRES iterations required to achieve the desired convergence.
![Observed run-time as function of $N$ for homogeneous media (green), smooth heterogeneous media (blue) and “fault” model (orange), with pure MPI (solid) and hybrid MPI-OpenMP (dashed) for $R=1$ right-hand sides. For comparison, theoretical scaling of polarized traces algorithm is given (solid black) as well as linear scaling (dashed black).[]{data-label="fig:not_seam_1"}](online_1.pdf){width="8.5cm"}
![Observed run-time as function of $N$ for homogeneous media (green), smooth heterogeneous media (blue) and “fault” model (orange), with pure MPI (solid) and hybrid MPI-OpenMP (dashed) for $R=L$ right-hand sides. For comparison, theoretical scaling of polarized traces algorithm is given (solid black) as well as linear scaling (dashed black).[]{data-label="fig:not_seam_R"}](online_L.pdf){width="8.5cm"}
N $50^3$ $100^3$ $100^3$ $200^3$ $200^3$ $400^3$ $400^3$ $400^3$
----------------------------- -------- --------- --------- --------- --------- --------- --------- ---------
L 5 10 10 20 20 40 40 40
MPI Tasks 5 10 10 80 80 640 640 640
OpenMP Threads per Task 1 1 2 1 2 1 2 3
Total Cores 5 10 20 80 160 640 1280 1920
Total Nodes 1 1 2 5 10 80 80 128
**Single rhs**
\# GMRES Iterations 4 4 4 5 5 6 6 6
Initialization \[s\] 0.2 1.0 0.9 6.9 4.4 18.9 18.9 18.4
Factorization \[s\] 4.1 41.1 21.9 153.2 78.3 320.5 200.1 148.6
Online \[s\] 4.0 39.2 22.6 182.0 109.7 696.6 401.4 315.5
Average GMRES \[s\] 0.9 8.4 4.8 32.0 19.2 103.5 59.3 46.6
**Pipelined rhs**
$R$ (number of rhs) 5 10 10 20 20 40 40 40
Online \[s\] 15.8 189.4 106.2 1255.5 668.5 3994.2 2654.4 1878.1
Average GMRES \[s\] 3.4 40.6 22.7 223.8 118.6 599.9 401.0 283.0
Online per rhs \[s\] 3.2 18.9 10.6 62.8 33.4 99.9 66.4 47.0
Average GMRES per rhs \[s\] 0.7 4.1 2.3 11.2 5.9 15.0 10.0 7.1
The solution at 64 Hz is provided in Fig. \[fig:constant\_model\]. The full results of the experiment are given in in Table \[table:test\_homogeneous\] and the observed run-times, compared to the theoretical run-times for $R=1$ and $R=L$ pipelined right-hand sides are shown in Fig. \[fig:not\_seam\_1\] and Fig. \[fig:not\_seam\_R\], respectively. Only when $R \ge L$ does the theoretical scalability break the linear threshold, however, in both cases, the method of polarized traces scales better than the theoretical scaling, which we attribute to optimizations and parallelism in the local solver. Of note in Table \[table:test\_homogeneous\], the number of GMRES iterations grows very slowly with the frequency, even when we do not scale $\alpha_{\text{pml}}$ optimally. In the experiments where hybrid parallelism (MPI-OpenMP) is used, the run-times are reduced almost linearly for medium-sized problems, however the improvements fade as the size of the problem increases. This behavior is due to the linear solver, where for large problems the memory access time in the triangular solves becomes dominant, reducing the parallelism.
Smooth heterogeneous media
--------------------------
Using the same configurations as above we solve the Helmholtz problem in the smoothed random media shown in Fig. \[fig:smooth\_model\] in order to demonstrate the effectiveness of the solver in heterogeneous media. This test demonstrates that the method is particularly robust to media where the rays can bend and develop caustics. The solution for the configuration equivalent to that of Fig. \[fig:constant\_model\] is given in Fig. \[fig:smooth\_model\_solution\]. In Fig. \[fig:smooth\_model\_solution\], it is clear that the features of the model are comparable to the wavelength used, thus the solution presents interference, caustics, non-spherical wavefronts. Table \[table:test\_smooth\] contains the complete experimental results, where we observe that the variation in the media has little real effect on the run-time or convergence properties. In this case, even if the rays bend, the preconditioner does a remarkable job at tracking the rays in the correct direction and propagating them accordingly. The timings are given in in Figs. \[fig:not\_seam\_1\] and Fig \[fig:not\_seam\_R\].
N $50^3$ $100^3$ $100^3$ $200^3$ $200^3$ $400^3$ $400^3$ $400^3$
----------------------------- -------- --------- --------- --------- --------- --------- --------- ---------
L 5 10 10 20 20 40 40 40
MPI Tasks 5 10 10 80 80 640 640 640
OpenMP Threads per Task 1 1 2 1 2 1 2 3
Total Cores 5 10 20 80 160 640 1280 1920
Total Nodes 1 1 2 5 10 80 80 128
**Single rhs**
\# GMRES Iterations 5 5 5 5 5 6 6 6
Initialization \[s\] 0.2 1.1 1.0 7.3 4.6 21.3 21.2 20.8
Factorization \[s\] 3.8 41.1 21.8 156.0 79.4 323.7 204.5 151.5
Online \[s\] 4.6 45.9 26.1 202.2 106.9 717.0 400.1 314.5
Average GMRES \[s\] 0.8 8.1 4.6 35.5 18.7 106.4 59.2 46.5
**Pipelined rhs**
$R$ (number of rhs) 5 10 10 20 20 40 40 40
Online \[s\] 17.1 225.1 118.8 1260.9 650.2 4085.0 2714.8 1872.1
Average GMRES \[s\] 3.0 39.8 20.9 223.6 115.6 613.3 409.2 281.9
Online per rhs \[s\] 3.4 22.5 11.9 63.0 32.5 102.1 67.9 46.8
Average GMRES per rhs \[s\] 0.6 4.0 2.1 11.2 5.8 15.3 10.2 7.0
Fault model
-----------
In general, iterative methods are very sensitive to discontinuous media. At high frequency, interaction with short-wavelength structures, such as discontinuities, increases the number of reflections. Each additional reflection requires additional iterations to convergence, hindering the efficiency of iterative methods.
Using the same configuration as for the homogeneous model, with the discontinuous velocity given in Fig. \[fig:fault\_model\], we demonstrate that the method of polarized traces deteriorates only marginally as a function of the frequency and number of subdomains. A solution at 64 Hz is given in Fig. \[fig:solution\_fault\_model\] and the run-time scalability is again given in Figs. \[fig:not\_seam\_1\] and \[fig:not\_seam\_R\]. As shown in Table \[table:test\_fault\], we observe the same behavior as in the previous cases and that the strong reflection is handed efficiently by the transmission and polarizing conditions.
N $50^3$ $100^3$ $100^3$ $200^3$ $200^3$ $400^3$ $400^3$ $400^3$
----------------------------- -------- --------- --------- --------- --------- --------- --------- ---------
L 5 10 10 20 20 40 40 40
MPI Tasks 5 10 10 80 80 640 640 640
OpenMP Threads per Task 1 1 2 1 2 1 2 3
Total Cores 5 10 20 80 160 640 1280 1920
Total Nodes 1 1 2 5 10 80 80 128
**Single rhs**
\# GMRES Iterations 4 5 5 5 5 6 6 6
Initialization \[s\] 0.4 1.1 1.0 7.3 4.7 20.4 20.3 21.0
Factorization \[s\] 3.8 40.4 22.1 152.2 79.9 317.6 199.5 152.5
Online \[s\] 3.7 46.2 26.2 188.5 109.8 713.2 395.8 315.6
Average GMRES \[s\] 0.8 8.1 4.6 33.0 19.2 106.2 58.7 46.5
**Pipelined rhs**
$R$ (number of rhs) 5 10 10 20 20 40 40 40
Online \[s\] 13.7 226.7 122.4 1222.7 647.1 4031.6 2710.6 1838.9
Average GMRES \[s\] 2.9 40.1 21.6 216.5 114.7 605.0 409.9 276.3
Online per rhs \[s\] 2.7 22.7 12.2 61.1 32.4 100.8 67.7 46.0
Average GMRES per rhs \[s\] 0.6 4.0 2.2 10.8 5.7 15.1 10.2 6.9
SEAM model
----------
Beyond mere sensitivity to discontinuities of the medium, iterative solvers are highly sensitive to the roughness and heterogeneity of the velocity model, due to the great amount of interactions, reflections, and drastic changes of direction of waves due to high gradients in the wavespeed. However, the performance method of polarized traces degrades only marginally for highly heterogeneous media, excepting resonant cavities. We demonstrate this desirable performance on the SEAM Phase I velocity model (Fig. \[fig:SEAM\_model\]). In this experiment, we test 3 problem sizes, $N=0.65$M, $5.16$M, and $41.2$M degrees of freedom, which use $L=12, 24,$ and $48$ layers, respectively. The remainder of the experimental setup is unchanged.
As seen in the data in Table \[table:test\_SEAM\] and plotted in Figs. \[fig:seam\_1\] and \[fig:seam\_R\], the run-times are sub-linear with respect to the total number of unknowns. Interestingly, we also observe a sub-linear run-time in the offline stages of the algorithm, which we attribute to the parallelism in the multi-frontal factorization. This is expected, as the factorization is more computationally intensive than memory intensive. In the more memory-intensive, and thus less parallel, solve phase, we still see an improvement over the theoretical curve, but the improvement is less pronounced.
Finally, for the largest test case, we demonstrate the impact of pipelining by comparing the scalability of our method with the theoretical scalability, as a function of $R$. As shown in Fig. \[fig:pipeline\_scaling\], experimental results indicate that we obtain the expected scalability. Slight divergence from the theoretical curve is expected once the pipeline is fully saturated because the theoretical curve does not take into account the cost of filling and flushing the pipeline.
N $6.51\cdot 10^5 $ $5.16\cdot 10^6$ $4.12 \cdot 10^7 $ $4.12 \cdot 10^7 $
----------------------------- ------------------- ------------------ -------------------- --------------------
L 12 24 48 48
MPI Tasks 12 48 384 384
OpenMP Threads per Task 1 2 2 3
Total Cores 12 96 768 1152
Total Nodes 1 6 77 77
**Single rhs**
\# GMRES Iterations 4 5 6 6
Initialization \[s\] 0.6 2.3 10.4 10.7
Factorization \[s\] 15.2 46.5 111.4 97.9
Online \[s\] 21.4 85.6 269.8 228.4
Average GMRES \[s\] 4.6 14.9 40.0 33.7
**Pipelined rhs**
$R$ (number of rhs) 12 24 48 48
Online \[s\] 106.3 474.8 1527.1 1415.4
Average GMRES \[s\] 22.8 83.9 229.4 212.9
Online per rhs \[s\] 8.8 19.8 31.8 29.5
Average GMRES per rhs \[s\] 1.9 3.5 4.8 4.4
![Observed runtime as function of $N$ for the SEAM model for $R=1$ right-hand side. For comparison, we show the theoretical scaling of the polarized traces algorithm (solid black), as well as a linear scaling (dashed black).[]{data-label="fig:seam_1"}](SEAM_online_1.pdf){width="8.5cm"}
![Observed runtime as function of $N$ for the SEAM model for $R=L$ right-hand sides. For comparison, we show the theoretical scaling of the polarized traces algorithm (solid black), as well as a linear scaling (dashed black).[]{data-label="fig:seam_R"}](SEAM_online_L.pdf){width="8.5cm"}
![Impact of scalability on pipelining for the SEAM model. $N$ is held constant and $R$ is increased. The observed runtime as function of $R$ is in green, and the dashed black line illustrates theoretical scalability.[]{data-label="fig:pipeline_scaling"}](SEAM_online_pipelining.pdf){width="8.5cm"}
Finally, Fig. \[fig:pipeline\_scaling\] depics the behavior of the pipelining as we increase the number of right-hand sides. As expected, as we add more and more righ-hand sides to be solved simultaneously the runtime per right-hand side decreases, until the pipeline is full, when the average runtime remains almost constant.
Conclusion
==========
We have presented a new and efficient solver for the $3$D high-frequency Helmholtz equation in heterogeneous media. The solver achieves a sub-linear runtimes by leveraging the solution of batches of right-hand sides properly pipelined and parallelism. The method presented in this paper broadens the applicability of parallel direct methods by embedding them in a domain decomposition method, whose rate of convergence is independent of the frequency.
Finally, the main limitations of the present method are large resonant cavities presenting high contrasts, for which the number of reflections can be large implying that the number of iterations for convergence can still be high.
Acknowledgments
===============
This worked was sponsored by Total SA. LD is also sponsored by AFOSR grant FA9550-17-1-0316, ONR grant N00014-16-1-2122, and NSF grant DMS-1255203. The authors are grateful to Mike Fehler and Total SA for their permission to use the 3D SEAM model.
Appendix: Matrix-free algorithms {#appendix:matrix-free}
================================
The application of the polarized system defined in Eq. \[eq:polarized\_SIE\] is achieved by applying each block as shown in Alg. \[alg:polarizedSystem\].
Application of $ \mathbf{\underline{\underline{M}}} $ \[alg:polarizedSystem\]
$(\underline{{\mathbf{v}}}^{\downarrow},\underline{{\mathbf{v}}}^{\uparrow}) = \underline{\underline{{\mathbf{v}}}}$ $\underline{{\mathbf{u}}}^{\downarrow} = \mathbf{\underline{D}}^{\downarrow} {\mathbf{v}}^{\downarrow} + \mathbf{\underline{U}} {\mathbf{v}}^{\uparrow}$ $\underline{{\mathbf{u}}}^{\uparrow} = \mathbf{\underline{D}}^{\uparrow} {\mathbf{v}}^{\uparrow} + \mathbf{\underline{L}} {\mathbf{v}}^{\downarrow}$ $\underline{\underline{{\mathbf{u}}}} = (\underline{{\mathbf{u}}}^{\downarrow},\underline{{\mathbf{u}}}^{\uparrow})$
To apply the blocks in a matrix-free fashion, we use Algs. \[alg:downwards\] and \[alg:upwardsReflections\] in line 4 of Alg. \[alg:polarizedSystem\], and we use Algs. \[alg:upwards\] and \[alg:downwardsReflections\] in line 5 of Alg. \[alg:polarizedSystem\]. All algorithms in this section are embarrasingly parallel at the level of the layers as depicted in Fig. \[fig:nodes\_load\].
Application of $ \mathbf{\underline{D}}^{\downarrow} $ \[alg:downwards\]
${\mathbf{u}}^{\downarrow,1}_{n^1} = -{\mathbf{v}}^{\downarrow,1}_{n^1}$ ${\mathbf{u}}^{\downarrow,1}_{n^1+1} = -{\mathbf{v}}^{\downarrow,1}_{n^1+1} $ $ {\widetilde}{{\mathbf{f}}}^{\ell} = \delta(z_{1}-z){\mathbf{v}}^{\downarrow,\ell-1}_{n^{\ell-1}} - \delta(z_{0}-z){\mathbf{v}}^{\downarrow,\ell-1}_{n^{\ell-1}+1} $ $ \mathbf{w}^{\ell} = (\mathbf{H}^{\ell})^{-1} {\widetilde}{{\mathbf{f}}}^{\ell} $ ${\mathbf{u}}^{\downarrow,\ell}_{n^{\ell}} = \mathbf{w}_{n^{\ell}} - {\mathbf{v}}^{\downarrow,\ell}_{n^{\ell}}$ ${\mathbf{u}}^{\downarrow,\ell}_{n^{\ell}+1} = \mathbf{w}_{n^{\ell}+1}-{\mathbf{v}}^{\downarrow,\ell}_{n^{\ell}+1} $ $\underline{{\mathbf{u}}}^{\downarrow} = \left ({\mathbf{u}}^{\downarrow,1}_{n^1} , {\mathbf{u}}^{\downarrow,1}_{n^1+1}, {\mathbf{u}}^{\downarrow,2}_{n^2}, ..., {\mathbf{u}}^{\downarrow,L-1}_{n^{L-1}}, {\mathbf{u}}^{\downarrow,L-1}_{n^{L-1}+1} \right)^{t} $
Upward sweep, application of $ ( \mathbf{\underline{D}}^{\uparrow} )^{-1}$ \[alg:upwards\]
${\mathbf{u}}^{\uparrow,L}_{0} = -{\mathbf{v}}^{\uparrow,L}_{0}$ ${\mathbf{u}}^{\uparrow,L}_{1} = -{\mathbf{v}}^{\uparrow,L}_{1} $ $ {\widetilde}{{\mathbf{f}}}^{\ell} = - \delta(z_{n^{\ell}+1}-z){\mathbf{v}}^{\uparrow,\ell+1}_{0} + \delta(z_{n^{\ell}}-z){\mathbf{v}}^{\uparrow,\ell+1}_{1} $ $ \mathbf{w}^{\ell} = (\mathbf{H}^{\ell})^{-1} {\widetilde}{{\mathbf{f}}}^{\ell} $ ${\mathbf{u}}^{\uparrow,\ell}_{1} = \mathbf{w}^{\ell}_{1} - {\mathbf{v}}^{\uparrow,\ell}_{1}$ ${\mathbf{u}}^{\uparrow,\ell}_{0} = \mathbf{w}^{\ell}_{0}-{\mathbf{v}}^{\uparrow,\ell}_{0} $ $\underline{{\mathbf{u}}}^{\uparrow} = \left ({\mathbf{u}}^{\uparrow,2}_{0} , {\mathbf{u}}^{\uparrow,2}_{1}, {\mathbf{u}}^{\uparrow,3}_{0}, ..., {\mathbf{u}}^{\uparrow,L}_{0}, {\mathbf{u}}^{\uparrow,L}_{1} \right)^{t} $
Downwards reflections, application of $ \mathbf{\underline{U}}$ \[alg:downwardsReflections\]
$ \begin{array}{ll} {\mathbf{f}}^{\ell} = & \delta(z_{1}-z){\mathbf{v}}^{\downarrow,\ell}_{0} - \delta(z_{0}-z){\mathbf{v}}^{\downarrow,\ell}_{1} \\
& - \delta(z_{n^{\ell}+1}-z){\mathbf{v}}^{\downarrow,\ell+1}_{1} + \delta(z_{n^{\ell}}-z){\mathbf{v}}^{\downarrow,\ell+1}_{0}
\end{array}$ $ \mathbf{w}^{\ell} = (\mathbf{H}^{\ell})^{-1} {\mathbf{f}}^{\ell} $ ${\mathbf{u}}^{\uparrow,\ell}_{1} = \mathbf{w}^{\ell}_{1} - {\mathbf{v}}^{\downarrow,\ell}_{1}$ ${\mathbf{u}}^{\uparrow,\ell}_{0} = \mathbf{w}^{\ell}_{0} $ $ {\mathbf{f}}^{L} = \delta(z_{1}-z){\mathbf{v}}^{\uparrow,L}_{0} - \delta(z_{0}-z){\mathbf{v}}^{\uparrow,L}_{1} $ $ \mathbf{w}^{L} = (\mathbf{H}^{L})^{-1} {\mathbf{f}}^{L} $ ${\mathbf{u}}^{\uparrow,L}_{1} = \mathbf{w}^{L}_{1} - {\mathbf{v}}^{\downarrow,L}_{1}$ ${\mathbf{u}}^{\uparrow,L}_{0} = \mathbf{w}^{L}_{0} $ $\underline{{\mathbf{u}}}^{\uparrow} = \left ({\mathbf{u}}^{\uparrow,2}_{0} , {\mathbf{u}}^{\uparrow,2}_{1}, {\mathbf{u}}^{\uparrow,3}_{0}, ..., {\mathbf{u}}^{\uparrow,L}_{0}, {\mathbf{u}}^{\uparrow,L}_{1} \right)^{t} $
[^1]: This hypothesis is critical for obtaining a quasi-linear complexity algorithm and is related to the complexity of solving a quasi-2D problem using multi-frontal methods (for further details see [@EngquistYing:Sweeping_PML; @Poulson_Engquist:a_parallel_sweeping_preconditioner_for_heteregeneous_3d_helmholtz]).
[^2]: For a review on classical Schwarz methods see [@Chan:Domain_decomposition_algorithms; @Toselli:Domain_Decomposition_Methods_Algorithms_and_Theory]; and for other applications of domain decomposition methods for the Helmholtz equations, see [@Bourdonnaye_Farhat_Roux:A_NonOverlapping_Domain_Decomposition_Method_for_the_Exterior_Helmholtz_Problem; @Ghanemi98adomain; @McInnes_Keyes:Additive_Schwarz_Methods_with_Nonreflecting_Boundary_Conditions_for_the_Parallel_Computation_of_Helmholtz_Problems; @Collino:Domain_decomposition_method_for_harmonic_wave_propagation_a_general_presentation; @Magoules:Application_of_a_domain_decomposition_with_Lagrange_multipliers_to_acoustic_problems_arising_from_the_automotive_industry; @Boubendir:An_analysis_of_the_BEM_FEM_non_overlapping_domain_decomposition_method_for_a_scattering_problem; @Astaneh_Guddati:A_two_level_domain_decomposition_method_with_accurate_interface_conditions_for_the_Helmholtz_problem].
[^3]: Nor is the method applicable only in the context of finite differences: finite elements [@Taus_Demanet_Zepeda:HDG_Helmholtz; @Fang_Qian_Zepeda_Zhao:Learning_Dominant_Wave_Directions_For_Plane_Wave_Methods_For_High_Frequency_Helmholtz_Equations] and integral equations [@ZepedaZhao:Fast_Lippmann_Schwinger_solver] approaches are also valid.
[^4]: With this renumbering the local depth index $z_k^{\ell}$ maps to the global depth index $z_{n_c^{\ell}+k}$ where $n_c^{\ell} = \sum_{j=1}^{\ell-1} n^j$.
[^5]: This scaling is empirically deduced , under the assumption that no large resonant cavities are present in the media.
[^6]: It uses a $ULV$ factorization when the compression is turned on [@Chandrasekaran:A_Fast_ULV_Decomposition_Solver_for_Hierarchically_Semiseparable_Representations; @Xia:Randomized_Sparse_Direct_Solvers]
[^7]: As it will be shown in the numerical experiments, this scaling can be further reduced due to the parallelism at the level of the multi-frontal solver.
[^8]: As a consequence, computation of imaging conditions can be performed without incurring in extra communication cost.
|
---
abstract: 'We show that any Pisot substitution on a finite alphabet is conjugate to a primitive proper substitution (satisfying then a coincidence condition) whose incidence matrix has the same eigenvalues as the original one, with possibly $0$ and $1$. Then, we prove also substitutive systems sharing this property and admitting “enough” multiplicatively independent eigenvalues (like for unimodular Pisot substitutions) are measurably conjugate to domain exchanges in Euclidean spaces which factorize onto minimal translations on tori. The combination of these results generalizes a well-known result of Arnoux-Ito to any unimodular Pisot substitution.'
address: 'Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 6140, Université de Picardie Jules Verne, 33 rue Saint Leu, 80000 Amiens, France.'
author:
- 'Fabien Durand, Samuel Petite'
bibliography:
- 'pisot.bib'
title: Unimodular Pisot Substitutions and Domain Exchanges
---
= 0cm
[^1]
Introduction
============
A classical way to tackle problems in geometric dynamics is to code the dynamics through a well-chosen finite partition to obtain a “nice” subshift which is easier to study (see the emblematic works [@Hadamard:1898] and [@Morse:1921]). The interesting aspects of the subshift could then be lifted back to the dynamical systems.
In the seminal paper [@Rauzy:1982], G. Rauzy proposed to go in the other way round: take your favorite subshift and try to give it a geometrical representation. He took what is now called the Tribonacci substitution given by $$\tau :1 \mapsto 12, 2 \mapsto 13 \hbox{ and } 3 \mapsto 1,$$
and proved that the subshift it generates is measure theoretically conjugate to a rotation on the torus $\mathbb{T}^2$. A similar result was already known for substitutions of constant length under some necessary and sufficient conditions [@Dekking:1978]. Later, in , the author show that subshifts whose block complexity is $2n+1$, and satisfy what is called the Condition (\*) (which includes the subshift generated by $\tau$), are measure theoretically conjugate to an interval exchange on 3 intervals.
The substitution $\tau$ has the specificity to be a unimodular (and irreducible) Pisot substitution, that is, its incidence matrix has determinant 1, its characteristic polynomial is irreducible and its dominant eigenvalue is a Pisot number (all its algebraic conjugates are, in modulus, strictly less than $1$). These properties provide key arguments to prove the main result in [@Rauzy:1982]. It naturally leads to what is now called the [**Pisot conjecture**]{} for symbolic dynamics:
[*Let $\sigma$ be a Pisot substitution. Then, the subshift it generates has purely discrete spectrum, [*i.e.*]{}, is measure theoretically conjugate to a translation on a group.*]{}
Many attempts have been done in this direction. The usual strategy is the same as the Rauzy’s one in [@Rauzy:1982]: show first that the substitutive system is measurably conjugate to a domain exchange (see Definition \[def:domainexchange\]). Then prove this system is measurably conjugate to a translation on a group.
A first important rigidity result, due to Host [@Host:1986], is that any eigenfunction of a primitive substitution is continuous. In a widely cited, but unpublished manuscript, Host also proved that the Pisot conjecture is true for unimodular substitutions defined on two letters, provided a condition called [*strong coincidence condition*]{} holds. This combinatorial condition first appeared in [@Dekking:1978] cited above. Barge an Diamond in , show then this condition is satisfied for any unimodular Pisot substitution on two letters. So the Pisot conjecture is true in this case .
Following the Rauzy’s strategy, but in a different way from the Host’s approach, Arnoux and Ito in , associate a self-affine domain exchange called [*Rauzy fractal*]{} to any unimodular Pisot substitution. They proved, this system is measurably conjugate to the substitutive system provided the substitution satisfies a combinatorial condition. Few time later, Host’s results were generalized by Canterini and Siegel in to any unimodular Pisot substitution and to the non-unimodular case [@Siegel:2003; @Siegel:2004], but without avoiding the strong coincidence condition. These works led to the development of a huge number of techniques to study the Rauzy fractals (see for instance [@Fogg:2002] and references therein). Let us mention also other fruitful geometrical approaches by using tilings in and more recently in [@Barge:2014] for the one-dimensional case.
In this paper, we show a similar result to and but skipping the combinatorial condition: any unimodular Pisot substitution is measurably conjugate to a self-affine domain exchange. Notice the domain exchange may, a priori, be different from the usual Rauzy fractal.
\[theo:main\] Let $\sigma$ be a unimodular Pisot substitution on $d$ letters and let $(\Omega, S)$ be the associated substitutive dynamical system. Then, there exist a self-affine domain exchange transformation $(E, \B, \tilde{\lambda}, T)$ in ${{\mathbb R}}^{d-1}$ and a continuous onto map $F \colon \Omega \to E$ which is a measurable conjugacy map between the two systems.
If $\pi \colon {{\mathbb R}}^{d-1} \to {{{{\mathbb R}}}}^{d-1}/{\mathbb Z}^{d-1}$ denotes the canonical projection, then the map $\pi \circ F$ defines, for some constant $r \ge 1$, an a.e. $r$-to-one factor map from $ (\Omega, S)$ to the dynamical system associated with a minimal translation on the torus ${{{{\mathbb R}}}}^{d-1}/{\mathbb Z}^{d-1}$.
The toral translation is explicitly described in (see also [@Fogg:2002]). To show the Pisot conjecture, one still have to show this domain exchange is conjugate to the toral translation.
We postpone to the next section the basic definitions and notions we use for dynamical systems, substitutive dynamics and Pisot substitutions. In Section \[sec:returnsub\], we prove by using the notion of return words, that any substitutive subshift is conjugate to a [*proper*]{} substitution ([*i.e.*]{}, having a nice combinatorial property implying, in particular, the strong coincidence condition). But, this new substitution may not be irreducible since the spectrum of its matrix contain the spectrum of a power of the older one but may also contain the values $0$ and $1$. We show then, in Section \[sec:domex\], that a such subshift, having enough multiplicatively independent eigenvalues (precised later), is measurably conjugate to a self-affine domain exchange. A byproduct of these two results gives us Theorem \[theo:main\]. The proof follows the same strategy as in . However, here, the standard property of irreducibility of Pisot substitutions are not used. We strongly need, instead, a condition on the eigenvalues which is precisely: the number of multiplicatively independent non trivial eigenvalues equals $\sum_{0< \vert \lambda \vert <1} \textrm{ dim }E_{\lambda}$ where $E_{\lambda}$ denotes the eigenspace associated with the eigenvalue $\lambda$ of the substitution matrix. This suggests a possible extension of these results to linearly recurrent symbolic systems like in .
Basic definitions {#sec:basicdef}
=================
Words and sequences
-------------------
An [*alphabet*]{} $A$ is a finite set of elements called [ *letters*]{}. Its cardinality is $|A|$. A [*word*]{} over $A$ is an element of the free monoid generated by $A$, denoted by $A^*$. Let $x = x_0x_1 \cdots x_{n-1}$ (with $x_i\in A$, $0\leq i\leq n-1$) be a word, its [*length*]{} is $n$ and is denoted by $|x|$. The [*empty word*]{} is denoted by $\epsilon$, $|\epsilon| = 0$. The set of non-empty words over $A$ is denoted by $A^+$. The elements of $A^{\mathbb{Z}}$ are called [*sequences*]{}. If $x=\dots x_{-1} x_0 x_1\dots$ is a sequence (with $x_i\in A$, $i\in \mathbb{Z}$) and $I=[k,l]$ an interval of $\mathbb{Z}$ we set $x_I = x_k x_{k+1}\cdots x_{l}$ and we say that $x_{I}$ is a [*factor*]{} of $x$. If $k = 0$, we say that $x_{I}$ is a [ *prefix*]{} of $x$. The set of factors of length $n$ of $x$ is written $\mathcal{L}_n(x)$ and the set of factors of $x$, or the [*language*]{} of $x$, is denoted by $\mathcal{L}(x)$. The [*occurrences*]{} in $x$ of a word $u$ are the integers $i$ such that $x_{[i,i + |u| - 1]}= u$. If $u$ has an occurrence in $x$, we also say that $u$ [*appears*]{} in $x$. When $x$ is a word, we use the same terminology with similar definitions.
A word $u$ is [*recurrent*]{} in $x$ if it appears in $x$ infinitely many times. A sequence $x$ is [*uniformly recurrent*]{} if it is recurrent and for each factor $u$, the difference of two consecutive occurrences of $u$ in $x$ is bounded.
Morphisms and matrices
----------------------
Let $A$ and $B$ be two finite alphabets. Let $\sigma$ be a [*morphism*]{} from $A^*$ to $B^*$. When $\sigma (A) = B$, we say $\sigma$ is a [*coding*]{}. We say $\sigma$ is [*non erasing*]{} if there is no $b\in A$ such that $\sigma (b)$ is the empty word. If $\sigma (A)$ is included in $B^+$, it induces by concatenation a map from $A^{\mathbb{Z}}$ to $B^{\mathbb{Z}}$: $\sigma (\dots x_{-1}. x_0 x_1 \dots ) = \dots \sigma (x_{-1}). \sigma (x_0) \sigma (x_1) \dots$, also denoted by $\sigma$. With the morphism $\sigma$ is naturally associated its [*incidence matrix*]{} $M_{\sigma} =
(m_{i,j})_{i\in B , j \in A }$ where $m_{i,j}$ is the number of occurrences of $i$ in the word $\sigma(j)$. Notice that for any positive integer $n$ we get $M_{\sigma^n} = M_{\sigma}^n$.
We say that an endomorphism is [*primitive*]{} whenever its incidence matrix is primitive ([*i.e.*]{}, when it has a power with strictly positive coefficients). The Perron’s theorem tells that the dominant eigenvalue is a real simple root of the characteristic polynomial and is strictly greater than the modulus of any other eigenvalue.
Substitutions and substitutive sequences
----------------------------------------
We say that an endomorphism $\sigma : A^* \rightarrow A^{*}$ is a [*substitution*]{} if there exists a letter $a \in A$ such that the word $\sigma(a)$ begins with $a$ and $\lim_{n\to+\infty}|\sigma^n(b)|=+\infty$ for any letter $b \in A$. In this case, for any positive integer $n$, $\sigma^n(a)$ is a prefix of $\sigma^{n+1}(a)$. Since $|\sigma^n(a)|$ tends to infinity with $n$, the sequence $(\sigma^n(\cdots aaa\cdots ))_{n\ge 0}$ converges (for the usual product topology on $A^\mathbb{Z}$) to a sequence denoted by $\sigma^\infty(a)$. The substitution $\sigma$ being continuous for the product topology, $\sigma^\infty(a)$ is a fixed point of $\sigma$: $\sigma (\sigma^\infty(a)) = \sigma^\infty(a)$.
A substitution $\sigma$ is [*left proper*]{} (resp. [*right proper*]{}) if all words $\sigma (b)$, $b \in A$, starts (resp. ends) with the same letter. For short, we say that a left and right proper substitution is [*proper*]{}.
The [*language*]{} of $\sigma : A^* \to A^*$, denoted by $\L (\sigma )$, is the set of words having an occurrence in $\sigma^n (b)$ for some $n\in \mathbb{N}$ and $b\in A$. Notice that we have $\L(\sigma^n) = \L(\sigma)$ for any positive integer $n$.
Dynamical systems and subshifts
-------------------------------
A [*measurable dynamical system*]{} is a quadruple $(X, \B, \mu, T)$ where $X$ is a space endowed with a $\sigma$-algebra $\B$, a probability measure $\mu$ and measurable map $T : X \to X$ that preserves the measure $\mu$, [*i.e.*]{}, $\mu(T^{-1}B) = \mu (B) $ for any $B \in \B$. This system is called [*ergodic*]{} if any $T$-invariant measurable set has measure $0$ or $1$. Two measurable dynamical systems $(X, \B, \mu, T)$ and $(Y, \B', \nu, S) $ are [*measure theoretically conjugate*]{} if we can find invariant subsets $X_{0} \subset X$, $Y_{0} \subset Y$ with $\mu(X_{0}) = \nu(Y_{0}) =1$ and a bimeasurable bijective map $\psi \colon X_{0} \to Y_{0}$ such that $S \circ \psi = \psi \circ T$ and $\mu(\psi^{-1} B) = \nu(B)$ for any $B \in \B'$.
By a [*topological dynamical system*]{}, or dynamical system for short, we mean a pair $(X,S)$ where $X$ is a compact metric space and $S$ a continuous map from $X$ to itself. It is well-known that such a system endowed with the Borel $\sigma$-algebra admits a probability measure $\mu$ preserved by the map $S$, and then form a measurable dynamical system. If the probability measure $\mu$ is unique, the system is said [*uniquely ergodic*]{}.
A [*Cantor system*]{} is a dynamical system $(X,S)$ where the space $ X $ is a Cantor space, [*i.e.*]{}, $ X $ has a countable basis of its topology which consists of closed and open sets and does not have isolated points. The system $(X,S)$ is [*minimal*]{} whenever $X$ and the empty set are the only $S$-invariant closed subsets of $X$. We say that a minimal system $(X,S)$ is [*periodic*]{} whenever $X$ is finite. A dynamical system $(Y,T)$ is called a [*factor*]{} of, or is [*semi-conjugate*]{} to, $(X,S)$ if there is a continuous and onto map $\phi: X \rightarrow Y$ such that $\phi \circ S = T \circ \phi$. The map $\phi $ is a [*factor map*]{}. If $\phi$ is one-to-one we say that $\phi $ is a [*conjugacy*]{}, and, that $(X,S)$ and $(Y,T)$ are [*conjugate*]{}.
For a finite alphabet $A$, we endow $A^{{\mathbb Z}}$ with the product topology. A [*subshift*]{} on $A$ is a pair $(X,S_{\mid X})$ where $X $ is a closed $S$-invariant subset of $A^{{\mathbb Z}}$ ($S(X) = X$) and $S$ is the [*shift transformation*]{}
----- --- ------------------------------ ---------------- ---------------------------------
$S$ : $A^{{\mathbb Z}}$ $\rightarrow $ $A^{{\mathbb Z}}$
$(x_n)_{ n \in {\mathbb Z}}$ $\mapsto$ $(x_{n+1})_{n\in {\mathbb Z}}$.
----- --- ------------------------------ ---------------- ---------------------------------
We call [*language*]{} of $X$ the set $\L (X) = \{ x_{[i,j]} ; x\in X, i\leq j\}$. A set defined with two words $u$ and $v$ of $A^{*}$ by $$[u.v]_X = \{ x\in X ; x_{[-|u|,|v|-1]} = uv \}$$ is called a [*cylinder set*]{}. When $u$ is the empty word we set $[u.v]_X = [v]_X$. The family of cylinder sets is a base of the induced topology on $X$. As it will not create confusion we will write $[u]$ and $S$ instead of $[u]_{X}$ and $S_{\mid X}$.
For $x$ a sequence on $A$, let $\Omega (x)$ be the set $\{ y \in A^{{{\mathbb N}}} ; y_{[i,j]} \in \L(x), \forall \ [i,j]\subset {\mathbb Z}\}$. It is clear that $(\Omega (x), S)$ is a subshift, it is called the [*subshift generated by*]{} $x$. Notice that $\Omega (x) = \overline{\{ S^n x ; n\in {\mathbb Z}\}}$. For a subshift $(X,S)$ on $A$, the following are equivalent:
1. $(X,S)$ is minimal;
2. For all $x\in X$ we have $X=\Omega (x)$;
3. For all $x\in X$ we have $\L(X)=\L(x)$.
We also have that $(\Omega (x), S)$ is minimal if and only if $x$ is uniformly recurrent. Note that if $(Y,S)$ is another subshift then, $\L(X) = \L(Y)$ if and only if $X=Y$.
Substitutive subshifts
----------------------
For primitive substitutions $\sigma$, all the fixed points are uniformly recurrent and generate the same minimal and uniquely ergodic subshift (for more details see [@Queffelec:1987]). We call it the [*substitutive subshift generated by $\sigma$*]{} and we denote it $(\Omega_\sigma , S)$.
There is another useful way to generate subshifts. For $\L$ a language on the alphabet $A$, define $X_\L\subset A^{\mathbb Z}$ to be the set of sequences $x = (x_n)_{n\in {\mathbb Z}}$ such that $\L(x) \subset \L$. The pair $(X_\L , T)$ is a subshift and we call it the [*subshift generated by $\L$*]{}. If $\sigma$ is a primitive substitution, then $\Omega_\sigma = X_{\L_{\sigma}}$ where $ \L_{\sigma} $ denotes the language of $\sigma$ [@Queffelec:1987]. It follows that for any positive integer $n$, $\sigma^n$ and $\sigma$ define the same subshift, that is $\Omega_{\sigma} = \Omega_{\sigma^n}$.
If the set $\Omega_{\sigma}$ is not finite, the substitution $\sigma$ is called [*aperiodic*]{}.
An algebraic number $\beta$ is called a [*Pisot-Vijayaraghan number*]{} if all its algebraic conjugates have a modulus strictly smaller than $1$.
\[def:Pisotsub\] Let $\sigma$ be a primitive substitution and let $P_{\sigma}$ denote the characteristic polynomial of the incidence matrix $M_{\sigma}$. We say that the substitution $\sigma$ is
- of [*Pisot type*]{} (or [*Pisot*]{} for short) if $P_{\sigma}$ has a dominant root $\beta >1$ and any other root $\beta'$ satisfies $0 < \vert \beta' \vert < 1$;
- of [*weakly irreducible Pisot type*]{} (or [*W. I. Pisot*]{} for short) whenever $P_\sigma$ has a real Pisot-Vijayaraghan number as dominant root, its algebraic conjugates, with possibly $0$ or roots of the unity as other roots;
- an [*irreducible*]{} substitution whenever $P_{\sigma}$ is irreducible over ${{\mathbb Q}}$;
- [*unimodular*]{} if $\det M_{\sigma} = \pm 1$.
For instance the Fibonacci substitution $0 \mapsto 01, 1 \mapsto 0$ and the Tribonacci substitution $1\mapsto 12, 2 \mapsto 13, 3 \mapsto 1$ are unimodular substitutions of Pisot type. Whereas the Thue-Morse substitution $0 \mapsto 01$, $1 \mapsto 10$ is a W. I. Pisot substitution. Notice that the notions of Pisot, W. I. Pisot, irreducible, unimodular depend only on the properties of the incidence matrix. So starting from a Pisot (resp. W. I. Pisot, irreducible, unimodular) substitution, we get many examples of Pisot (resp. W. I. Pisot, irreducible, unimodular) substitutions by permuting the letters of the initial one.
Standard algebraic arguments ensure that a Pisot substitution is an irreducible substitution, and of course, a Pisot substitution is of weakly irreducible Pisot type. In the following we will strongly use the fact that for any substitution of (resp. W. I. Pisot, irreducible, unimodular) Pisot type $\sigma$ and for every integer $n \ge 1$, the substitutions $\sigma^n$ are also of (resp. W. I. Pisot, irreducible, unimodular) Pisot type.
In , the authors prove that the fixed point of a unimodular substitution of Pisot type is non-periodic for the shift, thus the subshift generated is a non-periodic minimal Cantor system.
Dynamical spectrum of substitutive subshifts
---------------------------------------------
For a measurable dynamical system $(X, \B, \mu, T)$, a complex number $\lambda$ is an [*eigenvalue*]{} of the dynamical system $(X,\B, \mu, T)$ with respect to $\mu$ if there exists $f\in L^2(X,\mu)$, $f\not = 0$, such that $f\circ T = \lambda f$; $f$ is called an [*eigenfunction*]{} (associated with $\lambda$). The value $1$ is the [*trivial eigenvalue*]{} associated with a constant eigenfunction. If the system is ergodic, then every eigenvalue is of modulus 1, and every eigenfunction has a constant modulus $\mu$-almost surely. For a topological dynamical system, if the eigenfunction $f$ is continuous, $\lambda$ is called a [*continuous eigenvalue*]{}. The collection of eigenvalues is called the [*spectrum*]{} of the system, and form a multiplicative subgroup of the circle ${{\mathbb S}}=\{ z \in {{\mathbb C}}; \ \vert z\vert =1 \}$.
An important result for the spectrum is due to B. Host [@Host:1986]. It states that any eigenvalue of a substitutive subshift is a continuous eigenvalue. The following proposition, claimed in [@Host:1992] (see Proposition 7.3.29 in [@Fogg:2002] for a proof), shows that the spectrum of a unimodular substitution of Pisot type is not trivial.
\[prop:vpPisot\] Let $\sigma$ be a unimodular substitution of Pisot type and let $\alpha$ be a frequency of a letter in any infinite word of $\Omega_{\sigma}$. Then ${\rm exp}(2i\pi \alpha)$ is a continuous eigenvalue of the dynamical system $(\Omega_{\sigma}, S)$.
Recall that these frequencies are the coordinates of the right normalized eigenvector associated with the dominant eigenvalue of the incidence matrix of the substitution [@Queffelec:1987], and moreover for a unimodular Pisot substitution they are multiplicatively independent (Proposition 3.1 in ).
Notice the converse of the proposition is also true . For a proof, see the remark below Lemma \[lemme:condcont\] or Proposition 11 in . Actually, this is a general fact for any minimal Cantor system observed in [@Itza-Ortiz:2007]: given any continuous eigenvalue ${\rm exp}(2i\pi \alpha)$, $\alpha$ belongs to the additive subgroup of ${{\mathbb R}}$ generated by the intersection of sets of measures of clopen subsets for all the invariant probability measures. An other proof of that can be found in (Proposition 11) but it was not pointed out.
Domain exchange {#sec:domex}
---------------
Let us recall that a compact Euclidean set is said [*regular*]{} if it equals the closure of its interior.
\[def:domainexchange\] We call [*domain exchange transformation*]{} a measurable dynamical system $(E, \B, \tilde{\lambda}, T)$ where $E$ is a compact regular subset of an Euclidean space, $\tilde{\lambda}$ denotes the normalized Lebesgue measure on $E$ and $\mathcal B$ denotes the Borel $\sigma$-algebra, such that:
- there exist compact regular subsets $E_{1}$, $\ldots,$ $E_{n}$ such that $ E =E_{1}\cup \cdots \cup E_{n}$.
- The sets $E_{i}$ are disjoint in measure for the Lebesgue measure $\lambda$: $$\lambda(E_{i} \cap E_{j}) = 0 \hspace{0.5cm} \textrm{ when } i\neq j.$$
- For any index $i$, the map $T$ restricted to the set $E_{i}$, is a translation such that $T(E_{i}) \subset E$.
The domain exchange is said [*self-affine*]{}, if there is a finite number of affine maps $f_{1}, \ldots, f_{\ell}$ such that $ E = \bigcup_{i=1}^{\ell} f_{i}(E)$ and sharing the same linear part.
Matrix eigenvalues and return substitutions {#sec:returnsub}
===========================================
In this section, we recall the notion of return substitution introduced in [@Durand:1998a] and that any primitive substitutive subshift is conjugate to an explicit primitive and proper substitutive subshift without changing too much the eigenvalues of the associated substitution matrix [@Durand:1998b].
Let $A$ be an alphabet and $x \in A^{\mathbb Z}$ and let $u$ be a word of $x$. We call [*return word*]{} to $u$ of $x$ every factor $x_{[i,j-1]}$ where $i$ and $j$ are two successive occurrences of $u$ in $x$. We denote by $\R_{x,u}$ the set of return words to $u$ of $x$. Notice that for a return word $v$, $vu$ belongs to $\L(x)$ and $u$ is a prefix of the word $vu$. Suppose $x$ is uniformly recurrent. It is easy to check that for any word $u$ of $x$, the set $\R_{x,u}$ is finite. Moreover, for any sequence $y \in \Omega (x)$, we have $\R_{y,u} =\R_{x,u}$ . The sequence $x$ can be written naturally as a concatenation
$$x =\cdots m_{-1}m_{0} m_{1} \cdots, \hspace{1cm} m_{i} \in \R_{x, u}, \ i\in {\mathbb Z},$$ of return words to $u$, and this decomposition is unique. By enumerating the elements of $\R_{x, u}$ in the order of their first appearence in $(m_{i})_{i\ge 0}$, we get a bijective map $$\Theta_{x, u} \colon R_{x, u} \to \R_{x, u} \subset A^* ,$$ where $R_{x, u} =\{1, \ldots, \textrm {Card } (\R_{x, u} )\}$. This map defines a morphism. We denote by $ D_{u}(x)$ the unique sequence on the alphabet $R_{x,u}$ characterized by $$\Theta_{x,u} (D_{u}(x)) = x.$$
We call it the [*derived sequence of*]{} $x$ on $u$. Actually this sequence enables to code the dynamics of the induced system on the cylinder $[u]$. To be more precise, we need to introduce the following notions. A finite subset $\R \subset A^+$ is a [*code*]{} if every word $u \in A^+$ admits at most one decomposition in a concatenation of elements of $\R$.
We say that a code $\R$ is a [*circular code*]{} if for any words $$w_{1}, \ldots, w_{j}, w, w'_{1}, \ldots, w'_{k} \in \R; s\in A^+ \textrm{ and } t \in A^*$$ such that $$w=ts \textrm{ and } w_{1}\ldots w_{j} = sw'_{1} \ldots w'_{k}t$$ then $t$ is the empty word. It follows that $j=k+1$, $w_{i+1} =w'_{i'}$ for $1 \le i \le k$ and $w_{1}=s$.
\[mdretour\] Let $x$ be a uniformly recurrent sequence and let $u$ be a non empty prefix of $x$.
1. The set $\R_{x,u}$ is a circular code.
2. If $v$ is a prefix of $u$, then each return word on $u$ belongs to $\Theta_{x,v}(R^*_{x,v})$, [*i.e.*]{}, it is a concatenation of return words on $v$.
3. Let $v$ be a nonempty prefix of $D_{u}(x)$ and $w=\Theta_{x,u}(v)u$ then
- $w$ is a prefix of $x$,
- $D_{v} (D_{u}(x)) = D_{w}(x).$
- $\Theta_{x,u} \circ \Theta_{D_{u}(x) , v} = \Theta_{x,w}.$
The following proposition enables to associate to a substitution an other substitution on the alphabet $R_{x,u}$.
\[subst\_retour\] Let $x\in A^{{\mathbb N}}$ be a fixed point of the primitive substitution $\sigma$ which is not periodic for the shift and $u$ be a nonempty prefix of $x$. There exists a primitive substitution $\sigma_{u}$, defined on the alphabet $R_{x,u}$, characterized by $$\Theta_{x,u} \circ \sigma_{u} = \sigma \circ \Theta_{x,u}.$$
Even if this proposition is not stated for bi-infinite sequences, it follows that each derived sequence $D_{u}(x)$, where $u$ is a prefix of an aperiodic sequence $x \in A^{\mathbb Z}$ fixed by a primitive substitution $\sigma$, is a fixed point of the primitive substitution $\sigma_{u}$. To show this it is enough to check that $$\begin{aligned}
\Theta_{x,u} \circ \sigma_{u} (D_{u}(x)) = \sigma \circ \Theta_{x,u}( D_{u}(x) ) = \sigma x = x = \Theta_{x,u} \circ D_{u}(x).\end{aligned}$$ Since $\Theta_{x,u}(R_{x,u})$ is a circular code, we get that the sequence $D_{u}(x)$ is fixed by the substitution $\sigma_{u}$. This substitution, defined in the previous proposition, is called the [*return substitution*]{} ([*to* ]{} $u$). Moreover, we observe that for any integer $l >0$ $$(\sigma^l)_{u} = (\sigma_{u})^l.$$
Furthermore the incidence matrix of the return substitution has almost the same spectrum as the initial substitution. More precisely, we have:
\[subst\_retourvp\] Let $\sigma$ be a primitive substitution and let $u$ be a prefix of a fixed point $x$ which is not shift periodic. The incidence matrices $M_{\sigma }$ and $M_{\sigma_{u}}$ have the same eigenvalues, except perhaps zero and roots of the unity.
For instance for the Tribonacci substitution $\tau$, the induced substitution $\tau_{1}$ is the same as $\tau$. On the other hand, if we consider the substitution $$\sigma \colon 1 \mapsto 1123, 2\mapsto 211, \textrm{ and } 3 \mapsto 21,$$ it is also a substitution of Pisot type and the incidence matrix of the induced substitution $\sigma_{11}$ has $0$ as eigenvalue.
With the next property we obtain that if an induced system of a subshift $(X,S)$ is a proper substitutive subshift $(\Omega, S)$, then the system $(X,S)$ is conjugate to a proper substitutive subshift. The system $(X,S)$ is called an [*exduction*]{} of the system $(\Omega, S)$.
\[expansion\] Let $y = (y_{i})_{i\in {\mathbb Z}}$ be a fixed point of an aperiodic primitive substitution $\sigma$ on the alphabet $R$. Let $\Theta : R^* \to A^+$ be a non-erasing morphism, $x= \Theta(y)$ and $(X,S)$ be the subshift generated by $x$.
Then, there exist a primitive substitution $\xi$ on an alphabet $B$, an admissible fixed point $z$ of $\xi$, and a map $ \phi : B \to A$ such that:
1. $\phi(z) = x$;
2. If $\Theta(R)$ is a circular code, then $\phi$ is a conjugacy from $(\Omega_{\xi}, S)$ to $(X,S)$;
3. If $\sigma$ is proper (resp. right or left proper), then $\xi$ is proper (resp. right or left proper);
4. There exists a prefix $u \in B^+$ of $z$ such that $R_{y,y_{0}} = R_{z,u}$ and there is an integer $l \ge 1$ such that the return substitutions $\sigma_{y_{0}}^l$ and $\xi_{u}$ are the same.
Actually the first three statements of this proposition, correspond to Proposition 23 in . The substitution $\xi$ is explicit in the proof.
The statements 1), 2), 3), and the fact that $\xi$ is primitive, have been proven in . We will just give the proof of the first statement because we need it to prove the fourth statement.
Substituting a power of $\sigma$ for $\sigma$ if needed, we can assume that $\vert \sigma(j) \vert \ge \vert \Theta (j) \vert $ for any $j \in R$. For all $j \in R$, let us denote $m_{j} = \vert \sigma(j) \vert$ and $n_{j} = \vert \Theta(j) \vert$. We define
- An alphabet $B := \{ (j, p) ; j\in R, 1 \le p \le n_{j} \}$;
- A morphism $\phi \colon B^* \to A^*$ by $\phi(j,p) = (\Theta( j))_{p}$;
- A morphism $\psi \colon R^* \to B^*$ by $\psi(j) =(j,1)(j,2) \cdots (j, n_{j})$.
Clearly, we have $\phi \circ \psi = \Theta$. We define a substitution $\xi$ on $B$ by $$\begin{aligned}
\forall j \in R, \ 1 \le p \le n_{j} ; \ \xi(j,p )= \begin{cases} \psi((\sigma(j))_{p}) & \textrm{ if } 1 \le p < n_{j}\\ \psi((\sigma(j))_{[n_{j},m_{j}]}) & \textrm{ if } p= n_{j}. \end{cases} \end{aligned}$$
Thus for every $j \in R$, we have $\xi(\psi(j)) = \xi(j,1)\ldots \xi(j, n_{j}) = \psi(\sigma(j))$, [*i.e.*]{},
$$\label{eq:returnsub}
\xi \circ \psi = \psi \circ \sigma.$$
For $z = \psi(y)$ we obtain $\xi(z) = \psi(\sigma(y)) = \psi(y) = z$, that is $z$ is a fixed point of $\xi$. Moreover $\phi(z) = \phi(\psi(y)) = \Theta(y) =x$ and we get the point (1).
Let us prove the fourth statement.
Let $u =\psi(y_{0}) \in B^*$ where $y = \ldots y_{-1}. y_{0}y_{1}\ldots$, $y_i \in B$, $i\in \mathbb{Z}$. First, notice the morphism $\psi$ is one-to-one and then we have $\psi(\R_{y, y_{0}}) = \R_{\psi(y), \psi(y_{0})}$. It follows that $$R_{y,y_{0}} = R_{\psi(y), \psi(y_{0})} = R_{z,u},$$ and $$\psi \circ \Theta_{y,y_{0}} = \Theta_{\psi(y), \psi(y_{0})} = \Theta_{z,u}.$$
Therefore for the return substitution $\sigma_{y_{0}}$ to $y_{0}$, Proposition \[subst\_retour\] and Relation (\[eq:returnsub\]) give $$\Theta_{z, u} \circ \sigma_{y_{0}} = \psi \circ \Theta_{y,y_{0}} \circ \sigma_{y_{0}}
= \psi \circ \sigma \circ \Theta_{y,y_{0}}
= \xi \circ \psi \circ \Theta_{y,y_{0}}
=\xi \circ \Theta_{z, u}.$$
Consequently, we have $ \sigma_{y_{0}} = \xi_{u}$.
As a straightforward corollary of the propositions \[subst\_retour\], \[mdretour\], \[expansion\] and \[subst\_retourvp\], we get
\[coro:wPisot\] Let $\sigma $ be a primitive aperiodic substitution. Then there exists a proper primitive substitution $\xi$ on an alphabet $B$, such that
1. $(\Omega_{\sigma}, S)$ is conjugate to $(\Omega_{\xi}, S)$;
2. there exists $l\geq 1$ such that the substitution matrices $M_{\sigma}^l$ and $M_{\xi}$ have the same eigenvalues, except perhaps 0 and 1.
Let us fix a nonempty prefix $u$ of a fixed point $x$ of $\sigma$. Thus $x$ is not shift periodic. Substituting a power of $\sigma$ for $\sigma$ if needed, we can assume that the word $\Theta_{x,u}(1)u$ is a prefix of $\sigma(u)$. By the very definition of return word, for any letter $i \in R_{x,u}$, the word $\Theta_{x,u}(i)u$ has the word $u$ as a prefix. Then $\Theta_{x,u}(1)u$ is a prefix of the word $\sigma (\Theta_{x,u} (i) u)$. It follows from the equality in Proposition \[subst\_retour\], that $\Theta_{x,u}(1)u$ is also a prefix of the word $\Theta_{x,u}\circ \sigma_{u}(i)$. The uniqueness of the coding by $\Theta_{x,u} (R_{x,u})$, implies that the word $\sigma_{u}(i)$ starts with $1$, and the substitution $\sigma_{u}$ is left proper.
The propositions \[mdretour\] and \[expansion\] imply the existence of a left proper primitive substitution $\xi'$ such that $(\Omega_{\sigma}, S)$ is conjugate to $(\Omega_{\xi'}, S)$, moreover by Proposition \[subst\_retourvp\] there exists an integer $l>0$ such that the incidence matrices $M_{\sigma}^l$ and $M_{\xi'}$ share the same eigenvalues, except perhaps 0 and 1.
To obtain a proper substitution we need to modify $\xi'$. Let $a$ be the letter such that for all letter $b$, $\xi' (b)= aw(b)$ for some word $w(b)$. Now consider the substitution $\xi''$ defined by $\xi'' \colon b \mapsto w(b)a$. Then, $\xi'$ and $\xi''$ define the same language, so we have $\Omega_{\xi'} = \Omega_{\xi''}= \Omega_{\xi}$ where $\xi$ is the composition of substitutions $ \xi' \circ \xi''$ and is proper. We conclude observing that $M_{\xi} = M_{\xi'} M_{\xi''} = M_{\xi'}^2$.
In terms of Pisot substitutions, Corollary \[coro:wPisot\] becomes:
\[cor:Pisotsub\] Let $\sigma$ be an aperiodic substitution of Pisot type, then the substitutive subshift associated with $\sigma$ is conjugate to a substitutive subshift $(\Omega_{\xi}, S)$ where $\xi$ is a proper primitive substitution of weakly irreducible Pisot type.
The example after Proposition \[subst\_retourvp\] shows that the use of return substitutions seems to force to deal with W. I. Pisot substitutions. In fact, it is unavoidable to consider W. I. Pisot substitution to represent a substitutive subshift by a proper substitution. For instance, consider the non-proper substitution $\sigma : 0 \mapsto 001,$ $1 \mapsto 10$. The dimension group of the associated subshift, computed in [@Durand:1996], is of rank $3$. As a consequence, any proper substitution $\xi$ representing the subshift $\Omega_{\sigma}$ should be, at least, on $3$ letters (see for the details). Moreover Cobham’s theorem (see Theorem 14 in [@Durand:1998c]) for minimal substitutive subshifts implies that, taking powers if needed, $\xi$ and $\sigma$ share the same dominant eigenvalue. So, the substitution $\xi$ can not be irreducible.
Conjugacy with a domain exchange {#sec:domex}
================================
In this section we give sufficient conditions on a primitive proper substitution so that the associated substitutive system is measurably conjugate to a domain exchange in an Euclidean space.
Using Kakutani-Rohlin partitions {#subsec:KR}
--------------------------------
In this subsection, we will assume that $\xi$ is a primitive proper substitution on a finite alphabet $A$ equipped with a fixed order. First let us recall a structure property of the system $(\Omega_{\xi}, S)$ in terms of Kakutani-Rohlin towers.
\[substKR\] Let $\xi$ be a primitive proper substitution on a finite alphabet $A$. Then for every $n > 0$, $$\P_{n} =\{S^{-k} \xi^{n-1} ([a]) ; \ a \in A, \ 0 \le k \le \vert \xi^{n-1} (a) \vert -1 \}$$ is a clopen partition of $\Omega_{\xi}$ defining a nested sequence of Kakutani-Rohlin partition of $\Omega_{\xi}$, more precisely:
- The sequence of bases $(\xi^n(\Omega_{\xi}))_{n\ge 0}$ is decreasing and the intersection is only one point;
- For every $n>0$, $\P_{n+1}$ is finer than $\P_{n}$;
- The sequence $(\P_{n})_{n>0}$ spans the topology of $\Omega_{\xi}$.
To be coherent with the notations in , we take the conventions $\P_{0} = \{ \Omega_{\xi} \} $ and for an integer $n \ge 1$, $r_{n}(x) $ denotes the [*entrance time*]{} of a point $x \in \Omega_{\xi}$ in the base $\xi^{n-1}(\Omega_{\xi})$, that is $$r_n (x) = \min \{ k\geq 0 ; \ S^k x \in \xi^{n-1} (\Omega_\xi ) \}.$$
By minimality, this value is finite for any $x \in \Omega_{\xi}$ and the function $r_{n}$ is continuous.
The homeomorphism $ S_{\xi(\Omega_{\xi})} \colon \xi(\Omega_{\xi}) \ni x \mapsto S^{r_{2}(Sx)}(Sx) \in \xi(\Omega_{\xi})$ is then the induced map of the system $(X,S)$ on the clopen set $\xi(\Omega_{\xi})$. Since we have the relation $$\begin{aligned}
\label{eq:sysinduit}
\xi \circ S = S_{\xi(\Omega_{\xi})} \circ \xi,\end{aligned}$$ the induced system $(\xi(\Omega_{\xi}), S_{\xi(\Omega_{\xi})})$ is a factor of $(\Omega_{\xi},S)$ via the map $\xi$ (and in fact a conjugacy).
Note that for any integer $n>0$, $$\begin{aligned}
\label{eq:tpsretour}
r_{n}(Sx) - r_{n}(x) = \begin{cases} -1 & \textrm{if } x \not\in \xi^{n-1}(\Omega_{\xi}) \\
\vert \xi^{n-1}(a) \vert -1 & \textrm{if } x \in \xi^{n-1}([a]), a\in A.
\end{cases} \end{aligned}$$
More precisely, we can relate the entrance time and the incidence matrix by the following equality (see Lemma in ): For a primitive proper substitution $\xi$, we have for any $x \in \Omega_{\xi}$ and $n \ge 2$ $$\begin{aligned}
\label{eq:rn}
r_n(x) = \sum_{k=1}^{n-1}
\langle s_k (x) ,(M_\xi^t)^k H(1)\rangle\end{aligned}$$
where $\langle \cdot, \cdot \rangle$ denotes the usual scalar product, $M_{\xi}^t$ is the transpose of the incidence matrix, $H(1) =(1, \cdots, 1)^t $ and $s_k : \Omega_\xi \to \mathbb{Z}^{\#A}$ is a continuous function defined by $$s_k (x)_a = \# \{ r_k (x) < i \leq r_{k+1} (x) ; \ S^i x \in \xi^{k-1} ([a]) \}, \hspace{1cm} \textrm{for }a \in A.$$
In other words, the vector $s_{k}(x)$ counts, in each coordinate $a \in A$, the number of time that the positive iterates of $x$ meet the clopen set $\xi^{k-1}([a])$ before meeting for the first time the clopen set $\xi^{k}(\Omega_{\xi})$ and after meeting the clopen set $\xi^{k-1}(\Omega_{\xi})$.
The proof of the following lemma is direct from the definition and Proposition \[substKR\].
\[lemma:lemmeclassic\] For $\xi$ a primitive proper substitution, we have, for any $x \in \Omega_{\xi}$, $$\begin{aligned}
s_{1}(\xi x) = 0 \hspace{0.5cm} \textrm{ and } \hspace{0.5cm} \forall k >1, \
s_{k}(\xi x) = s_{k-1}(x).
$$ For any letter $a \in A$, $k\in {{\mathbb N}}^*$, we also have $ s_{k} (x)_{a} \le \sup_{b \in A} \vert \xi(b) \vert$.
From the ergodic point of view, it is well-known (see [@Queffelec:1987]) that subshifts generated by primitive substitutions are uniquely ergodic. We call $\mu$ the unique probability shift-invariant measure of $(\Omega_\xi , S)$. We have the following relations, for any positive integer $n$, $$\begin{aligned}
\label{eq:measure}
\vec{\mu}({n}) = M_{\xi} \vec{\mu}({n+1}), \hspace{0.5cm} \textrm{ and } \hspace{0.5cm} \langle H(1), \vec{\mu}(1) \rangle = 1,\end{aligned}$$ where $\vec{\mu}({n}) \in {{\mathbb R}}^{\sharp A}$ is the vector defined by $$\vec{\mu}(n)_{a} = \mu(\xi^{n-1}([a])), \hspace{1cm} \textrm{ for any letter } a \in A.$$
On the spectrum of a substitutive subshift
------------------------------------------
From this subsection, we assume that $\xi $ is a primitive proper substitution on a finite alphabet $A$. Taking a power of $\xi$ if needed, from classical results of linear algebra, there are $M_{\xi }^{t}$-invariant ${{\mathbb R}}$-vectorial subspaces $E^0, E^u, E^b$ and $E^s$ such that
1. ${{\mathbb R}}^{\#A} = E^0 \oplus E^s \oplus E^u \oplus E^b$,
2. $M_\xi^t v= 0$ for all $v \in E^0$,
3. $\lim_{ k \to + \infty}(M_\xi^{t})^k v =0$, $(M_{\xi}^t)^{n}v \not = 0$ for all $v\in E^s\setminus\{0\}$ and any $n\in {{\mathbb N}}$,
4. $\lim_{ k \to +\infty}||(M_\xi^{t})^k v|| = +\infty$ for all $v\in E^u\setminus\{0\}$ and
5. $((M_\xi^{t})^k v)_{k \in \mathbb{Z}}$ is bounded and $(M_{\xi}^{t})^{n}v \not = 0$ for all $v\in E^b\setminus\{0\}$ and $n \in {{\mathbb N}}$.
Let us apply some well-know facts to our context (see [@Host:1986] or for substitutions and for a wider context). Let $r_n$ and $s_n$ be as defined in Section \[subsec:KR\].
\[prop:condcont\] Let $\xi$ be a primitive proper substitution on an alphabet $A$. If $\lambda \in {{\mathbb S}}$ is an eigenvalue of the system $(\Omega_\xi , S)$, then $(\lambda^{-r_n}) _{ n\geq 1}$ converges uniformly to a continuous eigenfunction associated with $\lambda$. Moreover, $\sum_{n\ge 1} \max_{a\in A} |\lambda^{|\xi^n (a)|}-1|$ converges.
So if $\exp(2i\pi\alpha)$ is an eigenvalue of the substitutive system $(\Omega_{\xi}, S)$, for any letter $a$ of the alphabet $\vert \xi^{n} (a) \vert \alpha$ converges to $0$ mod ${\mathbb Z}$ as $n$ goes to infinity. In an equivalent way the vector $(M_{\xi}^t)^n \alpha (1,\cdots, 1)^t $ tends to $0$ mod ${\mathbb Z}^{\# A}$. The next lemma precises this for the usual convergence.
\[lemme:condcont\] Let $\lambda=\exp(2i\pi\alpha)$ be an eigenvalue of a substitutive system $(\Omega_\xi ,S)$ for a primitive proper substitution $\xi$ on a finite alphabet $A$. Then, there exist $m \in {{\mathbb N}}$, $v \in
{{\mathbb R}}^{\#A}$ and $w \in {\mathbb Z}^{\#A}$ such that $$\displaystyle \alpha H(1)=v + w, \hspace{1cm} (M_{\xi}^{t})^{m} w \in {\mathbb Z}^{\# A} \hbox{ and } (M_\xi^t)^{n} v
\rightarrow_{n\to \infty} 0,$$
where all entries of $H(1)$ are equal to $1$. Moreover
- The convergence is geometric: there exist $0 \le \rho < 1$ and a constant $C$ such that $$\vert\vert (M_\xi^t)^{n} v \vert \vert \le C \rho^n, \textrm{ for any } n\in {{\mathbb N}}.$$
- For any positive integer $n$, $$\langle v, \vec{\mu}(n) \rangle =0 \hspace{0.5cm} \textrm{ and } \hspace{0.5cm} \alpha = \langle (M^{t}_{\xi})^{n-1} w, \vec{\mu}(n) \rangle.$$
The first claim and item $i)$ comes from [@Host:1986]. We have just to show the item $ii)$. Notice that the relations (\[eq:measure\]) give us for any positive integer $$\begin{aligned}
\langle v ,\vec{\mu}(n) \rangle = \langle v, M_{\xi}^{p} \vec{\mu}(n+p) \rangle = \langle (M_{\xi}^{t})^{p} v, \vec{\mu}(n+p) \rangle \to_{ p\to +\infty} 0.
\end{aligned}$$ We deduce then $$\alpha = \alpha \langle H(1), \vec{\mu}(1) \rangle = \langle v, \vec{\mu}(1) \rangle + \langle w, \vec{\mu}(1) \rangle = \langle w, \vec{\mu}(1) \rangle = \langle (M^{t}_{\xi})^{n-1} w, \vec{\mu}(n) \rangle.$$
[**Remark.**]{} We get by Item $ii)$ of Lemma \[lemme:condcont\], that if $\exp(2i\pi\alpha)$ is an eigenvalue of a substitutive system, then $\alpha$ is in the subgroup of ${{\mathbb R}}$ generated by the component of the vector $\vec{\mu}(n)$, that is, in the subgroup generated by the frequency of occurrences of the words. This provides a converse to Proposition \[prop:vpPisot\].
If $\exp(2i\pi \alpha_{1} ), \ldots, \exp(2i \pi \alpha_{d-1})$ are $d-1$ eigenvalues of the substitutive system $(\Omega_{\xi}, S)$, from Proposition \[prop:condcont\] and Lemma \[lemme:condcont\] there exist $m \in {{\mathbb N}}$, $v (1), \dots v (d-1) \in
{{\mathbb R}}^{{\sharp A}}$ and $w (1) , \dots ,w (d-1) \in {\mathbb Z}^{\sharp A}$ such that for all $i \in \{ 1, \dots , d-1\}$:
$$\label{eq:wetv}
\alpha_i H(1)=v(i) + w(i), \hspace{0.5 cm} (M_\xi^{t})^m w(i) \in {\mathbb Z}^{\# A} \hbox{ and } \sum_{n\ge 1} (M_\xi^{t})^n v(i)
\textrm{ converges}.$$
Notice that up to take a power of $\xi$, if needed, we can assume that the constant $m =1$ and that any $v(i)$ has no component in $E^{0}$.
Let us recall Proposition \[prop:vpPisot\]: a unimodular Pisot substitutive subshift on $d$ letters admits $d-1$ non trivial eigenvalues $\exp(2i\pi \alpha_{1} ), \ldots, \exp(2i \pi \alpha_{d-1})$ that are [*multiplicatively independent*]{}, [*i.e.*]{}, $1, \alpha_{1}, \ldots, \alpha_{d-1}$ are rationally independent. This motivates the next proposition that interprets the arithmetical properties of the eigenvalues in terms of the vectors $v(i)$ and $w(i)$.
\[prop:linind\] If $\exp(2i\pi \alpha_{1} ), \ldots, \exp(2i \pi \alpha_{d-1})$ are $d-1$ multiplicatively independent eigenvalues of the substitutive system $(\Omega_{\xi}, S)$ for a proper primitive substitution $\xi$. Then, both families of vectors $\{M_{\xi}^tv(1),$ $\ldots, M_{\xi}^tv ({d-1}) \}$ and $\{M_{\xi}^tH(1),$ $M_{\xi}^t w({1}),$ $\ldots,$ $M_{\xi}^t w({d-1}) \}$ are linearly independent.
Notice it implies also that both family of vectors $\{v(1),$ $\ldots$, $v ({d-1}) \}$ and $\{H(1),$ $w({1})$, $\ldots$, $w({d-1}) \}$ are linearly independent.
The proof is similar to Proposition 10 in . We adapt it to our case.
Assume there exist reals $\delta_{0}, \delta_{1}, \ldots, \delta_{d-1} $, one being different from $0$, such that $\delta_{0} M^{t}_{\xi} H(1)+ \sum_{i=1}^{d-1} \delta_{i }M_{\xi} ^{t}w({i}) = 0$. Since all the vectors are in ${\mathbb Z}^{\sharp A}$, by an algebraic classical result, we can assume that any $\delta_{i}$ is an integer. Taking the inner product of this sum with the vector $\vec{\mu(2)}$, the normalization and recurrence relations of this vector (Relation (\[eq:measure\])) together with the normalization with respect to each $w({i})$ in item $ii)$ of Lemma \[lemme:condcont\], give us $\delta_{0}+ \sum_{i=1}^{d-1} \delta_{i} \alpha_{i} =0$. The rational independence of the numbers $1, \alpha_{1}, \ldots, \alpha_{d-1}$ implies any $\delta_{i} =0$. So the vectors $M_{\xi}^{t}H(1), M_{\xi}^{t}w({1}), \ldots, M_{\xi}^{t}w({d-1})$ are independent. Now, assume that there exist real numbers $\lambda_{i}$ such that $\sum_{i=1}^{d-1}\lambda_{i} M_{\xi}^t v({i}) =0$. We obtain $(\sum_{i=1}^{d-1}\lambda_{i} \alpha_{i})M_{\xi}^t H(1) - \sum_{i=1}^{d-1} \lambda_{i} M_{\xi}^t w({i}) =0$. The independence of the vectors $M_{\xi}^t H(1), M_{\xi}^t w({1}), \ldots$, $ M_{\xi}^t w({d-1})$ implies that $\lambda_{i} =0$ for any $i$. So the vectors $M_{\xi}^t v({1}), \ldots, M_{\xi}^t v({d-1})$ are independent.
The following property gives a bound on the number of multiplicatively independent eigenvalues for a substitutive subshift.
\[prop:ME\] Let $\xi$ be a proper primitive substitution. If the substitutive system $(\Omega_{\xi}, S)$ admits $d-1$ eigenvalues $\exp(2i\pi \alpha_{1})$, $\ldots$, $\exp(2i \pi \alpha_{d-1})$, then the vectorial space spanned by the vectors $v(i)$, $E_{\xi} = {\rm Vect}( v (1)$, $\ldots$, $v (d-1))$, is a subspace of $E^{s}$.\
Moreover if the eigenvalues are muliplicatively independent, then $d-1\le \textrm{dim } E^{s}$.
For $i\in \{1, \dots , d-1 \}$, the vector $v(i)$ can be decomposed using the ${{\mathbb R}}$-vectorial subspaces $E^0, E^u, E^b$ and $E^s$. From Lemma \[lemme:condcont\] it has no component in $E^u$ and $E^b$. From the choice we made in , it has no component in $E^0$. Thus $v(i)$ belongs to $E^s$. So we get $E_{\xi} \subset E^s$. The bound by the dimension is obtained with Proposition \[prop:linind\].
To construct the domain exchange of a Pisot substitution we will need the following direct corollary.
\[coro:ME\] Let $\xi$ be a proper primitive substitution. If the substitutive system $(\Omega_{\xi}, S)$ admits $\textrm{dim } E^{s}$ multiplicatively independent eigenvalues, then $ E_{\xi} := {\rm Vect}( v (1), \ldots, v (\textrm{dim } E^{s})) = E^{s}$. In particular, we have $M_{\xi}^{t} (E_{\xi}) = E_{\xi}$.
Notice that for a unimodular Pisot substitution $\sigma$, $\textrm{dim } E^{s} +1$ equals the degree of the associated Pisot number, or the number of letters in the alphabet. Thus, by Proposition \[prop:vpPisot\], the proper W. I. Pisot substitution $\xi$ associated to $\sigma$ in Corollary \[cor:Pisotsub\], fulfills the conditions of Corollary \[coro:ME\].
Semi-conjugacy with the domain exchange {#sec:proof}
---------------------------------------
We prove the main result, Theorem \[theo:main\], in this section. For this, we start recalling the very hypotheses we need to get the result.
[**Hypotheses `P`.**]{}
*Let $\xi$ be a primitive proper substitution on a finite alphabet $A$ such that:*
- \[hypi\] The characteristic polynomial $P_{\xi}$ admits a unique root greater than one in modulus.
- \[hypii\] The substitutive subshift $(\Omega_{\xi}, S)$ admits $\textrm{ dim } E^{s} = d-1$ eigenvalues $\exp(2i\pi \alpha_{1} ),$ $\ldots,$ $\exp(2i \pi \alpha_{d-1})$ such that $1,$ $\alpha_{1},$ $\ldots,$ $\alpha_{d-1}$ are rationally independent.
- \[hypiii\] Its Perron number $\beta$ satisfies $\beta \vert \det M^{t}_{\xi\vert E^{s}} \vert =1$.
For instance, all these hypotheses apply to the proper substitution $\xi$ of Corollary \[cor:Pisotsub\] associated with a unimodular Pisot substitution on $d$ letters: The statement $i)$ is obvious, the others come from the fact that the space $E^{s}$ is spanned by the eigenspaces associated with the algebraic conjugates $\beta_{1}, \ldots, \beta_{d-1}$ of the Pisot number leading eigenvalue $\beta$ of $M_{\xi}$. The unimodular hypothesis implies $\vert \beta \beta_{1} \cdots \beta_{d-1} \vert =1$.
From Hypotheses `P` $ii)$ and by a byproduct of the formula (\[eq:rn\]) on the entrance time $r_{n}$, with Formula (\[eq:wetv\]) on the vectors $v({i})$, up to consider a power of $\xi$, we get for any $i\in \{1, \ldots, d-1\}$ and $x\in \Omega_\xi$ $$\begin{aligned}
\alpha_i r_n (x) & = \sum_{k=1}^{n-1} \langle s_k (x),(M_\xi^{t})^k v (i)\rangle \mod {\mathbb Z}. \end{aligned}$$
Let $F_n = \left( \sum_{k=0}^{n-1} \langle s_k ,(M_\xi^{t})^k v (i)\rangle \right)^{t}_{1\leq i\leq d-1}$. The Proposition \[prop:condcont\] and Lemma \[lemme:condcont\] ensure the sequence $(F_n)_n$ uniformly converges to a continuous function $F \colon \Omega_{\xi} \to {{\mathbb R}}^{d-1}$, explicitly defined for $x \in \Omega_{\xi}$ by $$F(x) = \left(
\sum_{k=1}^{+\infty} \langle s_k(x) ,(M_\xi^t)^k v (i)\rangle
\right)^{t}_{1\leq i\leq d-1}.$$
Let $V$ be the matrix with rows $v (1)^{t}, \ldots, v (d-1)^{t}$. Then, the map $F$ may be written as $$F(x) = V\sum_{k=1}^{+\infty} M_\xi^{k} s_k(x).$$
\[lemma:deltaV\] Assume Hypotheses `P` $i), ii)$. There exist a continuous map $\Delta \colon \Omega_\xi \to {{\mathbb R}}^{\# A} $ and a bijective linear map $N \colon {{\mathbb R}}^{d-1} \to {{\mathbb R}}^{d-1}$ such that for $\alpha = (\alpha_{1}, \ldots, \alpha_{d-1})^{t}$ and for any $x \in \Omega_{\xi}$,
1. \[lemma:conjrot\] $F\circ S (x) = F (x) + \alpha \mod {\mathbb Z}^{d-1} $;
2. \[lemma:decompF\] $F (x) = V \Delta (x)$;
3. $M_\xi^t V^t = V^t N$;
4. the matrix $N$ is conjugated to the matrix $M^{t}_{\xi\vert E^{s}}$ restricted to the space $E^{s}$;
5. \[lemma:conjsub\] $F\circ \xi (x) = N^t ( F (x))$.
By the approximation property of the eigenfunctions in Proposition \[prop:condcont\] (see also Relation ), we get $F\circ S (x) = F (x) + \alpha \mod {\mathbb Z}^{d-1} $.
Let us prove Statement . We have $$\begin{aligned}
\label{eq:Fn}
F_n(x) = & V \left(\sum_{k=1}^{n-1} M_\xi^{k}s_{k}(x) \right) \\
=& V Proj \left(\sum_{k=1}^{n-1} M_\xi^{k}s_{k}(x) \right), \end{aligned}$$ where $Proj \colon {{\mathbb R}}^{\#A} \to E_{\xi}= \textrm{Vect }(v(1), \ldots, v(d-1))$ denotes the orthogonal projection onto $E_{\xi}$. Recall that by Corollary \[coro:ME\], $E_{\xi}$ has dimension $d-1$. Since $(F_n)_n$ uniformly converges (see Proposition \[prop:condcont\] and Lemma \[lemme:condcont\]), the projection $Proj (\sum_{k=1}^{n-1} M_\xi^{k}s_{k}(x))$ converges when $n$ goes to infinity to the vector $\Delta(x)$ belonging to $E_{\xi}$ for any $x \in \Omega_\xi$. Therefore, we obtain Statement .
Let us prove the other statements. The basic properties of $s_{n} \circ \xi$ (Lemma \[lemma:lemmeclassic\]) give for any $x \in \Omega_\xi$ and $n > 2$, $$\label{eq:Fnxi}
F_n\circ \xi = V M_\xi \left(\sum_{k=1}^{n-2} M_\xi^{k}s_{k} \right).$$
By the ${{\mathbb R}}$-independence of the vectors $v (i)$ (Proposition \[prop:linind\]), the linear map $V^t \colon {{\mathbb R}}^{d-1} \to E_{\xi}$ is bijective and since $M_\xi^{t} (E_{\xi}) = E_{\xi}$ (Corollary \[coro:ME\]), there exists a bijective linear map $N \colon {{\mathbb R}}^{d-1} \to{{\mathbb R}}^{d-1} $ such that
$$\begin{aligned}
\label{align:eigen}
M_\xi^{t} V^{t} = V^{t} N.\end{aligned}$$
This shows Statement (4). Therefore, using and , we obtain for $n>2$, $$F_{n}\circ \xi = V M_\xi \sum_{k=1}^{n-2} M_\xi^{k}s_{k} = N^{t } F_{n-1}.$$
Passing through the limit in $n$, we get and this achieves the proof.
From Lemma \[lemma:perron\] to Proposition \[prop:Finj\], we use the strategy developed in to tackle the Pisot conjecture. Recall that $\mu$ denotes the unique probability shift-invariant measure of the system $(\Omega_{\xi}, S)$, and $\lambda$ denotes the Lebesgue measure on $F (\Omega_\xi )$.
\[lemma:perron\] Assume Hypotheses `P` $i) -iii)$. There exists a constant $C$ such that for any letter $a \in A$ we have:
1. \[item:perronconstant\] $\lambda (F ([a])) = C \mu ( [a]) $,
2. \[item:xinperronconstant\] for any integer $n$ large enough, $F ([a])$ is the union of the measure theoretically disjoint sets $$F(S^{-k}\xi^n ([b])) , \hbox{ with } 0\leq k< |\xi^n (b)| , [a] \cap S^{-k} \xi^n ([b])\not = \emptyset,$$
3. \[item:borelperronconstant\] for any Borel set $B \subset [a]$, $$\lambda (F (B)) = C \mu (B) .$$
Let $G= F\circ S - F - (\alpha_{1}, \ldots, \alpha_{d-1})^t$. From the basic properties of the map $F$ (Lemma \[lemma:conjsub\]), it takes integer values. Being continuous, it is locally constant. Hence, there exists some integer $n_{0}\ge 0$ such that $G$ is constant on each sets $S^{-k} \xi^n([b])$, with $n > n_{0}$, $b\in A$ and $0\leq k< |\xi^n (b)|$ (see Proposition \[substKR\]).
Therefore, from Item of Lemma \[lemma:deltaV\], for any such $b$ and $k$, there exists a vector $\delta(k,b)\in {{\mathbb R}}^{d-1}$ such that $$F (S^{-k} \xi^n ([b])) = \delta (k,b) + F (\xi^n ([b])) = \delta (k,b) +(N^t)^n F ([b]).$$
By the very hypothesis $\texttt{P } iii)$, we have $\vert \det N^t \vert =1/ \beta$, so we get $$\lambda ( F (S^{-k} \xi^n ([b])) ) = \lambda ( (N^t)^n F ([b])) = \vert \det (N^t)^n \vert \lambda (F([b])) = \frac{1}{\beta^n} \lambda (F([b])).$$
Let $a\in A$, the partitions of $\Omega_{\xi}$ in Proposition \[substKR\] provide $$[a] = \bigcup_{\substack{0\leq k< |\xi (j)|, b\in A \\ [a] \cap S^{-k} \xi^n ([b])\not = \emptyset}} S^{-k} \xi^n([b]) .$$
Consequently, $$\begin{aligned}
\label{align:perronequality}
\lambda ( F ([a])) \leq & \sum_{\substack{k,b; 0\leq k< |\xi^n (b)| ,\\ [a] \cap S^{-k} \xi^n ([b])\not = \emptyset}} \frac{1}{\beta^n} \lambda (F([b]))
= \frac{1}{\beta^n }(M_\xi^n (\lambda ( F ([b])))^t_{b\in A})_a.\end{aligned}$$
From the Perron’s Theorem, the above inequality is an equality and $(\lambda ( F( [b])))^t_{b\in A}$ is a multiple of the eigenvector $( \mu ([a]))^t_{a\in A} = \vec{\mu}(1)$ of the dominant eigenvalue $\beta^n$ of $M_{\xi}^n$. This shows Item . Notice that the equality in also implies Item .
To prove Item , it is enough to use the partitions of $\Omega_{\xi}$ given in Proposition \[substKR\] and the ideas in the beginning of this proof. This part is similar to the proof of Proposition 4.3 in and we left it to the reader.
With the next proposition, we continue to follow the approach (and the proofs) in .
\[prop:injcyl\] Assume Hypotheses `P` $i)-iii)$. There exists a $\mu$-negligeable measurable subset $\mathcal{N} \subset \Omega_\xi$ such that $F$ is one-to-one on each cylinder set $[a]$: for any $x$ and $y$ in $[a]\setminus \mathcal{N}$ satisfying $F (x) = F(y)$, we have $x=y$.
Let $a\in A$. From Lemma \[lemma:perron\], the sets $$\mathcal{N}_a^{(\ell)} =
\bigcup_{\substack{(k_1,j_1)\not = (k_2,j_2) ;\\
0\leq k_1< |\xi^\ell (b_1)|, [a] \cap S^{-k_1} \xi^\ell ([b_1])\not = \emptyset \\
0\leq k_2< |\xi^\ell (b_2)|, [a] \cap S^{-k_2} \xi^\ell ([b_2])\not = \emptyset}}
F(S^{-k_1} \xi^\ell([b_1])) \cap F(S^{-k_2} \xi^\ell([b_2]))$$
have zero $\lambda$-measure, for any $\ell\in \mathbb{N}$ big enough. Item of Lemma \[lemma:perron\], gives furthermore, the sets $\mathcal{M}_a^{(\ell)} = F^{-1} (\mathcal{N}_a^{(\ell)})$ have zero measure with respect to $\mu$.
Let $x_1$ and $x_2$ be two distinct elements of $[a]$ such that $F(x_1) = F(x_2)$. It suffices to show that they belong to some $ \mathcal{M}_a^{(\ell)}$. Considering the partitions $\{\P_{\ell}\}_{l\ge 0}$ of Proposition \[substKR\], there exist infinitely many $\ell \in \mathbb{N}$ with two distinct couples $(k_1, b_1)$ and $(k_2,b_2)$, such that $0\leq k_1< |\xi^\ell (b_1)|$, $0\leq k_2< |\xi^\ell (b_2)|$, $x_1 \in S^{-k_1} \xi^\ell ([b_1])$ and $x_2 \in S^{-k_2} \xi^\ell ([b_2])$. Then, $x_1$ and $x_2$ belong to $\mathcal{M}_a^{(\ell)}$ for infinitely many $\ell$, which achieves the proof.
\[prop:Finj\] Assume Hypotheses `P` $i)-iii)$. The map $F$ is one-to-one except on a set of measure zero.
As $\xi$ is proper, there exists a letter $a$ such that $\xi (\Omega_\xi )$ is included in $[a]$. Therefore, from Proposition \[prop:injcyl\], $F$ is one-to-one on $\xi (\Omega_\xi )$ except on a set $ \mathcal{N}$ of zero measure. By the basic properties of the map $F$ (precisely Item of Lemma \[lemma:deltaV\]), if two points $x,y \in \Omega_{\xi}$ have the same image through $F$, then $F(\xi(x))= F(\xi(y))$, and hence $x,y \in \xi^{-1}({\mathcal{N}})$.
Recall that the induced system on $\xi(\Omega_{\xi})$ is a factor of $(\Omega_{\xi}, S)$ via the map $\xi$ (see Relation ). This implies that the measure $\mu (\xi^{-1}(\cdot))$ is invariant for the induced system $(\xi(\Omega_{\xi}), S_{\xi(\Omega_{\xi})})$. Since it is uniquely ergodic with respect to the induced probability measure, $\mu(\xi^{-1}({\mathcal{N}}))$ is proportional to $\mu({\mathcal N})$, so it is null. This achieves the proof.
The following proposition is a modification of the arguments in [@Kulesza:1995] Lemma 2.1.
Assume Hypotheses `P` $i), ii)$. \[prop:Regul\] For any clopen set $c$ in $\Omega_{\xi}$, the set $F(c)$ is regular, [*i.e.*]{}, $$\overline{{\rm int \ } F(c)} = F(c),$$ where ${\rm int\ } A$ denotes the interior of the set $A$ for the usual Euclidean topology.
First we show that ${\rm int \ } F(\Omega_{\xi}) \neq \emptyset$. Since $1, \alpha_{1}, \ldots, \alpha_{d-1}$ are rationally independant, by Lemma \[lemma:deltaV\], denoting by $\pi$ the canonical projection ${{\mathbb R}}^{d-1} \to {{\mathbb R}}^{d-1}/{\mathbb Z}^{d-1} = {{\mathbb T}}^{d-1}$, the map $\pi \circ F \colon \Omega_{\xi} \to {{\mathbb T}}^{d-1}$ has a dense image hence is onto. It follows that for any small $\epsilon$, there exist a finite family $\V$ of integer vectors such that $$B_{\epsilon}(0) \subset \bigcup_{p \in \V} F(\Omega_{\xi})+p.$$ By the Baire Category Theorem, the set $F(\Omega_{\xi})$ has a non empty interior.
Now let $\Omega^* =\Omega_{\xi} \setminus \bigcup \{O ; O \textrm{ is open and } {\rm int \ }F(O) = \emptyset \}$. From the previous remark it is a non empty compact set. Notice that $\Omega_{\xi} \setminus \Omega^* $ is the union of countably many open (and then $\sigma$-compact) subsets. The image $F(\Omega_{\xi} \setminus \Omega^* )$ is then a countable union of compact sets each of those with an empty interior. Again by the Baire Category Theorem, $F(\Omega^*)$ is dense in $F(\Omega_{\xi})$ and since $\Omega^*$ is compact, $F(\Omega^*)= F(\Omega_{\xi})$.
Let us show that $\Omega^*$ is $S$ invariant. Let $O$ be an open set in $\Omega_{\xi}$ such that ${\rm int\ }F(O)$ is empty. By Lemma \[lemma:deltaV\], the function $F\circ S - F - ( \alpha_{1}, \ldots, \alpha_{d-1}) \colon \Omega_{\xi} \to {\mathbb Z}$ is constant on a partition by clopen sets $\P$ of $\Omega_{\xi}$. For any atom $c$ of $\P$, $ {\rm int\ }F(c\cap O) = \emptyset$ and then $ {\rm int \ }F(S(c\cap O))$ is empty. We have $F(S O) = \cup_{c\in \P} F(S (c \cap O))$ is then a countable union of compact sets with empty interiors. Again by the Baire Category Theorem, $F(SO)$ has empty interior, and $\Omega^*$ is $S$-invariant.
By minimality, we get that $\Omega^* = \Omega_{\xi}$, so the image by $F$ of any open set has a non empty interior.
Finally, let $C$ be a clopen set, and assume that $A := F(C) \setminus \overline{{\rm int \ } F(C)}$ is not empty. From the previous assertion, $F( F^{-1}(A) \cap C) = A$ contains a ball and then $A$ intersects ${\rm int\ }F(C)$: a contradiction. This shows the statement of the proposition.
Let $\pi : {{\mathbb R}}^{d-1} \to {{\mathbb R}}^{d-1}/{\mathbb Z}^{d-1} ={{\mathbb T}}^{d-1}$ be the canonical projection.
\[prop:cst-one\] Assume Hypotheses `P` $i)-iii)$. The map $Z: \Omega_{\xi} \to \mathbb{Z}\cup\{\infty\}$ defined by $Z(x) = \# (\pi \circ F)^{-1} (\{ \pi \circ F (x) \} )$ is finite and constant $\mu$-a.e..
We claim $Z$ is measurable. For any $z\in \mathbb{Z}^{d-1}$, let $A_{z}$ be the set $A_z = \{ x\in \Omega_{\xi}; \ \exists y \in \Omega_{\xi}, F(x) = F(y) + z \}$. We have $A_z = F^{-1} (F(\Omega_{\xi}) + z)$, so it is a Borel set. Notice that for any integer $n$, $Z^{-1} (\{ n \} )$ is a finite intersection of such sets, so the claim is proved. By Proposition \[prop:Finj\], the map $F$ is a.e. one-to-one, and by compacity of the set $F(\Omega_{\xi})$, the projection $\pi \colon F(\Omega_{\xi}) \to {{\mathbb T}}^{d-1}$ is finite-to-one, so the map $Z$ is a.e. finite. It suffices to notice that $Z$ is $T$-invariant, to conclude by ergodicity.
Let $\xi$ be a unimodular Pisot substitution. By Corollary \[cor:Pisotsub\] and Proposition \[prop:vpPisot\], we can assume that $\xi$ satisfies the hypotheses `P` $i)-iii)$ (Subsection \[sec:proof\]). Let $E$ be the compact set $F(\Omega_{\xi})$. Proposition \[substKR\] and Lemma \[lemma:deltaV\] on the properties of the map $F$ both ensure the existence of an integer $n$ such that the map $F\circ S - F $ is constant on any set $E_{n,a,k,} :=F(S^{-k}\xi^{n}([a]))$ with $a \in A$ and $0 \le k < \vert \xi^{n}(a) \vert.$ Let $T$ be the transformation defined on $E_{n,a,k}$ by the translation of the vector $(F\circ S -F)_{\vert E_{n,a,k}}$. It follows from Lemma \[lemma:perron\] and Proposition \[prop:Regul\] that $E$ and $T$ define a domain exchange transformation on regular sets. Moreover, Item of Lemma \[lemma:deltaV\] provides it is self-affine with respect to the sets $E_{n,a,k}$ and the linear part $(N^{t})^{n}$. Finally, Proposition \[prop:Finj\] shows this domain exchange is measurably conjugate to the subshift $(\Omega_\xi, S)$ and Proposition \[prop:cst-one\] gives the map $\pi \circ F \colon \Omega_{\xi} \to {{\mathbb T}}^{d-1}$ is a.e. $Z$-to-one for some constant $Z$.
In the sequel, we denote by $Z$ the constant of Proposition \[prop:cst-one\]. We give here a characterization of this constant in term of the volume of the set $F(\Omega_{\xi})$.
Assume Hypotheses `P` $i)-iii)$. We have $\lambda(F(\Omega_{\xi})) = Z$.
The canonical projection $\pi : {{\mathbb R}}^{d-1} \to {{\mathbb T}}^{d-1}$ defines a factor map from the domain exchange to a minimal translation on the torus. So the image measure of the normalized measure $\frac{\lambda}{\lambda ( F(\Omega_{\xi}))}$ is the Lebesgue measure on the torus. For any integrable function $f : F(\Omega_{\xi}) \to {{\mathbb R}}$, the conditional expectation $ E(f \vert \pi^{-1} (\B_{{\mathbb T}}))$, with respect to the Borel $\sigma$-algebra of the torus $\B_{{\mathbb T}}$, is constant over any $\pi$-fiber. So it follows for a.e. points $y\in F(\Omega_{\xi})$, $$E(f \vert \pi^{-1} (\B_{{\mathbb T}}))( y) = \sum_{x \in F(\Omega_{\xi}); \ \pi(x) = \pi(y)} \gamma_{x,\pi(y)} f(x),$$ for some non negative measurable function $x\mapsto \gamma_{x,\pi(x)}$ such that
$$\begin{aligned}
\label{eq:norm}
\sum_{x; \pi(x) = \pi(y)} \gamma_{x,\pi(y)} =1 \hspace{0.5cm} \textrm{for a.e. } y. \end{aligned}$$
Since for any integrable function $f : F(\Omega_{\xi}) \to {{\mathbb R}}$ with support in a unit square $U$, we have $$\begin{aligned}
\frac1{\lambda F(\Omega_{\xi})}\int_{U} f {d \lambda} =& \int_{U} E(f \vert \pi^{-1} (\B_{{\mathbb T}})) d \lambda \\
= &\int_{ U\cap F(\Omega_{\xi}) } \gamma_{x,\pi(x)} f(x) d \lambda (x). \end{aligned}$$ We obtain that $\gamma_{x,\pi(x)} = \frac1{\lambda F(\Omega_{\xi})}$. We get the conclusion by the equation
[^1]: Both authors acknowledge the ANR Program SubTile. Fabien Durand also acknowledges the ANR Programs DynA3S and FAN
|
---
abstract: |
Realizations of scale invariance are studied in the context of a gravitational theory where the action (in the first order formalism) is of the form $S = \int L_{1} \Phi d^{4}x$ + $\int L_{2}\sqrt{-g}d^{4}x$ where $\Phi$ is a density built out of degrees of freedom, the “measure fields” independent of $g_{\mu\nu}$ and matter fields appearing in $L_{1}$, $L_{2}$. If $L_{1}$ contains the curvature, scalar potential $V(\phi)$ and kinetic term for $\phi$, $L_{2}$ another potential for $\phi$, $U(\phi)$, then the true vacuum state has zero energy density, when theory is analyzed in the conformal Einstein frame (CEF), where the equations assume the Einstein form. Global Scale invariance is realized when $V(\phi)$ = $f_{1}e^{\alpha\phi}$ and $U(\phi)$ = $f_{2}e^{2\alpha\phi}$. In the CEF the scalar field potential energy $V_{eff}(\phi)$ has in, addition to a minimum at zero, a flat region for $\alpha\phi
\rightarrow\infty$, with non zero vacuum energy, which is suitable for either a New Inflationary scenario for the Early Universe or for a slowly rolling decaying $\Lambda$-scenario for the late universe, where the smallness of the vacuum energy can be understood as a kind of see-saw mechanism.
author:
- |
E.I. Guendelman\
[*Physics Department, Ben-Gurion University, Beer-Sheva 84105, Israel*]{}
title: ' Scale Invariance, New Inflation and Decaying $\Lambda$-Terms\'
---
Introduction
============
Recent developments in cosmology have been influenced to a great extent by the idea of inflation$^1$, which provides an attractive scenario for solving some of the fundamental puzzles of the standard Big Bang model, like the horizon and the flatness problems as well as providing a framework for sensible calculations of primordial density perturbations.
However, although the inflationary scenario is very attractive, it has been recognized that a successful implementation requires some very special restrictions on the dynamics that drive inflation. In particular, in New Inflation$^2$, a potential with a large flat region, which then drops to zero (or almost zero) in order to reproduce the vacuum with almost zero (in Planck units) cosmological constant of the present universe, is required. It is hard to find a theory that gives a potential of this type naturally.
In addition to this, it is worthwhile pointing out that a potential with a very flat region, slowly approaching zero could be of use as a model for a decaying cosmological constant being considered as a model for the “accelerating universe”, now preferred by observations$^3$. This is of course, at a totally different scale to that of Inflation.
Here we want to see whether such shapes of potentials can be obtained from first principles, i.e. whether there is some fundamental principle that produces this type of behavior for a scalar field.
We find indeed that this is possible and the fundamental principle in question is none other than scale invariance. However, scale invariance has to be discussed in a more general framework then that of the standard Lagranian formulation of generally relativistic theories. Before going into the question of scale invariance it is necessary therefore to discuss first the general framework where this discussion will be set.
The Non Gravitating Vacuum Energy (NGVE) Theory. Strong and Weak Formulations.
==============================================================================
When formulating generally covariant Lagragian formulations of gravitational theories, we usually consider the form $$S_{1} = \int{L}\sqrt{-g} d^{4}x, g = det g_{\mu\nu}$$
As it is well known, $d^{4}x$ is not a scalar but the combination $\sqrt{-g} d^{4} x$ is a scalar. Inserting $\sqrt{-g}$, which has the transformation properties of a density, produces a scalar action (1), provided L is a scalar.
One could use nevertheless other objects instead of $\sqrt{-g}$, provided they have the same transformation properties and achieve in this way a different generally covariant formulation.
For example, given 4-scalars $\varphi_{a}$ (a = 1,2,3,4), one can construct the density $$\Phi = \varepsilon^{\mu\nu\alpha\beta} \varepsilon_{abcd}
\partial_{\mu} \varphi_{a} \partial_{\nu} \varphi_{b} \partial_{\alpha}
\varphi_{c} \partial_{\beta} \varphi_{d}$$ and consider instead of (1) the action$^4$ $$S_{2} = \int L \Phi d^{4} x.$$ L is again some scalar, which contains the curvature (i.e. the gravitational contribution) and a matter contribution, as is standard also in (1).
In the action (3) the measure carries degrees of freedom independent of that of the metric and that of the matter fields. The most natural and successful formulation of the theory is achieved when the connection is also treated as an independent degree of freedom. This is what is usually referred to as the first order formalism.
One can notice that $\Phi$ is the total derivative of something, for example, one can write $$\Phi = \partial_{\mu} ( \varepsilon^{\mu\nu\alpha\beta}
\varepsilon_{abcd} \varphi_{a}
\partial_{\nu} \varphi_{b}
\partial_{\alpha}
\varphi_{c} \partial_{\beta} \varphi_{d}).$$
This means that a shift of the form $$L \rightarrow L + constant$$ just adds the integral of a total divergence to the action (3) and it does not affect therefore the equations of motion of the theory. The same shift, acting on (1) produces an additional term which gives rise to a cosmological constant. Since the constant part of L does not affect the equations of motion, this theory is called the Non Gravitating Vacuum Energy (NGVE) Theory$^4$.
One can generalize this structure and allow both geometrical objects to enter the theory and consider $$S_{3} = \int L_{1} \Phi d^{4} x + \int L_{2} \sqrt{-g}d^{4}x$$
Now instead of (5), the shift symmetry can be applied only on $L_{1}$ ($L_{1} \rightarrow L_{1}$ + constant). Since the structure has been generalized, we call this formulation the weak version of the NGVE - theory. Here $L_{1}$ and $L_{2}$ are $\varphi_{a}$ independent.
There is a good reason not to consider mixing of $\Phi$ and $\sqrt{-g}$ , like for example using $$\frac{\Phi^{2}}{\sqrt{-g}}$$
This is because (6) is invariant (up to the integral of a total divergence) under the infinite dimensional symmetry $$\varphi_{a} \rightarrow \varphi_{a} + f_{a} (L_{1})$$ where $f_{a} (L_{1})$ is an arbitrary function of $L_{1}$ if $L_{1}$ and $L_{2}$ are $\varphi_{a}$ independent. Such symmetry (up to the integral of a total divergence) is absent if mixed terms (like (7)) are present. Therefore (6) is considered for the case when no dependence on the measure fields (MF) appears in $L_{1}$ or $L_{2}$.
In this paper we will see that the existence of two independent measures of integrations as in (6) allows new realizations of global scale invariance with most interesting consequences when the results are viewed from the point of view of cosmology.
Dynamics of a Scalar Field and the Requirement of Scale Invariance in the Weak NGVE - Theory
============================================================================================
The Action Principle
====================
We will study now the dynamics of a scalar field $\phi$ interacting with gravity as given by the following action $$S_{\phi} = \int L_{1} \Phi d^{4} x + \int L_{2} \sqrt{-g} d^{4} x$$ $$L_{1} = \frac{-1}{\kappa} R(\Gamma, g) + \frac{1}{2} g^{\mu\nu}
\partial_{\mu} \phi \partial_{\nu} \phi - V(\phi)$$ $$L_{2} = U(\phi)$$ $$R(\Gamma,g) = g^{\mu\nu} R_{\mu\nu} (\Gamma) , R_{\mu\nu}
(\Gamma) = R^{\lambda}_{\mu\nu\lambda}$$ $$R^{\lambda}_{\mu\nu\sigma} (\Gamma) = \Gamma^{\lambda}_
{\mu\nu,\sigma} - \Gamma^{\lambda}_{\mu\sigma,\nu} +
\Gamma^{\lambda}_{\alpha\sigma} \Gamma^{\alpha}_{\mu\nu} -
\Gamma^{\lambda}_{\alpha\nu} \Gamma^{\alpha}_{\mu\sigma}.$$
In the variational principle $\Gamma^{\lambda}_{\mu\nu},
g_{\mu\nu}$, the measure fields scalars $\varphi_{a}$ and the “matter” - scalar field $\phi$ are all to be treated as independent variables although the variational principle may result in equations that allow us to solve some of these variables in terms of others.
Global Scale Invariance
=======================
If we perform the global scale transformation ($\theta$ = constant) $$g_{\mu\nu} \rightarrow e^{\theta} g_{\mu\nu}$$ then (9) is invariant provided $V(\phi)$ and $U(\phi)$ are of the form $$V(\phi) = f_{1} e^{\alpha\phi}, U(\phi) = f_{2}
e^{2\alpha\phi}$$ and $\varphi_{a}$ is transformed according to $$\varphi_{a} \rightarrow \lambda_{a} \varphi_{a}$$ (no sum on a) which means $$\Phi \rightarrow \biggl(\prod_{a} {\lambda}_{a}\biggr) \Phi \\ \equiv \lambda
\Phi$$ such that $$\lambda = e^{\theta}$$ and $$\phi \rightarrow \phi - \frac{\theta}{\alpha}.$$
The Equations of Motion
=======================
We will now work out the equations of motion for arbitrary choice of $V(\phi)$ and $U(\phi)$. We study afterwards the choice (15) which allows us to obtain the results for the scale invariant case and also to see what differentiates this from the choice of arbitrary $U{\phi}$ and $V{\phi}$ in a very special way.
Let us begin by considering the equations which are obtained from the variation of the fields that appear in the measure, i.e. the $\varphi_{a}$ fields. We obtain then $$A^{\mu}_{a} \partial_{\mu} L_{1} = O$$ where $A^{\mu}_{a} = \varepsilon^{\mu\nu\alpha\beta}
\varepsilon_{abcd} \partial_{\nu} \varphi_{b} \partial_{\alpha}
\varphi_{c} \partial_{\beta} \varphi_{d}$ (21). Since it is easy to check that $A^{\mu}_{a} \partial_{\mu} \varphi_{a^{\prime}} =
\frac{\delta aa^{\prime}}{4} \Phi$, it follows that det $(A^{\mu}_{a}) =\frac{4^{-4}}{4!} \Phi^{3} \neq O$ if $\Phi\neq O$. Therefore if $\Phi\neq O$ we obtain that $\partial_{\mu} L_{1} = O$, or that $$L_{1} = \frac{-1}{\kappa} R(\Gamma,g) + \frac{1}{2} g^{\mu\nu}
\partial_{\mu} \phi \partial_{\nu} \phi - V = M$$ where M is constant.
Let us study now the equations obtained from the variation of the connections $\Gamma^{\lambda}_{\mu\nu}$. We obtain then $$-\Gamma^{\lambda}_{\mu\nu} -\Gamma^{\alpha}_{\beta\mu}
g^{\beta\lambda} g_{\alpha\nu} + \delta^{\lambda}_{\nu}
\Gamma^{\alpha}_{\mu\alpha} + \delta^{\lambda}_{\mu}
g^{\alpha\beta} \Gamma^{\gamma}_{\alpha\beta}
g_{\gamma\nu}\\ - g_{\alpha\nu} \partial_{\mu} g^{\alpha\lambda}
+ \delta^{\lambda}_{\mu} g_{\alpha\nu} \partial_{\beta}
g^{\alpha\beta}
- \delta^{\lambda}_{\nu} \frac{\Phi,_\mu}{\Phi}
+ \delta^{\lambda}_{\mu} \frac{\Phi,_ \nu}{\Phi} = O$$ If we define $\Sigma^{\lambda}_{\mu\nu}$ as $\Sigma^{\lambda}_{\mu\nu} =
\Gamma^{\lambda}_{\mu\nu} -\{^{\lambda}_{\mu\nu}\}$ where $\{^{\lambda}_{\mu\nu}\}$ is the Christoffel symbol, we obtain for $\Sigma^{\lambda}_{\mu\nu}$ the equation $$- \sigma, _{\lambda} g_{\mu\nu} + \sigma, _{\mu}
g_{\nu\lambda} - g_{\nu\alpha} \Sigma^{\alpha}_{\lambda\mu}
-g_{\mu\alpha} \Sigma^{\alpha}_{\nu \lambda}
+ g_{\mu\nu} \Sigma^{\alpha}_{\lambda\alpha} +
g_{\nu\lambda} g_{\alpha\mu} g^{\beta\gamma} \Sigma^{\alpha}_{\beta\gamma}
= O$$ where $\sigma = 1n \chi, \chi = \frac{\Phi}{\sqrt{-g}}$.
The general solution of (23) is $$\Sigma^{\alpha}_{\mu\nu} = \delta^{\alpha}_{\mu}
\lambda,_{\nu} + \frac{1}{2} (\sigma,_{\mu} \delta^{\alpha}_{\nu} -
\sigma,_{\beta} g_{\mu\nu} g^{\alpha\beta})$$ where $\lambda$ is an arbitrary function due to the $\lambda$ - symmetry of the curvature${}^{(5)}$ $R^{\lambda}_{\mu\nu\alpha} (\Gamma)$, $$\Gamma^{\alpha}_{\mu\nu} \rightarrow \Gamma^{\prime \alpha}_{\mu\nu}
= \Gamma^{\alpha}_{\mu\nu} + \delta^{\alpha}_{\mu}
Z,_{\nu}$$ Z being any scalar (which means $\lambda \rightarrow \lambda + Z$).
If we choose the gauge $\lambda = \frac{\sigma}{2}$, we obtain $$\Sigma^{\alpha}_{\mu\nu} (\sigma) = \frac{1}{2} (\delta^{\alpha}_{\mu}
\sigma,_{\nu} +
\delta^{\alpha}_{\nu} \sigma,_{\mu} - \sigma,_{\beta}
g_{\mu\nu} g^{\alpha\beta}).$$
Considering now the variation with respect to $g^{\mu\nu}$, we obtain $$\Phi = (\frac{-1}{\kappa} R_{\mu\nu} (\Gamma) + \frac{1}{2} \phi,_{\mu}
\phi,_{\nu}) - \frac{1}{2} \sqrt{-g} U(\phi) g_{\mu\nu} = O$$ solving for $R = g^{\mu\nu} R_{\mu\nu} (\Gamma)$ and introducing in (22), we obtain $$M + V(\phi) - \frac{2U(\varphi)}{\chi} = O$$ a constraint that allows us to solve for $\chi$, $$\chi = \frac{2U(\phi)}{M+V(\phi)}.$$
To get the physical content of the theory, it is convenient to go to the Einstein conformal frame where $$\overline{g}_{\mu\nu} = \chi g_{\mu\nu}$$ and $\chi$ given by (29b). In terms of $\overline{g}_{\mu\nu}$ the non Riemannian contribution $\Sigma^{\alpha}_{\mu\nu}$ dissappears from the equations, which can be written then in the Einstein form ($R_{\mu\nu} (\overline{g}_{\alpha\beta})$ = usual Ricci tensor) $$R_{\mu\nu} (\overline{g}_{\alpha\beta}) - \frac{1}{2}
\overline{g}_{\mu\nu}
R(\overline{g}_{\alpha\beta}) = \frac{\kappa}{2} T^{eff}_{\mu\nu}
(\phi)$$ where $$T^{eff}_{\mu\nu} (\phi) = \phi_{,\mu} \phi_{,\nu} - \frac{1}{2} \overline
{g}_{\mu\nu} \phi_{,\alpha} \phi_{,\beta} \overline{g}^{\alpha\beta}
+ \overline{g}_{\mu\nu} V_{eff} (\phi)$$ and $$V_{eff} (\phi) = \frac{1}{4U(\phi)} (V+M)^{2}.$$
In terms of the metric $\overline{g}^{\alpha\beta}$ , the equation of motion of the Scalar field $\phi$ takes the standard General - Relativity form $$\frac{1}{\sqrt{-\overline{g}}} \partial_{\mu} (\overline{g}^{\mu\nu}
\sqrt{-\overline{g}} \partial_{\nu}
\phi) + V^{\prime}_{eff} (\phi) = O.$$
Notice that if $V + M = O, V_{eff} = O$ and $V^{\prime}_{eff}
= O$ also, provided $V^{\prime}$ is finite and $U \neq O$ there. This means the zero cosmological constant state is achieved without any sort of fine tuning. This is the basic feature that characterizes the NGVE - theory and allows it to solve the cosmological constant problem$^{4}$.
In what follows we will study (33) for the special case of global scale invariance, which as we will see displays additional very special features which makes it attractive in the context of cosmology.
Notice that in terms of the variables $\phi$, $\overline{g}_{\mu\nu}$, the “scale” transformation becomes only a shift in the scalar field $\phi$, since $\overline{g}_{\mu\nu}$ is invariant (since $\chi \rightarrow \lambda^{-1} \chi$ and $g_{\mu\nu}
\rightarrow \lambda g_{\mu\nu}$) $$\overline{g}_{\mu\nu} \rightarrow \overline{g}_{\mu\nu}, \phi \rightarrow
\phi - \frac{\theta}{\alpha}.$$
Analysis of the Scale - Invariant Dynamics
==========================================
If $V(\phi) = f_{1} e^{\alpha\phi}$ and $U(\phi) = f_{2}
e^{2\alpha\phi}$ as required by scale invariance (14), (16), (17), (18), (19), we obtain from (33) $$V_{eff} = \frac{1}{4f_{2}} (f_{1} + M e^{-\alpha\phi})^{2}$$
Since we can always perform the transformation $\phi \rightarrow
- \phi$ we can choose by convention $\alpha > O$. We then see that as $\phi \rightarrow
\infty, V_{eff} \rightarrow \frac{f_{1}^{2}}{4f_{2}} =$ const. providing an infinite flat region. Also a minimum is achieved at zero cosmological constant for the case $\frac{f_{1}}{M} < O$ at the point $$\phi_{min} = \frac{-1}{\alpha} ln \mid\frac{f_1}{M}\mid.$$
Finally, the second derivative of the potential $V_{eff}$ at the minimum is $$V^{\prime\prime}_{eff} = \frac{\alpha^2}{2f_2} \mid{f_1}\mid^{2} > O$$ if $f_{2} > O$, there are many interesting issues that one can raise here. The first one is of course the fact that a realistic scalar field potential, with massive exitations when considering the true vacuum state, is achieved in a way consistent with the idea (although somewhat generalized) of scale invariance.
The second point to be raised is that there is an infinite region of flat potential for $\phi \rightarrow \infty$, which makes this theory an attractive realization of the improved inflationary model$^{2}$.
A peculiar feature of the potential (36), is that the constant M, provided it has the correct sign, i.e. that $f_{1}/M < O$, does not affect the physics of the problem. This is because if we perform a shift $$\phi \rightarrow \phi + \Delta$$ in the potential (36), this is equivalent to the change in the integration constant M $$M \rightarrow M e^{-\alpha\Delta}.$$
We see therefore that if we change M in any way, without changing the sign of M, the only effect this has is to shift the whole potential. The physics of the potential remains unchanged, however. This is reminiscent of the dilatation invariance of the theory, which involves only a shift in $\phi$ if $\overline{g}_{\mu\nu}$ is used (see eq. (35) ).
This is very different from the situation for two generic functions $U(\phi)$ and $V(\phi)$ in (34). There, M appears in $V_{eff}$ as a true new parameter that generically changes the shape of the potential $V_{eff}$, i.e. it is impossible then to compensate the effect of M with just a shift. For example M will appear in the value of the second derivative of the potential at the minimum, unlike what we see in eq. (38), where we see that $V^{\prime\prime}_{eff}$ (min) is M independent.
In conclusion, the scale invariance of the original theory is responsible for the non appearance (in the physics) of a certain scale, that associated to M. However, masses do appear, since the coupling to two different measures of $L_{1}$ and $L_{2}$ allow us to introduce two independent couplings $f_{1}$ and $f_{2}$, a situation which is unlike the standard formulation of globally scale invariant theories, where usually no stable vacuum state exists.
Notice that we have not considered all possible terms consistent with global scale invariance. Additional terms in $L_{2}$ of the form $e^{\alpha\phi} R$ and $e^{\alpha\phi} g^{\mu\nu} \partial_{\mu}\phi
\partial_{\nu}\phi$ are indeed consistent with the global scale invariance (14), (16), (17), (18), (19) but they give rise to a much more complicated theory, which will be studied in a separate publication. There it will be shown that for slow rolling and for $\phi \rightarrow \infty$ the basic features of the theory are the same as what has been studied here. Let us finish this section by comparing the appearance of the potential $V_{eff} (\phi)$, which has privileged some point depending on M (for example the minimum of the potential will have to be at some specific point), although the theory has the “translation invariance” (35), to the physics of solitons.
In fact, this very much resembles the appearance of solitons in a space-translation invariant theory: The soliton solution has to be centered at some point, which of course is not determined by the theory. The soliton of course breaks the space translation invariance spontaneously, just as the existence of the non trivial potential $V_{eff}
(\phi)$ breaks here spontaneously the translations in $\phi$ space, since $V_{eff}
(\phi)$ is not a constant.
Notice however, that the existence for $\phi \rightarrow \infty$, of a flat region for $V_{eff} (\phi)$ can be nicely described as a region where the symmetry under translations (35) is restored.
Cosmological Applications of the Model
======================================
Since we have an infinite region in which $V_{eff}$ as given by (36) is flat $(\phi \rightarrow \infty)$, we expect a slow rolling (new inflationary) scenario to be viable, provided the universe is started at a sufficiently large value of the scalar field $\phi$.
One should point out that the model discussed here gives a potential with two physically relevant parameters $\frac{f_1^{2}}{4f_{2}}$ , which represents the value of $V_{eff}$ as $\phi \rightarrow \infty$ , i.e. the strength of the false vacuum at the flat region and $\frac{\alpha^{2}f_ {1}^{2}}{2f_2}$ , representing the mass of the excitations around the true vacuum with zero cosmological constant (achieved here without fine tuning).
When a realistic model of reheating is considered, one has to give the strength of the coupling of the $\phi$ field to other fields. It remains to be seen what region of parameter space provides us with a realistic cosmological model.
Furthermore, one can consider this model as suitable for the very late universe rather than for the early universe, after we suitably reinterpret the meaning of the scalar field $\phi$.
This can provide a long lived almost constant vacuum energy for a long period of time, which can be small if $f_{1}^{2}/4f_{2}$ is small. Such small energy density will eventually disappear when the universe achieves its true vacuum state.
Notice that a small value of $\frac{f_{1}^{2}}{f_{2}}$ can be achieved if we let $f_{2} >> f_{1}$. In this case $\frac{f_{1}^{2}}{f_{2}} << f_{1}$, i.e. a very small scale for the energy density of the universe is obtained by the existence of a very high scale (that of $f_{2}$) the same way as a small fermion mass is obtained in the see-saw mechanism$^{6}$ from the existence also of a large mass scale.
Acknowledgement
===============
I would like to thank A. Davidson and A. Kaganovich for conversations on the subjects discussed here.
[99]{} For a non technical review and a good collection of further references on different aspects of inflation see A. Guth, “The Inflationary Universe”, Vintage, Random House (1998). For a more technical review see E.W. Kolb and M.S. Turner, “The Early Universe”, Addison Wesley (1990). A.D. Linde, Phys. Lett, 108B, (1982) 389; A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett, 48, (1982) 1220. For the review of this subject see M.S. Turner in the third Stromle Symposium: “The Galactic Halo”, ASP Conference Series, Vol 666, 1999, (eds) B.K. Gibson, T.S. Axelrod and M.E. Putman. E.I. Guendelman and A.B. Kaganovich, Phys. Rev., D53, (1996) 7020; E.I. Guendelman and A.B. Kaganovich, Proceedings of the third Alexander Friedmann International Seminar on Gravitation and Cosmology, ed. by Yu. N. Gneding, A.A. Grib and V.M. Mostepanenko (Friedmann laboratory Publishing, st. Petersburg, 1995); E.I. Guendelman and A.B. Kaganovich, Phys. Rev., D55, (1997) 5970; E.I. Guendelman and A.B. Kaganovich, Mod. Phys. Lett, A12, (1997) 2421; E.I. Guendelman and A.B. Kaganovich, Phys. Rev., D56, (1997) 3548; E.I. Guendelman and A.B. Kaganovich, Hadronic Journal, 21, (1998) 19; E.I. Guendelman and A.B. Kaganovich, Mod. Phys. Lett., A13, (1998) 1583; F. Gronwald, U. Muench and F.W. Hehl, Hadronic Journal, 21, (1998) 3; E.I. Guendelman and A.B. Kaganovich, Phys. Rev., D57, (1998) 7200; E.I. Guendelman and A.B. Kaganovich, “Gravity Cosmology and Particle Field Dynamics without the Cosmological Constant Problem”, to appear in the Proceedings of the sixth International Symposium on Particle, Strings and Cosmolgy, PASCO5-98; E.I. Guendelman and A.B. Kaganovich, “Field Theory Models without the Cosmological Constant problem”, Plenary talk (given by E.I. Guendelman) on the fourth Alexander Friedmann International Seminar on Gravitation and Cosmology, gr-qc/9809052. A. Einstein, “The Meaning of Relativity”, MJF books, NY (1956), see appendix II. M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, edited by D. Friedman (North Holland, Amsterdam, 1979)p. 315; T. Yanagida in Proceedings of the Workshop on “Unified Theory and Baryon Number in the Universe”, edited by O. Sawada and A. Sugamoto (KEK, Tsukuba, Japan, 1979); R. Mohapatra and G. Senjanovic, Phys. Rev. Lett., 44, (1980), 912 and Phys. Rev., D23, (1981) 165; A. Davidson and K.C. Wali, Phys. Rev. Lett., 59 (1987) 393.
|
---
abstract: 'Amperes law states that the magnetic moment of a ring is given by current times the area enclosed. Also from equilibrium statistical mechanics it is known that magnetic moment is the derivative of free energy with respect to magnetic field. In this work we analyze a quantum double ring system interacting with a reservoir. A simple S-Matrix model is used for system-reservoir coupling. We see complete agreement between the aforesaid two definitions when coupling between system and reservoir is weak, increasing the strength of coupling parameter however leads to disagreement between the two. Thereby signifying the important role played by the coupling parameter in mesoscopic systems.'
author:
- Colin Benjamin
- 'A. M. Jayannavar'
title: 'Equilibrium currents in quantum double ring system: A non-trivial role of system-reservoir coupling.'
---
Introduction
============
Mesoscopics, a fertile branch of Physics deals with systems and phenomena in the scale of nano- to micrometers. The distinguishing feature of these systems is that at extremely low temperatures an electron retains it’s phase coherence throughout the sample. These systems have revealed a new range of unexpected quantum phenomena, often counter-intuitive[@imry; @datta]. The notion of the usual ensemble averaged transport coefficients such as the conductivity, i.e., local and material specific, has to be replaced by that of conductance, i.e., global and operationally specific to the sample as well as the nature of probes of measurement. Some novel and hitherto unheard of features in classical physics, e.g., non-local current-voltage relationships[@webb], breakdown of Ohm’s law, absolute negative resistance (four probe)[@butiprl1986], normal-state Aharonov-Bohm effect[@webb], quantization of point-contact conductance (Landauer formalism)[@vanwees], universal conductance fluctuations (new form of ergodicity), persistent currents[@BIL], spin-polarized transport[@prinz; @datta1], coulomb blockade[@glazman] and many novel effects arising due to electron correlations, have been observed in these systems. Interpretation of these require full recognition of the wave nature of quasi particles and keeping track of their phase coherence over the entire sample including the measurement leads and probes (quantum measurement process). Even the equilibrium properties, are very sensitive to the nature of statistical ensemble used. The results differ qualitatively from one ensemble to another[@imry].
Motivation
==========
Recently it has been proposed that these systems will provide a testing ground for verifying the violation in the basic laws of thermodynamics[@TA]. This behavior has been explained by taking recourse to the effects of entanglement, through which the quantum system is so interlinked with the bath that the resulting behavior of the system alone cannot be treated within a conventional thermodynamic approach. Here the finite coupling between the bath and the system plays a crucial role. It should be noted that equilibrium thermodynamics of the super-system comprising of system (sub-system) plus bath, does not imply standard equilibrium thermodynamics for the sub-system alone. In-fact the thermodynamic equilibrium properties of the system depend on the coupling parameter, unlike equilibrium statistical mechanics. For example, it has been shown via an analytical treatment that the mean orbital magnetic moment, a thermodynamic property, is determined by the electrical resistivity (which is related to system-bath coupling parameter) of the material[@dattgup_prl]. What is crucial for dissipative diamagnetism is that system-bath interaction has to be treated exactly, there is no clear cut separation between what is the system and what is the bath- both are inexorably linked to one many-body system. In this work, motivated by the above results, we provide a simple example wherein the equilibrium properties are determined by the system-reservoir coupling parameter in a [**non-trivial manner**]{}. Our results follow from the consideration of the dephasing of a single particle quantum coherence while the earlier dynamical treatments required in addition to quantum coherence, entanglement between system and bath. It is in this spirit that we study the persistent current densities in a quantum double ring system coupled to a reservoir via a simple voltage probe method due to Büttiker[@buti]. We explicitly show that when the coupling parameter is very small there is perfect agreement between the magnetic moments calculated from the local currents (via Amperes law) and that from the derivative of free energy with respect to magnetic field, increasing the strength of coupling parameter however leads to disagreement between the two.
Background
==========
It is well known that spontaneous currents which never decay can flow in super-conducting systems. In 1983, Büttiker , Imry and Landauer[@BIL] first predicted that, a normal metal ring threaded by an Aharonov-Bohm flux $\phi$, in the phase coherent domain carries persistent currents . These arise because the magnetic flux breaks the time reversal symmetry thus inducing currents. This is a quantum effect and the total current flowing in the ring is related to the derivative of free energy with respect to flux[@cheungibm; @cheungprb]. For clean rings it is periodic in flux with period of $2\pi\phi/\phi_0$. The magnetic flux incidentally plays the same role as a periodic potential in the Bloch sense, and therefore one gets the band structure, elastic scattering if present in the loop induces gaps in the spectrum and reduces the magnitude of persistent currents. Later in 1985, Büttiker [@buti] investigated the effect of a reservoir coupled to a ring. This simply means breaking the phase coherence. Electrons enter the reservoir lose their phase memory and are re-injected again from the reservoir with an uncorrelated phase. The S-Matrix for coupling between system and reservoir is given by-
$$S_{J}=\left(\begin{array}{ccc}
-(a+b) & \sqrt{\epsilon}& \sqrt{\epsilon}\\
\sqrt{\epsilon} & a & b\\
\sqrt{\epsilon} & b & a
\end{array} \right)$$
wherein, $a=\frac{1}{2}(\sqrt{1-2\epsilon}-1)$ and $b=\frac{1}{2}(\sqrt{1-2\epsilon}+1)$, for $\epsilon\rightarrow 0$ the system and reservoir are decoupled while for $\epsilon\rightarrow 0.5$ the system and reservoir are strongly coupled. The above S-Matrix satisfies the conservation of current[@shapiro], and accounts for the possibility of strong to weakly coupled reservoirs through the coupler “$\epsilon$”. One of the important conclusions of the work was that the magnitude of persistent current flowing in the loop decreases with increasing coupling strength $\epsilon$ but without any change in their nature, i.e., diamagnetic or paramagnetic. This effect is solely due to exchange of carriers between reservoir and ring (dephasing), and also this is true if the lead connecting reservoir to ring has a charging energy much less than the level spacing. Experimentally persistent currents in both open and closed systems have been observed[@webb; @levy; @chandra; @maily] and these observations have given rise to a spurt in theoretical activities[@rmp_persis].
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Model
=====
Let us consider the double ring system as shown in FIG. 1. The static localized flux piercing the loop is necessary to break the time reversal symmetry and induce a persistent current in the system. The geometry we consider is a one-dimensional ring with an attached bubble and a lead connected to a reservoir at chemical potential $\mu$. The reservoir acts as an inelastic scatterer and as a source of energy dissipation. All the scattering processes in the leads including the loop are assumed to be elastic. Hence there is complete spatial separation between the elastic and inelastic processes. The loops J1J2aJ3J1 and J1J2bJ3J1 enclose the localized flux $\Phi$. However, the bubble J2aJ3bJ2 does not enclose the flux $\Phi$. The same S-Matrix coupler couples the double ring to the reservoir, i.e., $S_{J1}=S_J$, for the other two junctions we take symmetric couplers[@porod].
$$S_{J2}=S_{J3}=\left(\begin{array}{ccc}
-\frac{1}{3} & \frac{2}{3} & \frac{2}{3}\\
\frac{2}{3} & -\frac{1}{3} & \frac{2}{3}\\
\frac{2}{3} & \frac{2}{3} & -\frac{1}{3}
\end{array} \right)$$
The waves in the four arms of the system depicted in FIG. 1 are related as follows: The waves incident into the branches of the double ring system are related by the S-Matrices for J1J2 arm by-
$$\left(\begin{array}{c}
y_1\\
v_1\\
\end{array} \right) \ =\left(\begin{array}{cc}
0 & e^{ikl_1} e^{-i \alpha_1}\\
e^{ikl_1} e^{i \alpha_1} & 0 \\
\end{array} \right) \left(\begin{array}{c}
x_1\\
u_1
\end{array} \right)$$ for J2bJ3 arm by-
$$\left(\begin{array}{c}
y_2\\
v_2\\
\end{array} \right) \ =\left(\begin{array}{cc}
0 & e^{ikl_2} e^{-i \alpha_2} \\
e^{ikl_2} e^{i \alpha_2} & 0 \\
\end{array} \right) \left(\begin{array}{c}
x_2\\
u_2
\end{array} \right)$$
for J2aJ3 arm by-
$$\left(\begin{array}{c}
y_3\\
v_3\\
\end{array} \right) \ =\left(\begin{array}{cc}
0 & e^{ikl_3} e^{-i \alpha_3} \\
e^{ikl_3} e^{i \alpha_3} & 0 \\
\end{array} \right) \left(\begin{array}{c}
x_3\\
u_3
\end{array} \right)$$
for J3J1 arm by-
$$\left(\begin{array}{c}
y_4\\
v_4\\
\end{array} \right) \ =\left(\begin{array}{cc}
0 & e^{ikl_4} e^{i \alpha_4} \\
e^{ikl_4} e^{-i \alpha_4} & 0 \\
\end{array} \right) \left(\begin{array}{c}
x_4\\
u_4
\end{array} \right)$$
Here $kl_1$, $kl_2$, $kl_3$ and $kl_4$ are the phase increments of the wave function in the absence of flux. $\alpha_1$, $\alpha_2
$,$\alpha_3$, and $\alpha_4$ are the phase shifts due to flux in the arms of the considered double ring system. Clearly, $\alpha_{1}+\alpha_{2}+\alpha_{4}=\frac{2\pi\Phi}{\Phi_0} $, where $\Phi$ is the flux piercing the loop and $\Phi_0$ is the flux quantum $\frac{hc}{e}$, also $\alpha_{1}+\alpha_{3}+\alpha_{4}=\frac{2\pi\Phi}{\Phi_0}$, thus $\alpha_{2}=\alpha_{3}$[@colin_prb; @colin_ijmpb]. The current densities (in dimensionless form) in the various arms of the system are given as follows[@buti; @colin_prb]- $I_{1}=|x_1|^2-|y_1|^2$, $I_{2}=|x_2|^2-|y_2|^2$, $I_{3}=|x_3|^2-|y_3|^2$, $I_{4}=-|x_4|^2+|y_4|^2$, wherein complex amplitude for propagating waves $x_{1}, x_{2}, x_{3}, x_{4}, y_{1}, y_{2}, y_{3}, y_{4}, $ are as depicted in Fig. 1. The induced current densities in the various arms of the loop are assigned labels $I_{1}, I_{2}, I_{3}$ and $I_4$, while the arm lengths of various parts of the ring are $l_{1}, l_{2},
l_{3}$ and $l_{4}$.
Now a question may arise as to why we consider the persistent current densities in this special geometry of a double ring. This geometry exhibits the current magnification effect in equilibrium[@colin_prb] which is a quantum phenomena. The current magnification effect, a purely quantum phenomena arises in equilibrium and non-equilibrium systems[@mosk; @deo_cm; @colin_ijmpb] and leads to large orbital magnetic moments. A double ring system[@pareek_double] which does not exhibit current magnification effect will show [**no disagreement**]{} between the magnetic moments calculated from the local current densities (via Amperes law) and that from the derivative of the free energy, which we have separately verified[@cb_unpub].
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Theory
======
We know that magnetic moment is the derivative of free energy with respect to magnetic field, $\mu=-\frac{1}{c}\frac{\partial F}{\partial
H}$, wherein $F$ is the free energy and $H$ is the magnetic field enclosed by the system. From text books we know that a current carrying loop behaves as a magnet, in other words Amperes law which states- magnetic moment is current multiplied by the area enclosed. The orbital magnetic moment density for a system coupled to reservoir can be calculated in two ways, first from the formulation of Akkermans, et al, [@akker; @mello] one can calculate it as follows-
$$\begin{aligned}
d\mu=\frac{1}{2\pi i}\frac {\partial[ln det S]}{\partial \phi} dE\end{aligned}$$
Here, $d\mu$ is the differential contribution to the magnetic moment at energy E, and S is the on-shell scattering matrix. In the system considered in Fig. 1, the on-shell scattering matrix is just the complex reflection amplitude $r$. Also from Amperes law one can calculate the orbital magnetic moment density via the local currents in the system. The orbital magnetic moment density defined via local currents in a loop, depends strongly on the topology of the system, whereas the magnetic moment densities calculated from the eigen spectrum as also in the formulation of Akkermans, et al, do not. This is special to only one-dimensional geometry where the eigen-spectrum is independent of variation in topology of the system . In fact there are infinitely many topological structures possible[@cedraschi; @colin_ijmpb]. If we consider our system as depicted in Fig. 1, to be planar and lying in the x-y plane then the magnetic moment density ($\mu_1$) can be viewed as being generated by current density $I_1$ enclosing an area $A_r$ and by current density $I_3$ enclosing area $A_b$, i.e., $\mu_1=\frac{1}{c}(I_{1}A_{r}+I_{3}A_{b})$, wherein $A_{r}$ and $A_{b}$ are the areas enclosed by the ring ($J1J2aJ3J1$) and the bubble ($J2aJ3bJ2$) respectively. Another orientation of the system in which the arm $J2bJ3$ is in the x-z plane gives $\mu_2=\frac{1}{c}(I_{1}A_{r}+I_{2}A_{r})/2=I_c A_r/2$ wherein $I_{c}=I_{1}+I_{2}$ is said to be the most appropriate generalization of the equilibrium persistent current and which is consistent[@cb_unpub] with Eq. 1, see \[\] for further details. Several other orientations are possible, for example, if the bubble lies in x-y plane and the ring lies in x-z plane, then $\mu_{z}=\frac{1}{c}(I_{3}A_{r}-I_{2}A_{r})/2$ and $\mu_{y}=\frac{1}{c}I_{1}A_{r}$. Even when our system lies in the x-y plane for fixed $l_1, l_2, l_3$, by deforming their shapes we can have different values of magnetic moment density along the z direction. It is also worth mentioning that the total magnetic moment (at temperature $T=0$) of a representative system is obtained by integrating the magnetic moment densities up-to the Fermi wave-vector $k_f$.
Results
=======
From the Amperes law we calculate the orbital magnetic moment density for some length parameters. After calculating we plot the orbital magnetic moment densities obtained for the case of $\mu_{2}=\frac{1}{c}(I_{1}A_{r}+I_{2}A_{r})/2=I_c A_r/2$. In Fig. 2(a) we depict the plot of the energy-eigen values (normalised by $\pi^2$) of the closed system, and in Fig. 2(b) we plot the dimensionless orbital magnetic moment density $\mu_2$ obtained via the local persistent current densities as a function of the dimensionless Fermi wave-vector $kl$ for different coupling parameters. Now lets test the equivalence of these definitions as $\epsilon \rightarrow 0$, i.e., the reservoir and system are almost decoupled. We see complete agreement. It can be noted from Fig. 2(a) that the ground state carries diamagnetic current while 1st, 2nd and 3rd excited states carry paramagnetic current for small values of flux (which is obvious from their slopes), the 4th excited state carries diamagnetic currents while the fifth paramagnetic current. Similarly from the top most panel of Fig. 2(b), the $\epsilon \rightarrow 0$ limit, it can be seen that the ground state carries diamagnetic current (magnitude is negative) while 1st, 2nd and 3rd excited states carry paramagnetic currents (magnitude is positive), the 4th carries diamagnetic current while the fifth paramagnetic current. Indeed the two definitions are completely equivalent as far as the $\epsilon \rightarrow 0$ limit (henceforth referred to as the weak coupling case) is considered. Now as we increase $\epsilon$, we notice a dramatic change from the weak coupling case (see middle panel in FIG. 2(b)). There are paramagnetic-diamagnetic jumps at those Fermi energies wherein one would have expected pure diamagnetic or paramagnetic current. This behavior is more seen as one increases the coupling till the maximum ($\epsilon=0.5$) is reached, see lowest panel of Fig. 2(b). Looking closely at the middle panel of Fig. 2(b), one can notice that the peaks broaden, noticeably the ground, 1st, 3rd and 5th, while the nature of the currents carried at 2nd and 4th levels [**changes qualitatively**]{}. Also, as one approaches the $\epsilon=0.5$ limit the 1st, 3rd and 5th levels disappear completely, these are some of the other qualitative features which distinguish the cases depicted in Fig. 2(a) and (b). This reaffirms that predictions from equilibrium thermodynamics are in general not valid for a quantum system strongly coupled to a bath. In the conventional equilibrium treatment the properties of the bath appear only through a single parameter namely, the temperature $T$ and nowhere the coupling parameter appears in the magnitude of equilibrium physical quantities. Our results reaffirm that equilibrium properties are determined by the coupling parameter in the quantum domain. The finite coupling can lead to qualitative changes (and not just the broadening of levels or persistent current peaks) from that of the predictions of equilibrium statistical mechanics as shown above.
We have verified separately that the orbital magnetic moment density calculated by the formulation of Akkermans, et al, (Eq. 1) is similar to that in Fig. 2(b), for the geometry considered here[@cb_unpub]. In addition we also have calculated the orbital magnetic moment density for different topological configurations, e.g., $\mu_{1},\mu_{z}, \mu_{y}$ (see the [*Theory*]{} section for further details) and obtained results not in consonance with that calculated from the eigen-spectrum (equilibrium statistical mechanics), thus bolstering the fact that the orbital magnetic moment density calculated from the local current densities is inherently linked to the topology of the system. As already pointed out, the orbital magnetic moment density calculated through the local current densities is qualitatively different (nature) from that calculated from the closed system energy eigen-spectrum which is independent of topological variations.
We have seen these novel features not only for one set of length parameters but for many different set of length parameters. In Fig. 3(a), we plot the energy eigenvalues (normalised by $\pi^2$) as a function of flux, and in Fig. 3(b) the dimensionless orbital magnetic moment densities as a function of the dimensionless Fermi wave-vector $kl$ for different length parameters. Herein also it can be seen from Fig. 3(a) that the ground and the first excited states carry diamagnetic currents while 2nd and 3rd carry paramagnetic currents for small values of flux (which is obvious from their slopes), the 4th and 5th again carry diamagnetic currents. Similarly from the top most panel of Fig. 3(b), the $\epsilon \rightarrow 0$ limit, it can be seen that the ground and first excited state carry diamagnetic current (magnitude is negative) while 2nd and 3rd carry paramagnetic currents (magnitude is positive), the 4th and 5th carry diamagnetic currents. The two definitions are again completely equivalent as far as the $\epsilon \rightarrow 0$ limit is considered. Again as we increase $\epsilon$, we notice the same change from the weak coupling case (see middle panel in FIG. 3(b)). There are paramagnetic-diamagnetic jumps at those Fermi energies wherein one would have expected pure diamagnetic or paramagnetic current. This behavior is more seen as one increases the coupling till the maximum ($\epsilon=0.5$) is reached, see lowest panel of Fig. 3(b). Thus, in contrast, to the case of a single quantum ring coupled to a reservoir as was considered by Büttiker, wherein coupling only led to a broadening of energy levels[@buti], in the case of a quantum double ring system considered here, in addition to level broadening one also sees [**a change in nature**]{} of currents as one increases the strength of coupling to the reservoir.
One should also mention here that there is a drawback in modeling the inelastic effects by this model, if one changes the position of attachment of lead and the double ring system. Qualitatively, different results for the orbital magnetic moment density would be obtained as by definition it involves the length parameters of our system, These length parameters will of-course change with position of junction (J1) . Only in the very weakly coupled regime can the specific position of lead to reservoir attachment be ignored. Of-course this drawback does not exist for the system investigated by Büttiker in Ref. wherein the orbital magnetic moment density would be same irrespective of the position where lead is attached. However, this simple model for coupling between system and reservoir is enough in order to bring out the importance of finite coupling between system and bath, [*vis a vis*]{} equilibrium thermodynamics.
Conclusions
===========
In this work we have shown using a very simple model of system-reservoir coupling that the equilibrium properties are determined by the strength of coupling parameter. This is consistent with the recent findings[@TA; @dattgup_prl], that in the quantum domain, equilibrium properties of the sub-system are related to dissipative coefficients arising due to subsystem-bath coupling, unlike the predictions of conventional statistical mechanics. These fully dynamical studies in the quantum domain invoke coherence and entanglement between system and bath degrees of freedom to come to the same conclusion. However our treatment is simple and invokes the single particle coherence which is disrupted by the presence of the reservoir (without bringing the notion of quantum entanglement between system and bath). However these finite coupling induced qualitative changes can be observed only in hybrid rings which exhibit current magnification effect, a purely quantum phenomena, in equilibrium[@colin_prb]. We have verified separately that multiple ring structures which do not exhibit current magnification effect do not show these qualitative changes apart from features expected from broadening of energy levels[@cb_unpub]. These results can be verified experimentally by attaching a voltage probe coupled to a nanoscopic semi-conducting system as depicted in Fig. 1. Here the voltage probe serves the purpose of a reservoir. One can change the parameters of the junction between system and lead by applying appropriate gate voltage (to simulate the effects of coupling parameter) beneath the junction and measure the orbital magnetic response as function of the gate voltage.
Acknowledgments
===============
Authors thank Dr. P. S. Deo for useful discussions.
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---
abstract: 'In this paper, we determine new characterizations of nested and nested GVZ-groups, including character-free characterizations, but we additionally show that nested groups and nested GVZ-groups can be defined in terms of the existence of certain normal series.'
address:
- 'Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240, U.S.A.'
- 'Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240, U.S.A.'
author:
- 'Shawn T. Burkett'
- 'Mark L. Lewis'
title: Groups where the centers of the irreducible characters form a chain II
---
Introduction
============
All groups in this paper will be finite and when $G$ is a group, we will write $\irr G$ for the set of irreducible characters of $G$. In Problem \#24 of Research problems and themes I of [@YBGPPOV1], Berkovich asks for a description of the $p$-groups $G$ for which the centers (quasi-kernels) of the irreducible characters form a chain with respect to inclusion. In [@ML19gvz], the second author did just this. In this paper, we arrive at a different characterization of these groups using chains of subgroups that are defined for all groups.
Let $G$ be a group. We will define a characteristic subgroup $K (G)$ in terms of the centers of certain irreducible characters of $G$. Since the definition of this subgroup is technical, we postpone its statement until Section \[K section\]. We want to state some properties of this group.
\[intro Kprops\] Let $G$ be a nonabelian group, and let $N\lhd G$. Then either $K(G) \le N$ or $N \le Z(G)$ and $Z(G/N) = {Z}(G)/N$.
If $G$ is a group, then we define $Z_2$ by $Z_2/Z(G) = Z(G/Z(G))$. It turns out that the behavior of $K (G)$ depends on whether or not $Z_2 = Z(G)$ or $Z_2 > Z(G)$. We next state the situation when $Z_2 = Z(G)$.
\[intro Z2=Z\] Let $G$ be a nonabelian group. Assume $Z_2 = Z(G)$. Then the following are true:
1. $K(G)$ is the intersection of all the noncentral normal subgroups of $G$.
2. $K(G) \not\le Z(G)$ if and only if $G$ has a unique subgroup $N$ that is minimal among noncentral normal subgroups of $G$.
When $Z_2 > Z(G)$, we have the following properties.
\[intro Z\_2>Z\] Let $G$ be a group and suppose $Z_2 > Z(G)$. Then the following are true:
1. $K (G) \le Z (G)$ and $K (G)$ is an elementary abelian $p$-group for some prime $p$.
2. If $K(G) > 1$, then $Z_2/Z(G)$ is a $p$-group where $p$ is the prime dividing $|K (G)|$.
3. $Z(G/K(G)) > Z(G)/K(G)$ if and only if every irreducible character $\chi \in \irr {G/K(G)}$ satisfies $Z (\chi) > Z(G)$.
Following [@ML19gvz], we say that a group $G$ is [*nested*]{} if for all characters $\chi, \psi \in \irr G$ either $Z (\chi) \le Z (\psi)$ or $Z (\psi) \le Z(\chi)$. It is not difficult to see that a group $G$ is nested when the centers of the irreducible characters form a chain. In [@SBML1], we attach a subgroup to each conjugacy class, and we show that a group is nested if the subgroups attached to the conjugacy classes form a chain. In this paper, we find another way to determine whether a group is nested. Using the subgroup $K(G)$, we can determine when $G$ is nested.
\[intro K > 1\] Let $G$ be a group. Then $G$ is nested if and only if $K (G/N) > 1$ for every proper normal subgroup $N$.
We will define a chain of subgroups by defining $K_0 = 1$ and for $i \ge 1$, we set $K_i$ by $K_i/K_{i-1} = K (G/K_{i-1})$. Since $G$ is finite, this chain will terminate, and we write $K_\infty$ for the terminal term of this chain. In particular, we can use this chain to determine if $G$ is nested.
\[intro K\_infty\] Let $G$ be a group. Then $G$ is nested if and only if $K_\infty = G$.
When $G$ is nested, we will see that there is a correspondence between the groups that occur as the centers of irreducible characters and the $K_i$’s. In [@ML19gvz], the second author proved a number of results regarding the structure of the factors coming from the centers of characters. We show that these results hold for the $K_i$’s even when $G$ is not nested. We present one such result next. When $N$ is a normal subgroup of $G$, we define $Z_N$ by $Z_N/N = Z(G/N)$.
\[intro z\_k\] Let $G$ be a group. Suppose that there exists integers $1 \le j < k$ so that $Z_{K_i} > Z_{K_{i-1}}$ and $[K_{i},G] \le K_{i-1}$ for all integers $i$ with $j \le i \le k$. Then there is a prime $p$ so that $K_k/K_{j-1}$ and $Z_{K_k}/Z_{K_{j-1}}$ are $p$-groups. In particular, $K_i/K_{i-1}$ and $Z_{K_i}/Z_{K_{i-1}}$ are elementary abelian $p$-groups for every integer $i$ with $j \le i \le k$.
Note that the $K_i$’s are an ascending series of normal subgroups. Using the centers of characters in a different fashion, we construct a descending chain of normal subgroups $G = \delta_1 \ge \delta_2 \ge \dotsb\ge \delta_i$. We let $\delta_\infty$ be the terminal term in this series. We are able to show the following:
\[intro delta\] Let $G$ be a group. Then $G/[\delta_i,G]$ is a nested group for all integers $i$. In particular, $G$ is nested if and only if $\delta_\infty = 1$.
The study of nested groups was initiated by Nenciu in [@AN12gvz] and [@AN16gvz] under the additional hypothesis of GVZ-groups. A group $G$ is a [*GVZ-group*]{} if every irreducible character of $G$ vanishes off its center. We note that the definition of nested in [@AN12gvz] and [@AN16gvz] is different than the definition used in this paper, but the second author has shown in [@ML19gvz] that the definition in this paper is equivalent to Nenciu’s definition when $G$ is a GVZ-group.
Using the vanishing-off subgroup for subgroups in [@MLvos09] and [@NM14] that generalizes the vanishing-off subgroup of a character, we define the subgroup $U (G)$. To obtain results regarding $U (G)$, we will use results regarding Camina triples that were proved by Mlaiki in [@NM14]. In particular, we will see that $U (G)$ is always a subgroup of $K (G)$ and often equals $K (G)$, and thus, we will see that $U (G)$ shares many of the properties of $K (G)$. When $N$ is a normal subgroup of $G$, we define $\irr {G \mid N}$ to be the set of characters in $\irr G$ that do not have $N$ in their kernels. One property that is different is the following:
\[intro U prop\] Let $G$ be a group. Then $U (G)$ is the largest subgroup of $G$ so that every character in $\irr {G \mid U(G)}$ is fully ramified with respect to $Z (G)$.
Using $U (G)$ in a manner similar to $K (G)$, we will be able to determine when $G$ is a nested GVZ-group. In particular, we have the following theorem:
\[intro U > 1\] Let $G$ be a group. Then $G$ is a nested GVZ-group if and only if $U (G/N) > 1$ for every proper normal subgroup $N$ of $G$.
Finally, we will also use the vanishing-off subgroup to find a chain of subgroups $G = \epsilon_1 \ge \epsilon_2 \ge \dots \ge \epsilon_i$ so that $\epsilon_i \ge \delta_i$ and often $\epsilon_i = \delta_i$. We write $\epsilon_\infty$ for the terminal element of this series.
\[intro epsilon\] Let $G$ be a group. Then $G/[\epsilon_i,G]$ is a nested GVZ-group for every integer $i$. In particular, $G$ is a nested GVZ-group if and only if $\epsilon_\infty = 1$.
We close this section by noting that we initially defined the subgroup $U$ and the chain of $\epsilon$’s independently. We used them to characterize nested GVZ-groups in our preprint [@SBML2]. However, we realized that those definitions could be generalized, which led to the characterizations of nested groups in terms of the subgroup $K$ and the chain of $\delta$’s. We then obtained the results for $U$ and the chain of $\epsilon$’s as consequences of the more general work.
The subgroup $K(G)$ {#K section}
===================
Let $G$ be a group. Set $\mathcal{X} = \{ \chi \in \irr G \mid {Z} (\chi) > {Z} (G) \}$. Observe that if $\lambda$ is a linear character of $G$, then $Z(\lambda) = Z(G)$. Hence, if $G$ is abelian, then $\mathcal {X}$ is empty. On the other hand, if $G$ is nonabelian, then $Z (G) < G$ and $Z(1_G) = G$ and so, $1_G \in \mathcal {X}$. It follows that $\mathcal {X}$ is empty if and only if $G$ is abelian. When $G$ is abelian, we will set $K(G) = G$. When $G$ is nonabelian, define $K (G) = \bigcap_{\psi \in \mathcal{X}} \ker(\psi)$. Thus, $K (G)$ is the intersection of the kernels of these characters. We begin by showing when $G$ is nonabelian that $K (G)$ is always contained in the derived subgroup.
\[G’\] If $G$ is a nonabelian group, then $K (G) \le G'$.
Since $G$ is nonabelian, we have $Z(G) < G$. Recall that if $\lambda \in \irr {G/G'}$, then $Z(\lambda) = G > Z(G)$. This implies that $\irr {G/G'} \subseteq \mathcal {X}$ and so, we have $G' = \bigcap_{\lambda \in \irr {G/G'}} \ker (\lambda) \ge K(G)$.
Recall that a group $G$ is quasi-simple if $G$ is perfect (i.e. $G' = G$) and $G/Z(G)$ is nonabelian simple. We now show that abelian and quasi-simple groups are the only groups where $K (G) = G$.
\[K = G\] Let $G$ be a group. Then $K (G) = G$ if and only if $G$ is abelian or $G$ is quasi-simple.
By definition, $K (G) = G$ if $G$ is abelian. Now suppose $G$ is quasi-simple and consider $1_G \ne \chi \in \irr G$. Then $\ker (\chi) \le Z(G)$, and since $G/Z(G)$ is nonabelian simple, it follows that $Z(G/\ker (\chi)) = Z(G)/\ker (\chi)$. Hence, $Z(\chi) = Z(G)$, and so, $\mathcal {X} = \{ 1_G \}$. It follows that $K (G) = \ker (1_G) = G$. On the other hand, suppose that $G$ is not abelian and not quasi-simple. By Lemma \[G’\], we have $K(G) \le G'$. If $G' < G$, then we have the conclusion; so, we may assume $G$ is perfect. However, since $G$ is not quasimple, we know that $G/Z(G)$ is not simple, so there must exist a normal subgroup $M$ so that $Z(G) < M < G$. If $\mu \in \irr {G/M}$, then $Z(G) < M \le Z(\mu)$. It follows that $\irr {G/M} \subseteq \mathcal {X}$ and hence, $K (G) \subseteq \bigcap_{\mu \in \irr {G/M}} \ker (\mu) = M < G$. This proves the lemma.
The next observation is immensely useful. In fact, it allows us to define $K(G)$ in a different way, without mention of characters.
\[center kernel\] Let $G$ be a group and let $H \le G$. If $\chi \in \irr G$, then $[H,G] \le \ker (\chi)$ if and only if $H \le Z(\chi)$.
Assume that $[H,G] \le \ker(\chi)$. Then we have the quotient: $$H \ker(\chi)/\ker(\chi) \le Z (G/\ker(\chi)) = Z (\chi)/\ker (\chi),$$ and so it follows that $H \le Z(\chi)$. The converse is clear, since $Z (G/\ker(\chi)) = Z(\chi)/\ker (\chi)$.
Let $G$ be a group. Define $\gamma_G(g) = \{ [g,x] \mid x \in G\}$ for every element $g\in G$. We let $[g,G]$ denote the subgroup generated by $\gamma_G(g)$. The following lemma appeared in [@SBML]. Since the proof is only one line, we include it again here.
\[cen cond\] Let $G$ be a group. Fix an element $g \in G$ and a character $\chi \in \irr G$. Then $g \in Z(\chi)$ if and only if $[g,G] \le \ker (\chi)$.
This follows immediately from the definition that states: $Z(\chi)/\ker (\chi) = Z (G/\ker (\chi))$.
We now obtain a different characterization of $\mathcal {X}$ and this yields a description of $K(G)$ in terms of the groups $[g,G]$
\[K intersection\] Let $G$ be a nonabelian group. Then $\mathcal {X} = \bigcup_{g \in G \setminus Z(G)} \irr {G/[g,G]}$, and thus, $\displaystyle K (G) = \bigcap_{g \in G\setminus Z(G)} [g,G]$.
We work to prove the first conclusion. Suppose $g \in G \setminus Z(G)$ and $\chi \in \irr {G/[g,G]}$. By Lemma \[cen cond\], this implies that $g \in Z(\chi)$, and so, $Z(\chi) > Z(G)$. This leads to the containment: $\bigcup_{g \in G \setminus Z(G)} \irr {G/[g,G]} \subseteq \mathcal {X}$. Conversely, if $\chi \in \mathcal {X}$, then there exists some element $g \in Z(\chi) \setminus Z(G)$. Applying Lemma \[cen cond\], we see that $\chi \in \irr {G/[g,G]}$. This implies that $\mathcal {Z} \subseteq \bigcup_{g \in G \setminus Z(G)} \irr {G/[g,G]}$, and the first conclusion is proved.
For each element $g \in G$, we have that $[g,G] = \bigcap_{\chi \in \irr {G/[g,G]}} \ker (\chi)$. Hence, $\bigcap_{g \in G \setminus Z(G)} [g,G] = \bigcap_{g \in G \setminus Z(G)} \bigcap_{\chi \in \irr {G/[g,G]}} \ker(\chi)$. Using the claim from the first paragraph, we have $\bigcap_{g \in G \setminus Z(G)} \bigcap_{\chi \in \irr {G/[g,G]}} \ker(\chi) = \bigcap_{\chi \in \mathcal {X}} \ker (\chi) = K(G)$. This yields $K (G) = \bigcap_{g \in G \setminus Z(G)} [g,G]$, as desired.
Properties of $K(G)$
====================
We now survey some interesting properties of $K(G)$. Notice that this includes Theorem \[intro Kprops\].
\[Kprops\] Let $G$ be a nonabelian group. Set $K = K(G)$. Then the following hold:
1. If $N$ is normal in $G$, then either $K \le N$ or $N \le Z(G)$.
2. If $N \lhd G$ and $K \nleq N$, then $Z_N/N = {Z} (G/N) = {Z}(G)/N$.
3. If $N \lhd G$ and $K \nleq N$, then $KN/N \le K (G/N)$.
4. If $Z_K = Z(G)$, then $K(G/K) = 1$.
We begin by noting that $K \nleq N$ implies $N \le Z(G)$ if $N$ is a normal subgroup of $G$. Indeed if $K \nleq N$, then by the definition of $K$ there exists a character $\chi \in \irr{G/N}$ satisfying $Z(G) = Z(\chi)$. Since $N \le Z(\chi)$, this implies that $N \le Z(G)$ proving (1).
Now, let $N$ be a normal subgroup of $G$ satisfying $K \nleq N$, and assume that ${Z} (G/N) = Z_N/N > {Z}(G)/N$. Consider a character $\chi \in \irr {G/N}$. Since $N \le \ker(\chi)$, it follows that ${Z}(\chi) \ge Z_N > {Z}(G)$. In particular, $K \le \ker(\chi)$. Hence, we have $\irr{G/N} \subseteq \irr{G/K}$, from which it follows that $K \le N$. This contradiction establishes (2).
Next, we prove (3). Assume that $N \lhd G$ and $K \nleq N$. Then we have $Z_N = Z(G)$ by (2). So by Lemma \[K intersection\], we have $K (G/N) = \bigcap_{g \in G \setminus Z_N} [gN,G/N]$ and $K = \bigcap_{g \in G \setminus Z(G)} [g,G]$. Hence, $K \le [g,G]$ for all $g \in G \setminus Z(G)$, and thus, $KN/N \le \bigcap_{g \in G \setminus Z(G)}[g,G]N/N$. Notice that $\{aN \mid a \in \gamma_G (g) \} \subseteq \gamma_{G/N} (gN)$ for each $g \in G \setminus Z_N$ and $[g,G] \le N$ for each $g \in Z_N \setminus Z(G)$. It follows that $\bigcap_{g \in G \setminus Z(G)} [g,G]N/N \le \bigcap_{g \in G \setminus Z_N} [gN,G/N]$, and this gives the containment in (2).
Finally, we prove (4). We are assuming that $Z_K = Z(G)$, which implies that $Z(G/K) = Z(G)/K$. Following the definition of $\mathcal {X}$, we set $\mathcal {X}_K = \{ \chi \in \irr {G/K} \mid Z(\chi)/K > Z(G/K) \}$. We claim that $\mathcal {X} = \mathcal{X}_K$. Suppose that $\chi \in \irr {G/K}$ and $Z (\chi)/K > Z(G/K) = Z(G)/K$, and viewing $\chi$ as a character in $\irr G$, we have $Z (\chi) > Z(G)$. Conversely, suppose $\chi \in \irr G$ with $Z (\chi) > Z(G)$. By the definition of $K (G)$, we have $K (G) \le \ker (\chi)$ and $Z(G/K) = Z(G)/K \le Z(\chi)/K$. This implies that $\chi \in \irr {G/K}$. This prove the claim. Now, applying the definition of $K (G)$, we have $K (G) = \bigcap_{\chi \in \mathcal {X}} \ker (\chi)$. It follows that $K(G/K) = \bigcap_{\chi \in \mathcal {X}_K} \ker (\chi)/K = \bigcap_{\chi \in \mathcal {X}} \ker (\chi)/K = K(G)/K = 1$.
\[direct prods\] Let $M$ and $N$ be groups.
1. If $M$ and $N$ are both nonabelian, then $K (M \times N) = 1$.
2. If $M$ is nonabelian and $N$ is abelian, then $K (M \times N) = K(M)$.
We will let $G = M \times N$. Suppose $M$ and $N$ are both nonabelian. Then there exist elements $m \in M \setminus Z (M)$ and $n \in N \setminus Z(N)$. It follows that $1 \ne [m,G] \le M$ and $1 \ne [n,G] \le N$. By Lemma \[K intersection\], we have that $K (G) \le [m,G] \cap [n,G] \le M \cap N = 1$. Now, suppose that $M$ is nonabelian and $N$ is abelian. Given $g \in G \setminus Z(G)$, we have $g = mn$ for some $m \in M$ and $n \in N$. Observe that if $m \in Z(M)$, then $g \in Z(G)$, so we have $m \not\in Z(M)$. Observe that $[g,G] = [mn,MN] = [m,M]$ since $N$ is central in $G$. It follows that $\bigcap_{g \in G \setminus Z(G)} [g,G] = \bigcap_{m \in M \setminus Z(M)} [m,M]$, and applying Lemma \[K intersection\], we have $K (G) = K (M)$.
Let $G$ be a group. We let $Z_i$ denote the $i^{\rm th}$ member of the upper central series. That is, we set $Z_0 = 1$ and define $Z_i$ recursively by $Z_{i+1}/Z_i = Z(G/Z_i)$. It is not difficult to see that $Z_{i+1} = \{ g \in G \mid [g,G] \le Z_i\}$ for ever integer $i \ge 0$. We note that $G$ is nilpotent if and only if $G = Z_n$ for some integer $n$. We now study $K (G)$ in the situation where $Z_2 = Z(G)$. This theorem includes Theorem \[intro Z2=Z\].
Let $G$ be a nonabelian group. Assume $Z_2 = Z(G)$. Then the following are true:
1. $K(G)$ is the intersection of all the noncentral normal subgroups of $G$.
2. $K(G) \not\le Z(G)$ if and only if $G$ has a unique subgroup $N$ that is minimal among noncentral normal subgroups of $G$. In this case, $K (G) = N$ and $K(G) Z(G)/Z (G)$ is the unique minimal normal subgroup of $G/Z (G)$. Also, $K (G) = [n,G]$ for some noncentral element $n \in G$.
Observe that every noncentral normal subgroup contains a subgroup that is minimal among the noncentral normal subgroups of $G$. It is not difficult to see that this implies that the intersection of the noncentral normal subgroups of $G$ equals the intersection of the minimal noncentral normal subgroups of $G$. Hence, it suffices to prove that the intersection of the minimal noncentral normal subgroups of $G$ equals $K (G)$.
Let $N$ be minimal noncentral normal subgroup of $G$. Since $N$ is noncentral, there exists an element $n \in G \setminus Z(G)$. Because $n$ is noncentral, we have $[n,G] > 1$ and as $Z_2 = Z(G)$, we see that $[n,G] \not\le Z(G)$. On the other hand, $[n,G]$ is normal in $G$ and $[n,G] \le N$. The minimality of $N$ implies that $N = [n,G]$.
Now, if $g \in G \setminus Z(G)$, then $[g,G]$ will be a noncentral normal subgroup of $G$ since $Z_2 = Z(G)$. It follows that $[g,G]$ contains a minimal noncentral normal subgroup of $G$. Let $\mathcal {N} = \{ n \in G \setminus Z(G) \mid [n,G] {\rm ~is~minimal~noncentral~normal}\}$. It follows that if $g \in G \setminus Z(G)$, then there exists $n \in \mathcal {N}$ so that $[n,G] \le [g,G]$. It is not difficult to see that this implies that $\bigcap_{g \in G \setminus Z(G)} [g,G] = \bigcap_{n \in \mathcal {N}} [n,G]$ is the intersection of the minimal noncentral normal subgroups of $G$. Applying Lemma \[K intersection\], we obtain (1).
Suppose $K (G) \not\le Z(G)$. Then there exists $N \le K(G)$ so that $N$ is a minimal noncentral normal subgroup of $G$. Let $M$ be any minimal noncentral normal subgroup of $G$. By (1), we have $K (G) \le M$. Notice that we now have $N \le K (G) \le M$. Notice that this implies that $K (G) = N = M$, and hence, $N$ is the unique minimal noncentral normal subgroup of $G$. Notice that $K(G) Z(G) > Z (G)$, so $K (G) Z (G)/Z (G) > 1$. Let $L/Z(G)$ be a minimal normal subgroup of $G/Z(G)$. Then $L$ is a noncentral normal subgroup of $G$. Because $K (G)$ is the unique minimal noncentral subgroup of $G$, we have that $K (G) \le L$, and so, $K (G) Z (G) \le L$. Now, the fact that $L/Z(G)$ is minimal normal implies that $L = K(G) Z(G)$. Since $L$ was arbitrary, it follows that $K(G) Z(G)/Z(G)$ is the unique minimal normal subgroup of $G/Z(G)$. We apply the earlier paragraph to see that $K (G) = [n,G]$ for any $n \in K(G) \setminus Z (G)$.
Finally, suppose that $G$ has a unique minimal noncentral normal subgroup $N$. Then $N$ is contained in every noncentral normal subgroup of $G$, and so, $N \le K (G)$. On the other hand, by the second paragraph, we have $N = [n,G]$ for any $n \in N \setminus Z(G)$, and so, $N = [n,G] \ge \bigcap_{g \in G\setminus Z(G)} [g,G] = K (G)$ by Lemma \[K intersection\]. We conclude that $N = K(G)$ and $K (G) \not\le Z(G)$ as desired.
We next consider the case where $K(G)$ is central.
Let $G$ be a nonabelian group. If $K(G) > 1$ and $K(G) \le Z(G)$, then every minimal normal subgroup of $G$ is central.
Let $N$ be a minimal normal subgroup of $G$. By Lemma \[Kprops\], we have either $K(G) \le N$ or $N \le Z(G)$. Since $N$ is minimal normal, if $K (G) \le N$, then $N = K (G)$ and we are assuming $K (G) \le Z(G)$. This proves the result.
We now see that when $Z_2 > Z(G)$ that $K (G)$ must be central and elementary abelian. This includes Theorem \[intro Z\_2>Z\] (1) and (2).
\[Z\_2 > Z\] Let $G$ be a group. Then the following are true:
1. If $Z_2 > Z(G)$, then $K (G) \le Z (G)$ and $K (G)$ is an elementary abelian $p$-group for some prime $p$.
2. If $Z_2 > Z(G)$ and $K(G) > 1$, then $Z_2/Z(G)$ is a $p$-group where $p$ is the prime dividing $|K (G)|$.
3. If $Z_2 \ge Z_{K (G)} > Z(G)$, then $K(G)$ and $Z_{K (G)}/Z(G)$ are elementary abelian $p$-groups for some prime $p$ and $K(G) = [g,G]$ for all $g \in Z_{K(G)} \setminus Z(G)$.
Suppose that $Z_2 > Z(G)$. Then there exists $g \in Z_2 \setminus Z(G)$. It follows that $[g,G] \le Z(G)$. By Lemma \[K intersection\], we have $K (G) \le [g,G] \le Z(G)$. We can find a prime $p$ and element $x \in Z_2 \setminus Z(G)$ so that $x^p \in Z(G)$. Notice that $[x,G] = \gamma_G (x)$ since $[x,G] \le Z(G)$. Given an element $k \in K(G)$, there must exist an element $y \in G$ so that $k = [x,y]$. Now, $k^p = [x,y]^p = [x^p,y] = 1$ since $x^p \in Z(G)$. Since $k$ was arbitrary, this implies that $K(G)$ is an elementary abelian $p$-group. This proves (1).
Suppose in addition that $K(G) > 1$. Since $K(G) > 1$, we can find $1 \ne k \in K(G)$. Now, consider any $g \in Z_2 \setminus Z(G)$, and observe that $[g,G] = \gamma_G (g)$ and $K (G) \le [g,G]$ by Lemma \[K intersection\]. Thus, we can find an element $h \in G$ so that $[g,h] = k$. We see that $1 = k^p = [g,h]^p = [g^p,h]$. Thus, $g^p$ centralizes $h$ and $g$ does not centralize $h$. This implies that $p$ divides the order of $g Z(G)$ since otherwise $\langle g, Z(G) \rangle = \langle g^p,Z(G) \rangle \le C_G (h)$ and $g$ would centralize $h$ which is a contradiction. This implies that $p$ divides the order of every nontrivial element of $Z_2/Z(G)$ and thus, $Z_2/Z(G)$ must be a $p$-group. This implies (2).
Now, assume that $Z_{K (G)} > Z(G)$. Notice that $Z_{K (G)} \le Z_2 (G)$ and $K (G) > 1$, so by (2), $K (G)$ is an elementary abelian $p$-group. Let $g \in Z_{K(G)} \setminus Z(G)$. Then $[g,G] \le K (G)$ and $[g,G] \le K(G)$ by Lemma \[K intersection\]. Hence, we have $[g,G] = K(G)$. We see that $1 = [g,h]^p = [g^p,h]$ for all $h \in G$ and so, $g^p \in Z(G)$. This implies that $Z_{K(G)}/Z(G)$ is an elementary abelian $p$-group.
We will present an example a nonabelian $p$-group $G$ with $K (G) > 1$ where $Z_{K(G)} = Z (G)$. Our next two results show some interesting properties that do hold when $Z_{K(G)} > Z(G)$.
\[K equiv conditions\] Let $G$ be a group satisfying $K (G) > 1$ and $Z_2 > Z(G)$. Write $K = K (G)$. Then the following are equivalent:
1. $Z_{K} > Z(G)$.
2. There exists an element $g \in G$ satisfying $[g,G] = K$.
3. $K = [Z_{K},G]$.
By Lemma \[Z\_2 > Z\] (3), we see that if $Z_K > Z(G)$, then $K = [g,G]$ for every element $g \in Z_K \setminus Z(G)$. This shows (1) implies (2). Suppose there is an element $g \in G$ so that $[g,G] = K$. Then $g \in Z_K$ by the definition of $Z_K$. This implies that $K = [g,G] \le [Z_K,G] \le K$, and so, $K = [Z_K,G]$. We have (2) implies (3). Since $K > 1$, we see that if $[Z_K,G] = K$, then $Z_K > Z(G)$. Hence, (3) implies (1).
We note that this next result strongly uses the hypothesis that $K(G) > 1$. This includes Theorem \[intro Z2=Z\] (3).
\[K equiv cond 2\] Let $G$ be a group satisfying $K (G) > 1$ and $Z_2 > Z(G)$. Write $K = K(G)$. Then the following are equivalent:
1. $Z_K > Z(G)$.
2. $K (G/N) = KN/N$ for every normal subgroup $N$ of $G$ not containing $K$.
3. Every irreducible character $\chi \in \irr {G/K}$ satisfies $Z (\chi) > Z(G)$.
Assuming (1), we have that every character $\chi \in \irr {G/K}$ satisfies $Z(\chi) \ge Z_K > Z(G)$. Thus (3) holds. We now show that (3) implies (2). Assume that $Z(\chi) > Z(G)$ for every character $\chi \in \irr {G/K}$, and assume to the contrary that there exists a normal subgroup $N$ not containing $K$ and satisfying $K (G/N) \ne KN/N$. By Lemma \[Kprops\] (2), we must have that $K(G/N) > KN/N$. Then there exists a character $\psi \in \irr {G/N \mid K(G/N)}$ with $K$ in its kernel. Since $K$ is in the kernel of $\psi$, we have $Z(G) < Z(\psi)$ by our hypothesis. On the other hand, since $K(G/N)$ is not contained in the kernel of $\psi$, viewed as a character of $G/N$, it must be that $Z(\psi)/N = Z(G/N)$. Also, since $N$ is not contained in $K$, we use Lemma \[Kprops\] (1) to see that $Z (G/N) = Z (G)/N$. Combining these, we obtain $Z(G)/N < Z(\psi)/N = Z(G/N) = Z(G)/N$, which is a contradiction. Hence, (2) holds.
Finally, we show that (2) implies (1). Assume that (1) is not true, and let $G$ be a counterexample of minimal order. For every element $g \in Z_2$ that satisfies $g^p \in Z(G)$, it is not difficult to see that $[g,G]$ is elementary abelian. By Lemma \[K intersection\], we have $K \le [g,G]$. So $[g,G]$ is elementary abelian and contains $K$, and it follows that $[g,G] = K \times C_g$ for some subgroup $C_g$. Among the possible choices for $g$, we choose $g$ so that $|[g,G]|$ is minimal.
Since $G$ is a counterexample, we are assuming that $Z_K = Z(G)$. In light of Lemma \[K equiv conditions\], we cannot have that $[g,G] = K$. This implies that $[g,G] > K$, and hence, we must have $C_g > 1$. Let $N \le [g,G]$ be a minimal normal subgroup of $G$ satisfying $K \nleq N$. Note that $[g,G]/N = KN/N \times C_gN/N$.
We claim that $G/N$ satisfies (2). To that end, let $\overline{H}$ denote $HN/N$ for every subgroup $H \le G$, and suppose $\overline{H}$ is a normal subgroup of $\overline{G}$ not containing $K(\overline{G})$. Since $K(\overline{G}) = \overline{K}$, we deduce that $H$ does not contain $K$. So $K (\overline{G}/\overline{H}) \cong K(G/H) = KH/H \cong \overline{K}\overline{H}/\overline{H} = K(\overline{G})\overline{H}/\overline{H}$, as claimed.
Suppose $xN \in Z_{K (G/N)} \setminus Z(G/N)$. By Lemma \[Z\_2 > Z\] (3), we have $[xN,G/N] = K(G/N)$. On the other hand, using Lemma \[Kprops\] (2), we obtain $Z(G/N) = Z(G)/N$. Since $g \not\in Z(G)$, we see that $gN \not\in Z(G/N)$. Using Lemma \[K intersection\], we deduce that $K(G/N) \le [gN,G/N]$. Hence, $[xN,G/N] \le [gN,G/N]$.
Let $h \in G$. Suppose $a \in \gamma_G (h)$, then $aN \in \gamma_{G/N} (hN)$. We obtain $[h,G]N/N \le [hN,G/N]$. Also, as $\gamma_G (g) = [g,G]$ contains $N$, this yields $\gamma_{G/N} (gN) = \{aN \mid a \in \gamma_G (g) \}$ and so, $[g,G]/N = [gN,G/N]$. We now have $[x,G] \le [x,G]N \le [g,G]N = [g,G]$.
Observe that $Z_{K (G/N)} \le Z_2 ((G/N)/(Z (G)/N)$. It follows that $x \in Z_2$. By Lemma \[Z\_2 > Z\], we have that $(xN)^p \in Z(G/N)$ and so, $x^p \in Z(G)$. By the minimality of $g$, we have $\norm {[g,G]} \le \norm {[x,G]}$. This implies that $[x,G] = [g,G]$, and so, $[x,G]N/N \le K(G/N) = [xN,G/N] \le [gN,G/N] = [g,G]/N = [x,G]/N \le [x,G]N/N$. We conclude that $K(G/N) = [g,G]/N$.
Recall that we are assuming $K (G/N) = KN/N$. On the other hand, we have $K (G/N) = [g,G]/N = KN/N \times C_gN/N$. This implies that $C_g \le N$ and since $N$ is minimal normal, $N = C_g$. Certainly, $[g,G]$ contains a minimal normal subgroup $L$ different from $N$ and not containing $K$. The above argument shows that $C_g = L$, which is a contradiction. It follows that no such counterexample $G$ exists.
Suppose $G$ is a group, if the character $\chi \in \irr G$ satisfies $Z (\chi) > Z(G)$, then the definition of $K (G)$ implies that $K (G) \le \ker (\chi)$. Condition (2) of Lemma \[K equiv cond 2\] implies that $Z (\chi) > Z(G)$ for every character $\chi \in \irr {G/K(G)}$ is equivalent to $Z_{K(G)} > Z(G)$. This implies that if $Z_2 > Z(G)$ and $Z_K = Z(G)$, then there exists a character $\chi \in \irr {G/K}$ so that $Z(\chi)/K = Z(G)/K = Z(G/K)$. Note that there exists a group $L$ so that $Z (\chi) > Z(L)$ for all $\chi \in \irr L$. For example of such a group take $L$ to be `SmallGroup (32,27)` in the computer algebra package Magma [@magma]. Notice that if $G = L \times C_2$, where $C_2$ is the cyclic group of order 2, one obtains a group $G$ with a normal subgroup $N$ so that $Z(G/N) = Z(G)/N$ and every irreducible character $\chi \in \irr {G/N}$ satisfies $Z(\chi) > Z(G)$.
\[cen int\] Let $N$ and $M$ be normal subgroups of $G$. Then $Z_{N \cap M} = Z_N \cap Z_M$.
It is clear that $Z_{N \cap M} \le Z_N \cap Z_M$. Conversely, suppose $g \in Z_N \cap Z_M$, then $[g,G] \le N \cap M$. It follows that $g \in Z_{N \cap M}$, and so, we have $Z_N \cap Z_M \le Z_{N\cap M}$ as well.
As a corollary, we characterize $Z_{K (G)}$.
\[Z\_K\] If $G$ is a nonabelian group, then $Z_{K(G)} = \bigcap_{\psi \in \mathcal{X}} Z(\psi)$.
By Lemma \[cen int\], we see that $\bigcap_{\psi \in \mathcal{X}} Z_{\ker(\psi)} = Z_{\bigcap_{\psi \in \mathcal{X}} \ker (\psi)} = Z_{K(G)}$.
Nested groups {#nested section}
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This section is dedicated to characterizing nested groups in terms of $K$. Recall that a group is called [*nested*]{} if $\{ Z (\chi) \mid \chi \in \irr G\}$ is a chain with respect to inclusion. In this case, we write $\{ Z (\chi) \mid \chi \in\irr G \} = \{ X_0, X_1,\dotsb, X_n \}$, where $G = X_0 > X_1 > \dotsb > X_n \ge 1$. Following [@ML19gvz], we call this the [*chain of centers*]{} for $G$. We list here the results about nested groups from [@ML19gvz] that we require throughout this paper.
\[lewisgvz\] Let $G$ be a nested group with chain of centers $G = X_0 > X_1 > \dotsb > X_n \ge 1$. Then the following statements hold:
1. $X_n = {Z} (G)$.
2. $[X_i,G] < [X_{i-1},G]$ for each integer $1 \le i \le n$.
3. Fix a character $\chi \in \irr G$. Then ${Z} (\chi) = X_i$ if and only if $[X_i,G] \le \ker (\chi)$ and $[X_{i-1},G] \nleq \ker(\chi)$.
We begin by investigating the $K$-series. This next lemma will be the key in showing that the series defined by $K (G)$ is connected to determining if $G$ a nested group.
\[nested K\] Let $G$ be nested with chain of centers $G = X_0 > X_1 > \dotsb > X_n \ge 1$. Then $K(G) = [X_{n-1},G]$.
Observe that $$\bigcap_{\substack{\chi \in \irr G \\ Z(\chi) > Z(G)}} Z(\chi) = X_{n-1} > X_n = Z(G),$$ where the last equality holds by Lemma \[lewisgvz\] (1). We have $K (G) = \cap_{\chi \in \mathcal{X}} \ker (\chi)$. Recall that $\chi \in \mathcal{X}$ if and only if $Z(\chi) > Z(G) = X_n$. It follows that $\chi \in \mathcal {X}$ if and only if $X_{n-1} \le Z(\chi)$. By Lemma \[center kernel\], we see that $X_{n-1} \le Z (\chi)$ if and only if $[X_{n-1},G] \le \ker (\chi)$. It follows that $\mathcal X = \irr {G/[X_{n-1},G]}$, and since the intersection of the characters in $\irr {G/[X_{n-1},G]}$ will be $[X_{n-1},G]$, we conclude that $K (G) = [X_{n-1},G]$.
We define $K_i$ recursively by setting $K_0 = 1$ and defining $K_{i+1}/K_i = K(G/K_i)$. Observe, using Lemma \[G’\], that $K_{i+1} \le G'$ if $K_i < G'$ and, by definition, that $K_{i+1} = G$ if $K_i = G'$. Since $G$ is finite, there exists a positive integer $n$ so that $K_n = K_{n+1}$ and so, $k_i = K_n$ for all integers $i \ge n$. Using this value of $n$, we define $K_\infty = K_n$. This next result includes Theorem \[intro K\_infty\].
\[kappa series\] Let $G$ be a group. Then $K_\infty = G$ if and only if $G$ is nested. Moreover, if $G$ is nested with chain of centers $G = X_0 > X_1 > \dotsb > X_n \ge 1$, then $K_i = [X_{n-i},G]$ for every integer $i$ such that $0 \le i \le n$.
First let $G$ be nested with chain of centers $G = X_0 > X_1 > \dotsb > X_n \ge 1$. We show that $K_i = [X_{n-i},G]$ for each integer $i$ such that $0 \le i \le n$ by induction on $i$. We have just established the case $i = 1$ in Lemma \[nested K\]. The case $i = 0$ follows from the fact that $X_n = Z(G)$ (see Lemma \[lewisgvz\] (1)). So suppose $i \ge 1$, and assume that we have $K_i = [X_{n-i},G]$. Since $G$ is nested, $G/K_i$ is also nested. Because $Z(G/K_i) = X_{n-i}/K_i$, the chain of centers for $G/K_i$ is given by $G/K_i = X_0/K_i > \dotsb > X_{n-i}/K_i \ge K_i/K_i$ (see Lemma 2.7 of [@ML19gvz]). By Lemma \[nested K\], we have that $$K_{i+1}/K_i = K(G/K_i) = [X_{n-i-1}/K_i,G/K_i] = [X_{n-(i+1)},G]/K_i,$$ since $K_i = [X_{n-i},G] < [X_{n-i-1},G]$ (see Lemma \[lewisgvz\] (2)). This gives $K_{i+1} = [X_{n-(i+1)},G]$, as desired.
Now we prove the converse by induction on $\norm{G}$. By the inductive hypothesis, we have that $G/K(G)$ is nested. Since $K(G) \le \ker(\chi)$ for every character $\chi \in \irr G$ satisfying $Z(\chi) > Z(G)$, it follows that $\mathcal{C} = \{Z(\chi) \mid \chi\in\irr G\ \,\text{and}~ Z (\chi) > Z (G) \}$ is a chain. Hence, $\{Z (\chi) \mid \chi \in \irr G\} = \mathcal {C} \cup \{Z(G)\}$ is also a chain, which means that $G$ is nested.
As a corollary, we have Theorem \[intro K > 1\].
\[nested cor\] Let $G$ be a group. Then $G$ is nested if and only if $K (G/N) > 1$ for every proper normal subgroup $N$ of $G$.
Suppose $G$ is nested. If $N$ is a normal subgroup of $G$, then $G/N$ is nested. If $N$ is proper, then $G/N > 1$. Notice that if $K (G/N) = 1$, then $K_\infty (G/N) = 1$ violating Theorem \[kappa series\]. Conversely, suppose that $K (G/N) > 1$ for every proper normal subgroup $N$. We work by induction on $|G|$. If $|G| = 1$, then $G$ is nested. Thus, we may assume $G > 1$. By hypotheses, this implies that $K(G) > 1$. Note that $G/K(G)$ will satisfy the inductive hypothesis; so by induction we have that $G/K(G)$ is nested. Using Theorem \[kappa series\], this implies that $K_\infty (G/K (G)) = G/K (G)$. It is easy to see that this implies that $K_\infty = G$ and applying Theorem \[kappa series\] once again, we have that $G$ is nested.
We now prove that the factor groups for the $K$-series share the same properties that the factor groups for the chain of centers of a nested group have, even when the group is not nested. We start with a lemma.
\[Z\_[K\_i]{}\] Let $G$ be a group. If $Z_{K_i} > Z_{K_{i-1}}$ and $[K_i,G] \le K_{i-1}$ for some integer $i$, then $Z_2 (G/K_{i-1}) > Z(G/K_{i-1})$.
We begin by observing that $[K_i,G] \le K_{i-1}$ implies that $K_i \le Z_{K_{i-1}}$. Notice that $Z_{K_{i-1}}/K_{i-1} = Z(G/K_{i-1})$ and $K (G/K_{i-1}) = K_i/K_{i-1}$, so $Z_{K_i} > Z_{K_{i-1}}$ implies that $Z_2 (G/K_{i-1}) \ge Z_{K(G/K_{i-1})} = Z_{K_i}/K_{i-1} > Z(G/K_{i-1})$.
This next result which includes Theorem \[intro z\_k\] is a generalization of Theorem 1.4 (2) of [@ML19gvz].
Let $G$ be a group. Suppose that there exists integers $1 \le j < k$ so that $Z_{K_i} > Z_{K_{i-1}}$ and $[K_i,G] \le K_{i-1}$ for all integers $i$ with $j \le i \le k$. Then there is a prime $p$ so that $K_k/K_{j-1}$ and $Z_{K_k}/Z_{K_{j-1}}$ are $p$-groups. In particular, $K_i/K_{i-1}$ and $Z_{K_i}/Z_{K_{i-1}}$ are elementary abelian $p$-groups for every integer $i$ with $j \le i \le k$.
In light of Lemma \[Z\_[K\_i]{}\], we may apply Lemma \[Z\_2 > Z\] (2) and (3) to see that there exists a prime $p_i$ so that $K_{i}/K_{i-1}$ and $Z_{K_i}/Z_{K_{i-1}}$ are elementary abelian $p_i$-subgroups and $Z_2 (G/K_{i-1})$ is a $p_i$-subgroup. We need to show that the $p_i$’s are all equal. Notice that it suffices to show that $p_{i+1} = p_i$ for every each $i$. Thus, we fix an integer $i$. Notice that we may replace $G$ by $G/K_{i-1}$. Thus, we may assume that $i = 1$. It suffices to show that $p_1 = p_2$. By Lemma \[Kprops\] (4), we see that $Z_{K} > Z_{K_0} = Z(G)$ implies that $K_2 > K_1$. Since $Z_{K_2} > Z_{K_1}$ and $K_2 \le Z_{K_1}$, we have that $Z_2 (G/K_1) > Z(G/K_1)$, and so by Lemma \[Z\_2 > Z\], we have $K_2/K_1 = K(G/K_1) \le Z(G/K_1) = Z_{K_1}/K_1$. This implies that $K_2 \le Z_{K_1}$. If $K_2 \not\le Z(G)$, then $K_2/(K_2 \cap Z(G)) \cong K_2 Z(G)/Z(G)$ is a nontrivial subgroup of $Z_{K_1}/Z(G)$. This implies that $K_2/(K_2 \cap Z_1)$ is a $p_1$-group. On the other hand, $K_2/K_1$ is a $p_2$-group and $K_1 \le K_2 \cap Z(G)$. It follows that $p_2 = p_1$ in this case. The other possibility is that $K_2 \le Z(G)$. In this case, we have $Z (G) < Z_{K_1} < Z_{K_2} \le Z_2$. Since $Z_2/Z(G)$ is a $p_1$-group and $Z_{K_2}/Z_{K_1}$ is a $p_2$-group, it follows that $p_1 = p_2$.
We are not convinced that the hypothesis on conclusion (3) in this next lemma needs to be included. We would not be surprised if it were still true without the extra hypotheses.
Let $G$ be a group. Suppose that there exists integers $1 \le j < k$ so that $Z_{K_{i}} > Z_{K_{i-1}}$ and $[K_i,G] \le K_{i-1}$ for all integers $i$ with $j \le i \le k$. Then the following are true:
1. $[Z_{K_i},G] K_{j-1} = K_i$ for each integer $i$ with $j \le i \le k$.
2. $[Z_{K_k},G] \le Z_{K_{j-1}}$ if and only if $[K_k,G] \le K_{j-1}$.
3. In the situation of (2), the exponent of $Z_{K_k}/Z_{K_{j-1}}$ equals the exponent of $K_k/K_{j-1}$.
We prove (1) by induction. In light of Lemma \[Z\_[K\_i]{}\] we may apply Lemma \[K equiv conditions\] in $G/K_{j-1}$ to see that $[Z_{K_j},G] K_{j-1}/K_{j-1} = [Z_{K_j}/K_{j-1},G/K_{j-1}] = K_j/K_{j-1}$. This yields the equality $[Z_{K_j},G] K_{j-1} = K_j$. Suppose for some $i$ with $j < i \le k+1$ that $[Z_{K_{i-1}},G] Z_{K_{j-1}} = K_{i-1}$. Applying Lemma \[K equiv conditions\] in $G/K_{i-1}$, we obtain $K_i = [Z_{K_i},G] K_{i-1} = [Z_{K_i},G] ([Z_{K_{i-1}},G] K_{j-1}) = [Z_{K_i},G] K_{j-1}$. This proves (1).
By part (1), we have that $[Z_{K_k},G] K_{j-1} = K_k$. Observe that $[Z_{K_k},G] K_{j-1} = K_k \le Z_{K_{j-1}}$ if and only if $K_k/K_{j-1} \le Z(G/K_{j-1})$. And $K_k/K_{j-1} \le Z (G/K_{j-1})$ if and only if $[K_k,G] \le K_{j-1}$. This proves (2).
Let $p^e$ be the exponent of $Z_{K_k}/Z_{K_{j-1}}$ and let $p^f$ be the exponent of $K_k/K_{j-1}$. Suppose $x \in Z_{K_k}$; this implies that $x^{p^e} \in Z_{K_{j-1}}$ and $[x,g] \in K_k$ for all $g \in G$. We have that $[x,g]^{p^e} = [x^{p^e},g] \in K_{j-1}$. This implies that $e \le f$. On the other hand, we see that $[x^{p^f},g] = [x,g]^{p^f} \in K_{j-1}$ for all $g \in G$ implies that $x^{p^f} \in Z_{K_k}$ and so, $f \le e$. We conclude that $f = e$.
In [@ML19gvz], the second author shows that if $G$ is nested, then every member of the upper central series appears as the center of some irreducible character. The same turns out to be true about the lower central series. Moreover, the lower central series is a subseries of the $K$-series.
Let $G$ be a nested group with chain of centers $G = X_0 > X_1 > \dotsb > X_n \ge 1$. If $N$ is a normal subgroup of $G$, then $[N,G] = [X_i,G]$ for some integer $i$ with $0 \le i \ne n$. In particular, for every integer $i \ge 2$, there exists an integer $j$ with $0 \le j \le n$ such that $G_i = [X_j,G]$.
By Lemma 3.1 of [@ML19gvz], we have that $Z (G/[N,G]) = X_i/[N,G]$ for some integer $i$ with $0 \le i \le n$. So $[X_i,G] \le [N,G]$. But we also have $N \le X_i$, so $[N,G] \le [X_i,G]$. The second statement follows easily from the first.
Let $G$ be a group. Following the literature, we will say that a [*minimal class*]{} of $G$ is a non-central conjugacy class of $G$ whose size is minimal among the noncentral conjugacy classes of $G$. If $G$ is a nested $p$-group, we have the following consequence of Theorem \[Z\_2 > Z\], which determines the size of a minimal class of $G$.
\[min breadth\] If $G$ is a nested $p$-group with chain of centers $G = X_0 > X_1> \dotsb > X_n > 1$, then a minimal class of $G$ has size $\norm {[X_{n-1},G]}$.
Write $m = \min \{\norm {\mathrm{cl}_G (g)} \mid g \in G \setminus Z(G)\}$. By Theorem \[Z\_2 > Z\], any element of $X_{n-1} \setminus X_n$ has class size $\norm {[X_{n-1},G]}$. So $m \le \norm{[X_{n-1},G]}$. By [@LMM99], there exists an element $g \in Z_2$ lying in a minimal class. Since $\norm {\mathrm{cl}_G (g)} = \norm {[g,G]}$ and $[X_{n-1},G] = K (G) \le [g,G]$ by Lemma \[K intersection\], we have $m = \norm {[g,G]} \le \norm {[X_{n-1},G]} \le m$. This proves that $\norm {[g,G]} = m$ as desired.
The final theorem of this section shows that we can characterize nested $p$-groups by finding elements with certain properties.
\[last sect 4\] If $G$ is a nonabelian $p$-group then the following statements are equivalent:
1. $G$ is nested.
2. For every normal subgroup $N \le G$ with $G' \not\le N$, there exists an element $g_N \in G \setminus N$ depending on $N$ so that $\{\chi \in \irr {G/N} \mid Z(\chi) = Z_N \} = \{ \chi \in \irr {G/N} \mid g_N \notin Z(\chi)\}$.
3. For every normal subgroup $N \le G$ with $G' \not\le N$, there exists a normal subgroup $L_N$ depending on $N$ so that $N < L_N \le G$ so that $\{ \chi \in \irr {G/N} \mid Z (\chi) = Z_N \} = \irr {G/N \mid L_N/N }$.
Let $G$ be a nonabelian, nested $p$-group. We now show that (1) implies (2). Hence, we assume that $G$ is a nested group. For each normal subgroup $N \le G$ with $G' \not\le N$, we see that $G/N$ is also a nonabelian, nested $p$-group. Set $K_N/N = K(G/N)$ and observe that $K_N/N < G/N$. Since $G/K_N$ is nontrivial nilpotent, we have $Z_{K_N} > K_N$ and if $G/K_N$ is nonabelian, then we will have $Z_2 (G/N) > Z (G/N)$. Given a character $\chi \in \irr {G/N}$, we have that $Z(\chi) = Z_N$ if and only if $K_N \nleq \ker(\chi)$ by Lemma \[K equiv conditions\] (5). But by Lemma \[K equiv conditions\] (2), there exists an element $1 \ne g_N N\in G/N$ for which $[g_N,G]N/N = [g_N N,G/N] = K_N/N$. Since $K_N \le \ker(\chi)$ if and only if $g_N \in Z(\chi)$, we have (2). Since $[g_N,G]$ is a normal subgroup of $G$, we also have (3) by taking $L_N = [g_N,G]$.
To complete the proof, it remains only to show that (3) implies (1). Now assume (3); we prove that $G$ is nested by induction on $\norm{G}$. We start by showing that $K(G) > 1$. Indeed, there exists a nontrivial normal subgroup $L = L_1$ of $G$ so that $Z (\chi) = Z (G)$ if and only if $\chi \in \irr {G \mid L}$. Thus, $L \le \ker(\chi)$ for every character $\chi \in \irr G$ satisfying $Z(\chi) > Z(G)$. It follows that $L \le K(G)$ (in fact $L = K(G)$), which implies that $K(G) > 1$. Observe that (3) holds for any quotient of $G$, so if $N > 1$ is normal in $G$, then $G/N$ is nested by the inductive hypothesis and so, $K(G/N) > 1$. It follows from Corollary \[nested cor\] that $G$ is also nested.
If $G$ is nested and nilpotent, then [@ML19gvz Corollary 4.10] gives that $G = P \times Q$, where $P$ is a nested $p$-group for some prime $p$ and $Q$ is an abelian $p'$-group. In this case, the $K$-series is determined from the $p$-part of $G$.
We note the following consequence of Corollary 4.10 of [@ML19gvz] and Lemma \[direct prods\]. If $G$ is a nested nilpotent group, then the $K$-series gives a central series for $G'$ with elementary abelian $p$-quotients, for a fixed prime $p$. This fact was proved by the second author as [@ML19gvz Lemma 4.5] (using a different method).
Camina triples and Vanishing-off subgroups {#vanishing section}
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In this section, we introduce the subgroup $U (G)$, and we will see that $U(G)$ serves the same role for nested GVZ-groups as $K(G)$ serves for nested groups. We first define a subgroup $U (G \mid N)$ for every normal subgroup $N \le G$ that, in some sense, determines a set of characters of $G$ vanishing on $G \setminus N$. The subgroup $U (G)$ arises when $N = Z(G)$, and thus, $U(G)$ is related to fully ramified characters.
The following observation is useful.
\[irr sets\] Let $M$ and $N$ be normal subgroups of $G$. Then $M \le N$ if and only if $\irr {G \mid M} \subseteq \irr {G \mid N}$.
It is clear that $\irr {G\mid M} \subseteq \irr {G\mid N}$ when $M \le N$. Conversely, suppose that $\irr {G\mid M} \subseteq \irr {G\mid N}$. This implies that $\irr {G/N} \subseteq \irr {G/M}$, and so $$M = \bigcap_{\chi \in \irr {G/M}} \ker (\chi) \le \bigcap_{\chi \in \irr {G/N}} \ker(\chi) = N.$$
We begin with a review of vanishing-off subgroups of characters. Recall that the vanishing-off subgroup $V(\chi)$ of a character is defined by $$V(\chi) = \langle g \in G \mid \chi (g) \ne 0 \rangle.$$ (This subgroup is defined on page 200 of [@MI76].) In particular, $V (\chi)$ is the smallest subgroup $V$ such that $\chi$ vanishes on $G \setminus V$. The second author extends this definition to groups in [@MLvos09] by defining $V(G)$ to be the subgroup defined by $$V (G) = \langle g \in G \mid \chi (g) \ne 0 {\rm ~for~some~} \chi \in \irr G \rangle.$$ It is not difficult to see that $V (G)$ is the smallest subgroup of $G$ so that every nonlinear character in $\irr G$ vanishes on $G \setminus V(G)$.
Let $N$ be a normal subgroup of $G$. Following [@NM14], we define ${V} (G \mid N)$ by $$V (G \mid N) = \langle g \in G \mid \chi (g) \ne 0 {\rm ~for~some~} \chi \in \irr {G \mid N} \rangle.$$ Thus, ${V} (G \mid N)$ is the smallest subgroup $V$ of $G$ such that every character $\chi \in \irr {G \mid N}$ vanishes on $G \setminus V$. Observe that $V(G \mid G') = V(G)$. Note that if $N = 1$, then this product is empty and we follow the convention above that the empty product yields the trivial subgroup, so $V (G \mid 1) = 1$. We will need the following properties.
\[Vprops\] The following statements hold for every pair $H$ and $N$ of normal subgroups of $G$.
1. $N \le {V} (G \mid N)$.
2. $V (G \mid N) = \prod_{\chi \in \irr {G \mid N}} V (\chi)$.
3. ${V} (G \mid HN) = {V} (G \mid H) {V} (G \mid N)$.
4. If $N \le H$, then ${V} (G \mid N) \le {V} (G \mid H)$.
To see (1), suppose that there exists an element $n \in N \setminus {V} (G \mid N)$. Then $\chi (n) = 0$ for every character $\chi \in \irr {G \mid N}$, and $n$ lies in the kernel of every other irreducible character of $G$. By column orthogonality (e.g., see Theorem 2.18 of [@MI76]), one sees that $\norm {N} = \norm {\mathrm {cl}_G (n)}$, which is strictly less than $\norm {N}$. Thus, no such element $n$ exists.
We next show (2). Observe that $V (\chi) \le V (G \mid N)$ for all characters $\chi \in \irr {G \mid N}$. It follows that $\prod_{\chi \in \irr {G \mid N}} V(\chi) \le V (G \mid N)$. Conversely, if $g \in G$ satisfies $\chi (g) \ne 0$ for some character $\chi \in \irr {G \mid N}$, then $g \in V (\chi)$ and so, $g \in \prod_{\chi \in \irr {G \mid N}} V(\chi)$. It follows that the generators of $V( G \mid N)$ all lie in $\prod_{\chi \in \irr {G \mid N}}$; so, $V (G \mid N) \le \prod_{\chi \in \irr {G \mid N}} V(\chi)$ as desired.
We now show (3). To accomplish this, we show that $\irr {G \mid HN} = \irr {G \mid H} \cup \irr {G \mid N}$. First, fix a character $\chi \in \irr {G \mid HN}$. If $\chi \in \irr {G \mid H}$, then we have desired result. Thus, we may assume that $H \le \ker (\chi)$. Since $HN \not\le \ker(\chi)$, we cannot have $N$ contained in $\ker (\chi)$, so we must have $\chi \in \irr {G \mid N}$. The reverse containment is obvious. Thus, we have by (2) that ${V} (G \mid HN) = \prod_{\chi \in \irr {G\mid HN}} {V}(\chi)$ and ${V} (G \mid H) {V} (G \mid N) = \prod_{\chi \in \irr {G \mid H}} {V}(\chi) \prod_{\chi \in \irr {G \mid N}} {V}(\chi)$. The first observation implies that these products are equal.
Finally, (4) is immediate from (2) and the fact that $\irr {G \mid N} \subseteq \irr {G \mid H}$ whenever $N \le H$ by Lemma \[irr sets\].
For each normal subgroup $N$ of $G$, define ${U} (G \mid N) = \prod_{H \in \mathcal{H}} H$ where $\mathcal {H} = \{ H \lhd G \mid {V} (G \mid H) \le N \}$. Notice that if $N = G$, then $\mathcal {H}$ will be all normal subgroups of $G$ and so, $U (G \mid G) = G$. We will show when $N < G$ that $U = {U} (G \mid N)$ is the largest normal subgroup of $G$ for which every member of $\irr {G \mid U}$ vanishes on $G \setminus N$. In particular, if $H \nleq U$, then there exists a character $\chi \in \irr {G \mid H}$ that does not vanish on $G \setminus N$. Therefore, the subgroup ${U} (G \mid N)$ identifies a set of characters that, in some sense, is maximal with respect to vanishing on $G \setminus N$.
\[uiff\] Let $H$ and $N$ be normal subgroups of a group $G$. Then $H \le {U} (G \mid N)$ if and only if ${V} (G \mid H) \le N$.
If ${V} (G \mid H)\le N$, then it is clear that $H \le {U} (G \mid N)$. Conversely, if $H \le {U} (G \mid N)$, then ${V} (G \mid H) \le {V} (G \mid {U} (G \mid N)) = \prod_{K \in \mathcal {V}} {V} (G \mid K) \le N$ where $\mathcal {V} = \{ K \lhd G \mid {V} (G \mid K) \le N\}$.
Lemma \[uiff\] implies that the maps $N \mapsto {U} (G \mid N)$ and $N \mapsto {V} (G \mid N)$ give a (monotone) Galois connection from the lattice $\mathrm{Norm} (G)$ of normal subgroups of $G$ to itself. For more information on Galois connections, we refer the reader to [@galoisprimer].
We now present some basic properties of ${U} (G \mid N)$.
\[uproperties\] Let $G$ be a nonabelian group. The following hold.
1. For each normal subgroup $N \le G$, the subgroup ${U} (G \mid N)$ is the unique largest subgroup, $U \le G$, such that every character in $\irr {G \mid U}$ vanishes on $G \setminus N$.
2. For each normal subgroup $N \le G$ and every element $g \in G$, we have $g \in {U} (G \mid N)$ if and only if every character $\chi \in\irr G$ satisfying $g \notin \ker(\chi)$ vanishes on $G \setminus N$.
3. For each normal subgroup $N \le G$, we have ${U} (G \mid N) \le N \cap G'$.
4. If $N$ is characteristic in $G$, so is ${U} (G \mid N)$.
If every character in $\irr {G \mid H}$ vanishes on $G \setminus N$, then ${V} (G \mid N) \le H$, so $H \le {U} (G \mid N)$ by Lemma \[uiff\]. This establishes (1).
To show (2), first note that for an irreducible character $\chi \in \irr G$, we have $g \in \ker(\chi)$ if and only if $\inner{g}^G \le \ker (\chi)$, where $\inner {g}^G$ denotes the normal closure of $\inner {g}$. Hence, every character $\chi \in \irr G$ satisfying $g \notin \ker (\chi)$ vanishes on $G \setminus N$ if and only if every character $\chi \in \irr {G \mid \inner {g}^G}$ vanishes on $G \setminus N$. The latter happens if and only if ${V} (G \mid \inner{g}^G) \le N$, which happens if and only if $\inner {g}^G \le {U} (G \mid N)$. Finally, we note since ${U} (G \mid N)$ is normal in $G$ that ${U} (G \mid N)$ contains $g$ if and only if it contains $\inner {g}^G$.
Next, we prove (3). The fact that $U (G \mid N) \le N$ follows from Lemma \[uiff\] as $N \le V (G \mid N)$ by Lemma \[Vprops\]. The rest of statement (3) follows from the fact that no linear character can vanish on any element of $G$. In particular, this means that $\irr {G \mid U (G \mid N)} \subseteq \irr{G \mid G'}$.
Since $N$ uniquely determines $U (G \mid N)$, it follows that if $N$ is characteristic in $G$, then $U (G \mid N)$ will be characteristic.
We next consider how the map $N \mapsto {U} (G \mid N)$ interacts with quotients.
\[uquotients\] Let $H$ and $N$ be normal subgroups of $G$ that satisfy ${V} (G \mid N) \le H$. Then ${U} (G/N \mid H/N) = {U} (G \mid H)/N$.
Let $x \mapsto \overline{x}$ denote the canonical surjection $G \to G/N$. Define the sets $$\mathcal {C} = \{K \lhd G \mid {V} (G \mid K) \le H\}\ \,\text{and}\ \,\mathcal{D} = \{\overline {K} \lhd \overline {G} \mid {V} (\overline {G} \mid \overline {K}) \le \overline {H}\}.$$ We claim that $\mathcal {D} = \overline {\mathcal {C}}$. However, we first show that $\mathcal {D} = \overline {\mathcal {C}'}$, where $$\mathcal{C}' = \{K \lhd G \mid N \le K\ \,\text{and}\ \,{V} (G \mid K) \le H\}.$$ To that end, let $K \lhd G$ satisfy $N \le K$. Then $N \le {V} (G \mid K)$, and so we have $${V} (\overline {G} \mid \overline{K}) {V} (G \mid N)/N = {V} (G \mid K)/N.$$ So it follows that $\overline{K} \in \mathcal{D}$ if and only if $K \in \mathcal{C}'$, as claimed.
In particular, this gives $${U} (\overline{G} \mid \overline{H}) = \prod_{K \in \mathcal{D}} K = \prod_{K \in \mathcal{C}'} \overline {K} = \overline {\prod_{K \in \mathcal{C}'} K}.$$
Next note that since ${V} (G \mid N) \le H$, we have $K \in \mathcal{C}$ if and only if $KN \in \mathcal{C}'$. Hence $${U} (G \mid H) = \prod_{K \in \mathcal{C}} K = \prod_{K \in \mathcal{C}} KN = \prod_{K \in \mathcal{C}'} K.$$ The result now follows by taking quotients.
We see that the subgroups $V (G \mid N)$ and $U (G \mid N)$ are both related to zeros of irreducible characters. This next result was proved in [@SBML] and instead gives insight on which characters a specific element of $G$ vanishes.
\[basics\] Let $M$ be a normal subgroup of $G$ and let $g \in G \setminus M$. Then the following are equivalent:
1. $g$ is conjugate to every element in $gM$.
2. For every element $z \in M$, there exists an element $x \in G$ so that $[g,x] = z$.
3. $|C_G (g)| = |C_{G/M} (gM)|$.
4. $\chi (g) = 0$ for all $\chi \in \mathrm{Irr}(G \mid M)$.
If every element $g \in G \setminus M$ satisfies the equivalent conditions of Lemma \[basics\], we call $(G,M)$ a [*Camina pair*]{}. These objects were first considered by A. Camina in [@acamina], as a natural generalization of Frobenius groups. If there is a normal subgroup $N$ of $G$ containing $M$ so that every element $g \in G\setminus N$ satisfies the equivalent conditions of Lemma \[basics\], we call $(G,N,M)$ a [*Camina triple*]{}. These objects were first studied by Mattarei in his Ph.D. thesis [@SM92]. Many more properties of Camina triples were found by Mlaiki in [@NM14], where the following result appears.
\[equiv\] Let $M$ and $N$ be normal subgroups of $G$. Then the following are equivalent:
1. $(G,N,M)$ is a Camina triple.
2. For every element $g \in G \setminus N$ and every element $z \in M$, there exists an element $x \in G$ so that $[g,x] = z$.
3. $|C_G (g)| = |C_{G/M} (gM)|$ for all $g \in G \setminus N$.
4. $V (G \mid M) \le N$.
5. $\chi (g) = 0$ for every element $g \in G \setminus N$ and for every character $\chi \in \irr {G \mid M}$.
Since, by Lemma \[uiff\], we have that $V (G \mid M) \le N$ if and only if $M \le U (G \mid N)$, Lemma \[equiv\] yields the following result.
\[U camina\] Let $M$ and $N$ be normal subgroups of a group $G$. Then $(G,N,M)$ is a Camina triple if and only if $M \le {U} (G \mid N)$. In particular, $(G,N)$ is a Camina pair if and only if $N = {U} (G \mid N)$.
In the next section, we study the subgroup $U (G) = U (G \mid Z(G))$. It turns out that this subgroup is closely related to the subgroup $K(G)$ defined earlier. The next result gives an alternate description of $U (G \mid N)$ that, once specialized to the case $N = Z(G)$, foreshadows the connections between $K(G)$ and $U(G)$.
\[Ukernels\] Let $N < G$ be a normal subgroup. Then $U (G \mid N) = \bigcap\limits_{\chi \in \mathcal {U}} \ker(\chi)$ where $\mathcal {U} = \{ \chi \in \irr G \mid V (\chi) \nleq N\}$.
Write $U = U (G \mid N)$ and let $W = \bigcap_{\chi \in \mathcal {U}} \ker(\chi)$. Note that $V (G \mid U) \le N$, and so $V (\chi) \le N$ for all characters $\chi \in \irr {G \mid U}$. This implies that $U \le \ker(\chi)$ for every character $\chi \in \mathcal {U}$. We deduce that $U \le W$. Conversely, consider a character $\chi \in \irr {G \mid W}$. Since $W \nleq \ker(\chi)$, we must have $V (\chi) \le N$. In particular, $\chi$ vanishes on $G \setminus N$. It follows that $W \le U$, and hence also that $W = U$.
The subgroup $U (G)$ {#U section}
====================
Define ${U} (G) = {U} (G \mid {Z} (G))$. When $G$ is an abelian group, we have $Z(G) = G$ and so $U (G) = G$. We will study $U (G)$ when $G$ is nonabelian. Notice that Theorem \[intro U prop\] follows from Lemma \[uproperties\] in this case.
In this section, we will study the properties of the subgroup $U (G)$, many of which are shared by the subgroup $K(G)$. We will use these subgroups to tailor the results of $K(G)$ regarding nested groups specifically to nested GVZ-groups. Before doing this, we show that this subgroup is useful in describing existing properties of groups related to GVZ-groups.
A group $G$ is called a [*VZ-group*]{} if every nonlinear irreducible character vanishes on $G \setminus Z (G)$. These groups were studied in Fernández-Alcober and Moretó in [@VZ01] and also by the second author in [@MLIGT08]. We note that both nested groups and GVZ-groups generalize VZ-groups.
Let $G$ be a group. The following are equivalent:
1. $G$ is a VZ-group.
2. ${Z} (G) = {V} (G)$.
3. ${U} (G) = G'$.
Both (2) and (3) are equivalent to the condition that every nonlinear character $\chi \in \irr G$ vanishes off of $Z(G)$, which is equivalent to $G$ being a VZ-group.
An intimately related concept is a semi-extraspecial group. A $p$-group $G$ is called [*extraspecial*]{} if $G' = Z(G)$ is the socle of $G$, and $G$ is called [*semi-extraspecial*]{} if $G/N$ is extraspecial for every maximal subgroup $N$ of $Z(G)$. In particular, if $G$ is a VZ-group such that $Z (G) = G'$, then $G$ is a semi-extraspecial group. We showed in Lemma \[Kprops\] that $K(G) \le G'$, and we showed in Lemma \[Z\_2 > Z\] that $K(G)\le Z(G)$ when $Z_2>Z(G)$. Our first result about $U (G)$ shows that the same is true of $U (G)$. We actually show a bit more so that we may deduce a different description of semi-extraspecial groups.
\[properties\] Let $G$ be a nonabelian group. Then ${U} (G) \le G' \cap {Z} (G) \le G' {Z} (G) \le {V}(G)$.
We have $U (G) \le G' \cap Z (G)$ by Lemma \[uproperties\] (2). Since $Z(G) \le V (\chi)$ for all characters $\chi \in \irr G$, we have $Z (G) \le V (G)$, and so, also $G' Z(G) \le V (G)$.
If $G$ is a nontrivial group, then ${U} (G) = {V} (G)$ if and only if $G$ is a semi-extraspecial group.
Assume ${U} (G) = {V} (G)$. Then $G$ is not abelian, since if it were, we would have $U (G) = G > 1 = V (G)$. In light of Lemma \[properties\], we have ${U} (G) = {Z} (G) = G' = {V} (G)$. This means that $(G, {Z} (G))$ is a Camina pair, so $G$ is a $p$-group for some prime $p$, obviously of nilpotence class $2$. Since ${Z} (G) = G'$, we see that $G$ is in fact a Camina group. It is known that being a Camina group of nilpotence class $2$ is equivalent to being semi-extraspecial (e.g. see [@LVses Theorem 1.2]).
Suppose $G$ is a semi-extraspecial group; then $G' = {Z} (G)$ and every nonlinear irreducible character of $G$ vanishes on $G \setminus {Z} (G)$. Then ${V} (G) \le {Z} (G)$ and $G' \le {U} (G)$. Since the reverse containments always hold, we have ${U}(G) = G' = {Z}(G) = {V}(G)$.
We next find several descriptions of $U (G)$, the last of which was alluded to at the end of Section \[vanishing section\]. We must however introduce some new notation. For each element $g \in G \setminus Z(G)$, we let $D_G (g)/Z(G) = C_{G/Z(G)} (gZ(G))$; i.e. $D_G (g) = \{x \in G \mid [g,x] \in Z(G)\}$.
\[UDg\] Let $G$ be a group. The following hold.
1. ${U} (G) = \bigcap\limits_{g \in G \setminus {Z} (G)} [g,D_G(g)]$.
2. $U (G)$ is the largest normal subgroup of $G$ contained in $\bigcap\limits_{g \in G \setminus Z(G)} \gamma_G (g)$.
3. $U (G) = \bigcap\limits_{\chi \in \mathcal{V}} \ker(\chi)$ where $\mathcal {V} = \{\chi \in \irr G \mid V(\chi) > Z(G) \}$.
We first note that if $G$ is abelian, then each of these intersections is $G$, as is $U (G)$. We therefore assume that $G$ is nonabelian.
We begin by proving (1). Set $Z = {Z} (G)$, $U = {U} (G)$, and $D = \bigcap_{g \in G \setminus Z} [g,D_G(g)]$. Suppose first that $N \le [g,D_G(g)]$ for all $g \in G \setminus Z$. Let $\chi \in \irr {G \mid N}$. Take $\lambda$ to be the unique irreducible constituent of $\chi_Z$ and set $\nu = \lambda_N$. Observe that $\nu \ne \mathbbm{1}_N$. We see that $N \le [g,D_G(g)]$ for every element $g \in G \setminus Z$, that $\ker(\mu) < N$, and that $\ker(\mu) = \ker (\lambda) \cap N$. If there exists an element $g \in G \setminus Z$ so that $[g, D_G (g)] \le \ker(\lambda)$, then $N = [g, D_G (g)] \cap N \le \ker(\lambda) \cap N = \ker(\mu) < N$ which is a contradiction. Thus, we have $[g, D_G (g)] \nleq \ker(\lambda)$ for all $g \in G \setminus Z$. By Lemma 3.3 of [@SBML], $\lambda$ (and hence $\chi$) is fully ramified with respect to $G/Z$. Since $U$ is the unique largest subgroup of $G$ so that all characters in $\irr {G \mid U}$ vanish on $G \setminus Z$, we have $N \le U.$ Taking $N = D$, we obtain $D \le U$. If $U = 1$, then we must have $D = 1$ also. So, we may assume that $U > 1$. Let $g \in G \setminus Z$. Since $(G,Z,U)$ is a Camina triple, we have $gU \subseteq \mathrm{cl}_G(g)$. In particular, for each element $u\in U$, there exists an element $x_u \in G$ so that $[g, x_u]=u$. Since $x_u \in D_G(g)$, we have $U \le [g, D_G (g)]$. It follows that $U \le D$.
Now, we prove (2). Write $S = \bigcap_{g \in G \setminus Z(G)} \gamma_G(g)$. Let $H$ denote the largest normal subgroup of $G$ contained in $S$. Note that $U(G) \subseteq \gamma_G (g)$ for every $g\in G\setminus Z(G)$, and so $U(G)\le H$. Suppose that $N$ is a normal subgroup of $G$ contained in $S$. Then $gN \subseteq \mathrm {cl}_G (g)$ for every element $g\in G \setminus Z(G)$. By Lemma \[basics\], this means that every character $\chi \in \irr{G \mid N}$ vanishes off of $Z(G)$, so $N \le U(G)$. It follows that $H\le U(G)$. Finally, statement (3) is just Lemma \[Ukernels\] with $N = Z(G)$.
\[U le K\] Let $G$ be a group. Then ${U} (G) \le K (G)$.
The result is trivially true if $G$ is abelian, so assume that this is not the case. The fact that $U (G) \le K(G)$ follows immediately from any of the statements in Lemma \[UDg\].
\[U eq K\] If $Z_{U(G)} > Z(G)$, then $U (G) = K (G)$.
By Lemma \[U le K\], we know that $U(G) \le K(G)$. If $K(G) \nleq U(G)$, then $Z(G/U(G)) = Z(G)/U(G)$ by Lemma \[Kprops\] (2).
Combining Lemma \[U le K\] with Lemma \[direct prods\], we obtain the following:
Suppose $M$ and $N$ are groups. Then the following hold:
1. If $M$ and $N$ are nonabelian, then $U (M \times N) = 1$.
2. If $M$ is nonabelian and $N$ is abelian, then $U (M \times N) = U (M)$.
By Lemma \[U le K\], we have $U (M \times N) \le K (M \times N)$. Applying Lemma \[direct prods\] (1), we have $K(M \times N) = 1$, and it follows that $U (M \times N) = 1$ when $M$ and $N$ are nonabelian. When $M$ is nonabelian, and $N$ is abelian, we take $G = M \times N$. Observe that $\gamma_G (m,n) = \gamma_M (m)$ for all $m \in M$ and $n \in N$. It follows that $\cap_{(m,n) \in G \setminus Z(G)} \gamma_G (m,n) = \cap_{m \in M \setminus Z(M)} \gamma_M (m)$. Conclusion (2) now follows from Lemma \[UDg\] (2).
We will now show that a group $G$ satisfying ${U}(G) > 1$ is essentially a $p$-group in the sense that it is a $p$-group up to a central direct factor. To do so, we will appeal to the connection to Camina triples mentioned earlier. The following result can be found in [@NM14].
\[NM2\][(cf. [@NM14 Theorem 2.10])]{} Let $(G, N, M)$ be a Camina triple. If $G/N$ is not a $p$-group for any prime $p$, then $M \cap {Z}(G) = 1$.
Observe that if ${U}(G) > 1$, then Lemma \[U camina\] tells us that $(G,{Z}(G),{U}(G))$ is a Camina triple. It turns out that the structure of $G$ is quite restrictive in this case.
\[nontrivueqpq\] Let $G$ be a nonabelian group that satisfies ${U}(G) > 1$. Then the following are true:
1. $(G,Z(G),U(G))$ is a Camina triple.
2. $G = P \times Q$, where $p$ is a $p$-group for some prime $p$ and $Q$ is an abelian $p'$-group. In particular, $G$ is nilpotent.
3. $U (G)$ is an elementary abelian $p$-group for some prime $p$.
Since ${V} (G \mid {U} (G)) \le {Z}(G)$, we have that $(G, {Z}(G), {U}(G))$ is a Camina triple by Lemma \[U camina\]. By Lemma \[NM2\], $G/{Z}(G)$ must be a $p$-group since $1 < {U} (G) \le {Z} (G)$. Thus, if $Q$ is a complement for a Sylow $p$-subgroup of ${Z}(G)$, then $Q$ is direct factor of $G$. This proves (1). To prove (2), we have that $U (G) \le K (G)$ by Lemma \[U le K\]. Since $G$ is nonabelian and nilpotent, we have $Z_2 > Z(G)$. Thus, we may apply Lemma \[Z\_2 > Z\] to see that $U (G)$ is an elementary abelian $p$-group.
In [@MLvos09], the second author shows that if $G$ is nonabelian, nilpotent and satisfies ${V} (G) < G$, then $G = P\times Q$, where $P$ is a $p$-group and $Q$ is an abelian $p'$-group. In particular, Lemma \[nontrivueqpq\] may also be considered an analog of that result.
We now continue to study the subgroup $U (G)$. The next result shows that two more properties from Lemma \[Kprops\] satisfied by $K(G)$ are shared by $U (G)$. We will present examples to see that $U (G) > 1$, but $Z_{U(G)} = Z(G)$ does occur.
\[centerproperty\] Let $N \lhd G$, and assume that $U = {U} (G) \nleq N$. Then the following hold:
1. $N \le Z(G)$.
2. ${Z} (G/N) = {Z} (G)/N$.
3. $UN/N \le {U} (G/N)$. In particular, ${U} (G/N)$ is nontrivial.
4. If $Z_U = Z (G)$, then $U (G/U) = 1$.
By Lemma \[U le K\], we have $U \le K (G)$. Thus, if $U \not\le N$, then $K (G) \not\le N$. Now, (1) and (2) follow immediately from Lemma \[Kprops\] (1) and (2). The proof of (3) follows the same lines of the proof Lemma \[Kprops\] (3) where we substitute $[g, D_G (g)]$ for $[g,G]$. In view of Lemma \[UDg\] (4), we can prove (4) along the same lines as the proof of Lemma \[Kprops\] (4) where the property $Z(\chi) > Z(G)$ is replaced by $V(\chi) > Z(G)$.
It therefore suffices for our considerations to consider only $p$-groups when discussing groups $G$ satisfying $U(G) > 1$. We saw earlier that some nice properties hold whenever $Z_{K(G)} > Z(G)$. We now show that similar properties hold whenever $Z_{U(G)} > Z(G)$. In fact, we obtain a slightly stronger result than Lemma \[min breadth\] for $p$-groups satisfying $Z_{U(G)} > Z(G)$.
The following result will be useful; it shows that even more properties of $U(G)$ are mirrored by properties of $K(G)$.
\[equivconjs\] Let $G$ be a nonabelian $p$-group satisfying $U = {U}(G) > 1$. The following are equivalent:
1. ${Z}_U > {Z}(G)$.
2. There exists an element $g \in G$ satisfying $[g,G] = U$.
3. $U = [Z_U,G]$.
Suppose (1). By Corollary \[U eq K\], we have that $U = K (G)$, and using Lemma \[K equiv conditions\], we have (2). Next, suppose (2). In light of Lemma \[K intersection\], we have $K (G) \le [g,G] = U$, and by Lemma \[U le K\], we have $K (G) \le U(G)$, and so, $K (G) = U$, and $Z(G) < Z_U = Z_{K(G)}$ and applying Lemma \[K equiv conditions\], we have $U = K(G) = [Z_{K(G)},G] = [Z_U,G]$. The fact that (3) implies (1) is immediate.
Let $G$ be a group. Recall that a [*minimal class*]{} of $G$ is a non-central conjugacy class of $G$ whose size is minimal among the noncentral conjugacy classes of $G$.
\[mann sub\] Let $G$ be a $p$-group. If $Z_{U(G)} > Z(G)$, then every minimal class of $G$ lies in $Z_2$ and has size $\norm{U(G)}$. Moreover, we have $$Z_{U(G)} = \{g \in G \mid \norm{\mathrm{cl}_G(g)} \le \norm{U(G)} \} = Z(G) \cup \{g \in G \mid \mathrm{cl}_G (g) = g U (G) \}.$$
Let $m$ be the size of a minimal class of $G$. The assumption that $Z_{U (G)} > Z (G)$, implies $\norm {U(G)} > 1$. By Lemma \[nontrivueqpq\] (1), we know that $(G,Z(G),U(G))$ is a Camina triple, and by Lemma \[basics\], we see that every non-central class of $G$ is a union of $U(G)$-cosets, so $m$ divides the size of every non-central conjugacy class of $G$. Since $Z_{U(G)} > Z(G)$, there exists an element $g \in G \setminus Z(G)$ satisfying $[g,G] \le U(G)$ by Lemma \[equivconjs\]. Since $U (G) \le [g,G]$ by Lemma \[UDg\], we have $U (G) = [g,G]$ and hence $\mathrm{cl}_G (g) = gU(G)$. It follows that $m = \norm{U(G)}$. Thus, if $x \in G$ satisfies $\norm {\mathrm{cl}_G(x)} = m$, then $\mathrm{cl}_G(x) = xU(G)$. This means that if $x$ belongs to a minimal class, then $[x,G] = U(G)$. In particular, this means $x$ lies in $Z_2$ if $x$ belongs to a minimal class. Now, let $H = \{g \in G: \norm{\mathrm{cl}_G (g)} \le m \}$, and let $g, h \in H \setminus {Z}(G)$. Then $[gh,G] \le [g,G][h,G] = U$. It follows that $[gh,G]$ is either $U(G)$ or trivial, and so $gh \in H$.
We mention one consequence of Lemma \[mann sub\]. In [@mann1], Mann considers the subgroup generated by the minimal elements, and he proved when $G$ is a $p$-group, that this subgroup has nilpotence class at most $3$. In [@isma], Isaacs considered the subgroup $M (G)$ generated by the minimal elements and the central elements. He proved that if $G$ is supersolvable, then $M (G)$ is nilpotent of nilpotence class at most $3$. In [@mann2], Mann further generalized Isaacs’s result. In the situation of Lemma \[mann sub\], we have that $Z_{U(G)}$ is the [*Mann subgroup*]{} of $G$, and its nilpotence class is at most $2$.
\[U equiv\] Let $G$ be a nonabelian $p$-group and let $U = U (G) > 1$. The following are equivalent:
1. $Z_{U(G)} > Z(G)$.
2. $U(G/N) = UN/N$ for every normal subgroup $N$ of $G$ not containing $U$.
3. An irreducible character $\chi$ vanishes off of ${Z}(G)$ if and only if $\chi \in \irr {G \mid U}$.
Assume (1) holds. Since ${Z}(G/U) > {Z}(G)/U$, we have that ${Z}(\chi) > {Z}(G)$ for every character $\chi \in \irr {G/U}$. So if $\chi \in \irr {G/U}$, we must have ${V} (\chi) > {Z}(G)$. This implies that no character in $\irr {G/U}$ vanishes off of $Z (G)$. On the other hand, the definition of $U (G)$ implies that the characters in $\irr {G \mid U}$ all vanish off of $Z (G)$. Thus (3) follows.
Next, assume (3), and suppose that ${U} (G/N) > UN/N$. As $\irr {G/N\mid UN/N} = \irr {G\mid U} \cap \irr {G/N}$, there exists a character $\chi \in \irr {G/UN}$ that vanishes off of ${Z} (G/N)$. By Lemma \[centerproperty\], ${Z} (G/N) = {Z} (G)/N$, which implies that $\chi \in \irr {G/U}$ and vanishes off of ${Z}(G)$. As this is a contradiction, (2) holds.
Finally, the proof that (2) implies (1) follows the same lines as the proof that (2) implies (1) in Lemma \[K equiv cond 2\] where we use $U$ in place of $K$.
We now show that when $G$ is a GVZ-group, $U (G)$ and $K (G)$ are equal.
\[U = K\] Let $G$ be a GVZ-group. Then ${U} (G) = K (G)$. In particular, this holds if $G$ has nilpotence class 2.
If $K(G) \nleq \ker(\chi)$ for some character $\chi \in \irr G$, then ${Z} (\chi) = {Z} (G)$, so $\chi$ is fully ramified over ${Z}(G)$ since $G$ is a GVZ-group. This implies $K(G) \le {U}(G)$. The reverse inclusion comes from Lemma \[U le K\].
One may prove Lemma \[U = K\] in a variety of different ways. For example, the lemma follows directly from the definition of $K(G)$ and Lemma \[UDg\] (3), since $V (\chi) = Z (\chi)$ for all characters $\chi \in \irr G$ whenever $G$ is a GVZ-group. It also follows from a combination of Lemma \[UDg\] (2) and the fact that $\gamma_G (g) = [g,G]$ for all $g \in G$ when $G$ is a GVZ-group which is proved in [@SBML].
We now define a chain of subgroups $U_i$ by $U_0 = 1$ and $U_{i+1}/U_i = {U} (G/U_i)$ for each $i \ge 0$. Recall from Lemma \[properties\] that $U (G) \le G'$ when $G$ is nonabelian. It follows that $U_{i+1} \le G'$ when $U_i < G'$. On the other hand, we see that $U_{i+1} = G$ when $U_i = G'$. Observe that Lemma \[U = K\] implies that $U_i = K_i$ for all integers $i$ if $G$ is a GVZ-group, and Lemma \[U le K\] implies that we always have $U_i \le K_i$. Furthermore, one can see from Corollary \[U eq K\] and Lemma \[centerproperty\] (4) that $U_i < K_i$ implies that $U_i = U_{i+1}$. Note that we will write $U_\infty$ for the terminal term in this series.
\[nested gvz U thm\] Let $G$ be a nonabelian group. Then $G$ is a nested GVZ-group if and only if $U_\infty = G$. Moreover, in the event that $G$ is a nested GVZ-group with chain of centers $G = X_0 > X_1 > \dotsb > X_n > 1$, then $U_i = [X_{n-i},G]$ for all $i$.
If $G$ is a GVZ-group, then so is any epimorphic image of $G$. Also, we have that $U (G) = K (G)$ by Lemma \[U = K\]. Hence, $K_i = U_i$ for all $i$ when $G$ is a GVZ-group. Thus, $G$ being a nested GVZ-group implying $G = U_\infty$ follows from Theorem \[kappa series\]. If $G$ is a nested GVZ-group with chain of centers $G = X_0 > X_1 >\dotsb > X_n > 1$, then Theorem \[kappa series\] also gives $U_i = K_i = [X_{n-i},G]$.
Suppose $U_\infty = G$. Since $U_i \le K_i$, this implies $K_\infty = G$ and by Theorem \[kappa series\], this implies that $G$ is nested. Since $U_\infty = G$, we may apply Lemma \[Z(chi) = V(chi)\] to see that $G$ is a GVZ-group.
We now have the promised proof of Theorem \[intro U > 1\].
Let $G$ be a group. Then $G$ is a nested GVZ-group if and only if $U (G/N) > 1$ for every proper normal subgroup $N$ of $G$.
Notice that the proof of this will essentially be identical to the proof of Corollary \[nested cor\] with Theorem \[nested gvz U thm\] in place of Theorem \[kappa series\], so we do not repeat it.
We conclude this section by presenting the analog of Theorem \[last sect 4\] to nested GVZ-groups. Notice that we replace the condition that $Z(\chi) = Z_N$ with $\chi$ is fully ramified over $Z_N$ which we have seen is equivalent to $V (\chi) = Z (\chi) = Z_N$.
\[nested gvz thm 3\] Let $G$ be a nonabelian group. The following are equivalent.
1. $G$ is a nested GVZ-group.
2. For every normal subgroup $N \lhd G$, there exists an element $g_N \in G \setminus N$ so that $\{ \chi \in \irr {G/N} \mid \chi {\rm~is~fully~ramified~over~} Z_N \} = \{ \chi \in \irr {G/N} \mid g_N \notin Z(\chi) \}.$
3. For every normal subgroup $N\lhd G$, there exists a normal subgroup $N < L_N \lhd G$ so that $\{ \chi \in \irr {G/N} \mid \chi {\rm~is~fully~ramified~over~} Z_N \} = \{ \chi \in \irr {G/N \mid L_N/N}\}.$
Assume that $G$ is a nested GVZ-group. Observe that every quotient of a nested GVZ-group is also a nested GVZ-group. Therefore, statement (2) will follow if we show that whenever $G$ is a nested GVZ-group, there exists a nonidentity element $g \in G$ so that $\chi$ is fully ramified over $Z(G)$ if and only if $g \not\in Z(\chi)$. To that end, note that $U(G) > 1$ by Theorem \[nested gvz U thm\]. By that same theorem, we have $Z_{U(G)} > Z(G)$, for otherwise we would have $U_2 = U (G)$ by Lemma \[centerproperty\] (3). Hence by Lemma \[equivconjs\] there exists an element $g\in G$ so that $U (G) = [g,G]$; we also know from Lemma \[equivconjs\] that a character $\chi \in \irr G$ is fully ramified over $Z(G)$ if and only if $U(G) \nleq\ ker(\chi)$. Since $U (G) = [g,G] \le \ker(\chi)$ if and only if $g \in Z(\chi)$, statement (2) follows. Also, since $[g,G] \lhd G$, statement (3) follows.
Finally, assume that (3) holds. Then there exists a nontrivial normal subgroup $L$ of $G$ so that every character $\chi \in \irr {G \mid L}$ is fully ramified over $Z(G)$. This means that $L \le U (G)$. In particular, we have $U (G) > 1$. Note that (3) must hold for every quotient of $G$, hence for $G/U(G)$. Proceeding by induction on $\norm{G}$, we have that $G/U(G)$ is a nested GVZ-group. Thus, by Theorem \[nested gvz U thm\] we have that $U_\infty = G$, and so, $G$ is a nested GVZ-group.
The $\delta$–series and the $\epsilon$–series
=============================================
We see from Theorem \[kappa series\] that nonabelian nested groups can be defined by the existence of a certain ascending normal series, which is a central series whenever $G$ is nilpotent. We now show that nested groups can be characterized by the existence of a descending normal series, which is also central whenever $G$ is nilpotent. Define the subgroups $\delta_i$ by setting $\delta_1 = G$ and $\displaystyle \delta_{i+1} = \!\!\!\!\!\!\! \prod_{\chi \in \irr {G \mid [\delta_i,G]}} \!\!\!\!\!\!\!\!\! Z(\chi) $ for every integer $i\ge 1$. Note that if $[\delta_i,G] = 1$ (i.e, $\delta_i \le Z(G)$), then $\irr {G \mid [\delta_i,G]}$ is empty, and follow the convention that the empty product is the trivial subgroup.
\[contain\] If $i\ge 1$ is an integer, then $\delta_{i+1}$ is a subgroup of $\delta_i$.
We prove this by induction on $i$. It is clear that $\delta_2 \le \delta_1 = G$. Now assume for some integer $i \ge 1$ that $\delta_{i+1} \le \delta_i$. Then $\irr {G \mid [\delta_{i+1},G] } \subseteq \irr {G \mid [\delta_{i} , G]}$, and thus we have $\displaystyle \delta_{i+2} = \!\!\!\!\!\!\! \prod_{\chi \in \irr {G \mid [\delta_{i+1},G]}} \!\!\!\!\!\!\!\!\! Z(\chi) \le \!\!\!\!\!\!\! \prod_{\chi \in \irr {G \mid [\delta_{i},G]}} \!\!\!\!\!\!\!\!\! Z(\chi) = \delta_{i+1}$ as desired.
We now show that the $\delta_i$’s are a second normal series that determines if a group is nested.
\[nested basics\] Let $G$ be a group and let $i$ be a positive integer.
1. If $\mathcal{M}_i = \{ N ~{\rm is~normal~in~} G \mid [\delta_i,G] \not\le N \}$, then $\delta_{i+1} = \prod_{N \in \mathcal{M}_i} Z_N$.
2. If $N$ is a normal subgroup of $G$, then either $N \le Z_N \le \delta_{i+1}$ or $[\delta_i,G] \le N$.
3. If $N$ is a normal subgroup of $G$ and $i > 1$ is integer that satisfies $[\delta_i,G] \le N$ and $[\delta_{i-1},G] \not\le N$, then $Z_N = \delta_i$.
4. If $i > 1$ and $[\delta_i,G] < [\delta_{i-1},G]$, then $Z_{[\delta_i,G]} = \delta_i$.
5. $G/[\delta_i,G]$ is a nested group.
Observe that $\{ \ker (\chi) \mid \chi \in \irr {G \mid [\delta_i,G]} \} \subseteq \mathcal{M}_i$. Hence, we see that $\delta_{i+1} \le \prod_{N \in \mathcal{M}_i} Z_N$. On the other hand, if $N \in \mathcal{M}_i$, then $N < N[\delta_i,G]$ and so there exists a character $\chi \in \irr {G/N \mid N[\delta_i,G]/N}$. This implies that $\chi_{N[\delta_i/G]}$ has a nonprincipal irreducible constituent $\nu$. Since $N[\delta_i,G]/(N \cap [\delta_i,G]) = N/(N \cap [\delta_i,G]) \times [\delta_i,G]/(N \cap [\delta_i,G])$, we see that $\nu_{[\delta_i,G]}$ is irreducible and nonprincipal. This implies that $\chi \in \irr {G \mid [\delta_i,G]}$. It follows that $Z_N \le Z(\chi) \le \delta_{i+1}$. We conclude that $\prod_{N \in \mathcal{M}_i} Z_N \le \delta_{i+1}$. This gives the desired equality in (1).
To prove (2), let $N$ be a normal subgroup of $G$. If $[\delta_i,G] \le N$, then we are done. Otherwise, $[\delta_i,G] \not\le N$ and so $N \le Z_N \le \prod_{N \in \mathcal{M}_i} Z_N = \delta_{i+1}$. This proves (2).
We now work to prove (3). Suppose $[\delta_i,G] \le N$ and $[\delta_{i-1},G] \not\le N$. Since $[\delta_i,G] \le N$, we see that $\delta_i \le N$. On the other hand, since $[\delta_{i-1},G] \not\le N$, we have by (2) that $N \le \delta_i$. This proves $N = \delta_i$ and (3) is proved. Notice that (4) is just (3) with $N = [\delta_i,G]$.
To prove (5), suppose $N$ is a normal subgroup and $[\delta_i,G] \le N$. If $G' \le N$, then $Z_N = G$. Otherwise, we can find an integer $j \le i$ so that $[\delta_j,G] \le N$ and $[\delta_{j-1},G] \not\le N$. We then apply (3) to see that $Z_N = \delta_j$. It follows that $G/[\delta_i,G]$ is a nested group by Lemma 3.2 of [@ML19gvz].
Again, since $G$ is finite, there exists a positive integer $n$ so that $\delta_n = \delta_{n+1}$. We set $\delta_\infty = \delta_n$. This next result is Theorem \[intro delta\].
\[delta series\] Let $G$ be a nonabelian group. Then $\delta_\infty = 1$ if and only if $G$ is nested. Moreover, if $G$ is nested with chain of centers $G = X_0 > X_1 > \dotsb > X_n \ge 1$, then $\delta_i = X_{i-1}$ for every integer $1 \le i \le n+1$.
First, assume that $G$ is nested with chain of centers $G = X_0 > X_1 > \dotsb > X_n \ge 1$. We now show that $\delta_i = X_{i-1}$ by induction on $i$. The case $i = 1$ follows from the definitions since $X_0 = G = \delta_1$. Now assume that $\delta_i = X_{i-1}$ for some integer $i\ge 1$. Then $\displaystyle \delta_{i+1} = \!\!\!\!\!\!\! \prod_{\chi \in \irr {G \mid [\delta_{i},G]}} \!\!\!\!\!\!\!\!\! Z(\chi)$. Observe that if $\chi \in \irr {G \mid [\delta_i,G]}$, then $[\delta_i,G] = [X_{i-1},G] \not\le \ker (\chi)$, and thus, $X_{i-1} \not\le Z(\chi)$. Since $G$ is nested, this implies that the centers of the characters $\chi$ that we are taking the product over will run through $X_i, \dots, X_n$, and so, $\delta_{i+1} = X_{i} X_{i-1} \dotsb X_n = X_{i}$. This proves the claim. In particular, $\delta_{n+1} = X_n$ and $[X_n,G] = 1$ by Lemma \[lewisgvz\]. It follows that $\displaystyle \delta_{n+2} = \!\!\!\!\!\!\! \prod_{\chi \in \irr {G \mid 1}} \!\!\!\!\!\!\!\!\! Z(\chi)$. However, since the set $\irr {G \mid 1}$ is empty, we obtain $X_{n+2} = 1$, and it is easy to see this implies that $\delta_{\infty} = 1$.
Now assume that $\delta_\infty = 1$. By Lemma \[nested basics\], we see that $G$ is nested.
We also define the subgroups $\epsilon_i$ by $\epsilon_1 = G$ and $\epsilon_{i+1} = V (G \mid [\epsilon_i,G])$. Observe that $\delta_i = \epsilon_i$ when $G$ is a GVZ-group since $V (\chi) = Z (\chi)$ for all characters $\chi \in \irr G$. We have already seen that $\delta_{i+1} \le \delta_i$, so the $\epsilon_i$ is a chain when $G$ is a GVZ-group. We now show that this is always the case.
\[containment\] Let $G$ be a group. For each $i \ge 1$, $\delta_{i+1} \le \epsilon_{i+1} \le \epsilon_i$.
We first verify $\epsilon_{i+1} \le \epsilon_i$ by induction on $i$. If $i = 1$, it is clear. So let $i \ge 2$, and assume that $\epsilon_{i+1} \le \epsilon_{i}$. Let $\chi \in \irr {G \mid [\epsilon_{i+1},G]}$; then $\chi \in \irr {G \mid [\epsilon_{i},G]}$ by Lemma \[Vprops\] and so $\chi$ vanishes off $V (G \mid [\epsilon_{i},G]) = \epsilon_{i+1}$. This forces the relation $V (G \mid [\epsilon_{i+1},G]) \le \epsilon_{i+1}$, as required.
We now verify $\delta_{i+1} \le \epsilon_{i+1}$ by also induction on $i$. Observe that $\delta_1 = \epsilon_1 = G$. We may assume that $\delta_i \le \epsilon_i$ for some $i \ge 1$. It follows that $[\delta_i,G] \le [\epsilon_i,G]$. Hence, if $\chi \in \irr {G \mid [\delta_i,G]}$, then $\chi \in \irr {G \mid [\epsilon_i,G]}$. This implies that $Z(\chi) \le V (\chi) \le V (G \mid [\epsilon_i,G]) = \epsilon_{i+1}$. It follows that $\delta_{i+1} = \prod_{\chi \in [\delta_i,G]} Z(\chi) \le \epsilon_{i+1}$.
Observe that $\epsilon_i/\epsilon_{i+1} \le {Z} (G/\epsilon_{i+1})$, for each $i\ge 1$, since $[\epsilon_i,G] \le {V} (G \mid [\epsilon_i,G]) = \epsilon_{i+1}$. Also observe that if $\epsilon_i > {Z} (G)$ for some $i$, then $[\epsilon_i, G] > 1$. Therefore the set $\irr {G\mid [\epsilon_i,G]}$ is nonempty, and it follows that $\epsilon_{i+1} \ge {Z} (G)$. On the other hand, we see that if $\epsilon_i = Z (G)$, then $[\epsilon_i,G] = 1$ and so $\epsilon_{i+1} = V (G \mid 1) = 1$. Note that if $[\epsilon_{i-1},G] = [\epsilon_i,G]$, then $\epsilon_{i+1} = V (G \mid [\epsilon_i,G]) = V (G \mid [\epsilon_{i-1},G]) = \epsilon_i$. We write $\epsilon_\infty$ for the terminal term of this series.
\[Z(chi) = V(chi)\] Let $G$ be a group. If $\chi \in \irr {G \mid U(G)}$, then $Z (\chi) = V (\chi)$. In particular, if $\chi \in \irr {G \mid U_\infty}$, then $Z (\chi) = V (\chi)$.
Suppose $\chi \in \irr {G \mid U (G)}$. Notice that $U (G) \not\le \ker (\chi)$ implies that $\ker (\chi) \le Z (G)$ and $Z(G) = Z(\chi)$ by Lemma \[centerproperty\]. By Lemma \[uproperties\], we see that $\chi$ vanishes off of $Z(G) = Z(\chi)$. This implies $V (\chi) = Z (\chi)$. Now, suppose that $\chi \in \irr {G \mid U_\infty}$. Thus, we can find an integer $i$ so that $\chi \in \irr {G \mid U_i}$ and $\chi \not\in \irr {G \mid U_{i-1}}$. This implies that $\chi \in \irr {G/U_{i-1}}$. Hence, we may assume that $U_{i-1} = 1$. This implies that $i = 1$, and we are done by the first conclusion.
\[ind cond\] Let $G$ be a group. If $[\epsilon_{i+1},G] < [\epsilon_i,G]$, then $\delta_{i+1} = \epsilon_{i+1}$ and $[\epsilon_i,G]/[\epsilon_{i+1},G] \le U (G/[\epsilon_{i+1},G])$.
We work by induction on $i$ to prove $\delta_{i+1} = \epsilon_{i+1}$. Notice that $\delta_1 = \epsilon_1 = G$. Suppose $1 \le j < i$ and suppose that $\delta_j = \epsilon_j$. Notice that $[\epsilon_{j+1},G] = [\epsilon_j,G]$ implies that $\epsilon_{j+2} = V (G \mid [\epsilon_{j+1},G]) = V (G \mid [\epsilon_j,G]) = \epsilon_{j+1}$. It is not difficult to see that this would imply $\epsilon_i = \epsilon_{i+1}$ and we would have a contradiction. Thus, $[\epsilon_{j+1},G] < [\epsilon_j,G]$. Since $\delta_j = \epsilon_j$, we can apply Lemma \[nested basics\] (3) to see that $Z_{[\epsilon_{j+1}, G]} = \delta_{j+1}$. Notice that $\epsilon_{j+1} \le Z_{[\epsilon_{j+1},G]}$, so this implies that $\epsilon_{j+1} \le \delta_{j+1}$. On the other hand, Lemma \[containment\] implies that $\delta_{j+1} \le \epsilon_{j+1}$. This gives the equality $\delta_{j+1} = \epsilon_{j+1}$, and we get the conclusion by taking $j = i$.
We have that $\epsilon_{i+1} = \delta_{i+1} = Z_{[\delta_{i+1},G]} = Z_{[\epsilon_{i+1},G]}$. We also have $\delta_{i+1} = \epsilon_{i+1} = V (G \mid [\epsilon_i,G])$ by definition. This implies that $[\epsilon_i,G] \le U (G \mid \delta_{i+1})$. By definition $U/[\epsilon_{i+1},G] = U (G/[\epsilon_{i+1},G]) = U (G/[\epsilon_{i+1}, G] \mid Z (G/[\epsilon_{i+1}, G])) = U (G/[\epsilon_{i+1},G] \mid \delta_{i+1}/[\epsilon_{i+1},G]) = U(G \mid \delta_{i+1})/[\epsilon_{i+1},G]$ where the last equality follows from Lemma \[uquotients\]. Hence, we have $U = U(G \mid \delta_{i+1})$ and $[\epsilon_{i},G] \le U$ as desired.
We now prove the parallel result for nested GVZ-groups and the $\epsilon$-series that we proved for nested groups and the $\delta$-series.
\[nested gvz quo\] Let $G$ be a group. If $[\epsilon_{i+1},G] < [\epsilon_i,G]$, then $G/[\epsilon_{i+1},G]$ is a nested GVZ-group.
We know that $[\epsilon_{i+1},G] = [\delta_{i+1},G]$ by Lemma \[ind cond\]. By Lemma \[nested basics\] (5), this implies that $G/[\epsilon_{i+1},G]$ is nested. Let $\chi \in \irr {G/[\epsilon_{i+1},G]}$. Let $j$ be minimal so that $[\epsilon_{j+1},G] \le \ker (\chi)$. If $j = 0$, then $G' \le \ker (\chi)$ and $G = Z (\chi) = V(\chi)$. Suppose $j \ge 1$. Notice that $[\epsilon_j,G]$ will not be in the kernel of $\chi$. By Lemma \[ind cond\], we see that $[\epsilon_j,G]/[\epsilon_{j+1},G] \le U(G/[\epsilon_{j+1},G])$. Hence, we may then apply Lemma \[Z(chi) = V(chi)\] to see that $Z (\chi) = V (\chi)$. This implies that $G/[\epsilon_i,G]$ will be a GVZ-group also as desired.
We finally come to Theorem \[intro epsilon\].
Let $G$ be a nonabelian group. Then $G$ is a nested GVZ-group if and only if $\epsilon_\infty = 1$. Moreover, in the event that $G$ is a nested GVZ-group with chain of centers $G = X_0 > X_1 > \dotsb > X_n > 1$, then $\epsilon_{i+1} = X_i$ for every $0 \le i \le n$.
If $G$ is a GVZ-group, then so is any epimorphic image of $G$. We have $V (\chi) = Z (\chi)$ for every character $\chi \in \irr G$. Furthermore, we have from Lemma \[Vprops\] that $V (G \mid [\epsilon_i,G]) = \prod_{\chi} V(\chi)$, where the product is over all characters $\chi \in \irr {G \mid [\epsilon_i,G]}$. Hence, $\epsilon_i = \delta_i$ for all $i$ when $G$ is a GVZ-group and so the fact that $G$ being a nested GVZ-group implies that $\epsilon_\infty = 1$ follows from Theorem \[delta series\]. If $G$ is a nested GVZ-group with chain of centers $G = X_0 > X_1 >\dotsb > X_n > 1$, then Theorem \[delta series\] also yields $\epsilon_{i+1} = \delta_{i+1} = X_i$ for every $0 \le i\le n$. Conversely, if $\epsilon_\infty = 1$, then by Lemma \[nested gvz quo\], we see that $G$ is a nested GVZ-group.
Examples
========
In [@ML19gvz], the second author presents examples of nested GVZ-groups of arbitrarily large nilpotence class. Obviously, these groups provide examples where $U(G) > 1$ and $Z_{U (G)} > Z(G)$. In [@ML19gvz], the second author also showed that groups of maximal class are nested, and if they are of class greater than $2$, they are not GVZ-groups. In particular, when $G$ is a group of order $p^4$ and nilpotence class $3$, then one can see that $G$ is of maximal class, so $K (G) = Z(G) > 1$, but $|G:Z(G)| = p^3$, so no irreducible character is fully-ramified over $Z(G)$, and this implies that $U (G) = 1$. Similarly, one can show when $G = C_p \wr C_p$ for any odd prime $p$ that $G$ satisfies $K(G) > 1$ and $U(G) = 1$.
We continue by constructing groups $G$ where $U (G) > 1$ and $U (G) < Z(G)$.
Let $H$ and $K$ be $p$-groups for some prime $p$ and let $l \ge 1$ be integer. Suppose that $p^l \le |U (H)|, |U (K)|$ and $p^l < |Z(H)|, |H'|, |Z (K)|, |K'|$. Fix elements $x_1, \dots, x_l \in U(H)$ and $y_1, \dots, y_l \in U (K)$ be chosen so that $X = \langle x_1, \dots, x_l \rangle$ and $Y = \langle y_1, \dots y_l \rangle$ both have order $p^l$. (I.e., if we think of $ U(H)$ and $U (K)$ as vector spaces, then the $x_i$’s and the $y_i$’s each form linearly independent subsets.) We then take $N \le Z (H \times K)$ by $N = \langle (x_1,y_1), \dots, (x_l,y_l) \rangle$. Let $G = (H \times K)/N$ and observe that $Z (H \times K) = Z(H) \times Z (K)$, so $(Z(H) \times Z(K))/N \le Z(G)$.
Let $U/N = U(G)$. Take $M = (X \times Y)/N$. Observe that $G/M \cong ((H \times K)/N)/((X \times Y)/N) = H/X \times K/Y$. Since $|X| < |H'|$ and $|Y| < |K'|$, we see that $H/X$ and $K/Y$ are nonabelian, and we have that $U (G/M) = 1$. Since $UM/M \le U (G/M)$, we have $U \le M$.
Let $\sigma \in \irr H$ and $\tau \in \irr K$. We can find characters $\mu \in \irr X$ and $\nu \in \irr Y$ so that $\sigma_X = \sigma (1) \mu$ and $\tau_Y = \tau (1) \nu$. Hence, $(\sigma \times \tau)_M = \sigma (1) \tau (1) (\mu \times \nu)$. Observe that $N \le \ker (\sigma \times \tau)$ if and only if $\sigma (x_i) \tau (y_i) = \sigma (1) \tau (1)$ for all $i = 1, \dots, l$. This implies that $\mu (x_i) \nu (y_i) = 1$ for all $i$. Thus, $\irr G = \{ (\sigma, \tau) \in \irr {H \times K} \mid \mu (x_i) = \overline {\nu} (y_i) {\rm ~for~all~} i \in \{ 1, \dots, n\} \}$. Notice that $\mu = 1_X$ if and only if $\nu = 1_Y$. It follows that if $\chi \in \irr {G \mid M/N}$, then $\chi = \sigma \times \tau$ as above where $\mu \ne 1$ and $\nu \ne 1$. Since $X \le U (H)$ and $Y \le U (K)$, this implies that $\sigma \in \irr {H \mid U(H)}$ and $\tau \in \irr {K \mid (K)}$. We deduce that $\sigma$ is fully-ramified with respect to $H/Z(H)$ and $\tau$ is fully-ramified with respect to $K/Z(K)$. A quick check of degrees reveals that $\chi$ will be fully-ramified with respect to $(H \times K)/(Z(H) \times Z(K))$. Notice that one consequence of this is that $Z(G) \le (Z(H) \times Z(K))/N$ and so, $Z (G) = (Z(H) \times Z(K))/N$. A second consequence is that every character in $\irr {G \mid M/N}$ is fully-ramified with respect to $G/Z(G)$. We conclude that $U (G) = M/N$. Observe that $|U (G)| = |M:N| = p^{2l}/p^l = p^l$. Also, since $p^l < |Z(H)|, |Z(K)|$, we have that $U (G) < Z(G)$.
Next, we show that these examples can be extended to find groups where $Z_{U(G)} = Z(G)$. To do this, we need to add the assumption that $p^l < |U(H)|$ and $p^l < |U(K)|$. Thus, we can find $u \in U(H) \setminus X$ and $v \in U(K) \setminus Y$. Let $L = (N \times \langle (u,v)\rangle)/N$, and observe that $L$ is not contained $M$. It is not difficult to see that $G/L \cong (H \times K)/ (N \times \langle (u,v)\rangle)$, and by the last paragraph we see that $U (G/L) = (X \langle u \rangle \times Y \langle v \rangle)/L = (M \times \langle u \rangle \times \langle v \rangle)/L$ whereas $UL/L = ML/L = (M \times \langle (u,v) \rangle)/L$. This implies that $|U(G/L):M/L| = p^2$ and $|UL/L:M/L| = p$ and so, $UL/L < U(G/L)$. By Lemma \[U equiv\], this implies that $Z_{U(G)} = Z(G)$.
Now, we take $H$ and $K$ to be semi-extraspecial groups, then $H \times K$ has nilpotence class $2$, so $G$ has nilpotence class $2$. We will have that $U (G) = K (G)$, so we also obtain an example $K (G) < Z(G)$, and we will have examples where $Z_{K(G)} = Z(G)$. Note that we can find semi-extraspecial groups so that $p^l < |U(H)| = |Z(H)|$ and $p^l < |U(K)| = |Z(K)|$ for every prime $p$ and every positive integer $l$, so we can find groups where $|U(G)| = p^l$ and $Z_{U(G)} = Z(G)$ for all primes $p$ and all positive integers $l$.
Finally, using Magma [@magma], we have an examples of a $p$-groups $G$ satisfying $1 < U(G) < K(G)$.
p1 := PCGroup ([ 8, -3, 3, 3, 3, -3, 3, -3, 3, 2641, 52706, 3970,
16419, 15851, 1267, 61564, 59412, 5205, 8237 ]);
p2 := PCGroup ([ 8, -5, 5, 5, 5, -5, 5, -5, 5, 1875000, 1790081,
600009, 2175602, 1275130, 345018, 7504003, 68811, 1779, 84027,
15025004, 2005012, 251020, 15228, 15150005, 1530013, 726021,
109229 ]);
Each of these groups has order $p^8$ and nilpotence class $3$. In each group the center equals $K$ and has order $p^2$, but $U$ has order $p$. The first group is a $3$-group and the second group is a $5$-group. We also have found examples for $p = 7, 11, 13, 17, 19$, but we have not included these groups.
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abstract: |
Correlated pattern mining has increasingly become an important task in data mining since these patterns allow conveying knowledge about meaningful and surprising relations among data. Frequent correlated patterns were thoroughly studied in the literature.
In this thesis, we propose to benefit from both frequent correlated as well as rare correlated patterns according to the *bond* correlation measure. Nevertheless, a main moan addressed to correlated pattern extraction approaches is their high number which handicap their extensive utilizations. In order to overcome this limit, we propose to extract a subset without information loss of the sets of frequent correlated and of rare correlated patterns, this subset is called “Condensed Representation“. In this regard, we are based on the notions derived from the Formal Concept Analysis FCA, specifically the equivalence classes associated to a closure operator $f_{bond}$ dedicated to the *bond* measure, to introduce new concise representations of both frequent correlated and rare correlated patterns. We then design the new mining approach, called <span style="font-variant:small-caps;">Gmjp</span>, allowing the extraction of the sets of frequent correlated patterns, of rare correlated patterns and their associated concise representations. In addition, we present the <span style="font-variant:small-caps;">Regenerate</span> algorithm allowing the query of the $\mathcal{RCPR}$ condensed representation associated to the $\mathcal{RCP}$ set as well as the <span style="font-variant:small-caps;">RcpRegeneration</span> algorithm dedicated to the regeneration of the whole set of rare correlated patterns from the $\mathcal{RCPR}$ representation.
The carried out experimental studies highlight the very encouraging compactness rates offered by the proposed concise representations and prove the good performance of the <span style="font-variant:small-caps;">Gmjp</span> algorithm. To improve the obtained performance, we introduced and evaluated the optimized version of <span style="font-variant:small-caps;">Gmjp</span>. The latter shows much better performances than do the initial version of <span style="font-variant:small-caps;">Gmjp</span>. In order to prove the usefulness of the extracted condensed representation, we conduct a classification process based on correlated association rules derived from closed correlated patterns and their associated minimal generators. The obtained rules were applied to the context of intrusion detection and achieve encouraging results.
**Key Words:** [Formal Concept Analysis, Constraint Data Mining, Monotonicity, Anti-monotonicity, *bond* Correlation Measure, Itemset Extraction, Condensed Representation, Classification, Associative Rule.]{}
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<span style="font-variant:small-caps;">**Comité de Thèse**</span>
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<span style="font-variant:small-caps;">Faouzi MOUSSA</span> <span style="font-variant:small-caps;">Professeur, Faculté des Sciences de Tunis</span> <span style="font-variant:small-caps;">Président</span>
<span style="font-variant:small-caps;">Nadia ESSOUSSI</span> <span style="font-variant:small-caps;">Maitre de Conférences, F.S.E.G de Nabeul</span> <span style="font-variant:small-caps;">Rapporteur</span>
<span style="font-variant:small-caps;">Philippe LENCA</span> <span style="font-variant:small-caps;">Professeur, Telecom Bretagne</span> <span style="font-variant:small-caps;">Rapporteur</span>
<span style="font-variant:small-caps;">Amel TOUZI GRISSA</span> <span style="font-variant:small-caps;">Professeur, ESIG de Kairouan</span> <span style="font-variant:small-caps;">Examinateur</span>
<span style="font-variant:small-caps;">Sadok BEN YAHIA</span> <span style="font-variant:small-caps;">Professeur, Faculté des Sciences de Tunis</span> <span style="font-variant:small-caps;">Directeur</span>
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I would like to thank the members of my thesis committee. I thank Professor <span style="font-variant:small-caps;">Faouzi MOUSSA</span> for agreeing to chair my thesis committee.
I would like to thank Associate-Professor <span style="font-variant:small-caps;">Nadia ESSOUSSI</span> and Professor <span style="font-variant:small-caps;">Philippe LENCA</span> for accepting to review my thesis report, and for providing me with detailed corrections and interesting comments.
I would like to thank Professor <span style="font-variant:small-caps;">Amel GRISSA TOUZI</span> for participating to the thesis committee.
I want to express my deep thanks to my thesis supervisor Professor <span style="font-variant:small-caps;">Sadok BEN YAHIA</span> for trusting me and for allowing me to grow as a research scientist. During the whole period of study, Professor <span style="font-variant:small-caps;">BEN YAHIA</span> contributes by giving me intellectual freedom in my work, engaging me in new creative ideas, supporting my participation to various conference, and requiring a high quality of work in all my efforts.
I want to express my special thanks to Assistant-Professor <span style="font-variant:small-caps;">Tarek HAMROUNI</span> for collaborating in the realization of different phases of my thesis project. I am very grateful for all the offered efforts to ensure high-quality of this research. I greatly benefited from his scientific insight, his high-level of expertise in the field of Data-Mining and his ability to explore possible improvements in order to make a deeper development of our research.
My sincere thanks go to all the members of the <span style="font-variant:small-caps;">LIPAH</span> Laboratory of the Faculty of Sciences of Tunis, for the friendship, for the encouraging ambiance and emotional atmosphere during the last years.
I cannot finish without thanking my family. A special dedicate goes to my precious treasure, my mother <span style="font-variant:small-caps;">Radhia BENFRADJ BOUASKER</span>, for supporting and encouraging me during my studies. I also would like to thank my brothers, my sisters for providing assistance in numerous ways. I want to express my gratitude to my husband, dear <span style="font-variant:small-caps;">Mohamed</span>, without his comprehension and encouragements, I could not have accomplished this project. A particular thought goes for my lovely baby girl <span style="font-variant:small-caps;">Meriam</span>, my angel baby, the greatest joy of my life.
My Doctoral Graduation is dedicated to the memory of my beloved father, <span style="font-variant:small-caps;">Miled BOUASKER</span>. I am honored to have you as a father. Thank you for the high trust, for learning to me the strength and the patience, and for motivating me to always keep reaching for excellence.\
Thank you for the father you were.
La fouille des motifs corrélés est une piste de recherche de plus en plus attractive en fouille de données grâce à la qualité et à l’utilité des connaissances offertes par ces motifs. Plus précisément, les motifs fréquents corrélés ont été largement étudiés auparavant dans la littérature.
Notre objectif dans cette thèse est de bénéficier à la fois des connaissances offertes par les motifs corrélés fréquents ainsi que les motifs rares corrélés selon la mesure de corrélation *bond*. Cependant, un principal problème est lié à la fouille des motifs corrélés concerne le nombre souvent très élevé des motifs corrélés extraits. Un tel nombre handicape une exploitation optimale et aisée des connaissances encapsulées dans ces motifs. Pour pallier ce problème, nous nous intéressons dans cette thèse à l’extraction d’un sous-ensemble, sans perte d’information, de l’ensemble de tous les motifs corrélés. Ce sous-ensemble, le noyau d’itemsets, appelé “Représentations Concises”, à partir duquel tous les motifs redondants peuvent être régénérés sans perte d’informations. Le but d’une telle représentation est de minimiser le nombre de motifs extraits tout en préservant les connaissances cachées et pertinentes.
Afin de réaliser cet objectif, nous nous sommes basés sur les notions dérivées de l’analyse formelle de concepts AFC. Plus précisément, les représentations condensées, que nous proposons, sont issues des notions de classes d’équivalence induites par l’opérateur de fermeture $f_{bond}$ associé à la mesure de corrélation *bond*. Après la caractérisation des représentations condensées proposées, nous introduisons l’algorithme <span style="font-variant:small-caps;">Gmjp</span> dédié à l’extraction des motifs corrélés fréquents, des motifs corrélés rares ainsi que leurs représentations condensées associées. Nous présentons également l’algorithme <span style="font-variant:small-caps;">Regenerate</span> d’interrogation de la représentation $\mathcal{RCPR}$ associée à l’ensemble $\mathcal{RCP}$ des motifs corrélés rares et nous proposons aussi l’algorithme <span style="font-variant:small-caps;">RCPRegeneration</span> dédié à la régénération de l’ensemble total des motifs corrélés rares à partir de la représentation concise $\mathcal{RCPR}$.
L’évaluation expérimentale menée met en valeur les taux de compacités très intéressants offerts par les différentes représentations concises proposées et justifie également les performances encourageantes de l’approche <span style="font-variant:small-caps;">Gmjp</span>. Afin d’améliorer les performances de l’algorithme <span style="font-variant:small-caps;">Gmjp</span>, nous proposons une version optimisée de <span style="font-variant:small-caps;">Gmjp</span>. Cette version optimisée présente des temps d’exécution beaucoup plus réduits que la version initiale. De plus, nous avons conduit un processus de classification associative basé sur les règles associatives corrélées dérivées à partir des motifs corrélés fermés et de leurs générateurs minimaux. Les résultats de classification des données de détection d’intrusions, sont très encourageants et ont prouvé une grande utilité de la fouille des motifs corrélés.
**Mots Clés :** [Analyse Formelle de Concept, Fouille sous Contraintes, Monotonie, Anti-monotonie, Mesure *bond*, Extraction de motifs, Représentation concise, Classification, Règles associatives.]{}
Introduction {#ch_introduction}
============
Introduction and Motivations
----------------------------
The development of new information and communication technologies and the globalization of markets make the competition more and more increased among companies. In this sense, the need for access to an accurate information for decision-making is increasingly urgent. The actual problem is linked to lack of access to relevant information in the presence of the large amount of data. The collected data in various fields are becoming larger. This motivates the need to analyze and interpret data in order to extract useful knowledge.
In this context, the process of knowledge discovery from databases <span style="font-variant:small-caps;">(</span>KDD<span style="font-variant:small-caps;">)</span> is a complete process aiming to extract useful, hidden knowledge from huge amount of data [@Agra94]. Data Mining is one of the main steps of this process and is dedicated to offer the necessary tools needed for an optimal exploration of data. Many state of the art approaches were focused on frequent itemset extraction and association rule generation. Nevertheless, two main problems handicap the good use of the returned knowledge from the set of frequent itemsets. The first problem is related to the quality of the offered knowledge since the degree of correlation of the extracted itemsets may be not interesting for the end user. The second problem is related often to the huge quantity of the extracted knowledge.
To overcome these problems, many previous works propose to integrate the correlation measures within the mining process [@Brin97; @comine_Lee; @Omie03; @ccmine_Kim; @Xiong06hypercliquepattern]. Correlated pattern mining is then shown to be more complex but more informative than traditional frequent patterns mining. In fact, correlated patterns offer a precise information about the degree of apparition of the items composing a given itemset [@borgelt]. This key information specifies the simultaneous apparition frequency among items, *i.e.*, their co-occurrence, as well as their apparition frequency, *i.e.*, their occurrence.
Other state of the art approaches deal with the extraction of a subset, without information loss, of the whole set of correlated patterns. This subset, is named, “Condensed Representation“ and from which we are able to derive all the redundant correlated patterns. The condensed representations prove their high utility in different fields such as: bioinformatics [@pasquier2009] and data grids [@tarekJSS2015].
The main objective behind defining such a condensed representation is to reduce the number of the extracted patterns while preserving the same amount of pertinent knowledge. In addition to this, all of the extracted associated rules, derived from correlated patterns fulfilling a correlation measure such as *all-confidence* or *bond*, are valid with respect to minimal support and to minimal confidence thresholds [@Omie03].
Frequent correlated itemset mining was then shown to be an interesting task in data mining. Since its inception, this key task grasped the interest of many researchers since it meets the needs of experts in several application fields [@tarekds2010], such as market basket study. However, the application of correlated frequent patterns is not an attractive solution for some other applications, *e.g.*, intrusion detection, analysis of the genetic confusion from biological data, detection of rare diseases from medical data, to cite but a few [@livreIGIGlobal2010; @mahmood_less_frequent_patterns_vs_networks; @romero2010; @laszloIJSI2010; @haglin08]. As an illustration of the rare correlated patterns applications in the field of medicine, the rare combination of symptoms can provide useful insights for doctors.
To the best of our knowledge, there is no previous work that dealt with both frequent correlated as well as rare correlated patterns according to a specified correlation metric. Thus, motivated by this issue, we propose in this thesis to benefit from the knowledges returned from both frequent correlated as well as rare correlated patterns according to the *bond* correlation measure. To solve this challenging problem, we propose an efficient algorithmic framework, called <span style="font-variant:small-caps;">GMJP</span>, allowing the extraction of both frequent correlated patterns, rare correlated patterns as well as their associated concise representations.
Contributions
-------------
Our first contribution consists in defining and studying the characteristics of the condensed representations associated to frequent correlated as well as the condensed representations associated to rare correlated ones. In this respect, we are based on the notions derived from the Formal Concept Analysis <span style="font-variant:small-caps;">(FCA)</span> [@ganter99], specifically the equivalence classes associated to a closure operator $f_{bond}$ dedicated to the *bond* measure to introduce our new concise representations of both frequent correlated and rare correlated patterns. The first concise representation $\mathcal{RCPR}$ associated to the $\mathcal{RCP}$ set of rare correlated patterns, is composed by the maximal elements of the rare correlated equivalence classes, called “Closed Rare Correlated Patterns $\mathcal{CRCP}$ set“ union of their associated minimal generators called “Minimal Rare Correlated Patterns $\mathcal{MRCP}$ set“. Two other optimizations of the $\mathcal{RCPR}$ representation are also proposed. The first optimization is composed by the whole set $\mathcal{CRCP}$ of closed rare correlated patterns union of the minimal elements of the $\mathcal{MRCP}$ set. The second optimization is composed by the maximal elements of the $\mathcal{CRCP}$ of closed rare correlated patterns union of the whole $\mathcal{MRCP}$ set. We prove that both of these representations are also concise and exact. Our third optimized representation is a condensed approximate representation. The latter is composed by the maximal elements of the $\mathcal{CRCP}$ set union of the minimal elements of the $\mathcal{MRCP}$ set. According to the $\mathcal{FCP}$ set of frequent correlated patterns, the condensed exact representation is composed by the Closed Correlated Frequent Patterns. We prove the theoretical properties of accuracy and compactness of all the proposed representations.
Our second contribution is the design and the implementation of a new mining approach, called <span style="font-variant:small-caps;">Gmjp</span>, allowing the extraction of the sets of frequent correlated patterns, of rare correlated patterns and their associated concise representations. <span style="font-variant:small-caps;">Gmjp</span> is a sophisticated mining approach that allows a simultaneous integration of two opposite paradigms of monotonic and anti-monotonic constraints. In addition, we present the <span style="font-variant:small-caps;">Regenerate</span> algorithm allowing the query of the $\mathcal{RCPR}$ condensed representation associated to the $\mathcal{RCP}$ set as well as the <span style="font-variant:small-caps;">RcpRegeneration</span> algorithm dedicated to the regeneration of the whole set of rare correlated patterns from the $\mathcal{RCPR}$ representation.
Our third contribution consists in proposing an optimized version of <span style="font-variant:small-caps;">Gmjp</span>. The latter shows much better performance than the initial version of <span style="font-variant:small-caps;">Gmjp</span>. In order to prove the usefulness of the extracted condensed representation, we conduct a classification process based on correlated association rules derived from closed correlated patterns and their associated minimal generators. The obtained rules are applied to the context of intrusion detection and achieve promoting results.
The evaluation protocol of our approaches consists in experimental studies carried out over dense and sparse benchmark datasets commonly used in evaluating data mining contributions. The evaluation of the classification process is based on the <span style="font-variant:small-caps;">KDD 99</span> database of intrusion detection data. We also conduct the process of applying the $\mathcal{RCPR}$ representation on the extraction of rare correlated association rules from Micro-array gene expression data related to Breast-Cancer. The diverse obtained association-rules reveals a variety of relationship between up and down regulated gene-expressions.
Thesis Organization
-------------------
The remainder of this thesis is organized as follows:
**Chapter 2** introduces the basic notions related to the itemset search space and to itemset extraction. We also define two distinct categories of constraints: monotonic and anti-monotonic. We equally introduce the environment of Formal Concept Analysis <span style="font-variant:small-caps;">(FCA)</span> which offers the basis for the proposition of our approaches, specifically the notions of Closure Operator, Minimal Generator, Closed Pattern, Equivalence class and Condensed representation of a set of patterns.
**Chapter 3** offers an overview of the state of the art approaches dealing with correlated patterns mining. We start this chapter by defining the most used correlation measures. Then, we continue with the approaches related to frequent correlated patterns, followed by the state of the art of rare correlated patterns then the overview of the algorithms focusing on condensed representations of correlated patterns.
**Chapter 4** focuses on characterizing the $\mathcal{FCP}$ set of frequent correlated patterns as well as the $\mathcal{RCP}$ set of rare correlated patterns. It introduces the condensed exact and approximate representations associated to the $\mathcal{RCP}$ set as well as the concise exact representation associated to the $\mathcal{FCP}$ set. The main content of this chapter was published in [@rnti2012] and in [@ida2015].
**Chapter 5** introduces the <span style="font-variant:small-caps;">Gmjp</span> approach, allowing the extraction of the sets of frequent correlated patterns, of rare correlated patterns and their associated concise representations. The optimized version of <span style="font-variant:small-caps;">Gmjp</span>, named <span style="font-variant:small-caps;">Opt-Gmjp</span>, was also presented. This chapter also presents the theoretical complexity approximation of <span style="font-variant:small-caps;">Gmjp</span>. In addition, this chapter describes the <span style="font-variant:small-caps;">Regenerate</span> algorithm allowing the query of the $\mathcal{RCPR}$ condensed representation associated to the $\mathcal{RCP}$ set as well as the <span style="font-variant:small-caps;">RcpRegeneration</span> algorithm dedicated to the regeneration of the whole set of rare correlated patterns from the $\mathcal{RCPR}$ representation. The main content of this chapter was published in [@egc2012] and in [@ida2015].
**Chapter 6** focuses on the experimental validation of the proposed approaches. The evaluation process is based on two main axes, the first is related to the compactness rates of the condensed representations while the second axe concerns the running time. This chapter evaluates the optimized version of <span style="font-variant:small-caps;">Gmjp</span>, which presents much better performance than do <span style="font-variant:small-caps;">Gmjp</span> over different benchmark datasets. The content related to the optimizations and evaluations was published in [@sac2015].
**Chapter 7** describes the classification process based on correlated patterns. This chapter starts by presenting the framework of association rule extraction, it clarifies the properties of the generic bases of association rules. Then, we continue with the detailed presentation of the application of both frequent correlated and rare correlated patterns within the classification of some UCI benchmark datasets. We equally present the application of rare correlated patterns in the classification of intrusion detection data from the <span style="font-variant:small-caps;">KDD 99</span> dataset. The obtained results showed the usefulness of our proposed classification method over four different intrusion classes. This chapter is concluded with the application of the $\mathcal{RCPR}$ representation on the extraction of rare correlated association rules from Micro-array gene expression data. These extracted rules aims to identify relations among up and down regulated gene expressions. The main content of this chapter was published in [@pakdd2012] and in [@dexa2013].
**Chapter 8** concludes the thesis and sketches out our perspectives for future work.
Basic Notions {#ch2}
=============
Introduction {#se1}
------------
The extraction of correlated patterns is shown to be more complex but more informative than traditional frequent patterns mining. In fact, these correlated patterns present a strong link among the items they compose and they prove their high utility in many real life applications fields.
This chapter is dedicated to the introduction of the basic notions needed for the presentation of our approaches. The second section deals with the basic notions related to the search space as well as the itemsets’s extraction. Then, we link in the third section with the presentation of the foundations of the formal concepts analysis <span style="font-variant:small-caps;">(FCA)</span> framework [@ganter99]. The last section concludes the chapter.
Search Space
------------
We begin by presenting the key notions related to itemset extraction, that will be used thorough this thesis. First, let us define an extraction context.
### Extraction Context
\[definitionbasetransactions\] **Extraction Context**\
An extraction context <span style="font-variant:small-caps;">(</span>also called Context or Dataset<span style="font-variant:small-caps;">)</span> is represented by a triplet $\mathcal{C}$ = <span style="font-variant:small-caps;">(</span>$\mathcal{T},\mathcal{I},\mathcal{R}$<span style="font-variant:small-caps;">)</span> with $\mathcal{T}$ and $\mathcal{I}$ are, respectively, a finite sets of transactions <span style="font-variant:small-caps;">(</span>or objects<span style="font-variant:small-caps;">)</span> and of items <span style="font-variant:small-caps;">(</span>or attributes<span style="font-variant:small-caps;">)</span>, and $\mathcal{R}$ $\subseteq$ $\mathcal{T} \times \mathcal{I}$ is a binary relation between the transactions and the items. A couple <span style="font-variant:small-caps;">(</span>$t$, $i$<span style="font-variant:small-caps;">)</span> $\in$ $\mathcal{R}$ if $t$ $\in$ $\mathcal{T}$ contains $i$ $\in$ $\mathcal{I}$.
An example of an extraction context $\mathcal{C}$ $=$ $\textsc{(}$$\mathcal{T},\mathcal{I},\mathcal{R}$$\textsc{)}$ is given by Table \[Base\_transactions\]. In this context, the transaction set $\mathcal{T} = \{1, 2, 3, 4, 5\}$ <span style="font-variant:small-caps;">(</span>*resp.* the object set $\mathcal{O} = \{1, 2, 3, 4,
5\}$<span style="font-variant:small-caps;">)</span> and the items set $\mathcal{I}$ $=$ $\{$`A`, `B`, `C`, `D`, `E`,$\}$. The couple <span style="font-variant:small-caps;">(</span>2, B<span style="font-variant:small-caps;">)</span> $\in$ $\mathcal{R}$ since the transaction 2 $\in$ $\mathcal{T}$ contains the item B $\in$ $\mathcal{I}$.
We note, by sake of accuracy, that the notations of transactions database and extraction context have the same meaning thorough this thesis. They are denoted as $\mathcal{D}$ $=$ $\textsc{(}\mathcal{T}, \mathcal{I},
\mathcal{R}\textsc{)}$.
\[motif\] **Itemset or Pattern**\
A transaction $t$ $\in$ $\mathcal{T}$, having an identifier denoted by *TID* $\textsc{(}$Tuple IDentifier$\textsc{)}$, contains a non-empty set of items belonging to $\mathcal{I}$. A subset $I$ of $\mathcal{I}$ where $k$ $=$ $\vert I \vert $ is called a *$k$-pattern* or simply a *pattern*, and $k$ represents the cardinality of $I$. The number of transactions $t$ of a context $\mathcal{C}$ containing a pattern $I$, $\vert$$\{$ $t$ $ \in $ $\mathcal{D}$ $\vert $ $I$ $\subseteq $ $t$$\}$$\vert $, is called *absolute support* of $I$ and is denoted $Supp\textsc{(}\wedge I\textsc{)}$. The *relative support* of $I$ or the *frequency* of $I$, denoted $freq\textsc{(}I\textsc{)}$, is the quotient of the absolute support by the total number of the transactions of $\mathcal{D}$, *i.e.*, $freq\textsc{(}I\textsc{)}$ $=$ $\displaystyle\frac{\displaystyle{\vert {\{}t \in \mathcal{D} | I \subseteq t
{\}}\vert}}{\displaystyle{\vert \mathcal{T}\vert}}$.
We point that, thorough this thesis, we are mainly interested in itemsets *i.e.* the set of items as a kind of patterns. Consequently, we use a form without separators to denote an itemset. For example, `BD` stands for the itemset composed by the items `B` and `D`.
### Supports of a Pattern
To evaluate an itemset, many interesting measures can be used. The most common ones are presented by Definition \[definitionsupportmotif\].
\[definitionsupportmotif\] **Supports of a Pattern**\
Let $\mathcal{D}$=<span style="font-variant:small-caps;">(</span>$\mathcal{T}, \mathcal{I}, \mathcal{R}$<span style="font-variant:small-caps;">)</span> an extraction context and a non empty itemset $I$ $\subseteq$ $\mathcal{I}$. We distinguish three kinds of supports for an itemset $I$ :
- ***The conjunctive support:*** *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> = $\mid$$\{$$t$ $\in$ $\mathcal{T}$ $\mid$ $\forall$ $i$ $\in$ $I$ : <span style="font-variant:small-caps;">(</span>$t$, $i$<span style="font-variant:small-caps;">)</span> $\in$ $\mathcal{R}$$\}$$\mid$
- ***The disjunctive support:*** *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$I$<span style="font-variant:small-caps;">)</span> = $\mid$$\{$$t$ $\in$ $\mathcal{T}$ $\mid$ $\exists$ $i$ $\in$ $I$ : <span style="font-variant:small-caps;">(</span>$t$, $i$<span style="font-variant:small-caps;">)</span> $\in$ $\mathcal{R}$$\}$$\mid$, and,
- ***The negative support:*** *Supp*<span style="font-variant:small-caps;">(</span>$\neg$$I$<span style="font-variant:small-caps;">)</span> = $\mid$$\{$$t$ $\in$ $\mathcal{T}$ $\mid$ $\forall$ $i$ $\in$ $I$ : <span style="font-variant:small-caps;">(</span>$t$, $i$<span style="font-variant:small-caps;">)</span> $\notin$ $\mathcal{R}$$\}$$\mid$.
More explicitly, for an itemset $I$, the supports are defined as follows:
$\bullet$ *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span>: is equal to the number of transactions containing all the items of $I$.
$\bullet$ *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$I$<span style="font-variant:small-caps;">)</span>: is equal to the number of transactions containing at least one item of $I$.
$\bullet$ *Supp*<span style="font-variant:small-caps;">(</span>$\neg$$I$<span style="font-variant:small-caps;">)</span>: is equal to the number of transactions that do not contain any item of $I$.
It is important to note that the “De Morgan” law ensures the transition between the disjunctive and the negative support of an itemset $I$ as follows : *Supp*<span style="font-variant:small-caps;">(</span>$\neg$*I*<span style="font-variant:small-caps;">)</span> = $\mid\mathcal{T}\mid$ - *Supp*<span style="font-variant:small-caps;">(</span>$\vee$*I*<span style="font-variant:small-caps;">)</span>.
Let us consider the extraction context given by Table \[Base\_transactions\] that will be used thorough the different examples. We have *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$`AD`<span style="font-variant:small-caps;">)</span> = $\mid$$\{$1$\}$$\mid$ = $1$, *Supp*<span style="font-variant:small-caps;">(</span>$\vee$`AD`<span style="font-variant:small-caps;">)</span> = $\mid$$\{$ 1, 3, 5$\}$$\mid$ = $3$, and, *Supp*<span style="font-variant:small-caps;">(</span>$\neg$<span style="font-variant:small-caps;">(</span>`AD`<span style="font-variant:small-caps;"><span style="font-variant:small-caps;">)</span><span style="font-variant:small-caps;">)</span></span> = $\mid$$\{$2, 4$\}$$\mid$ = $2$ $^{\textsc{(}}$.
In the following, if there is no risk of confusion, the conjunctive support will be simply denoted by *support*. Note that *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$\emptyset$<span style="font-variant:small-caps;">)</span> = $|\mathcal{T}|$ since the empty set is included in all transactions, while *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$\emptyset$<span style="font-variant:small-caps;">)</span> = $0$ since the empty set does not contain any item. Moreover, $\forall$ $i$ $\in$ $\mathcal{I}$, *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$i$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$i$<span style="font-variant:small-caps;">)</span>, while in the general case, for $I$ $\subseteq$ $\mathcal{I}$ and $I$ $\neq$ $\emptyset$, *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> $\leq$ *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$I$<span style="font-variant:small-caps;">)</span>. A pattern $I$ is said to be *frequent* if *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> is greater than or equal to a user-specified minimum support threshold, denoted *minsupp* [@Agra94]. The following lemma shows the links that exist between the different supports of a non-empty pattern $I$. These links are based on the *inclusion-exclusion identities* [@galambos].
\[lemmaidentitésinclusionexclusion\] - **Inclusion-exclusion identities** - The inclusion-exclusion identities ensure the links between the conjunctive, disjunctive and negative supports of a non-empty pattern $I$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------
$ \textit{Supp}\textsc{(}\wedge I\textsc{)}\mbox{ }=\mbox{ }\displaystyle\sum_{\emptyset \subset I_1 \subseteq I} {\textsc{(}-1\textsc{)}^{\mbox{$\mid I_1\mid$\mbox{ - 1}}}\mbox{ <span style="font-variant:small-caps;">(</span>1<span style="font-variant:small-caps;">)</span>
}\textit{Supp}\textsc{(}\vee I_1\textsc{)}}$
$ \textit{Supp}\textsc{(}\vee I\textsc{)}\mbox{ }=\mbox{ }\displaystyle\sum_{\emptyset \subset I_1 \subseteq I} {\textsc{(}-1\textsc{)}^{ \mbox{$\mid I_1\mid $\mbox{ - 1}}}\mbox{ <span style="font-variant:small-caps;">(</span>2<span style="font-variant:small-caps;">)</span>
}\textit{Supp}\textsc{(}\wedge I_1\textsc{)}}$
$ \textit{Supp}\textsc{(}\neg I\textsc{)}\mbox{ } =\mbox{ } \mid \mathcal{T} \mid \mbox{ }-\mbox{ <span style="font-variant:small-caps;">(</span>3<span style="font-variant:small-caps;">)</span>
} \textit{Supp}\textsc{(}\vee I\textsc{)} \mbox{ }\textsc{(}\mbox{The De Morgan's law}\textsc{)}$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------
### Frequent Itemset - Rare Itemset - Correlated Itemset
Given a minimal threshold of support [@Agra94], we distinguish between two kinds of patterns, frequent patterns and infrequent patterns <span style="font-variant:small-caps;">(</span>also called Rare patterns<span style="font-variant:small-caps;">)</span>.
\[motiffréq\] **Frequent Itemset - Rare Itemset**\
Let an extraction context $\mathcal{C}$ = $\textsc{(}\mathcal{T}, \mathcal{I},\mathcal{R}\textsc{)}$, a minimal threshold of the conjunctive support *minsupp*, an itemset $I$ $\subseteq$ $\mathcal{I}$ is said *frequent* if *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> $\geq$ *minsupp*. Otherwise, $I$ is said *infrequent* or *rare*.
Let *minsupp* = 2. *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$`BCE`<span style="font-variant:small-caps;">)</span> = 3, the pattern `BCE` is a frequent pattern. However, the pattern `CD` is a rare pattern since *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$`CD`<span style="font-variant:small-caps;">)</span> = 1 $<$ 2.
In the following, we need to define the smallest rare patterns according to the relation of inclusion set. They correspond to rare patterns having all subsets frequent, and are defined as follows:
\[mrp\] **Minimal rare patterns**\
The $\mathcal{M}in \mathcal{RP}$ set of minimal rare patterns is composed of rare patterns having no rare proper subsets. This set is defined as: $\mathcal{M}in$$\mathcal{RP}$ = $\{I$ $\in$ $\mathcal{I} |$ $\forall$ $I_1 \subset I$: *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I_1$<span style="font-variant:small-caps;">)</span> $\geq$ *minsupp*$\}$.
\[example\_MRP\] Let us consider the extraction context sketched by Table \[Base\_transactions\]. For *minsupp* = 4, we have $\mathcal{M}in \mathcal{RP}$ = $\{$$A$, $D$, $BC$, $CE$$\}$. $A$ and $D$ are minimal rare items, $BC$ is a minimal rare itemset since it is composed by two frequent items: $B$ with *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$`B`<span style="font-variant:small-caps;">)</span> = 4 and $C$ with *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$`C`<span style="font-variant:small-caps;">)</span> = 4.
In fact, in order to reduce the high number of frequent itemsets and to improve the quality of the extracted frequent itemets, other interesting measures apart from the conjunctive support are introduced within the mining process. These latter are called “Correlation Measures”. The itemsets fulfilling a given correlation measure are called “Correlated Itemsets”. This latter type of itemsets is defined in a generic way in what follows:
\[motifCorr\] **Correlated Itemset**\
Let a correlation measure M, a minimal correlation threshold *minCorr*, an itemset $I$ $\subseteq$ $\mathcal{I}$ is said *correlated* according to the measure M, if *M*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\geq$ *minCorr*. $I$ is said *non correlated* otherwise.
### Categories of Constraints
Besides the minimal frequency constraint expressed by the *minsupp* threshold, other constraints can be integrated within the itemset’s extraction process. These constraints have two distinct types, “The monotonic constraints” and “The anti-monotonic constraints” [@luccheKIS05_MAJ_06].
\[anti-monotone\] **Anti-monotonic Constraint**\
A constraint $Q$ is *anti-monotone* if $\forall$ $I$ $\subseteq$ $\mathcal{I}$, $\forall$ $I_1$ $\subseteq$ $I$ : $I$ fulfills $Q$ $\Rightarrow$ $I_1$ fulfills $Q$.
\[monotone\] **Monotone Constraint**\
A constraint $Q$ is *monotone* if $\forall$ $I$ $\subseteq$ $\mathcal{I}$, $\forall$ $I_1$ $\supseteq$ $I$ : $I$ fulfills $Q$ $\Rightarrow$ $I_1$ fulfills $Q$.
The *frequency constraint*, i.e. having a support greater than or equal to *minsupp*, is an anti-monotonic constraint. In fact, $\forall$ $I$, $I_1$ $\subseteq$ $\mathcal{I}$, if $I_1$ $\subseteq$ $I$ and *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> $\geq$ *minsupp*, then *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I_1$<span style="font-variant:small-caps;">)</span> $\geq$ *minsupp* since *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I_1$<span style="font-variant:small-caps;">)</span> $\geq$ *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span>.
Dually, the *constraint of rarity*, i.e. having a support strictly lower than *minsupp*, is a monotonic constraint. In fact, $\forall$ $I$, $I_1$ $\subseteq$ $\mathcal{I}$, if $I_1$ $\supseteq$ $I$ and *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> $<$ *minsupp*, then *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I_1$<span style="font-variant:small-caps;">)</span> $<$ *minsupp* since *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I_1$<span style="font-variant:small-caps;">)</span> $\leq$ *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span>.
A set of itemset may fulfill different constraints simultaneously. Proposition \[conjCs\], whose proof is in [@lee2006], clarifies the conjunction of two constraints of the same nature.
\[conjCs\] The conjunction of anti-monotonic constraints $\textsc{(}$*resp.* monotonic$\textsc{)}$ is an anti-monotonic $\textsc{(}$*resp.* monotonic$\textsc{)}$ constraint.
Let us define now the dual notions of order-ideal and order-filter [@ganter99] defined on $\mathcal{P}\textsc{(}\mathcal{I}\textsc{)}$ and associated to the two kinds of constraints given by definitions \[anti-monotone\] et \[monotone\].
**Order Ideal**\
A subset $\mathcal{S}$ of $\mathcal{P}\textsc{(}\mathcal{I}\textsc{)}$ is an order ideal if it fulfills the following properties:
- If $I$ $\in$ $\mathcal{S}$, then $\forall$ $I_1$ $\subseteq$ $I$ : $I_1$ $\in$ $\mathcal{S}$.
- If $I$ $\notin$ $\mathcal{S}$, then $\forall$ $I$ $\subseteq$ $I_1$ : $I_1$ $\notin$ $\mathcal{S}$.
**Order Filter**\
A subset $\mathcal{S}$ of $\mathcal{P}\textsc{(}\mathcal{I}\textsc{)}$ is an order filter if it fulfills the following properties:
- If $I$ $\in$ $\mathcal{S}$, then $\forall$ $I_1$ $\supseteq$ $I$ : $I_1$ $\in$ $\mathcal{S}$.
- If $I$ $\notin$ $\mathcal{S}$, then $\forall$ $I$ $\supseteq$ $I_1$ : $I_1$ $\notin$ $\mathcal{S}$.
An anti-monotone constraint such as the frequency constraint induces an order ideal on the itemset lattice. Dually, a monotonic constraint as the rarity constraint induces an order filter on the itemset lattice. The set of itemsets fulfilling a given constraint is called *a Theory* [@mannila97]. This theory is delimited by two borders, the positive and the negative one, that are defined as follows:
\[bd\] **Negative/Positive Border** [@luccheKIS05_MAJ_06]\
When considering an anti-monotonic constraint $C_{am}$, the border corresponds to the set of itemsets whose all subsets fulfill this constraint and whose all super-sets do not fulfill. Let a set of itemsets $\mathcal{S}$$_{am}$ fulfilling an anti-monotonic constraint $C_{am}$, the border is formally defined as:
$\mathcal{B}d$<span style="font-variant:small-caps;">(</span>$\mathcal{S}$$_{am}$<span style="font-variant:small-caps;">)</span> = $\{$$X$ $|$ $\forall$ $Y$ $\subset$ $X$ : $Y$ $\in$ $\mathcal{S}$$_{am}$ and $\forall$ $Z$ $\supset$ $X$ : $Z$ $\notin$ $\mathcal{S}$$_{am}$$\}$
In the case of monotonic constraint $C_{m}$, the border corresponds to the set of patterns whose all supersets fulfills this constraint and whose all subsets do not fulfill.
Let a set of patterns $\mathcal{S}$$_{m}$ fulfilling a monotonic constraint $C_{m}$, the border is formally defined as follows:
$\mathcal{B}d$<span style="font-variant:small-caps;">(</span>$\mathcal{S}$$_{m}$<span style="font-variant:small-caps;">)</span> = $\{$$X$ $|$ $\forall$ $Y$ $\supset$ $X$ : $Y$ $\in$ $\mathcal{S}$$_{m}$ and $\forall$ $Z$ $\subset$ $X$ : $Z$ $\notin$ $\mathcal{S}$$_{m}$$\}$
However, we have to distinguish for a given constraint $C$ between positive and negative borders. Let a set of patterns $\mathcal{S}$ fulfilling a constraint $C$. The positive border is denoted by $\mathcal{B}{d}^{+}\textsc{(}\mathcal{S}$<span style="font-variant:small-caps;">)</span> and corresponds to the patterns belonging to the border $\mathcal{B}{d}\textsc{(}\mathcal{S}$<span style="font-variant:small-caps;">)</span> and fulfilling the constraint $C$. The negative border is denoted by $\mathcal{B}{d}^{-}\textsc{(}\mathcal{S}$<span style="font-variant:small-caps;">)</span> and corresponds to the set of patterns belonging to the border $\mathcal{B}{d}\textsc{(}\mathcal{S}$<span style="font-variant:small-caps;">)</span> and not fulfilling the constraint $C$. These two borders are formally expressed as follows:
$\mathcal{B}{d}^{+}\textsc{(}\mathcal{S}\textsc{)}$ = $\mathcal{B}{d}\textsc{(}\mathcal{S}$<span style="font-variant:small-caps;">)</span> $\cap$ $\mathcal{S}$,\
$\mathcal{B}{d}^{-}\textsc{(}\mathcal{S}\textsc{)}$ = $\mathcal{B}{d}\textsc{(}\mathcal{S}$<span style="font-variant:small-caps;">)</span> $\setminus$ $\mathcal{S}$.
In the next sub-section, we focus on the definition and the presentation of the notions related to condensed representations associated to a set of patterns.
### Condensed Representations of a set of Patterns
The extraction of interesting patterns may be a costly operation in execution time and in memory consumption. This is due to the high number of the generated candidates. In this regard, an interesting issue consists in extracting sets of patterns with more reduced sizes. From which it is possible to regenerate the whole sets of patterns. These reduced sets are called “‘Condensed Representations’’. In the case where the regeneration is performed in an exact way without information loss then the condensed representation is said *exact*. Otherwise, the condensed representation is said *approximative*. These representations are formally defined in what follows.
\[repConcise\]**Condensed Representations** [@mannila97]\
A concise representation of a set of interesting itemsets is a representative set allowing the characterization of the initial set in an exact or an approximative way.
Let $\mathcal{R}$ be a concise representation of a set of frequent patterns $\mathcal{E}$. $\mathcal{R}$ is said *concise exact representation*, if starting from $\mathcal{R}$, we are able to determine for a given pattern whether it is a frequent pattern or not and to determine its conjunctive support also. For example, the closed frequent patterns [@pasquier99_2005] constitute a concise exact representation of the set of frequent itemsets.
Otherwise, $\mathcal{R}$ is a *concise approximative representation* of a set of patterns $\mathcal{S}$ if it is not able to exactly determine the support values of all the itemsets belonging to the $\mathcal{S}$ set. The representation $\mathcal{R}$ returns approximate values of these supports. For example, the maximal frequent itemsets [@maxminer] constitute an approximative concise representation of the frequent patterns set. In fact, thanks to maximal frequent itemsets we are able to determine whether a given itemset is frequent or rare but it is not possible to exactly derive its conjunctive support value.
In general, a representation $\mathcal{R}$ constitutes “a perfect cover” if it fulfills the conditions established by the following definition:
\[Cover\] **Perfect Cover**\
A set $\mathcal{E}1$ is said a perfect cover of a set $\mathcal{E}$ if and only if $\mathcal{E}1$ allows to cover $\mathcal{E}$ without information loss and the size of $\mathcal{E}1$ never exceeds that of the set $\mathcal{E}$.
Various proposals aiming to reduce the size of a set of patterns $\mathcal{E}$ are based on the foundations of formal concepts analysis [@ganter99]. The next section is dedicated to the presentation of the formal concept analysis’s framework.
Formal Concepts Analysis
------------------------
### Introduction {#introduction}
The formal concept analysis initially introduced by Wille in 1982 [@wille82] treats formal concepts. A formal concept is a set of objects, *The Extension*, to which we applied a set of attributes, *The Intention*. The formal concept analysis provides a classification and an analysis tool whose principal element is the itemsets’s lattice defined as follows:
\[deftreillis\] **Itemsets’s Lattice**\
An itemsets’s lattice is a conceptual and hierarchical schema of patterns. It is also said lattice of set inclusion. In fact, the power set of $\mathcal{I}$ is ordered by set inclusion in the itemsets’ lattice.
This lattice shows the frequent itemsets, the rare ones as well as the $\mathcal{M}$in$\mathcal{RP}$ set of minimal rare patterns composing the positive border of the whole set of rare patterns.
### Galois Connection
**2.3.2.1. Closure Operator** In what follows, we present the fundamental basis of a closure operator.
\[definitionEnsembles ordonnés\] **Ordred Set**\
Let $E$ a set. A *Partial Order* over the set $E$ is a binary relation $\leq$ over the elements of $E$, such as for $x$, $y$, $z\in E$, the following properties holds [@davey02] :\
1. *Reflexivity* : $x\leq x$\
2. *Anti-symmetry* : $x\leq y$ and $y\leq x \Rightarrow x=y$\
3. *Transitivity* : $x\leq y$ and $y\leq z \Rightarrow x\leq z$
A set $E$ with a partial order $\leq$, denoted by $\textsc{(}$$E$,$\leq$<span style="font-variant:small-caps;">)</span>, is a *partially ordered set* [@davey02].
Through the following definition, we introduce the notion of closure operator.
\[ClosOp\]**Closure Operator** [@ganter99]\
Let a partially ordered set <span style="font-variant:small-caps;">(</span>$E$, $\leq$<span style="font-variant:small-caps;">)</span>. An application $f$ from <span style="font-variant:small-caps;">(</span>$E$, $\leq$<span style="font-variant:small-caps;">)</span> to <span style="font-variant:small-caps;">(</span>$E$, $\leq$<span style="font-variant:small-caps;">)</span> is a *closure operator*, if and only if $f$ fulfills the following properties. For all sub-sets $S, S'\subseteq E$ :
1\. *Isotonic* : $S\leq S' \Rightarrow f\textsc{(}S\textsc{)}
\leq f\textsc{(}S'\textsc{)}$
2\. *Extensive* : $S\leq f\textsc{(}S\textsc{)}$
3\. *Idempotency* : $f\textsc{(}f\textsc{(}S\textsc{)}\textsc{)}=f\textsc{(}S\textsc{)}$
We now define, the closure operator related to the conjunctive search space where the conjunctive support characterizes the associated patterns.\
**2.3.2.2. The Galois Connection**
\[Connexion de Galois\] **Galois Connection** [@ganter99]\
Let an extraction context $\mathcal{C}$ $=$ $\textsc{(}\mathcal{T}$, $\mathcal{I}$, $\mathcal{R}\textsc{)}$. Let $g_{c}$ the application from the power-set of $\mathcal{T}$ $^{\textsc{(}}$[^1]$^{\textsc{)}}$ to the power-set of items $\mathcal{I}$, and associate to the set of objects $T$ $\subseteq \mathcal{T}$ the set of items $i$ $\in$ $\mathcal{I}$ that are common to all the objects $t$ $\in$ $T$ :
$g_{c} : \mathcal{P}\textsc{(}\mathcal{T}\textsc{)}{}
\rightarrow \mathcal{P}\textsc{(}\mathcal{I}\textsc{)}{}$
$T\mapsto g_{c}\textsc{(}T\textsc{)}=\{i \in \mathcal{I}
| \forall\mbox{ } t\in T, \textsc{(}t,i\textsc{)}\in\mathcal{R}$ $\}$
Let $h_{c}$ the application, from the power-set of $\mathcal{I}$ to the power-set of $\mathcal{T}$, which associate to each set of items $\textsc{(}$commonly called pattern$\textsc{)}$ $I$ $\subseteq \mathcal{I}$ the set of objects $t$ $\subseteq \mathcal{T}$ containing all the items $i$ $\in$ $I$ :
$h_{c} : \mathcal{P}\textsc{(}\mathcal{I}\textsc{)}{}
\rightarrow \mathcal{P}\textsc{(}\mathcal{T}\textsc{)}{}$
$I\mapsto h_{c}\textsc{(}I\textsc{)}=\{t\in \mathcal{T}
| \forall\mbox{ } i\in I, \textsc{(}t,i\textsc{)}\in\mathcal{R}$ $\}$
The couple of applications $\textsc{(}$$g_{c}$,$h_{c}$$\textsc{)}$ is a *Galois connection* between the power-set of $\mathcal{T}$ and the power-set of $\mathcal{I}$.
The images of $\{1\}$ and of $\{1, 2\}$ by $g_{c}$ as well as those of $\{\textsl{A}, \textsl{E}\}$ and of $\{\textsl{C}, \textsl{D}\}$ by the application $h_{c}$ are :
$g_{c}$<span style="font-variant:small-caps;">(</span>$\{1\}$<span style="font-variant:small-caps;">)</span> $=$ $\{\textsl{A},\textsl{C}, \textsl{D}\}$ ; $g_{c}$<span style="font-variant:small-caps;">(</span>$\{2\}\textsc{)} =
\{\textsl{B}, \textsl{C}, \textsl{E}\}$ ; $g_{c}$<span style="font-variant:small-caps;">(</span>$\{1,
2\}\textsc{)} = \{\textsl{C}\}$.
$h_{c}\textsc{(}\{\textsl{A}, \textsl{E}\}\textsc{)} =
\{3, 5\}$ ; $h_{c}\textsc{(}\{\textsl{C},\textsl{D}\textsc{)} = \{1\}$.
[@ganter99]\
Given a Galois connection, the following properties are fulfilled: $\forall$ $I$, $I_{_{1}}$, $I_{_{2}}$ $\subseteq$ $\mathcal{I}$ and $T$, $T_{_{1}}$, $T_{_{2}}$ $\subseteq$ $\mathcal{T}$ :
1\. $I_{_{1}}$ $\subseteq $ $I_{_{2}}$ $\Rightarrow $ $h_{c}$<span style="font-variant:small-caps;">(</span>$I_{_{2}}$<span style="font-variant:small-caps;">)</span> $\subseteq $ $h_{c}$<span style="font-variant:small-caps;">(</span>$I_{_{1}}$<span style="font-variant:small-caps;">)</span>;
2\. $T_{_{1}}$ $\subseteq $ $T_{_{2}}$ $\Rightarrow $ $g_{c}$<span style="font-variant:small-caps;">(</span>$T_{_{2}}$<span style="font-variant:small-caps;">)</span> $ \subseteq $ $g_{c}$<span style="font-variant:small-caps;">(</span>$T_{_{1}}$<span style="font-variant:small-caps;">)</span>;
3\. $T$ $ \subseteq $ $h_{c}$<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\Leftrightarrow$ $I$ $ \subseteq $ $g_{c}$<span style="font-variant:small-caps;">(</span>$T$<span style="font-variant:small-caps;">)</span>.
Thanks to Definition \[Clos of Galois\], we introduce the closure operators associated to the Galois connection.
\[Clos of Galois\]**Closure Operators of the Galois Connection** [@ganter99]\
Lets consider the power-sets $\mathcal{P}\textsc{(}\mathcal{I}\textsc{)}$ and $\mathcal{P}\textsc{(}\mathcal{T}\textsc{)}$ provided with the inclusion set link $\subseteq$, *i.e*, the partially ordered sets <span style="font-variant:small-caps;">(</span>$\mathcal{P}\textsc{(}\mathcal{I}\textsc{)},$ $\subseteq$<span style="font-variant:small-caps;">)</span> and <span style="font-variant:small-caps;">(</span>$\mathcal{P}\textsc{(}\mathcal{T}\textsc{)},$ $\subseteq$<span style="font-variant:small-caps;">)</span>. The operators $f_{c}$ $^{\textsc{(}}$[^2]$^{\textsc{)}}$ and $O_{c}$ such as $f_{c} = g_{c} \circ h_{c}$ of <span style="font-variant:small-caps;">(</span>$\mathcal{P}\textsc{(}\mathcal{I}\textsc{)},$ $\subseteq$<span style="font-variant:small-caps;">)</span> in <span style="font-variant:small-caps;">(</span>$\mathcal{P}\textsc{(}\mathcal{I}\textsc{)},$ $\subseteq$<span style="font-variant:small-caps;">)</span> and $O_{c} = h_{c} \circ g_{c}$ of <span style="font-variant:small-caps;">(</span>$\mathcal{P}\textsc{(}\mathcal{T}\textsc{)},$ $\subseteq$<span style="font-variant:small-caps;">)</span> in <span style="font-variant:small-caps;">(</span>$\mathcal{P}\textsc{(}\mathcal{T}\textsc{)},$ $\subseteq$<span style="font-variant:small-caps;">)</span> are the *closure operators of the Galois connection*.
\[examplefoggof\] Let the extraction context illustrated by Table \[Base\_transactions\], we then have :\
$h_{c}\circ g_{c}$<span style="font-variant:small-caps;">(</span>$\{2\}$<span style="font-variant:small-caps;">)</span> $=$ $\{2, 3, 5\}$ ; $h_{c}\circ g_{c}$<span style="font-variant:small-caps;">(</span>$\{3\}$<span style="font-variant:small-caps;">)</span> $=$ $\{5\}$ ; $h_{c}\circ g_{c}$<span style="font-variant:small-caps;">(</span>$\{2, 3\}$<span style="font-variant:small-caps;">)</span> $=$ $\{2, 3 ,5\}$.\
$g_{c}\circ h_{c}$<span style="font-variant:small-caps;">(</span>$\{\textsl{B}\}$<span style="font-variant:small-caps;">)</span> $=$ $\{\textsl{B}, \textsl{E}\}$ ; $g_{c}\circ h_{c}$<span style="font-variant:small-caps;">(</span>$\{\textsl{D}\}$<span style="font-variant:small-caps;">)</span> $=$ $\{\textsl{D}\}$ ; $g_{c}\circ h_{c}$<span style="font-variant:small-caps;">(</span>$\{\textsl{A}, \textsl{D}\}$<span style="font-variant:small-caps;">)</span> $=$ $\{\textsl{D}\}$.
### Equivalence Classes, Closed Patterns and Minimal Generators
The application of the closure operator $\gamma$ induces an equivalence relation in the power-set $\mathcal{P}\textsc{(}\mathcal{I}$<span style="font-variant:small-caps;">)</span>, partitioning it on equivalence classes [@AyouniLYP10; @PASCAL00], denoted by $\gamma$-equivalence-class, defined as follows.
\[Cls-equiv\] **$\gamma$-Equivalence-Class**\
A $\gamma$-Equivalence-Class contains all the itemsets belonging exactly to the same transactions and sharing the same closure according to the $\gamma$ closure operator.
Within a $\gamma$-Equivalence-Class, the maximal element, according to the set inclusion, is said, “Closed Pattern” where as the minimal elements which are incomparable according to the set inclusion, are called “Minimal Generators”. They are defined in what follows.
\[DefMferme\] **Closed Pattern** [@PASCAL00]\
An itemset $I$ $\subseteq$ $\mathcal{I}$ is a closed itemset iff, $\gamma$<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span>=$I$.
\[DefGM\] **Minimal Generator** [@PASCAL00]\
An itemset $I1$ $\subseteq$ $\mathcal{I}$ is a minimal generator of a closed pattern $I$ if $\gamma$<span style="font-variant:small-caps;">(</span>$I1$<span style="font-variant:small-caps;">)</span>=$I$ and $\forall$ $I2$ $\subseteq$ $\mathcal{I}$, if $I2$ $\subseteq$ $I1$ and $\gamma$<span style="font-variant:small-caps;">(</span>$I2$<span style="font-variant:small-caps;">)</span>=$I$ then $I2$ = $I1$.
The following proposition introduces an interesting property of the minimal generators set.
[@titanic02] Let $\mathcal{GM}$ be the set of minimal generators extracted from a context $\mathcal{C}$, the $\mathcal{GM}$ set fulfills an order ideal property on the itemset lattice.
A *conjunctive equivalence class* is a set containing all the patterns having the same conjunctive closure. Thus, these patterns owns the same value of conjunctive support. The minimal generators are the smallest elements, according to the set inclusion property, in their equivalence classes. Whereas, the largest element in this class corresponds to the closed pattern. An example of a conjunctive equivalence class is given by Figure \[ClSeQuivConj\]. In this class, *ABCE* is the closed pattern whereas *AB* and *AE* are the associated minimal generators. All the elements belonging to this class share exactly the same conjunctive support, equal to 2.
At this level, we have presented the basic notions related to itemset’s extraction and to condensed representations.
Conclusion {#se4}
----------
Different approaches, derived from Formal Concept Analysis <span style="font-variant:small-caps;">(</span>FCA<span style="font-variant:small-caps;">)</span>, were proposed in order to reduce the size of the set of frequent itemsets. In addition, correlated pattern mining constitutes an interesting alternative to get more informative patterns with a manageable size and a high quality returned knowledge.\
The next chapter will be dedicated to the presentation, going from the general to the more specific, of the state of the art approaches related to correlated patterns mining. A Comparative study of these approaches will be also conducted.
Correlated Patterns Mining: Review of the Literature {#ch3}
====================================================
Introduction {#introChap3}
------------
In this chapter, we focus on presenting an overview of the literature approaches, which are related to our topic of mining correlated patterns. Our study goes from general to more specific. In this respect, we present in Section \[CsDM\] the approaches related to constraint-based data mining, we deal with the two kinds of constraints. Then, in Section \[CP-DM\], we specially concentrate on correlated pattern mining. We start by introducing the most common correlation measures, then we join with the state of the art of rare correlated patterns mining followed by frequent correlated patterns mining approaches. A synthetic summary of the studied approaches is proposed in Section \[seDisc\]. The chapter is concluded in Section \[ConcChap3\].
Constraint-based Itemset Mining {#CsDM}
-------------------------------
Within a process of pattern extraction, it is more difficult to localize the set of patterns fulfilling a set of constraints of different natures than to extract theories associated to a conjunction of constraints of the same nature [@luccheKIS05_MAJ_06]. Indeed, the opposite nature of the constraints makes that the reduction strategies are applicable to only a part of the constraints and not to all the constraints. Therefore, the extraction process will be more complicated and more expensive in terms of execution costs and memory greediness.
Many approaches have paid attention to the extraction of interesting patterns under constraints [@boulicautsurvey_contraintes]. One of the first algorithms belonging to this context is <span style="font-variant:small-caps;">DualMiner</span> [@Bucila03]. The latter allows the reduction of the search space while considering both of the monotonic and the anti-monotonic constraints. However, as highlighted by [@boley2009], <span style="font-variant:small-caps;">DualMiner</span> suffers from a main drawback related to the high cost of constraints evaluation.
In [@lee_constraint_06], the authors have proposed an approach of pattern extraction under constraints. The <span style="font-variant:small-caps;">ExAMiner</span> algorithm [@examiner_kais05] was also proposed in order to mine frequent patterns under monotonic constraints. It is important to mention that the effective reduction strategy adopted by <span style="font-variant:small-caps;">ExAMiner</span> could not be of use in the case of the monotonic constraint of rarity that we treat in this work, since this latter is sensitive to the changes in the transactions of the extraction context.
Many other works have also emerged. We cite for example, the <span style="font-variant:small-caps;">VST</span> algorithm [@vst] which allows the extraction of all the strings satisfying the set of monotonic and anti-monotonic constraints. Later, the <span style="font-variant:small-caps;">FAVST</span> algorithm [@favst] was introduced in order to improve the performance of the <span style="font-variant:small-caps;">VST</span> algorithm by reducing the number of scans of the database. Other approaches, belonging to this framework, have also been proposed such as the <span style="font-variant:small-caps;">DPC-COFI</span> algorithm and the <span style="font-variant:small-caps;">BifoldLeap</span> algorithm [@bifold]. The strategy of these approaches consists in extracting the maximal frequent itemsets which fulfill all of the constraints and from which the set of all the frequent valid itemsets will be derived. In [@miningzinc-2013], the authors proposed the <span style="font-variant:small-caps;">MiningZinc</span> framework dedicated to constraint programming for itemset mining. The constraints are defined, within the <span style="font-variant:small-caps;">MiningZinc</span> system, in a declarative way close to mathematical notations. The solved tasks within the proposed system concerns closed frequent itemset mining, cost-based itemset mining, high utility itemset mining and discriminative patterns mining. In a more generic way, in [@Guns2016], the author presented a generic overview of methods devoted to bridge the gap between the two fields of constraint-based itemset mining and constraint programming.
Correlated Pattern Mining {#CP-DM}
-------------------------
This section is dedicated to the study of the correlated pattern mining. First, we start by introducing the commonly used correlation measures, presenting their properties and comparing them.
### Correlation Measures {#MesCorr}
The integration of the correlation measures within the mining process allows to reduce the number of the extracted patterns while improving the quality of the retrieved knowledge. The quality is expressed by the degree of correlation between the items composing the result itemsets. To achieve this goal, different correlation measures were proposed in the literature, we start with the *bond* measure.
#### 3.3.1.1 The *bond* measure
The *bond* measure [@Omie03] is mathematically equivalent to *Coherence* [@comine_Lee], *Tanimoto-coefficient* [@Tanimoto1958], and *Jaccard*. In [@tarekds2010], the authors propose a new expression of *bond* in Definition \[La mesure bond\].
\[La mesure bond\] **The *bond* measure**\
The *bond* measure of a non-empty pattern $I$ $\subseteq$ $\mathcal{I}$ is defined as follows:
$\textit{bond}\textsc{(}\textit{I}\textsc{)} = \frac{\displaystyle
\textit{Supp}\textsc{(}\wedge\textit{I}\textsc{)}}{\displaystyle
\textit{Supp}\textsc{(}\vee\textit{I}\textsc{)}}$
This measure conveys the information about the correlation of a pattern $I$ by computing the ratio between the number of co-occurrences of its items and the cardinality of its universe, which is equal to the transaction set containing a non-empty subset of $I$. It is worth mentioning that, in the previous works dedicated to this measure, the disjunctive support has never been used to express it.
The use of the disjunctive support allows to reformulate the expression of the *bond* measure in order to bring out some pruning conditions for the extraction of the patterns fulfilling this measure. Indeed, as shown later, the *bond* measure fulfills several properties that offer interesting pruning strategies allowing to reduce the number of generated pattern during the extraction process. Note that the value of the *bond* measure of the empty set is undefined since its disjunctive support is equal to $0$. However, this value is positive since $\lim_{ I\mapsto\emptyset}$ *bond* <span style="font-variant:small-caps;">(</span>*I*<span style="font-variant:small-caps;"><span style="font-variant:small-caps;">)</span></span> = $\frac{\displaystyle |\mathcal{T}|}{\displaystyle 0}$ = $+\infty$. As a result, the empty set will be considered as a correlated pattern for any minimal threshold of the *bond* correlation measure.
It has been proved, in [@tarekds2010], that the *bond* measure fulfills other interesting properties. In fact, *bond* is: <span style="font-variant:small-caps;">(</span>$i$<span style="font-variant:small-caps;">)</span> *Symmetric* since we have $\forall$ $I$, $J$ $\subseteq$ $\mathcal{I}$, *bond*<span style="font-variant:small-caps;">(</span>$IJ$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$JI$<span style="font-variant:small-caps;">)</span>; <span style="font-variant:small-caps;">(</span>$ii$<span style="font-variant:small-caps;">)</span> *descriptive* *i.e.* is not influenced by the variation of the number of the transactions of the extraction context.
In addition, it has been shown in [@HanDMKD2010] that it is desirable to select a descriptive measure which is not influenced by the number of transactions that contain none of pattern items. The symmetric property fulfilled by the *bond* measure makes it possible not to treat all the combinations induced by the precedence order of items within a given pattern. Noteworthily, the anti-monotony property, fulfilled by the *bond* measure as proven in [@Omie03], is of interest. Indeed, all the subsets of a correlated pattern are also necessarily correlated. Then, we can deduce that any pattern having at least one uncorrelated proper subset is necessarily uncorrelated. It will thus be pruned without computing the value of its *bond* measure. In the next definition, we introduce the relationship between the *bond* measure and the cross-support property.
**Cross-support property of the *bond* measure** [@Xiong06hypercliquepattern]\
Thanks to the cross-support property, having a minimal threshold *minbond* and an itemset $I$ $\subseteq$ $\mathcal{I}$, if $\exists$ $x$ and $y$ $\in$ $I$ such as $\frac{\displaystyle\textit{Supp}\textsc{(}\wedge
x\textsc{)}}{\displaystyle\textit{Supp}\textsc{(}\wedge y\textsc{)}}
< \textit{minbond}$ then $I$ is not correlated since *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> < *minbond*;
We continue, in what follows, with the presentation of the *all-confidence* measure.
#### 3.3.1.2 The *all-confidence* measure
The *all-confidence* measure [@Omie03] is defined as follows:
**The *all-confidence* measure**\
The *all-confidence* measure [@Omie03] is defined for any non-empty set $I$ $\subseteq$ $\mathcal{I}$ as follows:
*all-conf*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{\displaystyle\textit{Supp}\textsc{(}\wedge
I\textsc{)}}{\displaystyle \textit{max} \{
\textit{Supp}\textsc{(}\wedge i\textsc{)} | i\in I \} }$
*All-confidence* conserves the anti-monotonic property [@Omie03] as well as the cross-support property [@Xiong06hypercliquepattern].
Let us consider the extraction context given by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>*cf.* page <span style="font-variant:small-caps;">)</span> . For a minimal threshold of *all-confidence* equal to *0.4*. We have *all-confidence*<span style="font-variant:small-caps;">(</span>`ABCE`<span style="font-variant:small-caps;">)</span> = $$\displaystyle\frac{\displaystyle
\textit{Supp}\textsc{(}\wedge \texttt{ABCE}\textsc{)}}{\displaystyle
\textit{max} \{ \textit{Supp}\textsc{(}\wedge \texttt{A}\textsc{)},
\textit{Supp}\textsc{(}\wedge \texttt{B}\textsc{)},
\textit{Supp}\textsc{(}\wedge \texttt{C}\textsc{)},
\textit{Supp}\textsc{(}\wedge \texttt{E}\textsc{)}\}}$$ = $$\displaystyle\frac{\displaystyle 2}
{\displaystyle \textit{max}\{3, 4\}}$$ = *0.50*. The *ABCE* itemset is correlated according to the *all-confidence* measure. All the direct subsets of *ABCE* are also correlated. We have *all-confidence*<span style="font-variant:small-caps;">(</span>`ABE`<span style="font-variant:small-caps;">)</span> = *all-confidence*<span style="font-variant:small-caps;">(</span>`ACE`<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{\displaystyle 2}
{\displaystyle 4}$ = *0.50*, *all-confidence*<span style="font-variant:small-caps;">(</span>`BCE`<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{\displaystyle 3}
{\displaystyle 4}$ = *0.75*.
For the itemset `AD`, we have $\displaystyle\frac{\displaystyle
\textit{Supp}\textsc{(}\wedge\textsl{D}\textsc{)}}{\displaystyle
\textit{Supp}\textsc{(}\wedge\textsl{A}\textsc{)}}$ $=$ $\displaystyle\frac{\displaystyle 1}
{\displaystyle 3}$ = *0.33* $<$ *0.4* and we have *all-confidence*<span style="font-variant:small-caps;">(</span>`AD`<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{\displaystyle 1}
{\displaystyle 3}$ = *0.33*. The `AD` itemset does not fulfill the cross-support property, thus it is a non-correlated itemset. This example illustrates the conservation of the anti-monotonicity and the cross-support properties of the *all-confidence* measure.
We continue in what follows with the [hyper-confidence]{} measure.
#### 3.3.1.3 The *hyper-confidence* measure
The *hyper-confidence* measure denoted by *h-conf* of an itemset $I$ $\subseteq$ $\mathcal{I}$ is defined as follows.
**The *hyper-confidence* measure**\
The *hyper-confidence* measure of an itemset $I$ $=$ $\{$*$i_{1}$*, *$i_{2}$*, $\ldots$, *$i_{m}$*$\}$ is equal to:
*h-conf*<span style="font-variant:small-caps;">(</span>$X$<span style="font-variant:small-caps;">)</span>=min$\{$*Conf*<span style="font-variant:small-caps;">(</span> $i_{1}$ $\Rightarrow$ $i_{2}$, $i_{3}$, $\ldots$, $i_{m}$ <span style="font-variant:small-caps;">)</span>, $\ldots$, *Conf*<span style="font-variant:small-caps;">(</span>$i_{m}$ $\Rightarrow$ $i_{1}$, $i_{2}$, $\ldots$, $i_{m-1}$ <span style="font-variant:small-caps;">)</span>$\}$,
where *Conf* stands for the *Confidence* measure associated to association rules.
The *hyper-confidence* measure is equivalent to the *all-confidence* measure, it thus fulfills the anti-monotonicity and the cross-support properties.
We continue in what follows with the *any-confidence* measure.
#### 3.3.1.4 The *any-confidence* measure
This measure is defined, for any non empty set $I$ $\subseteq$ $\mathcal{I}$ as follows:
**The *any-confidence* measure**\
*any-conf*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = $\displaystyle
\frac{\displaystyle
\textit{Supp}\textsc{(}\wedge I\textsc{)}}{\displaystyle
\textit{min} \{ \textit{Supp}\textsc{(}\wedge i\textsc{)} | i\in I \}}$
The *any-confidence* measure [@Omie03] does not preserve nor the anti-monotonicity neither the cross-support properties.
Let us consider the extraction context given by Table \[Base\_transactions\]. For a minimal correlation threshold equal to *0.80*. The *any-confidence* value of *AB* is equal to, *any-confidence*<span style="font-variant:small-caps;">(</span>`AB`<span style="font-variant:small-caps;">)</span> = $\displaystyle
\frac{\displaystyle
\textit{Supp}\textsc{(}\wedge\texttt{AB}\textsc{)}}{\displaystyle
\textit{min}\{\textit{Supp}\textsc{(} \wedge \texttt{A}\textsc{)},
\textit{Supp}\textsc{(}\wedge \texttt{B}\textsc{)} \}}$ = $\displaystyle
\frac{\displaystyle 2}
{\displaystyle \textit{min}\{3, 4\}}$ = *0.66*. *AB* do not fulfill the minimal threshold of correlation, thus it is a non-correlated itemset according to the *any-confidence* measure. Whereas, the *AD* itemset is correlated and its correlation value is equal to 1. We also have, $\displaystyle
\frac{\displaystyle
\textit{Supp}\textsc{(}\wedge\textsl{A}\textsc{)}}{\displaystyle
\textit{Supp}\textsc{(}\wedge\textsl{C}\textsc{)}}$ $=$ $\displaystyle
\frac{\displaystyle 3}
{\displaystyle 4}$ = *0.75* $<$ *0.80*, however, *any-confidence*<span style="font-variant:small-caps;">(</span>*AD*<span style="font-variant:small-caps;">)</span> $=$ *1* $>$ *0.80*. This example illustrates the non preservation of the anti-monotonicity as well as the cross-support properties.
We present in what follows the *$\chi^2$* Coefficient.
#### 3.3.1.5 The *$\chi^2$* Coefficient
The *$\chi^2$* coefficient is defined as follows :
**The *$\chi^2$* Coefficient** [@Brin97]\
The *$\chi^2$* coefficient of an itemset $Z$ $=$ $xy$, with $x$ and $y$ $\in$ $\mathcal{I}$, is defined as follows:
*$\chi^2$*<span style="font-variant:small-caps;">(</span>$Z$<span style="font-variant:small-caps;">)</span> $=$ $|\mathcal{T}|$ $\times$ $\frac{\displaystyle
\textsc{(}\textit{Supp}\textsc{(}\wedge xy\textsc{)} -
\textit{Supp}\textsc{(}\wedge
x\textsc{)}\times\textit{Supp}\textsc{(}\wedge
y\textsc{)}\textsc{)}^{2}}{\displaystyle\textit{Supp}\textsc{(}\wedge
x\textsc{)}\times\textit{Supp}\textsc{(}\wedge
y\textsc{)}\times\textsc{(}1 - \textit{Supp}\textsc{(}\wedge
x\textsc{)}\textsc{)}\times\textsc{(}1 -
\textit{Supp}\textsc{(}\wedge y\textsc{)}\textsc{)}}$
Some relevant properties of the $\chi^2$ coefficient are given by the following proposition.
The $\chi^2$ coefficient is a statistic and symmetric measure [@Brin97].
Other correlation measures are also of use in the literature, we mention for example the *cosine* measure, the *lift* measure [@Brin97], the $\phi$ coefficient also named the Pearson coefficient [@XiongKDD2004].
#### 3.3.1.6 Synthesis
We recapitulate the different properties of the presented measures in Table \[tab\_recapitulatif\_prop\]. The “$\checkmark$” symbol indicates that the measure fulfills the property.
In our previous study, we specifically focused on correlation measures which are most used in correlated patterns mining. Withal, the *cosine* and the *kulczynski* measures were not studied since these two measures are rarely used on correlated patterns mining due to the non conservation of the anti-monotonicity property [@HanDMKD2010]. The *lift* measure is used within the association rule evaluation.
We conclude, according to this overview, that the most interesting measures are *bond* and *all-confidence*. This is justified by the fact that these two measures fulfilled the pertinent properties of anti-monotonicity and cross-support.
We present, in what follows, the state of the art approaches dealing with correlated patterns mining. We precisely start with rare correlated pattern mining.
### Rare Correlated Patterns Mining {#EdeAMCR}
Various approaches devoted to the extraction of correlated patterns under constraints have been proposed. However, the recuperation of all the patterns that are both highly correlated and infrequent is based on the naive idea to extract the set of all frequent patterns for a very low threshold *minsupp* and then to filter out these patterns by a measure of correlation.
Another idea is to extract the whole set of the correlated patterns without any integration of the rarity constraint. The obtained set contains obviously all the frequent correlated as well as the rare correlated patterns. It is relevant to note that the application of these two ideas is very expensive in execution time and in memory consumption due to the explosion of the number of candidates to be evaluated.
The approach proposed in [@Cohen_mcr_2000] is based on the previous principle. This approach allows to extract the items’s pairs correlated according to the *Similarity* measure but without computing their support. In fact, the *Similarity* measure allows to evaluate the similarity between two items and corresponds to the quotient of the number of the simultaneous appearance divided by the number of the complementary appearance. Consequently, the *Similarity* measure is semantically equivalent to the *bond* measure. However, any analysis of this measure have been conducted.\
In fact, this approach proposes to assign to each item a signature composed by the identifier list of the transactions to which the item belongs. Then, the *Similarity* is computed and it corresponds to the number of the intersections of their signatures divided by the union of their signatures. We conclude that the frequency constraint was not integrated in order to recuperate the highly correlated itemsets with a weak support. From these patterns, the association rules with a high confidence and a weak support are generated.
In this same context, we mention the <span style="font-variant:small-caps;">DiscoverMPatterns</span> algorithm [@ma-icdm2001]. In fact, this latter is devoted to the extraction of the correlated patterns based on the *all-confidence* measure. Nevertheless, a first version of the approach was dedicated to the extraction of all the correlated patterns without any restriction of the support value in order to specifically get the rare correlated itemsets. Then, within the second version of the approach, the minimum support threshold constraint was integrated. Consequently, this constraint integration allows to extract the frequent correlated patterns.
Another principle of the resolution of the rare correlated patterns extraction consists in extracting all the frequent patterns for a very weak minimal support threshold. Evidently, the obtained set contains a subset of the infrequent correlated patterns. Xiong et al. relied on this idea to introduce the <span style="font-variant:small-caps;">Hyper-CliqueMiner</span> algorithm [@Xiong06hypercliquepattern]. The output of this algorithm is the set of frequent correlated patterns for a very low *minsupp* value. It is to note, that the good performances of this algorithm are justified by the use of the anti-monotonic property of the correlation measure as well as the *cross-support* property which allows to reduce significantly the evaluated candidates and thus to reduce the time needed.
The approach proposed in [@thomo_2010] stands also within this principle. This approach allows to extract the frequent and frequently correlated 2-itemsets. It is judged as a naive approach that is based on the extraction of all the solution set for a very low *minsupp* values. Then, a post processing is performed in order to maintain only the high correlated itemsets. The <span style="font-variant:small-caps;">FT-Miner</span> algorithm [@ptminer] outputs the correlated infrequent itemsets according to the *N-Confidence* semantically equivalent to *all-Confidence*. The *all-Confidence* measure was also treated in the <span style="font-variant:small-caps;">Partition</span> algorithm [@Omie03], which allows to extract the correlated patterns according to both *all-Confidence* and *bond* measures. The choice of the measure to be considered depends on the user’s input preferences.
The approach proposed in [@MCROkubo] also belongs to the same trend of approaches dealing with correlated infrequent itemsets. Indeed, it is based on the principle that the patterns which are weakly correlated according to the *bond* correlation measure are generally rare in the extraction context. The expressed constraint corresponds to a restriction of the maximum correlation value. This is a monotonic constraint since it corresponds to the opposite of the anti-monotonic constraint of minimal correlation. In order to get rid from rare patterns that represent exceptions, and they are not informative, a minimal frequency constraint was also integrated. The idea consists then in extracting the top$-N$ rare patterns which are the most informative ones.
The problem of integrating constraints during the process of correlated pattern mining was also studied in the works, respectively, proposed in [@Brin97] and in [@grahne_correlated_2000]. These approaches deal with constrained correlated pattern mining, they rely on the $\chi^2$ correlation coefficient. They exploit the various pruning opportunities offered by these constraints and benefit from the selective power of each type of constraints. However, the coefficient $\chi^2$ does not fulfill the anti-monotonic constraint as does the *bond* measure. Besides, these approaches are limited to the extraction of a small subset which is composed only by minimal valid patterns *i.e.* the minimal patterns which fulfill all of the imposed constraints. Furthermore, the authors do not propose any concise representation of the extracted correlated patterns.
Also, in [@surana2010], a study of different properties of interesting measures was conducted in order to suggest a set of the most adequate properties to consider while mining rare associations rules.
It is deduced that for all these approaches, the monotonic constraint of rarity was never included within the mining process in order to retrieve all the rare highly correlated patterns.
### Frequent Correlated Patterns Mining {#EdeAMCF}
In [@comine_Lee], the authors proposed the <span style="font-variant:small-caps;">CoMine</span> approach which is dedicated to the extraction of frequent correlated patterns according to the *all-confidence* and to the *bond* measures. We distinguish two different versions of the <span style="font-variant:small-caps;">CoMine</span> approach. The first version treats the *bond* measure while the second treats the *all-Confidence* measure. <span style="font-variant:small-caps;">CoMine</span> also constitute the core of the <span style="font-variant:small-caps;">I-IsCoMine-AP</span> and <span style="font-variant:small-caps;">I-IsCoMine-CT</span> algorithms [@IsComine2011].
Also, the *bond* measure was studied in [@LeBras2011], the authors proposed an apriori-like algorithm for mining classification rules. Moreover, the authors in [@borgelt] proposed a generic approach for frequent correlated pattern mining. Indeed, the *bond* correlation measure and eleven other correlation measures were used. All of them fulfill the anti-monotonicity property. Correlated patterns mining was then shown to be more complex and more informative than frequent pattern mining [@borgelt].
Many other works have also emerged. In [@HanDMKD2010], the authors provide a unified definition of existing null-invariant correlation measures and propose the <span style="font-variant:small-caps;">GAMiner</span> approach allowing the extraction of frequent high correlated patterns according to the *Cosine* and to the *Kulczynsky* measures. In this same context, the <span style="font-variant:small-caps;">NICOMiner</span> algorithm was also proposed in [@Kimpkdd2011] and it allows the extraction of correlated patterns according to the *Cosine* measure. We highlight that the *Cosine* measure has the specificity of being not monotonic neither anti-monotonic.
In this same context, we also cite the <span style="font-variant:small-caps;">Atheris</span> approach [@skypattern2011] which allows the extraction of condensed representation of correlated patterns according to user’s preferences. In [@FlipPattern2012], the authors introduced the concept of flipping correlation patterns according to the *Kulczynsky* measure. However, the *Kulczynsky* measure does not fulfill the interesting anti-monotonic property as the *bond* measure.
The *all-confidence* measure was handled within the work proposed in [@Karim-etal-2012]. The approach outputs the correlated patterns <span style="font-variant:small-caps;">(</span>also called the associated patterns<span style="font-variant:small-caps;">)</span>, the non correlated patterns <span style="font-variant:small-caps;">(</span>also called the independent patterns<span style="font-variant:small-caps;">)</span>. Also, in [@pakdd2013] the authors propose a method to extract *all-confidence* frequent correlated patterns and they also discuss the impact of fixing the *minsupp* threshold value over the quality of the obtained itemsets and propose to fix a minimal correlation threshold for each item.
In the next subsection, we study the approaches of extracting the condensed representations of frequent correlated patterns.
### Condensed Representations of Correlated Patterns Mining
The problem of mining concise representations of correlated patterns was not widely studied in the literature. We mention the <span style="font-variant:small-caps;">Ccmine</span> [@ccmine_Kim] approach of mining closed correlated patterns according to the *all-confidence* measure which constitute a condensed representation of frequent correlated patterns. We also precise that the authors in [@tarekds2010] proposed the <span style="font-variant:small-caps;">CCPR-Miner</span> algorithm allowing the extraction of closed frequent correlated patterns according to the *bond* measure.
In this context, we also cite the $\textsc{Jim}$ approach [@borgelt]. In fact, $\textsc{Jim}$ allows to extract the closed correlated frequent patterns which constitute a perfect cover of the whole set of frequent correlated patterns. The choice of the considered correlation measure is fixed by the user’s parameters within the $\textsc{Jim}$ approach.
In fact, the $\textsc{Jim}$ approach is, on the one hand the most efficient state of the art approach extracting condensed representation of frequent correlated patterns according to the *bond* measure. On the other hand, $\textsc{Jim}$ is the unique approach which dealt with the same kind of patterns as we treat in our mining approach, that we present in the following chapters. In this sense, in our experimental study, we will focus on comparing our mining approach by the $\textsc{Jim}$ approach.
Discussion {#seDisc}
----------
Based on the previous review of the literature, we conclude that most of the approaches dealt with the *bond* and the *all-confidence* measures. These latter fulfill the interesting anti-monotonic property, that allows to reduce the search space by early pruning irrelevant candidates. Therefore, the frequent correlated set of patterns results from the conjunction of both constraints of the same type: the correlation and the frequency.
In fact, the recuperation of all the patterns that are both highly correlated and infrequent is based on the naive idea to extract the set of all frequent patterns for a very low threshold *minsupp* and then to filter out these patterns by a measure of correlation. Another resolution strategy consists in extracting the whole set of the correlated patterns without any integration of the rarity constraint. Then, a post-processing is performed in order to uniquely retrieve the rare correlated itemsets.
In other words, the monotonic constraint of rarity was never integrated within the mining process and thus the exploration of the search space of candidates that does not fulfill the rarity constraint is obviously barren. In addition, another problem is related to the high consuming of the memory and the CPU resources due to the combinatorial explosion of the number of candidates depending on the size of the mined dataset. We highlight, that <span style="font-variant:small-caps;">Jim</span> [@borgelt] is the unique approach that dealt with different anti-monotonic correlation measures. However, <span style="font-variant:small-caps;">Jim</span> is limited to frequent correlated patterns and do not consider the rare correlated ones.
Table \[TabEdeA\] recapitulates the characteristics of the different visited approaches. This table summarizes the following properties:
1. **The correlation measure:** This property describes the considered correlation measure.
2. **The kind of the extracted patterns:** This property describes the kind of patterns outputted by the mining algorithm
3. **The nature of constraints:** This property describes the nature of the constraints included within the algorithm: anti-monotonic or monotonic.
To the best of our knowledge, no previous work was dedicated to the extraction of concise representations of patterns under the conjunction of constraints of distinct types. This problem is then a challenging task in data mining, which strengthens our motivation for the treatment of this problematic. Therefore, the work proposed in this thesis is the first one that puts the focus on mining concise representations of both frequent and rare correlated patterns according to the anti-monotonic *bond* measure.
Conclusion {#ConcChap3}
----------
In this chapter, we proposed an overview of the state of the art approaches dealing with correlated patterns mining preceded by a presentation of the different correlation measures. We deduced that, there is no previous work that dealt with both frequent correlated as well as rare correlated patterns according to a specified correlation metric. Thus, motivated by this issue, we propose in this thesis to benefit from the knowledge returned from both frequent correlated as well as rare correlated patterns according to the *bond* measure. To tackle this challenging task, we propose in the next chapter the characterization of both frequent correlated patterns, rare correlated patterns and their associated concise representations.
Condensed Representations of Correlated Patterns {#ch_4}
================================================
Introduction {#introChap4}
------------
The main moan that can be related to frequent pattern mining approaches stands in the fact that the latter do not offer the information concerning the correlation degree among the items in the extraction context. This stands behind our motivation to provide to the user the key information about the correlation between items as well as the frequency of their occurrence. This aim is reachable thanks to the integration of the correlation measures within the mining process.
The correlation measure, that we treat throughout this thesis, is *bond*. Our motivations behind the choice of this measure is explicitly described in Section \[S2chap4\]. In Section \[sebd\], we focus on the correlated patterns associated to the *bond* measure, we characterize this set of patterns. Section \[sec\_FBond\] is devoted to the presentation of the closure operator associated to *bond*. We introduce the associated exact condensed representations in Section \[section\_RC\] and in Section \[section\_RCfcp\]. Section \[ConcChap4\] concludes the chapter.
Motivations behind our choice of the *bond* measure {#S2chap4}
---------------------------------------------------
Based on the study of the state of the art approaches proposed in the previous chapter, we find that the almost of the existing approaches are dealing with the *bond* and the *all-confidence* measures. The *bond* measure fulfills the anti-monotony property which is an interesting property. Indeed, the latter reduce the search space when pruning the non potential candidates, therefore optimizing the extraction time as well as the memory consumption. It has been proved in the literature that the *bond* measure presents many interesting properties. In fact, the *bond* measure is:
1. *symmetric* since $\forall$ $I$, $J$ $\subseteq$ $\mathcal{I}$, *bond*<span style="font-variant:small-caps;">(</span>$IJ$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$JI$<span style="font-variant:small-caps;">)</span>;
2. *descriptive* since it is not influenced by the number of transactions that contain none of the items composing the pattern;
3. fulfills the *cross-support* property [@Xiong06hypercliquepattern]. Thanks to this property, given a minimal threshold *minbond* and an itemset $I$ $\subseteq$ $\mathcal{I}$, if $\exists$ $x$ and $y$ $\in$ $I$ such as $\frac{\displaystyle\textit{Supp}\textsc{(}\wedge
x\textsc{)}}{\displaystyle\textit{Supp}\textsc{(}\wedge y\textsc{)}}
< \textit{minbond}$ then $I$ is not correlated since *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $<$ *minbond*;
4. induces an *anti-monotonic* constraint for a fixed minimal threshold *minbond*. In fact, $\forall$ $I$, $I_1$ $\subseteq$ $\mathcal{I}$, if $I_1$ $\subseteq$ $I$, then *bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span> $\geq$ *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span>. Therefore, the set $\mathcal{CP}$ of correlated patterns forms an order ideal. Indeed, all the subsets of a correlated pattern are necessarily correlated ones.
We present in the following an interesting relation between the value of the *bond* measure and the conjunctive and disjunctive supports values for each couple of two patterns $I$ and $I_1$ such as $I$ $\subseteq$ $I_1$ [@tarekds2010].
\[PropBond\] Let $I$, $I_1$ $\subseteq$ $\mathcal{I}$ and $I$ $\subseteq$ $I_1$. If *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span>, then *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I_1$<span style="font-variant:small-caps;">)</span> and *Supp*<span style="font-variant:small-caps;">(</span>$\vee I$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\vee I_1$<span style="font-variant:small-caps;">)</span>.
According to the previous proposal, if *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span>, then *Supp*<span style="font-variant:small-caps;">(</span>$\neg I$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\neg I_1$<span style="font-variant:small-caps;">)</span>. In fact, both $I$ and $I_1$ have the same conjunctive support and, according to the Morgan law, we build the following relation between the disjunctive and the negative supports of a pattern: *Supp*<span style="font-variant:small-caps;">(</span>$\neg I$<span style="font-variant:small-caps;">)</span> = $|\mathcal{T}|$ - *Supp*<span style="font-variant:small-caps;">(</span>$\vee I$<span style="font-variant:small-caps;">)</span>. On the other hand, if *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\neq$ *bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span>, then *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> $\neq$ *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I_1$<span style="font-variant:small-caps;">)</span> or *Supp*<span style="font-variant:small-caps;">(</span>$\vee I$<span style="font-variant:small-caps;">)</span> $\neq$ *Supp*<span style="font-variant:small-caps;">(</span>$\vee I_1$<span style="font-variant:small-caps;">)</span> <span style="font-variant:small-caps;">(</span>*i.e.* one of the two supports is different or both<span style="font-variant:small-caps;">)</span>.\
In this context, we propose to study the *bond* correlation measure in an integrated mining process aiming to extract both frequent and rare correlated patterns as well as their associated condensed representations. In this regard, we present in the next section the specification of the frequent correlated patterns as well as the rare correlated patterns according to the *bond* measure.
Characterization of the Correlated patterns according to the *bond* measure {#sebd}
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### Definitions and Properties
The *bond* measure [@Omie03] is mathematically equivalent to *Coherence* [@comine_Lee], *Tanimoto coefficient* [@Tanimoto1958], and *Jaccard* [@jaccard_1901]. It was redefined in [@tarekds2010] as follows:
\[Defbond\] **The *bond* measure**\
The *bond* measure of a non-empty pattern $I \subseteq \mathcal{I}$ is defined as follows:
$\textit{bond}\textsc{(}\textit{I}\textsc{)} = \frac{\displaystyle
\textit{Supp}\textsc{(}\wedge\textit{I}\textsc{)}}{\displaystyle
\textit{Supp}\textsc{(}\vee\textit{I}\textsc{)}}$
The *bond* measure takes its values within the interval $[0,1]$. While considering the universe of a pattern $\mathcal{I}$ [@comine_Lee], *i.e.*, the set of transactions containing a non empty subset of $I$, the *bond* measure represents the simultaneous occurrence rate of the items of the pattern $I$ in its universe. Thus, the higher the items of the pattern are dispersed in its universe, <span style="font-variant:small-caps;">(</span>*i.e.* weakly correlated<span style="font-variant:small-caps;">)</span>, the lower the value of the *bond* measure is, as *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> is smaller than *Supp*<span style="font-variant:small-caps;">(</span>$\vee I$<span style="font-variant:small-caps;">)</span>. Inversely, the more the items of $I$ are dependent from each other, <span style="font-variant:small-caps;">(</span>*i.e.* strongly correlated<span style="font-variant:small-caps;">)</span>, the higher the value of the *bond* measure is, since *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> would be closer to *Supp*<span style="font-variant:small-caps;">(</span>$\vee I$<span style="font-variant:small-caps;">)</span>.
The set of correlated patterns associated to the *bond* measure is defined as follows.
\[defCP\] **Correlated patterns**\
Considering a correlation threshold *minbond*, the set of correlated patterns, denoted $\mathcal{CP}$, is equal to: $\mathcal{CP}$ = $\{I$ $\subseteq \mathcal{I}|$ *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\geq$ *minbond*$\}$.
Let us consider the dataset given by Table \[Base\_transactions\]. For *minbond* = 0.5, we have *bond*<span style="font-variant:small-caps;">(</span>$AB$<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{2}{5}$ = 0.4 $<$ 0.5. The itemset $AB$ is then not a correlated one. Whereas, since *bond*<span style="font-variant:small-caps;">(</span>$BCE$<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{3}{5}$ = 0.6 $\geq$ 0.5, the itemset $BCE$ is a correlated one.
In the following, we define the set composed by the maximal correlated patterns as follows:
\[bdpos\] **Maximal correlated patterns**\
The set of maximal correlated patterns constitutes the positive border of correlated patterns and is composed by correlated patterns having no correlated proper superset. This set is defined as: $\mathcal{M}ax$$\mathcal{CP}$ = $\{$$I$ $\in$ $\mathcal{CP}$$\mid$ $\forall$ $I_1$ $\supset$ $I$: $I_1$ $\notin$ $\mathcal{CP}$$\}$, or equivalently: $\mathcal{M}ax$$\mathcal{CP}$ = $\{$$I$ $\in$ $\mathcal{CP}$$\mid$ $\forall$ $I_1$ $\supset$ $I$: *bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span> $<$ *minbond*$\}$.
\[example\_MCMax\] Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(Page )</span>. For *minbond* = 0.2, we have $\mathcal{M}ax$$\mathcal{CP}$ = $\{$$ACD$, $ABCE$$\}$.
As far as we integrate the frequency constraint with the correlation constraint, we can distinguish between two sets of correlated patterns, which are the “Frequent correlated patterns” set and the “Rare correlated patterns” set. These two distinct sets will be characterized separately in the remainder.
### Frequent Correlated Patterns
\[defCFP\] **The set of frequent correlated patterns**\
Considering the support threshold *minsupp* and the correlation threshold *minbond*, the set of frequent correlated patterns, denoted $\mathcal{FCP}$, is equal to: $\mathcal{FCP}$ = $\{$$I$ $\subseteq$ $\mathcal{I}$ $|$ *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> $\geq$ *minsupp* and *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\geq$ *minbond*$\}$.
In fact, the $\mathcal{FCP}$ set is composed by the patterns fulfilling at the same time the correlation and the frequency constraints. A pattern is said to be “Frequent Correlated” if its support exceeds the minimal frequency threshold *minsupp* and its correlation value also exceeds the minimal correlation threshold *minbond*. The $\mathcal{FCP}$ set corresponds to the conjunction of two anti-monotonic constraints of correlation and of frequency. Thus, it induces an order ideal on the itmeset lattice.
\[example\_set\_FCP\] Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(Page )</span>. For *minsupp* = 4 and *minbond* = 0.2, the $\mathcal{FCP}$ set consists of the following patterns where each triplet represents the pattern, its conjunctive support value and its *bond* value: $\mathcal{FCP}$ = $\{$<span style="font-variant:small-caps;">(</span>$B$, 4, $\displaystyle\frac{4}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$C$, 4, $\displaystyle\frac{4}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$E$, 4, $\displaystyle\frac{4}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BE$, 4, $\displaystyle\frac{4}{4}$<span style="font-variant:small-caps;">)</span>$\}$.
### Rare Correlated Patterns
The set of rare correlated patterns associated to the *bond* measure is defined as follows.
\[defCRP\] **The set of rare correlated patterns**\
Considering the support threshold *minsupp* and the correlation threshold *minbond*, the set of rare correlated patterns, denoted $\mathcal{RCP}$, is equal to: $\mathcal{RCP}$ = $\{I$ $\subseteq$ $\mathcal{I}$ $|$ *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> $<$ *minsupp* and *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\geq$ *minbond*$\}$.
\[example\_set\_CRP\] Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(Page )</span>. For *minsupp* = 4 and *minbond* = 0.2, the set $\mathcal{RCP}$ consists of the following patterns where each triplet represents the pattern, its conjunctive support value and its *bond* value: $\mathcal{RCP}$ = $\{$<span style="font-variant:small-caps;">(</span>$A$, 3, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$D$, 1, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AB$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AC$, 3, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AD$, 1, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BC$, 3, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CD$, 1, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CE$, 3, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABC$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACD$, 1, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BCE$, 3, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABCE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>$\}$. This associated $\mathcal{RCP}$ set as well as the $\mathcal{FCP}$ set of the previous example are depicted by Figure \[figure\_CRP\]. The support shown at the top left of each frame represents the conjunctive one. As shown in Figure \[figure\_CRP\], the rare correlated patterns are localized below the border induced by the anti-monotonic constraint of correlation and over the border induced by the monotonic constraint of rarity.
We deduce from Definition \[defCRP\] that the $\mathcal{RCP}$ set corresponds to the intersection between the set $\mathcal{CP}$ of correlated patterns and the set $\mathcal{RP}$ of rare patterns, *i.e.*, $\mathcal{RCP}$ = $\mathcal{CP}$ $\cap$ $\mathcal{RP}$. The following proposition derives from this result.
\[prop\_set\_CRP\] Let $I$ $\in$ $\mathcal{RCP}$. We have:
- Based on the order ideal of the set $\mathcal{CP}$ of correlated patterns, we have $\forall$ $I_1$ $\subseteq$ $I$: $I_1$ $\in$ $\mathcal{CP}$
- Based on the order filter of the set $\mathcal{RP}$ of rare patterns, we have $\forall$ $I_1$ $\supseteq$ $I$: $I_1$ $\in$ $\mathcal{RP}$.
The proof follows from the properties induced by the constraints of rarity and correlation. The set $\mathcal{RCP}$, whose elements fulfill the constraint “being a rare correlated pattern”, results from the conjunction between two theories corresponding to both constraints of distinct types. So, the set $\mathcal{RCP}$ is neither an order ideal nor an order filter. The search space of this set is delimited by: <span style="font-variant:small-caps;">(</span>i<span style="font-variant:small-caps;">)</span> The maximal correlated elements which are also rare, *i.e.* the rare patterns among *the set $\mathcal{M}$$ax$$\mathcal{CP}$* of maximal correlated patterns <span style="font-variant:small-caps;">(</span>*cf.* Definition \[bdpos\]<span style="font-variant:small-caps;">)</span> and; <span style="font-variant:small-caps;">(</span>ii<span style="font-variant:small-caps;">)</span> The minimal rare elements which are correlated, *i.e.* the correlated patterns among *the set $\mathcal{M}$$in$$\mathcal{RP}$* of minimal rare patterns <span style="font-variant:small-caps;">(</span>*cf.* Definition \[mrp\]<span style="font-variant:small-caps;">)</span>. Therefore, each rare correlated pattern is necessarily included between an element from each set of the two aforementioned sets.
Therefore, the localization of these elements is more difficult than the localization of theories corresponding to constraints of the same nature. Indeed, the conjunction of anti-monotonic constraints <span style="font-variant:small-caps;">(</span>*resp.* monotonic<span style="font-variant:small-caps;">)</span> is an anti-monotonic one <span style="font-variant:small-caps;">(</span>*resp.* monotonic<span style="font-variant:small-caps;">)</span> [@luccheKIS05_MAJ_06]. For example, the constraint “being a correlated frequent pattern” is anti-monotonic, since it results from the conjunction of two anti-monotonic constraints namely, “being a correlated pattern” and “being a frequent pattern”. This constraint induces, then, an order ideal on the itemset lattice [@tarekds2010]. However, the constraint “being a rare and a not correlated pattern” is monotonic, since it results from the conjunction of two monotonic constraints namely, “being a not correlated pattern” and ”being a rare pattern”. This constraint induces, then, an order filter on the itemset lattice.
In order to assess the size of the $\mathcal{RCP}$ set, and given the nature of the two constraints induced by the minimal thresholds of rarity and correlation respectively *minsupp* and *minbond*, the size of the $\mathcal{RCP}$ set of rare correlated patterns varies as shown in the following proposition.
**a<span style="font-variant:small-caps;">)</span>** Let *minsupp*$_1$ and *minsupp*$_2$ be two minimal thresholds of conjunctive support and $\mathcal{RCP}_{s1}$ and $\mathcal{RCP}_{s2}$ be the two sets of patterns associated to each threshold for the same value of *minbond*. We have: if *minsupp*$_1$ $\leq$ *minsupp*$_2$, then $\mathcal{RCP}_{s1}$ $\subseteq$ $\mathcal{RCP}_{s2}$ and consequently $|\mathcal{RCP}_{s1}|$ $\leq$ $|\mathcal{RCP}_{s2}|$.\
**b<span style="font-variant:small-caps;">)</span>** Let *minbond*$_1$ and *minbond*$_2$ be two minimal thresholds of *bond* measure and let $\mathcal{RCP}_{b1}$ and $\mathcal{RCP}_{b2}$ be the two sets of patterns associated to each threshold for the same value of *minsupp*. We have: if *minbond*$_1$ $\leq$ *minbond*$_2$, then $\mathcal{RCP}_{b2}$ $\subseteq$ $\mathcal{RCP}_{b1}$, consequently $|\mathcal{RCP}_{b2}|$ $\leq$ $|\mathcal{RCP}_{b1}|$.
- The proof of **a<span style="font-variant:small-caps;">)</span>** derives from the fact that for $I$ $\subseteq$ $\mathcal{I}$, if *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> $<$ *minsupp*$_1$, then *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> $<$ *minsupp*$_2$. Therefore, $\forall$ $I$ $\in$ $\mathcal{RCP}_{s1}$, $I$ $\in$ $\mathcal{RCP}_{s2}$. As a result, $\mathcal{RCP}_{s1}$ $\subseteq$ $\mathcal{RCP}_{s2}$.\
- The proof of **b<span style="font-variant:small-caps;">)</span>** derives from the fact that for $I$ $\subseteq$ $\mathcal{I}$, if *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\geq$ *minbond*$_2$, then *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\geq$ *minbond*$_1$. Therefore, $\forall$ $I$ $\in$ $\mathcal{RCP}_{b2}$, $I$ $\in$ $\mathcal{RCP}_{b1}$. As a result, $\mathcal{RCP}_{b2}$ $\subseteq$ $\mathcal{RCP}_{b1}$.\
We can then deduce that the size of the set $\mathcal{RCP}$ is proportional to *minsupp* and inversely proportional to *minbond*. However, in the general case, we cannot decide about the size of the set $\mathcal{RCP}$ when both thresholds vary simultaneously. The next section is dedicated to the presentation of the closure operator associated to the *bond* measure. This operator characterizes the correlated patterns through the induced equivalence classes.
The $f_{bond}$ closure operator {#sec_FBond}
-------------------------------
The $f_{bond}$ closure operator associated to the *bond* measure is defined as follows [@tarekds2010]:
\[closure\_fbond1\] **The operator $f_{bond}$**\
[ $$\begin{aligned}
\large f_{bond}:
\mathcal{P}\textsc{(}\mathcal{I}\textsc{)}&
\rightarrow &
\mathcal{P}\textsc{(}\mathcal{I}\textsc{)}\\
I & \mapsto &
f_{bond}\textsc{(}I\textsc{)} =
I\cup\{i\in\mathcal{I}\setminus I | \ \textit{bond}\textsc{(}I\textsc{)} = \textit{bond}\textsc{(} I\cup\{i\}\textsc{)}\}
\end{aligned}$$]{}
The operator $f_{bond}$ has been shown to be a closure operator [@tarekds2010]. Indeed, it fulfills the extensitivity, the isotony and the idempotency properties [@ganter99]. The closure of a pattern $I$ by $f_{bond}$, *i.e.* $f_{bond}\textsc{(}I$<span style="font-variant:small-caps;">)</span>, corresponds to the maximal set of items containing $I$ and sharing the same *bond* value with $I$.
Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(Page )</span>. For *minbond* = 0.2, we have *bond*<span style="font-variant:small-caps;">(</span>$AB$<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{2}{5}$, *bond*<span style="font-variant:small-caps;">(</span>$ABC$<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{2}{5}$ and *bond*<span style="font-variant:small-caps;">(</span>$ABE$<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{2}{5}$. Thus, $C$ $\in$ $f_{bond}$<span style="font-variant:small-caps;">(</span>$AB$<span style="font-variant:small-caps;">)</span>, and $E$ $\in$ $f_{bond}$<span style="font-variant:small-caps;">(</span>$AB$<span style="font-variant:small-caps;">)</span>. Contrariwise, *bond*<span style="font-variant:small-caps;">(</span>$ABD$<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{0}{5}$ = 0. Thus, $D$ $\notin$ $f_{bond}$<span style="font-variant:small-caps;">(</span>$AB$<span style="font-variant:small-caps;">)</span>. Consequently, we have $f_{bond}$<span style="font-variant:small-caps;">(</span>$AB$<span style="font-variant:small-caps;">)</span> = $ABCE$.
Let us illustrate the different properties of the $f_{bond}$ closure operator:
1. For the Extensitivity property: we have, for example, $f_{bond}$<span style="font-variant:small-caps;">(</span>`CD`<span style="font-variant:small-caps;">)</span> = `ACD`, `CD` $\subseteq$ $f_{bond}$<span style="font-variant:small-caps;">(</span>`CD`<span style="font-variant:small-caps;">)</span>.
2. For the Isotony property: we have, for example, `AB` $\supset$ `B`, $f_{bond}$<span style="font-variant:small-caps;">(</span>`AB`<span style="font-variant:small-caps;">)</span> = `ABCE` and $f_{bond}$<span style="font-variant:small-caps;">(</span>`B`<span style="font-variant:small-caps;">)</span> = `BE`.
3. For the Idempotency property: we have, the example of the closed itemset `ABCE`, $f_{bond}$<span style="font-variant:small-caps;">(</span>$f_{bond}$<span style="font-variant:small-caps;">(</span>`ABCE`<span style="font-variant:small-caps;">)</span><span style="font-variant:small-caps;">)</span> = `ABCE`.
The closure operator $f_{bond}$ induces an equivalence relation on the power-set of the set of items $\mathcal{I}$, splitting it into disjoint *$f_{bond}$* equivalence classes which are formally defined as follows:
\[EquivClsbond\] **Equivalence class associated to the closure operator $f_{bond}$**\
An equivalence class associated to the $f_{bond}$ closure operator is composed by all the patterns having the same closure by the operator $f_{bond}$.
In each class, all the elements have the same $f_{bond}$ closure and the same value of *bond*. The minimal patterns of a *bond* equivalence class are the smallest incomparable members, *w.r.t.* set inclusion, while the $f_{bond}$ closed pattern is the largest one. These sets are formally defined in the following:
\[ClosedCorr\] **Closed correlated patterns**\
The set $\mathcal{CCP}$ of closed correlated patterns by $f_{bond}$ is equal to: $\mathcal{CCP}$ = $\{$$I$ $\in$ $\mathcal{CP}$$\mid$ $\nexists$ $I_{1}$ $\supset$ $I$: *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$I_{1}\textsc{)}\}$, or equivalently: $\mathcal{CCP}$ = $\{$$I$ $\in$ $\mathcal{CP}$$\mid$ $\nexists$ $I_{1}$ $\supset$ $I$: $f_{bond}\textsc{(}I$<span style="font-variant:small-caps;">)</span> = $f_{bond}\textsc{(}I_{1}\textsc{)}\}$.
\[MinCorr\] **Minimal correlated patterns**\
The set $\mathcal{MCP}$ of minimal correlated patterns is equal to: $\mathcal{MCP}$ = $\{$$I$ $\in$ $\mathcal{CP}$$\mid$ $\nexists$ $I_{1}$ $\subset$ $I$: *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$I_{1}\textsc{)}\}$, or equivalently: $\mathcal{MCP}$ = $\{$$I$ $\in$ $\mathcal{CP}$$\mid$ $\nexists$ $I_{1}$ $\subset$ $I$: $f_{bond}\textsc{(}I$<span style="font-variant:small-caps;">)</span> = $f_{bond}\textsc{(}I_{1}\textsc{)}\}$.
The set $\mathcal{MCP}$ of minimal correlated patterns forms an order ideal. In fact, this set is composed by the patterns which fulfill the anti-monotonic constraint “Being minimal in the equivalence class and being correlated”. Indeed, this constraint corresponds to the conjunction between the two following anti-monotonic constraints, “being minimal” and “being correlated”.
The following proposal presents the common properties of patterns belonging to the same $f_{bond}$ equivalence class.
\[proprietes\_CEq\_f\_bond\] Let $\mathcal{C}$ be an equivalence class associated to the closure operator $f_{bond}$ and $I$, $I_1$ $\in$ $\mathcal{C}$. We have: **a<span style="font-variant:small-caps;">)</span>** $f_{bond}\textsc{(}I$<span style="font-variant:small-caps;">)</span> = $f_{bond}\textsc{(}I_1$<span style="font-variant:small-caps;">)</span>, **b<span style="font-variant:small-caps;">)</span>** *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span>, **c<span style="font-variant:small-caps;">)</span>** *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I_1$<span style="font-variant:small-caps;">)</span>, **d<span style="font-variant:small-caps;">)</span>** *Supp*<span style="font-variant:small-caps;">(</span>$\vee I$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\vee I_1$<span style="font-variant:small-caps;">)</span>, and, **e<span style="font-variant:small-caps;">)</span>** *Supp*<span style="font-variant:small-caps;">(</span>$\neg I$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\neg I_1$<span style="font-variant:small-caps;">)</span>.
a<span style="font-variant:small-caps;">)</span>
: Thanks to Definition \[EquivClsbond\], $I$ and $I_1$ share the same closure by $f_{bond}$. Let $F$ be this closure.
b<span style="font-variant:small-caps;">)</span>
: Since the closure operator preserves the value of the *bond* measure <span style="font-variant:small-caps;">(</span>*cf.* Definition \[closure\_fbond1\]<span style="font-variant:small-caps;">)</span>, and since $I$ and $I_1$ have the same closure $F$, we have so *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$F$<span style="font-variant:small-caps;">)</span>, and *bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$F$<span style="font-variant:small-caps;">)</span>. Therefore, *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span>.
c<span style="font-variant:small-caps;">)</span>, d<span style="font-variant:small-caps;">)</span>, and e<span style="font-variant:small-caps;">)</span>
: As $I$ $\subseteq$ $F$ and *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$F$<span style="font-variant:small-caps;">)</span>, according to Proposition \[PropBond\], both of $I$ and $F$ share the same conjunctive, disjunctive and negative supports. It is the same case for $I_1$ and $F$. Therefore, both $I$ and $I_1$ have the same conjunctive, disjunctive and negative supports.
Therefore, all the patterns belonging to the same equivalence class induced by $f_{bond}$, appear exactly in the same transactions <span style="font-variant:small-caps;">(</span>thanks to the equality of the conjunctive support<span style="font-variant:small-caps;">)</span>. Besides, the items associated to the patterns of the same class characterize the same transactions. In fact, each class necessarily contains a non empty subset of every pattern of the class <span style="font-variant:small-caps;">(</span>thanks to the equality of the disjunctive support<span style="font-variant:small-caps;">)</span>. This closure operator links the conjunctive search space to the disjunctive one. In this respect, we begin the next section by the study of the characteristics of the *rare correlated equivalence classes*, induced by the $f_{bond}$ closure operator.
Condensed representations of rare correlated patterns {#section_RC}
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Before introducing our condensed representations, we highlight that the condensed representations prove their high utility in various fields such as: bioinformatics [@pasquier2009] and data grids [@tarekJSS2015].
### Characterization of the rare correlated equivalence classes {#sub_sec_EquivCls}
The equivalence classes permit to retain only the non-redundant patterns. Indeed, among all the patterns of a given equivalence class, only the patterns which are necessary for the regeneration of the whole set of rare correlated patterns, are maintained. Doing so, it considerably reduces the redundancy among the extracted knowledge. The notion of equivalence classes also facilitates the exploration of the search space. Indeed, the application of the $f_{bond}$ closure operator allows switching from the minimal elements of a class to its maximal element without having to sweep through the intermediate levels.
Each equivalence class, induced by the $f_{bond}$ closure operator, contains the patterns sharing the same $f_{bond}$ closure, and thus they are characterized by the same conjunctive, disjunctive supports as well as the same *bond* value <span style="font-variant:small-caps;">(</span>*cf.* Proposition \[proprietes\_CEq\_f\_bond\]<span style="font-variant:small-caps;">)</span>. Therefore, the elements of the same equivalence class have the same behavior towards the correlation and the rarity constraints. In fact, for a correlated equivalence class, *i.e.* a class which contains correlated patterns, all of them are rare or frequent. It is also the same for a non-correlated equivalence class, *i.e.* which contains non-correlated patterns. Therefore we can deduce that, for an equivalence class induced by $f_{bond}$, it is sufficient to evaluate the correlation and the rarity constraints for just one pattern in order to get information about all the other elements of this class. In this respect, we distinguish four different types of equivalence classes: <span style="font-variant:small-caps;">(</span>**i**<span style="font-variant:small-caps;">)</span> correlated frequent classes; <span style="font-variant:small-caps;">(</span>**ii**<span style="font-variant:small-caps;">)</span> non-correlated frequent classes; <span style="font-variant:small-caps;">(</span>**iii**<span style="font-variant:small-caps;">)</span> rare correlated classes; and <span style="font-variant:small-caps;">(</span>**iv**<span style="font-variant:small-caps;">)</span> rare non-correlated classes. The main characteristic of equivalence classes induced by the $f_{bond}$ operator is very interesting. Indeed, this is not the case for all the closure operators. For example, the application of the conjunctive closure operator associated to the conjunctive support induces equivalence classes where the behavior of a given pattern towards the correlation constraint is not representative of the behavior of all the patterns of this class. For each class, each pattern must be independently tested from the other patterns in the same class to check whether it fulfills the correlation constraint or not. It results from the above that, the application of the $f_{bond}$ provides a more selective process to extract rare correlated patterns.
Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. For *minsupp* = 4 and *minbond* = 0.2, Figure \[ExpEquivCls\] shows the obtained rare correlated equivalence classes. We enumerate for example the class which contains the patterns $AB$, $AE$, $ABC$, $ABE$, $ACE$, and $ABCE$. Their respective conjunctive supports are equal to 2 and their *bond* value is equal to $\displaystyle\frac{2}{5}$. The pattern $ABCE$ is the closed correlated one of this class.
The $\mathcal{RCP}$ set of rare correlated patterns is then split into disjoint equivalence classes, the rare correlated equivalence classes. In each class, the closed pattern is the largest one with respect to the inclusion set relation. On the other hand, the smallest incomparable patterns are the minimal rare correlated patterns *w.r.t.* the inclusion set relation. The set of minimal and set of closed patterns are formally defined as follows:
\[CRCP\] **Closed rare correlated patterns**\
The $\mathcal{CRCP}$ $^{\textsc{(}}$[^3]$^{\textsc{)}}$ set of closed rare correlated patterns is equal to: $\mathcal{CRCP}$ = $\{$$I$ $\in$ $\mathcal{RCP}$$|$ $\forall$ $I_{1}$ $\supset$ $I$: *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $>$ *bond*<span style="font-variant:small-caps;">(</span>$I_{1}\textsc{)}\}$.
The $\mathcal{CRCP}$ set corresponds to the intersection between the rare correlated patterns set and the set of closed correlated patterns. We have so, $\mathcal{CRCP}$ = $\mathcal{RCP}$ $\cap$ $\mathcal{CCP}$.
\[MRCP\] **Minimal rare correlated patterns**\
The $\mathcal{MRCP}$ $^{\textsc{(}}$[^4]$^{\textsc{)}}$ set of minimal rare correlated patterns is equal to: $\mathcal{MRCP}$ = $\{$$I$ $\in$ $\mathcal{RCP}$$|$ $\forall$ $I_{1}$ $\subset$ $I$: *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $<$ *bond*<span style="font-variant:small-caps;">(</span>$I_{1}\textsc{)}\}$.
The $\mathcal{MRCP}$ set corresponds to the intersection between the set of rare correlated patterns and the set of minimal correlated patterns. Thus, we have, $\mathcal{MRCP}$ = $\mathcal{RCP}$ $\cap$ $\mathcal{MCP}$.
\[example\_MCRP\_and\_CCRP\] Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(Page )</span>. For *minsupp* = 4 and *minbond* = 0.2, we have $\mathcal{CRCP}$ = $\{$$A$, $D$, $AC$, $AD$, $ACD$, $BCE$, $ABCE$$\}$ and $\mathcal{MRCP}$ = $\{$$A$, $D$, $AB$, $AC$, $AD$, $AE$, $BC$, $CD$, $CE$$\}$.
An accurate representation of rare correlated patterns should determine, for an arbitrary pattern, whether it is rare correlated or not. If it is a rare correlated one, then this representation must allow drifting without information loss the values of its support and its *bond* measure. In this respect, the proposed representations in this work will be later shown to be perfect: their respective sizes never exceed that of the whole set of rare correlated patterns. In addition, since they are information lossless, these representations allow, whenever of need, the derivation of the whole set of rare correlated patterns efficiently.
To define our concise exact representations of rare correlated patterns, we are based on the notion of equivalence classes.
The first intuitive idea when defining a concise exact representation of the rare correlated patterns is to study whether the minimal elements or maximal ones of the equivalence classes would constitute an exact concise representation of the $\mathcal{RCP}$ set.
In this respect, it is important to remind that the $\mathcal{RCP}$ set results from the intersection of the order ideal of correlated patterns and the order filter of rare patterns. Thus, the $\mathcal{RCP}$ set does not induce neither an order ideal nor an order filter. In this situation, we take independently each set, to check whether the $\mathcal{CRCP}$ set or the $\mathcal{MRCP}$ set can provide a concise exact representation of the $\mathcal{RCP}$ set.
Let us analyze, in the following, each of the two sets separately:
- Let us begin with the $\mathcal{MRCP}$ set composed by the minimal elements of the rare correlated equivalence classes: In fact, due to the nature of its elements - minimal of their equivalence classes - this set allows for a given pattern $I$ to evaluate it towards the constraint of rarity. Indeed, it is enough to find an element $J$ $\in$ $\mathcal{MRCP}$ *s.t.* $J$ $\subseteq$ $I$ to decide whether $I$ is a rare pattern or not. If it is not the case, then $I$ is not a rare pattern. However, the set $\mathcal{MRCP}$ cannot determine, in the general case, whether $I$ is correlated or not <span style="font-variant:small-caps;">(</span>this is possible only if $I$ $\in$ $\mathcal{MRCP}$<span style="font-variant:small-caps;">)</span>. Even if it exists $J$ $\in$ $\mathcal{MRCP}$ *s.t.* $J$ $\subset$ $I$, and even knowing that $J$ is correlated, we cannot confirm the correlation of nature of $I$ since this constraint is an anti-monotonic one <span style="font-variant:small-caps;">(</span>the fact that $J$ is correlated does not imply that $I$ is also correlated<span style="font-variant:small-caps;">)</span>. Thus, the $\mathcal{MRCP}$ set, taken alone, cannot be an exact representation of the $\mathcal{RCP}$ set.
- Let us now treat the case of the $\mathcal{CRCP}$ of maximal elements of the rare correlated equivalence classes: Dually to the previous analysis of $\mathcal{MRCP}$, the $\mathcal{CRCP}$ set allows determining the nature of correlation for a given pattern $I$. If it is included in just one pattern $J$ $\in$ $\mathcal{CRCP}$, then $I$ is correlated. Otherwise, it is not a correlated one. However, due to their nature, the patterns composing the $\mathcal{CRCP}$ cannot in the general case derive the information about the status of rarity of a given pattern $I$ <span style="font-variant:small-caps;">(</span>this is possible only if $I$ $\in$ $\mathcal{CRCP}$<span style="font-variant:small-caps;">)</span>. Even if it exists $J$ $\in$ $\mathcal{CRCP}$, *s.t.* $I$ $\subset$ $J$ and even if we already know that $J$ is rare, we cannot decide whether $I$ is rare or not since the constraint of rarity is monotone <span style="font-variant:small-caps;">(</span>the fact that $J$ is rare does not imply that $I$ is also rare<span style="font-variant:small-caps;">)</span>. Thus, the $\mathcal{CRCP}$ set, taken alone, cannot be an exact representation of the $\mathcal{RCP}$ set. Nevertheless, it is proved from the previous analysis that the union of the $\mathcal{MRCP}$ set and the $\mathcal{CRCP}$ set would constitute an accurate concise representation of the $\mathcal{RCP}$ set of rare correlated patterns. The first alternative will be studied in the next sub-section, and will be then followed by two other optimizations in order to retain only the key elements for the lossless regeneration of the $\mathcal{RCP}$ set.
Based on this study, we introduce in the following subsections our new concise exact and approximate representations.
### The **$\mathcal{RCPR}$** concise exact representation
The first representation, that we introduce, is defined as follows:
\[rmcr\] **The $\mathcal{RCPR}$ representation**\
Let $\mathcal{RCPR}$ be the concise exact representation of the $\mathcal{RCP}$ set based on the $\mathcal{CRCP}$ set and on the $\mathcal{MRCP}$ set of the minimal rare correlated patterns. The $\mathcal{RCPR}$ representation is equal to: $\mathcal{RCPR}$ = $\mathcal{CRCP}$ $\cup$ $\mathcal{MRCP}$. The support, *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span>, and the *bond* value, *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> of each pattern $I$ of $\mathcal{RCPR}$ are exactly determined.
\[exprep1\] Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. For *minsupp* = 4 and *minbond* = 0.2, while considering the $\mathcal{CRCP}$ set and $\mathcal{MRCP}$ set <span style="font-variant:small-caps;">(</span>*cf.* Example \[example\_MCRP\_and\_CCRP\]<span style="font-variant:small-caps;">)</span>, the $\mathcal{RCPR}$ set is equal to: $\{$<span style="font-variant:small-caps;">(</span>$A$, $3$, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$D$, $1$, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AB$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AC$, $3$, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AD$, $1$, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BC$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CD$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACD$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BCE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span> and <span style="font-variant:small-caps;">(</span>$ABCE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>$\}$.
The following theorem proves that the $\mathcal{RCPR}$ representation is a lossless concise representation of the $\mathcal{RCP}$ set.
\[representation1\_exacte\] The $\mathcal{RCPR}$ representation is a concise exact representation of the $\mathcal{RCP}$ set of rare correlated patterns.
Let $I$ $\subseteq$ $\mathcal{I}$. We distinct between three different cases:
**a<span style="font-variant:small-caps;">)</span>** If $I$ $\in$ $\mathcal{RCPR}$, then $I$ is a rare correlated pattern and we have its support and its *bond* values.
**b<span style="font-variant:small-caps;">)</span>** If $\nexists$ $J$ $\in$ $\mathcal{RCPR}$ as $J$ $\subseteq$ $I$ and $\nexists$ $Z$ $\in$ $\mathcal{RCPR}$ as $I$ $\subseteq$ $Z$, then $I$ $\notin$ $\mathcal{RCP}$ since $I$ does not belong to any rare correlated equivalence class.
**c<span style="font-variant:small-caps;">)</span>** If $I$ $\in$ $\mathcal{RCP}$. In fact, according to Proposition \[prop\_set\_CRP\], $I$ is correlated since it is included in a correlated pattern, namely $Z$. It is also rare, since it contains a rare pattern, namely $J$. In this case, it is sufficient to localize the $f_{bond}$ closure of $I$ namely $F$. Then, the closed pattern $F$ belongs then to $\mathcal{RCPR}$ since $I$ is rare correlated and $\mathcal{RCPR}$ includes the $\mathcal{CRCP}$ set of closed rare correlated patterns. Therefore, $F$ = $min_{\subseteq}${$I_1$ $\in$ $\mathcal{RCPR} |$ $I$ $\subseteq$ $I_1$}. Since the $f_{bond}$ closure operator conserves the *bond* value and thus the conjunctive support <span style="font-variant:small-caps;">(</span>*cf.* Proposition \[proprietes\_CEq\_f\_bond\]<span style="font-variant:small-caps;">)</span>, we have: *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$F$<span style="font-variant:small-caps;">)</span> and *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\wedge F$<span style="font-variant:small-caps;">)</span>.
\[example\_3\_cas\_RMCR\] Consider the $\mathcal{RCPR}$ representation illustrated by the previous example. Let us consider each case separately. The pattern $AD$ $\in$ $\mathcal{RCPR}$. Thus, we have its support equal to 1 and its *bond* value equal to $\displaystyle\frac{1}{3}$. Even though, the pattern $BE$ is included in two patterns from the $\mathcal{RCPR}$ representation, namely $BCE$ and $ABCE$, $BE$ $\notin$ $\mathcal{RCP}$ since no element of $\mathcal{RCPR}$ is included in $BE$. Consider now the pattern $ABC$. It exists two patterns of $\mathcal{RCPR}$ proving that the pattern $ABC$ is a rare correlated one, namely $AB$ and $ABCE$, since $AB$ $\subseteq$ $ABC$ $\subseteq$ $ABCE$. The smallest pattern in $\mathcal{RCPR}$ covering $ABC$, *i.e.* its closure, is $ABCE$. Then, *bond*<span style="font-variant:small-caps;">(</span>$ABC$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$ABCE$<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{2}{5}$, and *Supp*<span style="font-variant:small-caps;">(</span>$\wedge ABC$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$
\wedge ABCE $<span style="font-variant:small-caps;">)</span> = 2.
We show through the following proposition that the $\mathcal{RCPR}$ representation is a *perfect cover* of the $\mathcal{RCP}$ set of rare correlated patterns.
The $\mathcal{RCPR}$ representation is a *perfect cover* of the $\mathcal{RCP}$ set.
In fact, the size of the $\mathcal{RCPR}$ representation *does never exceed* that of the $\mathcal{RCP}$ set whatever the extraction context, the *minsupp* and the *minbond* values. Indeed, it is always true that <span style="font-variant:small-caps;">(</span>$\mathcal{CRCP}$ $\cup$ $\mathcal{MRCP}$<span style="font-variant:small-caps;">)</span> $\subseteq$ $\mathcal{RCP}$. Furthermore, knowing the conjunctive support of a given pattern and its *bond* value, we can compute the disjunctive support and thus the negative support. The interrogation of the representation $\mathcal{RCPR}$ can be based on the proof of Theorem \[representation1\_exacte\]. Thus, for a given pattern, thanks to the $\mathcal{RCPR}$ representation, we can determine whether it is rare correlated or not. If it is correlated rare, then its support as well as its *bond* value will be derived using the mechanism described by the previous theorem. The regeneration process of the whole set of rare correlated patterns can also be based on this theorem. This process starts by the smallest rare correlated patterns namely the minimal rare correlated patterns <span style="font-variant:small-caps;">(</span>constituting the $\mathcal{MRCP}$ set<span style="font-variant:small-caps;">)</span>. These patterns belong to $\mathcal{RCPR}$ and we have therefore all their required information. It is then sufficient to localize for each minimal $M$ its closure $F$ which belongs to $\mathcal{RCPR}$ <span style="font-variant:small-caps;">(</span>$F$ $\in$ $\mathcal{CRCP}$ and this set is included in $\mathcal{RCPR}$<span style="font-variant:small-caps;">)</span>. All the patterns which are included between $M$ and $F$ share the same support and *bond* value as $M$ and $F$ since they belong to the same rare correlated equivalence class.
It is important to mention that it is necessary to maintain for each pattern $I$ of the representation, at the same time *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> as well as *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span>. On the one hand, *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> is equal to the ratio between the conjunctive and the disjunctive support of $I$ and cannot determine the conjunctive support of $I$. On the other hand, knowing only the conjunctive support of $I$ is not sufficient to compute the *bond* value. The disjunctive support can be derived by the inclusion-exclusion identities only if we know all the conjunctive supports values of all the subsets of $I$ [@galambos]. Nevertheless, if $I$ is a rare correlated pattern, then its subsets are not necessarily rare correlated. Therefore, we don’t have the values of their conjunctive supports. Thus, we must keep track of the conjunctive support and the *bond* value for each element of the $\mathcal{RCPR}$ representation.
In this case, the closed and minimal patterns of the equivalence classes constitute, as shown previously, an interesting solution in order to represent with a concise and exact manner the $\mathcal{RCP}$ set. In fact, the localization of these patterns requires a limited neighborhood, *i.e.*, just the strict supersets and subsets, and not all their subsets. In addition to this, the derivation of the support of the whole set of patterns from the closed and minimal non derivable can be done directly.
\[remak\_manage\_MCRP\_vs\_CCRP\] It is also interesting to mention that the fact to consider in the $\mathcal{RCPR}$ set, the union of the two sets $\mathcal{MRCP}$ and $\mathcal{CRCP}$ allows avoiding redundancy, because of the duplication of some patterns, which may appear in the representation when we consider the $\mathcal{MRCP}$ set and the $\mathcal{CRCP}$ set separately. For example, if we consider the example \[exprep1\], we note that the elements <span style="font-variant:small-caps;">(</span>$A$, 3, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span> , <span style="font-variant:small-caps;">(</span>$D$, 1, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AC$, 3, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span> and <span style="font-variant:small-caps;">(</span>$AD$, 1, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span> belong to both sets $\mathcal{MRCP}$ and $\mathcal{CRCP}$. However, one advantage of saving each set separately allows the reduction of some tests of inclusion in the extraction of the representation. In fact, to make the choice between tolerating some duplication or benefiting from a potential reduction of the regeneration cost depends on the nature of the application where we can privilege either the optimization of the memory space or the derivation cost.
We propose in the following two refined versions of the $\mathcal{RCPR}$ representation, in order to further reduce the size of this representation.
### The $\mathcal{MM}$$ax$$\mathcal{CR}$ concise exact representation
The first refinement is based on the fact that the $\mathcal{MRCP}$ set of minimal rare correlated patterns increased only by the maximal patterns according to the inclusion set, among the $\mathcal{CRCP}$ set of closed rare correlated patterns is sufficient to faithfully represent the $\mathcal{RCP}$ set. In this respect, we define the $\mathcal{M}$$ax$$\mathcal{CRCP}$ set of maximal closed rare correlated patterns as follows:
\[defMF\] **The $\mathcal{M}$$ax$$\mathcal{CRCP}$ set of maximal closed rare correlated patterns**\
The $\mathcal{M}$$ax$$\mathcal{CRCP}$ set is composed by the patterns which are closed correlated rare ones <span style="font-variant:small-caps;">(</span>*cf.* Definition \[CRCP\], page <span style="font-variant:small-caps;">)</span> and at the same time they are maximal correlated <span style="font-variant:small-caps;">(</span>*cf.* Definition \[bdpos\], page <span style="font-variant:small-caps;">)</span>. Then, we have $\mathcal{M}$$ax$$\mathcal{CRCP}$ = $\mathcal{CRCP}$ $\cap$ $\mathcal{M}ax$$\mathcal{CP}$.
The $\mathcal{M}$$ax$$\mathcal{CRCP}$ set is then limited to the elements of the $\mathcal{M}ax$$\mathcal{CP}$ set which are also rare, in addition of being the largest correlated patterns.
\[example\_MaxCCRP\] Consider the extraction context sketched by Table \[Base\_transactions\]<span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. For *minsupp* = 4 and *minbond* = 0.2, we have $\mathcal{M}$$ax$$\mathcal{CRCP}$ = $\{$$ACD$, $ABCE$$\}$.
We define in the following the representation based on the $\mathcal{M}$$ax$$\mathcal{CCRP}$ set.
\[rep2concise exacte def\] **The $\mathcal{MM}$$ax$$\mathcal{CR}$ representation**\
Let $\mathcal{MM}$$ax$$\mathcal{CR}$ be the representation based on the $\mathcal{MRCP}$ set and the $\mathcal{M}$$ax$$\mathcal{CRCP}$. We have, $\mathcal{MM}$$ax$$\mathcal{CR}$ = $\mathcal{MRCP}$ $\cup$ $\mathcal{M}$$ax$$\mathcal{CRCP}$. For each element $I$ of the $\mathcal{MM}ax \mathcal{CR}$, the support, *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span>, and the *bond* value, *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> are computed.
\[exprep2\] Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. For *minsupp* = 4 and *minbond* = 0.2, the representation $\mathcal{MM}ax \mathcal{CR}$ is equal to: $\{$<span style="font-variant:small-caps;">(</span>$A$, $3$, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$D$, $1$, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AB$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AC$, $3$, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AD$, $1$, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BC$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CD$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACD$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, and <span style="font-variant:small-caps;">(</span>$ABCE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>$\}$. We remark that, for this example, the unique element of the $\mathcal{RCPR}$ representation that does not belong to the $\mathcal{MM}$$ax$$\mathcal{CR}$ representation is, the pattern $BCE$. In fact, the closed patterns that were removed, *i.e.*, $A$, $D$, $AC$ and $AD$, are also minimal. Indeed, the $\mathcal{MM}$$ax$$\mathcal{CR}$ representation would be more optimized than do the $\mathcal{RCPR}$ representation, if the sets $\mathcal{MRCP}$ and $\mathcal{CRCP}$ were saved separately <span style="font-variant:small-caps;">(</span>*cf.* Remark \[remak\_manage\_MCRP\_vs\_CCRP\]<span style="font-variant:small-caps;">)</span>. In fact, the duplicate storage of the patterns $A$, $D$, $AC$ and $AD$ is avoided.
Theorem \[representation2\_exacte\] proves that the $\mathcal{MM}$ax$\mathcal{CR}$ set is a lossless representation of the $\mathcal{RCP}$ set.
\[representation2\_exacte\] The $\mathcal{MM}$ax$\mathcal{CR}$ set is an exact concise representation of the $\mathcal{RCP}$ set of rare correlated patterns.
Let a pattern $I$ $\subseteq$ $\mathcal{I}$. We distinguish between three cases:
**a<span style="font-variant:small-caps;">)</span>** If $I$ $\in$ $\mathcal{MM}$$ax$$\mathcal{CR}$, then $I$ is a rare correlated pattern and we have its support and *bond* values.
**b<span style="font-variant:small-caps;">)</span>** If $\nexists$ $J$ $\in$ $\mathcal{MM}$$ax$$\mathcal{CR}$ as $J$ $\subseteq$ $I$ or $\nexists$ $Z$ $\in$ $\mathcal{MM}$$ax$$\mathcal{CR}$ as $I$ $\subseteq$ $Z$, then $I$ $\notin$ $\mathcal{RCP}$ since $I$ do not belong to any rare correlated equivalence class.
**c<span style="font-variant:small-caps;">)</span>** Else, $I$ $\in$ $\mathcal{RCP}$. In fact, according to Proposition \[prop\_set\_CRP\], $I$ is a correlated pattern since it is included in a correlated pattern, say $Z$. It is also rare since it contains a rare pattern, say $J$. Since $I$ is a rare correlated pattern and the representation $\mathcal{MM}$$ax$$\mathcal{CR}$ includes the $\mathcal{MRCP}$ set containing the minimal elements of the different rare correlated equivalence classes, this representation has at least one element of the equivalence class of $I$, particularly all the minimal patterns of its class.\
Since both the conjunctive support and the *bond* measure decrease as far the size of the patterns is lowered, the support and bond values of $I$ are equal to the minimal values of the measures associated to its subsets belonging to the $\mathcal{MM}$$ax$$\mathcal{CR}$ representation. We deduce then that:\
$\bullet$ *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> = $min$$\{$*Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I_1$<span style="font-variant:small-caps;">)</span>$|$ $I_1$ $\in$ $\mathcal{MM}$$ax$$\mathcal{CR}$ and $I_1$ $\subseteq$ $I$$\}$; and\
$\bullet$ *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = $min$$\{$*bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span>$|$ $I_1$ $\in$ $\mathcal{MM}$$ax$$\mathcal{CR}$ and $I_1$ $\subseteq$ $I$$\}$.
Consider the $\mathcal{MM}$$ax$$\mathcal{CR}$ presented in Example \[exprep2\]. The treatment of the first and second cases is similar to the first two cases of the $\mathcal{RCPR}$ representation <span style="font-variant:small-caps;">(</span>*cf.* Example \[example\_3\_cas\_RMCR\]<span style="font-variant:small-caps;">)</span>. The case of the pattern $ABE$ is illustrative of the third alternative. In fact, it exists two patterns, from the $\mathcal{MM}$$ax$$\mathcal{CR}$ representation, which makes $ABE$ a rare correlated pattern, namely $AB$ and $ABCE$ <span style="font-variant:small-caps;">(</span>$AB$ $\subseteq$ $ABE$ $\subseteq$ $ABCE$<span style="font-variant:small-caps;">)</span>. The elements of the $\mathcal{MM}$$ax$$\mathcal{CR}$ representation which are included in $ABE$ are $AB$ and $AE$. Consequently, *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$ABE$<span style="font-variant:small-caps;">)</span> = $min$$\{$*Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$AB$<span style="font-variant:small-caps;">)</span>, *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$AE$<span style="font-variant:small-caps;">)</span>$\}$ = $min$$\{$2, 2$\}$ = 2, and *bond*<span style="font-variant:small-caps;">(</span>$ABE$<span style="font-variant:small-caps;">)</span> = $min$$\{$*bond*<span style="font-variant:small-caps;">(</span>$AB$<span style="font-variant:small-caps;">)</span>, *bond*<span style="font-variant:small-caps;">(</span>$AE$<span style="font-variant:small-caps;">)</span>$\}$ = $min$$\{$$\displaystyle\frac{2}{5}$, $\displaystyle\frac{2}{5}$$\}$ = $\displaystyle\frac{2}{5}$.
Since the $\mathcal{MM}$$ax$$\mathcal{CR}$ representation is included in the $\mathcal{RCPR}$ representation, and the latter is a perfect cover of the $\mathcal{RCP}$ set, then we deduce that the $\mathcal{MM}$$ax$$\mathcal{CR}$ representation is also a perfect cover of the $\mathcal{RCP}$ set. In the next sub-section, we present another refinement of the $\mathcal{RCPR}$ representation.
### The **$\mathcal{M}$$in$$\mathcal{MCR}$** concise exact representation
Dually to the previous definition, it is sufficient to maintain in the $\mathcal{RCPR}$ representation, just the minimal elements, according to the inclusion set, among the $\mathcal{MRCP}$ set. The pruning of the other elements from the $\mathcal{MRCP}$ set will be shown to be information lossless during the regeneration of the whole set of rare correlated patterns. The $\mathcal{M}$$in$$\mathcal{MRCP}$ set of minimal elements among the $\mathcal{MRCP}$, is thus defined as follows:
\[defMM\] **The $\mathcal{M}$$in$$\mathcal{MRCP}$ set of the minimal elements of the $\mathcal{MRCP}$ set**\
The $\mathcal{M}$$in$$\mathcal{MRCP}$ set is composed by the patterns which are minimal rare correlated patterns <span style="font-variant:small-caps;">(</span>*cf.* Definition \[MRCP\], page <span style="font-variant:small-caps;">)</span> and at the same time minimal rare patterns <span style="font-variant:small-caps;">(</span>*cf.* Definition \[mrp\], page <span style="font-variant:small-caps;">)</span>. Thus, $\mathcal{M}in \mathcal{MRCP}$ = $\mathcal{MRCP}$ $\cap$ $\mathcal{M}in \mathcal{RP}$.
The $\mathcal{M}in \mathcal{MRCP}$ set is then limited by the minimal rare correlated patterns which are also minimal rare <span style="font-variant:small-caps;">(</span>In addition to being the smallest rare patterns<span style="font-variant:small-caps;">)</span>.
\[expMM\] Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. For *minsupp* = 4 and *minbond* = 0.2, we have $\mathcal{M}in$$\mathcal{MRCP}$ = $\{$$A$, $D$, $BC$, $CE\}$.
The following definition introduces the representation based on the $\mathcal{M}in\mathcal{MCRP}$ set.
\[rep3concise exacte def\] **The $\mathcal{M}in \mathcal{MCR}$ representation**\
Let $\mathcal{M}in\mathcal{MCR}$ be the representation based on the $\mathcal{M}in \mathcal{MCRP}$ set and the $\mathcal{CRCP}$ set. We have $\mathcal{M}in \mathcal{MCR}$ = $\mathcal{CRCP}$ $\cup$ $\mathcal{M}in\mathcal{MRCP}$. For each element $I$ of $\mathcal{M}in\mathcal{MCR}$, its support *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> and its *bond* value *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> are computed.
\[exprep3\] Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. For *minsupp* = 4 and *minbond* = 0.2, we have the $\mathcal{M}$$in$$\mathcal{MCR}$ = $\{$<span style="font-variant:small-caps;">(</span>$A$, 3, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$D$, 1, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AC$, 3, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AD$, 1, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BC$, 3, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CE$, 3, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACD$, 1, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BCE$, 3, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, and <span style="font-variant:small-caps;">(</span>$ABCE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>$\}$. We remark that, this representation has three elements less than the $\mathcal{RCPR}$ representation, namely $AB$, $AE$ and $CD$.
The following theorem proves that this representation is also a lossless reduction of the $\mathcal{RCP}$ set.
\[representation3\_exacte\] The $\mathcal{M}$$in$$\mathcal{MCR}$ representation is a concise exact representation of the $\mathcal{RCP}$ set of rare correlated patterns.
Let $I$ $\subseteq$ $\mathcal{I}$. We distinguish between three different cases:
**a<span style="font-variant:small-caps;">)</span>** If $I$ $\in$ $\mathcal{M}$$in$$\mathcal{MCR}$, then $I$ is a rare correlated pattern and we know its support as well as its *bond* value.
**b<span style="font-variant:small-caps;">)</span>** If $\nexists$ $J$ $\in$ $\mathcal{M}$$in$$\mathcal{MCR}$ as $J$ $\subseteq$ $I$ or $\nexists$ $Z$ $\in$ $\mathcal{M}$$in$$\mathcal{MCR}$ as $I$ $\subseteq$ $Z$, then $I$ $\notin$ $\mathcal{RCP}$ since $I$ does not belong to any rare correlated equivalence class.
**c<span style="font-variant:small-caps;">)</span>** Otherwise, $I$ $\in$ $\mathcal{RCP}$. In fact, according to Proposition \[prop\_set\_CRP\], $I$ is correlated since it is included in a correlated pattern, namely $Z$. It is also rare since it includes a rare pattern, namely $J$. Since the $\mathcal{CRCP}$ set belongs to $\mathcal{M}$$in$$\mathcal{MCR}$, it is enough to localize the closed pattern associated to $I$, namely $F$, equal to: $F$ = $min_{\subseteq}${$I_1$ $\in$ $\mathcal{M}$$in$$\mathcal{MCR}$ $|$ $I$ $\subseteq$ $I_1$}. Then, *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$F$<span style="font-variant:small-caps;">)</span> and *Supp*<span style="font-variant:small-caps;">(</span>$\wedge I$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\wedge F$<span style="font-variant:small-caps;">)</span>.\
The treatment of these three cases is similar to those of the $\mathcal{RCPR}$ representation, <span style="font-variant:small-caps;">(</span>*cf.* Example \[example\_3\_cas\_RMCR\] page <span style="font-variant:small-caps;">)</span>. The $\mathcal{M}$$in$$\mathcal{MCR}$ representation also constitutes a perfect cover of the $\mathcal{RCP}$ set, since it is included in the $\mathcal{RCPR}$ representation.
After the introduction of our exact condensed representations, we deal in the following with the approximate concise representation.
### The $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$ concise approximate representation
The approximate concise representation, that we introduce, is defined as follows:
**The $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$ representation** \[rep4concise approxdef\]\
Let $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$ be the representation based on the $\mathcal{M}ax$$\mathcal{CRCP}$ set and the $\mathcal{M}in$$\mathcal{MRCP}$ set. We have $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$ = $\mathcal{M}ax$$\mathcal{CRCP}$ $\cup$ $\mathcal{M}in$$\mathcal{MRCP}$. For each element $I$ of $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$, the support *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> and the *bond* value *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> are computed.
\[expRC4\] We have, $\mathcal{M}in$$\mathcal{MRCP}$ = $\{$$A$, $D$, $BC$, $CE$$\}$ <span style="font-variant:small-caps;">(</span>*cf.* Example \[expMM\]<span style="font-variant:small-caps;">)</span> and $\mathcal{M}ax$$\mathcal{CRCP}$ = $\{$$ACD$, $ABCE$$\}$ <span style="font-variant:small-caps;">(</span>*cf.* Example \[example\_MaxCCRP\]<span style="font-variant:small-caps;">)</span>. Therefore, $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$ = $\{$<span style="font-variant:small-caps;">(</span>$A$, 3, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$D$, 1, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BC$, 3, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CE$, 3, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACD$, 1, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABCE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>$\}$.
In the previous example, the $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$ representation has six elements less than the $\mathcal{RCPR}$ set, eleven elements less than the $\mathcal{MM}$$ax$$\mathcal{CR}$ set and one element less than $\mathcal{M}$$in$$\mathcal{MCR}$. However, this representation can not exactly derive the support and the *bond* values of a given rare correlated pattern.
\[representation4\_approx\] The $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$ is an **approximate** concise representation of the $\mathcal{RCP}$ set of the rare correlated patterns.
For a given pattern $I$ $\subseteq$ $\mathcal{I}$, we can determine thanks to this representation whether $I$ is rare correlated or not. It suffices to find two patterns $J$ and $Z$ belonging to $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$ such as $J$ $\subseteq$ $I$ $\subseteq$ $Z$. If $J$ or $Z$ can not be found, then $I$ $\notin$ $\mathcal{RCP}$. However, the support and *bond* values can be exactly derived only if $I$ $\in$ $\mathcal{M}$$in$$\mathcal{MM}$$ax$$\mathcal{CR}$. Otherwise, this representation can not offer an exact derivation of these values, since it may not contain any representative element of the equivalence class of $I$ <span style="font-variant:small-caps;">(</span>*i.e.* neither the closed pattern if it does not belong to the $\mathcal{M}ax$$\mathcal{CRCP}$ set nor to the associated minimal if they don’t belong to the $\mathcal{M}in$$\mathcal{MRCP}$ set<span style="font-variant:small-caps;">)</span>. We propose in this case an approximate process in order to get these values. We define, in this regard, the maximal and minimal borders of the conjunctive, the disjunctive and the *bond* value of a correlated rare pattern $I$. Let,
$\bullet$ *R1* = $\max$$\{$Supp<span style="font-variant:small-caps;">(</span>$\wedge$$F$<span style="font-variant:small-caps;">)</span>, $F$ $\in$ $\mathcal{M}ax$$\mathcal{CRCP}$ $|$ $I$ $\subseteq$ $F$$\}$,
$\bullet$ *R2* = $\min$$\{$Supp<span style="font-variant:small-caps;">(</span>$\wedge$$G$<span style="font-variant:small-caps;">)</span>, $G$ $\in$ $\mathcal{M}in$$\mathcal{MRCP}$ $|$ $G$ $\subseteq$ $I$$\}$,
$\bullet$ *R3* = $\min$$\{$Supp<span style="font-variant:small-caps;">(</span>$\vee$$F$<span style="font-variant:small-caps;">)</span>, $F$ $\in$ $\mathcal{M}ax$$\mathcal{CRCP}$ $|$ $I$ $\subseteq$ $F$$\}$ and
$\bullet$ *R4* = $\max$$\{$*Supp*<span style="font-variant:small-caps;">(</span>$\vee$$G$<span style="font-variant:small-caps;">)</span>, $G$ $\in$ $\mathcal{M}in$$\mathcal{MRCP}$ $|$ $G$ $\subseteq$ $I$$\}$.
We define, therefore, the minimal and maximal borders in terms of *R1* and of *R2* as follows. Let *MinConj* be the minimal border of the conjunctive support of the pattern $I$, *i.e.*, *MinConj* = $\min$<span style="font-variant:small-caps;">(</span>*R1*, *R2*<span style="font-variant:small-caps;">)</span>. Let *MaxConj* be the maximal border of the conjunctive support of the pattern $I$, *i.e.*, *MaxConj* = $\max$<span style="font-variant:small-caps;">(</span>*R1*, *R2*<span style="font-variant:small-caps;">)</span>.
According to the disjunctive support of a given pattern $I$, we define the maximal and minimal borders in terms of *R3* and of *R4* as follows. Let *MinDisj* be the minimal border of the disjunctive support, *MinDisj* = $\min$<span style="font-variant:small-caps;">(</span>*R3*, *R4*<span style="font-variant:small-caps;">)</span> and let *MaxDisj* be the maximal border of the disjunctive support, *MaxDisj* = $\max$<span style="font-variant:small-caps;">(</span>*R3*, *R4*<span style="font-variant:small-caps;">)</span>.
Consequently, the conjunctive support of a rare correlated pattern $I$ will be included between *MinConj* and *MaxConj*. In the same way, the disjunctive support will be included between *MinDisj* and *MaxDisj*. Formally, we have *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> $\in$ $[$*MinConj*, *MaxConj*$]$ and *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$I$<span style="font-variant:small-caps;">)</span> $\in$ $[$*MinDisj*, *MaxDisj*$]$.
Therefore, we define the minimal and maximal borders of the *bond* value of a rare correlated pattern $I$ in terms of *MinConj*, *MinDisj*, *MaxConj* and *MaxDisj* as follows.
Since *MinDisj* $\leq$ *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$I$<span style="font-variant:small-caps;">)</span> $\leq$ *MaxDisj*, then we have $\displaystyle\frac{1}{\textit{MaxDisj}}$ $\leq$ $\displaystyle\frac{1}{\textit{Supp}\textsc{(}\vee I\textsc{)}}$ $\leq$ $\displaystyle\frac{1}{\textit{MinDisj}}$.
As *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$I<span style="font-variant:small-caps;">)</span> $>$ 0 then we can deduce that, $\displaystyle\frac{\textit{Supp}\textsc{(}\wedge I\textsc{)}}{\textit{MaxDisj}}$ $\leq$ $\displaystyle\frac{\textit{Supp}\textsc{(}\wedge I\textsc{)}}{\textit{Supp}\textsc{(}\vee I\textsc{)}}$ $\leq$ $\displaystyle\frac{\textit{Supp}\textsc{(}\wedge I\textsc{)}}{\textit{MinDisj}}$. This is equivalent to, $\displaystyle\frac{\textit{Supp}\textsc{(}\wedge I\textsc{)}}{\textit{MaxDisj}}$ $\leq$ *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\leq$ $\displaystyle\frac{\textit{Supp}\textsc{(}\wedge I\textsc{)}}{\textit{MinDisj}}$. Already, *MinConj* $\leq$ *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$ $I$<span style="font-variant:small-caps;">)</span> , then $\displaystyle\frac{\textit{MinConj}}{\textit{MaxDisj}}$ $\leq$ *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span>. Besides, *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$I$<span style="font-variant:small-caps;">)</span> $\leq$ *MaxConj* then *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\leq$ $\displaystyle\frac{\textit{MaxConj}}{\textit{MinDisj}}$. We then have, $\displaystyle\frac{\textit{MinConj}}{\textit{MaxDisj}}$ $\leq$ *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\leq$ $\displaystyle\frac{\textit{MaxConj}}{\textit{MinDisj}}$.
We conclude then that, *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\in$ $[$*Minbond*, *Maxbond*$]$, with *Minbond* = $\displaystyle\frac{\textit{MinConj}}{\textit{MaxDisj}}$ and *Maxbond* = $\displaystyle\frac{\textit{MaxConj}}{\textit{MinDisj}}$.
With respect to Example \[expRC4\], we have $ABE$ is a correlated rare pattern since <span style="font-variant:small-caps;">(</span>$A$ $\subset$ $ABE$ $\subset$ $ABCE$<span style="font-variant:small-caps;">)</span>. The conjunctive, disjunctive and the *bond* value of $ABE$ are approximated as follows:\
$\bullet$ *R1* = Supp<span style="font-variant:small-caps;">(</span>$\wedge$$ABCE$<span style="font-variant:small-caps;">)</span> = 2,\
$\bullet$ *R2* = Supp<span style="font-variant:small-caps;">(</span>$\wedge$$A$<span style="font-variant:small-caps;">)</span> = 3,\
$\bullet$ *R3* = Supp<span style="font-variant:small-caps;">(</span>$\vee$$ABCE$<span style="font-variant:small-caps;">)</span> = 5,\
$\bullet$ *R4* = Supp<span style="font-variant:small-caps;">(</span>$\vee$$A$<span style="font-variant:small-caps;">)</span> = 5.\
Thus, we have *MinConj* = $\min$<span style="font-variant:small-caps;">(</span>*R1*, *R2*<span style="font-variant:small-caps;">)</span> = $\min$<span style="font-variant:small-caps;">(</span>2, 3<span style="font-variant:small-caps;">)</span> = 2, *MaxConj* = $\max$<span style="font-variant:small-caps;">(</span>*R1*, *R2*<span style="font-variant:small-caps;">)</span> = $\max$<span style="font-variant:small-caps;">(</span>2, 3<span style="font-variant:small-caps;">)</span> = 3, and $\min$<span style="font-variant:small-caps;">(</span>*R3*, *R4*<span style="font-variant:small-caps;">)</span> = $\max$<span style="font-variant:small-caps;">(</span>*R3*, *R4*<span style="font-variant:small-caps;">)</span> = $\min$<span style="font-variant:small-caps;">(</span>5, 5<span style="font-variant:small-caps;">)</span> = $\max$<span style="font-variant:small-caps;">(</span>5, 5<span style="font-variant:small-caps;">)</span> = 5. We have, therefore, *MinDisj* = *MaxDisj* = 5. This implies that, *Minbond* = $\displaystyle\frac{\textit{MinConj}}{\textit{MaxDisj}}$ = $\displaystyle\frac{2}{5}$ and *Maxbond* = $\displaystyle\frac{\textit{MaxConj}}{\textit{MinDisj}}$ = $\displaystyle\frac{3}{5}$.
Consequently, we have Supp<span style="font-variant:small-caps;">(</span>$\wedge$$ABE$<span style="font-variant:small-caps;">)</span> $\in$ $[$2, 3$]$, Supp<span style="font-variant:small-caps;">(</span>$\vee$$ABE$<span style="font-variant:small-caps;">)</span> $\in$ $[$5, 5$]$ so Supp<span style="font-variant:small-caps;">(</span>$\vee$$ABE$<span style="font-variant:small-caps;">)</span> = 5 and bond<span style="font-variant:small-caps;">(</span>$ABE$<span style="font-variant:small-caps;">)</span> $\in$ $[$$\displaystyle\frac{2}{5}$, $\displaystyle\frac{3}{5}$$]$.
We remark, according to the extraction context illustrated by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>, that the conjunctive, disjunctive and the *bond* values of the pattern $ABE$ corresponds respectively to 2, 5 and $\displaystyle\frac{2}{5}$. These values does not contradict the previously obtained approximate values. We affirm that the approximation mechanism offered by the approximate concise representation $\mathcal{RM}$$in$$\mathcal{MM}$$ax$$\mathcal{F}$ is valid.
After presenting the condensed representations associated to the rare correlated patterns, we focus on the next section on the condensed representation associated to the $\mathcal{FCP}$ set of frequent correlated patterns
Condensed representation of frequent correlated patterns {#section_RCfcp}
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Based on the $f_{bond}$ closure operator, a condensed representation which cover the frequent correlated patterns was proposed in [@tarekds2010]. This representation is based on the frequent closed correlated patterns. The proposed representation is considered more concise than the representation based on minimal correlated patterns thanks to the fact that a $f_{bond}$ equivalence class always contains only one closed pattern, but potentially several minimal patterns. Before introducing the representation, let us define the two discussed sets of frequent closed correlated patterns and of the frequent minimal correlated pattern associated to the $f_{bond}$ operator.
**Frequent closed correlated pattern**\
The set $\mathcal{FCCP}$ of frequent closed correlated patterns is equal to: $\mathcal{FCCP}$ = $\{$ $I$ $\in$ $\mathcal{CCP}$ $|$ *Supp*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\geq$ *minsupp*$\}$.
\[définitionMotif minimal\] **Frequent minimal correlated pattern**\
Let $\textit{I}$ $\in$ $\mathcal{FCP}$. The pattern *I* is said to be minimal if and only if $\forall$ $i$ $\in$ $\textit{I}$, *bond*<span style="font-variant:small-caps;">(</span>*I*<span style="font-variant:small-caps;">)</span> $<$ *bond*<span style="font-variant:small-caps;">(</span>*I*$\backslash$$\{i\}$<span style="font-variant:small-caps;">)</span> or, equivalently, $\nexists$ $I_1$ $\subset$ $I$ such that $f_{bond}\textsc{(}I\textsc{)}$ = $f_{bond}\textsc{(}I_1\textsc{)}$.
Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. For *minsupp* = *4* and *minbond* = *0.20*, the $\mathcal{FCCP}$ set of frequent closed correlated pattern is equal to: $\{$<span style="font-variant:small-caps;">(</span> <span style="font-variant:small-caps;">(</span>*C*, 4, 4<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>*BE*, 4, 4<span style="font-variant:small-caps;">)</span>$\}$ while the frequent minimal correlated pattern are the items <span style="font-variant:small-caps;">(</span>*B*, 4, 4<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>*C*, 4, 4<span style="font-variant:small-caps;">)</span> and <span style="font-variant:small-caps;">(</span>*E*, 4, 4<span style="font-variant:small-caps;">)</span>.
Now, let us define the new concise representation of frequent correlated patterns based on the frequent closed correlated patterns associated to the *bond* measure.
\[rep concise exacte def\] The representation $\mathcal{RFCCP}$ based on the set of frequent closed correlated patterns associated to $f_{bond}$ is defined as follows:
$\mathcal{RFCCP} = \{$<span style="font-variant:small-caps;">(</span>$\textit{I}$, *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$*I*<span style="font-variant:small-caps;">)</span>, *Supp*<span style="font-variant:small-caps;">(</span>$\vee$*I*<span style="font-variant:small-caps;">)</span><span style="font-variant:small-caps;">)</span> $|$ $I\in\mathcal{FCCP}$ $\}$.
Consider the extraction context sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. For *minsupp* = *4* and *minbond* = *0.20*, the representation $\mathcal{RFCCP}$ of the $\mathcal{FCP}$ set is equal to: $\{$<span style="font-variant:small-caps;">(</span> <span style="font-variant:small-caps;">(</span>*C*, 4, 4<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>*BE*, 4, 4<span style="font-variant:small-caps;">)</span>$\}$.
The next theorem proves that the proposed $\mathcal{RFCCP}$ representation is a condensed exact representation of the $\mathcal{FCP}$ set of frequent correlated patterns.
The representation $\mathcal{RFCCP}$ constitutes an exact concise representation of the $\mathcal{FCP}$ set.
*Thanks to a reasoning by recurrence, we will demonstrate that, for an arbitrary pattern $I\subseteq \mathcal{I}$, its $f_{bond}$ closure, $f_{bond}$<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span>, belongs to $\mathcal{FCCP}$ if it is frequent correlated. In this regard, let $\mathcal{FMCP}_k$ be the set of frequent minimal correlated patterns of size $k$ and $\mathcal{FCCP}_k$ be the associated set of closures by $f_{bond}$. The hypothesis is verified for single items $i$ inserted in $\mathcal{FMCP}_1$, and their closures $f_{bond}\textsc{(}i\textsc{)}$ are inserted in $\mathcal{FCCP}_1$ if $\textit{Supp}\textsc{(}\wedge i\textsc{)} \geq$ *minsupp* <span style="font-variant:small-caps;">(</span>since $\forall$ $i$ $\in$ $\mathcal{I}$, $\textit{bond}\textsc{(}i\textsc{)}$ $=$ $1$ $\geq$ *minbond*<span style="font-variant:small-caps;">)</span>. Thus, $f_{bond}\textsc{(}i\textsc{)}\in\mathcal{FCCP}$. Now, suppose that $\forall I\subseteq \mathcal{I}$ such as $|I| = n$. We have $f_{bond}\textsc{(}I\textsc{)}\in\mathcal{FCCP}$ if $I$ is frequent correlated. We show that, $\forall I\subseteq \mathcal{I}$ such as $|I|$ $=$ <span style="font-variant:small-caps;">(</span>$n
+ 1$<span style="font-variant:small-caps;">)</span>, we have $f_{bond}\textsc{(}I\textsc{)}\in\mathcal{FCCP}$ if $I$ is frequent correlated. Let $I$ be a pattern of size <span style="font-variant:small-caps;">(</span>$n + 1$<span style="font-variant:small-caps;">)</span>. Three situations are possible:\
**<span style="font-variant:small-caps;">(</span>a<span style="font-variant:small-caps;">)</span>** if $I \in \mathcal{FCCP}$, then necessarily $f_{bond}\textsc{(}I\textsc{)}\in\mathcal{FCCP}$ since $f_{bond}$ is idempotent.\
**<span style="font-variant:small-caps;">(</span>b<span style="font-variant:small-caps;">)</span>** if $I$ $\in \mathcal{FMCP}_{n+1}$, then $f_{bond}\textsc{(}I\textsc{)}\in\mathcal{FCCP}_{n+1}$ and, hence, $f_{bond}\textsc{(}I\textsc{)}\in\mathcal{FCCP}$.\
**<span style="font-variant:small-caps;">(</span>c<span style="font-variant:small-caps;">)</span>** if $I$ is neither closed nor minimal – $I \notin \mathcal{FCCP}$ and $I \notin \mathcal{FMCP}_{n+1}$ – then $\exists I_1 \subset I$ such as $|I_1|$ $=$ $n$ and *bond*<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $=$ *bond*<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span>. In fact, $f_{bond}$<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $=$ $f_{bond}$<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span>, and $I$ is then frequent correlated. Moreover, using the hypothesis, we have $f_{bond}$<span style="font-variant:small-caps;">(</span>$I_1$<span style="font-variant:small-caps;">)</span> $\in$ $\mathcal{FCCP}$ and, hence, $f_{bond}$<span style="font-variant:small-caps;">(</span>$I$<span style="font-variant:small-caps;">)</span> $\in$ $\mathcal{FCCP}$.*
Conclusion {#ConcChap4}
----------
In this chapter, we studied the frequent correlated and the rare correlated patterns according to the *bond* correlation measure. We equally described the equivalence classes induced by the $f_{bond}$ closure operator associated to the *bond* measure. Then, we introduced the condensed exact and approximate representations associated to rare correlated patterns and also to frequent correlated ones. We proved their theoretical properties of accuracy and compactness. This chapter was concluded with the condensed representation associated to the $\mathcal{FCP}$ set of frequent correlated patterns. In the next chapter, we propose a mining approach, called <span style="font-variant:small-caps;">GMJP</span>, allowing the extraction of both frequent correlated patterns, rare correlated patterns and their associated concise representations.
Extraction Approach of Correlated Patterns and associated Condensed Representations {#ch_5}
===================================================================================
Introduction {#introduction-1}
------------
This chapter is dedicated to the presentation of our approach called <span style="font-variant:small-caps;">Gmjp</span>. Section \[SeCs\] is devoted to the analysis of the different integration mechanism of the constraints of frequency and of correlation. Section \[section\_algo\] presents the description of the <span style="font-variant:small-caps;">Gmjp</span> approach, going from general to specificities. We describe the three different steps of <span style="font-variant:small-caps;">Gmjp</span>, then we present <span style="font-variant:small-caps;">Opt-Gmjp</span> the optimized version of the <span style="font-variant:small-caps;">Gmjp</span> algorithm in Section \[se-optgmjp\]. We compute the theoretical approximate time complexity of <span style="font-variant:small-caps;">Gmjp</span> in Section \[se-comp\]. In Section \[sec\_Regeneration\], we describe the regeneration strategy of the rare correlated patterns from the $\mathcal{RCPR}$ representation. We conclude the chapter in Section \[ConcChap5\].
Integration mechanism of the constraints {#SeCs}
----------------------------------------
This chapter is dedicated to the presentation of the extraction approach of both frequent correlated and rare correlated itemsets as well as their associated condensed representations. For the case of the frequent correlated patterns, the extraction process is straightforward since the set of frequent correlated patterns induces an ideal order on the itemset lattice and fulfills an anti-monotonic constraint. Whereas, for the rare correlated patterns, we have to handle two constraints of distinct types: monotonic and anti-monotonic. The evaluation order of the constraints is of paramount importance given the opposite nature of the handled constraints of rarity and of correlation. Thus, we distinguish two different possible scenarios:
$\bullet$ First Scenario: We first apply the rarity constraint and the associated conjunctive closure operator, then we apply the correlation constraint.\
$\bullet$ Second Scenario: We apply the correlation constraint and the associated $f_{bond}$ closure operator, then we integrate the rarity constraint.
These two scenarios will be analyzed in what follows in order to justify our choice of the adequate scenario in our proposed extraction approach.
### First Scenario
In this case, we firstly extract the rare patterns. Then, we filter the retained rare patterns by the correlation constraint. Thus, only the rare correlated patterns are retained. In this situation, in order to reduce the redundancy among the rare correlated itemsets, we apply the conjunctive closure operator associated to the conjunctive support [@ganter99]. This latter splits the itemset lattice into disjoint equivalence classes where, for each pattern, this closure preserve only the conjunctive support. Consequently, the whole set of the itemsets that appear in the same transactions are merged into the same equivalence class. They share the same conjunctive support, the same conjunctive closure, but they have eventually different disjunctive supports. Thereby, these rare equivalence classes, *i.e*, those containing just rare patterns, are evaluated by the anti-monotonic constraint of correlation. The rare patterns belonging to the same class, are then divided into rare correlated patterns and rare non-correlated ones as shown by Figure \[Exp1\].
Let us consider the extraction context given by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. For *minsupp* = 3, we distinguish two rare equivalence classes $\mathcal{C}_1$ and $\mathcal{C}_2$ shown in Figure \[Exp1\] and composed by the following elements :
- $\mathcal{C}_1$ contains the itemsets `D`, `AD`, `CD` and `ACD`. $\mathcal{C}_1$ has the value of conjunctive support equal to 1 and the conjunctive closed pattern is `ACD`.
- $\mathcal{C}_2$ contains the itemsets `AB`, `AE`, `ABC`, `ABE`, `ACE`, and `ABCE`. $\mathcal{C}_2$ has the value of conjunctive support equal to 2 and the conjunctive closed pattern is `ABCE`.
Let us apply the correlation constraint for a minimal threshold *minbond* = 0.3. For the $\mathcal{C}_1$ equivalence class, the patterns $\{$<span style="font-variant:small-caps;">(</span>`D`, 1, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>`AD`, 1, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span>$\}$ are rare correlated whereas <span style="font-variant:small-caps;">(</span>`CD`, 1, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>`ACD`, 1, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span> are rare non-correlated itemsets. This is explained by the fact that the elements of the equivalence class $\mathcal{C}_1$ do not share the same disjunctive support, consequently they do not share the same *bond* value. On the other side, all of the elements of the equivalence class $\mathcal{C}_2$ are rare correlated.
### Second Scenario
The second scenario consists in extracting all of the correlated patterns and partitioning them into equivalence classes thanks to the $f_{bond}$ closure operator, then filtering out the obtained equivalence classes by the rarity constraint. In fact, all the itemsets belonging to the same equivalence class share obviously the same conjunctive, disjunctive supports and the same *bond* measure. Consequently, when considering the anti-monotonic constraint of correlation, we can distinguish two kinds of classes namely: correlated classes and non-correlated ones. The good question that we have to think about it is what will be the effect of applying the monotonic constraint of rarity within this classes? In other words, how these equivalence classes will be affected by the interception of the rarity constraint?
In fact, the $f_{bond}$ closure operator preserves the *bond* value, the conjunctive, the disjunctive as well as the negative supports in the same equivalence class. Consequently, all the itemsets belonging to the same equivalence class present the same behavior regarding to the constraints of rarity and those of correlation. In this respect, the correlated patterns of an equivalence class, are either rare correlated or they are frequent correlated. This property also hold for the non correlated equivalence classes. Therefore, these classes are not affected by the application of the rarity constraint. As shown by Figure \[Exp2\], we have correlated frequent classes, correlated rare classes, non correlated frequent classes and non correlated rare classes.
### Summary
The equivalence classes induced by the $f_{bond}$ closure operator present pertinent characteristics. In fact, this privilege is not offered by the conjunctive closure operator [@ganter99]. In fact, the state of a given pattern in an equivalence class induced by the conjunctive closure operator is not representative of the state of the other patterns of its same class. In this regard, we are motivated for the application of the second scenario within the design of our mining approach.
We introduce, in the next section, our new <span style="font-variant:small-caps;">Gmjp</span> approach $^{\textsc{(}}$[^5]$^{\textsc{)}}$.
The <span style="font-variant:small-caps;">Gmjp</span> approach {#section_algo}
---------------------------------------------------------------
We introduce in this section the <span style="font-variant:small-caps;">Gmjp</span> approach which allows, according to the user’s input parameters, the extraction of the desired output. As shown by Figure \[new\_overview\], four different scenarios are possible for running the <span style="font-variant:small-caps;">Gmjp</span> approach:\
$\bullet$ **First Scenario**: outputs the whole set $\mathcal{FCP}$ of frequent correlated patterns,\
$\bullet$ **Second Scenario**: outputs the $\mathcal{RFCCP}$ concise exact representation of the $\mathcal{FCP}$ set,\
$\bullet$ **Third Scenario**: outputs the whole set $\mathcal{RCP}$ of rare correlated patterns,\
$\bullet$ **Fourth Scenario**: outputs the $\mathcal{RCPR}$ concise exact representation of the $\mathcal{RCP}$ set. The <span style="font-variant:small-caps;">Gmjp</span> algorithm takes as an input a dataset $\mathcal{D}$, a minimal support threshold *minsupp* and a minimal correlation threshold *minbond*. We mention that <span style="font-variant:small-caps;">Gmjp</span> determines exactly the *support* and the *bond* values of each pattern of the desired output according to the user’s parameters.
![Overview of <span style="font-variant:small-caps;">Gmjp</span> approach.[]{data-label="new_overview"}](gmjp.eps)
### Overview of the approach
We illustrate the different steps of <span style="font-variant:small-caps;">Gmjp</span> when running the fourth script aiming to extract the $\mathcal{RCPR}$ representation. Our choice of this fourth scenario is motivated by the fact that the extraction of the $\mathcal{RCPR}$ representation corresponds to the most challenging mining task for <span style="font-variant:small-caps;">Gmjp</span>.
In fact, $\mathcal{RCPR}$ is composed by the set of rare correlated patterns which results from the intersection of two theories [@mannila97] induced by the constraints of correlation and rarity. So, this set is neither an order ideal nor an order filter. Therefore, the localization of the elements of the $\mathcal{RCPR}$ representation is more difficult than the localization of theories corresponding to the conjunction of constraints of the same nature. Indeed, the conjunction of anti-monotonic constraints <span style="font-variant:small-caps;">(</span>*resp.* monotonic<span style="font-variant:small-caps;">)</span> is an anti-monotonic constraint <span style="font-variant:small-caps;">(</span>*resp.* monotonic<span style="font-variant:small-caps;">)</span> [@luccheKIS05_MAJ_06]. For example, the constraint “being a correlated frequent pattern” is anti-monotonic, since it results from the conjunction of two anti-monotonic constraints namely, “being a correlated pattern” and “being a frequent pattern”. This constraint induces, then, an order ideal on the itemset lattice.
In fact, the <span style="font-variant:small-caps;">Gmjp</span> algorithm mainly operates in three steps as depicted by Figure \[figure\_overview\]. The pseudo-code of <span style="font-variant:small-caps;">Gmjp</span> is given by Algorithm \[AlgoCRPR\].
1. A first scan of the dataset is performed in order to extract all the items and assigning to each item the set of transactions in which it appears. Then, a second scan of the dataset is carried out in order to identify, for each item, the list of the co-occurrent items <span style="font-variant:small-caps;">(</span>*cf.* Line 1 Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
2. The second step consists in integrating both of the constraints rarity as well as correlation within a mining process of $\mathcal{RCPR}$. In this situation, this problem is split into independent chunks since each item is separately treated. In fact, for each item <span style="font-variant:small-caps;">(</span>*cf.* Line 2 Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>, a set of candidates is generated <span style="font-variant:small-caps;">(</span>*cf.* Line (b) Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>. Once obtained, these candidates are pruned using the following pruning strategies:\
<span style="font-variant:small-caps;">(</span>***a***<span style="font-variant:small-caps;">)</span> **The pruning of the candidates which check the cross-support property** <span style="font-variant:small-caps;">(</span>*cf.* Line (i) Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>[@tarekds2010]. In fact, as defined in section \[S2chap4\] <span style="font-variant:small-caps;">(</span>*cf.* Chapter 4, page <span style="font-variant:small-caps;">)</span>, the cross-support property allows to prune non-correlated candidates. More clearly, any pattern, containing two items fulfilling the cross-support property w.r.t. a minimal threshold of correlation, is not correlated. Thus, this property avoids the computation of its conjunctive and disjunctive supports, required to evaluate its *bond* value.\
<span style="font-variant:small-caps;">(</span>***b***<span style="font-variant:small-caps;">)</span> **The pruning based on the order ideal of the correlated patterns** <span style="font-variant:small-caps;">(</span>*cf.* Line (ii) Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>. Recall that the set of correlated patterns induces an order ideal property. Therefore, each correlated candidate, having a non correlated subset, will be pruned since it will not be a correlated pattern.
Then, the conjunctive, disjunctive supports and the *bond* value of the retained candidates are computed <span style="font-variant:small-caps;">(</span>*cf.* Line (iii) Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>. Thus, the uncorrelated candidates are also pruned. At the level $n$, the local minimal rare correlated patterns of size $n$ are determined among the retained candidates <span style="font-variant:small-caps;">(</span>*cf.* Line (iv) Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>. The local closed rare correlated patterns of size $n-1$ are also filtered <span style="font-variant:small-caps;">(</span>*cf.* Line (v) Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>. This process comes to an end when there is no more candidates to be generated <span style="font-variant:small-caps;">(</span>*cf.* Line (c) Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
3. The third and last step consists of filtering the global minimal rare correlated patterns <span style="font-variant:small-caps;">(</span>*cf.* Line 3 Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span> and the global rare correlated patterns among the two sets of local minimal rare correlated patterns and local closed ones <span style="font-variant:small-caps;">(</span>*cf.* Line 4 Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
In what follows, we will explain more deeply these different steps of <span style="font-variant:small-caps;">Gmjp</span>.
![Overview of <span style="font-variant:small-caps;">Gmjp</span> when extracting the $\mathcal{RCPR}$ representation.[]{data-label="figure_overview"}](Fig3b.eps)
\[AlgoCRPR\]
### First Step: The power of the bit vectors and of co-occurrent vectors
Initially, the dataset is scanned in order to extract the items and to build, for each item, the bitset called here “BSVector”. In fact, a bitset is a container that can store a huge number of bits while optimizing the memory consumption <span style="font-variant:small-caps;">(</span>For example, 32 elements are stored in a memory block of 4 bytes<span style="font-variant:small-caps;">)</span>. Each block of memory is treated in just one `CPU` operation by a 32 bits processor. Therefore, we were motivated for these kinds of structures within the <span style="font-variant:small-caps;">Gmjp</span> algorithm in order to optimize the conjunctive and the disjunctive supports computations.
Then, the dataset is scanned again in order to identify, for each item $I$, the list of the co-occurrent items which corresponds to the items occurring in the same transactions as the item $I$. These latter ones are stored in a vector of integers, called here “COVector”. We note that one of the main challenges of the <span style="font-variant:small-caps;">Gmjp</span> algorithm is that it allows pushing two constraints of distinct types and to deliver the output with only two scans of the dataset. We also uphold that the bitsets, when incorporated into the mining process within the <span style="font-variant:small-caps;">Gmjp</span> algorithm, sharply decrease the size of the memory required to store immediate results and significantly save execution costs.
### Second Step: Getting the Local Minimal and the Local Closed Rare Correlated Patterns without closure computations
Worth of mention, the main thrust of the <span style="font-variant:small-caps;">Gmjp</span> algorithm is to break the search space into independent sub-spaces. In fact, for each item $I$, a level-wise mining process is performed using the COVector containing the co-occurrent items of $I$. At each level $n$, starting by the second level, a set of candidates is generated, then pruned according to the different pruning strategies described previously. The minimal rare correlated patterns of size $n$, associated to the item $I$, are called **Local Minimal Rare Correlated Patterns** and they are determined by comparing their *bond* values versus those of their respective immediate subsets. Similarly, the closed rare correlated patterns of size $n-1$ associated to the item $I$ are called **Local Closed Rare Correlated Patterns**, and they are determined by comparing their *bond* values to those of their respective immediate supersets.
It is also important to mention that the implementation of the different stages of this second step <span style="font-variant:small-caps;">(</span>candidate generation, evaluation and pruning<span style="font-variant:small-caps;">)</span> was based on simple vectors of integers. Thus, we do not require more complex data structure during the implementation of the <span style="font-variant:small-caps;">Gmjp</span> algorithm. This feature makes <span style="font-variant:small-caps;">Gmjp</span> a practical approach for handling both monotonic and anti-monotonic constraints even for large datasets.
One of the major challenges in the design of the <span style="font-variant:small-caps;">Gmjp</span> algorithm is how to perform subset and superset checking to efficiently identify Local Minimal and Local Closed patterns$?$ The answer is to construct and manage a multi-map hash structure, $^{\textsc{(}}$[^6]$^{\textsc{)}}$ in order to store at each level $n$ the rare correlated patterns of size $n$. This technique has been shown to be very powerful since it makes the subset and the superset checking practical even on dense datasets.
Thus, our proposed efficient solution <span style="font-variant:small-caps;">(</span>as we prove it experimentally later<span style="font-variant:small-caps;">)</span> is to integrate both of the monotonic constraint of rarity and the anti-monotonic constraint of correlation within the mining process and to identify the local closed rare correlated patterns without closure computing.
### Third Step: Filtering the Global Minimal and the Global Closed Rare Correlated patterns
After identifying the local minimal and the local closed rare correlated patterns associated to each item $I$ of the dataset $\mathcal{D}$, the third step consists in filtering the $\mathcal{MRCP}$ set of Global Minimal Rare Correlated patterns and the $\mathcal{CRCP}$ set of Global Closed Rare Correlated patterns. This task is performed using two distinct multi-map hash structures. In fact, for each local minimal rare correlated pattern $LM$ previously identified, we check whether it has a direct subset <span style="font-variant:small-caps;">(</span>belonging to the whole set of local minimal patterns<span style="font-variant:small-caps;">)</span> with the same *bond* value. If it is not the case, then the local minimal pattern $LM$ is a global minimal rare pattern and it is added to the $\mathcal{MRCP}$ set. Similarly, for each local closed rare correlated pattern $LC$ previously identified, we check whether it has a direct superset <span style="font-variant:small-caps;">(</span>belonging to the whole set of local closed patterns<span style="font-variant:small-caps;">)</span> with the same *bond* value. If it is not the case, then the local closed pattern $LC$ is a global closed rare pattern and it is added to the $\mathcal{CRCP}$ set of Closed rare correlated patterns.
We note that we are limited to the description of the extraction of the $\mathcal{RCPR}$ representation since the post-processing operation of the representations $\mathcal{MM}$ax$\mathcal{CR}$, $\mathcal{M}$in$\mathcal{MCR}$ from the $\mathcal{RCPR}$ representation is obviously done and we prove that the needed execution time is negligible.
In what follows, we illustrate with a running example of the <span style="font-variant:small-caps;">Gmjp</span> algorithm.
### Running example
Let us consider the extraction context $\mathcal{C}$ sketched by Table \[Base\_transactions\] <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>. First, the BSVectors and the COVectors associated to each item of this dataset are constructed, as we plot by Figure \[figure\_COVector\].
These BSVectors are next used to compute the conjunctive and the disjunctive supports. We have, for example, the item $A$ which belongs to the transactions $\{1, 3, 5\}$ and the item $C$ which belongs to the transactions $\{1, 2, 3, 5\}$. We, then, have *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$*AC*<span style="font-variant:small-caps;">)</span> = 3 and *Supp*<span style="font-variant:small-caps;">(</span>$\vee$*AC*<span style="font-variant:small-caps;">)</span><span style="font-variant:small-caps;">)</span> = 4.
The local minimal and the local closed correlated rare patterns associated to each item $I$ of the dataset $\mathcal{D}$, are extracted. A detailed example of the process of the item $A$ is given by Figure \[figure\_ExpItemA\].
The finally obtained $\mathcal{RCPR}$ representation, for *minsupp* = 4 and for *minbond* = 0.20, is composed by the following global minimal and global closed correlated patterns: $\mathcal{RCPR}$ = $\{$ <span style="font-variant:small-caps;">(</span>$A$, $3$, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$D$, $1$, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AB$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AC$, $3$, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AD$, $1$, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BC$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CD$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACD$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BCE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span> and <span style="font-variant:small-caps;">(</span>$ABCE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>$\}$.
Last, it is important to notice that <span style="font-variant:small-caps;">Gmjp</span> is not an exclusive approach in the sense that it can be coupled with other efficient approaches to mine statistically significant patterns.
In the next section, we present <span style="font-variant:small-caps;">Opt-Gmjp</span> the optimized version of the <span style="font-variant:small-caps;">Gmjp</span> algorithm.
<span style="font-variant:small-caps;">Opt-Gmjp</span>: The optimized version of <span style="font-variant:small-caps;">Gmjp</span> {#se-optgmjp}
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In this section, we present <span style="font-variant:small-caps;">Opt-Gmjp</span> the optimized version of <span style="font-variant:small-caps;">Gmjp</span> approach. Our improvements cover the four different scenarios S1, S2, S3 and S4 of the <span style="font-variant:small-caps;">Gmjp</span> approach. Nevertheless, we describe specifically the optimization of the third scenario S3. In fact, the latter deals with two constraints of distinct types namely the anti-monotonic constraint of correlation and the monotonic constraint of rarity.
We start by presenting a generic overview of the <span style="font-variant:small-caps;">Opt-Gmjp</span> algorithm which is illustrated by Figure \[figure-optgmjp\]. The pseudo code of <span style="font-variant:small-caps;">Opt-Gmjp</span>, when running the third scenario S3, is given by Algorithm \[AlgoCori\] while all of the used notations are illustrated in table \[notationsTab\]. We note that the “MC“ notation stands for “Monotonic Constraint“ while the “AMC“ notation stands for “Anti-Monotonic Constraint“.
In fact, the proposed optimizations are of two types: the transformation of the initial extraction context and the reduction of the number of distinct constraints evaluation. In fact, the measurement of the impact of pushing the monotonic constraint of rarity and the anti-monotonic constraint of correlation early within the <span style="font-variant:small-caps;">Opt-Gmjp</span> algorithm helps to measure the selectivity power of each type of constraint during the mining process.
1. **Transformation of the initial extraction context**\
Initially, the extraction context is scanned once to build the new transformed extraction context and to construct an in-memory structure <span style="font-variant:small-caps;">(</span>*cf.* Line 1 Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>. In fact, we assign to each item a bitset, each column of this bitset indicates the presence or the absence of the item in a specified transaction. For example, if the third column of this list contains 0, then the item $I$ is not present in the third transaction. The transformed extraction context associated to the initial context of Table \[BD\] is given by Table \[BDnew\].
2. **Initialization of the tree-data structure**\
Initially, the $\mathcal{RCP}$ set of rare correlated pattern is set to the empty-set <span style="font-variant:small-caps;">(</span>*cf.* Line 2 Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>. Then, we compute the conjunctive support of the items. The items are then sorted in an ascending order of their support <span style="font-variant:small-caps;">(</span>*cf.* Line 3 Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>. All the items are added to the nodes of the first level of our tree structure. Thus, they constitute the set of 1-itemsets candidates $\mathcal{CAND}_{1}$ <span style="font-variant:small-caps;">(</span>*cf.* Line 4 Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>. All the items are evaluated according to the monotone constraint of rarity: The rare items are printed to the output set, <span style="font-variant:small-caps;">(</span>*cf.* Line <span style="font-variant:small-caps;">(</span>b<span style="font-variant:small-caps;">)</span> Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>, while frequent ones are not pruned, *i.e.*, they are maintained.
3. **Solving recursively the mining problem**\
This step is the main optimization of our algorithm. The idea consists in dividing our mining problem into sub-problems. For each item `I`, a sub-tree is constructed and a depth-first traversal is therefore performed. The candidates of size $n$ are generated by building intersection of itemsets of size $n-1$ <span style="font-variant:small-caps;">(</span>*cf.* Line <span style="font-variant:small-caps;">(</span>d<span style="font-variant:small-caps;">)</span> Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>. They are then evaluated as follows:
- **Evaluation of the anti-monotone constraint of correlation**: if the candidate is correlated, we have to distinguish two possible cases:
- **if the candidate is rare correlated**, then it is added to the result set <span style="font-variant:small-caps;">(</span>*cf.* Line <span style="font-variant:small-caps;">(</span>i.1.<span style="font-variant:small-caps;">)</span> Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>.
- **if the candidate is frequent correlated**, then it will not be pruned. In fact, a frequent pattern can have rare supersets. In this case, the candidate is maintained and we continue to develop its sub-tree.
- **if the candidate is not correlated**, then all its supersets will not be correlated according to the monotonicity property of the non-correlation constraint. In this case, the candidate and the associated sub-tree are pruned <span style="font-variant:small-caps;">(</span>*cf.* Line <span style="font-variant:small-caps;">(</span>i.2.<span style="font-variant:small-caps;">)</span> Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>.
This generation and evaluation process is continued, the dedicated procedure is recursively called while there is a number of candidates to be generated <span style="font-variant:small-caps;">(</span>*cf.* Line <span style="font-variant:small-caps;">(</span>g<span style="font-variant:small-caps;">)</span> Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>. Finally, the used memory is freed and the $\mathcal{RCP}$ is outputted <span style="font-variant:small-caps;">(</span>*cf.* Line 6 Algorithm \[AlgoCori\]<span style="font-variant:small-caps;">)</span>. The used data structure is illustrated by Figure \[figure-exp-tree\].
4. **A running Example**
The example shown in Figure \[figure-exp-tree\], illustrates how <span style="font-variant:small-caps;">Opt-Gmjp</span> works. The tree is made from five sorted items, `D`, `A`, `B`, `C`, and `E`, with their respective supports 1, 3, 4, 4 and 4 respectively. We begin by the node containing the item having the lowest support: which is `D`. The anti-monotone constraint ’Amc‘ is having the *bond* value $\geq$ 0.20 and the monotone constraint ’Mc‘ is having the conjunctive support $<$ 4. Intersecting `D` with `A`, `B`, `C`, and `E` produces `DA`, `DB`, `DC` and `DE`. The candidates `DB` and `DE` are pruned since they have null support. Thus, their supersets will also have null support. The candidates `DA` and `DC` are rare correlated, thus they pass both constraints Amc and Mc. Their intersection produces `DAC` with support 1 and bond value equal to $\displaystyle\frac{1}{4}$, which also fulfills both constraints. The rare correlated itemsets are added to the output set and the `D`-sub-tree is deleted from our tree. The `A`-sub-tree, `B`-sub-tree, `C`-sub-tree and `E`-sub-tree are successively built. The same process is repeated recursively until there’s no more candidates to be generated and evaluated. Finally, the obtained result set of correlated rare itemsets is composed by: <span style="font-variant:small-caps;">(</span>$D$, $1$, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$A$, $3$, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AB$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AC$, $3$, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BC$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$DA$, $1$, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$DC$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABC$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACE$, $3$, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BCE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$DAC$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, and <span style="font-variant:small-caps;">(</span>$ABCE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>.
\[AlgoCori\]
In the next section, we present our analysis of the theoretical time complexity of the <span style="font-variant:small-caps;">Gmjp</span> algorithm.
Theoretical Time Complexity {#se-comp}
---------------------------
Proposition \[comp\] gives the theoretical time complexity of the <span style="font-variant:small-caps;">Gmjp</span> algorithm when running the fourth scenario dedicated to the extraction of the $\mathcal{RCPR}$ representation.
\[comp\] The worst case time complexity of the first step is bounded by $O\textsc{(}N \times M\textsc{)}$, that of the second step is bounded by $O\textsc{(}\textsc{(}N^3 + \textsc{(}N^2 \times M\textsc{)}\textsc{)}\times 2^N\textsc{)}$, while that of the third step is bounded by $O\textsc{(}N^2\textsc{)}$, where $M$ = $|\mathcal{T}|$ and $N$ = $|\mathcal{I}|$. The theoretical complexity in the worst case of the <span style="font-variant:small-caps;">Gmjp</span> algorithm is bounded by the sum of those of its three steps.
First of all, let us recall the respective roles of the distinct steps of the <span style="font-variant:small-caps;">Gmjp</span> algorithm.
- **The first step:** *Scanning the extraction context twice in order to build the bitset vector and the co-occurring vector associated to each item $I$.*\
The complexity $\mathcal{C}_{_{1}}$ of this step, is equal to, $\mathcal{C}_{_{1}}$ = 2 $\times$ $O\textsc{(}N \times M\textsc{)}$ $\approx$ $O\textsc{(}N \times M\textsc{)}$.
- **The second step:** *Extracting the local minimal and the local closed rare correlated patterns.*\
The cost of this step is equal to those of its associated instructions which are as follows:
1. The cost of the initialization of the integer $n$ carried out in line <span style="font-variant:small-caps;">(</span>a<span style="font-variant:small-caps;">)</span> <span style="font-variant:small-caps;">(</span>*cf.* Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span> is in $O\textsc{(}1\textsc{)}$.
2. The generation of the candidates of size $n$, <span style="font-variant:small-caps;">(</span>*cf.* line <span style="font-variant:small-caps;">(</span>b<span style="font-variant:small-caps;">)</span> in Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>, is done in $O\textsc{(}N-1\textsc{)}$ since in the worst case the number of generated candidates is $N-1$.
3. The cost of the pruning of candidates *w.r.t.* the cross-support property of the bond measure is done in $O\textsc{(}N^2\textsc{)}$ <span style="font-variant:small-caps;">(</span>*cf.* line <span style="font-variant:small-caps;">(</span>i<span style="font-variant:small-caps;">)</span> in Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
4. The cost of the pruning of candidates *w.r.t.* the ideal order property of correlated patterns is done in $O\textsc{(}N^2\textsc{)}$ <span style="font-variant:small-caps;">(</span>*cf.* line <span style="font-variant:small-caps;">(</span>ii<span style="font-variant:small-caps;">)</span> in Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
5. The cost of the computation of the conjunctive and the disjunctive supports of the itemset candidates is bounded by $O\textsc{(}N \times M\textsc{)}$ <span style="font-variant:small-caps;">(</span>*cf.* line <span style="font-variant:small-caps;">(</span>iii<span style="font-variant:small-caps;">)</span> in Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
6. The checking of the constraints of rarity and of correlation is done in $O\textsc{(}1\textsc{)}$, while the checking of the local minimality of the set of candidates is done in $O\textsc{(}N^2\textsc{)}$ and the updating of the $\mathcal{MRCP}$ set of minimal rare correlated patterns is done in $O\textsc{(}1\textsc{)}$ <span style="font-variant:small-caps;">(</span>*cf.* line <span style="font-variant:small-caps;">(</span>iv<span style="font-variant:small-caps;">)</span> in Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
7. The extraction of the local closed rare correlated patterns of size $n-1$ is bounded by $O\textsc{(}N^2\textsc{)}$ <span style="font-variant:small-caps;">(</span>*cf.* line <span style="font-variant:small-caps;">(</span>v<span style="font-variant:small-caps;">)</span> in Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
8. The cost of increasing the integer $n$ is done in $O\textsc{(}1\textsc{)}$ <span style="font-variant:small-caps;">(</span>*cf.* line <span style="font-variant:small-caps;">(</span>vi<span style="font-variant:small-caps;">)</span> in Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
9. There are, in the worst case, $\textsc{(}2^{N} - N - 1\textsc{)}$ candidates to be generated using the <span style="font-variant:small-caps;">Apriori-Gen</span> procedure <span style="font-variant:small-caps;">(</span>*cf.* line <span style="font-variant:small-caps;">(</span>vii<span style="font-variant:small-caps;">)</span> in Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>. The cost of this step is bounded by $O\textsc{(}2^{N} - N\textsc{)}$.
Consequently, the cost $\mathcal{C}_{_{2}}$ of this second step, is approximatively equal to:
$\mathcal{C}_{_{2}}$ = $O\textsc{(}1\textsc{)}$ $+$ $O$<span style="font-variant:small-caps;">(</span>$N$ - $1$<span style="font-variant:small-caps;">)</span> $\times$ <span style="font-variant:small-caps;">\[</span> $O$<span style="font-variant:small-caps;">(</span>$2^{N}$ - $N$<span style="font-variant:small-caps;">)</span> $\times$ <span style="font-variant:small-caps;">\[</span> $O\textsc{(}N^2\textsc{)}$ $+$ $O\textsc{(}N^2\textsc{)}$ $+$ $O\textsc{(}N \times M\textsc{)}$ $+$ $O\textsc{(}1\textsc{)}$ $+$ $O\textsc{(}N^2\textsc{)}$ $+$ $O\textsc{(}1\textsc{)}$ $+$ $O\textsc{(}N^2\textsc{)}$ $+$ $O\textsc{(}1\textsc{)}$ <span style="font-variant:small-caps;">\]</span><span style="font-variant:small-caps;">\]</span>
$\approx$ $O\textsc{(}1\textsc{)}$ $+$ $O$<span style="font-variant:small-caps;">(</span>$N$ - $1$<span style="font-variant:small-caps;">)</span> $\times$ <span style="font-variant:small-caps;">\[</span> $O$<span style="font-variant:small-caps;">(</span>$2^{N}$<span style="font-variant:small-caps;">)</span> $\times$ <span style="font-variant:small-caps;">\[</span> $O\textsc{(}N^2\textsc{)}$ $+$ $O\textsc{(}N \times M\textsc{)}$ <span style="font-variant:small-caps;">\]</span> <span style="font-variant:small-caps;">\]</span>
$\approx$ $O$<span style="font-variant:small-caps;">(</span>$2^{N}$<span style="font-variant:small-caps;">)</span> $\times$ <span style="font-variant:small-caps;">(</span> $O\textsc{(}N^3\textsc{)}$ $+$ $O\textsc{(}N^2 \times M\textsc{)}$<span style="font-variant:small-caps;">)</span>
$\approx$ $O$<span style="font-variant:small-caps;">(</span><span style="font-variant:small-caps;">(</span>$N^3$ + <span style="font-variant:small-caps;">(</span>$N^2$ $\times$ $M$<span style="font-variant:small-caps;">)</span><span style="font-variant:small-caps;">)</span> $\times$ $2^{N}$<span style="font-variant:small-caps;">)</span>.
- **The third step:** *Filtering the global minimal and the global closed patterns among the local ones*.
In fact, this step consists in checking for each local minimal <span style="font-variant:small-caps;">(</span>*resp.* closed<span style="font-variant:small-caps;">)</span> pattern, whether it has a subset <span style="font-variant:small-caps;">(</span>*resp.* superset<span style="font-variant:small-caps;">)</span> with the same *bond* value or not. The complexity, $\mathcal{C}_{_{3}}$, of this step is then bounded by $O\textsc{(}N^2\textsc{)}$ <span style="font-variant:small-caps;">(</span>*cf.* lines $3$ and $4$ in Algorithm \[AlgoCRPR\]<span style="font-variant:small-caps;">)</span>.
Consequently, the complexity of the <span style="font-variant:small-caps;">Gmjp</span> algorithm is equal to: $\mathcal{C}_{_{1}}$ $+$ $\mathcal{C}_{_{2}}$ $+$ $\mathcal{C}_{_{3}}$
$\approx$ $O{\textsc{(}}N \times M\textsc{)}$ $+$ $O$<span style="font-variant:small-caps;">(</span><span style="font-variant:small-caps;">(</span>$N^3$ + <span style="font-variant:small-caps;">(</span>$N^2$ $\times$ $M$<span style="font-variant:small-caps;">)</span><span style="font-variant:small-caps;">)</span> $\times$ $2^{N}$<span style="font-variant:small-caps;">)</span> $+$ $O\textsc{(}N^2\textsc{)}$
$\approx$ $O$<span style="font-variant:small-caps;">(</span><span style="font-variant:small-caps;">(</span>$N^3$ + <span style="font-variant:small-caps;">(</span>$N^2$ $\times$ $M$<span style="font-variant:small-caps;">)</span><span style="font-variant:small-caps;">)</span> $\times$ $2^{N}$<span style="font-variant:small-caps;">)</span> $\approx$ $2^{N}$.
It is important to mention that the complexity in the worst case of the <span style="font-variant:small-caps;">Gmjp</span> algorithm is not reachable in practice. Indeed, there is not a context that simultaneously gives the respective worst case complexities of the three steps. Hence, the worst case complexity of <span style="font-variant:small-caps;">Gmjp</span> is roughly bounded by the sum of those of its three steps.
In the next section, we describe the query process of the $\mathcal{RCPR}$ representation and the regeneration of the $\mathcal{RCP}$ set from the $\mathcal{RCPR}$ representation.
The query and the regeneration strategies {#sec_Regeneration}
-----------------------------------------
We begin the first sub-section with the querying strategy of the $\mathcal{RCPR}$ representation.
### Querying of the $\mathcal{RCPR}$ representation {#sub_sec_query}
In the following, we introduce the <span style="font-variant:small-caps;">Regenerate</span> algorithm, whose pseudo code is given by Algorithm \[Rege1\], dedicated to the query of the $\mathcal{RCPR}$ representation. In fact, the interrogation of the $\mathcal{RCPR}$ representation allows determining the nature of a given pattern. If it is a rare correlated pattern, then, its conjunctive, disjunctive, negative supports as well as its *bond* value are faithfully derived from the $\mathcal{RCPR}$ representation. $\bullet$ **Description of the <span style="font-variant:small-caps;">Regenerate</span> algorithm**
The <span style="font-variant:small-caps;">Regenerate</span> algorithm takes as an input the number of the transactions $|\mathcal{T}|$, the $\mathcal{RCPR}$ representation and an arbitrary itemset $I$ and it proceeds in two distinct ways depending on the state of $I$:
- If the itemset $I$ belong to the $\mathcal{RCPR}$ representation <span style="font-variant:small-caps;">(</span>*cf.* Line 2 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>, then $I$ is a rare correlated itemset. In this regard, we have the conjunctive support and the *bond* values, thus we compute the disjunctive and the negative supports. The disjunctive support is equal to the ratio of the conjunctive support by the *bond* value, <span style="font-variant:small-caps;">(</span>*cf.* Line 3 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>. The negative support is equal to the number of transactions $|\mathcal{T}|$ minus the disjunctive support <span style="font-variant:small-caps;">(</span>*cf.* Line 4 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>. The algorithm returns the values of the different supports as well as the *bond* value of the rare correlated itemset $I$, <span style="font-variant:small-caps;">(</span>*cf.* Line 5 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>.
- If the itemset $I$ do not belong to the $\mathcal{RCPR}$ representation, then we distinguish two different cases:
- If it exists two itemsets $J$ and $Z$ belonging to $\mathcal{RCPR}$ such as, $J$ $\subset$ $I$ and $Z$ $\supset$ $I$, <span style="font-variant:small-caps;">(</span>*cf.* Line 7 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>, then $I$ is a rare correlated pattern and we have to determine its supports and correlation values. The closed itemset $F$ associated to $I$ is the minimal itemset, according to the inclusion set, that covers $I$; <span style="font-variant:small-caps;">(</span>*cf.* Line 9 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>. The conjunctive support and the *bond* value of $I$ are equal to those of its closure $F$, <span style="font-variant:small-caps;">(</span>*cf.* Lines 10 and 11 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>. The disjunctive and the supports are computed in the same manner as the first case, <span style="font-variant:small-caps;">(</span>*cf.* Lines 12 and 13 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>. Finally the algorithm outputs the values of the different supports as well as the *bond* value of the rare correlated itemset $I$, <span style="font-variant:small-caps;">(</span>*cf.* Line 14 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>.
- If it not exists two itemsets $J$ and $Z$ belonging to $\mathcal{RCPR}$ such as, $J$ $\subset$ $I$ and $Z$ $\supset$ $I$, then the algorithm outputs the emptyset to indicates that the itemset $I$ is not a rare correlated pattern, <span style="font-variant:small-caps;">(</span>*cf.* Line 16 Algorithm \[Rege1\]<span style="font-variant:small-caps;">)</span>.
\[exp\_regeneration1\] Let us consider the $\mathcal{RCPR}$ representation given by Example \[exprep1\], <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>, for *minsupp* = 4 and *minbond* = 0.2. Consider the pattern $ACE$. When comparing the pattern $ACE$ with the elements of the $\mathcal{RCPR}$ representation, we remark that $AE$ $\subset$ $ACE$ and $ACE$ $\subset$ $ABCE$. Then, the pattern $ACE$ is a rare correlated pattern and the associated closed pattern is $ABCE$. Consequently, *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$ACE$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\wedge$$ABCE$<span style="font-variant:small-caps;">)</span> = 2, *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$ACE$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$ABCE$<span style="font-variant:small-caps;">)</span> = 5, *Supp*<span style="font-variant:small-caps;">(</span>$\neg$$ACE$<span style="font-variant:small-caps;">)</span> = $|\mathcal{T}|$ - *Supp*<span style="font-variant:small-caps;">(</span>$\vee$$ACE$<span style="font-variant:small-caps;">)</span> = 5 - 5 = 0 and *bond*<span style="font-variant:small-caps;">(</span>$ACE$<span style="font-variant:small-caps;">)</span> = *bond*<span style="font-variant:small-caps;">(</span>$ABCE$<span style="font-variant:small-caps;">)</span> = $\displaystyle\frac{2}{5}$. Consider the pattern $BE$. In fact, $BE$ $\notin$ $\mathcal{RCPR}$ and there is no element of $\mathcal{RCPR}$ which is included in $BE$. Therefore, the <span style="font-variant:small-caps;">Regenerate</span> algorithm returns the empty set and indicates that the pattern $BE$ is not a rare correlated pattern.
In what follows, we introduce the strategy of regeneration of the $\mathcal{RCP}$ set, *i.e.* the set of all rare correlated patterns, from this representation.
### Regeneration of the $\mathcal{RCP}$ set from the $\mathcal{RCPR}$ representation {#sub_sec_regeneration}
The regeneration of the $\mathcal{RCP}$ set from the $\mathcal{RCPR}$ representation is achieved through the <span style="font-variant:small-caps;">RcpRegeneration</span> algorithm which pseudo-code is given by Algorithm \[CRPRegeneration\]. This latter algorithm inputs the $\mathcal{RCPR}$ representation and provides the $\mathcal{RCP}$ set of rare correlated patterns. The conjunctive support and the *bond* value of each pattern are exactly determined. $\bullet$ **Description of the <span style="font-variant:small-caps;">RcpRegeneration</span> algorithm**
The <span style="font-variant:small-caps;">RcpRegeneration</span> algorithm takes as an input the number of the transactions $|\mathcal{T}|$ and the $\mathcal{RCPR}$ representation which is composed by the two sets $\mathcal{MRCP}$ and $\mathcal{CRCP}$. The algorithm generates the $\mathcal{RCP}$ set as described in the following:
1. Initially, the $\mathcal{RCP}$ set is assigned with the empty set, <span style="font-variant:small-caps;">(</span>*cf.* Line 2 Algorithm \[CRPRegeneration\]<span style="font-variant:small-caps;">)</span>. Then, all the itemsets of the $\mathcal{RCPR}$ representation <span style="font-variant:small-caps;">(</span>The elements of the $\mathcal{MRCP}$ and the $\mathcal{CRCP}$ sets<span style="font-variant:small-caps;">)</span> are added to the $\mathcal{RCP}$ set, <span style="font-variant:small-caps;">(</span>*cf.* Lines 4 and 5 Algorithm \[CRPRegeneration\]<span style="font-variant:small-caps;">)</span>.
2. For each minimal generator $M$ of the $\mathcal{MRCP}$ set, we determine its closure $F$ among the $\mathcal{CRCP}$ set. In fact, $F$ corresponds to the minimal itemset, according to inclusion set, that covers $M$ <span style="font-variant:small-caps;">(</span>*cf.* Line 9 Algorithm \[CRPRegeneration\]<span style="font-variant:small-caps;">)</span>.
3. At this step, we derive all the patterns that are included between the minimal generator $M$ and its closure $F$. Thus, we need an intermediate itemset $D$ that contains the set of items belonging to $F$ and not to $M$: $D$ = $F$$\backslash$$M$, <span style="font-variant:small-caps;">(</span>*cf.* Line 11 Algorithm \[CRPRegeneration\]<span style="font-variant:small-caps;">)</span>. Then, each item $j$ included in the itemset $D$, is concatenated with the generator $M$ in order to form a new itemset $X$: $X$ = $M$ $\cup$ $j$, <span style="font-variant:small-caps;">(</span>*cf.* Line 13 Algorithm \[CRPRegeneration\]<span style="font-variant:small-caps;">)</span>. The conjunctive support and the *bond* value of $X$ are equal to those of its closure $F$ <span style="font-variant:small-caps;">(</span>*cf.* Lines 14 and 15 Algorithm \[CRPRegeneration\]<span style="font-variant:small-caps;">)</span>.
4. Finally, the rare correlated itemset $X$ is added to the $\mathcal{RCP}$ set after the non-redundancy checking <span style="font-variant:small-caps;">(</span>*cf.* Line 18 Algorithm \[CRPRegeneration\]<span style="font-variant:small-caps;">)</span>. The whole $\mathcal{RCP}$ set is outputted in Line 19 of algorithm \[CRPRegeneration\].
\[CRPRegeneration\]
Consider the $\mathcal{RCPR}$ representation illustrated by example \[exprep1\] , <span style="font-variant:small-caps;">(</span>Page <span style="font-variant:small-caps;">)</span>, for *minsupp* = 4 and *minbond* = 0.2. The regeneration of the $\mathcal{RCP}$ set is carried out as follows. Firstly, the $\mathcal{RCP}$ set is initialized to the empty set. Then, the elements of the $\mathcal{RCPR}$ representation are appended to the $\mathcal{RCP}$ set. We have so, $\mathcal{RCP}$ = $\{$<span style="font-variant:small-caps;">(</span>$A$, $3$, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$D$, $1$, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AB$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AC$, $3$, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AD$, $1$, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BC$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CD$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACD$, $1$, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BCE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABCE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>$\}$. Then, the patterns $ABE$ and $ABC$ included between the minimal pattern <span style="font-variant:small-caps;">(</span>$AB$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span> and its closure <span style="font-variant:small-caps;">(</span>$ABCE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span> are generated. In addition, the pattern $ACE$, included between the minimal pattern <span style="font-variant:small-caps;">(</span>$AE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span> and its closure <span style="font-variant:small-caps;">(</span>$ABCE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, is also derived. The patterns $ABE$, $ABC$ and $ACE$ are also generated, as they share the same conjunctive support and *bond* value of the closed pattern $ABCE$, are then inserted in the $\mathcal{RCP}$ set. This latter is updated and contains all the rare correlated patterns: $\mathcal{RCP}$ = $\{$<span style="font-variant:small-caps;">(</span>$A$, $3$, $\displaystyle\frac{3}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$D$, 1, $\displaystyle\frac{1}{1}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AB$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AC$, $3$, $\displaystyle\frac{3}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AD$, 1, $\displaystyle\frac{1}{3}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$AE$, $2$, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$BC$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CD$, 1, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$CE$, $3$, $\displaystyle\frac{3}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABC$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACD$, 1, $\displaystyle\frac{1}{4}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ACE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>, <span style="font-variant:small-caps;">(</span>$ABCE$, 2, $\displaystyle\frac{2}{5}$<span style="font-variant:small-caps;">)</span>$\}$.
Conclusion {#ConcChap5}
----------
We introduced, in this chapter, the <span style="font-variant:small-caps;">Gmjp</span> extraction approach to mine correlated patterns in a generic way <span style="font-variant:small-caps;">(</span>i.e., with two types of constraints: anti-monotonic constraint of frequency and monotonic constraint of rarity<span style="font-variant:small-caps;">)</span>. Our approach is based on the key notion of bitsets codification that supports efficient correlated patterns computation thanks to an adequate condensed representation of patterns. In the next chapter, we present our experimental evaluation of <span style="font-variant:small-caps;">Gmjp</span> and of <span style="font-variant:small-caps;">Opt-Gmjp</span> according to both quantitative and qualitative aspects.
Experimental Validation {#ch_6}
=======================
Introduction {#IntroChap6}
------------
This chapter is devoted to the experimental evaluation of the proposed <span style="font-variant:small-caps;">Gmjp</span> algorithm. Our evaluation is performed on two principal axes. In Section \[se1Chap6\], we present the experimental environment, specifically the characteristics of the used datasets as well as the experimental protocol. Then, we present in Section \[sec\_XP1\] the qualitative evaluation of the proposed condensed representations which is measured by the compactness rates offered by each proposed exact and approximate concise representation. Section \[sec\_XP2\], is dedicated to the evaluation of the performance of both <span style="font-variant:small-caps;">Gmjp</span>, as well as the optimized version that highlights the important measured improvements.
Experimental Environment {#se1Chap6}
------------------------
In this chapter, we aim to show, through extensive carried out experiments, that the different proposed concise representations provide interesting compactness rates compared to the whole set of correlated patterns. In addition to this, we aim to prove the efficiency of the proposed <span style="font-variant:small-caps;">Gmjp</span> approach. In our experiments, we used two evaluation measures: the conjunctive support to measure the frequency <span style="font-variant:small-caps;">(</span>respectively the rarity<span style="font-variant:small-caps;">)</span> and the *bond* measure to evaluate the correlation of a pattern.
All experiments were carried out on a PC equipped with a $2.4$ GHz Intel Core TM i3 processor and $4$ GB of main memory, running the Linux Ubuntu 12.04. The used datasets are described in what follows.
### Datasets
The experiments were carried out on different dense and sparse benchmark datasets $^{\textsc{(}}$[^7]$^{\textsc{)}}$. Table \[Caract\_bases\_benchmark\] summarizes the characteristics of the considered datasets. A brief description of the content of each dataset is given below:
- **<span style="font-variant:small-caps;">Connect</span>**: This dataset contains all legal positions in the game of connect-4 in which neither player has won yet, and in which the next move is not forced.
- **<span style="font-variant:small-caps;">Mushroom</span>**: This dataset includes descriptions of hypothetical samples corresponding to 23 species of gilled mushrooms.
- **<span style="font-variant:small-caps;">Pumsb</span>**: This dataset contains Census Data from PUMS <span style="font-variant:small-caps;">(</span>Public Use Microdata Samples<span style="font-variant:small-caps;">)</span>. Each object represents the answers to a census questionnaire.
- **<span style="font-variant:small-caps;">Pumsb\*</span>**: This dataset is obtained after deleting all frequent items for a minimum support threshold set to 80$\%$ in the original <span style="font-variant:small-caps;">Pumsb</span>.
- **<span style="font-variant:small-caps;">Retail</span>**: The <span style="font-variant:small-caps;">Retail</span> dataset contains information about Market Basket of clients in a Belgian Supermarket.
- **<span style="font-variant:small-caps;">Accidents</span>**: This dataset represents traffic accidents obtained from the National Institute of Statistics <span style="font-variant:small-caps;">(</span>NIS<span style="font-variant:small-caps;">)</span> for the Region of Flanders <span style="font-variant:small-caps;">(</span>Belgium<span style="font-variant:small-caps;">)</span>.
- **<span style="font-variant:small-caps;">T10I4D100K</span>**: This is a synthetic dataset generated using the generator from the IBM Almaden Quest Research Group. The goal of this generator is to create objects similar to those obtained in a supermarket environment.
- **<span style="font-variant:small-caps;">T40I10D100K</span>**: Identically to <span style="font-variant:small-caps;">T10I4D100K</span>, this dataset is also generated from the IBM generator. The differences between this dataset and <span style="font-variant:small-caps;">T10I4D100K</span> are the number of items and the average size of the objects.
[lrrrr]{} **Dataset**&**Property** &**Number** & **Number of**& **Average length**\
& &**of items** &**transactions**& **of transactions**\
<span style="font-variant:small-caps;">Connect</span> & **Dense** &[129]{} & [67, 557]{} & [43.00]{}\
<span style="font-variant:small-caps;">Chess</span> & **Dense** &[75]{} & [3 196]{} & [37.00]{}\
<span style="font-variant:small-caps;">Mushroom</span> & **Dense** &[119]{} & [8, 124]{} & [23.00]{}\
<span style="font-variant:small-caps;">Pumsb</span>& **Dense** &[7, 117]{} & [49, 046]{} & [74.00]{}\
<span style="font-variant:small-caps;">Pumsb\*</span>& **Dense** & [7, 117]{} &[49, 046]{} & [50.00]{}\
<span style="font-variant:small-caps;">Retail</span> & **Sparse** &[16, 470]{} &[88, 162]{} & [10.00]{}\
<span style="font-variant:small-caps;">Accidents</span>& **Sparse**& [468]{} &[340, 183]{} & [33.81]{}\
<span style="font-variant:small-caps;">T10I4D100K</span>& **Sparse** & [870]{} & [100, 000]{} &[10.10]{}\
<span style="font-variant:small-caps;">T40I10D100K</span>& **Sparse**& [942]{} &[100, 000]{} &[39.61]{}\
### Experimental Protocol
Our objective is to prove, through extensive carried out experiments, the efficiency of the proposed <span style="font-variant:small-caps;">Gmjp</span> algorithm.
Our first batch of experiments focus on evaluating the compactness rates by different condensed representations of rare correlated patterns. We also build a quantitative comparison between the $\mathcal{FCP}$, the $\mathcal{RCP}$ sets and their associated condensed representations.
Our second batch of experiments focus on studying running times of the proposed <span style="font-variant:small-caps;">Gmjp</span> algorithm while running the four different scenarios.
Evaluation of the compactness rates offered by the proposed representations {#sec_XP1}
---------------------------------------------------------------------------
The compactness rate offered by a concise representation measures the reduction of the size of the representation compared to the size of the whole set of patterns. For example, for the $\mathcal{RCPR}$ representation, the compactness rate is equal to: 1 - $\frac{|\mathcal{RCPR}|}{|\mathcal{RCP}|}$. Worth of cite, our experimental study confirms that the $\mathcal{RCPR}$ representation is a perfect cover of the $\mathcal{RCP}$ set. In fact, the obtained results show that the size of $\mathcal{RCPR}$ is always smaller than that of the $\mathcal{RCP}$ set over the entire range of the considered support and *bond* thresholds. Our study concerns both dense and sparse datasets.
### Effect of *minsupp* variation
For example, considering the <span style="font-variant:small-caps;">Mushroom</span> dataset for *minsupp* = *35*$\%$ and *minbond* = *0.15*: $|\mathcal{RCPR}|$ = *1, 810* while $|\mathcal{RCP}|$ = *100, 156*, with a reduction rate reaching approximately *98*$\%$. This is explained by the nature of the induced equivalence classes. In fact, we have in this case, $|\mathcal{MRCP}|$ = *1, 412* and $|\mathcal{CRCP}|$ = *652*. Since the $\mathcal{RCPR}$ representation corresponds to the union without redundancy of the $\mathcal{MRCP}$ and the $\mathcal{CRCP}$ sets, we have always $|\mathcal{RCPR}|$ $\leq$ $|\mathcal{MRCP}|$ + $|\mathcal{CRCP}|$.
In this respect, Figure \[Fct\_minsupp\] shows that all the compression rates proportionally vary to *minsupp* and disproportionately with respect to *minbond* values. This is due to the fact that, the size of the different representations increases as far as *minsupp* increases and decreases whenever *minbond* increases.
We also find that the respective sizes of the concise exact representations $\mathcal{MM}$$ax$$\mathcal{CR}$ and $\mathcal{M}$$in$$\mathcal{MCR}$ never exceed the size of the $\mathcal{RCPR}$ representation. This is justified by the nature of the elements composing both representations. In fact, the $\mathcal{MM}$$ax$$\mathcal{CR}$ is composed by the $\mathcal{MRCP}$ set of minimal rare correlated patterns and the $\mathcal{M}ax$$\mathcal{CRCP}$ set of maximal closed rare correlated patterns. Nevertheless, we confirm that the size of the $\mathcal{M}ax$$\mathcal{CRCP}$ set is always lower than that of the $\mathcal{CRCP}$ set. According to the $\mathcal{M}$$in$$\mathcal{MCR}$ representation, it is based on the $\mathcal{M}in$$\mathcal{MRCP}$ set of the minimal elements of the $\mathcal{MRCP}$ set and on the $\mathcal{CRCP}$ set. In fact, we remark that the size of the $\mathcal{M}in$$\mathcal{MRCP}$ set never exceeds the size of the $\mathcal{MRCP}$ set.
In what follows, we evaluate the compactness of the representations based on the variation of the correlation threshold *minbond*.
### Effect of *minbond* variation
Let us consider the results depicted by Figure \[Fct\_minbond\]. The first intuition is that all the compression rates vary disproportionately to *minbond* values. For the <span style="font-variant:small-caps;">Pumsb\*</span> dataset, for *minsupp* = *80*$\%$ and for *minbond* = *0.30*, we have $|\mathcal{RCP}|$ = *65, 536* $>$ $|\mathcal{RCPR}|$ $=$ $|$$\mathcal{RMM}$$ax$$\mathcal{F}|$ $=$ $|\mathcal{M}in$$\mathcal{MRCP}|$ = *2, 048*. Now, while increasing *minbond* from *0.30* to *0.60*, we remark that the size of the $\mathcal{RCP}$ increase as well the $|\mathcal{RCPR}|$ representation, whereas the size of $\mathcal{RMM}$$ax$$\mathcal{F}$ and of $\mathcal{M}in$$\mathcal{MRCP}$ are unchanged. Thus, these representations offer constant reduction rates in spite of *minbond* variation. For the same example, we have $|\mathcal{RCP}|$ = *130, 000* $>$ $|\mathcal{RCPR}|$ = *4, 096* $>$ $|$$\mathcal{RMM}$$ax$$\mathcal{F}|$ = *2, 048* $=$ $|\mathcal{M}in$$\mathcal{MRCP}|$ = *2, 048*.
We sketch in Table \[labelsize1\], the experimental results associated to both the $\mathcal{FCP}$ set of frequent correlated patterns and to the $\mathcal{RCP}$ set of rare correlated patterns. To summarize, the concise representations $\mathcal{FCCPR}$ and $\mathcal{RCPR}$ present very encouraging reduction rates over several datasets and for different ranges of *minsupp* and *minbond* thresholds. We note that, the ‘gain‘ corresponds to the “reduction rate“ said also “compactness rate“.
[lclrrrcrrcr]{} **Dataset**& ***minsupp*** & ***minbond***& **$\#$ $\mathcal{FCP}$** & **$\#$ $\mathcal{FCCPR}$** & **Gain of** & &**$\#$ $\mathcal{RCP}$** & **$\#$ $\mathcal{RCPR}$** & &**Gain of**\
& & & & & **$\mathcal{FCCPR}$** & &**** & & &**$\mathcal{RCPR}$**\
<span style="font-variant:small-caps;">Mushroom</span> &[30$\%$]{} & [0.15]{} & [2, 701]{} & [427]{}& [**84.19$\%$**]{}& &[98, 566]{}& [1, 704]{} & &[**98.27$\%$**]{}\
&[45$\%$]{} & [0.15]{} & [307]{} & [83]{} & [**72.96$\%$**]{}& &[100, 960]{}& [1, 985]{} & &[**98.03$\%$**]{}\
<span style="font-variant:small-caps;">Pumsb\*</span> &[40$\%$]{} & [0.45]{} & [10, 674]{} & [1646]{} & [**84.57$\%$**]{}& &[448, 318]{}& [3, 353]{}& & [**99.25$\%$**]{}\
&[40$\%$]{} & [0.50]{} & [9, 760]{} & [1325]{} & [**86.42$\%$**]{}& & [82, 413]{}& [3, 012]{}& &[**96.34$\%$**]{}\
<span style="font-variant:small-caps;">Connect</span> &[10$\%$]{} & [0.80]{} & [534, 026]{} & [15, 152]{} & [**97.16$\%$**]{}& & [56]{}& [56]{} & &[**0$\%$**]{}\
&[50$\%$]{} & [0.80]{} & [533, 991]{} & [15, 117]{} & [**97.16$\%$**]{}& & [91]{}& [91]{}& &[**0$\%$**]{}\
<span style="font-variant:small-caps;">Accidents</span>& [40$\%$]{} & [0.30]{}&[32, 529]{} &[32, 528]{} & [**0$\%$**]{} & & [117, 805]{} &[1, 722]{}& &[**98.53$\%$**]{}\
& [60$\%$]{} & [0.30]{}&[2, 057]{} &[2, 047]{} & [**0$\%$**]{}& &[148, 259]{} &[2, 743]{}& &[**98.14$\%$**]{}\
At this stage, we have analyzed the variation of the size of the different concise exact representations according to both *minsupp* and *minbond* variations. In the next section, we put the focus on the evaluation of the running time of the proposed <span style="font-variant:small-caps;">Gmjp</span> approach.
Evaluation of the running time of <span style="font-variant:small-caps;">Gmjp</span> {#sec_XP2}
------------------------------------------------------------------------------------
### Overall Performance Evaluation of <span style="font-variant:small-caps;">Gmjp</span>
We emphasize that, according to the results given by Table \[labelCPUTime1\] $^\textsc{(}$[^8]$^{\textsc{)}}$ , that the execution time varies depending on the number of distinct items of the considered dataset. This is explained by the principle of <span style="font-variant:small-caps;">Gmjp</span> which is based on the idea of processing each item separately and based on the list of the co-occurrent of each item.
For example, the computation costs are relatively high for the <span style="font-variant:small-caps;">T40I10D100K</span> dataset, and they are low for the <span style="font-variant:small-caps;">Mushroom</span> dataset. This is explained by the fact that, the <span style="font-variant:small-caps;">Mushroom</span> dataset only contains *119* items while the <span style="font-variant:small-caps;">T40I10D100K</span> dataset contains *942* items. We also note that the highest execution times are obtained with the <span style="font-variant:small-caps;">Retail</span> dataset, since this latter contains a high number of distinct items, equal to [16, 470]{}.
[lrrrrrrrr]{} **Dataset**& **Number** & **Average** & **Average**& **Average** & **Average** & **Average** & **Average**\
& **of Items** & ***minsupp*** & ***minbond***& **Time S1** & **Time S2** & **Time S3** & **Time S4**\
<span style="font-variant:small-caps;">Mushroom</span> &[119]{} & [58$\%$]{} & [0.30]{} & [7]{} & [11.4]{} & [20]{}& [19.6]{}\
& & [40$\%$]{} & [0.57]{} & [3.75]{} & [5.25]{} & [11]{}& [709]{}\
<span style="font-variant:small-caps;">Accidents</span> &[468]{} & [7.8$\%$]{} & [0.50]{} & [709]{} & [703]{} &[793]{}&[784.2]{}\
<span style="font-variant:small-caps;">Retail</span> &[16, 470]{} & [25.83$\%$]{} & [0.50]{} & [5.83]{} & [13.16]{} &[1903]{}&[1902]{}\
<span style="font-variant:small-caps;">T10I4D100K</span> &[870]{} & [5$\%$]{} & [0.20]{} & [2]{} & [3]{} &[163]{}&[163]{}\
<span style="font-variant:small-caps;">T40I10D100K</span> &[942]{} & [8.2$\%$]{} & [0.50]{} & [148]{} & [182.6]{} &[491]{}&[490.4]{}\
It is worth of mention that the computation time of the fourth scenario dedicated to the extraction of the $\mathcal{RCPR}$ representation are the highest ones among the other scenarios. This can be explained by the fact that the extraction of the $\mathcal{RCPR}$ representation is the most complex mining task within the <span style="font-variant:small-caps;">Gmjp</span> approach. Thereby, we focus on the next subsection on studying the cost of the three different steps of <span style="font-variant:small-caps;">Gmjp</span> when extracting the $\mathcal{RCPR}$ representation.
### Performance Evaluation of the $\mathcal{RCPR}$ representation Mining
We study, in this subsection, the running time of need for the extraction of the $\mathcal{RCPR}$ representation. It is worth noting that the running times of the <span style="font-variant:small-caps;">Gmjp</span> algorithm vary according to the characteristics of the dataset. For example, the computation costs are relatively high for the <span style="font-variant:small-caps;">T10I10D100K</span> dataset, and they are low for the <span style="font-variant:small-caps;">Mushroom</span> dataset. This is due to the difference in the characteristics of these two datasets. In fact, the <span style="font-variant:small-caps;">Mushroom</span> dataset only contains *119* items and *8, 124* transactions while the <span style="font-variant:small-caps;">T10I10D100K</span> dataset contains *870* items and *100, 000* transactions. We also note, for the different datasets, that the extraction costs are slightly sensitive to the changes of the *minsupp* and *minbond* values.
We present, in Tables \[labelRCPR1\] and \[labelRCPR2\], the CPU time corresponding to each step of the <span style="font-variant:small-caps;">Gmjp</span> algorithm and depending respectively on the variation of *minsupp* and on the variation of *minbond*.
[lrrrrrr]{} **Dataset**& ***minbond*** & ***minsupp*** & **First Step** & **Second Step** & **Third Step**\
<span style="font-variant:small-caps;">Mushroom</span> &[0.3]{}& [20$\%$]{} & [6]{} & [13]{} &[0]{}\
& & [40$\%$]{} & [6]{} & [13]{} &[0]{}\
& & [60$\%$]{} & [6]{} & [14]{} &[0]{}\
& & [80$\%$]{} & [6]{} & [15]{} &[0]{}\
<span style="font-variant:small-caps;">Retail</span> &[0.5]{}& [5$\%$]{} &[276]{} &[1, 627]{} &[0]{}\
& & [10$\%$]{}&[273]{} &[1, 627]{} &[0]{}\
& & [30$\%$]{}&[274]{} &[1, 628]{} &[0]{}\
& & [50$\%$]{}&[275]{} &[1, 628]{} &[0]{}\
<span style="font-variant:small-caps;">Accidents</span>&[0.5]{}& [1$\%$]{} &[724]{} &[67]{} &[0]{}\
& & [3$\%$]{}&[714]{} &[67]{} &[0]{}\
& & [5$\%$]{}&[716]{} &[67]{} &[0]{}\
& & [10$\%$]{}&[715]{} &[67]{} &[0]{}\
& & [15$\%$]{}&[717]{} &[67]{} &[0]{}\
& & [20$\%$]{}&[717]{} &[67]{} &[0]{}\
We conclude, according to these results, that the obtained execution times are slightly sensitive to the variation of the *minbond* and *minsupp* values.
[lrrrrrr]{} **Dataset**& ***minsupp*** & ***minbond***& **First Step** & **Second Step** & **Third Step**\
<span style="font-variant:small-caps;">Mushroom</span> &[50$\%$]{} & [0.4]{} & [6]{} & [4]{} &[0]{}\
& & [0.7]{} & [6]{} & [1]{} &[0]{}\
& & [1]{} & [6]{} & [1]{} &[0]{}\
<span style="font-variant:small-caps;">T10I4D100K</span>& [5$\%$]{} & [0.2]{}&[25]{} &[137]{} &[0]{}\
& & [0.4]{}&[26]{} &[138]{} &[0]{}\
& & [0.6]{}&[26]{} &[138]{} &[0]{}\
& & [0.8]{}&[25]{} &[138]{} &[0]{}\
& & [1]{}&[25]{} &[137]{} &[0]{}\
We have, for example, the CPU time needed for the execution of the first step for the <span style="font-variant:small-caps;">T10I4D100K</span> dataset is about *26* seconds, while for the <span style="font-variant:small-caps;">Retail</span> dataset, the execution of the first step needs about *275* seconds. This is justified by the fact that the <span style="font-variant:small-caps;">T10I4D100K</span> dataset contains only *870* items while the <span style="font-variant:small-caps;">Retail</span> dataset contains *16, 470* items. We also remark, for the <span style="font-variant:small-caps;">Accidents</span> dataset, that the execution times are relatively high compared to the other datasets. This is justified by its high number of transactions, equal to *340, 183* transactions which induces that the first step becomes more costly. In this regard, the first step of the building of the BSVector and the COVector of the items needs about *720* seconds and it lasts more than the second step, which needs only *67* seconds.
It is also important to mention that the CPU time dedicated to the third step, allowing to filter the global minimal and the global closed patterns among the sets of the identified local minimal and local closed ones, is negligible and equal to null. This confirms the very good choice of the suitable multimap data structures during the third step.
We also note, that the execution times needed for the post-processing of the representations $\mathcal{MM}$ax$\mathcal{CR}$, $\mathcal{M}$in$\mathcal{MCR}$ from the $\mathcal{RCPR}$ representation are negligible.
We have, thus, evaluated the performance of the <span style="font-variant:small-caps;">Gmjp</span> approach while running the four different execution scenarios. We focused specially on the fourth scenario dedicated to the extraction of the $\mathcal{RCPR}$ representation. In the next section, we evaluate <span style="font-variant:small-caps;">Opt-Gmjp</span> the optimized version of the <span style="font-variant:small-caps;">Gmjp</span> approach.
Optimizations and Evaluations {#sec_Opt}
-----------------------------
Our aim in the next subsections is to evaluate the impact of varying the thresholds of both the correlation and the rarity constraints.
In the remainder, we study the impact of the rarity constraint threshold variation on the execution time of the <span style="font-variant:small-caps;">Opt-Gmjp</span> version.
### Effect of *minsupp* variation
Table \[labeltimexp2\] presents our results while fixing the *minbond* threshold and varying the monotone constraint of rarity threshold, *minsupp*. We consider as an example the <span style="font-variant:small-caps;">Mushroom</span> dataset, while varying *minsupp* from [20$\%$]{} to [80$\%$]{}, the size of the result set |$\mathcal{RCP}$| varies from 261 itemsets to 3352 itemsets while the CPU-time and the memory consumption underwent a slight variation. Whereas, for the <span style="font-variant:small-caps;">T40I10D100K</span> dataset, the variation of *minsupp* from [2$\%$]{} to [15$\%$]{} induces an increase in the CPU-time of the second and third steps from [60.09]{} to [79.49]{} seconds. The size of the output result increase also from [341]{} to [932]{} itemsets.
The <span style="font-variant:small-caps;">Chess</span> dataset presents a specific behavior according to *minsupp* variation. The variation of *minsupp* from [30$\%$]{} to [50$\%$]{} induces an increase in the CPU-time of the second and third steps from [5.604]{} to [300.163]{} seconds. The size of the $\mathcal{RCP}$ set increase in a very significant way from [618]{} to [36010, 648]{} itemsets.
### Effect of *minbond* variation
Table \[labelbondxp3\] presents the results obtained when varying the correlation threshold *minbond* for a fixed *minsupp* threshold. In this experiment, we found that for the <span style="font-variant:small-caps;">Mushroom</span> dataset, the *minbond* threshold was chosen to be increasingly selective, from [0.2]{} to the highest value, equal to [1]{}. This variation affects very slightly the CPU-time and the memory consumption, while the size of the output set decreases sharply from [54, 395]{} to 126 itemsets. Whereas, for the sparse <span style="font-variant:small-caps;">T10I4D100K</span> dataset, the *minbond* variation act slightly on the execution time. It increases just by 2 seconds, and the output’s size decreases from 915 to 860 itemsets. However, for the <span style="font-variant:small-caps;">Chess</span> dataset, the size of $\mathcal{RCP}$ set and the CPU-time are very sensitive to the *minbond* variation. For example, a slight variation of *minbond* from 0.40 to 0.45 induces an important decrease of the $\mathcal{RCP}$ set from 5167, 090 to 1560, 073 itemsets. The CPU-time is also lowered from 40.124 to 0.451 seconds when *minbond* decrease from 0.4 to 0.5.
The most interesting observation we found from the previous experiments was that the choice of very selective correlation threshold do not affect significantly the CPU-time and the memory consumption, while it affects the size of our result set. Whereas, the fact of pushing more selective the rarity constraint increases the execution time needed for the second and the third steps. This confirms that monotone and anti-monotone constraints are mutually of use in the selectivity. It is also important to mention that, the first step of transforming the database is not affected by both constraints variation.
In the next sub-section, we evaluate the proposed optimization and we compare the optimized version of <span style="font-variant:small-caps;">Gmjp</span> *vs.* the <span style="font-variant:small-caps;">Jim</span> approach [@borgelt].
\[htbp\]
[lrrrrll]{} **Dataset** & ***minbond*** & ***minsupp*** & **|$\mathcal{RCP}$|** & **CPU Time** & **CPU Time** & **Avg. Memory**\
& & & & **Step 1** & **Steps 2 and 3** & **Consumption (Ko)**\
<span style="font-variant:small-caps;">Mushroom</span> &[0.30]{} & [20$\%$]{} & [261]{} & [0.20]{} & **[0.208]{}** &\
& & [40$\%$]{} & [2810]{} & [0.20]{} & **[0.246]{}** & [18, 850]{}\
& & [80$\%$]{} & [3352]{} & [0.20]{} & **[0.247]{}** &\
<span style="font-variant:small-caps;">T40I10D100K</span> & [0.50]{} & [2$\%$]{} & [341]{} & [16]{} &**[60.09]{}** &\
& & [11$\%$]{} & [889]{} & [16]{} &**[63.028]{}** & [131,516]{}\
& & [15$\%$]{} & [932]{} & [16]{} &**[79.49]{}** &\
<span style="font-variant:small-caps;">Chess</span> &[0.30]{} & [10$\%$]{} & [16]{} & [0.068]{} & **[0.116]{}** &\
& & [30$\%$]{} & [618]{} & [0.068]{} & **[5.604]{}** & [13,509]{}\
& & [50$\%$]{} & [36010, 648]{} & [0.068]{} & **[300.163]{}** &\
\[htbp\]
[lrrrlll]{} **Dataset** & ***minsupp*** & ***minbond*** & **|$\mathcal{RCP}$|** & **CPU Time** & **CPU Time** & **Avg. Memory**\
& & & & **Step 1** & **Steps 2 and 3** & **Consumption (Ko)**\
<span style="font-variant:small-caps;">Mushroom</span> & [40$\%$]{} & [0.2]{} & [54, 395]{} & [0.20]{} & **[0.977]{}** & [18, 590]{}\
& & [1]{} & [126]{} &[0.20]{} & **[0.198]{}** &\
<span style="font-variant:small-caps;">Chess</span> & [50$\%$]{} & [0.40]{} & [5167, 090]{} & [0.068]{} & **[40.124]{}** &\
& & [0.45]{} & [1560, 073]{} &[0.068]{} & **[12.127]{}** &\
& & [0.50]{} & [162]{} &[0.068]{} & **[0.451]{}** & [13, 556]{}\
& & [0.60]{} & [40]{} &[0.068]{} & **[0.073]{}** &\
& & [1]{} & [38]{} &[0.068]{} & **[0.054]{}** &\
<span style="font-variant:small-caps;">T10I4D100K</span> & [5$\%$]{} & [0.40]{} & [915]{} & [16]{} &**[47.95]{}** & [131, 572]{}\
& & [1]{} & [860]{} &[16]{} &**[49.48]{}** &\
### Performance of <span style="font-variant:small-caps;">Opt-Gmjp</span> *vs.* <span style="font-variant:small-caps;">Gmjp</span>
**6.5.4.1 Comparison of <span style="font-variant:small-caps;">Opt-Gmjp</span> *vs.* <span style="font-variant:small-caps;">Gmjp</span>**\
The goal of our evaluation is to compare the scalability level of <span style="font-variant:small-caps;">Opt-Gmjp</span> *vs.* <span style="font-variant:small-caps;">Gmjp</span>. In fact, scalability is an important criteria for constrained itemset mining approaches. Our <span style="font-variant:small-caps;">Opt-Gmjp</span> algorithm demonstrates good scalability as far as we increase the size of the datasets according to two dimensions: the number of transactions |$\mathcal{T}$| and the number of items |$\mathcal{I}$|. While <span style="font-variant:small-caps;">Gmjp</span> reached a point where it consumed about seven times more CPU-time than <span style="font-variant:small-caps;">Opt-Gmjp</span>. Tables \[Cmp1\] and \[CORI2\] reported our results while varying respectively *minsupp* and *minbond*.
As example, while testing the <span style="font-variant:small-caps;">Mushroom</span> dataset containing 8, 124 transactions, <span style="font-variant:small-caps;">Opt-Gmjp</span> finishes in average in [0.432]{} seconds while <span style="font-variant:small-caps;">Gmjp</span> needs in average $11$ seconds. As another experiment example, we tested also on <span style="font-variant:small-caps;">Accidents</span> with 340, 183 transactions and 468 items. <span style="font-variant:small-caps;">Opt-Gmjp</span> finishes in $47.617$ seconds while <span style="font-variant:small-caps;">Gmjp</span> needs $793$ seconds. <span style="font-variant:small-caps;">Gmjp</span> finished the <span style="font-variant:small-caps;">Mushroom</span> dataset with about 8K transactions in $20$ seconds while <span style="font-variant:small-caps;">Opt-Gmjp</span> finished, in average, in $0.278s$, $52.591s$ and $47.617s$ for the 8K, 100K and 340K transactions datasets, respectively.
\[htbp\]
[lrrrlr]{} **Dataset** &***minsupp***& ***minbond***&**CPU Time**& **CPU Time**\
& & & **<span style="font-variant:small-caps;">Gmjp</span>** & **<span style="font-variant:small-caps;">Opt-Gmjp</span>**\
<span style="font-variant:small-caps;">Mushroom</span> & [20$\%$]{} & [0.30]{} & [20]{} & **[0.14]{}**\
& [40$\%$]{} & & [19]{} & **[0.18]{}**\
& [60$\%$]{} & & [19]{} & **[0.18]{}**\
& [80$\%$]{} & & [21]{} & **[0.18]{}**\
<span style="font-variant:small-caps;">Accidents</span> & [1$\%$]{} & [0.50]{} &[802]{} &**[22]{}**\
& [3$\%$]{} & &[802]{} &**[22]{}**\
& [5$\%$]{} & &[790]{} &**[21]{}**\
& [10$\%$]{} & &[783]{} &**[21]{}**\
& [12$\%$]{} & &[788]{} &**[22]{}**\
<span style="font-variant:small-caps;">T40I10D100K</span> & [2$\%$]{} & [0.50]{} & [489]{} &**[51]{}**\
& [5$\%$]{} & & [494]{} &**[53]{}**\
& [8$\%$]{} & & [493]{} &**[51]{}**\
& [11$\%$]{} & & [489]{} &**[51]{}**\
& [15$\%$]{} & & [490]{} &**[51]{}**\
\[htbp\]
[lrrrlr]{} **Dataset** &***minsupp***& ***minbond***&**CPU Time**& **CPU Time**\
& & & **<span style="font-variant:small-caps;">Gmjp</span>** & **<span style="font-variant:small-caps;">Opt-Gmjp</span>**\
<span style="font-variant:small-caps;">T10I4D100K</span> & [5$\%$]{} & [0.20]{} & [163]{} &**[39]{}**\
& & [0.40]{} & [164]{} &**[40]{}**\
& & [0.60]{} & [163]{} &**[39]{}**\
& & [0.80]{} & [163]{} &**[39]{}**\
& & [1]{} & [163]{} &**[39]{}**\
<span style="font-variant:small-caps;">Mushroom</span> & [40$\%$]{} & [0.20]{} & [21]{} &**[0.90]{}**\
& & [0.40]{} & [9]{} &**[0.14]{}**\
& & [0.70]{} & [7]{} &**[0.13]{}**\
& & [1]{} & [7]{} &**[0.13]{}**\
= cmbx7.5 scaled
[|l|c|c|rr|rr|rr|rr|]{} **Dataset**& **Avg**& **Avg**& & & &\
& ***minsupp*** & ***minbond***& & & & & & & &\
& & & **Gmjp**& **Opt-Gmjp** & **Gmjp**& **Opt-Gmjp** & **Gmjp**& **Opt-Gmjp** & **Gmjp**& **Opt-Gmjp**\
<span style="font-variant:small-caps;">Mushroom</span> & [58$\%$]{} & [0.30]{} & [7]{} & [0.114]{} & [11.4]{} & [0.052]{}& [20]{} & [0.172]{}& [19.6]{} & [0.206]{}\
& [40$\%$]{} & [0.57]{} & [3.75]{} & [0.096]{} & [5.25]{} & [0.058]{} & [11]{} & [0.325]{}& [709]{} & [0.525]{}\
<span style="font-variant:small-caps;">Accidents</span> &[7.8$\%$]{} & [0.50]{} & [709]{} &[7.094]{} & [703]{} &[7.978]{} &[793]{} &[22.034]{} &[784.2]{} &[22.430]{}\
<span style="font-variant:small-caps;">T10I4D100K</span> & [5$\%$]{} & [0.20]{} & [2]{} &[0.132]{} & [3]{} &[0.14]{}&[163]{} & [39.804]{}&[163]{} & [39.424]{}\
<span style="font-variant:small-caps;">T40I10D100K</span> & [8.2$\%$]{} & [0.50]{} & [148]{} & [4.222]{}& [182.6]{} & [7.566]{} &[491]{} & [51.798]{}&[490.4]{} & [51.740]{}\
We highlight that <span style="font-variant:small-caps;">Opt-Gmjp</span> outperformed <span style="font-variant:small-caps;">Gmjp</span> in the different evaluated bases. This is dedicated to the efficient integration of the monotone and anti-monotone constraints in an early stages of the mining process. We also present in Table \[addlabel\] a summarized comparison of the performances of <span style="font-variant:small-caps;">Gmjp</span> *vs.* **Optimized <span style="font-variant:small-caps;">Gmjp</span>** over the four different scenarios S1, S2, S3 and S4. We thus conclude that the optimized version of <span style="font-variant:small-caps;">Gmjp</span> offers important reduction of the running time over all the tested benchmark datasets and for wide range of constraints threshold. In what follows, we evaluate our optimized version *vs.* the <span style="font-variant:small-caps;">Jim</span> approach [@borgelt].\
**6.5.4.2 Comparison of <span style="font-variant:small-caps;">Opt-Gmjp</span> *vs.* <span style="font-variant:small-caps;">Jim</span>**\
The goal of these experiments is to prove the competitive performances of <span style="font-variant:small-caps;">Opt-Gmjp</span> compared to other state-of the art approaches dealing with frequent correlated itemsets. Our comparative study covers the <span style="font-variant:small-caps;">Jim</span> approach [@borgelt] which is implemented in the C language and is publicly available. = cmbx7.5 scaled
---------------------------------------------------------- --------------- --------------- ------------------------------------------------------------ ------------------------------------------------------- ------------------------------------------------------------ -------------------------------------------------------
**Dataset** ***minsupp*** ***minbond*** **Opt <span style="font-variant:small-caps;">Gmjp</span>** **<span style="font-variant:small-caps;">Jim</span>** **Opt <span style="font-variant:small-caps;">Gmjp</span>** **<span style="font-variant:small-caps;">Jim</span>**
**S1** **S2**
<span style="font-variant:small-caps;">T10I4D100K</span> [5$\%$]{} [0.20]{} **0.133** 0.20 **0.15** 0.19
[0.40]{} **0.133** 0.19 **0.13** 0.18
[0.60]{} **0.132** 0.18 **0.13** 0.18
[0.80]{} **0.129** 0.18 **0.13** 0.18
[1]{} **0.135** 0.19 **0.13** 0.19
<span style="font-variant:small-caps;">Mushroom</span> [20$\%$]{} [0.30]{} 0.082 **0.06** 0.140 **0.03**
[40$\%$]{} **0.029** 0.03 0.060 **0.03**
[60$\%$]{} 0.200 **0.02** 0.020 0.02
[80$\%$]{} 0.200 **0.02** 0.023 **0.02**
[90$\%$]{} 0.210 **0.02** **0.019** 0.02
<span style="font-variant:small-caps;">Retail</span> [5$\%$]{} [0.50]{} **0.249** 0.25 0.46 **0.25**
[10$\%$]{} 0.250 0.23 0.36 **0.24**
[20$\%$]{} 0.249 0.22 0.26 **0.22**
[40$\%$]{} 0.240 0.23 0.25 **0.22**
[50$\%$]{} 0.240 0.22 0.36 **0.20**
---------------------------------------------------------- --------------- --------------- ------------------------------------------------------------ ------------------------------------------------------- ------------------------------------------------------------ -------------------------------------------------------
: Performance comparison of our Improved **<span style="font-variant:small-caps;">Opt-Gmjp</span>** *vs.* <span style="font-variant:small-caps;">Jim</span> [@borgelt] <span style="font-variant:small-caps;">(</span>Time in seconds<span style="font-variant:small-caps;">)</span>.[]{data-label="TabCmpJim"}
We report in Table \[TabCmpJim\] $^\textsc{(}$[^9]$^{\textsc{)}}$ a comparison between our improved <span style="font-variant:small-caps;">Gmjp</span> approach with the <span style="font-variant:small-caps;">Jim</span> approach [@borgelt]. Our comparative study is restricted to the first Scenario S1 and second scenario S2, since the <span style="font-variant:small-caps;">Jim</span> approach does not consider the rare correlated patterns. Therefore, we are not able to compare the third and the fourth scenarios S3 and S4. We highlight that our running time are competitive to those achieved by <span style="font-variant:small-caps;">Jim</span> for different ranges of frequency and correlation thresholds. Note-worthily, for the <span style="font-variant:small-caps;">T10I4D100K</span> dataset, our obtained results are even better than <span style="font-variant:small-caps;">Jim</span> for both first and second scenarios. While, for the <span style="font-variant:small-caps;">Mushroom</span> dataset, the results of the first scenario are very close to those of <span style="font-variant:small-caps;">Jim</span>. Whereas, <span style="font-variant:small-caps;">Jim</span> outperformed our <span style="font-variant:small-caps;">Gmjp</span> in the second scenario when extracting the frequent closed correlated itemsets.
Conclusion {#ConcChap6}
----------
We presented in this chapter the experimental evaluation of our <span style="font-variant:small-caps;">Gmjp</span> mining approach. The evaluation is based on two main axes, the first is related to the compactness rates of the condensed representations while the second axe concerns the running time. We measured the global performance of <span style="font-variant:small-caps;">Gmjp</span> then we focused on the performance of the fourth execution scenario S4. The optimized version <span style="font-variant:small-caps;">Opt-Gmjp</span> presents much better performance than <span style="font-variant:small-caps;">Gmjp</span> over different benchmark datasets. The two main features which constitute the thrust of the improved version: (*i*) only one scan of the database is performed to build the new transformed dataset; (*ii*) it offers a resolution of the problem of handling both rarity and correlation constraints. In the next chapter, we present the classification process based on correlated patterns.
Associative-Classification Process based on Correlated Patterns {#ch_7}
===============================================================
Introduction {#IntroChap7}
------------
In this chapter, we put the focus on the classification process based on correlated patterns. The second section is devoted to the description of the framework of the association rules. We continue in the third section with a specific kind of association called “Generic Bases of Association Rules“. The fourth section presents the description of the associative-classification based on correlated patterns. We evaluate the classification accuracy of frequent correlated patterns *vs.* rare correlated patterns. In Section \[se-IDS\], we present the application of rare correlated patterns on the classification of intrusion detection data derived from the <span style="font-variant:small-caps;">KDD 99</span> dataset. In Section \[SecGene\], we propose the process of applying the $\mathcal{RCPR}$ representation on the extraction of rare correlated association rules from Micro-array gene expression data.
Overview of association rules {#se2}
-----------------------------
The extraction of association rules is one of the most important techniques in data mining [@BoukerSYN14; @GasmiYNB07]. The leading approach of generating association rules is based on the extraction of frequent patterns [@Agra94]. We clarify the basic notions related to association rules through the following definitions.
**Association Rule**\
An association rule $R$ is a relation between itemsets, in the form $R$ : $A$ $\Rightarrow$ $B$$\backslash$$A$, with $A$ and $B$ are two itemsets and $A$ $\subset$ $B$. The itemset $A$ is called ‘Premise‘ <span style="font-variant:small-caps;">(</span>or ‘Antecedent‘<span style="font-variant:small-caps;">)</span> whereas the itemset $B$$\backslash$$A$ is called ‘Conclusion‘ <span style="font-variant:small-caps;">(</span>or ‘Consequent‘<span style="font-variant:small-caps;">)</span> of the association rule $R$.
Each association rule, $R$ : $A$ $\Rightarrow$ $B$$\backslash$$A$, is characterized by:
1. **The value of the Support:** Corresponding to the number of times where the association holds reported by the number of occurrence of the itemset $B$. The support metric assesses the frequency of the association rule.
2. **The value of the Confidence:** Corresponding to the number of times where the association holds reported by the number of occurrence of the itemset $A$. The confidence expresses the reliability of the rule.
The support and the confidence are formally defined as follows:
**Support, Confidence of an association rule**\
Let an association rule $R$ : $A$ $\Rightarrow$ $B$$\backslash$$A$, its support, denoted by *Supp*<span style="font-variant:small-caps;">(</span>$R$<span style="font-variant:small-caps;">)</span> = *Supp*<span style="font-variant:small-caps;">(</span>$B$<span style="font-variant:small-caps;">)</span>, where as the confidence, denoted by, *Conf*<span style="font-variant:small-caps;">(</span>$R$<span style="font-variant:small-caps;">)</span> = $\displaystyle \displaystyle\frac{\textit{Supp}\textsc{(}B\textsc{)}}{\textit{Supp}\textsc{(}A\textsc{)}}$.
**Valid, Exact and Approximative Association Rule**\
An association rule $R$ is said Valid whenever:\
$\bullet$ The value of the confidence is greater than or equal to the minimal threshold of confidence *minconf*, and\
$\bullet$ The value of its support is greater than or equal to the minimal threshold of support *minsupp*. If the confidence of the rule $R$, Conf<span style="font-variant:small-caps;">(</span>R<span style="font-variant:small-caps;">)</span>, is equal to 1 then the rule $R$ is said an Exact association rule, otherwise it is said approximative.
The extraction of the association rules consists in determining the set of valid rules *i.e.*, whose support and confidence are at least equal, respectively, to a minimal threshold of support *minsupp* and a minimal threshold of confidence *minconf* predefined by the user. This problem is decomposed into two subproblems [@Agra94] as follows:
1\. Extraction of frequent itemsets;
2\. Generation of valid association rules based on the frequent extracted itemset set: the generated rules are in the form $R$ : $A$ $\Rightarrow$ $B$$\backslash$$A$, with $A$ $\subset$ $B$ and Conf<span style="font-variant:small-caps;">(</span>$R$<span style="font-variant:small-caps;">)</span>$\geq$ *minconf*.
The association rule extraction problem suffers from the high number of the generated association rules from the frequent itemset set. In fact, the number of the extracted frequent itemsets can be exponential in function of the number of items $|\mathcal{I}|$. In fact, from a frequent itemset $F$, we can generate $2^{|F|}-1$ association rules. The huge number of association rules leads to a deviation regarding to the principal objective namely, the discovery of reliable knowledge and with a manageable size. To palliate this problem, many techniques derived from the Formal Concept Analysis <span style="font-variant:small-caps;">(</span>FCA<span style="font-variant:small-caps;">)</span>, were proposed. These techniques aimed to reduce, without information loss, the set of association rules. The main idea is to determine a minimal set of association rules allowing to derive the redundant association rules, this set is called “‘Generic bases of association rules’’.
Extraction of the generic bases of association rules {#se3}
----------------------------------------------------
The approaches derived from the <span style="font-variant:small-caps;">FCA</span> allows to extract the generic bases of association rules. These generic bases allow to derive the set of redundant association rules without information loss. In fact, these bases constitute a compact set of association rules easily interpretable by final user. Every generic base constitutes an information lossless representation of the whole set of association rules if it fulfills the following properties [@marzena022]:
- **Lossless:** The generic base must enable the derivation of all valid association rules,
- **Sound:** The generic base must forbid the derivation of the non valid association rules, and,
- **Informative:** The generic base must allow to exactly retrieve the support and confidence values of all the generated rules.
The majority of the generic bases of association rules express implications between minimal generators and closed frequent itemsets [@marzena022; @tarekcla06_paper_2_revised_version; @pasquierIGIBook09]. In this thesis, we focus on the $\mathcal{IGB}$ generic base [@IGB] defined in what follows.
**The $\mathcal{IGB}$ Generic Base** [@IGB]\
Let $\mathcal{FCP}$ be the set of frequent closed patterns and let $\mathcal{FMG}$ be the set of frequent minimal generators of all the frequent closed itemsets include or equal to a frequent closed itemset $F$. The $\mathcal{IGB}$ base is defined as follows:\
$\mathcal{IGB}$ = $\{$$R$: $fmg$ $\Rightarrow$ <span style="font-variant:small-caps;">(</span>$F$$\backslash$$fmg$<span style="font-variant:small-caps;">)</span> $\lvert$ $F$ $\in$ $\mathcal{FCP}$, $fmg$ $\in$ $\mathcal{FMG}$, <span style="font-variant:small-caps;">(</span>$F$$\backslash$$fmg$<span style="font-variant:small-caps;">)</span> $\neq$ $\emptyset$, Conf<span style="font-variant:small-caps;">(</span>$R$<span style="font-variant:small-caps;">)</span> $\geq$ *minconf*, $\nexists$ $g_{1}$ $|$ $g_{1}$ $\in$ $\mathcal{FMG}$ and Conf<span style="font-variant:small-caps;">(</span>$g_{1}$ $\Rightarrow$ <span style="font-variant:small-caps;">(</span>$F$$\backslash$$g_{1}$<span style="font-variant:small-caps;">)</span><span style="font-variant:small-caps;">)</span> $\geq$ *minconf*.$\}$
Thus, the generic rules of the $\mathcal{IGB}$ generic base represent implications between the minimal premises, according to the size on number of items, and the maximal conclusions.
Association rule-based classification process {#chap7se4}
---------------------------------------------
### Description {#subse-descrip}
We present in the following, the application of the $\mathcal{RCPR}$ and the $\mathcal{RFCCP}$ representations in the design of an association rules based classifier. In fact, we used the $\mathcal{MRCP}$ and the $\mathcal{CRCP}$ sets, composing the $\mathcal{RCPR}$ representation, within the generation of the generic $^\textsc{(}$[^10]$^{\textsc{)}}$ rare correlated rules. The $\mathcal{RFCCP}$ representation is used to generate generic frequent correlated rules, of the form $Min$ $\Rightarrow$ $Closed$ $\setminus$ $Min$, with $Min$ is a minimal generator and $Closed$ is a closed pattern. The procedure allowing the extraction of the generic correlated association rules is an adapted version of the original <span style="font-variant:small-caps;">GEN-IGB</span> [@IGB] that we implemented as a <span style="font-variant:small-caps;">C++</span> program.
Then, from the generated set of the generic rules, only the classification rules will be retained, *i.e.*, those having the label of the class in its conclusion part. Subsequently, a dedicated associative-classifier is fed with these rules and has to perform the classification process and returns the accuracy rate for each class.
The aim of the evaluation of the classification process is the comparison of the effectiveness of frequent correlated patterns *vs.* rare correlated patterns within the classification process. The comparison is carried out through two directions:\
$\bullet$ Study of the impact of *minbond* variation\
$\bullet$ Study of the impact of *minconf* variation.
### Effect of *minbond* variation
The accuracy rate of the classification, is equal to $\frac{NbrCcTr}{TotalNbrTr}$, with $NbrCcTr$ stands for the number of the correctly classified transactions and $TotalNbrTr$ is equal to the whole number of the classified transactions. The classification results reported in Table \[minbondcls1\] corresponds to the variation of the correlation constraint for a fixed *minsupp* and *minconf* thresholds, with *minconf* corresponds to the minimum threshold of the confidence measure [@Agra94].
We remark, for the frequent correlated patterns, that as far as we increase the *minbond* threshold, the number of exact and approximate association rules decreases while maintaining always an important accuracy rate. Another benefit for the *bond* correlation measure integration, is the improvement of the response time, that varies from *1000* to *0.01* seconds. Whereas, for the rare correlated patterns, we highlight that the increase of the *minbond* threshold induces a reduction in the accuracy rate. This is explained by a decrease in the number of the obtained classification rules.
[lrrrrrrrrr]{} **Dataset**& ***minsupp*** & ***minconf***& ***minbond*** & **$\#$ Exact** & **$\#$ Approximate** & **$\#$ Classification** & **Accuracy** & **Response** & **Property of**\
& & & & **Rules** & **Rules** & **Rules** & **rate** & **Time <span style="font-variant:small-caps;">(</span>sec<span style="font-variant:small-caps;">)</span>** & **Patterns**\
<span style="font-variant:small-caps;">Wine</span> & [1$\%$]{}& [0.60]{} & [0]{} & [387]{} & [5762]{} &[650]{} & [97.75$\%$]{} & [1000]{} &[Frequent]{}\
& & & [0.10]{} & [154]{} &[2739]{} &[340]{} & [95.50$\%$]{} &[13.02]{}&[Frequent]{}\
& & & [0.20]{} & [60]{} &[1121]{} &[125]{} & [94.38$\%$]{} & [1.00]{}&[Frequent]{}\
& & & [0.30]{} & [20]{} &[319]{} &[44]{} & [87.07$\%$]{} & [0.01]{}&[Frequent]{}\
<span style="font-variant:small-caps;">Zoo</span> & [50$\%$]{}& [0.70]{} & [0.30]{} & [486]{} & [2930]{} &[235]{} & [89.10$\%$]{} & [40]{}&[Rare]{}\
& & & [0.40]{} & [149]{} &[436]{} &[45]{} & [89.10$\%$]{} &[3]{}&[Rare]{}\
& & & [0.50]{} & [38]{} &[88]{} &[11]{} & [83.16$\%$]{} & [0.01]{}&[Rare]{}\
& & & [0.60]{} & [12]{} &[31]{} &[6]{} & [73.26$\%$]{} & [0.01]{}&[Rare]{}\
<span style="font-variant:small-caps;">TicTacToe</span>& [10$\%$]{} &[0.80]{}& [0]{} & [0]{} & [16]{} &[16]{} & [69.40$\%$]{}&-&[Frequent]{}\
&&&[0.05]{} & [0]{} & [16]{} &[16]{} & [69.40$\%$]{}&-&[Frequent]{}\
&&&[0.07]{} & [0]{} & [8]{} &[8]{} & [63.25$\%$]{}&-&[Frequent]{}\
&&&[0.1]{} & [0]{} & [1]{} &[1]{} & [60.22$\%$]{}&-&[Frequent]{}\
&&&[0]{} & [1, 033]{} & [697]{} &[192]{} & [100.00$\%$]{}&-&[Rare]{}\
&&&[0.05]{} & [20]{} & [102]{} &[115]{} & [100.00$\%$]{}&-&[Rare]{}\
&&&[0.07]{} & [8]{} & [66]{} &[69]{} & [97.07$\%$]{}&-&[Rare]{}\
&&&[0.1]{} & [2]{} & [0]{} &[1]{} & [65.34$\%$]{}&-&[Rare]{}\
[lrrrrrrrr]{} **Dataset**& ***minbond*** & ***minsupp***& ***minconf*** & **$\#$ Exact** & **$\#$ Approximate** & **$\#$ Classification** & **Accuracy** & **Property of**\
& & & & **Rules** & **Rules** & **Rules** & **rate** & ***Correlated***\
& & & & & & & & **patterns**\
<span style="font-variant:small-caps;">Wine</span> &[0.1]{}& [20$\%$]{} & [0.60]{} & [7]{} & [274]{} &[25]{} & [76.40$\%$]{} & [Frequent]{}\
& & & [0.80]{} & [7]{} &[86]{} &[10]{} & [86.65$\%$]{} &[Frequent]{}\
& & & [0.90]{} & [7]{} &[30]{} &[4]{} & [84.83$\%$]{} & [Frequent]{}\
&[0.1]{}& [20$\%$]{} &[0.60]{} &[91]{} &[1516]{} &[168]{} & [**95.50$\%$**]{} & [**Rare**]{}\
& & &[0.80]{} &[91]{} &[449]{} &[84]{} & [92.69$\%$]{} & [Rare]{}\
& & &[0.90]{} &[91]{} &[100]{} &[48]{} & [91.57$\%$]{} & [Rare]{}\
<span style="font-variant:small-caps;">Iris</span> &[0.15]{}& [20$\%$]{} & [0.60]{} & [3]{} & [22]{} &[7]{} & [**96.00$\%$**]{}&[**Frequent**]{}\
& & & [0.95]{} & [3]{} &[6]{} &[3]{} & [95.33$\%$]{}&[Frequent]{}\
&[0.15]{}& [20$\%$]{} & [0.60]{} & [17]{} & [32]{} &[8]{} & [80.06$\%$]{}&[Rare]{}\
& & & [0.95]{} & [17]{} &[7]{} &[5]{} & [80.00$\%$]{}&[Rare]{}\
&[0.30]{}& [20$\%$]{} &[0.60]{} &[3]{} &[22]{} &[7]{} & [**96.00$\%$**]{}&[**Frequent**]{}\
& & &[0.95]{} &[3]{} &[6]{} &[3]{} & [95.33$\%$]{}&[Frequent]{}\
&[0.30]{}& [20$\%$]{} &[0.60]{} &[8]{} &[14]{} &[4]{} & [70.00$\%$]{}&[Rare]{}\
& & &[0.95]{} &[8]{} &[6]{} &[3]{} & [69.33$\%$]{}&[Rare]{}\
### Effect of *minconf* variation
We note according to the results sketched by Table \[minconfCls2\], that for the datasets <span style="font-variant:small-caps;">Wine</span> and <span style="font-variant:small-caps;">TicTacToe</span>, the highest values of the accuracy rate are achieved with the rare correlated rules. Whereas, for the <span style="font-variant:small-caps;">Iris</span> dataset, the frequent correlated rules performed higher accuracy than the rare ones. In this regard, we can conclude that for some datasets, the frequent correlated patterns have better informativity than rare ones. Whereas, for other datasets, rare correlated patterns bring more rich knowledge. This confirms the beneficial complementarity of our approach in inferring new knowledge from both frequent and rare *correlated* patterns.
In the next section, we present the application of the rare correlated associative rules on intrusion detection data.
Classification of Intrusion Detection Data {#se-IDS}
------------------------------------------
The intrusion detection problem [@brahmiYAP10; @BrahmiYAP11] is a common problem. In this context, We present, in this section, the experimental evaluation of the correlated classification association rules, previously extracted in Section \[chap7se4\], when applied to the <span style="font-variant:small-caps;">KDD 99</span> dataset of intrusion detection data.
### Description of the <span style="font-variant:small-caps;">KDD 99</span> Dataset {#DataDescrip}
In the <span style="font-variant:small-caps;">KDD 99</span> dataset $^\textsc{(}$[^11]$^\textsc{)}$, each line or connexion represents a data stream between two defined instants between a source and a destination, each of them identified by an IP address under a given protocol(TCP, UDP). Every connection is labeled either normal or attack and has 41 discrete and continuous attributes that are divided into three groups [@farid2010]. The first group of attributes is the basic features of network connection, which include the duration, prototype, service, number of bytes from IP source addresses or from destination IP addresses. The second group of features is composed by the content features within a connection suggested by domain knowledge. The third group is composed by traffic features computed using a two-second time window.
<span style="font-variant:small-caps;">KDD 99</span> defines 38 attacks categories partitioned into four `Attack` classes, which are <span style="font-variant:small-caps;">Dos</span>, <span style="font-variant:small-caps;">Probe</span>, <span style="font-variant:small-caps;">R2L</span> and <span style="font-variant:small-caps;">U2R</span>, and one <span style="font-variant:small-caps;">Normal</span> class. These categories are described in [@NahlaSac2004] and in [@farid2010] as follows:\
$\bullet$ **Denial of Service Attacks (DOS)**: in which an attacker overwhelms the victim host with a huge number of requests. Such attacks are easy to perform and can cause a shutdown of the host or a significant slow in its performance. Some examples of DOS attack: Neptune, Smurf, Apache2 and Pod.
$\bullet$ **User of Root Attacks (U2R)**: in which an attacker or a hacker tries to get the access rights from a normal host in order, for instance, to gain the root access to the system. Some examples of U2R attack: Httptunnel, Perl, Ps, Rootkit.
$\bullet$ **Remote to User Attacks (R2L)**: in which the intruder tries to exploit the system vulnerabilities in order to control the remote machine through the network as a local user. Some examples of R2L attack: Ftp-write, Imap, Named, Xlock.
$\bullet$ **Probe**: in which an attacker attempts to gather useful information about machines and services available on the network in order to look for exploits. Some examples of Probe attack: Ipsweep, Mscan, Saint, Nmap. The <span style="font-variant:small-caps;">KDD 99</span> dataset contains 4, 940, 190 objects in the learning set. We consider 10$\%$ of the training set in the construction step of the classifier, containing 494, 019 objects. The learning set contains 79.20$\%$ <span style="font-variant:small-caps;">(</span>respectively, 0.83$\%$, 19.65$\%$, 0.22$\%$ and 0.10$\%$<span style="font-variant:small-caps;">)</span> of <span style="font-variant:small-caps;">Dos</span> <span style="font-variant:small-caps;">(</span>respectively, <span style="font-variant:small-caps;">Probe</span>, <span style="font-variant:small-caps;">Normal</span>, <span style="font-variant:small-caps;">R2L</span> and <span style="font-variant:small-caps;">U2R</span><span style="font-variant:small-caps;">)</span>.
### Experimentations and Discussion of Obtained Results {#RecapXp}
Table \[TabNewKDD3\] summarizes the obtained results, where <span style="font-variant:small-caps;">AR</span> and <span style="font-variant:small-caps;">DR</span>, respectively, denote “Association Rule” and “Detection Rate”, with Detection Rate = $\frac{NbrCcCx}{TotalNbrCx}$, with $NbrCcCx$ stands for the number of the correctly classified connections and $TotalNbrCx$ is equal to the whole number of the classified connections, while *minconf* is the minimum threshold of the confidence measure [@Agra94].
In addition, by “Construction step”, we mean that the step associated to the extraction of the $\mathcal{RCPR}$ representation while “Classification step” represents the step in which the classification association rules are derived starting from $\mathcal{RCPR}$ and applied for detecting intrusions.
We note that the highest value of the detection rate is achieved for the classes <span style="font-variant:small-caps;">Normal</span> and <span style="font-variant:small-caps;">Dos</span>. In fact, this is related to the high number of connections of these two classes. This confirms that our proposed approach presents interesting performances even when applied to voluminous datasets. We also remark that the detection rate varies from an attack class to another one. In fact, for the <span style="font-variant:small-caps;">U2R</span> class, this rate is relatively low when compared to the others classes.
To sum up, according to Table \[TabNewKDD3\], the computational cost varies from one attack class to another one. It is also worth noting that, for all the classes, the construction step is much more time-consuming than the classification step. This can be explained by the fact that the extraction of the $\mathcal{RCPR}$ concise representation is a sophisticate problem.
Furthermore, the results shown by Table \[TabCmpNahla\] prove that our proposed classifier is more competitive than the decision trees as well as the Bayesian networks [@NahlaSac2004]. In fact, our approach presents better results for the attack classes <span style="font-variant:small-caps;">Dos</span>, <span style="font-variant:small-caps;">R2L</span> and <span style="font-variant:small-caps;">U2R</span> than these two approaches. For the <span style="font-variant:small-caps;">Normal</span> class, the obtained results using our approach are close to those obtained with the decision trees. The Bayesian networks based approach presents better detection rate only for the <span style="font-variant:small-caps;">Probe</span> attack class. We thus deduce that the proposed rare correlated association rules constitute an efficient classification tool when were applied to the intrusion detection in a computer network. = cmbx7. scaled
------------------------------------------------------ -------------------------------------------------------------------------------------------------------- --------------- --------------- ------------------------------------------------------- ------------------------------------------------------- ---------------------------------------------------------- --------------------------------------------------------------------------------------------------------------
**Attack** ***minsupp*** ***minbond*** ***minconf*** **$\#$ of generic** **$\#$ of generic** **$\#$ of generic** **CPU Time**
**class** **<span style="font-variant:small-caps;">(</span>$\%$<span style="font-variant:small-caps;">)</span>** **exact** **approximate** **<span style="font-variant:small-caps;">AR</span>s of** **<span style="font-variant:small-caps;">(</span>in seconds<span style="font-variant:small-caps;">)</span>**
**<span style="font-variant:small-caps;">AR</span>s** **<span style="font-variant:small-caps;">AR</span>s** **classification**
<span style="font-variant:small-caps;">Dos</span> [80]{} [0.95]{} [0.90]{} [4]{} [31]{} [17]{} [121]{}
<span style="font-variant:small-caps;">Probe</span> [60]{} [0.70]{} [0.90]{} [232]{} [561]{} [15]{} [56]{}
<span style="font-variant:small-caps;">Normal</span> [85]{} [0.95]{} [0.95]{} [0]{} [10]{} [3]{} [408]{}
<span style="font-variant:small-caps;">R2L</span> [80]{} [0.90]{} [0.70]{} [2]{} [368]{} [1]{} [1, 730]{}
<span style="font-variant:small-caps;">U2R</span> [60]{} [0.75]{} [0.75]{} [106]{} [3]{} [5]{} [33]{}
------------------------------------------------------ -------------------------------------------------------------------------------------------------------- --------------- --------------- ------------------------------------------------------- ------------------------------------------------------- ---------------------------------------------------------- --------------------------------------------------------------------------------------------------------------
= cmbx7.5 scaled
------------------------------------------------------ --------------------------------------------------------------- -------------------- -----------------------
**Attack class** **Rare correlated** **Decision trees** **Bayesian networks**
**generic <span style="font-variant:small-caps;">AR</span>s** [@NahlaSac2004] [@NahlaSac2004]
<span style="font-variant:small-caps;">Dos</span> **98.68** 97.24 96.65
<span style="font-variant:small-caps;">Probe</span> 70.69 77.92 **88.33**
<span style="font-variant:small-caps;">Normal</span> [**100.00**]{} 99.50 97.68
<span style="font-variant:small-caps;">R2L</span> [**81.52**]{} 0.52 8.66
<span style="font-variant:small-caps;">U2R</span> [**38.46**]{} 13.60 11.84
------------------------------------------------------ --------------------------------------------------------------- -------------------- -----------------------
Application of the $\mathcal{RCPR}$ representation on Micro-array gene expression data {#SecGene}
--------------------------------------------------------------------------------------
We present, in this section, the application of the $\mathcal{RCPR}$ condensed representation of rare correlated patterns on Micro-array gene expression data. In fact, the $\mathcal{RCPR}$ representation <span style="font-variant:small-caps;">(</span>*cf.* Definition \[rmcr\] Page <span style="font-variant:small-caps;">)</span>, is composed by the $\mathcal{CRCP}$ set of Closed Rare Correlated Patterns as well as the associated $\mathcal{MRCP}$ set of Minimal Rare Correlated Patterns. From these two sets, we extract the generic rare correlated associated rules, as described in Sub-section \[subse-descrip\] <span style="font-variant:small-caps;">(</span>*cf.* Page <span style="font-variant:small-caps;">)</span>. The extracted association rules will be then analyzed in order to evaluate the relevance of the obtained biological knowledge.
### Our Motivations
Since many years, gene expression technologies have offered a huge amount of micro-array data by measuring expression levels of thousands of genes under various biological experimental conditions. The micro-array datasets present specific characteristics which is the high density of data. These datasets are in the form of <span style="font-variant:small-caps;">(</span>N x M<span style="font-variant:small-caps;">)</span> matrix with N represents the rows <span style="font-variant:small-caps;">(</span>the conditions or the experiments<span style="font-variant:small-caps;">)</span> and M represents the columns <span style="font-variant:small-caps;">(</span>the genes<span style="font-variant:small-caps;">)</span>. In this regard, the key task in the interpretation of biological knowledge is to identify the differentially expressed genes. In this respect, we are based on rare correlated patterns in order to identify up and down regulated genes.
Several related works [@top-interval; @mcrMiner; @pasquier2009; @goodrule; @Article3] were focused on the extraction of frequent patterns and the generation of frequent association rules in order to analyze micro-array data. Our motivation behind the choice of biological data is based on the review of the existing literature that confirms that there is no previous work that addresses the issue of analysis of gene expressions from rare correlated patterns. Our proposed association-rules based process can be classified as an expression-based interpretation approach for biological associations. In fact, we are based on gene expression profiles varying under hundreds of biological conditions. In what follows, we provide the description of the used micro-array dataset.
### Description of the Micro-array gene expression data
For the application of our approach, we used the breast cancer 2 <span style="font-variant:small-caps;">GSE1379</span> dataset $^\textsc{(}$[^12]$^{\textsc{)}}$. The original data is composed by 60 samples and 22, 575 genes. We present in Table \[sample1\] a sample of the <span style="font-variant:small-caps;">GSE1379</span> dataset containing only 5 genes on columns and 5 samples on rows.
Table \[GenName\] illustrates examples of some relevant genes of the <span style="font-variant:small-caps;">GSE1379</span> dataset enriched with their description.
### The Discretization process
The discretization aimed to transform the continuous data into discrete data. We performed a discretization process based on the R.Basic package of the R statistical framework $^\textsc{(}$[^13]$^{\textsc{)}}$. First of all, we apply the Z-Normalization [@zNormalization] over the whole dataset in order to transform the initial data distribution to a normal distribution. The second step consists in determining the over-expressed cutoff $O_{c}$ and the under-expressed cutoff $U_{c}$. In fact, according to the Z-Normal distribution table, when considering a confidence level $1-\alpha$ equal to 95$\%$, we have:
$\bullet$ The over-expressed cutoff $O_{c}$ = Z($\alpha$/2) = 1.96
$\bullet$ The under-expressed cutoff $U_{c}$ = -Z($\alpha$/2) = -1.96
Thus, we have for each sample $i$ and for each gene $j$, $V_{ij}$ corresponds to the value of the gene expression $j$ within the sample $i$. The $V_{ij}$ expression is evaluated as follows:
- if $V_{ij}$ $\leq$ $U_{c}$ then $V_{ij}$ is under-expressed $\downarrow$
- if $V_{ij}$ $\geq$ $O_{c}$ then $V_{ij}$ is over-expressed $\uparrow$
- if $U_{c}$ < $V_{ij}$ > $O_{c}$ then $V_{ij}$ is unexpressed
We present in Table \[DistData\], a sample of the discretized data, where the over-expressed genes are referenced by the value of 1 whereas the under-expressed genes are referenced by the value of 0. The ‘$-$‘ symbol represent unexpressed gene expressions which are not relevant for our analysis.
After the discretization process, we apply a substitution function in order to transform the discretized gene expression values in the adequate input format for the mining process. Consequently, we apply our substitution function $\theta$ as follows:
- if $V_{ij}$ is over-expressed $\uparrow$ then $V_{ij}$ $\leftarrow$ $Id_{j}$, with $Id_{j}$ corresponds to the unique identifier of gene $j$
- if $V_{ij}$ is under-expressed $\downarrow$ then $V_{ij}$ $\leftarrow$ ‘$Id_{j}$ + $\arrowvert$M$\arrowvert$‘, with $Id_{j}$ is the unique identifier of gene $j$ and $\arrowvert$M$\arrowvert$ corresponds to the number of the distinct genes, $\arrowvert$M$\arrowvert$ = 22, 575 in our tested dataset.
We present in Table \[SubsData\], a sample of the final substituted data. This sample is in the adequate input format of our mining algorithm <span style="font-variant:small-caps;">Opt-Gmjp</span>.
### Experimental results
We conducted several experiments over the <span style="font-variant:small-caps;">GSE1379</span> dataset in order to extract the most relevant exact and approximate association rules. The <span style="font-variant:small-caps;">GSE1379</span> dataset was preprocessed with the <span style="font-variant:small-caps;">GEO2R</span> tool in order to identify genes that are differentially expressed across experimental conditions. The Results obtained by the <span style="font-variant:small-caps;">GEO2R</span> tool are presented as a table of genes ordered by significance. Thus, we maintain the 550 most relevant genes from 22, 575 initial genes. For these experiments, the <span style="font-variant:small-caps;">Opt-Gmjp</span> algorithm was applied to the <span style="font-variant:small-caps;">GSE1379</span> with $|\mathcal{T}|$ = 60 and with $|\mathcal{I}|$ = 1, 100 distinct items values.
[|rrrr||rrrc|]{} ***minsupp*** & ***minbond***& $|\mathcal{MRCP}|$ &$|\mathcal{CRCP}|$ & ***minconf*** & **$\#$ Exact** & **$\#$ Approximate** & **CPU**\
& & & & & **Rules** & **Rules** & **Time <span style="font-variant:small-caps;">(</span>sec<span style="font-variant:small-caps;">)</span>**\
& [0.30]{} & [120]{} & [146]{} & [0.70]{} &[19]{} & [3]{} & [0.0405]{}\
& & & & [0.50]{} &[19]{} & [17]{} &[0.0405]{}\
& & & & [0.30]{} &[19]{} & [20]{} & [0.0405]{}\
[50$\%$]{} & [0.30]{} & [157]{} & [244]{} & [0.70]{} &[26]{} & [77]{} & [0.0754]{}\
& & & & [0.50]{} &[26]{} & [128]{} &[0.0754]{}\
& & & & [0.30]{} &[26]{} & [134]{} &[0.0754]{}\
& [0.50]{} & [79]{} & [72]{} & [0.30]{} &[7]{} & [6]{} & [0.0595]{}\
& [0.70]{} & [59]{} & [56]{} & [0.30]{} &[3]{} & [0]{} & [0.0463]{}\
These experiments were conducted in order to assess the scalability of our <span style="font-variant:small-caps;">Opt-Gmjp</span> algorithm when applied to very dense biological dataset and to evaluate the impact of varying the *minsupp*, the *minbond* and the *minconf* thresholds on the number of the extracted association rules. We report in Table \[geneAssRule1\] the execution times as well as the number of the approximate and exact extracted association rules. We can draw theses conclusions:
- The sizes of the $\mathcal{MRCP}$ set of minimal correlated rare patterns as well as that of the $\mathcal{CRCP}$ set of closed rare correlated patterns depends only on the variation of *minsupp* and *minbond* thresholds. We deduce that, $|\mathcal{MRCP}|$ and $|\mathcal{CRCP}|$ decrease when increasing *minbond* from [0.30]{} to [0.70]{}.
- The execution times are not affected by the variation of <span style="font-variant:small-caps;">minconf</span> threshold. In fact, the reported execution times corresponds to the CPU-time needed for extracting the $\mathcal{RCPR}$ representation. The CPU-time needed for the derivation of the association rules is negligible in all the performed experiments.
- The number of the extracted association rules decreases while increasing the *minconf* threshold. For example, for *minsupp* = [50$\%$]{} , *minbond* = [0.30]{} and *minconf* = 0.30, we have $|Approximate-Rules|$ = 134, while for *minconf* = 0.70, $|Approximate-Rules|$ = 77. It’s obviously that the number of the exact rules is insensitive to the variation of the *minconf* threshold since the confidence of exact rules is equal to [100$\%$]{}.
- The increase of the *minbond* threshold value from 0.30 to 0.70, induce a decrease in the size of the $\mathcal{MRCP}$ and $\mathcal{CRCP}$ sets. This reflects that the used dataset do not present important correlation degree among the items. The items are dispersed in the universe due to the low-level of co-expression of the mined genes.
### Biological significance of Extracted Association rules
Table \[ExpAR\] shows different examples of association rules extracted by a dedicated procedure previously described in sub-section \[subse-descrip\]. In Table \[ExpAR\], supports are expressed in number of transactions while confidence are given in percentages. The association rules show groups of genes that are over-expressed or under-expressed in a set of conditions.
To determine the functional relationship among the obtained gene sets, we used the <span style="font-variant:small-caps;">STRING 10</span> $^\textsc{(}$[^14]$^{\textsc{)}}$ resource [@string] which is a database of known and predicted protein-protein interaction.
In this regard, the gene sets obtained within the association rules were uploaded into <span style="font-variant:small-caps;">STRING</span> and the following prediction methods were employed: co-expression, co-occurrence with a medium confidence score equal to 40$\%$. This analysis shows the interactions among the gene sets as shown in Figures \[fig-string1\] and \[fig-string2\]. This finding support the hypothesis that the returned gene sets thank to our rare correlated association rules, show an important degree of biological interrelatedness.
In figure \[fig-string2\], we highlight just the most relevant genes reported in the biological literature and related to the analysis of breast cancer [@breast1]. These genes are: HOXB13, ABCC11, CHDH, ESR1 and IL17BR [@Article3].
According to Table \[ExpAR\], Rule 0 reflects that the estrogen receptor 1 which is a Nuclear hormone receptor and is expressed by ESR1 when it is over-expressed in this experiment induces an over-expression of the glutathione S-transferase alpha 2 traduced by gene GSTA2. Rule 3 highlights that if the HOXB13 and the BLOC1S6 genes are down-expressed then the CHDH and the CRISPLD2 genes are over-expressed. In fact, the HOXB13 gene refers to Sequence-specific transcription factor which is part of a developmental regulatory system that provides cells with specific positional identities on the anterior-posterior axis. While, the CRISPLD2 gene is cysteine-rich secretory protein LCCL domain and the CHDH gene expresses the choline dehydrogenase. Rule 6 present an interesting relation between the IL17BR gene, reflecting the interleukin 17 receptor B and playing a role in controlling the growth and differentiation of hematopoietic cells, and the PFKP gene corresponding to phospho-fructokinase, platelet. The PFKP gene catalyzes the phosphorylation of fructose-6-phosphate (F6P) by ATP to generate fructose-1, 6-bisphosphate (FBP) and ADP.
Almost of the obtained association rules highlights important relationship of the HOXB13 and the IL17BR genes. In fact, the analysis of these two genes expression may be useful for identifying patients appropriate for alternative therapeutic regimens in early-stage breast cancer [@Article3]. In summary, we conclude that the diverse obtained rare correlated association-rules reveals a variety of relationship between up and down gene-expression which proves that breast cancer is an interesting biologically heterogeneous research field. Thus to deduce that rare correlated patterns present good results when applied to the context of biological data since they are able to reveal hidden and surprising relations among genes properties.
Conclusion {#ConcChap7}
----------
This chapter was dedicated to the description of the associative classification process based on the correlated patterns. For this purpose, we started by presenting the framework of association rules extraction, we clarify the properties of the generic bases of association rules. We continued with the detailed description and presentation of the application of both frequent correlated and rare correlated patterns within the classification of some UCI benchmark datasets. We equally present the application of rare correlated patterns in the classification of intrusion detection data from the <span style="font-variant:small-caps;">KDD 99</span> dataset. The effectiveness of the proposed classification method has been experimentally proved. The chapter was concluded with the application of the $\mathcal{RCPR}$ representation on the extraction of biologically relevant associations among Micro-array gene expression data. A better classification accuracy may be achieved while thinking about missing-values treatment [@OthmanY06].
Conclusion and Perspectives {#ch_conc}
===========================
Conclusion {#conclusion}
----------
In this thesis, we were mainly interested to two complementary classes of patterns namely rare correlated patterns and frequent correlated patterns according to the *bond* correlation measure. In fact, the $\mathcal{FCP}$ set of frequent correlated patterns result from the intersection of the set of frequent patterns and the set of correlated patterns. The $\mathcal{FCP}$ set is then the result of the conjunction of two anti-monotonic constraints of frequency and of correlation. Consequently this $\mathcal{FCP}$ set induces an order ideal on the itemset lattice. Nevertheless, the $\mathcal{RCP}$ set of rare correlated patterns result from the conjunction of two constraints of distinct types namely the monotonic constraint of rarity and the anti-monotonic constraint of correlation. Thus, the localization of the $\mathcal{RCP}$ set is more difficult and the extraction process is more costly. This characteristic constitute one of the challenges to deal with through this thesis.
This thesis report was partitioned into four different parts. The first part was dedicated to the review of correlated patterns mining. In this regard, we started the first chapter of this part by introducing the basic notions related to the itemset search space, to itemset extraction. We defined the two distinct categories of constraints. We introduced equally the environment of Formal Concept Analysis FCA which offer the basis for the proposition of our approaches, specifically the notions of Closure Operator, Minimal Generator, Closed Pattern, Equivalence class and Condensed representation of a set of patterns. Thereafter, we studied in the second chapter of this first part, the state of the art approaches dealing with correlated patterns mining. Our study covers the frequent correlated patterns mining, the rare correlated patterns as well as the approaches focusing on condensed representations of correlated itemsets.
The second part was dedicated to the presentation of our approaches. The first chapter of this part was devoted to the characterization of both frequent correlated and rare correlated patterns and the introduction of their associated condensed representations. We deeply defined the properties of the $f_{bond}$ closure operator associated to the *bond* measure and we describe the structural specificities of the induces equivalence classes. In fact, the condensed representations associated to the $\mathcal{RCP}$ set of rare correlated are composed by the union of the closed correlated rare patterns and their associated minimal generators. Nevertheless, for the case of frequent correlated patterns, the closed correlated frequent patterns constitute a condensed concise representation of the $\mathcal{FCP}$ set. In the second chapter, we focused on the presentation of our <span style="font-variant:small-caps;">Gmjp</span> extraction approach. In fact, <span style="font-variant:small-caps;">Gmjp</span> is the first approach to mine *bond* correlated patterns in a generic way <span style="font-variant:small-caps;">(</span>i.e., with two types of constraints: anti-monotonic constraint of frequency and monotonic constraint of rarity<span style="font-variant:small-caps;">)</span>. Our mining approach was based on the key notion of bitsets codification that supports efficient correlated patterns computation thanks to an adequate condensed representation of patterns. The deeply description of the whole steps of <span style="font-variant:small-caps;">Gmjp</span> as well as the theoretical complexity approximation and a running example were equally detailed. This fifth chapter was concluded by the algorithms of interrogation and of regeneration of the condensed representation associated to rare correlated patterns.
The third part of this report was dedicated to the experimental validation of our <span style="font-variant:small-caps;">Gmjp</span> as well as the presentation and evaluation of the associative-classification process. In the first chapter of this third part we focused on the experimental evaluation of <span style="font-variant:small-caps;">Gmjp</span>. The evaluation process was based on two main axes, the first is related to the compactness rates of the condensed representations while the second axe concerns the running time. We equally proposed an optimized version of <span style="font-variant:small-caps;">Gmjp</span> which present much better performance than <span style="font-variant:small-caps;">Gmjp</span> over different benchmark datasets. The two main keys which constitute the thrust of the improved version: (*i*) only one scan of the database is performed to build the new transformed dataset; (*ii*) it offers a resolution of the problem of handling both monotonic and anti-monotonic constraints within a unique mining process. In fact, opposite constraint mining is classified as an NP-Hard problem [@boley2009]. But, our goal was optimally achieved without relying on the border’s extraction. This constitute a strong added-value to <span style="font-variant:small-caps;">Gmjp</span>, since many approaches are based on border’s identification in order to extract such difficult set of patterns.
In the second chapter of this third part, we presented the classification process based on correlated patterns. Since the classification process that we proposed was based on associative rules, thus we started the chapter by presenting the framework of association rules extraction, we clarified the properties of the generic bases of association rules. We continued with the detailed presentation of the application of both frequent correlated and rare correlated patterns within the classification of some UCI benchmark datasets. In addition, we reported in this chapter the application of rare correlated patterns in the classification of intrusion detection data from the <span style="font-variant:small-caps;">KDD 99</span> dataset. The obtained results showed the usefulness of our proposed classification method over four different intrusion classes. We concluded the chapter with the application of rare correlated associative rules on Micro-array gene expression data. The obtained rules helped to identify potential relations among up and down regulated gene expressions related to Breast Cancer.
The fourth and final part concluded the thesis report.
Perspectives
------------
The obtained results in this thesis opens many perspectives from which we quote:
The extraction of generalized association rules starting from rare correlated patterns also from frequent correlated patterns. In addition, we plan to extend our approach to other correlation measures [@Kimpkdd2011; @borgelt; @surana2010; @Omie03] through classifying them into classes of measures sharing the same properties. An important direction is to propose a generic way allowing the extraction of the sets of frequent correlated patterns and rare correlated patterns as well as their associated concise representations. Pieces of new knowledge in the form of exact or approximate correlated generalized association rules can then be derived.
The extension of the extraction of correlated patterns to the extraction of both frequent and rare sequential correlated patterns. A promoting area for applying sequential patterns is: opinion mining. In fact, Opinion Mining is an important research area [@Ohana2011] which is based on the extraction of opinions and the sentiment analysis from text data <span style="font-variant:small-caps;">(</span>Text Mining<span style="font-variant:small-caps;">)</span>. Opinion Mining is a fruitful field since it is concerned with many real life application fields such as: Financial analysis, market estimation, customer behavior detection. In fact, the evaluation of new products and services nearby customers is based on the comments and advices of web visitors. Consequently, the derivation of association rules and their application to opinion mining [@Jindal2010] is a potentially interesting research axe.
Another Fruitful perspective consists in addressing the issue of correlated patterns mining from big datasets. In fact, big data mining is a new challenging task since computational requirements are difficult to provide. An interesting solution is to exploit parallel frameworks, such as MapReduce [@mapred-2012] that offer the opportunity to make powerful computing and storage. Consequently, mining condensed representations of correlated patterns from big real life datasets thank to the MapReduce environment is an up to date challenging mining task.
Publication List
----------------
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\[1\] , Hamrouni Tarek, Ben Yahia Sadok. Motifs Corrélés rares: Caractérisation et nouvelles représentations concises exactes . Appeared in the Revue of New Information Technologies RNTI: Quality of Data and Knowledge: Measure and Evaluate the Quality of Data and Knowledge, <span style="font-variant:small-caps;">(</span>MQDC 2012<span style="font-variant:small-caps;">)</span>, pages 89-116.**\[indexed DBLP\]**.
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\[2\] , Hamrouni Tarek, Ben Yahia Sadok. Efficient Mining of New Concise representations of Rare Correlated Patterns. Appeared in the IDA journal ’Intelligent Data Analysis’ 2015, Volume 19, pages 359-390. **<span style="font-variant:small-caps;">(</span>Impact factor in 2014 = 0.50<span style="font-variant:small-caps;">)</span>**.
$\bullet$ ****\
\[3\] , Hamrouni Tarek, Ben Yahia Sadok. Algorithmes d’extraction et d’interrogation d’une représentation concise exacte des motifs corrélés rares. In proceedings of the 12th international francophone conference on extraction and managing knowledges <span style="font-variant:small-caps;">(</span>EGC 2012<span style="font-variant:small-caps;">)</span>, Bordeaux, France, pages 225-230. **\[indexed DBLP\], rank C**.
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\[4\] , Hamrouni Tarek, Ben Yahia Sadok. New Exact Concise Representation of Rare Correlated Patterns: Application to Intrusion Detection. In proceedings of the 16th Pacific Asia conference <span style="font-variant:small-caps;">(</span>PAKDD 2012<span style="font-variant:small-caps;">)</span>, Kuala Lumpur, Malaysia, pages 61-72. **\[indexed IEEE, DBLP\], rank A**.
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\[5\] , Ben Yahia Sadok. Inferring New Knowledge from Concise Representations of both Frequent and Rare Jaccard Itemsets. In proceedings of the 24th International conference of Database and Expert Systems Applications <span style="font-variant:small-caps;">(</span>DEXA 2013<span style="font-variant:small-caps;">)</span>, Prague, Check Republic, pages 109-123. **\[indexed IEEE, DBLP\], rank B**.
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\[6\] , Ben Yahia Sadok. Key Correlation Mining by Simultaneous Monotone and Anti-monotone Constraints Checking. In proceedings of the 30th ACM Symposium on Applied Computing <span style="font-variant:small-caps;">(</span>SAC 2015<span style="font-variant:small-caps;">)</span>, Salamanca, Spain. **\[indexed ACM, IEEE, DBLP\], rank B**.
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[^1]: The power-set of a set $\mathcal{T}$, is constituted by the sub-sets of $\mathcal{T}$, is denoted by $\mathcal{P}$<span style="font-variant:small-caps;">(</span>$\mathcal{T}$<span style="font-variant:small-caps;">)</span>.
[^2]: We use the index **c** since the closure operator gathers itemsets sharing the same common **c**onjunctive support.
[^3]: $\mathcal{CRCP}$ stands for **C**losed **R**are **C**orrelated **P**atterns.
[^4]: $\mathcal{MRCP}$ stands for **M**inimal **R**are **C**orrelated **P**atterns.
[^5]: <span style="font-variant:small-caps;">Gmjp</span> stands for **G**eneric **M**ining of **J**accard **P**atterns. We note, by sake of accuracy, that the notation of *Jaccard* measure corresponds to the *bond* measure.
[^6]: We used in our implementation the <span style="font-variant:small-caps;">C++</span> STL Standard Template Library multi-map.
[^7]: Available at *http://fimi.cs.helsinki.fi/data*.
[^8]: We note that ‘S1‘ stands for the First Scenario, ‘S2‘ stands for the Second Scenario, ‘S3‘ stands for the Third Scenario and ‘S4‘ stands for the Fourth Scenario.
[^9]: We note that “S1” stands for the First Scenario and “S2” stands for the Second Scenario.
[^10]: By “generic”, it is meant that these rules are with minimal premises and maximal conclusions, w.r.t. set-inclusion.
[^11]: The <span style="font-variant:small-caps;">KDD 99</span> dataset is available at the following link: *http://kdd.ics.uci.edu/databases/kddcup99/kddcup99.html*.
[^12]: The breast cancer dataset is publicly available and downloaded from http://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE1379 .This dataset is submitted on May 2004 and updated on March 2012.
[^13]: The R Project for Statistical Computing is downloaded from https://www.r-project.org.
[^14]: <span style="font-variant:small-caps;">STRING</span> stands for the **S**earch **T**ool for the **R**etrieval of **IN**teracting **G**enes/Proteins and is publicly available at http://string-db.org.
|
---
address:
- |
Institute of Mathematics NAS of Ukraine\
Tereschenkivska str. 3\
01601 Kiev\
Ukraine
- |
Institute of Mathematics NAS of Ukraine\
Tereschenkivska str. 3\
01601 Kiev\
Ukraine
author:
- Roman Grushevoi
- Kostyantyn Yusenko
date: 'April 11, 2008'
title: 'Unitarization of linear representations of non-primitive posets '
---
[^1]
0. Introduction {#introduction .unnumbered}
===============
The representation theory of partially ordered sets (posets) in linear vector spaces has been studied extensively and found to be of great importance for studying indecomposable representations of group and algebras, Cohen-Macaulay modules and many others algebraical objects (see [@Drozd; @GabrielRoiter; @Kleiner1; @Kleiner2; @Simson1992] and many others). A representation of a given poset $\mathcal P$ in some vector space $V$ is a collection $(V;V_i),\ i\in \mathcal P$ of vector subspaces $V_i\subset V$ such that $V_i\subset V_j$ as soon as $i\prec j$ in $\mathcal P$. Usually such representations are studied up to equivalence (which is given by linear bijections between two spaces that bijectively map the corresponded subspaces). M. Kleiner and L. Nazarova (see [@Kleiner1; @Nazarova]) completely classified all posets into three classes: finite type posets (posets that have finite number of indecomposable nonequivalent representation), tame posets (posets that have at most one-parametric family of indecomposable representations in each dimension) and wild posets (the classification problem of their indecomposable representations contains as a subproblem a problem of classification up to conjugacy classes a pair of two matrices).
It is also possible to develop a similar theory over Hilbert spaces. By representation we understand a collection $(H;H_i)$ of Hilbert subspaces in some Hilbert space $H$ such that $H_i\subset H_j$ as soon as $i\prec j$. The equivalence between two system of Hilbert subspaces is given by unitary operator which bijectively maps corresponding subspaces. It turns out that in this case the classification problem becomes much more harder: even the poset $\mathcal P=\{a,b_1,b_2\}, b_1\prec b_2$ becomes a $*$-wild poset (it is impossible to classify all representation of this poset in a reasonable way see [@KruglyaSamoilenko2]). We add an “extra” relation $$\label{orthoscalarity}
\alpha_1P_1+\ldots+\alpha_nP_n=\gamma I,$$ between the projections $P_i:H\mapsto H_i$ on corresponding subspaces for some weight $\chi=(\alpha_1,\ldots,\alpha_n) \in
\mathbb R^n_+$ (this relation will be called *orthoscalarity condition*). When the system of subspaces is a so-called $m$-filtration, this relation plays an important role in different areas of mathematics (see [@Totaro1994; @Klyachko] and references therein) and this is actually one of the original motivations to investigate such representations of posets in Hilbert space.
The interconnection between linear and Hilbert representations of the posets is given by *unitarization* which asks whether for given linear representation $(V,V_i)$ it is possible to provide a hermitian structure in $V$ so that the linear relation (\[orthoscalarity\]) holds for some weight $\chi$. In [@GrushevoyYusenko] for the case when $\mathcal P$ is a *primitive* poset we proved that a poset $\mathcal P$ is of finite orthoscalar type (has finitely many irreducible representations with orthoscalarity condition up to the unitary equivalence) if and only if it is of finite (linear) type. Also there were proved that each indecomposable representation of poset of finite (linear) type could be unitarized with some weight and for each representation we described all appropriated for unitarization weights.
In this paper we will prove the same for *non-primitive* posets. The approach given in [@GrushevoyYusenko] does not work longer for *non-pirmitive* case. We will use instead the notion of $\chi$-stable representation from Geometric Invariant Theory of the product of the grassmannians of $V$ (see [@Hu2004; @Totaro1994] and references therein). It turns out that for indecomposable representation $\chi$-unitarizability is essentially the same as $\chi$-stability. This gives a machinery to compute all appropriated for unitarization weights.
The main results of the paper are the following theorems.
\[mainthmFin\] A partially ordered set $\mathcal P$ has finite number of irreducible finite-dimensional Hilbert representations with orthoscalarity condition if and only if it does not contain subsets of the following form $(1,1,1,1)$, $(2,2,2)$, $(1,3,3)$, $(1,2,5)$, $(N,4)$, where $(n_1,\ldots,n_s)$ denotes the cardinal sum of linearly ordered set $\mathcal L_1,\ldots,\mathcal L_s$, whose orders equal $n_1,\ldots,n_s$, respectively, and $(N,4)$ is the set $\{a_1, a_2, b_1, b_2, c_1, c_2, c_3, c_4\}$, with the order $a_1
\prec a_2, b_1 \prec b_2, b_1 \prec a_2, c_1 \prec c_2 \prec c_3
\prec c_4$, and no other elements are comparable.
\[mainthmUnit\] Each indecomposable linear representation of the poset of finite type can be unitarized with some weight.
**Acknowledgments.** We would like to thank Prof. Yu.S. Samoilenko and Prof. V.L. Ostrovskii for the statement of the problem and Prof. Ludmila Turowska for stimulating discussion. The second author thanks Chalmers University of Technology for hospitality and for the fruitful environment and Thorsten Weist for helpful remarks. The second author was partially supported by the Swedish Institute and by Ukrainian Grant for Young Scientists.
Preliminaries.
==============
In this section we will briefly recall some basic facts concerning partially ordered sets, their representations and unitarization of linear representations.
Posets and Hasse quivers
------------------------
Let $(\mathcal P,\prec)$ be a finite partially ordered set (or poset for short) which for us will be $\{a_1,\ldots,a_n\}$. By the width of the poset $\mathcal P$ we understand the cardinality of the largest antichain of $\mathcal P$, i.e. the cardinality of a subset of $\mathcal P$ where any two element are incomparable.
A poset $\mathcal P$ of the width $s$ is called *primitive* and denoted by $(n_1,\ldots,n_s)$ if this poset is the cardinal sum of $s$ linearly ordered sets $\mathcal L_1,\ldots,\mathcal L_s$ of orders $n_1,\ldots,n_s$. Otherwise the poset is called *non-primitive*.
We will use the standard graphic representations for the poset $\mathcal{P}$ called *Hasse quiver*. This representation associates to each elements $x \in \mathcal P$ a vertex $x$ and a unique arrow $x\rightarrow y,\ y\in \mathcal P$ if $x\prec y$ and if there is no $z \in \mathcal P$ such that $x\prec z \prec y$. For example, let $\mathcal P= (N,2)=\{a_1,a_2,b_1,b_2,c_1,c_2\}$ with the following order $$a_1\prec a_2, \quad b_1\prec b_2, \quad c_1\prec c_2, \quad b_1\prec a_2,$$ then the corresponding Hasse quiver is the following:
(50,60) (0,15)[$a_1$]{} (20,15)[$b_1$]{} (40,15)[$c_1$]{} (2,24)[(0,1)[15]{}]{} (22,24)[(0,1)[15]{}]{} (42,24)[(0,1)[15]{}]{} (0,44)[$a_2$]{} (20,44)[$b_2$]{} (40,44)[$c_2$]{} (19,24)[(-1,1)[15]{}]{} (50,25)[$.$]{}
Linear representations of posets. Indecomposability and Bricks
--------------------------------------------------------------
By a linear representation $\pi$ of a given poset $\mathcal P$ in a complex vector space $V$ we understand a rule that to each element $i \in \mathcal P$ associates a subspace $V_i
\subseteq V$ in such a way that $i \prec j$ implies $V_i \subseteq
V_j$. We will often think of $\mathcal P$ as a set $\{1,2,\ldots,n\}$, where $n$ is the cardinality of $\mathcal P$ and write $\pi=(V;V_1,\ldots,V_n)$ or $\pi=(V;V_i),$ notation $\pi(i)=V_i$, $\pi_0=V$ also will be used.
By the dimension vector $d_\pi$ of the representation $\pi$ we understand a vector $d_\pi=(d_0;d_1,\ldots,d_n)$, where $d_0=\dim
V$, $d_i=\dim (V_i/\sum_{j\prec i} V_j$). There is qudratic form $Q_{\mathcal P}$ on $\mathbb Z^{card(\mathcal P)+1}$ given by $$Q_{\mathcal P}(x_0, x_1,\ldots, x_n) =x_0^2+\sum_{i\in \mathcal P} x_i^2+ \sum_{a,b \in \mathcal{P}, a\prec b} x_a x_b - \sum_{a\in \mathcal P} x_0x_a.$$
Throughout the paper we denote by $e_i$ the $i$-th coordinate vector $e_i=(\delta_{ij})$, by $e_{i_1\ldots i_k}$ we understand the vector $e_{i_1}+\ldots+e_{i_k}$, and by $\langle
x_1,\ldots,x_n \rangle$ the complex vector space spanned by vectors $x_1,\ldots,x_n \in V$. We will use a graphical picture for the representation of posets. For example the following picture describes the representation $\pi=(\mathbb C \langle e_1,e_2
\rangle;\langle e_1 \rangle,\langle e_2 \rangle,\langle e_{12}
\rangle)$ for the poset $(1,1,1)$
(100,70) (0,15)[$\langle e_1 \rangle$]{} (40,15)[$\langle e_2 \rangle$]{} (80,15)[$\langle e_{12} \rangle$]{} (48,24)[(0,1)[30]{}]{} (9,24)[(1,1)[32]{}]{} (90,24)[(-1,1)[32]{}]{} (45,57)[$\mathbb C^2$]{}
In fact the set of all linear representations of a poset $\mathcal
P$ forms the additive category $\rm{Rep}(\mathcal P)$, where the set of morphisms $\rm{Mor}(\pi_1,\pi_2)$ between two representations $\pi_1=(V;V_1,\ldots,V_n)$ and $\pi_2=(W;W_1,\ldots,W_n)$ consists of linear maps $C:V \rightarrow W$, such that $C(V_i)\subset W_i$. Two representations $\pi_1$ and $\pi_2$ of $\mathcal P$ are isomorphic (or equivalent) if there exists an invertible morphism $C
\in \rm{Mor}(\pi_1,\pi_2)$, i.e. there exist an invertible linear map $C:V \rightarrow W$ such that $C(V_i)=W_i$.
One can define a direct sum $\pi=\pi_1 \oplus \pi_2$ of two objects $\pi_1, \pi_2
\in \mathcal P$ in the following way: $$\pi=(V \oplus W; V_1 \oplus
W_1,\ldots,V_n \oplus W_n).$$ Using the notion of direct sum it is natural to define *indecomposable* representations as the representations that are not isomorphic to the direct sum of two non-zero representations, otherwise representations are called *decomposable*. It is easy to show that a representation $\pi$ is indecomposable if and only if there is no non-trivial idempotents in endomorphism ring $\rm{End}(\pi)$. A representation $\pi$ is called *brick* if there is no non-trivial endomorphism of this representation (or equivalently when the ring $\rm{End}(\pi)$ is one-dimensional). It is obvious that if a representation is *brick* then it is *indecomposable*. But there exist *indecomposable* representations of posets which are not *brick*, for example $$A_\alpha= \left(
\begin{array}{cc}
1 & \alpha \\
0 & 1 \\
\end{array}
\right), \quad \alpha \in \mathbb R \backslash \{0\},$$ the representation $\pi_{\alpha}=(V;V_1,V_2,V_3,V_4)$ of the poset $\mathcal N=(1,1,1,1)$ $$\begin{aligned}
V=\mathbb C^2 \oplus \mathbb C^2; \quad V_1=\mathbb
C^2 \oplus 0, \quad V_2=0 \oplus \mathbb C^2, \\
V_3=\left\{ (x,x)\in \mathbb C^4\ |\ x \in \mathbb C^2\right \}, \quad
V_4=\left\{ (x,A_\alpha x)\in \mathbb C^4\ |\ x \in \mathbb C^2\right \},\end{aligned}$$ is indecomposable but is not brick.
& & & &e\_1,e\_2,e\_3,e\_4 & & &\
&& && & & &\
e\_1,e\_2 & &e\_3,e\_4 & & &e\_1+e\_2,e\_1+e\_2 & & e\_1+e\_2,e\_1+e\_2\
&&&&&&&\
e\_1,e\_2 &&e\_3,e\_4 & &e\_1+e\_2,e\_1+e\_2 &&& e\_1+e\_2,e\_1+e\_2\
Recall that a poset $\mathcal P$ is called a poset of *finite (linear) type* if there exist only finitely many non-isomorphic indecomposable representation of $\mathcal P$ in the category $\rm{Rep}(\mathcal P)$. Result obtained by M.M.Kleiner [@Kleiner1] gives a complete description of the posets of finite type.
(see [@Kleiner1], Theorem 1) A poset $\mathcal P$ is a set of finite type if and only if it does not contain subsets of the form $(1,1,1,1)$, $(2,2,2)$, $(1,3,3)$, $(1,2,5)$ and $(N,4)$, where $$\begin{gathered}
(N,4)=\{a_1,a_2,b_1,b_2,c_1,c_2,c_3,c_4\}, \\ a_1\prec
a_2, \ b_1\prec b_2,\ b_1\prec a_2,\ c_1\prec c_2 \prec c_3 \prec
c_4.\end{gathered}$$
The posets in previous theorem will be called critical henceforth.
Kleiner described also all indecomposable representations of posets of finite type up to equivalence using the notion of *sincere* representations (see [@Kleiner2]).
We call a representation $\pi$ *sincere* if it is indecomposable and the components of the dimension vector $d_\pi=(d_0;d_1,\ldots,d_n)$ satisfy $d_i\not=0;$ otherwise we say that the representation is *degenerated*.
A poset is called *sincere* if it has at least one *sincere* representation. It is easy to see that any indecomposable representation of a poset of finite type actually is a sincere representation of its some sincere subposet. To describe all indecomposable representations of a fixed poset $\mathcal P$ of finite type one needs to describe all sincere representations of its all sincere subposets $\mathcal R$ (including itself if $\mathcal P$ is sincere) and to add zero spaces $V_i/\sum_{j\prec i} V_j$ if $i\not\in \mathcal R$.
Unitary representations of posets
---------------------------------
In the spirit of a number of previous articles we study representation theory of posets over Hilbert spaces. Denote by $\rm{Rep}(\mathcal P, H)$ a sub-category in $\rm{Rep}(\mathcal P)$, defined as follows: its set of objects consists of finite-dimensional Hilbert spaces and two objects $\pi=(H,H_i)$ and $\tilde{\pi}=(\tilde H,\tilde H_i)$ are equivalent in $\rm{Rep}(\mathcal P, H)$ if there exists a unitary operator $U:H\rightarrow\tilde H$ such that $U(H_i)= \tilde H_i$ (unitary equivalent). Representation $\pi\in\rm{Rep}(\mathcal P, H)$ is called *irreducible* iff the $C^*$-algebra generated by set of orthogonal projections $\{P_i\}$ on the subspaces $\{H_i\}$ is irreducible. Let us remark that indecomposability of a representation $\pi$ in $\rm{Rep}(\mathcal P)$ implies irreducibility of $C^*(\{P_i, i\in\mathcal P\})$ but the converse is false.
The problem of classification all irreducible objects in the category $\rm{Rep}(\mathcal P, H)$ becomes much harder. Even for the primitive poset $\mathcal P=(1,2)$ it is hopeless to describe in a reasonable way all its irreducible representations: indeed this lead us to classify up to unitary equivalence three subspace in a Hilbert space, two of which are orthogonal, but it is well-known due to [@KruglyaSamoilenko2] that such problem is $*$-wild. Hence it is natural to consider some additional relation.
Let us consider those objects $\pi \in \rm{Rep}(\mathcal P, H)$, $\pi=(H;H_1,\ldots,H_n)$, for which the following linear relation holds: $$\alpha_1P_1+\ldots+\alpha_nP_n=\gamma I, \label{linear_rel}$$ where $\alpha_i,\gamma$ are some positive real numbers, and $P_i$ are the orhoprojections on the subspaces $H_i$. These objects form a category which will be also denoted by $\rm{Rep}(\mathcal P, H)$. Such representations will be called *orthoscalar representations*.
Unitarization
-------------
Obviously there exists a forgetful functor from $\rm{Rep}(\mathcal
P,H)$ to $\rm{Rep}(\mathcal P)$ which maps each system of Hilbert spaces to its underlying system of vector spaces. We ask whether there exists "functor in reverse direction”?
We say that a given representation $\pi \in \rm{Rep}(\mathcal P)$, $\pi=(V;V_1,\ldots,V_n)$ of the poset $\mathcal P$ can be unitarized with a weight (or is unitarizable) $\chi=(\alpha_1,\ldots,\alpha_n)$ if it is possible to choose hermitian structure $\langle\cdot,\cdot\rangle_{\mathbb C}$ in $V$, so that the corresponding projections $P_i$ onto subspace $V_i$ satisfy the following relation: $$\alpha_1P_1+\ldots+\alpha_nP_n=\lambda_\chi(\pi) I,$$ where $\lambda_\chi(\pi)$ is equal to $\frac{1}{\dim
V}\sum_{i=1}^n\alpha_i \dim V_i$.
For a given linear representation $\pi$ of the poset $\mathcal P$ by $\triangle_\pi^{\mathcal P}$ we denote the set of those weights $\chi$ that are appropriated for unitarization. And correspondingly we say that representation $\pi$ can be *unitarized* if the set $\triangle_\pi^{\mathcal P}$ is nonempty.
In [@GrushevoyYusenko] we showed that each indecomposable non-degenerated representation of primitive poset of finite type can be unitarized and for each such representation we completely described the sets $\triangle_\pi^{\mathcal P}$. The approach provided in [@GrushevoyYusenko] does not work longer for *non-pirmitive* case. In this paper we will use rather different approach which comes from Geometric Invariant Theory and gives exact criteria for unitarization.
Balanced metric and stable representations of posets
====================================================
Let $\pi=(V;V_1,\ldots,V_n)$ be a system of subspaces (in particular it can be a representation of some poset $\mathcal P$) in a complex vector space $V$ and let $\chi=(\alpha_1,\ldots,\alpha_n)$ be some weight, i.e. the vector from $\mathbb R^n_+$. Denote by $\lambda_\chi(\pi)$ the number defined by $$\lambda_\chi(\pi)=\frac{1}{\dim V}\sum_{i=1}^n \alpha_i \dim V_i.$$ If $U$ is a subspace of $V$ one can form another system of subspaces $\pi \cap U$ generated by $\pi$ and $U$ $$\pi \cap U=(U;V_1\cap U,\ldots,V_n \cap U).$$ By $\lambda_\chi(\pi \cap U)$ we will understand the number given by $$\lambda_\chi(\pi \cap U)=\frac{1}{\dim U}\sum_{i=1}^n \alpha_i \dim (V_i\cap U).$$
We say that a system of subspace $\pi=(V;V_1,...,V_n)$ is $\chi$-stable if for each proper subspace $U\subset V$ the following inequality holds $$\lambda_\chi(\pi\cap U)<\lambda_\chi(\pi).$$
Suppose for a moment that for a system $\pi=(V;V_1,\ldots,V_n)$ we have chosen a sesquilinear form $\langle\cdot,\cdot\rangle$ on $V$ so that $$\chi_1 P_{V_1}+\ldots+\chi_n P_{V_n}=\lambda_\chi(\pi)I_V,$$ following [@Hu2004] we will call such form *$\chi$-balanced* metric. For a system $\pi$ to possess a $\chi$-balanced metric is essentially the same as to be unitarized with the weight $\chi$. The list of necessarily restrictions on weight $\chi$ can be obtained by the following lemma
If the indecomposable system of subspaces $\pi$ possesses $\chi$-balanced metric then $\pi$ is $\chi$-stable.
The proof of these statement can be obtained by taking the trace from linear equation. Indeed let $\pi=(V;V_1,\ldots,V_n)$ possesses a $\chi$-balanced metric, i.e. a metric $\langle\cdot,\cdot\rangle$ on $V$ such that $$\label{chi-m}
\alpha_1 P_{V_1}+\ldots+\alpha_n P_{V_n}=\lambda_\chi(\pi)I_V.$$ By taking the trace from the left and right hand sides of (\[chi-m\]) we can write $\lambda_\chi(\pi)$ as $$\lambda_\chi(\pi)=\frac{tr(\alpha_1 P_{V_1}+\ldots+\alpha_n
P_{V_n})}{tr(I_V)}=\frac{1}{tr(I_V)}\sum_{i=1}^n\alpha_i
tr(P_{V_i}).$$ Let $U$ be some proper subspace of $V$. Denote by $P_U$ an orthogonal projection on $U$. Multiplying (\[chi-m\]) by $P_U$ from the left we obtain $$\chi_1 P_{V_1}P_U+\ldots+\chi_n P_{V_n}P_U=\lambda_\chi(\pi)P_U,$$ then taking trace of the last we get $$\lambda_\chi(\pi)=\frac{1}{tr(P_U)}\sum_{i=1}^n\alpha_i tr(P_{V_i}P_U).$$ Observe that $tr(P_{V_i}P_U)\geq tr(P_{V_i\cap U})$ (this can be proved using spectral theorem for the pair of projection and then by restriction to two-dimensional representation). This gives us $$\lambda_\chi(\pi)=\frac{1}{tr(P_U)}\sum_{i=1}^n\alpha_i tr(P_{V_i}P_U)\geq \frac{1}{tr(P_U)}\sum_{i=1}^n\alpha_i tr(P_{V_i\cap
U})=\lambda_\chi(\pi\cap U).$$ It remains to prove that the inequality is strict. Indeed assume that $tr(P_{V_i}P_U)=tr(P_{V_i\cap U})$ for all $i$. Then it is easy to see the all $P_{V_i}$ commutes with $P_U$ (again using spectral theorem for the pair of two projections) hence the subspace $U$ is invariant with respect to projections $P_i$ which means that the representation $\pi$ is decomposable. Therefore $\lambda_\chi(\pi\cap U) < \lambda_\chi(\pi)$ for all proper subspaces $U$, and hence $\pi$ is $\chi$-stable.
A natural question arises whether the reverse statement is true, i.e. does every $\chi$-stable system $\pi$ possesses $\chi$-balanced metric?
When $\pi$ is a collection of filtrations (recall that a filtration is a chain of subspaces $V_0\subset\ldots\subset V_m )$ this assertion was proved by Totaro ([@Totaro1994]) and Klyachko ([@Klyachko]). If fact this can be proved for any configuration of subspaces and any weight $\chi$. Here we will reproduce shortly what was done in [@Hu2004].
Let $V$ be a complex vector space and let $\chi \in \mathbb N^n$. Consider the product of Grassmanians $$\textrm{Gr}(k_1,V)\times\ldots\times\textrm{Gr}(k_n,V).$$ Any system of subspaces $\pi=(V;V_1,\ldots,V_n)$ of vector space $V$ with dimension vector equal to $d=(\dim V;k_1,\ldots,k_n)$ can be considered as a point of $\prod_{i=1}^{n}\textrm{Gr}(k_i,V)$. We equip $\prod_{i=1}^{n}\textrm{Gr}(k_i,V)$ with simplectic form $\delta$, which is the skew bilinear form $$\begin{split}
\delta&:\prod_{i=1}^{n}\textrm{Gr}(k_i,V) \times
\prod_{i=1}^{n}\textrm{Gr}(k_i,V) \rightarrow \mathbb C \\
\delta&:(\pi,\tilde \pi)\mapsto\sum_i \chi_i tr(A_i \tilde A_i^*),
\end{split}$$ where $A_i$ and $\tilde A_i$ is a matrix representation of $V_i$ and $\tilde V_i$ (their columns form an orthonormal bases for $V_i$ and $\tilde V_i$ correspondingly), and $*$ is adjoint correspondingly to standart hermitian metric $\langle\cdot,\cdot\rangle$ on $V$.
As Lie group $SU(V)$ acts diagonally on $\prod_{i=1}^{m}\textrm{Gr}(k_i,V)$ preserving symplectic form $\sigma$, the action is given by operating on $V$ (via its linear representation). The corresponding moment map $\Phi:\prod_{i=1}^{m}\textrm{Gr}(k_i,V) \rightarrow su^*(V)$. $su^*(V)$ the dual of Lie algebra of $SU(V)$ which is given by the algebra of traceless Hermitian matrices over $V$. This moment map is given by $$\Phi(\pi)=\sum_i\chi_i A_iA_i^*-\lambda_\chi(\pi)I,$$
Assuming that $\pi$ is $\chi$-stable is possible to find (see [@Hu2004]) such $g \in SL(V)$ that $\Phi(g\cdot\pi)=0$. Then correspondently to hermitian metric $g\langle\cdot,\cdot\rangle=\langle g \cdot, g \cdot\rangle$ the following holds $$\sum_i \chi_i A_i A_i^*=\lambda_\omega(\pi) I.$$ Taking into account that $A_iA_i^*$ is an orthogonal projection on $V_i$ correspondinly to $g\langle\cdot,\cdot\rangle$ we get desirable result.
GIT stabiliby of $\pi$ with respect to $SL(V)$-lineralization is the same as $\chi$-stability (Theorem 2.2, [@Hu2004]). Taking the action of corresponding moment map $$\Phi: \textrm{Gr}(k_1,V)\times\ldots\times\textrm{Gr}(k_n,V)
\rightarrow \sqrt{-1}su(V)$$ of the diagonal action of $SU(V)$ ($su(V)$ is corresponding Lie algebra of $SU(V)$) it can be written as follows $$\Phi(\pi)=\sum_i\omega_i A_iA_i^*-\lambda_\omega(\pi)I,$$ where $A_i$ is a matrix preresentation of $V_i$ (its columns form an orthonormal bases for $V_i$).
Assuming that $\pi$ is $\chi$-stable is possible to find (see [@Hu2004]) such $g \in SL(V)$ that $\Phi(g\cdot\pi)=0$. Then correspondently to hermitian metric $g\langle\cdot,\cdot\rangle=\langle g \cdot, g \cdot\rangle$ the following holds $$\sum_i \omega_i A_i A_i^*=\lambda_\omega(\pi) I,$$ (here $\langle\cdot,\cdot\rangle$ denotes standard hermitian metric on $V$).
To conclude it remains to note that for $\chi \in \mathbb R^n_+$ one can find appropriated sequence of rational $\chi_n$ that tends to $\chi$ and that $\pi$ is $\chi_n$-stable (it is possible because stable condition is open) and then one can make use (for example) Shulman’s lemma about representation of limit relation.
Summing up the following theorem holds ([@Hu2004]).
Let $\chi$ be the weight. For the indecomposable system of subspaces $\pi$ the following conditions are equivalent:
1. $\pi$ can be unitarized with weight $\chi$;
2. $\pi$ is $\chi$-stable system.
This theorem gives exact criteria of unitarization of any linear indecomposable representation of partially ordered set with the weight $\chi$. On practice in order to check the $\chi$-stability for a given representation $\pi=(V;V_1,\ldots,V_n)$ of some poset with dimension vector $d=(\dim V_0;\dim V_1,\ldots,\dim V_n)$ one can describe all possible subdimension vectors $d^\prime=(\dim
U;\dim(V_1\cap U),\ldots,\dim(V_n\cap U))$, $U\subset V$ and to check for these vector stability condition $$\frac{1}{d^\prime_0}\sum\chi_i d^\prime_i<\frac{1}{d_0}\sum\chi_i
d_i,$$ Let us remark that on subdimension vectors there exist a natural coordinate partial order. That is evident that to check the stability condition one should check inequality above for maximal vectors.
Proof of the Theorem \[mainthmFin\]
===================================
*Sufficiently.* Let $\mathcal P$ be the poset which does not contain any of critical posets. Assume that these posets could have infinite number of unitary inequivalent Hilbert space representations with fixed weight. If two Hilbert space representations with the same weight are unitary non-equivalent then they are linearly inequivalent (due to [@KruglyaNazarovaRoiter], Theorem 1). Hence each such poset has infinite number of indecomposable linear representation. But this contradicts Kleiner’s theorem.
*Necessity.* Our aim is for each critical poset $\mathcal P$ to build infinite series of indecomposable pairwise nonequivalent Hilbert representations of this poset with the same weight. For the primitive case this can be done using the connection between Hilbert orthoscalar representations of the posets with the representations of some certain class of $*$-algebras that connected with star-shaped graphs (see for example [@OstrovskyiSamoilenko]). But this approach does not work for the nonprimitive case (namely for the set $(N,4)$). Here we consider quite different approach which is based on unitarization.
Let $\mathcal P$ be a poset and let $I \subset \mathcal P$ ($I$ can be empty). Define an extended poset $\mathcal {\tilde P}_I$ by adding to $\mathcal P$ an element $\tilde p$ subject to the relations $\tilde p \prec i$, $i\in I$. Let $\pi=(V;V_i)$ be a representation of the poset $\mathcal P$. Assume that there are two linearly independent vectors $v_1,v_2\in V$ such that the following conditions are satisfied $$\dim((\sum_{i\in I} V_i + \langle v_1+\lambda v_2\rangle) \cap
(\sum_{i\in I} V_i + \langle v_1+\tilde \lambda
v_2\rangle))=\dim(\sum_{i\in I} V_i),\quad$$ where $\lambda\neq\tilde\lambda \in D$, and $D$ is dense in $\mathbb
C$. One can show that such vectors exist if $\dim(\sum_{i\in I}
V_i)+1<\dim(V)$. We can define a family of representation $\tilde\pi_{\lambda}$ of the $\mathcal {\tilde P}_{I}$, by letting $$\tilde \pi_\lambda(x)=\left\{
\begin{array}{c}
\pi(x), \quad x\neq\tilde p, \\
\sum_{i\in I}\pi(i)+ \langle v_1+\lambda v_2\rangle, \quad x=\tilde p. \\
\end{array}
\right.$$
The following proposition is straightforward.
Assume that $\pi=(V;V_i)$ is a brick representation of the poset $\mathcal P$ and $\tilde \pi_\lambda(\tilde p)$ is the corresponding family of representations of $\mathcal P_I$ for appropriated choosen $v_1,v_2 \in V$. Then the following holds
1. The representation $\tilde\pi _\lambda$ is brick ($End(\tilde\pi_\lambda)\cong\mathbb C$) for each $\lambda \in \mathbb C$.
2. If $\lambda\neq\lambda^\prime$ then $\tilde\pi_\lambda$ is not equivalent to $\tilde\pi_{\lambda^\prime}$.
3. If $\lambda \neq
\lambda^\prime$ and $\pi_\lambda$, $\pi_{\lambda^\prime}$ are $\chi$-stable then the corresponding systems of projection (after unitarization) are unitary inequivalent.
Using the construction above for each pre-critical poset (that is critical poset without one element) we will build its extended poset that coincide with critical poset and we will choose appropriated representation of pre-critical what allows to build extended representations of critical posets. These family of extended representation are given in dimension $d^\mathcal P$ which is the imaginary root of the corresponding quadratic form $Q_\mathcal{P}$, $Q_\mathcal{P}(d^\mathcal P)=0$. We will prove that these representations are stable for all $\lambda \in \mathbb C, \lambda
\neq0,1$ (see Appendix B for the description of subdimension vectors) with the same weight $\chi^\mathcal{P}$ which is defined by the dimension vector in the following way: $\chi^\mathcal{P}_i=d^\mathcal{P}_i$ and $\lambda_\chi=d^\mathcal{P}_0$.
*1) Case $(1,1,1,1)$*.
For $\mathcal P=(1,1,1)$, $I=\emptyset$ the extended poset $\mathcal{\tilde P}_I$ is equal to $(1,1,1,1)$ and for the representation $\pi=(\mathbb C^2;\langle e_1\rangle;\langle
e_2\rangle; \langle e_1+e_2\rangle)$ its extended representation has the following form.
(150,20) (0,10)[$\langle e_1 \rangle$]{} (30,10)[$\langle e_2 \rangle$]{} (60,10)[$\langle e_1+e_2 \rangle$]{} (110,10)[$\langle e_1+\lambda e_2 \rangle$]{}
The dimension vector is equal to $d^{(1,1,1,1)}=(2;1;1;1;1)$ and the representations are $(1,1,1,1)$–stable.
*2) Case $(2,2,2)$.*
For $\mathcal P=(1,2,2)$, $I=\{a_1\}$ the extended poset $\mathcal{\tilde P}_I$ is equal to $(2,2,2)$ and for the representation $\pi=(\mathbb C^3;\langle e_{123}\rangle;\langle
e_2\rangle, \langle e_1,e_2\rangle;\langle e_2\rangle, \langle
e_2,e_3\rangle)$ its extended representation has the following form
(150,51) (30,5)[$\langle e_{123} \rangle$]{} (97,5)[$\langle e_1 \rangle$]{} (145,5)[$\langle e_3 \rangle$]{} (42,16)[(0,1)[25]{}]{} (105,16)[(0,1)[25]{}]{} (153,16)[(0,1)[25]{}]{} (10,45)[$\langle e_{123}, e_1+\lambda e_3 \rangle$]{} (90,45)[$\langle e_1, e_2 \rangle$]{} (138,45)[$\langle e_2, e_3 \rangle$]{}
The dimension vector is equal to $d^{(2,2,2)}=(3;1,1;1,1;1,1)$ and the representations are $(1,1,1,1,1,1)$–stable.
*3) Case $(1,3,3)$.*
For $\mathcal P=(1,2,3)$, $I=\{b_2\}$ the extended poset $\mathcal{\tilde P}_I$ is equal to $(1,3,3)$ and for the representation $\pi=(\mathbb C^4;\langle
e_{123},e_{24}\rangle;\langle e_4\rangle, \langle e_1,e_4\rangle;
\langle e_3\rangle,\langle e_2,e_3\rangle, \langle
e_1,e_2,e_3\rangle)$ its extended representation has the following form
(150,91) (15,5)[$\langle e_{123}, e_{24} \rangle$]{} (92,5)[$\langle e_4 \rangle$]{} (155,5)[$\langle e_3 \rangle$]{} (100,16)[(0,1)[25]{}]{} (163,16)[(0,1)[25]{}]{} (85,45)[$\langle e_1, e_4 \rangle$]{} (148,45)[$\langle e_2, e_3 \rangle$]{} (100,56)[(0,1)[25]{}]{} (163,56)[(0,1)[25]{}]{} (60,85)[$\langle e_1, e_4, e_2+\lambda e_3 \rangle$]{} (143,85)[$\langle e_1, e_2, e_3 \rangle$]{}
Dimension vector is equal to $d^{(1,3,3)}=(4;2;1,1,1;1,1,1)$ and the representations are $(2,1,1,1,1,1,1)$–stable.
*4) Case $(1,2,5)$.*
For $\mathcal P=(1,2,4)$, $I=\{c_4\}$ the extended poset $\mathcal{\tilde P}_I$ is equal to $(1,2,5)$ and for the representation $$\begin{gathered}
\pi=(\mathbb C^6;\langle e_{123},e_{245},e_{16}\rangle;\langle
e_5,e_6\rangle, \langle e_1,e_2,e_5,e_6\rangle;\\ \langle
e_4\rangle,\langle e_3,e_4\rangle, \langle
e_2,e_3,e_4\rangle,\langle e_1,e_2,e_3,e_4\rangle)\end{gathered}$$ its extended representation has the following form
(150,155) (15,5)[$\langle e_{123},e_{245},e_{16}\rangle$]{} (99,5)[$\langle e_5, e_6 \rangle$]{} (165,5)[$\langle e_4 \rangle$]{} (113,16)[(0,1)[20]{}]{} (173,16)[(0,1)[20]{}]{} (85,40)[$\langle e_1, e_2, e_5, e_6 \rangle$]{} (158,40)[$\langle e_3, e_4 \rangle$]{} (173,51)[(0,1)[20]{}]{} (153,75)[$\langle e_2, e_3, e_4 \rangle$]{} (173,86)[(0,1)[20]{}]{} (145,110)[$\langle e_1, e_2, e_3, e_4 \rangle$]{} (173,121)[(0,1)[20]{}]{} (125,145)[$\langle e_1, e_2, e_3, e_4, e_5+\lambda e_6 \rangle$]{}
In this case the dimension vector is equal to $d^{(1,2,5)}=(6;3;2,2;1,1,1,1,1)$ and the representations are $(3,2,2,1,1,1,1,1)$–stable.
*5) Case $(N,4)$.*
For $\mathcal P=(1,2,4)$, $I=\{a_1,b_1\}$ its extended poset $\mathcal{\tilde P}_I$ is equal to $(N,4)$ and for the representation $$\pi=(\mathbb C^5;\langle
e_{235},e_{134}\rangle;\langle e_5\rangle, \langle
e_1,e_2,e_5\rangle;\langle e_4\rangle, \langle
e_3,e_4\rangle,\langle e_2,e_3,e_4\rangle,\langle
e_1,e_2,e_3,e_4\rangle)$$ its extended representations has the following form
(150,120) (28,5)[$\langle e_{235}, e_{134}\rangle$]{} (125,5)[$\langle e_5 \rangle$]{} (180,5)[$\langle e_4 \rangle$]{} (50,16)[(0,1)[20]{}]{} (133,16)[(0,1)[20]{}]{} (188,16)[(0,1)[20]{}]{} (0,40)[$\langle e_{235},e_{134}, e_5, e_3+\lambda e_4\rangle$]{} (112,40)[$\langle e_1, e_2, e_5\rangle$]{} (173,40)[$\langle e_3, e_4 \rangle$]{} (188,51)[(0,1)[20]{}]{} (168,75)[$\langle e_2, e_3, e_4 \rangle$]{} (188,86)[(0,1)[20]{}]{} (160,110)[$\langle e_1, e_2, e_3, e_4 \rangle$]{} (129,16)[(-4,1)[75]{}]{}
The dimension vector is equal to $d^{(\mathcal
N_4)}=(5;2,1;1,2;1,1,1,1)$ and the representations are $(2,1,1,2,1,1,1,1)$–stable.
So, for each critical posets $\mathcal{ \tilde P}$ we build the infinite family of pairwise inequivalent $d^{\mathcal P}$–stable brick representation $\pi_\lambda$ in the dimension $d^{\mathcal P}$ (see appendix for the proof of stability). Hence each critical poset has infinite number of pairwise unitary inequivalent representations satysfying $$d^{\mathcal P}_1 P_1+\ldots+d^{\mathcal P}_n P_n=d^{\mathcal P}_0
I,$$ i.e. each critical posets has infinite Hilbert space representable type. This completes the proof of Theorem \[mainthmFin\].
Theorem 1 can be reformulated in the following way — a poset has finite orthoscalar Hilbert type if and only if it has finite linear type.
Quite sincere representations and the proof of Theorem \[mainthmUnit\]
======================================================================
Quite sincere representations
-----------------------------
To describe all irreducible orthoscalar representations of posets of finite type we need a new notion of sincere representation which we will call quite sincerity.
The following is for linear representations of posets.
Let $\mathcal P=\{1,\ldots,n\}$ be a poset. We call a representation $\pi$ of $\mathcal P$ *quite sincere* if it is indecomposable and the following conditions holds for all $i=1,\dots,n$:
- $\pi(i)\not=0;$
- $\pi(i)\not=\pi(0)$;
- $\pi(i)\not=\pi(j)$ as $i\prec j$.
We say that the poset is *quite sincere* if it has at least one *quite sincere* representation.
The following theorem describes all quite sincere posets of finite type and gives all their quite sincere representations.
The set of quite sincere posets of finite type consists of four primitive posets $(1,1,1)$, $(1,2,2)$, $(1,2,3)$, $(1,2,4)$ and following non-primitive posets:
(50,60) (0,15)[$a_1$]{} (20,15)[$b_1$]{} (40,15)[$c_1$]{} (2,24)[(0,1)[15]{}]{} (22,24)[(0,1)[15]{}]{} (42,24)[(0,1)[15]{}]{} (0,44)[$a_2$]{} (20,44)[$b_2$]{} (40,44)[$c_2$]{} (19,24)[(-1,1)[15]{}]{} (48,30)[;]{} (15,0)[$\mathcal P_1$]{}
(50,75) (0,15)[$a_1$]{} (20,15)[$b_1$]{} (40,15)[$c_1$]{} (2,24)[(0,1)[15]{}]{} (22,24)[(0,1)[15]{}]{} (42,24)[(0,1)[15]{}]{} (0,44)[$a_2$]{} (20,44)[$b_2$]{} (40,44)[$c_2$]{} (42,52)[(0,1)[15]{}]{} (40,71)[$c_3$]{} (19,24)[(-1,1)[15]{}]{} (48,30)[;]{} (15,0)[$\mathcal P_2$]{}
(50,75) (0,15)[$a_1$]{} (20,15)[$b_1$]{} (40,15)[$c_1$]{} (22,24)[(0,1)[15]{}]{} (42,24)[(0,1)[15]{}]{} (20,44)[$b_2$]{} (40,44)[$c_2$]{} (22,52)[(0,1)[15]{}]{} (42,52)[(0,1)[15]{}]{} (20,71)[$b_3$]{} (40,71)[$c_3$]{} (39,24)[(-1,3)[14]{}]{} (48,30)[;]{} (15,0)[$\mathcal P_3$]{}
(50,90) (0,15)[$a_1$]{} (20,15)[$b_1$]{} (40,15)[$c_1$]{} (2,24)[(0,1)[15]{}]{} (22,24)[(0,1)[15]{}]{} (42,24)[(0,1)[15]{}]{} (0,44)[$a_2$]{} (20,44)[$b_2$]{} (40,44)[$c_2$]{} (22,52)[(0,1)[15]{}]{} (42,52)[(0,1)[15]{}]{} (20,71)[$b_3$]{} (40,71)[$c_3$]{} (19,24)[(-1,1)[15]{}]{} (4,24)[(1,3)[14]{}]{} (48,30)[;]{} (15,0)[$\mathcal P_4$]{}
(50,110) (0,15)[$a_1$]{} (20,15)[$b_1$]{} (42,15)[$c_1$]{} (2,24)[(0,1)[15]{}]{} (22,24)[(0,1)[15]{}]{} (44,24)[(0,1)[15]{}]{} (0,44)[$a_2$]{} (20,44)[$b_2$]{} (42,44)[$c_2$]{} (44,52)[(0,1)[15]{}]{} (40,71)[$c_3$]{} (44,80)[(0,1)[15]{}]{} (42,98)[$c_4$]{} (24,24)[(1,4)[17.5]{}]{} (20,24)[(-1,1)[15]{}]{} (50,30)[;]{} (15,0)[$\mathcal P_5$]{}
(50,110) (0,15)[$a_1$]{} (20,15)[$b_1$]{} (40,15)[$c_1$]{} (2,24)[(0,1)[15]{}]{} (22,24)[(0,1)[15]{}]{} (42,24)[(0,1)[15]{}]{} (0,44)[$a_2$]{} (20,44)[$b_2$]{} (40,44)[$c_2$]{} (42,52)[(0,1)[15]{}]{} (40,71)[$c_3$]{} (42,80)[(0,1)[15]{}]{} (40,98)[$c_4$]{} (4,24)[(1,1)[15]{}]{} (39,24)[(-1,1)[15]{}]{} (48,30)[.]{} (15,0)[$\mathcal P_5^*$]{}
Complete list of all quite sincere representations of these posets are given in the Appendix A.
Let $\pi$ be a quite sincere representation of a poset $\mathcal P$. The following two possibilities can occur
- $\pi$ is a sincere representation of $\mathcal P$ with $\pi(i)\neq\pi(0)$;
- there is $k\in\mathcal P$ such that $\pi(k)=\sum_{i\prec k}\pi(i)$.
In the first case the representation (and obviously the corresponding poset) is in Kleiner’s list of sincere posets and their representations [@Kleiner2]. In the second the representation $\pi$ generates an indecomposable representation of poset the $\mathcal P_1=\mathcal P\setminus {k}$ which we denote by $\pi_1$. It is clear that $\pi_1$ is a quite sincere representation of $\mathcal P_1$ and it again satisfies one of the two above possibilities. Proceeding in this way we will obtain a sincere representation of some poset $\mathcal P_k$ with condition $\pi(i)\neq\pi(0)$ which is in Kleiner’s list.
Summing up we have the following algorithm for calculation of all quite sincere posets and their quite sincere representations:
1. all sincere posets which have a sincere representation such that $\pi(i)\neq\pi(0)$ are quite sincere and Kleiner’s list gives all. These posets are $(1,1,1)$, $(1,2,2)$, $(1,2,3)$, $(1,2,4)$, $\mathcal{P}_2$ with representations listed in Appendix A (for $\mathcal{P}_2$ marked with \*);
2. Let $\mathcal P$ be a quite sincere poset and and $I\subset \mathcal
P$ such that $\sum_{i\in I}\pi(i)\neq \pi(0)$. Let $\mathcal {\tilde
P}_I$ be the corresponding extended poset defined as in Section 3. Let $\tilde \pi_I$ be a representation of $\mathcal {\tilde P}_I$ given by $\tilde\pi_I(j)=\pi(j)$ for all $j\in\mathcal P$ and $\tilde\pi_I(\tilde p)=\sum_{i\in I}\pi(i)$. This is evident that these representations are quite sincere representations of corresponding posets $\mathcal {\tilde P}_I$.
In this way by induction one can obtain all quite sincere posets and all their quite sincere representations. The above procedure terminates because the dimensions of representations are bounded.
Let us remark that the unitarization of a quite sincere representation is equivalent to unitarization of all indecomposable representation of poset of finite type.
Proof of Theorem \[mainthmUnit\]
--------------------------------
As it was mentioned before in order to prove that all primitive posets of finite type can be unitarized it is enough to see that all quite sincere posets and their quite sincere representations are unitarized. All such representations are listed in Appendix A (second column in the table). The fact that all quite sincere representation of primitive poset (from Appendix A) can be unitarized with some weight is due to [@GrushevoyYusenko].
To prove that quite sincere representation 1)-6) of the poset $\mathcal P_2$ from Appendix A can be unitarized we show their stability with the the weight which is equal to the dimension of representation. For this one can describe all possible subdimension for each representation and then check the stability condition. Description of all possible subdimension is straightforward (this is done in Appendix D) and this is a routine to check the stability condition for all these vectors.
Representations of $\mathcal P_1, \mathcal P_3-\mathcal P_5$ and representations $7)$, $8)$ of $\mathcal P_2$ can be obtained by adding one (or several) subspace(s) to either a representation of primitive posets or one of the representation $1)-6)$ of $\mathcal
P_2$. Their unitarization follows from the following observation.
Let $\pi=(V;V_i)$ be a $\chi$-stable system of subspaces. Then for each subspace $U\subset V$ there exists a weight $\tilde \chi$ such that the system of subspaces $\tilde \pi$ generated by $\pi$ and $U$, $$\tilde \pi=(V;V_1,\ldots,V_n,U)$$ is $\tilde
\chi$-stable.
Let $K\subset V$ be a subspace of $V$ that the difference $\lambda_\chi(\pi)-\lambda_\chi(\pi \cap K)$ is minimal. Let $R=\lambda_\chi(\pi)-\lambda_\chi(\pi \cap K)$. Since $\pi$ is stable, $R>0$ and there exist $\epsilon>0$ that $R-\epsilon>0$. Define $\tilde \chi$ in the folowing way $$\tilde \chi_i=\chi_i, \quad i=1,\dots,n, \quad
\tilde \chi_{n+1}=R-\epsilon.$$ Our claim is that $\tilde \pi$ is $\tilde\chi$-stable. Indeed, let $M\subset V$ be an arbitrary subspace of $V$ then we have $$\begin{split}
\frac{1}{\dim M}\sum_{i=1}^{n+1}\tilde \chi_i \dim
(\tilde \pi(i)\cap M)&=\frac{1}{\dim M}\sum_{i=1}^{n}\chi_i \dim
(V_i\cap M)+\frac{\tilde\chi_{n+1} \dim(U\cap M)}{\dim M}\\
&\leq \frac{1}{\dim
V}\sum_{i=1}^{n}\chi_i\dim V_i-R+ \frac{ (R-\epsilon) \dim(U\cap M)}{\dim
M}\\
& < \frac{1}{\dim
V}\sum_{i=1}^{n}\chi_i\dim V_i < \frac{1}{\dim
V}\sum_{i=1}^{n+1}\tilde \chi_i\dim \tilde \pi(i).
\end{split}$$ Hence $\tilde \pi$ is $\tilde\chi$-stable.
If indecomposable system of subspaces $\pi=(V;V_1,\ldots,V_n)$ is unitarizable then for arbitrary collections of subspaces $U_j \subset V$, $j=1,\ldots,m$ the system $\pi=(V;V_1,\ldots,V_n,U_1,\ldots,U_m)$ is also unitarizable.
Let us note that when a preliminary version of this article was ready the authors were informed that the same result was independently obtained in [@Ya2010].
It remains to prove that the representation of $\mathcal P^*_5$ from the Appendix A is unitarized. Since it is dual to representation of $\mathcal P_5$, this follows from the following lemma.
Let $\pi=(V;V_i)$ be unitarizable with the weight $\chi$ and let $\pi^\prime=(V;V^\prime_i)$ be indecomposable dual system (each $V^\prime_i$ is a complement to $V_i$) assume also that the dimension vector of $\pi$ is a real root, i.e. $Q_\mathcal P(d_\pi)=1$. Then $\pi^\prime$ is also unitarizable with the weight $\chi$.
As $\pi$ is unitarizable with $\chi$, we have $\sum\chi_i
P_{V_i}=\lambda_\pi I$ for appropriated choice of scalar product. It is not hard to check the dimension vector $d_\pi^\prime$ is also a real root. Undecomposability of $\pi^\prime$ implies that it is linearly equivalent to the system $(V;\textrm{Im}(I-P_{V_i}))$ (because there exist only one indecomposable representation with dimension vector $d_\pi^\prime$). The latter system is obviously unitarized with the weight $\chi$ due to $\sum\chi_i(I-P_{V_i})=(\sum\chi_i-\lambda_\pi) I$.
Since $\pi^\prime$ is dual system one has $$\frac{1}{\dim U}\left(\sum_{i=1}^{n}\chi_i\big(\dim(V_i\cup U)+\dim(V^\prime_i\cup
U)\big)\right)\leq\sum_{i=1}^{n}\chi_i,$$ for each subspace $U\subset V$. Using $\chi$-stability of $\pi$ we have $$\begin{split}
\frac{1}{\dim U}\sum_{i=1}^{n}\chi_i\dim(V_i^\prime\cup
U)&<\sum_{i=1}^{n}\chi_i-\frac{1}{\dim
V}\sum_{i=1}^{n}\chi_i\dim(V_i)\\
&=\frac{1}{\dim
V}\sum_{i=1}^{n}\chi_i\dim(V^\prime_i).
\end{split}$$ Hence $\pi^\prime$ is $\chi$-stable.
Now Theorem 2 is completely proved.
Appendix A. Quite sincere representation and weights appropriated for unitarization. {#appendix-a.-quite-sincere-representation-and-weights-appropriated-for-unitarization. .unnumbered}
====================================================================================
In this appendix you can find a complete description of all quite sincere representations of finite posets and the weights appropriated for the unitarization. To simplify the notation we will denote by $V_{i_1,\ldots,i_j}$ vector space spanned by vectors $e_{i_1},\ldots,e_{i_j}$.
**Poset** **Representation** $\pi=(V;\pi(a_i);\pi(b_i);\pi(c_i))$ **Weight $\chi$**
------------------ ------------------------------------------------------------------------------------- -----------------------
$(1,1,1)$ $(\mathbb{C}^2;V_1;V_2; V_{12})$ $(1,1,1)$
$(1,2,2)$ 1\) $(\mathbb{C}^3;V_{123};V_1, V_{1,2};V_{3}, V_{2,3})$ $(1,1,1,1,1)$
2\) $(\mathbb{C}^3; V_{12,13};V_{1}, V_{1,2};V_{3}, V_{2,3})$ $(2,1,1,1,1)$
$(1,2,3)$ 1\) $(\mathbb{C}^4;V_{123,24};V_4, V_{1,4};V_3, V_{2,3}, V_{1,2,3})$ $(2,1,1,1,1,1)$
2\) $(\mathbb{C}^4; V_{124,13};V_4, V_{1,2,4};V_{3}, V_{2,3}, V_{1,2,3})$ $(2,1,2,1,1,1)$
3\) $(\mathbb{C}^4; V_{123,24}; V_{1,4}, V_{1,2,4}; V_{3}, V_{2,3}, V_{1,2,3})$ $(2,2,1,1,1,1)$
$(1,2,4)$ 1\) $(\mathbb{C}^5;V_{134,235};V_{5}, V_{1,2,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,1,2,1,1,1,1)$
2\) $(\mathbb{C}^5; V_{123,245};V_{1,5}, V_{1,2,5};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,2,1,1,1,1,1)$
3\) $(\mathbb{C}^5; V_{124,235};V_{1,5}, V_{1,2,3,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,2,2,1,1,1,1)$
4\) $(\mathbb{C}^5; V_{124,23,15}; V_5, V_{1,2,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(3,1,2,1,1,1,1)$
5\) $(\mathbb{C}^5; V_{13,234,45};V_{1,5}, V_{1,2,5};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(3,2,1,1,1,1,1)$
6\) $(\mathbb{C}^5; V_{12,234,45}; V_{1,5}, V_{1,2,3,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(3,2,2,1,1,1,1)$
7\) $(\mathbb{C}^6; V_{123,245,16}; V_{5,6}, V_{1,2,5,6};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(3,2,2,1,1,1,1)$
8\) $(\mathbb{C}^6; V_{125,234,46}; V_{1,6}, V_{1,2,3,6};$
$ V_5, V_{4,5}, V_{3,4,5}, V_{1,2,3,4,5})$ $(3,2,2,1,1,1,2)$
9\) $(\mathbb{C}^6; V_{125,134,46}; V_{1,6}, V_{1,2,3,6};$
$ V_{5}, V_{4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(3,2,2,1,1,2,1)$
10\) $(\mathbb{C}^6; V_{125,234,46}; V_{1,6}, V_{1,2,3,6};$
$ V_5, V_{3,4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(3,2,2,1,2,1,1)$
11\) $(\mathbb{C}^6; V_{135,124,46};V_{1,6},V_{1,2,3,6};$
$V_{4,5}, V_{3,4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(3,2,2,2,1,1,1)$
$\mathcal P_1$ $(\mathbb{C}^3;V_{123}, V_{23,1};V_1, V_{1,2};V_{3},V_{2,3})$ $(1,0.1,1,1,1,1)$
$\mathcal P_2$ 1\) $(\mathbb{C}^4;V_{14}, V_{1,2,4}; V_4, V_{4,123};V_3, V_{2,3}, V_{1,2,3})^*$ $(1,1,1,1,1,1,1)$
2\) $(\mathbb{C}^4;V_{14}, V_{1,2,4}; V_4, V_{4,12,23}; V_3, V_{2,3}, V_{1,2,3})^*$ $(1,1,1,2,1,1,1)$
3\) $(\mathbb{C}^5; V_{1,25}, V_{1,2,3,5}; V_5, V_{123,24,5};$
$ V_{3,4}, V_{2,3,4},V_{1,2,3,4})^*$ $(2,1,1,2,2,1,1)$
4\) $(\mathbb{C}^5; V_{1,25}, V_{1,2,3,5}; V_5, V_{13,234,5};$
$ V_4, V_{2,3,4}, V_{1,2,3,4})^*$ $(2,1,1,2,1,2,1)$
5\) $(\mathbb{C}^5; V_{1,25}, V_{1,2,3,5}; V_5, V_{123,24,5};$
$ V_4, V_{3,4}, V_{1,2,3,4})^*$ $(2,1,1,2,1,1,2)$
6\) $(\mathbb{C}^5; V_{15,4}, V_{1,2,4,5}; V_5, V_{123,24,5};$
$ V_3, V_{2,3}, V_{1,2,3})^*$ $(2,1,1,2,1,1,1)$
7\) $(\mathbb{C}^4; V_{123,24}, V_{13,2,4}; V_4, V_{1,4};$
$V_3, V_{2,3}, V_{1,2,3})$ $(2,0.1,1,1,1,1,1)$
8\) $(\mathbb{C}^4; V_{124,13}, V_{12,13,4}; V_4, V_{1,2,4};$
$V_3, V_{2,3}, V_{1,2,3})$ $(2,0.1,1,2,1,1,1)$
$\mathcal{P}_3$ $(\mathbb{C}^4;V_{123,24};V_4, V_{1,4}, V_{1,3,4}; V_3, V_{2,3}, V_{1,2,3})$ $(2,1,1,0.1,1,1,1)$
$\mathcal P_4$ $(\mathbb{C}^4;V_{14}, V_{1,2,4}; V_4, V_{4,123}, V_{1,23,4};$
$ V_3, V_{2,3}, V_{1,2,3})$ $(1,1,1,1,0.1,1,1,1)$
$\mathcal P_5$ $(\mathbb{C}^5; V_{15,4}, V_{1,2,4,5}; V_5, V_{123,24,5};$
$ V_3, V_{2,3}, V_{1,2,3}, V_{1,2,3,5})$ $(2,1,1,2,1,1,1,0.1)$
$\mathcal P_5^*$ $(\mathbb{C}^5;V_{5}, V_{1,2,5}; V_{134,235}, V_{13,23,4,5};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(1,2,2,0.1,1,1,1,1)$
An interesting phenomena that each quite sincere representation can be unitarized with the weight that equal to the dimension vector (for primitive case) or with the weight that an arbitrary closed to the dimension vector (for non-primitive poset) and this weight in some sense is the most stable weight.
Appendix B. The sets $\triangle_{\pi}^{\mathcal P}$ and several examples. {#appendix-b.-the-sets-triangle_pimathcal-p-and-several-examples. .unnumbered}
=========================================================================
It is routine to describe the set $\triangle_{\pi}^{\mathcal P}$ for an arbitrary linear representation $\pi$ of $\mathcal P$. Instead we will give an algorithm of its description.
$\triangle_{\pi}^{\mathcal P}$ is convex.
One can see that if $\chi \in \triangle_{\pi}^{\mathcal P}$ then $(1-t)\chi \in \triangle_{\pi}^{\mathcal P}$ for each $t\in [0,1]$, because if $\pi$ is stable with $\chi$ then $\pi$ is also stable with $(1-t)\chi$. Hence $\triangle_{\pi}^{\mathcal P}$ is convex and connected.
Stability conditions for the system of subspaces $\pi=(V;V_1,\ldots,V_n)$ define some $(m\times n)$–matrix $A_\pi\in
M_{m,n}(\mathbb R)$. Namely this matrix is defined in the following way. For any vector of the form $d=(d_0;d_1,\ldots,d_n)\in \mathbb
Z^{n+1}$ with $d_0>0$ by $n(d)$ we denote normalized vector $n(d)=\left(\frac{d_1}{d_0},\ldots,\frac{d_n}{d_0}\right)$. Let $\dim(\pi)=(\dim \pi_0;\dim \pi_i)$ be dimension vector of $\pi$ and $Sub(\pi)=\{d_{\pi,i}\ |\ i\in \{1,\ldots,m\} \}$ be the set of maximal subdimension vectors for $\pi$. Then the matrix $A_\pi$ has the following form $$A_\pi= \left(
\begin{array}{c}
n(d_{\pi,1})-n(\dim(\pi)) \\
\vdots \\
n(d_{\pi,m})-n(\dim(\pi)) \\
\end{array}
\right) \oplus -I_n,$$ here by $\oplus$ be understand The set $\triangle_{\pi}^{\mathcal
P}$ thus can be defined as the set of those $\chi=(\alpha_1,\ldots,\alpha_n)$ that $A_\pi\chi<0$. Using the standard methods concerning to systems of linear inequalities (see for example [@PadbergManfred]) these sets can be described in terms of extreme points and extremal rays.
Let $P \subset \mathbb R^n$ be some subset. A point $x \in P$ is called an extreme point of P if for all $x_1,x_2 \in P$ and every $0<\mu<1$ such that $x=\mu x_1 +(1-\mu) x_2$, we have $x=x_1=x_2$.
The set of extreme points can be determined by the set of those points $x_0 \in \mathbb R^n$ that $A_\pi^\prime x_0=0$ for some $(n\times n)$ full rank submatrix $A_\pi^\prime$ (see [@PadbergManfred]), hence this set contains the only element $x_0=0$.
1. A set $C \subset \mathbb R^n$ is a cone if for every pair of points $x_1,x_2 \in C$ we have $\lambda_1x_1+\lambda_2x_2 \in C$ for all $\lambda_1,\lambda_2\geq0$.
2. A half-line $y=\{\lambda x \ | \ \lambda \geq 0, \ x
\in \mathbb R^n\}$ is an extremal ray of $C$ if $y \in C$ and $-y \notin C$ and if for all $y_1,y_2 \in C$ and $0<\mu<1$ with $y=(1-\mu)y_1+\mu y_2$ we have $y=y_1=y_2$.
Obviously $\triangle_{\pi}^{\mathcal P}$ is a cone. The following proposition describes how to determine all extermal rays of $\triangle_{\pi}^{\mathcal P}$.
(see [@PadbergManfred]) $x \in \overline{\triangle_{\pi}^{\mathcal P}}$ is an extremal ray if and only if there exist $rank(A_\pi)-1$ linear independent row vectors $a_1,\ldots,a_{rank(A_\pi)-1}$ of $A_\pi$ such that $$\left(
\begin{array}{c}
a_1 \\
\vdots \\
a_{rank(A_\pi)-1} \\
\end{array}
\right) \cdot x =\vec{0}.$$
(see [@PadbergManfred]) The sets $\triangle_{\pi}^{\mathcal P}$ are described in the following way $$\triangle_{\pi}^{\mathcal P}=cone(A),$$ where $A$ is the set of extremal rays, $cone(X)$ is an open cone of $X$ defined by $$cone(x_1,\ldots,x_n)=\left\{\sum_{i=1}^n \mu_i x_i\ |\ \mu_i>0\right\}.$$
We consider two examples:
*1)*. Let $\mathcal P=(1,1,1)$, and $\pi=(\mathbb
C^2;V_1,V_2,V_{12})$. Matrix $A_{\pi}$ is given by $$A_\pi= \left(
\begin{array}{ccc}
\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\
-\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\
-\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\
\end{array}
\right)\oplus -I_3.$$ The set $\triangle^{(1,1,1)}_{\pi}$ has the only extreme point $(0,0,0)$ and three extremal rays $(1,1,0),\ (1,0,1),\ (0,1,1)$. Hence the whole set is given by $$\triangle^{(1,1,1)}_{(\mathbb
C^2;V_1,V_2,V_{12})}=\{(\alpha_1+\alpha_2,\alpha_1+\alpha_3,\alpha_2+\alpha_3)
\ \mid \ \alpha_i \in \mathbb R_+\};$$
*2)*. Let us take the poset $\mathcal
P=\mathcal(N,2)=\{a_1,a_2,b_1,b_2,c_1,c_2\}$, $a_1\prec a_2$, $b_1\prec b_2$, $b_1\prec a_2$, $c_1\prec c_2$. It has the only quite sincere representation $\pi$:
(150,50) (4,5)[$\langle e_{123}\rangle$]{} (55,5)[$\langle e_1 \rangle$]{} (98,5)[$\langle e_3 \rangle$]{} (15,16)[(0,1)[20]{}]{} (63,16)[(0,1)[20]{}]{} (106,16)[(0,1)[20]{}]{} (0,40)[$\langle e_1,e_{123}\rangle$]{} (48,40)[$\langle e_1, e_2\rangle$]{} (91,40)[$\langle e_2, e_3 \rangle$]{} (59,16)[(-2,1)[35]{}]{}
Let $\chi=(\alpha_1,\alpha_2,\beta_1,\beta_2,\gamma_1,\gamma_2)$. The normalized dimension vector of $\pi$ is equal to $\left(\frac{1}{3},\frac{2}{3},\frac{1}{3},\frac{2}{3},\frac{1}{3},\frac{2}{3}\right)$. The set of maximal subdimension vectors is $$\begin{aligned}
Sub(\pi)=\{&(1;0,0;0,0;1,1),\ (1;0,0;0,1;0,1),\ (1;0,1;0,0;0,1),\
(1;0,1;1,1;0,0),\\ &(1;1,1;0,0;0,0),\ (2;0,1;0,1;1,2),\
(2;0,1;1,1;1,1),\ (2;0,1;1,2;0,1),\\ &(2;1,1;0,1;1,1),\
(2;1,2;1,1;0,1)\}.\end{aligned}$$ The corresponding matrix $A_\pi$ has the following form $$A_\pi= \left(
\begin{array}{cccccc}
-\frac{1}{3} & -\frac{2}{3}
&-\frac{1}{3}&-\frac{2}{3}&\frac{2}{3}&\frac{1}{3}\\
-\frac{1}{3} & -\frac{2}{3}
&-\frac{1}{3}& \frac{1}{3}&-\frac{1}{3}&\frac{1}{3}\\
-\frac{1}{3} & \frac{1}{3}
&-\frac{1}{3}&-\frac{2}{3}&-\frac{1}{3}&\frac{1}{3}\\
-\frac{1}{3} & \frac{1}{3}
&\frac{2}{3}& \frac{1}{3}&-\frac{1}{3}&-\frac{2}{3}\\
\frac{2}{3} & \frac{1}{3}
&-\frac{1}{3}&-\frac{2}{3}&-\frac{1}{3}&-\frac{2}{3}\\
-\frac{1}{3} & -\frac{1}{6}
&-\frac{1}{3} & -\frac{1}{6}&\frac{1}{6}&\frac{1}{3}\\
-\frac{1}{3} & -\frac{1}{6}
&\frac{1}{6} & -\frac{1}{6}&\frac{1}{6}&-\frac{1}{6}\\
-\frac{1}{3} & -\frac{1}{6}
&\frac{1}{6} & \frac{1}{3}&-\frac{1}{3}&-\frac{1}{6}\\
\frac{1}{6} & -\frac{1}{6}
&-\frac{1}{3} & -\frac{1}{6}&\frac{1}{6}&-\frac{1}{6}\\
\frac{1}{6} & \frac{1}{3}
&\frac{1}{6} & -\frac{1}{6}&-\frac{1}{3}&-\frac{1}{6}\\
\end{array}
\right )\oplus -I_6.$$ The set $\triangle^{(N,2)}_{\pi}$ has nine extremal rays $$\begin{gathered}
(0,0,0,1,1,1),\ (0,0,1,0,0,1),\ (0,1,0,0,1,0),\ (0,1,0,1,0,1),\\
(0,2,1,1,3,1),\ (1,0,0,0,0,1),\ (2,0,0,1,1,2),\ (1,0,0,1,1,1),\\
(1,0,1,0,1,0).\end{gathered}$$ Hence the whole set is given by $$\begin{aligned}
\triangle^{(N,2)}_{\pi}=\{(&\alpha_6+2\alpha_7+\alpha_8+\alpha_9,\alpha_3+\alpha_4+2\alpha_5,\alpha_2+\alpha_5+\alpha_9,\\
&\alpha_1+\alpha_4+\alpha_5+\alpha_7+\alpha_8,\alpha_1+\alpha_3+3\alpha_5+\alpha_7+\alpha_8+\alpha_9,\\
&\alpha_2+\alpha_4+\alpha_5+\alpha_6+2\alpha_7+\alpha_8) \ \mid \ \alpha_i \in \mathbb
R_+\}.\end{aligned}$$
One can see that $(1,1,1,1,1,1) \notin \triangle_{\pi}^{(N,2)}$ (this vector is semistable but is not stable) which means that given linear representation of the poset $(N,2)$ can not be obtained as the spectral filtration of three partial reflections $A_i=A_i^*=A_i^3$ sum of which is zero $A_1+A_2+A_3=0$.
Appendix C. Missing details in the proof of Theorem 1. {#appendix-c.-missing-details-in-the-proof-of-theorem-1. .unnumbered}
======================================================
The following lists contains all possible subdimension for extended representations used in the proof of Theorem 1.
[c|p[4cm]{}|p[6cm]{}]{} **[Poset]{} & **[Subdimensions]{}****
$d=(\dim U;\dim(V_i\cup U))$ & **[Subspace $U$]{}\
$(1,1,1,1)$ & $(1;1;0;0;0)$ & $\langle e_1 \rangle$\
& $(1;0;1;0;0)$ & $\langle e_2 \rangle$\
& $(1;0;0;1;0)$ & $\langle e_1+e_2 \rangle$\
& $(1;0;0;0;1)$ & $\langle e_1+\lambda e_2 \rangle$\
$(2,2,2)$ & $(1;0,0;0,0;1,1)$ & $\langle e_3\rangle$\
& $(1;0,0;0,1;0,1)$ & $\langle e_2\rangle$\
& $(1;0,1;0,0;0,1)$ & $\langle e_2+(\lambda-1) e_3\rangle$\
& $(1;0,0;1,1;0,0)$ & $\langle e_1\rangle$\
& $(1;0,1;0,1;0,0)$ & $\langle (\lambda-1)e_1+\lambda e_2\rangle$\
& $(1;1,1;0,0;0,0)$ & $\langle e_1+e_2+e_3\rangle$\
& $(2;0,1;0,1;1,2)$ & $\langle e_3,e_2\rangle$\
& $(2;0,1;1,1;1,1)$ & $\langle e_3,e_1\rangle$\
& $(2;0,1;1,2;0,1)$ & $\langle e_2,e_1\rangle$\
& $(2;1,1;0,1;1,1)$ & $\langle e_3,e_1+e_2+e_3\rangle$\
& $(2;1,1;1,1;0,1)$ & $\langle e_1,e_1+e_2+e_3\rangle$\
& $(2;1,2;0,1;0,1)$ & $\langle e_1+\lambda e_3,e_1+e_2+e_3\rangle$\
$(1;3;3)$**
& $(1;0;0,0,0;1,1,1)$ & $\langle e_3\rangle$\
& $(1;0;0,0,1;0,1,1)$ & $\langle e_2+\lambda e_3\rangle$\
& $(1;0;0,1,1;0,0,1)$ & $\langle e_1\rangle$\
& $(1;0;1,1,1;0,0,0)$ & $\langle e_4\rangle$\
& $(1;1;0,0,0;0,0,1)$ & $\langle e_1+e_2+e_3\rangle$\
& $(1;1;0,0,1;0,0,0)$ & $\langle \lambda e_1+e_2+ \lambda e_3+(1-\lambda) e_4\rangle$\
& $(2;0;0,0,1;1,2,2)$ & $\langle e_3,e_2\rangle$\
& $(2;0;0,1,1;1,1,2)$ & $\langle e_3,e_1\rangle$\
& $(2;0;0,1,2;0,1,2)$ & $\langle e_2+\lambda e_3,e_1\rangle$\
& $(2;0;1,1,1;1,1,1)$ & $\langle e_4,e_3\rangle$\
& $(2;0;1,1,2;0,1,1)$ & $\langle e_4,e_2+\lambda e_3\rangle$\
& $(2;0;1,2,2;0,0,1)$ & $\langle e_4,e_1\rangle$\
& $(2;1;0,0,1;1,1,2)$ & $\langle e_3,e_1+e_2+e_3\rangle$\
& $(2;1;0,1,1;0,1,2)$ & $\langle e_1,e_1+e_2+e_3\rangle$\
& $(2;1;1,1,1;0,1,1)$ & $\langle e_4,e_2\rangle$\
& $(2;1;1,1,2;0,0,1)$ & $\langle \lambda e_1+e_2+ \lambda e_3+(1-\lambda) e_4, e_4 \rangle$\
& $(2;1;0,1,2;0,1,1)$ & $\langle \lambda e_1+e_2+ \lambda e_3+(1-\lambda) e_4, e_2+\lambda e_3 \rangle$\
& $(2;1;0,1,1;1,1,1)$ & $\langle e_3,e_1+e_2-e_4 \rangle$\
& $(2;2;0,0,1;0,0,1)$ & $\langle e_2+e_4,e_1+e_2+e_3\rangle$\
& $(3;1;0,1,2;1,2,3)$ & $\langle e_3,e_2,e_1\rangle$\
& $(3;1;1,1,2;1,2,2)$ & $\langle e_4,e_3,e_2\rangle$\
& $(3;1;1,2,2;1,1,2)$ & $\langle e_4,e_3,e_1\rangle$\
& $(3;1;1,2,3;0,1,2)$ & $\langle e_4,e_2+\lambda e_3,e_1\rangle$\
& $(3;2;0,1,2;1,1,2)$ & $\langle e_3,e_2+e_4,e_1+e_2+e_3\rangle$\
& $(3;2;1,1,2;0,1,2)$ & $\langle e_4,e_2,e_1+e_2+e_3\rangle$\
$(1,2,5)$
& $(1;0;0,0;1,1,1,1,1)$ & $\langle e_4\rangle$\
& $(1;0;0,1;0,0,1,1,1)$ & $\langle e_2\rangle$\
& $(1;0;1,1;0,0,0,0,1)$ & $\langle e_5+2e_6\rangle$\
& $(1;1;0,0;0,0,0,1,1)$ & $\langle e_1+e_2+e_3\rangle$\
& $(1;1;0,1;0,0,0,0,0)$ & $\langle e_1+e_6\rangle$\
& $(2;0;0,0;1,2,2,2,2)$ & $\langle e_4,e_3\rangle$\
& $(2;0;0,1;1,1,2,2,2)$ & $\langle e_4,e_2\rangle$\
& $(2;0;0,2;0,0,1,2,2)$ & $\langle e_2,e_1\rangle$\
& $(2;0;1,1;1,1,1,1,2)$ & $\langle e_5+\lambda e_6,e_4\rangle$\
& $(2;0;1,2;0,0,1,1,2)$ & $\langle e_5+\lambda e_6,e_2\rangle$\
& $(2;0;2,2;0,0,0,0,1)$ & $\langle e_6,e_5\rangle$\
& $(2;1;0,0;1,1,1,2,2)$ & $\langle e_4,e_1+e_2+e_3\rangle$\
& $(2;1;0,1;0,1,1,2,2)$ & $\langle e_3,e_1+e_2+e_3\rangle$\
& $(2;1;0,1;1,1,1,1,1)$ & $\langle e_4,e_2+e_4+e_5\rangle$\
& $(2;1;0,2;0,0,1,1,1)$ & $\langle e_2,e_1+e_6\rangle$\
& $(2;1;1,1;0,0,0,1,2)$ & $\langle e_5+\lambda e_6,e_1+e_2+e_3\rangle$\
& $(2;1;1,1;0,0,1,1,1)$ & $\langle e_5,e_2+e_4+e_5\rangle$\
& $(2;1;1,2;0,0,0,1,1)$ & $\langle e_6,e_1\rangle$\
& $(2;2;0,1;0,0,0,1,1)$ & $\langle e_1+e_6,e_1+e_2+e_3\rangle$\
& $(3;0;0,1;1,2,3,3,3)$ & $\langle e_4,e_3,e_2\rangle$\
& $(3;0;0,2;1,1,2,3,3)$ & $\langle e_4,e_2,e_1\rangle$\
& $(3;0;1,1;1,2,2,2,3)$ & $\langle e_5+\lambda e_6,e_4,e_3\rangle$\
& $(3;0;1,2;1,1,2,2,3)$ & $\langle e_5+\lambda e_6,e_4,e_2\rangle$\
& $(3;0;1,3;0,0,1,2,3)$ & $\langle e_5+\lambda e_6,e_2,e_1\rangle$\
& $(3;0;2,2;1,1,1,1,2)$ & $\langle e_6,e_5,e_4\rangle$\
& $(3;0;2,3;0,0,1,1,2)$ & $\langle e_6,e_5,e_2\rangle$\
& $(3;1;0,1;1,2,2,3,3)$ & $\langle e_4,e_3,e_1+e_2+e_3\rangle$\
& $(3;1;0,2;0,1,2,3,3)$ & $\langle e_3,e_2,e_1\rangle$\
& $(3;1;1,1;1,1,1,2,3)$ & $\langle e_5+\lambda e_6,e_4,e_1+e_2+e_3\rangle$\
& $(3;1;1,2;0,1,1,2,3)$ & $\langle e_5+\lambda e_6,e_3,e_1+e_2+e_3\rangle$\
& $(3;1;1,2;1,1,2,2,2)$ & $\langle e_5,e_4,e_2\rangle$\
& $(3;1;1,3;0,0,1,2,2)$ & $\langle e_6,e_2,e_1\rangle$\
& $(3;1;2,2;0,0,1,1,2)$ & $\langle e_6,e_5,e_2+e_4+e_5\rangle$\
& $(3;1;2,3;0,0,0,1,2)$ & $\langle e_6,e_5,e_1\rangle$\
& $(3;2;0,1;1,1,1,2,2)$ & $\langle e_4,e_2+e_4+e_5,e_1+e_2+e_3\rangle$\
& $(3;2;0,2;0,1,1,2,2)$ & $\langle e_3,e_1+e_6,e_1+e_2+e_3\rangle$\
& $(3;2;0,2;1,1,1,1,2)$ & $\langle e_4,e_2+e_4+e_5,e_1+e_6\rangle$\
& $(3;2;1,2;0,0,1,2,2)$ & $\langle e_6,e_1,e_1+e_2+e_3\rangle$\
& $(3;3;0,1;0,0,0,1,2)$ & $\langle e_2+e_4+e_5,e_1+e_6,e_1+e_2+e_3\rangle$\
& $(4;1;0,2;1,2,3,4,4)$ & $\langle e_4,e_3,e_2,e_1\rangle$\
& $(4;1;1,2;1,2,3,3,4)$ & $\langle e_5+\lambda e_6,e_4,e_3,e_2\rangle$\
& $(4;1;1,3;1,1,2,3,4)$ & $\langle e_5+\lambda e_6,e_4,e_2,e_1\rangle$\
& $(4;1;2,2;1,2,2,2,3)$ & $\langle e_6,e_5,e_4,e_3\rangle$\
& $(4;1;2,3;1,1,2,2,3)$ & $\langle e_6,e_5,e_4,e_2\rangle$\
& $(4;1;2,4;0,0,1,2,3)$ & $\langle e_6,e_5,e_2,e_1\rangle$\
& $(4;2;0,2;1,2,2,3,3)$ & $\langle e_4,e_3,e_2+e_4+e_5,e_1+e_2+e_3\rangle$\
& $(4;2;1,2;1,1,2,3,3)$ & $\langle e_6,e_4,e_1,e_1+e_2+e_3\rangle$\
& $(4;2;1,3;0,1,2,3,3)$ & $\langle e_6,e_3,e_2,e_1\rangle$\
& $(4;2;1,3;1,1,2,2,3)$ & $\langle e_5,e_4,e_2,e_1+e_6\rangle$\
& $(4;2;2,3;0,0,1,2,3)$ & $\langle e_6,e_5,e_2+e_4+e_5,e_1\rangle$\
& $(4;3;0,2;1,1,1,2,3)$ & $\langle e_4,e_2+e_4+e_5,e_1+e_6,e_1+e_2+e_3\rangle$\
& $(4;3;1,2;0,0,1,2,3)$ & $\langle e_6,e_2+e_4+e_5,e_1,e_1+e_2+e_3\rangle$\
& $(5;2;1,3;1,2,3,4,5)$ & $\langle e_5+\lambda e_6,e_4,e_3,e_2,e_1\rangle$\
& $(5;2;2,3;1,2,3,3,4)$ & $\langle e_6,e_5,e_4,e_3,e_2\rangle$\
& $(5;2;2,4;1,1,2,3,4)$ & $\langle e_6,e_5,e_4,e_2,e_1\rangle$\
& $(5;3;1,3;1,2,2,3,4)$ & $\langle e_4,e_3,e_2+e_4+e_5,e_1+e_6,e_1+e_2+e_3\rangle$\
& $(5;3;2,3;0,1,2,3,4)$ & $\langle e_6,e_5,e_2+e_4+e_5,e_1,e_1+e_2+e_3\rangle$\
$(N,4)$ & $(1;0,0;0,0;1,1,1,1)$ & $\langle e_4\rangle$\
& $(1;0,0;0,1;0,0,1,1)$ & $\langle e_2\rangle$\
& $(1;0,1;0,0;0,1,1,1)$ & $\langle e_3+\lambda e_4\rangle$\
& $(1;0,1;1,1;0,0,0,0)$ & $\langle e_5\rangle$\
& $(1;1,1;0,0;0,0,0,1)$ & $\langle e_1+e_3+e_4\rangle$\
& $(2;0,1;0,0;1,2,2,2)$ & $\langle e_4,e_3\rangle$\
& $(2;0,1;0,1;1,1,2,2)$ & $\langle e_4,e_2\rangle$\
& $(2;0,1;0,2;0,0,1,2)$ & $\langle e_2,e_1\rangle$\
& $(2;0,1;1,1;1,1,1,1)$ & $\langle e_5,e_4\rangle$\
& $(2;0,1;1,2;0,0,1,1)$ & $\langle e_5,e_2\rangle$\
& $(2;0,2;1,1;0,1,1,1)$ & $\langle e_5,e_3+\lambda e_4\rangle$\
& $(2;1,1;0,0;1,1,1,2)$ & $\langle e_4,e_1+e_3+e_4\rangle$\
& $(2;1,1;0,1;0,1,1,2)$ & $\langle e_1,e_1+e_3+e_4\rangle$\
& $(2;1,2;0,0;0,1,1,2)$ & $\langle e_3+\lambda e_4,e_1+e_3+e_4\rangle$\
& $(2;1,2;1,1;0,0,1,1)$ & $\langle e_5,e_2+e_3+e_5\rangle$\
& $(2;2,2;0,0;0,0,0,1)$ & $\langle e_2+e_3+e_5,e_1+e_3+e_4\rangle$\
& $(3;0,2;0,1;1,2,3,3)$ & $\langle e_4,e_3,e_2\rangle$\
& $(3;0,2;0,2;1,1,2,3)$ & $\langle e_4,e_2,e_1\rangle$\
& $(3;0,2;1,1;1,2,2,2)$ & $\langle e_5,e_4,e_3\rangle$\
& $(3;0,2;1,2;1,1,2,2)$ & $\langle e_5,e_4,e_2\rangle$\
& $(3;0,2;1,3;0,0,1,2)$ & $\langle e_5,e_2,e_1\rangle$\
& $(3;1,2;0,1;1,2,2,3)$ & $\langle e_4,e_3,e_1\rangle$\
& $(3;1,2;0,2;0,1,2,3)$ & $\langle e_2,e_1,e_1+e_3+e_4\rangle$\
& $(3;1,2;1,1;1,1,2,2)$ & $\langle e_5,e_4,e_2+e_3+e_5\rangle$\
& $(3;1,2;1,2;0,1,2,2)$ & $\langle e_5,e_3,e_2\rangle$\
& $(3;1,3;1,1;0,1,2,2)$ & $\langle e_5,e_3+\lambda e_4,e_2+e_3+e_5\rangle$\
& $(3;2,2;0,1;1,1,1,2)$ & $\langle e_4,e_2+e_3+e_5,e_1+e_3+e_4\rangle$\
& $(3;2,3;0,1;0,1,1,2)$ & $\langle e_3+\lambda e_4,e_2+e_3+e_5,e_1+e_3+e_4\rangle$\
& $(3;2,3;1,1;0,0,1,2)$ & $\langle e_5,e_2+e_3+e_5,e_1+e_3+e_4\rangle$\
& $(4;1,3;0,2;1,2,3,4)$ & $\langle e_4,e_3,e_2,e_1\rangle$\
& $(4;1,3;1,2;1,2,3,3)$ & $\langle e_5,e_4,e_3,e_2\rangle$\
& $(4;1,3;1,3;1,1,2,3)$ & $\langle e_5,e_4,e_2,e_1\rangle$\
& $(4;2,3;0,2;1,2,2,3)$ & $\langle e_4,e_3,e_2+e_3+e_5,e_1\rangle$\
& $(4;2,3;1,2;1,1,2,3)$ & $\langle e_5,e_4,e_2+e_3+e_5,e_1+e_3+e_4\rangle$\
& $(4;2,4;1,2;0,1,2,3)$ & $\langle e_5,e_3+\lambda e_4,e_2+e_3+e_5,e_1+e_3+e_4\rangle$\
It is routine to check that for the corresponding representations stability conditions (where weight is taken to be dimension) holds for all maximal subdimension listed above, hence representation are unitarizable.
Appendix D. Missing details in the proof of Theorem 2. {#appendix-d.-missing-details-in-the-proof-of-theorem-2. .unnumbered}
======================================================
The following lists contain all possible subdimension for representations 1)-6) of the poset $\mathcal P_2$.
[c|p[4cm]{}|p[6cm]{}]{} **[No.]{} & **[Subdimensions]{}****
$d=(\dim U;\dim(V_i\cup U))$ & **[Subspace $U$]{}\
$1)$ & $(1;0,0;0,0;1,1,1)$ & $\langle e_3\rangle$\
& $(1;0,0;0,1;0,0,1)$ & $\langle e_1+e_2+e_3\rangle$\
& $(1;0,1;0,0;0,1,1)$ & $\langle e_2\rangle$\
& $(1;0,1;1,1;0,0,0)$ & $\langle e_4\rangle$\
& $(1;1,1;0,0;0,0,0)$ & $\langle e_1+e_4\rangle$\
& $(2;0,1;0,0;1,2,2)$ & $\langle e_3,e_2\rangle$\
& $(2;0,1;0,1;1,1,2)$ & $\langle e_3,e_1+e_2+e_3\rangle$\
& $(2;0,1;1,1;1,1,1)$ & $\langle e_4,e_3\rangle$\
& $(2;0,1;1,2;0,0,1)$ & $\langle e_4,e_1+e_2+e_3\rangle$\
& $(2;0,2;0,0;0,1,2)$ & $\langle e_2,e_1\rangle$\
& $(2;0,2;1,1;0,1,1)$ & $\langle e_4,e_2\rangle$\
& $(2;1,1;0,0;1,1,1)$ & $\langle e_3,e_1+e_4\rangle$\
& $(2;1,2;0,0;0,1,1)$ & $\langle e_2,e_1+e_4\rangle$\
& $(2;1,2;1,1;0,0,1)$ & $\langle e_4,e_1\rangle$\
& $(3;0,2;0,1;1,2,3)$ & $\langle e_3,e_2,e_1\rangle$\
& $(3;0,2;1,1;1,2,2)$ & $\langle e_4,e_3,e_2\rangle$\
& $(3;0,2;1,2;1,1,2)$ & $\langle e_4,e_3,e_1+e_2+e_3\rangle$\
& $(3;1,2;0,1;1,2,2)$ & $\langle e_3,e_2,e_1+e_4\rangle$\
& $(3;1,2;1,1;1,1,2)$ & $\langle e_4,e_3,e_1\rangle$\
& $(3;1,2;1,2;0,1,2)$ & $\langle e_4,e_1,e_1+e_2+e_3\rangle$\
& $(3;1,3;1,1;0,1,2)$ & $\langle e_4,e_2,e_1\rangle$\
$2)$ & $(1;0,0;0,0;1,1,1)$ & $\langle e_3\rangle$\
& $(1;0,0;0,1;0,1,1)$ & $\langle e_2+e_3\rangle$\
& $(1;0,1;0,0;0,1,1)$ & $\langle e_2\rangle$\
& $(1;0,1;0,1;0,0,1)$ & $\langle e_1+e_2\rangle$\
& $(1;0,1;1,1;0,0,0)$ & $\langle e_4\rangle$\
& $(1;1,1;0,0;0,0,0)$ & $\langle e_1+e_4\rangle$\
& $(2;0,1;0,1;1,2,2)$ & $\langle e_3,e_2\rangle$\
& $(2;0,1;0,2;0,1,2)$ & $\langle e_2+e_3,e_1+e_2\rangle$\
& $(2;0,1;1,1;1,1,1)$ & $\langle e_4,e_3\rangle$\
& $(2;0,1;1,2;0,1,1)$ & $\langle e_4,e_2+e_3\rangle$\
& $(2;0,2;0,1;0,1,2)$ & $\langle e_2,e_1\rangle$\
& $(2;0,2;1,1;0,1,1)$ & $\langle e_4,e_2\rangle$\
& $(2;0,2;1,2;0,0,1)$ & $\langle e_4,e_1+e_2\rangle$\
& $(2;1,1;0,1;1,1,1)$ & $\langle e_3,e_1+e_4\rangle$\
& $(2;1,2;0,1;0,1,1)$ & $\langle e_2,e_1+e_4\rangle$\
& $(2;1,2;1,1;0,0,1)$ & $\langle e_4,e_1\rangle$\
& $(3;0,2;0,2;1,2,3)$ & $\langle e_3,e_2,e_1\rangle$\
& $(3;0,2;1,2;1,2,2)$ & $\langle e_4,e_3,e_2\rangle$\
& $(3;0,2;1,3;0,1,2)$ & $\langle e_4,e_2+e_3,e_1+e_2\rangle$\
& $(3;1,2;0,2;1,2,2)$ & $\langle e_3,e_2,e_1+e_4\rangle$\
& $(3;1,2;1,2;1,1,2)$ & $\langle e_4,e_3,e_1\rangle$\
& $(3;1,3;1,2;0,1,2)$ & $\langle e_4,e_2,e_1\rangle$\
$3)$ & $(1;0,0;0,1;0,1,1)$ & $\langle e_2+e_4\rangle$\
& $(1;0,1;0,0;1,1,1)$ & $\langle e_3\rangle$\
& $(1;0,1;0,1;0,0,1)$ & $\langle e_1+e_2+e_3\rangle$\
& $(1;0,1;1,1;0,0,0)$ & $\langle e_5\rangle$\
& $(1;1,1;0,0;0,0,1)$ & $\langle e_1\rangle$\
& $(2;0,1;0,0;2,2,2)$ & $\langle e_4,e_3\rangle$\
& $(2;0,1;0,1;1,2,2)$ & $\langle e_4,e_2\rangle$\
& $(2;0,1;0,2;0,1,2)$ & $\langle e_2+e_4,e_1+e_2+e_3\rangle$\
& $(2;0,1;1,2;0,1,1)$ & $\langle e_5,e_2+e_4\rangle$\
& $(2;0,2;0,0;1,2,2)$ & $\langle e_3,e_2\rangle$\
& $(2;0,2;0,1;1,1,2)$ & $\langle e_3,e_1+e_2+e_3\rangle$\
& $(2;0,2;1,1;1,1,1)$ & $\langle e_5,e_3\rangle$\
& $(2;0,2;1,2;0,0,1)$ & $\langle e_5,e_1+e_2+e_3\rangle$\
& $(2;1,1;0,1;1,1,1)$ & $\langle e_4,e_2+e_5\rangle$\
& $(2;1,2;0,0;1,1,2)$ & $\langle e_3,e_1\rangle$\
& $(2;1,2;0,1;0,1,2)$ & $\langle e_1,e_1+e_2+e_3\rangle$\
& $(2;1,2;1,1;0,1,1)$ & $\langle e_5,e_2\rangle$\
& $(2;2,2;0,0;0,0,1)$ & $\langle e_2+e_5,e_1\rangle$\
& $(3;0,2;0,1;2,3,3)$ & $\langle e_4,e_3,e_2\rangle$\
& $(3;0,2;1,1;2,2,2)$ & $\langle e_5,e_4,e_3\rangle$\
& $(3;0,2;1,3;0,1,2)$ & $\langle e_5,e_2+e_4,e_1+e_2+e_3\rangle$\
& $(3;1,2;0,1;2,2,3)$ & $\langle e_4,e_3,e_1\rangle$\
& $(3;1,2;0,2;1,2,3)$ & $\langle e_2+e_4,e_1,e_1+e_2+e_3\rangle$\
& $(3;1,2;1,2;1,2,2)$ & $\langle e_5,e_4,e_2\rangle$\
& $(3;1,3;0,1;1,2,3)$ & $\langle e_3,e_2,e_1\rangle$\
& $(3;1,3;1,1;1,2,2)$ & $\langle e_5,e_3,e_2\rangle$\
& $(3;1,3;1,2;1,1,2)$ & $\langle e_5,e_3,e_1+e_2+e_3\rangle$\
& $(3;2,3;0,1;1,1,2)$ & $\langle e_3,e_2+e_5,e_1\rangle$\
& $(3;2,3;1,1;0,1,2)$ & $\langle e_5,e_2,e_1\rangle$\
& $(4;1,3;0,2;2,3,4)$ & $\langle e_4,e_3,e_2,e_1\rangle$\
& $(4;1,3;1,2;2,3,3)$ & $\langle e_5,e_4,e_3,e_2\rangle$\
& $(4;1,3;1,3;1,2,3)$ & $\langle e_5,e_4,e_2,e_1+e_2+e_3\rangle$\
& $(4;2,3;0,2;2,2,3)$ & $\langle e_4,e_3,e_2+e_5,e_1\rangle$\
& $(4;2,4;1,2;1,2,3)$ & $\langle e_5,e_3,e_2,e_1\rangle$\
$4)$ & $(1;0,0;0,0;1,1,1)$ & $\langle e_4\rangle$\
& $(1;0,0;0,1;0,1,1)$ & $\langle e_2+e_3+e_4\rangle$\
& $(1;0,1;0,0;0,1,1)$ & $\langle e_3\rangle$\
& $(1;0,1;0,1;0,0,1)$ & $\langle e_1+e_3\rangle$\
& $(1;0,1;1,1;0,0,0)$ & $\langle e_5\rangle$\
& $(1;1,1;0,0;0,0,1)$ & $\langle e_1\rangle$\
& $(2;0,1;0,1;1,2,2)$ & $\langle e_4,e_2+e_3+e_4\rangle$\
& $(2;0,1;0,2;0,1,2)$ & $\langle e_2+e_3+e_4,e_1+e_3\rangle$\
& $(2;0,1;1,1;1,1,1)$ & $\langle e_5,e_4\rangle$\
& $(2;0,1;1,2;0,1,1)$ & $\langle e_5,e_2+e_3+e_4\rangle$\
& $(2;0,2;0,0;0,2,2)$ & $\langle e_3,e_2\rangle$\
& $(2;0,2;1,2;0,0,1)$ & $\langle e_5,e_1+e_3\rangle$\
& $(2;1,1;0,0;1,1,2)$ & $\langle e_4,e_1\rangle$\
& $(2;1,2;0,1;0,1,2)$ & $\langle e_3,e_1\rangle$\
& $(2;1,2;1,1;0,1,1)$ & $\langle e_5,e_2\rangle$\
& $(2;2,2;0,0;0,0,1)$ & $\langle e_2+e_5,e_1\rangle$\
& $(3;0,2;0,1;1,3,3)$ & $\langle e_4,e_3,e_2\rangle$\
& $(3;0,2;0,2;1,2,3)$ & $\langle e_4,e_2+e_3+e_4,e_1+e_3\rangle$\
& $(3;0,2;1,2;1,2,2)$ & $\langle e_5,e_4,e_2+e_3+e_4\rangle$\
& $(3;0,2;1,3;0,1,2)$ & $\langle e_5,e_2+e_3+e_4,e_1+e_3\rangle$\
& $(3;1,2;0,1;1,2,3)$ & $\langle e_4,e_3,e_1\rangle$\
& $(3;1,2;0,2;0,2,3)$ & $\langle e_3,e_2+e_3+e_4,e_1\rangle$\
& $(3;1,2;1,1;1,2,2)$ & $\langle e_5,e_4,e_2\rangle$\
& $(3;1,2;1,2;0,2,2)$ & $\langle e_5,e_2,e_2+e_3+e_4\rangle$\
& $(3;1,3;0,1;0,2,3)$ & $\langle e_3,e_2,e_1\rangle$\
& $(3;1,3;1,1;0,2,2)$ & $\langle e_5,e_3,e_2\rangle$\
& $(3;1,3;1,2;0,1,2)$ & $\langle e_5,e_3,e_1\rangle$\
& $(3;2,2;0,1;1,1,2)$ & $\langle e_4,e_2+e_5,e_1\rangle$\
& $(3;2,3;1,1;0,1,2)$ & $\langle e_5,e_2,e_1\rangle$\
& $(4;1,3;0,2;1,3,4)$ & $\langle e_4,e_3,e_2,e_1\rangle$\
& $(4;1,3;1,2;1,3,3)$ & $\langle e_5,e_4,e_3,e_2\rangle$\
& $(4;1,3;1,3;1,2,3)$ & $\langle e_5,e_4,e_2+e_3+e_4,e_1+e_3\rangle$\
& $(4;2,3;1,2;1,2,3)$ & $\langle e_5,e_4,e_2,e_1\rangle$\
& $(4;2,4;1,2;0,2,3)$ & $\langle e_5,e_3,e_2,e_1\rangle$\
$5)$ & $(1;0,0;0,0;1,1,1)$ & $\langle e_4\rangle$\
& $(1;0,1;0,0;0,1,1)$ & $\langle e_3\rangle$\
& $(1;0,1;0,1;0,0,1)$ & $\langle e_1+e_2+e_3\rangle$\
& $(1;0,1;1,1;0,0,0)$ & $\langle e_5\rangle$\
& $(1;1,1;0,0;0,0,1)$ & $\langle e_1\rangle$\
& $(2;0,1;0,0;1,2,2)$ & $\langle e_4,e_3\rangle$\
& $(2;0,1;0,2;0,0,2)$ & $\langle e_1+e_4,e_1+e_2+e_3\rangle$\
& $(2;0,1;1,1;1,1,1)$ & $\langle e_5,e_4\rangle$\
& $(2;0,2;0,1;0,1,2)$ & $\langle e_3,e_1+e_2+e_3\rangle$\
& $(2;0,2;1,1;0,1,1)$ & $\langle e_5,e_3\rangle$\
& $(2;0,2;1,2;0,0,1)$ & $\langle e_5,e_1+e_2+e_3\rangle$\
& $(2;1,1;0,1;1,1,2)$ & $\langle e_4,e_1\rangle$\
& $(2;1,2;0,0;0,1,2)$ & $\langle e_3,e_1\rangle$\
& $(2;1,2;0,1;0,0,2)$ & $\langle e_1,e_1+e_2+e_3\rangle$\
& $(2;1,2;1,1;0,0,1)$ & $\langle e_5,e_2\rangle$\
& $(2;2,2;0,0;0,0,1)$ & $\langle e_2+e_5,e_1\rangle$\
& $(3;0,2;1,1;1,2,2)$ & $\langle e_5,e_4,e_3\rangle$\
& $(3;0,2;1,3;0,0,2)$ & $\langle e_5,e_1+e_4,e_1+e_2+e_3\rangle$\
& $(3;1,2;0,1;1,2,3)$ & $\langle e_4,e_3,e_1\rangle$\
& $(3;1,2;0,2;1,1,3)$ & $\langle e_4,e_1,e_1+e_2+e_3\rangle$\
& $(3;1,2;1,2;1,1,2)$ & $\langle e_5,e_4,e_1\rangle$\
& $(3;1,3;0,1;0,1,3)$ & $\langle e_3,e_2,e_1\rangle$\
& $(3;1,3;1,2;0,1,2)$ & $\langle e_5,e_3,e_1+e_2+e_3\rangle$\
& $(3;2,2;0,1;1,1,2)$ & $\langle e_4,e_2+e_5,e_1\rangle$\
& $(3;2,3;0,1;0,1,2)$ & $\langle e_3,e_2+e_5,e_1\rangle$\
& $(3;2,3;1,1;0,0,2)$ & $\langle e_5,e_2,e_1\rangle$\
& $(4;1,3;0,2;1,2,4)$ & $\langle e_4,e_3,e_2,e_1\rangle$\
& $(4;1,3;1,2;1,2,3)$ & $\langle e_5,e_4,e_3,e_2\rangle$\
& $(4;1,3;1,3;1,1,3)$ & $\langle e_5,e_4,e_1,e_1+e_2+e_3\rangle$\
& $(4;2,3;0,2;1,2,3)$ & $\langle e_4,e_3,e_2+e_5,e_1\rangle$\
& $(4;2,3;1,2;1,1,3)$ & $\langle e_5,e_4,e_2,e_1\rangle$\
& $(4;2,4;1,2;0,1,3)$ & $\langle e_5,e_3,e_2,e_1\rangle$\
$6)$ & $(1;0,0;0,0;1,1,1)$ & $\langle e_3\rangle$\
& $(1;0,0;0,1;0,0,1)$ & $\langle e_1+e_2+e_3\rangle$\
& $(1;0,1;0,0;0,1,1)$ & $\langle e_2\rangle$\
& $(1;0,1;1,1;0,0,0)$ & $\langle e_5\rangle$\
& $(1;1,1;0,0;0,0,0)$ & $\langle e_4\rangle$\
& $(2;0,1;0,0;1,2,2)$ & $\langle e_3,e_2\rangle$\
& $(2;0,1;0,1;1,1,2)$ & $\langle e_3,e_1+e_2+e_3\rangle$\
& $(2;0,1;1,1;1,1,1)$ & $\langle e_5,e_3\rangle$\
& $(2;0,1;1,2;0,0,1)$ & $\langle e_5,e_1+e_2+e_3\rangle$\
& $(2;0,2;0,0;0,1,2)$ & $\langle e_2,e_1\rangle$\
& $(2;0,2;1,1;0,1,1)$ & $\langle e_5,e_2\rangle$\
& $(2;0,2;1,2;0,0,0)$ & $\langle e_5,e_2+e_4\rangle$\
& $(2;1,1;0,0;1,1,1)$ & $\langle e_4,e_3\rangle$\
& $(2;1,2;0,1;0,1,1)$ & $\langle e_4,e_2\rangle$\
& $(2;1,2;1,1;0,0,1)$ & $\langle e_5,e_1\rangle$\
& $(2;2,2;0,0;0,0,0)$ & $\langle e_4,e_1+e_5\rangle$\
& $(3;0,2;0,1;1,2,3)$ & $\langle e_3,e_2,e_1\rangle$\
& $(3;0,2;1,1;1,2,2)$ & $\langle e_5,e_3,e_2\rangle$\
& $(3;0,2;1,2;1,1,2)$ & $\langle e_5,e_3,e_1+e_2+e_3\rangle$\
& $(3;0,2;1,3;0,0,1)$ & $\langle e_5,e_2+e_4,e_1+e_2+e_3\rangle$\
& $(3;1,2;0,1;1,2,2)$ & $\langle e_4,e_3,e_2\rangle$\
& $(3;1,2;1,1;1,1,2)$ & $\langle e_5,e_3,e_1\rangle$\
& $(3;1,2;1,2;0,1,2)$ & $\langle e_5,e_1,e_1+e_2+e_3\rangle$\
& $(3;1,3;1,1;0,1,2)$ & $\langle e_5,e_2,e_1\rangle$\
& $(3;1,3;1,2;0,1,1)$ & $\langle e_5,e_4,e_2\rangle$\
& $(3;2,2;0,1;1,1,1)$ & $\langle e_4,e_3,e_1+e_5\rangle$\
& $(3;2,3;0,1;0,1,1)$ & $\langle e_4,e_2,e_1+e_5\rangle$\
& $(3;2,3;1,1;0,0,1)$ & $\langle e_5,e_4,e_1\rangle$\
& $(4;1,3;1,2;1,2,3)$ & $\langle e_5,e_3,e_2,e_1\rangle$\
& $(4;1,3;1,3;1,1,2)$ & $\langle e_5,e_3,e_2+e_4,e_1+e_2+e_3\rangle$\
& $(4;2,3;0,2;1,2,2)$ & $\langle e_4,e_3,e_2,e_1+e_5\rangle$\
& $(4;2,3;1,2;1,1,2)$ & $\langle e_5,e_4,e_3,e_1\rangle$\
& $(4;2,4;1,2;0,1,2)$ & $\langle e_5,e_4,e_2,e_1\rangle$\
**
$$\begin{xy}
\xymatrix@R10pt@C20pt{
&1\\1\ar[ru]^{i}\ar[rd]^{j}\\&1\\1\ar[ru]^k\ar[rd]^l&\\&1 }
\end{xy}$$
**Poset** **Representation** $\pi=(V;\pi(a_i);\pi(b_i);\pi(c_i))$ **Weight $\chi$**
--------------------- ------------------------------------------------------------------------------------------ --------------------------
$(1,1,1)$ $(\mathbb{C}^2;V_1;V_2; V_{12})$ $(1,1,1)$
$(1,2,2)$ 1\) $(\mathbb{C}^3;V_{12,13};V_1, V_{1,2};V_{3}, V_{2,3})$ $(1,1,1,1,1)$
2\) $(\mathbb{C}^3; V_{123};V_{1}, V_{1,2};V_{3}, V_{2,3})$ $(2,1,1,1,1)$
\[0.5ex\] $(1,2,3)$ 1\) $(\mathbb{C}^4;V_{123,24};V_4, V_{1,4};V_3, V_{2,3}, V_{1,2,3})$ $(2,1,1,1,1,1)$
2\) $(\mathbb{C}^4; V_{124,13};V_4, V_{1,2,4};V_{3}, V_{2,3}, V_{1,2,3})$ $(2,1,2,1,1,1)$
3\) $(\mathbb{C}^4; V_{123,24}; V_{1,4}, V_{1,2,4}; V_{3}, V_{2,3}, V_{1,2,3})$ $(2,2,1,1,1,1)$
$(1,2,4)$ 1\) $(\mathbb{C}^5;V_{134,235};V_{5}, V_{1,2,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,1,2,1,1,1,1)$
2\) $(\mathbb{C}^5; V_{123,245};V_{1,5}, V_{1,2,5};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,2,1,1,1,1,1)$
3\) $(\mathbb{C}^5; V_{123,245};V_{1,5}, V_{1,2,3,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,2,1,1,1,1,0.4)$
4\) $(\mathbb{C}^5; V_{123,234,15}; V_2, V_{1,2,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $?$
5\) $(\mathbb{C}^5; V_{13,234,45};V_{1,5}, V_{1,2,5};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(3,2,1,1,1,1,1)$
6\) $(\mathbb{C}^5; V_{12,234,45}; V_{1,5}, V_{1,2,3,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(3,2,2,1,1,1,1)$
7\) $(\mathbb{C}^6; V_{123,245,16}; V_{5,6}, V_{1,2,5,6};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(3,2,2,1,1,1,1)$
8\) $(\mathbb{C}^6; V_{125,234,46}; V_{1,6}, V_{1,2,3,6};$
$ V_5, V_{4,5}, V_{3,4,5}, V_{1,2,3,4,5})$ $(3,2,2,1,1,1,2)$
9\) $(\mathbb{C}^6; V_{125,134,46}; V_{1,6}, V_{1,2,3,6};$
$ V_{5}, V_{4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(3,2,2,1,1,2,1)$
10\) $(\mathbb{C}^6; V_{125,4,23}; V_{1,6}, V_{1,2,3,6};$
$ V_5, V_{3,4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $?$
11\) $(\mathbb{C}^6; V_{135,124,46};V_{1,6},V_{1,2,3,6};$
$V_{4,5}, V_{3,4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(3,2,2,2,1,1,1)$
$\mathcal P_1$ $(\mathbb{C}^3;V_{123}, V_{123,1};V_1, V_{1,2};V_{3},V_{2,3})$ $(1,0.1,1,1,1,1)$
$\mathcal P_2$ 1\) $(\mathbb{C}^4;V_{14}, V_{1,2,4}; V_4, V_{4,123};V_3, V_{2,3}, V_{1,2,3})$ \* $(1,1.9,1,1,1,1,1)$
2\) $(\mathbb{C}^4;V_{14}, V_{1,2,4}; V_4, V_{4,12,23}; V_3, V_{2,3}, V_{1,2,3})$ \* $(1,1.9,1,2,1,1,1)$
3\) $(\mathbb{C}^4;V_{123,24}, V_{13,2,4}; V_{4}, V_{1,4}; V_3, V_{2,3}, V_{1,2,3})$ $(2,0.1,1,1,1,1,1)$
4\) $(\mathbb{C}^4; V_{124,13}, V_{12,13,4};V_4, V_{1,2,4};V_3, V_{2,3}, V_{1,2,3})$ $(2,0.1,1,2,1,1,1)$
$\mathcal P_3$ 1\) $(\mathbb{C}^4; V_4, V_{1,4}; V_{123,24}, V_{12,24,3};V_3, V_{2,3}, V_{1,2,3})$ $(1,1,2,0.1,1,1,1)$
2\) $(\mathbb{C}^4;V_4, V_{1,2,4}; V_{124,13}, V_{24,1,3}; V_3, V_{2,3}, V_{1,2,3})$ $(1,2,2,0.1,1,1,1)$
3\) $(\mathbb{C}^4;V_{1,4}, V_{1,2,4}; V_{123,24}, V_{12,24,3};$
$ V_3, V_{2,3}, V_{1,2,3})$ $(2,1,2,0.1,1,1,1)$
$\mathcal P_4$ $(\mathbb{C}^4; V_{123,24}; V_4, V_{1,4}, V_{1,3,4}; V_3, V_{2,3}, V_{1,2,3})$ $(2,1,1,0.1,1,1,1)$
$\mathcal P_5$ $(\mathbb{C}^4; V_{14}, V_{1,2,4}; V_4, V_{4,123}, V_{1,4,23}; V_3, V_{2,3}, V_{1,2,3})$ $(1,1,1,1,1,1,1,1.1)$
$\mathcal P_6$ 1\) $(\mathbb{C}^4; V_{14}, V_{1,2,4};V_4, V_{4,123}, V_{4,3,12};$
$V_3, V_{2,3}, V_{1,2,3})$ $(1,1,1,1,0.1,1,1,1)$
2\) $(\mathbb{C}^4;V_{123,24}, V_{13,2,4};V_4, V_{1,4}, V_{1,3,4};$
$V_3, V_{2,3}, V_{1,2,3})$ $(2,0.1,1,1,0.1,1,1,1)$
$\mathcal P_7$ 1\) $(\mathbb{C}^5;V_5, V_{1,2,5};V_{134,235}, V_{4,13,235};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(1,2,2,0.1,1,1,1,1)$
2\) $(\mathbb{C}^5; V_{1,5},V_{1,2,5};V_{123,245}, V_{4,123,25};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,1,2,0.1,1,1,1,1)$
3\) $(\mathbb{C}^5;V_{1,5},V_{1,2,3,5};V_{123,245}, V_{4,123,25};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(?)$
4\) $(\mathbb{C}^5;V_5,V_{1,2,5};V_{123,234,15}, V_{4,123,23,15};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(?)$
5\) $(\mathbb{C}^5;V_{1,5},V_{1,2,5};V_{13,234,45}, V_{13,23,4,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,1,3,0.1,1,1,1,1)$
6\) $(\mathbb{C}^5;V_{1,5},V_{1,2,3,5};V_{12,234,45}, V_{12,23,4,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,2,3,0.1,1,1,1,1)$
7\) $(\mathbb{C}^6;V_{5,6},V_{1,2,5,6};V_{123,245,16}, V_{4,123,25,16};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,2,3,0.1,1,1,1,1)$
8\) $(\mathbb{C}^6;V_{1,6},V_{1,2,3,6};V_{125,234,46}, V_{5,12,234,46};$
$V_5, V_{4,5}, V_{3,4,5}, V_{1,2,3,4,5})$ $(2,2,3,0.1,1,1,1,2)$
9\) $(\mathbb{C}^6;V_{1,6},V_{1,2,3,6};V_{125,134,46}, V_{5,12,134,46};$
$V_5, V_{4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(2,2,3,0.1,1,1,2,1)$
10\) $(\mathbb{C}^6;V_{1,6}, V_{1,2,3,6};V_{125,4,234}, V_{5,12,4,234};$
$V_5, V_{3,4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(?)$
11\) $(\mathbb{C}^6;V_{1,6}, V_{1,2,3,6};V_{135,124,46}, V_{13,12,4,5,6};$
$V_{4,5}, V_{3,4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(2,2,3,0.1,2,1,1,1)$
$\mathcal P_8$ 1\) $(\mathbb{C}^5;V_5, V_{1,2,5};V_{134,235},V_{1,3,4,25};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $?$
2\) $(\mathbb{C}^5;V_{1,5}, V_{1,2,5};V_{123,245},V_{3,4,12,25};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,1,2,0.1,1,1,1,1)$
3\) $(\mathbb{C}^5;V_{1,5}, V_{1,2,3,5};V_{123,245},V_{3,4,12,25};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,1,2,0.1,1,1,1,0.4)?$
4\) $(\mathbb{C}^6;V_{5,6}, V_{1,2,5,6};V_{123,245,16},V_{3,4,12,25,16};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,2,3,0.1,1,1,1,1)$
5\) $(\mathbb{C}^6;V_{1,6}, V_{1,2,3,6};V_{125,234,46},V_{4,5,12,23,6};$
$V_5, V_{4,5}, V_{3,4,5}, V_{1,2,3,4,5})$ $(2,2,3,0.9,1,1,1,2)$
6\) $(\mathbb{C}^6;V_{1,6}, V_{1,2,3,6};V_{125,134,46},V_{4,5,12,13,6};$
$V_5, V_{4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(2,2,3,0.9,1,1,2,1)$
$\mathcal P_9$ 1\) $(\mathbb{C}^5;V_{134,235};V_5, V_{1,2,5}, V_{1,2,4,5};$
$ V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(2,1,2,0.1,1,1,1,1)$
2\) $(\mathbb{C}^5;V_{123,234,15};V_5, V_{1,2,5}, V_{1,2,4,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(?)$
3\) $(\mathbb{C}^5;V_{13,234,45};V_{1,5}, V_{1,2,5}, V_{1,2,4,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(3,2,1,0.1,1,1,1,1)$
4\) $(\mathbb{C}^6;V_{123,245,16};V_{5,6}, V_{1,2,5,6}, V_{1,2,4,5,6};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(3,2,2,0.1,1,1,1,1)$
5\) $(\mathbb{C}^6;V_{125,234,46};V_{1,6}, V_{1,2,3,6}, V_{1,2,3,5,6};$
$V_5, V_{4,5}, V_{3,4,5}, V_{1,2,3,4,5})$ $(3,2,2,0.1,1,1,1,2)$
6\) $(\mathbb{C}^6;V_{125,134,46};V_{1,6}, V_{1,2,3,6}, V_{1,2,3,5,6};$
$V_5, V_{4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(3,2,2,0.1,1,1,2,1)$
7\) $(\mathbb{C}^6;V_{125,4,23};V_{1,6}, V_{1,2,3,6}, V_{1,2,3,5,6};$
$V_5, V_{3,4,5}, V_{2,3,4,5}, V_{1,2,3,4,5})$ $(?)$
$\mathcal P_{10}$ 1\) $(\mathbb{C}^5;V_5, V_{1,2,5}; V_{134, 235}, V_{13,23,4,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(1,2,2,0.1,1,1,1,1)$
2\) $(\mathbb{C}^5;V_5, V_{1,2,5};V_{123,234,15}, V_{1,23,4,5};$
$V_4, V_{3,4}, V_{2,3,4}, V_{1,2,3,4})$ $(?)$
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[^1]: This work has been partially supported by the Scientific Program of National Academy of Sciences of Ukraine, Project No 0107U002333.
|
---
author:
- 'M. Yuan'
- 'K. Biermann'
- 'S. Takada'
- 'C. Bäuerle'
- 'P. V. Santos'
title: 'Remotely pumped GHz anti-bunched emission from single exciton states'
---
Flying are particularly interesting for quantum communication since the photon coherence can be preserved over several kilometers [@Marcikic_N421_509_03]. Photons are thus ideal particles for the such as entanglement, non-localization and teleportation [@Bouwmeester_N390_575_97; @Bouwmeester_PL82_1345_99]. The technical challenges associated with the manipulation of photonic states are, however, formidable due to the difficulty of bringing two photons into interaction within a short distance. Opto-electronic excitations in the solid state are, in contrast, much easier to manipulate. Here, are flying opto-electronic qubits, which can be used for the exchange of quantum information between remote sites [@Bauerle_RPP81_56503_18]. Recently, flying qubits based on hybrid surface acoustic wave (SAW) structures on semiconductor platforms are attracting increasingly attention [@PVS326]. One prominent advantage of SAWs is the ability to provide mobile strain and piezoelectric potentials to modulate, confine, and transfer particles between remote locations. Researchers have coupled SAWs to a variety of systems, including, for example, electrons [@Hermelin_N477_435_11; @McNeil_N477_439_11; @PVS321], superconducting qubits [@Gustafsson_S346_207_14; @Satzinger_N563_661_18], diamond NV centers [@Golter_PRX6_41060_16], and excitons [@PVS177]. SAWs of $\mu$m-sized wavelengths have also been used to populate two-level systems with charged carriers as well as to induce the emission of anti-bunched photons at frequencies [@PVS218; @PVS246; @Hsiao_NC11_917_20].
{width="85.00000%"}
Flying excitons with their natural inter-conversion to photons offer several advantages for opto-electronic control as well as for interfacing electronic and photonic excitations. Especially suitable for these applications are the long-living spatially indirect (or dipolar) excitons (IXs) in a double quantum well (DQW) structure subjected to a transverse electric field $F_z$ \[cf. Fig \[IXFigSetup\](a)\]. These excitons are formed by the Coulomb binding of electrons and holes driven to different quantum wells (QWs) by the applied field, which controls both the lifetime and the emission energy of the IXs via the quantum confined Stark effect. Analogously to the piezoelectric transport of charged particles [@PVS156], the charge-neutral IXs can be confined and transported by the mobile band-gap modulation produced by the SAW strain field, as illustrated in Figs. \[IXFigSetup\](b) and \[IXFigSetup\](c). The long-range transport of IXs enabled by their long lifetime has so far only been demonstrated in wide transport channels using SAWs with wavelengths of a few $\mu$m [@PVS177; @PVS260; @PVS266].
A main challenge for the implementation of flying excitonic qubits is the creation of two-level excitonic states interconnected by a transport channel, which can store single particles and convert them to photons. In this work, we realize a major step towards this goal by demonstrating the manipulation and remote pumping of two-level excitonic states by flying IXs propelled by GHz-SAWs in a GaAs-based semiconductor platform. The single states used here consist of excitons bound to single shallow impurities (denoted as D$_\mathrm{B}$) in a DQW structure. We have recently reported that these states can be spectrally isolated and resonantly excited by appropriately biasing the DQW structure [@PVS314]. We demonstrate the pumping of individual D$_\mathrm{B}$ centers by IXs driven along a narrow transport channel by a SAW. The oscillating SAW strain field modulates the narrow emission lines of the D$_\mathrm{B}$ centers, which can be used as a sensitive probe of the local strain amplitudes. Time-resolved spectroscopic studies shows that the recombination lifetime of the D$_\mathrm{B}$ states is sufficiently short to follow the 3.5 GHz SAW pumping rate. More importantly, photon correlation investigations reveal that the acoustic pumping of these centers the emission of anti-bunched photons with a repetition rate corresponding to the SAW frequency, which shows that the center acts as a single photon source operating at very high frequencies.
Results {#results .unnumbered}
=======
Exciton energy modulation by SAWs {#exciton-energy-modulation-by-saws .unnumbered}
---------------------------------
The ${\mathrm{D_B}}$ center can be resonantly activated under weak optical excitation by biasing the DQW structure with a voltage $V_t$ close to the onset of IX formation [@PVS314]. Under these conditions, the photoexcited electron-hole pairs bind to free residual carriers to form trions (Ts). The conversion of trions to IXs via electron tunneling through the DQW barrier requires the excitation of a free electron to the band states. As illustrated in Fig. \[IXFigSetup\](a) the tunneling to single D$_{\rm B}$ becomes energetically favorable to the IX formation. The emission from these centers is characterized by a narrow line \[with full-width-at-half-maximum (FWHM) of 0.25 meV, cf. lowest spectrum in Figure \[IXFigFWHM\](a)\] spectrally isolated from the IX and direct exciton (DX, excitons whose electron and hole reside in the same QW) transitions.
{width="90.00000%"}
The spectroscopic studies were carried out using the setup of Figs. \[IXFigSetup\](b) and \[IXFigSetup\](c) using SAWs with a wavelength of ${{\lambda_{\mathrm{SAW}}}}=800$ nm (corresponding to a frequency of ${f_\mathrm{SAW}}=3.58$ GHz). The photoluminescence (PL) spectra of Fig. \[IXFigFWHM\](a) show that under an increasing SAW field the ${\mathrm{D_B}}$ line initially broadens and eventually splits into two. The SAW strain field periodically modulates the excitonic transition energies $E_\mathrm{C}(t)$ (C = DX, T, ${\mathrm{D_B}}$) according to:[@PVS107]
$$E_\mathrm{C}(t)=E_\mathrm{C,0}+\frac{\Delta E_\mathrm{C}}{2}\sin\left(\frac{2\pi}{{T_\mathrm{SAW}}}t\right) ,
\label{EqEc}$$
where $\Delta E_\mathrm{C}$ is the peak-to-peak modulation amplitude and ${T_\mathrm{SAW}}=1/{f_\mathrm{SAW}}$ the SAW period.
For energy shifts $\Delta E_\mathrm{C}$ smaller than the linewidth, the modulation manifests itself as an apparent broadening of the time-integrated PL lines. For larger modulation amplitudes, the time-averaged PL develops a camel-like shape with peaks at energies $E_\mathrm{C,0}\pm \Delta E_\mathrm{C}/2$ corresponding to the maximum and minimum band-gaps under the SAW field, thus reproducing the behavior observed in Fig. \[IXFigFWHM\](a). Figure \[IXFigFWHM\](b) displays the dependence of the peak-to-peak modulation amplitude for the ($\Delta E_\mathrm{B}$) and DX ($\Delta E_\mathrm{DX}$) transitions determined from fits of the measured spectra to a model for the time-integrated PL line shape under a SAW described in detail in Sec. \[SM\_MOD\]. The dashed line yields the corresponding strain-induced band-gap modulation determined using the GaAs deformation potentials and the SAW fields in the DQW calculated from the applied rf-power (cf. Sec. \[SM1\]). As expected for a shallow center, the D$_{\rm B}$ energy modulation amplitude follows closely the one for the DX states, which increases proportionally to the SAW amplitude (and, thus, to $\sqrt{{P_\ell}}$, ${P_\ell}$ being the linear power density of the SAW). It is worthwhile to emphasize that narrow D$_{\rm B}$ linewidths of the bound exciton states enables the quantitative determination of very small strain levels.
Long-range IX transport {#SM_TRANSPORT .unnumbered}
-----------------------
The lower panel of Fig. \[IXFigTrans\](a) displays a spectrally resolved PL map of the exciton distribution under the electrostatic stripe gates under optical excitation by a focused laser spot at $y=0$ \[cf. sketch of Fig. \[IXFigTrans\](e)\]. This map was recorded in the absence of a SAW The PL around the excitation spot (thin blue line in the upper panel, integrated for $|y|<3~\mu$m) shows the characteristic emission line from DX, T, and IXs superimposed on a broad PL background from the doped layers and emission centers in the substrate (note that the intensity of the DX and T lines become strongly suppressed under the applied transverse field). Away from the generation area the PL becomes dominated by the emission from IXs, which, due to the long recombination lifetime, can diffuse up to the top region of the guard gate (thick orange line). Note also that the diffusing IXs can easily cross the narrow gap between the stripe and guard gate at $y=13~\mu$m.
{width="99.00000%"}
![[**Emission dynamics and photon autocorrelation from single ${\mathrm{D_B}}$.**]{} (a) Time-resolved laser (blue) and PL emission from a ${\mathrm{D_B}}$ states remotely excited a ${P_\mathrm{rf}}=-2$ dBm SAW (red). The traces were collected by placing the excitation spot $8~\mu$m away from the ${\mathrm{D_B}}$ center. ${T_\mathrm{SAW}}$ and $T_\mathrm{laser}$ denote the SAW period and laser repetition delay, respectively. The inset show a close up of the signal with a time constan t of 110 ps (red line). (b) Second-order photon autocorrelation $g^{(2)}(\tau_c)$ for the ${\mathrm{D_B}}$ state under acoustic excitation (${P_\mathrm{rf}}=-3$ dBm). The excitation spot was $1~\mu$m away. The dashed envelopes are guides to the eye. The inset shown the averaged around the delays corresponding to multiples of the laser period.[]{data-label="IXFigTR"}](Figure4){width=".95\columnwidth"}
Figure \[IXFigTrans\](b) displays a PL map recorded under the same conditions as in Fig. \[IXFigTrans\](a), but now under a SAW propagating along the $y$ direction. The acoustic field pushes the IXs upwards leading to a strong increase of the IX PL for positive $y$ (cf. orange line in the upper panel) and a reduction for negative $y$. The recombination energy and location of the transported IXs can be controlled by changing the bias applied to the gates. As an example, Fig. \[IXFigTrans\](c) shows a map recorded by increasing the guard bias by $0.08$ V relative to $V_t$. The IX emission energy blueshifts as IXs enter the guard gate as well as when the particles are pushed by the SAW beyond the guard, where they become converted to DXs or trions. The additional energy for the blueshift is provided by the moving SAW field.
The IX transport over tens of ${{\lambda_{\mathrm{SAW}}}}$ can remotely activate ${\mathrm{D_B}}$, as shown in Fig. \[IXFigTrans\](d). Here, the SAW amplitude and gate biasing conditions were selected to enhance the emission of a ${\mathrm{D_B}}$ center under the guard gate approx. 13 $\mu$m (corresponding to 16${{\lambda_{\mathrm{SAW}}}}$) away from the excitation spot (dashed circle).
Photoluminescence dynamics and autocorrelation {#photoluminescence-dynamics-and-autocorrelation .unnumbered}
----------------------------------------------
The excitation of the ${\mathrm{D_B}}$ centers by GHz SAW fields induces a strong time modulation of their optical emission. The red line in Fig. \[IXFigTR\](a) displays the time-resolved PL trace of a center recorded on a ${\mathrm{D_B}}$ center located about $\ell\sim8$ $\mu$m away from the laser excitation spot. The blue curve reproduces, for comparison, a time-resolved profile of the exciting laser spot (not to scale), which consist of pulses with a FWHM of about $0.28$ ns and a repetition time of 9 ns. The short-period oscillations in the ${\mathrm{D_B}}$ response (red curve) correspond to the SAW period ${T_\mathrm{SAW}}=0.28$ ns. A close up of the oscillations (upper inset) reveals that the D$_\mathrm{B}$ emission decays with a time constant of approx. 110 ps, thus demonstrating that the PL from these centers can follow the fast varying acoustic field. The pulses create a high density cloud of IX, which partially screen the modulation potential around the excitation spot. As a consequence, these oscillations persist over times longer than the laser repetition period. The envelope of the oscillations features two maxima: the first appears immediately after the laser pulse and the second at a delay of approximately $\tau_{\ell}\sim2.6$ ns . The first maximum is attributed to the drift of hot excitons propelled by repulsive exciton interaction within the high-density cloud around the excitation spot [@PVS279]. The second maximum is assigned to carriers transported by the SAW, which reach the D$_\mathrm{B}$ after a delay $\tau_{\ell}=\ell/v_\mathrm{SAW}$, where ${v_\mathrm{SAW}}=2960$ m/s is the SAW velocity.
The photon emission statistics of the $ {\mathrm{D_B}}$ centers was addressed by recording photon autocorrelation ($ g^{2} $) histograms under acoustic excitation using a Hanbury-Brown and Twiss setup. The results obtained after correcting the background emission from IX states (see details of the data treatment in Sec. \[SM3\]) are shown in Fig. \[IXFigTR\](b). As in Fig. \[IXFigTR\](a) the short and long period oscillations are associated with the repetition periods of the SAW and the laser pulses, respectively. The histogram shows a clear suppression of the coincidence rate at the time delay $\tau_c=0$. In order to confirm the selective suppression at $\tau_c=0$, the inset displays the averaged value of the laser repetition period. The latter shows that the suppression of coincidences at $\tau_c=0$ is well below the statistical fluctuations, thus proving the emission of anti-bunched photons. The autocorrelation $ g^{2}(0)=0.75\pm 0.03 $, which corresponds to the simultaneous emission of photons, is probably an upper limit determined by photon collection from the neighboring areas from the center.
Discussion {#discussion .unnumbered}
==========
In conclusion, we have investigated the dynamic modulation and transport of excitons by high-frequency, sub-micron-wavelength SAWs on GaAs DQW structures. show that GHz-SAW field can pump single exciton states bound to impurities, which act as two level states emitting anti-bunched photons.
Methods {#methods .unnumbered}
=======
[ **Sample structure:** ]{} The studies were carried out an (Al,Ga)As DQW structure consisting of two coupled GaAs QWs grown by molecular beam epitaxy on a n-type doped GaAs(001) substrate \[cf. Fig. \[IXFigSetup\](a)\]. The QWs are $16$ nm-wide and separated by a 4 nm-thick Al$_{0.33}$Ga$_{0.67}$As barrier. The electric field $F_z$ induced by the bias $V_{\rm t}$ applied across the structure drives photoexcited electrons into QW$_2$ and holes into QW$_1$, thus increasing the recombination lifetime. Due to the narrow barrier width, the overlap of the electron and hole wavefunctions in the adjacent QWs is still sufficiently strong to maintain the Coulomb correlation required for IX formation. The DQW structure can thus hold both direct (or intra-QW, DX) and indirect (inter QW) excitons with transition energies indicated by the and arrows in Fig. \[IXFigSetup\](a), respectively.
[**Generation of SAWs:**]{} SAWs with a wavelength of $\lambda_{\rm SAW}=800$ nm (corresponding to a SAW frequency ${f_\mathrm{SAW}}=3.58$ GHz at 4 K) were generated by split-finger aluminum interdigital transducers (IDTs) deposited on the sample surface \[cf. Fig. \[IXFigSetup\](b)\]. The depth of the DQW was chosen to yield a type II modulation under the SAW excitation [@PVS177]. The IDTs are oriented along a $\langle110\rangle$ surface direction with a length and width of 150 $\mu$m and 28 $\mu$m, respectively. The SAW intensity is quantified in terms of either the nominal radio-frequency (rf) power applied to the IDT (${P_\mathrm{rf}}$) or the SAW linear power density ${P_\ell}$, which is defined as the ratio between the acoustic power and the width of the SAW beam. The latter is obtained by using the measured rf-scattering parameters of the IDTs to determine the fraction of the input rf-power coupled to the acoustic mode.
[**Electrostatic channels for IX transport:**]{} The IX acoustic transport channel is defined by a semi-transparent Ti stripe placed on the SAW path and biased with a voltage $V_t$ with respect to the doped substrate (cf. cross-section diagram of Fig. \[IXFigSetup\]). The stripe is 2 $\mu$m wide and ends on a small trap (diameter of 0.9 $\mu$m) surrounded by a guard gate with an external diameter of 7.5 $\mu$m. The guard gate, which is biased by a separate voltage $V_g$, reduces lateral stray electric fields in the narrow regions of the stripe, which can dissociate IXs [@Schinner_PRB83_165308_11]. As shown in the experiments, IX can easily tunnel over the small separation region (approx. $0.2~\mu$m) between the stripe and guard gate.
[**Optical spectroscopy:**]{} Optically detected IX transport experiments were carried out at 4 K with a spatial resolution of approx. $1~\mu$m. The excitons were excited by a spot from a pulsed laser (wavelength of 770 nm, pulse width of 280 ps) focused by a microscope objective on the semitransparent stripe. The photoluminescence (PL) from IXs emitted along the transport path is collected by the same objective and spectrally analysed by a monochromator with a charge-coupled-device (CCD) detector. Spatially and spectrally resolved PL maps of the IX distribution are obtained by aligning the transport path with the input slit of the spectrometer. The time-resolved PL studies were performed by triggering the laser pulses at a subharmonic (${f_\mathrm{SAW}}/32$) of the rf-frequency applied to the IDTs. The PL was in this case spectrally filtered by a band-pass filter and detected by a superconducting photon detector coupled to a time correlator with a combined time resolution of 40 ps.
[**Authors contributions:** ]{} M.Y., P.V.S., and C.B. have conceived this project. M.Y. and P.V.S. carried out the optical and electrical measurements and analyzed the data. K.B. designed and fabricated the layer structures using molecular beam epitaxy. M.Y. and S. T. have fabricated the acoustic devices. M.Y. and P.V.S. have equally contributed to the analysis of the results as well as to the preparation of the manuscript.
[**Acknowledgements:** ]{} We thank M. Lopes and A. Hern[á]{}ndez-M[í]{}nguez for helpful discussions and suggestions as well as S. Meister, S. Rauwerdink and A. Tahraoui for the expertise in sample fabrication. S.T. and C.B. acknowledge financial support from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 654603 and JSPS KAKENHI Grant Number JP18K14082. M.Y., C.B. and P.V.S. acknowledge financial support from the French National Agency (ANR) and Deutsche Forschungsgesellschaft (DFG) in the frame of the the International Project on Collaborative Research SingleEIX Project No. ANR-15-CE24/DFG SA-598-12/1.
[***Competing interests:*** ]{} Authors declare no competing interests.
[22]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](https://doi.org/10.1038/nature01376) [****, ()](https://doi.org/https://doi.org/10.1038/37539) [****, ()](https://doi.org/10.1103/PhysRevLett.82.1345) [****, ()](https://doi.org/10.1088/1361-6633/aaa98a) [****, ()](https://doi.org/10.1088/1361-6463/ab1b04) [****, ()](http://dx.doi.org/10.1038/nature10416) [****, ()](http://dx.doi.org/10.1038/nature10444) [****, ()](https://doi.org/10.1038/s41467-019-12514-w) [****, ()](https://doi.org/10.1126/science.1257219) [****, ()](https://doi.org/10.1038/s41586-018-0719-5) [****, ()](https://doi.org/10.1103/PhysRevX.6.041060) [****, ()](https://doi.org/10.1103/PhysRevLett.99.047602) [****, ()](https://doi.org/10.1038/NPHOTON.2009.191) [****, ()](https://doi.org/10.1021/nl203461m), [****, ()](https://doi.org/10.1038/s41467-020-14560-1) [****, ()](https://doi.org/10.1088/0034-4885/68/7/r02) [****, ()](https://doi.org/10.1103/PhysRevB.89.085313) [****, ()](https://doi.org/10.1088/1367-2630/16/3/033035) [****, ()](https://arxiv.org/abs/1807.10102) [****, ()](https://doi.org/10.1103/PhysRevB.63.121307) [****, ()](https://doi.org/10.1103/PhysRevB.91.125302) ****, [10.1103/PhysRevB.83.165308](https://doi.org/10.1103/PhysRevB.83.165308) ()
Supplementary Material {#supplementary-material .unnumbered}
======================
[**Remotely activated GHz anti-bunched photon source from single exciton states**]{}
M. Yuan$^1$, K. Biermann$^1$, S. Takada$^{2,3}$, C. Bäuerle$^2$, and P. V. Santos$^1$
$^{1}$[Paul-Drude-Institut f[ü]{}r Festk[ö]{}rperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V., Hausvogteiplatz 5-7, 10117 Berlin, Germany]{}
$^{2}$[Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N[é]{}el, 38000 Grenoble, France]{}
$^{3}$[National Institute of Advanced Industrial Science and Technology (AIST), National Metrology Institute of Japan (NMIJ), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan]{}
Calculation of the SAW-induced energy modulation {#SM1}
================================================
We calculate in this section the bandgap modulation induced by the SAW modulation. For that purpose, we compare the measured electrical response of the IDTs with numerical calculation of the SAW fields carried out by solving the elasticity equations for the layer structure of the sample [@PVS156]. Fig. \[s11\] displays the rf scattering coefficient $S_{11}$ measured for the IDT, which yields the fraction of the applied rf power reflected by the transducers. From the amplitude of the dip at the resonance frequency ${f_\mathrm{SAW}}=3.54$ GHz, we extract that for a typical applied power of 30 W/m, the transmitted SAW power propagating along one direction of the IDT is ${P_\ell}=0.4$ W/m. The SAW strain modulates the energy of both the conduction, $E_e$ and the valence bands (heavy hole), $E_{hh}$. The calculated combined bandgap modulation $E_e-E_{hh}$ at the depth of the DQW is shown in Fig. \[sawmod\](a). We plot in Fig. \[sawmod\](b) the calculated piezoelectric field $E_p$ in the DQW plane
![Radio-frequency scattering parameter $S_{11}$ (corresponding to the electric reflection coefficient) measured at room temperature for the IDT used to generate the SAWs in the experiments.[]{data-label="s11"}](FigureSM1.pdf){width="50.00000%"}
![(a) Calculated bandgap strain-induced modulation $E_\mathrm{e}-E_\mathrm{hh}$ at the depth of the DQW calculated for ${P_\ell}=0.4$ W/m. (b) The corresponding piezoelectric field $E_p$. []{data-label="sawmod"}](FigureSM2.pdf){width="50.00000%"}
Acoustic modulation of the transition energies {#SM_MOD}
==============================================
{width="50.00000%"}
In addition to the studies of the dependency of the bound exciton linewidths on acoustic intensity, we also investigated how the acoustic fields impact the DX and trion lines. Figure \[free\_width\_P\](a) displays PL spectra recorded under flat-band conditions for increasing SAW intensities (quantified by the nominal power ${P_\mathrm{rf}}$ applied to the IDT). Each spectrum displays two lines associated with the excitation of DXs and trions (T, a DX bound to a free carrier, studied in Ref [@PVS314]). The spectrum for the lowest ${P_\mathrm{rf}}$ essentially corresponds to the PL response in the absence of acoustic excitation. With increasing SAW intensity, both lines slightly broaden and the overall emission intensity decreases.
In order to extract the effects of the acoustic field, we assume that PL lines have a Gaussian shape with width $w$ and that their central energy is modulated by the SAW according to Eq. \[EqEc\] of the main text. Under these assumptions, the time-integrated PL spectrum can be expressed by the following integral over one SAW period
$$I_C(E)= \frac{1}{{T_\mathrm{SAW}}}\frac{I_{C,0}}{w\sqrt{\pi/2}}\int_0^{{T_\mathrm{SAW}}}\exp\left[-2\left(\frac{E-E_C(t)}{w}\right)^2\right]dt.
\label{EqIC}$$
For energy shifts $\Delta E_C<<w$, the modulation manifests itself as a broadening of the lines with increasing SAW amplitude. For high modulation amplitudes, the time-averaged PL line splits into two peaks with energies $E_{C,0}\pm \Delta E_C/2$ corresponding to the maximum and minimum band-gaps under the SAW field.
The lines superimposed on Fig. \[free\_width\_P\](a) are fits of the measured PL data to Eq. [\[EqIC\]]{}. From the fit we extract the SAW modulation amplitudes $\Delta E_C$ for DX and trion, plotted as symbols in Fig. \[free\_width\_P\](b) for different SAW amplitudes. The latter is quantized in terms of the SAW linear power density ${P_\ell}$, defined as the acoustic power carried by the SAW mode per unit length perpendicular to the SAW beam. The expected linear dependence on $\sqrt{{P_\ell}}$ is revealed when the SAW-induced apparent broadening exceeds the unperturbed (i.e., in the absence of a SAW) width of the PL line, as shown by the dashed lines. For ${P_\ell}^{1/2}>0.7$ (W/m)$^{1/2}$ one observes a nonlinear increase of $\Delta E_C$ with SAW amplitude.
The modulation amplitude determined from the fits correlates well with the band-gap modulation expected from the deformation potential mechanism shown in Fig. \[SM1\]. As an example, the circled red cross in Fig. \[free\_width\_P\]b marks $\Delta E_C$ for ${P_\ell}=0.4$ W/m extracted from the measured data, which is in good agreement with the analysis shown in Fig. \[sawmod\](a). It indicates that the simple model of adding the respective modulation of the conduction band and the valance band, used for the calculation here, is not suitable for trion due to the binding to an extra charge.
Background correction of the autocorrelation data {#SM3}
=================================================
Due to their long lifetimes, IX can diffuse and accumulate around the ${\mathrm{D_B}}$ centers. As a result, IX emission is also present within the spectral window of the autocorrelation measurements. The repopulation time of the IX around the ${\mathrm{D_B}}$ center is very long: as a consequence, the SAW modulation of IX intensity in the measured $g^{(2)}$ can be ignored. To correct for the background emission from IXs, we diverted the detection spot away from the ${\mathrm{D_B}}$ center to a position, where the PL from the center disappeared but the IX signal remained similar. A background autocorrelation $[C_{\rm bgd}]^2$ was recorded at this position. The measured total autocorrelation $[C_{\rm total}]^2$ can be expressed as $[C_{\rm bgd}+C_{\rm center}]^2$, where $[C_{\rm center}]^2$ is the autocorrelation from the ${\mathrm{D_B}}$ center that is sought after. From the measurements we extract $[C_{\rm bgd}]^2+2C_{\rm bgd}C_{\rm center}\approx0.85[C_{\rm total}]^2$. We fit the measured $g^{(2)}$ to a smooth (i.e., without SAW induced oscillations) multi-peak profile (Lorentzian), each peak sharing the same height and width, and use this profile to approximate $[C_{\rm total}]^2$. Finally $0.85[C_{\rm total}]^2$ is subtracted from the measured $g^{(2)}$.
|
---
abstract: 'We present a fully microscopic approach to the transition rate of two exciton-photon polaritons. The non-trivial consequences of the polariton composite nature — here treated exactly through a development of our composite-exciton many-body theory — lead to results noticeably different from the ones of the conventional approaches in which polaritons are mapped into elementary bosons. Our work reveals an appealing fundamental scattering which corresponds to a photon-assisted exchange — in the absence of Coulomb process. This scattering being dominant when one of the scattered polaritons has a strong photon character, it should be directly accessible to experiment. In the case of microcavity polaritons, it produces a significant enhancement of the polariton transition rate when compared to the one coming from Coulomb interaction. This paper also contains the crucial tools to securely tackle the many-body physics of polaritons, in particular towards its possible BEC.'
author:
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$^1$\
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$^2$\
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title: |
**Polariton-polariton scattering :\
exact results through a novel approach**
---
PACS :
71.36.+c Polaritons
71.35.-y Excitons and related phenomena
71.35.Lk Collective effects (Bose effects, phase space filling, and excitonic phase transitions)
Almost 50 years ago, J.J. Hopfield \[1\] has shown that photons can be dressed by semiconductor excitations to form a mixed state, the polariton. As this mixed state is the exact eigenstate of the coupled photon-semiconductor Hamiltonian, semiconductor should not absorb any photon. Actually, photon absorption is observed when the exciton broadening is large compared to the exciton-photon coupling. The absorption then follows the Fermi golden rule in spite of the fact that the transition is not towards a continuum, but a discrete state, namely, the exciton with a center-of-mass momentum equal to the photon momentum. This can be physically understood by saying that the exciton broadening acts as a continuum around the exciton discrete level \[2\].
In the most basic approach to polaritons \[3,4\], the excitons are considered as non-interacting elementary bosons: The polariton effect then is a bare one-photon one-exciton effect, independent from the laser intensity. This has to be contrasted with the exciton optical Stark effect discovered 25 years ago by D. Hulin and coworkers \[5,6\], in which the observed exciton line shift, proportional to the laser intensity, only comes from excitons differing from elementary bosons. This apparent contradiction in problems quite similar, namely, photons interacting with a semiconductor, has been tackled by one of us \[7\] : Through the derivation of the polariton effect and the exciton optical Stark effect within the same framework, it is possible to show that the polariton picture with excitons as non-interacting elementary bosons, is valid at low laser intensity, while the composite nature of the excitons becomes crucial when the laser intensity gets large.
We have recently constructed a many-body theory for excitons in which the composite nature of the particles is treated exactly. It has, up to now, been developed through more than 30 publications. This leads us to consider that the reader now has some knowledge of this many-body theory, at least through the short review paper \[8\] or, for more details, through the appendices of ref. \[9\]. The most important result of this theory is the fact that excitons interact not only through Coulomb scatterings for interactions between their carriers, but also through “Pauli scatterings” for *carrier exchanges in the absence of Coulomb process*. It turns out that these dimensionless pure carrier exchanges — which are by construction missed in effective Hamiltonians for boson excitons, due to a bare dimensional argument, — dominate *all* semiconductor optical nonlinearities. Among these nonlinear effects, we can cite the exciton optical Stark shift \[6\] extensively studied in the 80’s. We can also cite the Faraday rotation in photoexcited semiconductors that we have recently suggested \[10\]. More generally, through a novel approach to the nonlinear susceptibility \[11\], we have proved that the large detuning behavior of $\chi^{(3)}$ is *entirely* controlled by these pure carrier exchanges between composite excitons. Processes in which Coulomb interactions take place only enter as detuning corrections.
An “ideal” polariton made of non-interacting boson excitons and (non-interacting boson) photons, would behave as a non-interacting boson. In order to explain the observed scatterings between polaritons, it is necessary to take into account the fact that excitons do interact. To possibly treat their interactions through available many-body procedures — valid for elementary quantum particles only \[12,13\], — excitons are commonly considered as elementary bosons, their composite nature being hidden through the Coulomb exchange term of the exciton-exciton scattering in the exciton effective Hamiltonian generated by the “bosonization” procedure. In previous works, we have shown that not only the effective exciton-exciton scattering up to now used \[14\] should have been rejected long ago because it induces an unphysical non hermiticity in the effective exciton Hamiltonian \[15\], but, worse, there is no way to properly treat the interactions between excitons through an effective potential between elementary bosons, whatever the exciton-exciton scatterings are \[9,16\]. All our works on exciton many-body effects end with the same conclusion : It is not possible to forget the composite nature of the excitons by reducing them to elementary bosons, whatever the bosonization procedure is.
The present work on polariton-polariton scatterings once more supports this conclusion. Our composite-exciton many-body theory reveals the existence of an appealing “photon-assisted exchange” channel in the scattering of two polaritons which directly comes from the exciton composite nature (see fig.1(a)) : In this channel, the excitonic parts $i_1$ and $i_2$ of the polaritons $P_1$ and $P_2$ exchange their carriers without any Coulomb process. In a second step, one of the resulting excitons, let us say $i'_1$, transforms into the photon part $n'_1$ of the polariton $P'_1$ through the vacuum Rabi coupling, so that this photon-assisted exchange scattering also is an energy-like quantity in spite of the absence of Coulomb process. When compared to the two other channels of the standard approach to polariton-polariton scattering, associated to direct and exchange Coulomb interaction between the excitonic parts of the polaritons (see figs.1(b,c))), this photon-assisted exchange scattering is obviously going to be dominant when one of the four polaritons has a strong photon character, the photon-polariton overlap being then larger than the exciton-polariton overlap. This makes it directly accessible to experiments.
**Microscopic formalism**
The Hamiltonian of a semiconductor coupled to a photon field splits as $H=H_{\mathrm{ph}}+H_\mathrm{sc}+H_{\mathrm{ph-sc}}$. The photon part reads $H_\mathrm{ph}=\sum_n\omega_n
a_n^\dag a_n$ where $a_n^\dag$ creates a photon in a mode $n$. The semiconductor part, $H_\mathrm{sc}=H_e+H_h+V_{ee}+V_{hh}+V_{eh}$, contains kinetic and Coulomb contributions for *free* carriers. Its one-electron-hole-pair eigenstates, $(H_\mathrm{sc}-E_i)B_i^\dag|v\rangle=0$, are the excitons, their bound and extended states forming a complete basis for one-pair states. For photons close to the exciton resonance, the photon-semiconductor coupling, $H_\mathrm{ph-sc}=W+W^\dag$, can be reduced to its resonant terms, $W=\sum_{n,i}\Omega_{n i}\,\alpha_n^\dag B_i$, where $\Omega_{n i}$ is the vacuum Rabi coupling between the $n^\mathrm{th}$ photon mode and the $i^\mathrm{th}$ exciton: $W$ creates a photon while destroying an exciton.
Polaritons $C_P^\dag|v\rangle$ are the $H$ eigenstates in the subspace made of one photon coupled to one exciton, $(H-\mathcal{E}_P)C_P^\dag|v\rangle=0$. They thus form a complete orthogonal basis for this subspace. Consequently, we can write photons and excitons in terms of polaritons as $$a_n^\dag=\sum_PC_P^\dag\,\alpha_{Pn}\ ,\ \ \ \
B_i^\dag=\sum_PC_P^\dag\,\beta_{Pi}\ ,$$ while polaritons read in terms of photons and excitons as $$C_P^\dag=\sum_n
a_n^\dag\,\alpha_{nP}+\sum_iB_i^\dag\,\beta_{iP}\ .$$ The prefactors in these expansions are nothing but the Hopfield coefficients, *i.e.*, the photon-polariton overlap $\alpha_{nP}=\langle v|a_nC_P^\dag|v\rangle=\alpha_{Pn}^\ast$ and the exciton-polariton overlap $\beta_{iP}=\langle v|B_iC_P^\dag|v\rangle=\beta_{Pi}^\ast$. They can be made large or small depending on the polariton character.
**Elementary scatterings between polaritons**
Due to their excitonic components, the polaritons are not exact bosons. This is readily seen from $[C_{P'},C_P^\dag]=\delta_{P',P}-\tilde{D}_{P'P}$, where $\tilde{D}_{P'P}$ is the “polariton deviation-from-boson operator”. Using eqs. (1,2), $\tilde{D}_{P'P}$ is just the exciton deviation-from-boson operator $D_{i'i}$ of the composite-exciton many-body theory (see eq.(1.3) of ref. \[9\]), dressed by photons through the exciton-polariton overlaps, namely, $\tilde{D}_{P'P}=\sum_{i',i}\beta_{P'i'}\,\,D_{i'i}
\,\beta_{iP}$. The Pauli scatterings of two polaritons $\tilde{\lambda}$, resulting from fermion exchanges as shown in fig.1(d), appear through $$\left[\tilde{D}_{P'_1P_1},C_{P_2}^\dag\right]=\sum_{P'_2}C_{P'_2}^\dag
\left\{\tilde{\lambda}\left(^{P'_2\ P_2}_{P'_1\
P_1}\right)+(P_1\leftrightarrow P_2)\right\}\ .$$ These scatterings are just the exciton Pauli scatterings $\lambda$ of the composite-exciton many-body theory (see eq. (1.2) of ref.), dressed by photons $$\label{eq:lambdaScattPol}
\tilde{\lambda}\left(^{P'_2\ P_2}_{P'_1\
P_1}\right)=\sum_{i'_1,i'_2,i_1,i_2}\beta_{P'_1i'_1}\beta_{P'_2i'_2}
\,\lambda\left(^{i'_2\ i_2}_{i'_1\ i_1}\right)\,
\beta_{i_2P_2}\beta_{i_1P_1}\ .$$
As for excitons \[9\], it is not possible to describe the interactions between polaritons through a potential, due to the composite nature of the excitonic part of these polaritons. The clean way to overcome this difficulty is to introduce the “creation-potential” of the polariton $P$, defined as $$[H,C_P^\dag]-\mathcal{E}_PC_P^\dag=\tilde{V}_P^\dag-X_P^\dag\ .$$ Its first part $\tilde{V}_P^\dag$ barely is the exciton creation-potential $V_i^\dag$ coming from Coulomb interaction between excitons (see eq. (1.7) of ref. \[9\]), dressed by photons, $\tilde{V}_P^\dag=\sum_iV_i^\dag\,\beta_{iP}$. The second part $X_P^\dag$ is conceptually new. It comes from the composite nature of the excitons, through the exciton deviation-from-boson operator $D_{i'i}$ $$X_P^\dag=\sum_{n,i',i}a_{n}^\dag\,\Omega_{ni'}\,D_{i'i}\,\beta_{iP}
\ .$$
These two creation-potentials give rise to two physically different scatterings. The ones associated to $\tilde{V}_P^\dag$, $$\left[\tilde{V}_{P_1}^\dag,C_{P_2}^\dag\right]=\sum_{P'_1,P'_2}
C_{P'_1}^\dag C_{P'_2}^\dag\,\tilde{\xi}^\mathrm{dir}
\left(^{P'_2\ P_2}_{P'_1\ P_1}\right)\ ,$$ shown in Fig.1(b), are naïve. They just correspond to the exciton direct Coulomb scatterings $\xi^\mathrm{dir}$ (see eq. (1.8) in ref. \[9\]), dressed by photons as in eq. (4), $$\label{eq:directScattPol}
\tilde{\xi}^\mathrm{dir}\left(^{P'_2\ P_2}_{P'_1\
P_1}\right)=\sum_{i'_1,i'_2,i_1,i_2}\beta_{P'_1i'_1}\beta_{P'_2i'_2}
\,\xi^\mathrm{dir}\left(^{i'_2\ i_2}_{i'_1\ i_1}\right)\,
\beta_{i_2P_2}\beta_{i_1P_1}\ .$$
The ones associated to the second creation-potential $X_P^\dag$ are more interesting. They correspond to the photon-assisted exchange scatterings dexcribed in the introduction and shown in fig.1(a). Being defined through $$\left[X_{P_1}^\dag,C_{P_2}^\dag\right]=\sum_{P'_1,P'_2}C_{P'_1}^\dag
C_{P'_2}
^\dag\left\{\chi\left(^{P'_2\ P_2}_{P'_1\ P_1}\right)+(P_1\leftrightarrow
P_2)\right\}\ ,$$ their mathematical expression is $$\begin{aligned}
\chi\left(^{P'_2\ P_2}_{P'_1\ P_1}\right)=\sum_{
n'_1,i'_1,i'_2,i_1,i_2}\hspace{4cm}\nonumber\\
\beta_{P'_2i'_2}\alpha_{P'_1n'_1}\Omega_{n'_1i'_1}
\lambda\left(^{i'_2\ i_2}
_{i'_1\ i_1}\right)\,
\beta_{i_1P_1}\beta_{i_2P_2}\ .\end{aligned}$$ They read in terms of the exciton pure exchange Pauli scatterings $\lambda$, *without any Coulomb process*.
The two energy-like scatterings $\tilde{\xi}^\mathrm{dir}\left(^{P'_2\
P_2}_{P'_1\ P_1}\right)$, $\chi\left(^{P'_2\ P_2}_{P'_1\ P_1}\right)$ and the dimensionless Pauli scattering $\tilde{\lambda}
\left(^{P'_2\ P_2}_{P'_1\ P_1}\right)$ this procedure generates, constitute the crucial tools to, in the future, tackle any problem dealing with the many-body physics of polaritons, with the exciton composite nature included in an exact way. This is going to be of importance in view of the claimed recent observation of the polariton BEC \[17,18\].
**Polariton transition rate**
To get the polariton transition rate, we barely follow the procedure we have already used to get the exciton transition rate \[9,16\]. When $H$ does not split as $H_0+V$, the standard form of the Fermi golden rule cannot be used. The transition rate from a normalized state $|\phi_i\rangle$ to a normalized state $|\phi_f\rangle$, is then obtained through \[16\] $$\frac{t}{T_{i\rightarrow f}}=\left|\langle\phi_f|F_t(\hat{H})P_{\perp}H|
\phi_i\rangle\right|^2\ ,$$ with $P_{\perp}=1-|\phi_i\rangle\langle\phi_i|$ and $\hat{H}=H-\langle\phi_i|H|\phi_i\rangle$, while $|F_t(E)|^2=2\pi\,t\,\delta_t(E)$ where $\delta_t(E)=(\sin Et/2)/\pi E$ is a peaked function of width $2/t$.
We here consider the transition rate of two identical polaritons $P$ towards two polaritons $(P_1,P_2)$. To normalize the initial state $C_P^{\dag 2}|v\rangle$ and the scattered state $C_{P_1}^\dag C_{P_2}^\dag |v\rangle$, we use $$\begin{aligned}
\langle v|C_{P'_1}C_{P'_2}C_{P_2}^\dag C_{P_1}^\dag|v\rangle=
\delta_{P'_1,P_1}\,\delta_{P'_2,P_2}-\tilde{\lambda}
\left(^{P'_2\ P_2}_{P'_1\ P_1}\right)\nonumber\\
+\ (P_1\leftrightarrow P_2)\ ,\end{aligned}$$ which follows from eq. (3).
The transition rate from $(P,P)$ to $(P_1,P_2)$, given in eq.(11), makes use of $P_{\perp}HC_P^{\dag 2}|v\rangle$. To get it, we note that, due to eqs. (5,7,9), $|S_P\rangle=(H-2\mathcal{E}_P)C_P^{\dag
2}|v\rangle$ reads in terms of the two elementary scatterings of the polariton many-body theory as $$|S_P\rangle=
\sum_{P'_1,P'_2}\left\{\tilde{\xi}^\mathrm{dir}\left(^{P'_2\ P}
_{P'_1\ P}\right)-2\chi\left(^{P'_2\ P}_{P'_1\ P}\right)\right\}
C_{P'_1}^\dag
C_{P'_2}^\dag|v\rangle\ .$$ As $P_{\perp}C_P^{\dag 2}|v\rangle=0$, this makes $P_{\perp}HC_P^{\dag 2}|v\rangle$, equal to $P_{\perp}|S_P\rangle$, linear in polariton scatterings — its precise value being $[|S_P\rangle
-|S'_P\rangle]$ with $|S'_P\rangle=\xi_PC_P^{\dag 2}|v\rangle$, where $\xi_P=\langle v|C_P^2|S_P\rangle\langle v|C_P^2C_P^{\dag
2}|v\rangle^{-1}$. Consequently, to get the transition rate from $(P,P)$ to $(P_1,P_2)\neq (P,P)$ at lowest order in the interactions, we just have, in eq. (11), to replace $\hat{H}$ by its zero order contribution, i.e., $\hat{H}$ by $(\mathcal{E}_{P_1}+\mathcal{E}_{P_2}-2\mathcal{E}_P)$ in $\langle v|C_{P_2}C_{P_1}F_t(\hat{H})$. We then note that, for a large sample, the $\tilde{\lambda}$’s, as the exciton Pauli scatterings $\lambda$, are small compared to 1. This makes the $\tilde{\lambda}$’s in the normalization factors negligible as well as the contribution coming from $|S'_P\rangle$. This allows to show that the transition rate from polaritons $(P,P)$ to polaritons $(P_1,P_2)$ reduces to \[19\] $$\begin{aligned}
\frac{1}{T_{PP\rightarrow
P_1P_2}}\simeq 4\pi\,\delta_t(\mathcal{E}_{P_1}
+\mathcal{E}_{P_2}-2\mathcal{E}_P)\hspace{3cm}\nonumber\\ \times
\left|\tilde{\xi}^\mathrm{dir}\left(^{P_2\
P} _{P_1\ P}\right)-\tilde{\xi}^\mathrm{in}\left(^{P_2\ P} _{P_1\
P}\right)-\chi\left(^{P_2\ P}_{P_1\ P}\right) -\chi\left(^{P_1\
P}_{P_2\ P}\right)\right|^2\ .\end{aligned}$$ $\tilde{\xi}^\mathrm{in}$, the diagram of which is shown in fig.(1c), is the exciton “in” Coulomb exchange scattering $\xi^\mathrm{in}$ of the composite-exciton many-body theory (see eq.(1.10) in ref. \[9\]), dressed by photons as in eq. (8).
Equation (14) for the polariton transition rate is one of the key results of the paper. This transition rate conserves energy at the scale $1/t$, as expected. Its amplitude contains direct and “in” exchange Coulomb scatterings similar to the ones we found for the exciton transition rate \[9,16\]. However, it contains in addition a contribution from the photon-assisted exchange channel, independent from any Coulomb process. This physically appealing contribution is directly linked simultaneously to the exciton composite nature and to the partly photon nature of the polariton.
**State of the art**
In the most naïve approach to polariton-polariton scattering, one just replaces the excitons in the photon-semiconductor coupling $W$ by elementary bosons $\bar{B}_i$ with $[\bar{B}_i,\bar{B}_j^\dag]=\delta_{i,j}$ and the semiconductor Hamiltonian by an effective exciton-exciton Hamiltonian. If the effective scattering $\xi^\mathrm{dir}\left(^{n\
\,j}_{m\ i}\right)-
\xi^\mathrm{out}\left(^{n\ \,j}_{m\ i}\right)$, derived by Haug and Schmitt-Rink \[14\], were used, the bracket in the polariton transition rate (14) would be $\tilde{\xi}^\mathrm{dir}\left(^{P'_2\ P}_{P'_1\
P}\right)-\tilde{\xi}^\mathrm{out}\left(^{P'_2\ P}_{P'_1\ P}\right)$. Besides the fact that $\xi^\mathrm{in}\neq \xi ^\mathrm{out}$ (for a complete discussion, see ref. \[9\]), this procedure totally misses the photon-assisted exchange scattering $\chi$ which is dominant when one of the two scattered polaritons has a strong photon character.
A more elaborate approach, which relies on a truncated Usui’s bosonization procedure, has been proposed by the Quattropani’s group \[20,21\]. Their Coulomb contribution is now correct (the prefactors of eq. (18) in ref. \[20\], given in their eqs.(19,20), being nothing but $\xi^\mathrm{dir}-\xi^\mathrm{in}$). They also find a second contribution called therein “anharmonic saturation term” (see eq. (15) in ref. \[20\] or the third equation of ref. \[21\]). Due to the $a^\dag
\bar{B}^\dag \bar{B}\bar{B}$ structure of the interaction from which it appears, we could think it to be the photon-assisted exchange channel we find. However, when considered carefully, the physics it bares is at odd. This can be seen seen from the prefactor $Y$ of the interaction term, eq. (15) of ref. \[20\], explicitly given in eq. (16): It contains the *cube* of the exciton relative motion wave function. The three wave functions barely come from the three boson-exciton operators of $a^\dag \bar{B}^\dag \bar{B}\bar{B}$. Since the exciton-photon vacuum Rabi coupling also depends on the exciton wave function, their exciton saturation density $n_{sat}$ as defined in ref. \[21\], ends by reading $$\frac{1}{n_{sat}L^2}=\frac{2\sum_{\v k}\phi_0^3(\v k)}{\sum_{\v
k}\phi_0(\v k)}=\frac{4\pi}{7}\left(\frac{a_X}{L}\right)^2\ ,$$ where $\phi_0(\v k)=\langle\v
k|\nu_0\rangle$ is the 2D ground state exciton wave function equal to $\sqrt{2\pi}[1+(ka_X/2)^2]^{-3/2}a_X/L$, with $a_X$ being the 3D Bohr radius and $L^2$ the well area.
On the opposite, in our photon-assisted exchange scattering $\chi$, enters the exciton Pauli scattering. For ground state exciton with zero center-of-mass momentum, it reads \[22\] $$\lambda\left(^{\nu_0\v 0\ \nu_0\v 0}_{\nu_0\v 0\ \nu_0\v
0}\right)=\sum_{\v k}\phi_0^4(\v
k)=\frac{4\pi}{5}\left(\frac{a_X}{L}\right)^2\ .$$ Therefore we see that the photon-assisted exchange scattering $\chi$ has to definitely contain the $4^{\mathrm{th}}$ power of the exciton wave function, due to the *four* excitons involved in the carrier exchange linked with this channel. Since ref. \[20\] is very elliptical with respect to the derivation of their interaction Hamiltonian, we cannot point out the origin of the incorrectness.
**Microcavity polaritons**
We end this work on polaritons by considering a specific example, microcavity polaritons, as they are of high current interest for the possible observation of Bose-Einstein condensation of semiconductor excitations \[17,18\].
In microcavities, the excitons are localized in quantum well. In the strong confinement regime, they are quasi-2D excitons; their energy simply reads $E_{\nu\v q}=(E_g+\varepsilon_{ze}+\varepsilon_{zh}+\epsilon_{\nu}+
\hbar^2q^2/2M_X)$, where $\v q$ is the 2D exciton center-of-mass momentum. To get the polariton scatterings from the exciton scatterings, we first need to determine the exciton-polariton and photon-polariton overlaps. To do so, we note that, for a single photon mode $n=\v q$ close to the ground state exciton resonance, we can forget the higher quantum well subbands as well as the exciton excited levels, so that the excitons of interest reduce to $i=(\nu_0,\v q)$. Due to momentum conservation in the exciton-photon coupling, the eigenvalue equation for the polariton $(P_{\v
q},\mathcal{E}_{\v q})$ then reduces to a $2\times 2$ matrix $$\left( \begin{array}{cc} E_{\nu_0\v q}-\mathcal{E}_{\v q} &
\Omega^\ast_{\v q}\\
\Omega_{\v q} & \omega_{\v
q}-\mathcal{E}_{\v q}
\end{array} \right)
\left( \begin{array}{c} \beta_{\nu_0\v q,P_{\v q}} \\ \alpha_{\v q,P_{\v
q}}
\end{array}
\right) =0\ ,$$ where $\omega_{\v
q}=\sqrt{(k^2+q^2)c^2/\epsilon}$ is the cavity photon energy with $k=\pi/L_{cav}$ for an optical cavity length $L_{cav}$. The vacuum Rabi coupling between the cavity photon with energy $\omega_{\v q}$ and the ground state exciton $(\nu_0\v q)$ located in the well reduces to $\Omega_{\v q}=\Omega_0\sqrt{\omega_{\v 0}/\omega_{\v
q}}$. Typical values for InGaAs/GaAs microcavities are $\omega_{\v 0}\simeq E_{\nu_0\v 0}\simeq$ 1.5 eV, $\Omega_0\simeq 4$ meV, while $m_e\simeq 0.067$, $m_h\simeq 0.34$ and $\epsilon\simeq 13$. This gives a 3D Bohr radius $a_X\simeq 12$ nm while $e^2/\epsilon a_X\simeq 8.8$ meV. These parameters, used in the present paper, correspond to Savvidis *et al.*’s experimental conditions \[23\].
The resulting frequencies for the upper and lower polariton branches are given by $$\mathcal{E}_{\v q}^{\pm}=(\omega_{\v q}+E_{\nu_0\v q}\pm\sqrt{T_{\v
q}})/2
\ ,$$ where $T_{\v q}=(\omega_{\v q}-
E_{\nu_0\v q})^2+4|\Omega_{\v q}|^2$, the corresponding overlaps being $$\begin{aligned}
\beta_{\nu_0\v q,P_{\v q}}^- &=& \alpha_{P_{\v q},\v
q}^+=2\Omega^\ast_{\v q}/N_{\v q }\nonumber\\
\alpha_{\v q,P_{\v q}}^- &=& -\beta_{\nu_0\v q,P_{\v q}}^+=(\omega_{\v
q}-E_{\nu_0\v q}-\sqrt{T_{\v q}}) /N_{\v q}\ ,\end{aligned}$$ with $N_{\v q}^2=(\omega_{\v q}-E_{\nu_0\v q}-\sqrt{T_{\v q}})^2+
4|\Omega_{\v q}|^2$.
We use these results to calculate the Coulomb and photon-assisted exchange scatterings for a pair of identical cavity polaritons $(\v q,\v q)$ in the lower branch, scattered into a signal polariton $(\v q=\v 0)$ at the zone center and an idler polariton $(2\v q)$. Figure 2(a) shows the dispersion relation $\mathcal{E}_{\v q}^{\pm}$ of the two polariton branches, while fig.2(b) shows the energy difference $\Delta_{\v q}=
(2\mathcal{E}_{\v q}^- -\mathcal{E}_{\v 0}^- -\mathcal{E}_{2\v q}^-)$ associated to the $(\v q,\v q)\rightarrow (\v 0,2\v q)$ transition. Energy conservation in the transition rate given in eq. (14) imposes $q=q^\ast$ with $q^\ast\simeq 0.025/a_X$. This momentum inside the well corresponds to work at the “magic” angle $\theta=15^\circ$ for the laser beam outside the cavity. As the photon momenta are small on the $a_X^{-1}$ exciton scale, the momenta $(\v q,\v q)$ and $(\v 0,2\v q)$ involved in this transition are essentially zero on this scale. The exciton Coulomb and Pauli scatterings appearing in the polariton scatterings can thus be replaced by their values for zero center-of-mass momentum. Consequently, the $\v q$ dependence of these cavity-polariton scatterings only comes from the polariton overlaps. By noting that $\xi^\mathrm{dir}\left(^{n\ j}_{i\ i}\right)=0$, as seen from eq. (B.18) in ref. \[9\], — which makes the direct Coulomb channel reducing to zero, — the ratio of the contributions to the polariton transition rate from $(\v q,\v q)$ to $(\v 0,2\v q)$ coming from the photon-assisted exchange channel and from the naïve Coulomb channel,then reads $$\begin{aligned}
R_{\v q}&=&\frac{\chi\left(^{2\v q\ \v q}_{\,\v 0\ \,\,\v q}\right)+
\chi\left(_{2\v q\ \v q}^{\v 0\ \ \v q}\right)}{\tilde{\xi}^\mathrm{in}
\left(^{2\v q\ \v q}_{\,\v 0\ \,\,\v q}\right)}\nonumber\\
&\simeq&
\frac{\lambda\left(^{\nu_0\v 0\
\nu_0\v 0}_{\nu_0\v 0\ \nu_0\v 0}\right)}{\xi^\mathrm{in}
\left(^{\nu_0\v 0\
\nu_0\v 0}_{\nu_0\v 0\ \nu_0\v 0}\right)}
\left(\Omega_{\v 0}\frac{
\alpha_{P_{\v 0},\v 0}^-}{\beta_{P_{\v 0},\nu_0\v 0}^-}
+\Omega_{2\v q}\frac{\alpha_{P_{2\v q},2\v q}^-}
{\beta_{P_{2\v q},\nu_02\v q}^-}\right)\ ,\end{aligned}$$ The diagonal “in” Coulomb scattering for 2D ground state excitons with zero center-of-mass momentum can be calculated analytically as $$\xi^\mathrm{in}\left(^{\nu_0\v 0\
\nu_0\v 0}_{\nu_0\v 0\ \nu_0\v
0}\right)=-(4\pi-315\pi^3/1024)(a_X/L)^2(e^2/\epsilon a_X)\ ,$$ *i.e.*, $\xi^\mathrm{in} \simeq
-3.0(a_X/L)^2(e^2/\epsilon a_X)$. By using the Pauli scattering given in eq. (16) and the overlaps given in eq. (19), we obtain the ratio $R_{\v q}$ shown in fig.2(c).
It can be of interest to note that, for resonant photon $\omega_{\v 0}=
E_{\nu_0\v 0}$, $$R_{q=0}=-2\Omega_0\,\frac{\lambda\left(^{\nu_0\v 0\
\nu_0\v 0}_{\nu_0\v 0\ \nu_0\v 0}\right)}{\xi^\mathrm{in}
\left(^{\nu_0\v 0\
\nu_0\v 0}_{\nu_0\v 0\ \nu_0\v 0}\right)}=2R_{q\rightarrow\infty}\ ,$$ since, due to eq. (19), $\alpha_{P_{\v 0},\v 0}^-=-\beta_
{P_{\v 0},\nu_0\v 0}^-=-1/\sqrt{2}$, while $\alpha_{P_{2\v q},2\v q}^-$ goes to 0 when $q\rightarrow\infty$, assuming an infinite exciton mass.
We see from fig.2(c) that the photon-assisted exchange channel produces a significant enhancement of the polariton transition rate, when compared to the one coming from the naïve Coulomb interactions between the excitonic components of the polaritons: For typical experimental conditions as the ones considered here, the increase of the transition rate at the “magic” value $q=q^\ast$, $$\begin{aligned}
\frac{\left|\tilde{\xi}^\mathrm{in}\left(^{2\v q\ \v q}_{\v 0\ \ \v
q}\right)+\chi\left(^{2\v q\ \v q}_{\v 0\ \ \v q}\right)+\chi
\left(^{\v 0\ \ \v q}_{2\v q\ \v q}\right)\right|^2}
{\left|\tilde{\xi}^\mathrm{in}\left(^{2\v q\ \v q}_{\v 0\ \ \v
q}\right)\right|^2}\nonumber\\
=
\left(1+R_{\v q}\right)^2\simeq 2.1\ ,\end{aligned}$$ is found to be slightly larger than 2.
**Conclusion**
Through a fully microscopic procedure, which makes use of the composite-exciton many-body theory we have recently proposed, it is now possible to approach the strong coupling of photons and semiconductor excitations in an exact way, *i.e.*, without mapping the excitons into a boson subspace at any stage. We show that the composite nature of the polaritons gives rise to a photon-assisted exchange scattering, free from Coulomb process. The results of this exact approach disagree with the ones obtained by using bosonized excitons, even through the elaborate procedure proposed by the Quattropani’s group. This photon-assisted exchange channel, dominant when one of the scattered polaritons has a strong photon character, produces a significant enhancement of the transition rate for a pair of pump microcavity polaritons scattered into an idler and a signal at the zone center. This paper also contains crucial tools to tackle the many-body physics of polaritons, towards their possible Bose Einstein condensation.
[99]{}
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To get eq. (14), we have used eq. (12) and the fact that $\sum_{P'_1,P'_2}\tilde{\lambda}\left(^{P''_2\ P'_2}
_{P''_1\ P'_1}\right)\,\chi\left(^{P'_2\ P_2}_{P'_1\ P_1}\right)$ is equal to zero, which follows from $\langle v|B_i\alpha_n^\dag|v\rangle=0$, the sums over $P$’s being performed through closure relations.
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|
**INTEGRAL OPERATORS WITH INFINITELY SMOOTH BI-CARLEMAN KERNELS OF MERCER TYPE**
.5cm
Igor M. Novitskii
Khabarovsk Division\
Institute of Applied Mathematics\
Far-Eastern Branch of the Russian Academy of Sciences\
54, Dzerzhinskiy Street, Khabarovsk 680 000, RUSSIA\
e-mail: [email protected]
[**Abstract:**]{} With the aim of applications to solving general integral equations, we introduce and study in this paper a special class of bi-Carleman kernels on $\mathbb{R}\times\mathbb{R}$, called $K^\infty$ kernels of Mercer type, whose property of being infinitely smooth is stable under passage to certain left and right multiples of their associated integral operators. An expansion theorem in absolutely and uniformly convergent bilinear series concerning kernels of this class is proved extending to a general non-Hermitian setting both Mercer’s and Kadota’s Expansion Theorems for positive definite kernels. Another theorem proved in this paper identifies families of those bounded operators on a separable Hilbert space $\mathcal{H}$ that can be simultaneously transformed by the same unitary equivalence transformation into bi-Carleman integral operators on $L^2(\mathbb{R})$, whose kernels are $K^\infty$ kernels of Mercer type; its singleton version implies in particular that any bi-integral operator is unitarily equivalent to an integral operator with such a kernel. [**AMS Subject Classification:**]{} 47G10, 45P05, 45A05, 47B33, 47N20, 47A10
[**Key Words:**]{} Hilbert space operator, bounded integral linear operator, characterization theorems for integral operators, bi-Carleman kernel, bilinear series expansions of kernels, Mercer’s Theorem, Kadota’s Theorem, linear integral equation
Introduction and the Main Results
=================================
In the theory of general linear integral equations in $L^2$ spaces, the equations with bounded infinitely differentiable bi-Carleman kernels (termed $K^\infty$ kernels) should and do lend themselves well to solution by approximation and variational methods. The question of whether a second-kind integral equation with arbitrary kernel can be reduced to an equivalent one with a $K^\infty$ kernel was positively answered using a unitary-reduction method by the author [@nov:IJPAM2]. In the present paper, we enrich the $K^\infty$ kernels by imposing an extra condition (Definition \[MerKernel\]) that guarantees the kernels and their partial and strong derivatives to be expandable in absolutely and uniformly convergent bilinear series which may also be relevant when solving integral equations. We then show that that condition can always be achieved by means of a unitary reduction which involves no loss of generality in the study of integral equations of the second kind.
Before we can write down our main results, we need to fix the terminology and notation and to give some definitions and preliminary material. Throughout this paper, $\mathcal{H}$ is a complex, separable, infinite-dimensional Hilbert space with the inner product $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ and the norm $\left\|\cdot\right\|_{\mathcal{H}}$. If $L\subset\mathcal{H}$, we write $\overline{L}$ for the norm closure of $L$ in $\mathcal{H}$, and $\mathrm{Span}(L)$ for the norm closure of the set of all linear combinations of elements of $L$. Let $\mathfrak{R}({\mathcal{H}})$ be the Banach algebra of all bounded linear operators acting on ${\mathcal{H}}$. For an operator $A$ of $\mathfrak{R}(\mathcal{H})$, $A^*$ stands for the adjoint to $A$ with respect to $\langle\cdot,\cdot\rangle_{\mathcal{H}}$, $\mathrm{Ran\,}A=\left\{Af\mid f\in\mathcal{H}\right\}$ for the range of $A$. An operator $A\in\mathfrak{R}(\mathcal{H})$ is said to be *invertible* if it has an inverse which is also in $\mathfrak{R}(\mathcal{H})$, that is, if there is an operator $B\in\mathfrak{R}(\mathcal{H})$ for which $BA=AB=I_{{\mathcal{H}}}$ where $I_{{\mathcal{H}}}$ is the identity operator on ${\mathcal{H}}$; $B$ is denoted by $A^{-1}$. An operator $P\in\mathfrak{R}(\mathcal{H})$ is called *positive* and denoted by $P\ge0$ if $\langle Px,x\rangle_{\mathcal{H}}\ge 0$ for all $x\in\mathcal{H}$. Also recall that a normal operator $A\in\mathfrak{R}(\mathcal{H})$ ($AA^*=A^*A$) is called *diagonal*(*izable*) if $\mathcal{H}$ admits an orthonormal basis consisting of eigenvectors of $A$. If $T\in\mathfrak{R}({\mathcal{H}})$, define the operator families $\mathcal{M}(T)$ and $\mathcal{M}^+(T)$ by $$\begin{gathered}
\mathcal{M}(T)=\left(T\mathfrak{R}({\mathcal{H}})\cup
T^*\mathfrak{R}({\mathcal{H}})\right)\cap
\left(\mathfrak{R}({\mathcal{H}})T\cup\mathfrak{R}({\mathcal{H}})T^*\right),
\label{MT}\\
\mathcal{M}^{+}(T)=\{P\in\mathcal{M}(T)\mid P\ge0\}\label{Pc},\end{gathered}$$ where $S\mathfrak{R}({\mathcal{H}})=\{SV\mid V\in \mathfrak{R}({\mathcal{H}})\}$, $\mathfrak{R}({\mathcal{H}})S=\{VS\mid V\in \mathfrak{R}({\mathcal{H}})\}$. The following are some simple remarks about families just defined.
\[remmt\] First note from that $\mathcal{M}(S)=\mathcal{M}(S^*)$, and that the set $\mathcal{M}(S)$ is alternatively defined as $$\begin{gathered}
\begin{split}
\mathcal{M}(S)=&
\left(S\mathfrak{R}(\mathcal{H})\cap\mathfrak{R}(\mathcal{H})S\right)\cup
\left(S^*\mathfrak{R}(\mathcal{H})\cap\mathfrak{R}(\mathcal{H})S^*\right)
\\
&\cup\left(S\mathfrak{R}(\mathcal{H})\cap\mathfrak{R}(\mathcal{H})S^*\right)
\cup\left(S^*\mathfrak{R}(\mathcal{H})\cap\mathfrak{R}(\mathcal{H})S\right),
\end{split}\end{gathered}$$ which is the same as saying that an operator $A$ belongs to $\mathcal{M}(S)$ if and only if there exist two operators $M$, $N\in\mathfrak{R}(\mathcal{H})$ such that at least one of the four relations $$\label{relations}
\begin{split}
A=SM&=NS,\quad A=S^*M=NS^*, \\&
A=SM=NS^*,\quad \text{or}\ A=S^*N=MS,
\end{split}$$ holds. The family $\mathcal{M}^{+}(S)$ can then be characterized as the set of all positive operators $P\in\mathfrak{R}(\mathcal{H})$ that are expressible as $P=SB$, or as $P=BS$, where $B\in\mathfrak{R}(\mathcal{H})$, that is to say, $\mathcal{M}^{+}(S)=
\left\{P\in S\mathfrak{R}(\mathcal{H})\cup\mathfrak{R}(\mathcal{H})S\mid P
\ge 0\right\}$. If, finally, $S\in\mathcal{M}(T)$, then, by , there are four operators $K$, $L$, $Q$, $R\in\mathfrak{R}(\mathcal{H})$ such that $
\mathcal{M}(S)=\left(TK\mathfrak{R}(\mathcal{H})\cup T^*L\mathfrak{R}(\mathcal{H})\right)\cap
\left(\mathfrak{R}(\mathcal{H})QT^*\cup \mathfrak{R}(\mathcal{H})RT)\right)
$, whence it follows via that $\mathcal{M}(S)\subseteq\mathcal{M}(T)$, and, consequently, $\mathcal{M}^{+}(S)\subseteq\mathcal{M}^{+}(T)$.
\[mfac\] A factorization of an operator $T\in\mathfrak{R}({\mathcal{H}})$ into the product $$\label{f1}
T = WV^*,$$ provided that the operators $V$, $W\in\mathfrak{R}({\mathcal{H}})$ are subject to the provisos $$\label{f2}
VV^*\in\mathcal{M}^{+}(T),\quad WW^*\in\mathcal{M}^{+}(T),$$ is called an $\mathcal{M}$ *factorization for $T$*.
\[TWV\] An ingenuous example of an $\mathcal{M}$ factorization for any $T\in\mathfrak{R}(\mathcal{H})$ is easy to come by. Just put $W=UP$ and $V=P$, where $P$ is the positive square root of $|T|=(T^*T)^{\frac1{2}}$ and $U$ is the partially isometric factor in the polar decomposition $T=U|T|$; since $T=WV^*$, $WW^*=U|T|U^*=TU^*=UT^*\in\mathcal{M}^{+}(T)$, and $VV^*=|T|=T^*U=U^*T\in\mathcal{M}^{+}(T)$, it follows that the requirements , are satisfied.
The set of *regular values* for the operator $T\in\mathfrak{R}(\mathcal{H})$, denoted by $\Pi(T)$, is the set of complex numbers $\lambda$ such that the operator $I_{{\mathcal{H}}}-\lambda T$ is invertible, that is, it has an inverse $R_\lambda(T)=\left(I_{{\mathcal{H}}}-\lambda T\right)^{-1}
\in\mathfrak{R}({\mathcal{H}})$ that satisfies $$\label{eqress}
\left(I_{{\mathcal{H}}}-\lambda T\right)R_\lambda(T)=R_\lambda(T)
\left(I_{\mathcal{H}}-\lambda T\right)=I_{\mathcal{H}}.$$ The operator $$\label{eqresf}
T_\lambda:= TR_\lambda(T)\ (=R_\lambda(T)T)$$ of $\mathcal{M}(T)$ is then referred to as the *Fredholm resolvent* of $T$ at $\lambda$. Remark that if $\lambda$ is a regular value for $T$, then, for each fixed $g\in\mathcal{H}$, the (unique) solution $f\in\mathcal{H}$ to the second-kind equation $f-\lambda Tf=g$ may be written as $$\label{equnsol}
f=g+\lambda T_\lambda g.$$
Let $\mathbb{R}$ be the real line $(-\infty,+\infty)$ equipped with the Lebesgue measure, and let $L^2=L^2(\mathbb{R})$ be the Hilbert space of (equivalence classes of) measurable complex-valued functions on $\mathbb{R}$ equipped with the inner product $\langle f,g\rangle_{L^2}=\int_{\mathbb{R}}f(s)\overline{g(s)}\,ds$ and the norm $\|f\|_{L^2}=\langle f,f\rangle^{\frac{1}2}$. A linear operator $T\colon L^2\to L^2$ is *integral* if there is a complex-valued measurable function $\boldsymbol{T}$ (*kernel*) on the Cartesian product $\mathbb{R}^2=\mathbb{R}\times\mathbb{R}$ such that $$(Tf)(s)=\int_{\mathbb{R}} \boldsymbol{T}(s,t)f(t)\,dt$$ for every $f\in L^2$ and almost every $s\in\mathbb{R}$. Recall [@Halmos:Sun Theorem 3.10] that integral operators are bounded, and need not be compact. A kernel $\boldsymbol{T}$ on $\mathbb{R}^2$ is said to be *Carleman* if $\boldsymbol{T}(s,\cdot)\in L^2$ for almost every fixed $s$ in $\mathbb{R}$. To each Carleman kernel $\boldsymbol{T}$ there corresponds a *Carleman function* $\boldsymbol{t}\colon \mathbb{R}\to L^2$ defined by $\boldsymbol{t}(s)=\overline{\boldsymbol{T}(s,\cdot)}$ for all $s$ in $\mathbb{R}$ for which $\boldsymbol{T}(s,\cdot)\in L^2$. The Carleman kernel $\boldsymbol{T}$ is called *bi-Carleman* in case its conjugate transpose kernel $\boldsymbol{T}^{\boldsymbol{\prime}}$ ($\boldsymbol{T}^{\boldsymbol{\prime}}(s,t)=\overline{\boldsymbol{T}(t,s)}$) is also a Carleman kernel. Associated with the conjugate transpose $\boldsymbol{T}^{\boldsymbol{\prime}}$ of every bi-Carleman kernel $\boldsymbol{T}$ there is therefore a Carleman function $\boldsymbol{t}^{\boldsymbol{\prime}}\colon \mathbb{R}\to L^2$ defined by $\boldsymbol{t}^{\boldsymbol{\prime}}(s)
=\overline{\boldsymbol{T}^{\boldsymbol{\prime}}(s,\cdot)}
\left(=\boldsymbol{T}(\cdot,s)\right)$ for all $s$ in $\mathbb{R}$ for which $\boldsymbol{T}^{\boldsymbol{\prime}}(s,\cdot)\in L^2$. With each bi-Carleman kernel $\boldsymbol{T}$, we therefore associate the pair of Carleman functions $\boldsymbol{t}$, $\boldsymbol{t}^{\boldsymbol{\prime}}\colon \mathbb{R}\to L^2$, both defined, via $\boldsymbol{T}$, as above. An integral operator whose kernel is Carleman (resp., bi-Carleman) is referred to as the *Carleman* (resp., *bi-Carleman*) operator. The integral operator $T$ is called *bi-integral* if its adjoint $T^*$ is also an integral operator; in that case if $\boldsymbol{T}^{\boldsymbol{\ast}}$ is the kernel of $T^*$ then, in the above notation, $\boldsymbol{T}^{\boldsymbol{\ast}}(s,t)=\boldsymbol{T}^{\boldsymbol{\prime}}(s,t)$ for almost all $(s,t)\in\mathbb{R}^2$ (see, e.g., [@Halmos:Sun Theorem 7.5]). A bi-Carleman operator is always a bi-integral operator, but not conversely. Henceforth in this paper, kernels and Carleman functions will always be denoted by bold-face uppercase and bold-face lowercase letters, respectively.
\[rem1\] From the viewpoint of the foundations of integral equation theory, the bad news about Fredholm resolvents is that the property of being an integral operator is not shared in general by Fredholm resolvents of integral operators; an example of an integral operator whose Fredholm resolvent at any non-zero regular value is not an integral operator can be found in [@Kor:nonint1], or in [@Kor:alg Section 5, Theorem 8]. But, fortunately, this phenomenon can never be extended to Carleman operators due to an algebraic property of these operators, a property which is the content of the so called “Right-Multiplication Lemma” (see [@Halmos:Sun Theorem 11.6] or [@Kor:book1 Corollary IV.2.8]):
\[rimlt\] Let $T$ be a Carleman operator, let $\boldsymbol{t}$ be the Carleman function associated with the inducing Carleman kernel of $T$, and let $A\in\mathfrak{R}\left(L^2\right)$ be arbitrary. Then the product operator $TA$ is also a Carleman operator, and the composition function $A^*(\boldsymbol{t}(\cdot))\colon \mathbb{R}\to L^2$ is the Carleman function corresponding with its kernel.
Thus, by , the Fredholm resolvent of a Carleman (resp., bi-Carleman) operator at its every regular value is always a Carleman (resp., bi-Carleman) operator. Furthermore, if the operator $S$ is bi-Carleman, then it follows from the above proposition and that the corresponding family $\mathcal{M}(S)$ must consist of bi-Carleman operators only. In the general theory of integral equations of the second kind in $L^2$, that is, equations of the form $$\label{skequ}
f(s)-\lambda\int_{\mathbb{R}} \boldsymbol{T}(s,t)f(t)\,dt =g(s)\quad
\text{for almost all $s\in\mathbb{R}$},$$ it is customary to call the kernel $\boldsymbol{T}_\lambda$ (when it exists) of the Fredholm resolvent $T_\lambda$ of a bi-integral operator $T$, induced on $L^2$ by the kernel $\boldsymbol{T}$, a *resolvent kernel for $\boldsymbol{T}$ at $\lambda$*. Once the resolvent kernel $\boldsymbol{T}_\lambda$ comes to be known, one can express the $L^2$ solution $f$ to equation in a direct fashion as $$\label{solution}
f(s)=g(s)+\lambda\int_{\mathbb{R}} \boldsymbol{T}_\lambda(s,t)g(t)\,dt,$$ regardless of the particular choice of the function $g$ of $L^2$ (cf. ). Therefore, in the case when the kernel $\boldsymbol{T}$ is Carleman and $\lambda\in\Pi(T)$ is arbitrary, the problem of solving equation may be reduced to the problem of explicitly constructing (in terms of $\boldsymbol{T}$) the resolvent kernel $\boldsymbol{T}_\lambda$ which is a priori known to exist. For a precise formulation of this latter problem (not solved here) and for comments to the solution of some of its special cases we refer to the works by Korotkov [@Kor:problems], [@Kor:alg] (in both the references, see Problem 4 in §5). Here we only recall that as long as a measurable kernel $\boldsymbol{T}$ of is bi-Carleman but otherwise unrestricted, there seems to be as yet no analytic machinery for explicitly constructing its resolvent kernel $\boldsymbol{T}_\lambda$ at all $\lambda\in\Pi(T)$. With the objective of making problems like this more tractable, we propose in the present paper to treat some analytical restrictions which can be imposed on bi-Carleman kernels $\boldsymbol{T}$ without loss of generality as far as the solving of such equations as is concerned. These restrictions will be introduced presently by means of Definitions \[Kmkernel\] and \[MerKernel\], and the generality will be preserved using unitary reductions by means of Theorem \[infsmooth\] below.
Throughout this paper, the symbols $\mathbb{C}$, $\mathbb{N}$, and $\mathbb{Z}$, refer to the complex plane, the set of all positive integers, and the set of all integers, respectively, and each of the letters $i$ and $j$ is reserved for all non-negative integers. $C(X,B)$, where $B$ is a Banach space (with norm $\|\cdot\|_B$), denotes the Banach space (with the norm $\|f\|_{C(X,B)}=\sup_{x\in X}\,\|f(x)\|_B$) of continuous $B$-valued functions defined on a locally compact space $X$ and *vanishing at infinity* (that is, given any $f\in C(X,B)$ and $\varepsilon>0$, there exists a compact subset $X(\varepsilon,f) \subset X$ such that $\|f(x)\|_{B}<\varepsilon$ whenever $x\not\in X(\varepsilon,f)$). In addition, we introduce the following notation: if an equivalence class $f\in L^2$ contains a function belonging to $C(\mathbb{R},\mathbb{C})$, we write $[f]$ to mean that function, and we denote the $i$-th derivative of $[f]$, if exists, by $[f]^{(i)}$. We also say that the series $\sum_n f_n$ is *$B$-absolutely convergent in $C(X,B)$* if $f_n\in C(X,B)$ ($n\in\mathbb{N}$) and the series $\sum_n \|f_n(x)\|_B$ converges in $C(X,\mathbb{R})$ (the sum notation $\sum_n$ will always be used instead of the more detailed symbol $\sum_{n=1}^\infty$).
\[Kmkernel\] A bi-Carleman kernel $\boldsymbol{T}\colon \mathbb{R}^2\to\mathbb{C}$ is called a *$K^\infty$ kernel* (see [@nov:CEJM]) if it satisfies the three generally independent conditions:
\(i) the function $\boldsymbol{T}$ and all its partial derivatives of all orders are in $C\left(\mathbb{R}^2,\mathbb{C}\right)$,
\(ii) the Carleman function $\boldsymbol{t}$, $\boldsymbol{t}(s)=\overline{\boldsymbol{T}(s,\cdot)}$, and its (strong) derivatives, $\boldsymbol{t}^{(i)}$, of all orders are in $C\left(\mathbb{R},L^2\right)$,
\(iii) the Carleman function $\boldsymbol{t}^{\boldsymbol{\prime}}$, $\boldsymbol{t}^{\boldsymbol{\prime}}(s)=
\overline{\boldsymbol{T}^{\boldsymbol{\prime}}(s,\cdot)}=
\boldsymbol{T}(\cdot,s)$, and its (strong) derivatives, $(\boldsymbol{t}^{\boldsymbol{\prime}})^{(i)}$, of all orders are in $C\left(\mathbb{R},L^2\right)$.
To deal with $K^\infty$ kernels, we let $D_r^i$ denote the $i$-th order partial derivative operator with respect to the $r$-th variable, and we let $D_{r_1,r_2}^{i}$ denote the product (in the operator sense) of $i$ factors, each of which is the mixed second-order partial derivative operator $D_{r_2}^1D_{r_1}^1$, so, for instance, $D_{r_1,r_2}^{2}=D_{r_2}^1D_{r_1}^1D_{r_2}^1D_{r_1}^1$.
\[MerKernel\] Let $\boldsymbol{T}$ be a $K^\infty$ kernel and let $T$ be the integral operator it induces on $L^2$. We say that the $K^\infty$ kernel $\boldsymbol{T}$ is of *Mercer type* if every operator belonging to $\mathcal{M}(T)$ is an integral operator with a $K^\infty$ kernel.
The idea of a counter-example in [@Nov:Lon Section 2] may be used to show that there might be $K^\infty$ kernels which are not of Mercer type. The (in a sense algebraic) condition on a $K^\infty$ kernel for being of Mercer type is of course tailored with specific applications in mind. One of these is a theorem, the first main result of the present paper, which asserts, among other things, that any $K^\infty$ kernel of Mercer type, along with all its partial and strong derivatives, is entirely recoverable from the knowledge of at least one $\mathcal{M}$ factorization for its associated integral operator, by means of bilinear series formulae universally applicable on arbitrary orthonormal bases of $L^2$:
\[mfactor\] Let $T\in\mathfrak{R}\left(L^2\right)$ be an integral operator, with a kernel $\boldsymbol{T}_0$ that is a $K^\infty$ kernel of Mercer type. Then at each regular value $\lambda\in\Pi(T)$
\(a) the Fredholm resolvent $T_\lambda$ of $T$ is also an integral operator and its kernel, the resolvent kernel $\boldsymbol{T}_\lambda$ for $\boldsymbol{T}_0$, is also a $K^\infty$ kernel of Mercer type, and, moreover,
\(b) for any $\mathcal{M}$ factorization $T = WV^*$ for $T$ and for any orthonormal basis $\{u_n\}$ for $L^2$, the following formulae hold $$\begin{gathered}
\left(D_2^jD_1^i\boldsymbol{T}_\lambda\right)(s,t)=
\sum_n\left[R_\lambda(T)Wu_n\right]^{(i)}(s)
\overline{\left[Vu_n\right]^{(j)}(t)}, \label{meijT1}
\\
\begin{split} \label{meijkt2}
\boldsymbol{t}_\lambda^{(i)}(s)&=\sum_n\overline{\left[R_\lambda(T)Wu_n\right]^{(i)}(s)}Vu_n,
\\
\left(\boldsymbol{t}^{\boldsymbol{\prime}}_\lambda\right)^{(j)}(t)&=
\sum_n \overline{\left[Vu_n\right]^{(j)}(t)}R_\lambda(T)Wu_n,
\end{split}
\\
\left[T_\lambda f\right]^{(i)}(s)=
\sum_n\left\langle f,Vu_n\right\rangle_{L^2}\left[R_\lambda(T)Wu_n\right]^{(i)}(s)\label{meijkTf3},\end{gathered}$$ for all $i$, $j$, all $s$, $t\in\mathbb{R}$, and all $f\in L^2$, where the series of converges $\mathbb{C}$-absolutely in $C\left(\mathbb{R}^2,\mathbb{C}\right)$, the two series of , in which $\boldsymbol{t}_\lambda$, $\boldsymbol{t}^{\boldsymbol{\prime}}_\lambda$ denote the associated Carleman functions of $\boldsymbol{T}_\lambda$, both converge in $C\left(\mathbb{R},L^2\right)$, and the series of converges $\mathbb{C}$-absolutely in $C(\mathbb{R},\mathbb{C})$.
For the case $\lambda=0$, the results of the theorem have been announced without proofs in [@nov:IJPAM1 Theorem 4]. In this particular case, $R_\lambda(T)=I_{L^2}$, $\boldsymbol{T}_\lambda=\boldsymbol{T}_0$, and the bilinear formula reminds one of Mercer’s (see [@Porter Theorem 4.24]) and Kadota’s (see [@Kadota1]) Theorems; the first theorem, recall, is about absoluteness and uniformity of convergence of bilinear (orthogonal) eigenfunction expansions for continuous compactly supported kernels of positive integral operators, and the second is about term-by-term differentiability of those expansions while retaining the absolute and the uniform convergence. The similarity is closest when Theorem \[mfactor\] is applied to a diagonal operator $T$ by taking as $T=WV^*$ the $\mathcal{M}$ factorization of Example \[TWV\] and as $\{u_n\}$ any orthonormal basis with respect to which $T$ is diagonalizable, because then formula (with $\lambda=0$) reduces, after a computation, to a bilinear eigenfunction expansion, $$\left(D_2^jD_1^i\boldsymbol{T}_0\right)(s,t)=
\sum_n\lambda_n\left[u_n\right]^{(i)}(s)
\overline{\left[u_n\right]^{(j)}(t)},$$ converging $\mathbb{C}$-absolutely in $C\left(\mathbb{R}^2,\mathbb{C}\right)$. Applied within the same setting, formula looks like: $
\left[Tf\right]^{(i)}(s)=
\sum_n\lambda_n \left\langle f,u_n\right\rangle_{L^2}\left[u_n\right]^{(i)}(s)
$ in the sense of $\mathbb{C}$-absolute convergence in $C\left(\mathbb{R},\mathbb{C}\right)$, and gives something very akin to Schmidt’s Theorem, see [@Porter Theorem 4.22]. In the general non-diagonalizable case, it is therefore natural to view formulae , as a “necessary substitute” for the above classical diagonal ones, justifying the name chosen for the kernels defined in Definitions \[Kmkernel\] and \[MerKernel\].
Formulae , do not, of course, solve the problem (mentioned at the end of Remark \[rem1\]) of explicitly constructing the resolvent kernel $\boldsymbol{T}_\lambda$ for $\boldsymbol{T}_0$, but succeed in reducing it to one of explicitly finding at most countably many functions $R_\lambda(T)Wu_n$, $Vu_n$ ($n\in\mathbb{N}$), which we do not yet know how to resolve. Nevertheless, the formulae in Theorem \[mfactor\] may (hopefully) prove quite interesting from a theoretical perspective in terms of developing a general scheme for bilinearly representing integral kernels via the use of the operators that they define but ignoring, if needed, the explicit knowledge about the spectra of those operators.
The proof of Theorem \[mfactor\] is given in Section \[prmfactor\] below and actually proves a somewhat looser version of it, which will be formulated later, in Section \[faccol\], as Corollary \[mfactor3\]. This corollary is of the same sort as Theorem \[mfactor\], but, among others, we have relaxed the $\mathcal{M}$ factoring assumption about the operators $W$, $V$ being used in formulae -. In Section \[faccol\], we also try to give a unified view of various bilinear expansion theorems which involve the use of canonical forms of operators, by deriving them as consequences of (the proof of) Theorem \[mfactor\].
At first glance it may seem that the conditions defining $K^\infty$ kernel of Mercer type are not only hardly verifiable, but also very artifical and rather contrived, and also that for most applications it is too restrictive to confine oneself to such kernels. It is somewhat surprising, therefore, to discover the fact that any bi-integral operator can be made to have as its kernel a $K^\infty$ kernel of Mercer type. For the interpretation of the corresponding result, it is helpful to recall the notion of a unitary equivalence. A bounded linear operator $U\colon {\mathcal{H}}\to L^2$ is said to be *unitary* if $\mathrm{Ran\,}U=L^2$ and $\langle Uf,Ug\rangle_{L^2}=\langle f,g\rangle_{\mathcal{H}}$ for all $f$, $g\in {\mathcal{H}}$. An operator $S\in \mathfrak{R}({\mathcal{H}})$ is said to be *unitarily equivalent* to an operator $T\in \mathfrak{R}\left(L^2\right)$ if a unitary operator $U\colon {\mathcal{H}}\to L^2$ exists such that $T=USU^{-1}$. It is also relevant to mention the fact that a necessary and sufficient condition that an operator $S\in\mathfrak{R}({\mathcal{H}})$ be unitarily equivalent to a (general) bi-Carleman integral operator is that there exist an orthonormal sequence $\left\{e_n\right\}$ in $\mathcal{H}$ such that $$\label{kh}
\left\|Se_n\right\|_{\mathcal{H}}\rightarrow 0,\quad
\left\|S^*e_n\right\|_{\mathcal{H}}\rightarrow 0\quad
\text{as $n\rightarrow\infty$}$$ (or, equivalently, that $0$ belong to the essential spectrum of $SS^*+S^*S$). This fact was first stated by von Neumann in [@Neu] for self-adjoint operators and was then extended by Korotkov to the general case (see [@Kor:book1 Theorem III.2.7], [@Halmos:Sun Theorem 15.14]). Recall [@Halmos:Sun Theorem 15.11] that the class of operators satisfying includes all bi-integral operators when the Hilbert space $\mathcal{H}$ is $L^2$, or in general $L^2(Y,\mu)$ associated with a positive, $\sigma$-finite, separable, and not purely atomic, measure $\mu$. The bi-integral operators, on the other hand, are generally involved in second-kind integral equations (like ) in $L^2(Y,\mu)$, as the adjoint equations to such equations are customarily required to be integral.
It is pleasant to know that the same condition as proves necessary and sufficient for the operator $S$ to be unitarily equivalent to a bi-Carleman operator generated by a $K^\infty$ kernel of Mercer type. The second principal result of the present paper, Theorem \[infsmooth\] below, both states this fact and characterizes families incorporating those operators in $\mathfrak{R}({\mathcal{H}})$ that can be simultaneously transformed by the same unitary equivalence transformation into bi-Carleman integral operators having as kernels $K^\infty$ kernels of Mercer type.
\[infsmooth\] Suppose that for an operator family $\mathcal{S}=
\left\{S_\gamma\right\}_{\gamma\in\mathcal{G}}\subset\mathfrak{R}({\mathcal{H}})$ with an index set of arbitrary cardinality there exists an orthonormal sequence $\left\{e_n\right\}$ in $\mathcal{H}$ such that $$\label{1.2}
\lim\limits_{n\to\infty}\sup\limits_{\gamma\in\mathcal{G}}\left\|S_\gamma e_n\right\|_{\mathcal{H}}=0,
\quad
\lim_{n\to\infty}\sup_{\gamma\in\mathcal{G}}\left\|\left(S_\gamma\right)^* e_n\right\|_{\mathcal{H}}=0.$$ Then there exists a unitary operator $U\colon \mathcal{H}\to L^2$ such that all the operators $T_\gamma=U S_\gamma U^{-1}$ $(\gamma\in\mathcal{G})$ and their linear combinations are bi-Carleman operators on $L^2$, whose kernels are $K^\infty$ kernels of Mercer type.
This result has recently been published without proof in [@nov:IJPAM1 Theorem 3]. Section \[intrepr\] of the present paper is entirely devoted to proving Theorem \[infsmooth\]. The method of proof yields a technique for constructing that unitary operator $U\colon \mathcal{H}\to L^2$ whose existence the theorem asserts. The technique uses no spectral properties of the operators $S_\gamma$, other than their joint property imposed in , to determine the action of $U$ by specifying two orthonormal bases, of $\mathcal{H}$ and of $L^2$, one of which is meant to be the image by $U$ of the other, the basis for $L^2$ may be chosen to be an infinitely smooth wavelet basis.
Proof of Theorem \[mfactor\] {#prmfactor}
============================
\(a) Use , , and the invertibility of $R_\lambda(T)$, to see that
$$\begin{gathered}
\mathcal{M}(T_\lambda)=\left(TR_\lambda(T)\mathfrak{R}\left(L^2\right)
\cup T^*\left(R_\lambda(T)\right)^*\mathfrak{R}\left(L^2\right)\right)
\\
\cap \left(\mathfrak{R}\left(L^2\right)R_\lambda(T)T\cup
\mathfrak{R}\left(L^2\right)\left(R_\lambda(T)\right)^*T^*\right)=
\mathcal{M}(T),\end{gathered}$$
for each $\lambda\in\Pi(T)$. Now the first assertion in the theorem is immediate from Definition \[MerKernel\].
\(b) This part of the proof proves the second assertion in the theorem, and is divided into four steps. The first three steps are to establish formulae - for the case in which $\lambda=0$. Step 4 takes care of the case of an arbitrary regular $\lambda$.
Throughout what follows let $\left\{u_n\right\}$ be an arbitrary but fixed orthonormal basis for $L^2$, and let $W$, $V$ be arbitrary but likewise fixed operators of $\mathfrak{R}\left(L^2\right)$ such that both $F=VV^*$ and $G=WW^*$ are in $\mathcal{M}^{+}(T)$, and $T=WV^*$. We then let $\boldsymbol{F}$, $\boldsymbol{G}$ denote the $K^\infty$ kernels of the integral positive operators $F$, $G$, respectively.
*Step 1.* A convenient way to begin the proof of assertion (b) for $\lambda=0$ is to assume for the moment that the following properties of the function systems $\{Vu_n\}$, $\{Wu_n\}$ hold true:
\(A) $\left[Vu_n\right]^{(i)}$, $\left[Wu_n\right]^{(i)}\in C(\mathbb{R},\mathbb{C})$, for all $n\in\mathbb{N}$ and all $i$,
\(B) the series $\sum_n\left|\left[Vu_n\right]^{(i)}\right|^2$, $\sum_n\left|\left[Wu_n\right]^{(i)}\right|^2$ converge in $C(\mathbb{R},\mathbb{C})$, for all $i$.
Having made these assumptions, the first thing to do is to establish the existence of a $K^\infty$ kernel $\boldsymbol{H}$, with associated Carleman functions $\boldsymbol{h}$ and $\boldsymbol{h}^{\boldsymbol{\prime}}$, such that, for all $i$, $j$, $$\label{firsts}
\left(D_2^jD_1^i\boldsymbol{H}\right)(s,t)=
\sum_n\left[Wu_n\right]^{(i)}(s)\overline{\left[Vu_n\right]^{(j)}(t)}$$ with the series converging $\mathbb{C}$-absolutely in $C\left(\mathbb{R}^2,\mathbb{C}\right)$, and $$\label{seconds}
\boldsymbol{h}^{(i)}(s)=\sum_n\overline{\left[Wu_n\right]^{(i)}(s)}Vu_n,
\quad\left(\boldsymbol{h}^{\boldsymbol{\prime}}\right)^{(j)}(t)=\sum_n \overline{\left[Vu_n\right]^{(j)}(t)}Wu_n,$$ in the sense of convergence in $C\left(\mathbb{R},L^2\right)$. For this purpose, invoke the inequalities $$\begin{gathered}
\left({\sum_{n=p}^r\left|\left[Wu_n\right]^{(i)}(s)
\overline{\left[Vu_n\right]^{(j)}(t)}\right|}\right)^2
\leq \sum_{n=p}^r\left|\left[Wu_n\right]^{(i)}(s)\right|^2\sum_{n=p}^{r}
\left|\left[Vu_n\right]^{(j)}(t)\right|^2,
\\
\left\|\sum_{n=p}^r\overline{\left[Wu_n\right]^{(i)}(s)}Vu_n\right\|_{L^2}^2\le\|V\|^2
\sum_{n=p}^r\left|\left[Wu_n\right]^{(i)}(s)\right|^2,
\\
\left\|\sum_{n=p}^r \overline{\left[Vu_n\right]^{(j)}(t)}Wu_n\right\|_{L^2}^2\le\|W\|^2
\sum_{n=p}^r\left|\left[Vu_n\right]^{(j)}(t)\right|^2
\end{gathered}$$ to infer, via (A) and (B), that the series of and of do indeed converge in the senses above. Then apply the corresponding theorems on termwise differentiation of series to conclude that functions $\boldsymbol{H}$ of $C\left(\mathbb{R}^2,\mathbb{C}\right)$ and $\boldsymbol{h}$, $\boldsymbol{h}^{\boldsymbol{\prime}}$ of $C\left(\mathbb{R},L^2\right)$, defined as $$\label{me3.21}
\begin{gathered}
\boldsymbol{H}(s,t)=\sum_n\left[Wu_n\right](s)\overline{\left[Vu_n\right](t)},
\\
\boldsymbol{h}(s)=\overline{\boldsymbol{H}(s,\cdot)}=\sum_n\overline{
\left[Wu_n\right](s)}Vu_n,
\\
\boldsymbol{h}^{\boldsymbol{\prime}}(t)=\boldsymbol{H}(\cdot,t)=
\sum_n \overline{\left[Vu_n\right](t)}Wu_n,
\end{gathered}$$ have the desired expansions , for all $i$, $j$, and hence that $$\label{meconi}
D_2^jD_1^i\boldsymbol{H}\in C\left(\mathbb{R}^2,\mathbb{C}\right),\quad
\boldsymbol{h}^{(i)},\ \left(\boldsymbol{h}^{\boldsymbol{\prime}}\right)^{(j)}\in
C\left(\mathbb{R},L^2\right)$$ for all $i$, $j$. Since, moreover, each mixed partial derivative of $\left[Wu_n\right](s)\overline{\left[Vu_n\right](t)}$ is everywhere independent of the sequence in which the partial differentiations with respect to $s$ and $t$ are carried out, it follows that not only those of the form as in but also all other partial derivatives of $\boldsymbol{H}$ belong to $C\left(\mathbb{R}^2,\mathbb{C}\right)$, hereby showing conclusively that $\boldsymbol{H}$ is a $K^\infty$ kernel.
Now fix any $f\in L^2$, and observe then that $$\label{Tf}
Tf=WV^*f=\sum_n\left\langle f,Vu_n\right\rangle_{L^2}Wu_n,$$ where the series converges to $Tf$ in $L^2$. On the other hand, the convergence properties of the series of make it possible to write, for each temporarily fixed $s\in\mathbb{R}$, the following chain of relations $$\begin{gathered}
\label{Kf}
\sum_n\left\langle f,Vu_n\right\rangle_{L^2}\left[Wu_n\right](s)
=\left\langle f,\sum_n\overline{\left[Wu_n\right](s)}Vu_n\right\rangle_{L^2}\\
=
\int_{\mathbb{R}}\left(\sum_n\left[Wu_n\right](s)\overline{\left[Vu_n\right](t)}\right)f(t)\,dt=
\int_{\mathbb{R}}\boldsymbol{H}(s,t)f(t)\,dt.\end{gathered}$$ Because of the assumptions (A), (B) made about the functions $\left[Wu_n\right]$, and also because of the inequality $$\left(\sum_{n=p}^r\left|\left\langle f,Vu_n\right\rangle_{L^2}\left[Wu_n\right]^{(i)}(s)\right|\right)^2
\leq\sum_{n=p}^r\left|\left\langle V^*f,u_n\right\rangle_{L^2}\right|^2
\sum_{n=p}^r\left|\left[Wu_n\right]^{(i)}(s)\right|^2,$$ the first series of is $\mathbb{C}$-absolutely convergent in $C(\mathbb{R},\mathbb{C})$ and can be differentiated termwise any number of times while retaining this type of convergence. Comparison of with shows then that the series representation holds with $\lambda=0$ for all $i$, and that $$(Tf)(s)=\int_{\mathbb{R}}\boldsymbol{H}(s,t)f(t)\,dt\quad
\text{for almost every $s\in\mathbb{R}$}.$$ Since $f\in L^2$ was arbitrary, the latter equality means that the operator $T$ (which is equal to the Fredholm resolvent $T_{\lambda}$ at $\lambda=0$) is an integral operator with the function $\boldsymbol{H}$ as its kernel, so, by the uniqueness of the kernel, $\boldsymbol{H}=\boldsymbol{T}_0$ in the $C\left(\mathbb{R}^2,\mathbb{C}\right)$ sense, and $\boldsymbol{h}=\boldsymbol{t}_0$, $\boldsymbol{h}^{\boldsymbol{\prime}}=\boldsymbol{t}^{\boldsymbol{\prime}}_0$ in the $C\left(\mathbb{R},L^2\right)$ sense. The final conclusion is thus that, when $\lambda=0$, assertion (b) in the theorem will follow once we show that the functions $Vu_n$, $Wu_n$ ($n\in\mathbb N$) do in fact enjoy properties (A) and (B). This is the object of the next two steps.
*Step 2.* This step consists of proving that, under the assumptions made about $V$ and $W$ prior to Step 1, the series convergence properties stated in (B) always hold whenever the function properties stated in (A) are met, that is, that (A) implies (B). The proof is further given only for the first series of (B), as the proof for the second series, $\sum_n\left|\left[Wu_n\right]^{(i)}\right|^2$, is entirely similar.
If $\ell$ is a non-negative integer, $m$ is a positive integer, and $Q=[a,b]\times[c,d]$ is a compact rectangle of $\mathbb{R}^2$, we define the three quantities $$\begin{gathered}
q_1(Q,m;\ell)=\iint\limits_{Q}
\left(\left(D_{1,2}^\ell\boldsymbol{F}\right)(s,t)-
\sum_{n\le m}\left[Vu_n\right]^{(\ell)}(s)\overline{\left[Vu_n\right]^{(\ell)}(t)}\right)\,ds\,dt,
\label{main1}
\\
q_2(Q,m;\ell)=\sum_{n>m} \int\limits_a^b\left[Vu_n\right]^{(\ell)}(s)\,ds
\int\limits_c^d\overline{\left[Vu_n\right]^{(\ell)}(t)}\,dt,
\label{main2}
\\
q_3(Q,m;\ell)=\iint\limits_{Q}\left(\sum_{n>m} \left[Vu_n\right]^{(\ell)}(s)
\overline{\left[Vu_n\right]^{(\ell)}(t)}\right)\,ds\,dt,
\label{main3}\end{gathered}$$ and prove that, for each $\ell$, $$\label{q1q2}
q_1(Q,m;\ell)=q_2(Q,m;\ell)\quad \text{for all $Q$ and all $m$.}$$ First, for this purpose, utilize the $L^2$ representation $$Ff=VV^*f=\sum_n\left\langle f,Vu_n\right\rangle_{L^2}Vu_n\quad(f\in L^2)$$ in order to write, for each $Q$ and $m$, $$\begin{gathered}
q_1(Q,m;0)
\\
=\sum_n\left\langle Vu_n,\chi_{[a,b]}\right\rangle_{L^2}
\left\langle\chi_{[c,d]},Vu_n\right\rangle_{L^2}-
\sum_{n\le m} \left\langle Vu_n,\chi_{[a,b]}\right\rangle_{L^2}
\left\langle\chi_{[c,d]},Vu_n\right\rangle_{L^2}
\\
=\sum_{n>m}\left\langle Vu_n,\chi_{[a,b]}\right\rangle_{L^2}
\left\langle\chi_{[c,d]},Vu_n\right\rangle_{L^2}
=q_2(Q,m;0),\end{gathered}$$ where $\chi_E$ denotes the characteristic function of a set $E$. This implies that the identity holds with $\ell=0$. Proceeding by induction over $\ell$, suppose identity to be satisfied for some fixed $\ell$. The stage is now set for the induction step.
It is first necessary to remark that the integrand in must be non-negative on the main diagonal of $\mathbb{R}^2$, that is, denoting the integrand by $\boldsymbol{F}_m^{\ell}(s,t)$: $$\label{nonneg}
\boldsymbol{F}_m^{\ell}(s,s)=\left(D_{1,2}^\ell\boldsymbol{F}\right)(s,s)-
\sum_{n\le m}\left|\left[Vu_n\right]^{(\ell)}(s)\right|^2
\ge0\quad\text{for all $s\in\mathbb{R}$.}$$ Indeed, if it were not so, there would exist a square $Q^\prime$, with centre at some point on the main diagonal of $\mathbb{R}^2$ and sides parallel to the coordinate axes, such that its corresponding quantity $q_1(Q^\prime,m;\ell)$ would be negative, contradicting the non-negativity of $q_2(Q^\prime,m;\ell)$ which in turn would be implied by coincidence of the integration intervals on the right side of . In more detail, if $\boldsymbol{F}_m^{\ell}(s_0,s_0)=-2\delta$ for some $s_0$ and $\delta>0$, then by the continuity of $\boldsymbol{F}_m^{\ell}$ on $\mathbb{R}^2$, which follows from that of $D_{1,2}^\ell\boldsymbol{F}$ and from property (A), there is a square $Q^\prime$, $$Q^\prime=\left\{(s,t)\in\mathbb{R}^2: |s-s_0|\le\varepsilon/2,\,|t-s_0|\le
\varepsilon/2\right\}
\quad(\varepsilon>0)$$ in which $\mathrm{Re}\,\boldsymbol{F}_m^{\ell}(s,t)<-\delta$. By the induction hypothesis $$\label{Q1Q2}
\iint\limits_{Q^\prime}\boldsymbol{F}_m^{\ell}(s,t)\,ds\,dt=
q_1(Q^\prime,m;\ell)=q_2(Q^\prime,m;\ell)\geq0$$ so $0\leq\iint\limits_{Q^\prime}\mathrm{Re}\,\boldsymbol{F}_m^{\ell}(s,t)\,ds\,dt
<-\delta\varepsilon^2<0,$ a contradiction. (Note: that the diagonal value, $\boldsymbol{F}_m^{\ell}(s,s)$, of the integrand in at any point $s\in\mathbb{R}$ and the value of the integral in taken over any square like $Q^\prime=[a,b]\times[a,b]$ both have, for any $\ell$ and $m$, a zero imaginary part comes from the Hermiticity of $\boldsymbol{F}_m^{\ell}$, $\boldsymbol{F}_m^{\ell}(s,t)=\overline{\boldsymbol{F}_m^{\ell}(t,s)}$ for all $s$, $t\in\mathbb{R}$. This latter property is mainly inherited from that of $\boldsymbol{F}$ ($\boldsymbol{F}(s,t)=\overline{\boldsymbol{F}(t,s)}$ for all $s$, $t\in\mathbb{R}$ because of the self-adjointness of $F$) thanks to assumption (i) about $\boldsymbol{F}$, as follows: $
\left(D_{1,2}^\ell\boldsymbol{F}\right)(s,t)=
\overline{\left(D_{2,1}^\ell\boldsymbol{F}\right)(t,s)}=
\overline{\left(D_{1,2}^\ell\boldsymbol{F}\right)(t,s)}
$ for all $s$, $t\in\mathbb{R}$.)
Further, it is seen from that, for each $m$, there is the inequality $$\label{mainineq}
\sum_{n\le m}\left|\left[Vu_n\right]^{(\ell)}(s)\right|^2\le
\left(D_{1,2}^\ell\boldsymbol{F}\right)(s,s)\le C_\ell=
\sup_{s\in\mathbb{R}}\left(D_{1,2}^\ell\boldsymbol{F}\right)(s,s)
\quad(s\in\mathbb{R}),$$ from which it follows in particular that the series in the integrand of is termwise integrable, implying (via and ) that $q_1(Q,m;\ell)=q_3(Q,m;\ell)$ for all $m$ and all $Q$. In turn, this new identity implies that $$\label{lFst}
\left(D_{1,2}^\ell\boldsymbol{F}\right)(s,t)=\sum_n\left[Vu_n\right]^{(\ell)}(s)
\overline{\left[Vu_n\right]^{(\ell)}(t)}$$ almost everywhere in $\mathbb{R}^2$. The series here converges by uniformly in each variable separately for all values of the other, and its sum-function, denoted by $\boldsymbol{S}_\ell$, is therefore a continuous function of either argument. One can now appeal directly to a subtle Example 2 from [@Zaanenb Ch. 14, pp. 545-546], as applied to functions $f_1=D_{1,2}^\ell\boldsymbol{F}$ and $f_2=\boldsymbol{S}_\ell$, in order to be sure that $$\label{Dl12Fss}
\left(D_{1,2}^\ell\boldsymbol{F}\right)(s,s)
=\sum\limits_n\left|\left[Vu_n\right]^{(\ell)}(s)\right|^2\quad \text{for all $s\in\mathbb{R}$.}$$ Since $\left(D_{1,2}^\ell\boldsymbol{F}\right)(s,s)\to 0$ as $|s|\to\infty$, Dini’s Monotone Convergence Theorem may now be applied to the 1-point compactification of $\mathbb{R}$, to yield the conclusion that the series of does converge in $C(\mathbb{R},\mathbb{C})$ (compare this result with property (B)). In particular, it follows that the series of is converging in $C\left(\mathbb{R}^2,\mathbb{C}\right)$ to $D_{1,2}^\ell\boldsymbol{F}$. That, in turn, justifies the following computation $$\begin{gathered}
q_1(Q,m;\ell+1)
\\=\left(D_{1,2}^\ell\boldsymbol{F}\right)(b,d)-\left(D_{1,2}^\ell\boldsymbol{F}\right)(b,c)-
\left(D_{1,2}^\ell\boldsymbol{F}\right)(a,d)+\left(D_{1,2}^\ell\boldsymbol{F}\right)(a,c)\\
-\sum_{n\le m}\left(\left[Vu_n\right]^{(\ell)}(b)-\left[Vu_n\right]^{(\ell)}(a)\right)
\left(\overline{\left[Vu_n\right]^{(\ell)}(d)-\left[Vu_n\right]^{(\ell)}(c)}\right)
\\
=\sum_n\left(\left[Vu_n\right]^{(\ell)}(b)\overline{\left[Vu_n\right]^{(\ell)}(d)}
-\left[Vu_n\right]^{(\ell)}(b)\overline{\left[Vu_n\right]^{(\ell)}(c)}\right.
\\ \left. -\left[Vu_n\right]^{(\ell)}(a)\overline{\left[Vu_n\right]^{(\ell)}(d)}
+\left[Vu_n\right]^{(\ell)}(a)\overline{\left[Vu_n\right]^{(\ell)}(c)}\right)
\\-\sum_{n\le m}\left(\left[Vu_n\right]^{(\ell)}(b)-\left[Vu_n\right]^{(\ell)}(a)\right)
\left(\overline{\left[Vu_n\right]^{(\ell)}(d)-\left[Vu_n\right]^{(\ell)}(c)}\right)
\\=\sum_{n>m}\left(\left[Vu_n\right]^{(\ell)}(b)-\left[Vu_n\right]^{(\ell)}(a)\right)
\left(\overline{\left[Vu_n\right]^{(\ell)}(d)-\left[Vu_n\right]^{(\ell)}(c)}\right)
\\=q_2(Q,m;\ell+1),\end{gathered}$$ which proves that with $\ell+1$ instead of $\ell$ holds true for all $m$ and $Q=[a,b]\times[c,d]$. Therefore, by induction, the identity is true for every non-negative integer $\ell$. Hence, as implies the $C(\mathbb{R},\mathbb{C})$ convergence of the series of by what has just been seen in the course of the induction step, the first series of (B) converges in $C(\mathbb{R},\mathbb{C})$ for each fixed $i$.
*Step 3.* In this step, the proof of assertion (b) will be completed for $\lambda=0$, by showing that, under the conditions laid on $V$ and $W$ at the beginning of the proof, property (A) always holds. We shall restrict ourselves to dealing only with the functions $Vu_n$ ($n\in\mathbb{N}$), because the proof of (A) for $Wu_n$ ($n\in\mathbb{N}$) can be obtained in a similar way, but using respectively $\mathrm{Ran\,}W^*$ and $\boldsymbol{G}$ instead of $\mathrm{Ran\,}V^*$ and $\boldsymbol{F}$.
Choose $\left\{v_k\right\}$ to be an orthonormal basis for the subspace $\overline{\mathrm{Ran\,} V^*}$, with the property that $\left\{v_k\right\}\subset\mathrm{Ran\,}V^*$, and let $\left\{\widetilde{u}_n\right\}$ be any orthonormal basis for $L^2$ such that $\left\{v_k\right\}\subset\left\{\widetilde{u}_n\right\}$. Observe that $v_k\in\mathrm{Ran\,}V^*$ implies $Vv_k=Ff_k$ for some $f_k\in L^2$. Therefore, by property (ii) for the Carleman function $\boldsymbol{f}(s)=\overline{\boldsymbol{F}(s,\cdot)}$, there is in the equivalence class $Vv_k$ a function of $C(\mathbb{R},\mathbb{C})$, namely $\left[Vv_k\right](s)=\left\langle f_k,\boldsymbol{f}(s)\right\rangle_{L^2}$ ($s\in\mathbb{R}$), such that its every derivative, $\left[Vv_k\right]^{(i)}(\cdot)=
\left\langle f_k,\boldsymbol{f}^{(i)}(\cdot)\right\rangle_{L^2}$, is also in $C(\mathbb{R},\mathbb{C})$. Then, since every $V\widetilde{u}_n$ is equal either to $Vv_k$, for some $k$, or to the zero function and therefore all $V\widetilde{u}_n$ satisfy (A), the reasoning of Step 2 can be applied to the function system $\{V\widetilde{u}_n\}$ in place of $\{Vu_n\}$ in order to arrive at the conclusion that, for each $i$, the series $\sum_n\left|\left[V\widetilde{u}_n\right]^{(i)}\right|^2$ converges in $C(\mathbb{R},\mathbb{C})$ (cf. (B)). After that, one can conclude immediately that each series of the form $\sum_n\left\langle f,\widetilde{u}_n\right\rangle_{L^2}\left[V\widetilde{u}_n\right]^{(i)}$ $\left(f\in L^2\right)$ also converges in $C(\mathbb{R},\mathbb{C})$, and hence defines a $C(\mathbb{R},\mathbb{C})$ function, which is nothing else than $\left[Vf\right]^{(i)}$ because $Vf=\sum_n\left\langle f,\widetilde{u}_n\right\rangle_{L^2}V\widetilde{u}_n$ in the sense of convergence in $L^2$. In particular, $\left[Vu_n\right]^{(i)}\in C(\mathbb{R},\mathbb{C})$, for all $n\in\mathbb{N}$ and all $i$.
*Step 4.* Now let $\lambda$ be an arbitrary non-zero regular value for $T$, and factorize the Fredholm resolvent $T_\lambda$ of $T$ at $\lambda$ in the form $T_\lambda=W_\lambda V^*$ where $W_\lambda=R_\lambda(T)W$ (see , ). This factorization need not be an $\mathcal{M}$ factorization for $T_\lambda$. If, however, the operator $G_\lambda=W_\lambda\left(W_\lambda\right)^*$ is known to be integral with a $K^\infty$ kernel, then the previous three steps of the proof may easily be adapted, with $W_\lambda$ written instead of $W$, to show that the formulae - all hold exactly as stated in the theorem.
Let us therefore focus attention on the operator $G_\lambda$. That this operator is Carleman follows form the representation $$\begin{gathered}
\label{frres}
G_\lambda=W_\lambda \left(W_\lambda\right)^*
=G +\lambda TR_\lambda(T)G+\bar\lambda G\left(R_\lambda(T)\right)^*T^*
\\
+\left|\lambda\right|^2TR_\lambda(T) G\left(R_\lambda(T)\right)^*T^*\end{gathered}$$ which may be established with the aid of the equality $R_\lambda(T)=I_{L^2}+\lambda TR_\lambda(T)$ (cf. ), and in which each term is a Carleman operator by the right-multiplication lemma. In addition, if $\boldsymbol{g}$ is the Carleman function corresponding with the $K^\infty$ kernel $\boldsymbol{G}$ of the integral operator $G=WW^*$, then the function $\boldsymbol{g}_\lambda\colon \mathbb{R}\to L^2$ defined by $$\begin{gathered}
\boldsymbol{g}_\lambda(\cdot)=\boldsymbol{g}(\cdot)+\bar\lambda
\left(R_\lambda(T) G\right)^*\left(\boldsymbol{t}_0(\cdot)\right)+
\lambda\left(\left(R_\lambda(T)\right)^*T^*\right)^*
\left(\boldsymbol{g}(\cdot)\right)
\\+
\left|\lambda\right|^2\left(R_\lambda(T) G\left(R_\lambda(T)\right)^*T^*\right)^*
\left(\boldsymbol{t}_0(\cdot)\right)\end{gathered}$$ can be regarded as a Carleman function associated with the Carleman kernel of $G_\lambda$, by Proposition \[rimlt\] again. Since, for each $i$, $$\label{mecfder}
\boldsymbol{g}^{(i)}, \boldsymbol{t}_0^{(i)},
\left(\boldsymbol{t}^{\boldsymbol{\prime}}_0\right)^{(i)}\in C\left(\mathbb{R},L^2\right),$$ it follows that $$\begin{gathered}
\label{mederqp}
\boldsymbol{g}_\lambda^{(i)}(\cdot)=\boldsymbol{g}^{(i)}(\cdot)+\bar\lambda
\left(R_\lambda(T) G\right)^*\left(\boldsymbol{t}_0^{(i)}(\cdot)\right)+
\lambda\left(\left(R_\lambda(T)\right)^*T^*\right)^*
\left(\boldsymbol{g}^{(i)}(\cdot)\right)
\\+
\left|\lambda\right|^2\left(R_\lambda(T)G\left(R_\lambda(T)\right)^*T^*\right)^*
\left(\boldsymbol{t}_0^{(i)}(\cdot)\right)\in C\left(\mathbb{R},L^2\right),\end{gathered}$$ because the operators $R_\lambda(T)$, $G$, and $T$, are bounded. Analogously, for each $i$, $$\begin{gathered}\label{meforpart}
\left(\left(R_\lambda(T)\right)^*(\boldsymbol{t}_0(\cdot))\right)^{(i)}=
\left(R_\lambda(T)\right)^*\left(\boldsymbol{t}_0^{(i)}(\cdot)\right)
\in C\left(\mathbb{R},L^2\right),\\
\left(R_\lambda(T)(\boldsymbol{g}(\cdot))\right)^{(i)}=
R_\lambda(T)\left(\boldsymbol{g}^{(i)}(\cdot)\right)\in
C\left(\mathbb{R},L^2\right),\\
R_\lambda(T) G\left(R_\lambda(T)\right)^*
\left(\boldsymbol{t}_0(\cdot))\right)^{(i)}=
R_\lambda(T) G\left(R_\lambda(T)\right)^*
\left(\boldsymbol{t}_0^{(i)}(\cdot)\right)
\in C\left(\mathbb{R},L^2\right).
\end{gathered}$$ Note also that each of the last three terms in is a product of two Carleman operators. From this it becomes possible to express the inducing Carleman kernel $\boldsymbol{G}_\lambda$ of $G_\lambda$ by means of convolutions as follows: $$\begin{gathered}
\boldsymbol{G}_\lambda(s,t)=
\boldsymbol{G}(s,t)+\lambda\left\langle\boldsymbol{g}(t),
\left(R_\lambda(T)\right)^*\left(\boldsymbol{t}_0(s)\right)\right\rangle_{L^2}+\bar\lambda
\left\langle\boldsymbol{t}^{\boldsymbol{\prime}}_0(t),R_\lambda(T)(\boldsymbol{g}(s))\right\rangle_{L^2}
\\+
\left|\lambda\right|^2\left\langle\boldsymbol{t}^{\boldsymbol{\prime}}_0(t),R_\lambda(T) G
\left(R_\lambda(T)\right)^*\left(\boldsymbol{t}_0(s)\right)\right\rangle_{L^2}.\end{gathered}$$ Then, for each fixed $i$, $j$, a partial differentiation of $\boldsymbol{G}_\lambda$ yields, after the equalities of are taken into account, the following equality on $\mathbb{R}^2$: $$\begin{gathered}
\left(D_2^jD_1^i\boldsymbol{G}_\lambda\right)(s,t)=
\left(D_2^jD_1^i\boldsymbol{G}\right)(s,t)
+
\lambda\left\langle\boldsymbol{g}^{(j)}(t),
\left(R_\lambda(T)\right)^*\left(\boldsymbol{t}_0^{(i)}(s)\right)\right\rangle_{L^2}
\\+
\bar\lambda\left\langle\left(\boldsymbol{t}^{\boldsymbol{\prime}}_0\right)^{(j)}(t),R_\lambda(T)
\left(\boldsymbol{g}^{(i)}(s)\right)\right\rangle_{L^2}
\\+
\left|\lambda\right|^2\left\langle\left(\boldsymbol{t}^{\boldsymbol{\prime}}_0\right)^{(j)}(t),
R_\lambda(T) G\left(R_\lambda(T)\right)^*\left(\boldsymbol{t}_0^{(i)}(s)\right)
\right\rangle_{L^2};\end{gathered}$$ clearly we are free to compute each term on the right-hand side here by alternatively applying $D_2^1$ and $D_1^1$ in any order we please. Hence, according to , , and by the continuity of the inner product, the partial derivatives of $\boldsymbol{G}_\lambda$ are all in $C\left(\mathbb{R}^2,\mathbb{C}\right)$. This acquisition in conjunction with implies that $\boldsymbol{G}_\lambda$ is a $K^\infty$ kernel, thereby completing the proof of assertion (b) of the theorem.
Corollaries and Applications {#faccol}
============================
\[moreuse\] An easier way to see that $G_\lambda=W_\lambda\left(W_\lambda\right)^*$ in is an integral operator with a $K^\infty$ kernel is by making more use of the assumption in Theorem \[mfactor\] that the $K^\infty$ kernel $\boldsymbol{T}_0$ of $T$ is of Mercer type, that is, that the operator family $\mathcal{M}(T)$ consists only of integral operators with $K^\infty$ kernels. The argument might be as follows: the first and last terms in the right-hand side of clearly both belong to $\mathcal{M}^{+}(T)$ (see and Remark \[remmt\]). The second term $\lambda TR_\lambda(T)G$, and hence its adjoint $\bar\lambda G\left(R_\lambda(T)\right)^*T^*$ (which is just the third one), does belong to $\mathcal{M}(T)$, because $\lambda TR_\lambda(T)G=
\lambda TR_\lambda(T)B^*T^*\in T\mathfrak{R}\left(L^2\right)
\cap\mathfrak{R}\left(L^2\right)T^*\subset\mathcal{M}(T)$ or $\lambda TR_\lambda(T)G=\lambda TR_\lambda(T)BT\in
T\mathfrak{R}\left(L^2\right)\cap\mathfrak{R}\left(L^2\right)T\subset
\mathcal{M}(T)$ according as $G=TB$ or $G=BT$ (see again Remark \[remmt\]). Each summand on the right of is thus an integral operator with a $K^\infty$ kernel and hence so is $G_\lambda$ itself.
In addition to Remark \[moreuse\] let us confess that also in the course of the whole proof of assertion (b) in Theorem \[mfactor\] we have not used the full strength of the assumption on the $K^\infty$ kernel $\boldsymbol{T}_0$ of being of Mercer type, but only a consequence of it, namely the existence of inducing $K^\infty$ kernels for $F=VV^*$ and $G=WW^*$, whenever $T=WV^*$ is an $\mathcal{M}$ factorization for $T$. So what that proof really proves is the following, more general, result intended for the case where $T=WV^*$ is not necessarily an $\mathcal{M}$ factorization for, at first, not necessarily integral $T$.
\[mfactor3\] Suppose that an operator $T\in\mathfrak{R}\left(L^2\right)$ has a factorization into a product $T=WV^*$ $(V, W\in \mathfrak{R}\left(L^2\right))$ such that both $F=VV^*$ and $G=WW^*$ are integral operators with $K^\infty$ kernels. Then at each regular value $\lambda\in\Pi(T)$ the Fredholm resolvent $T_\lambda$ of $T$ is an integral operator and its kernel $\boldsymbol{T}_\lambda$ is a $K^\infty$ kernel. Moreover, for any orthonormal basis $\{u_n\}$ for $L^2$, the following formulae hold $$\begin{gathered}
\left(D_2^jD_1^i\boldsymbol{T}_\lambda\right)(s,t)=
\sum_n\left[R_\lambda(T)Wu_n\right]^{(i)}(s)
\overline{\left[Vu_n\right]^{(j)}(t)}, \label{meijT13}
\\
\begin{split} \label{meijkt23}
\boldsymbol{t}_\lambda^{(i)}(s)&=\sum_n\overline{\left[R_\lambda(T)Wu_n\right]^{(i)}(s)}Vu_n,
\\
\left(\boldsymbol{t}^{\boldsymbol{\prime}}_\lambda\right)^{(j)}(t)&=
\sum_n \overline{\left[Vu_n\right]^{(j)}(t)}R_\lambda(T)Wu_n,
\end{split}
\\
\left[T_\lambda f\right]^{(i)}(s)=
\sum_n\left\langle f,Vu_n\right\rangle_{L^2}\left[R_\lambda(T)Wu_n\right]^{(i)}(s)\label{meijkTf33},\end{gathered}$$ for all $i$, $j$, all $s$, $t\in\mathbb{R}$, and all $f\in L^2$, where the series of converges $\mathbb{C}$-absolutely in $C\left(\mathbb{R}^2,\mathbb{C}\right)$, the two series of both converge in $C\left(\mathbb{R},L^2\right)$, and the series of converges $\mathbb{C}$-absolutely in $C(\mathbb{R},\mathbb{C})$.
The proof is exactly the same as the proof given in the previous section for assertion (b) of Theorem \[mfactor\]. Corollary \[mfactor3\] also opens a slightly different, but equivalent, way to define a $K^\infty$ kernel of Mercer type, involving families $\mathcal{M}^{+}(\cdot)$ in place of $\mathcal{M}(\cdot)$ (cf. Definition \[MerKernel\]):
\[cormer\] An operator $T\in\mathfrak{R}\left(L^2\right)$ is an integral operator with a $K^\infty$ kernel of Mercer type if and only if the family $\mathcal{M}^{+}(T)$ consists only of integral operators with $K^\infty$ kernels.
The “only if” is immediate from the inclusion $\mathcal{M}^{+}(T)\subset\mathcal{M}(T)$. For the proof of the “if”, assume that every operator $A\in\mathcal{M}^{+}(T)$ is an integral operator with a $K^\infty$ kernel. If $S\in\mathcal{M}(T)$, it is to be proved that $S$ is an integral operator with a $K^\infty$ kernel. For this, let $S=WV^*$ be an $\mathcal{M}$ factorization for $S$, where, according to and Remark \[remmt\], $VV^*$, $WW^*\in\mathcal{M}^{+}(S)\subseteq\mathcal{M}^{+}(T)$. Then it follows by assumption that both $F=VV^*$ and $G=WW^*$ are integral operators with $K^\infty$ kernels. Apply Corollary \[mfactor3\] to the operator $S=WV^*$ and when $\lambda=0$ conclude that $S$ is an integral operator with a $K^\infty$ kernel. The corollary is proved.
\[rDost\] The next result, Dostanić’s [@Dost89]-[@Dost93] extension of Mercer’s Theorem to a class of continuous non-Hermitian kernels on $[0,1]^2$, deserves mention because it turns out to also fit into the general bilinear expansion scheme proved in Section \[prmfactor\].
\[pureDost\] If $T=H\left(I_{L^2(0,1)}+S\right)$, where $H\ge0$, $S=S^*$ are integral operators induced on $L^2(0,1)$ by continuous kernels on $[0,1]^2$, and if either [(I)]{} $I_{L^2(0,1)}+S$ is positive and invertible, or [(II)]{} $I_{L^2(0,1)}+S$ is invertible and $H$ is one-to-one, then the kernel $\boldsymbol{T}_0$ of the integral operator $T$ is represented on $[0,1]^2$ by the absolutely and uniformly convergent series $$\label{Dost:form}
\boldsymbol{T}_0(s,t)=\sum_n \lambda_n\psi_n(s)\overline{\varphi_n(t)},$$ where $\{\psi_n\}$ and $\{\varphi_n\}$ are biorthogonal systems of eigenfunctions for $T$ and for $T^*$, respectively: $T\psi_n=\lambda_n\psi_n$, $T^*\varphi_m=\lambda_m\varphi_m$, $\langle\psi_n,\varphi_m\rangle_{L^2(0,1)}=\delta_{nm}$ (Kronecker delta).
The proof of this result rests on an eigenvalue-eigenfunction analysis (see, e.g., [@Gohberg:Krejn Chapter 5, §8]) of compact operators that are self-adjoint with respect to the definite or indefinite inner product $[f,g]_{L^2(0,1)}=\langle(I_{L^2(0,1)}+S)f,g\rangle_{L_2(0,1)}$ according as case (I) or case (II) is in question. Alternatively, the result may be proved without direct recourse to that analysis by using a factorization argument similar to that used for Theorem \[mfactor\](b); a possible outline of a proof may be roughly sketched as follows. First note that the integral operator $T$ in Proposition \[pureDost\] has an $\mathcal{M}$ factorization $T=WV^*$ such that both $VV^*$ and $WW^*$ are integral operators with continuous kernels on $[0,1]^2$. Indeed, define the factors by $V=(I_{L^2(0,1)}+S)^{\frac12}\Lambda^{\frac12}$, $W=(I_{L^2(0,1)}+S)^{-\frac12}\Lambda^{\frac12}$, where $\Lambda=(I_{L^2(0,1)}+S)^{\frac12}H(I_{L^2(0,1)}+S)^{\frac12}$, in case (I), or by $V=(I_{L^2(0,1)}+S)H^{\frac12}$, $W=H^{\frac12}$ in case (II); in both cases, then, $VV^*=(I_{L^2(0,1)}+S)T$, $WW^*=T(I_{L^2(0,1)}+S)^{-1}=H$, and $T=WV^*$, where the integral operators $H$, $S$, and $T$, are known to have continuous kernels. Having thus factorized the integral operator $T$, it can be proved by adapting arguments in Section \[prmfactor\] that any series of the form $\sum_nWu_n(s)\overline{Vu_n(t)}$, where $\{u_n\}$ is an orthonormal basis in $L^2(0,1)$, converges absolutely and uniformly to the kernel $\boldsymbol{T}_0$ of $T$. Thus, in order to prove the desired convergence behavior of representation in Proposition \[pureDost\], it suffices to prove that $\lambda_n\psi_n(s)\overline{\varphi_n(t)}=Wu_n(s)\overline{Vu_n(t)}$ ($n\in\mathbb{N}$) for some orthonormal basis $\{u_n\}$. It is a straightforward calculation to show that in case (I) (resp., (II)) such a $\{u_n\}$ can be chosen to be an orthonormal basis in $L^2(0,1)$ with respect to which the compact, self-adjoint operator $(I_{L^2(0,1)}+S)^{\frac12}H(I_{L^2(0,1)}+S)^{\frac12}$ (resp., $H^{\frac12}(I_{L^2(0,1)}+S)H^{\frac12}$) diagonalizes (see proof of Corollary \[gDost\] below for more details).
The following generalization of Proposition \[pureDost\] is included as an application of Corollary \[mfactor3\].
\[gDost\] Let $T=H\left(I_{L^2}+S\right)$, where $0\le H\in\mathfrak{R}\left(L^2\right)$, $S\in\mathfrak{R}(L^2)$, and both $H$ and $(I_{L^2}+S^*)H(I_{L^2}+S)$ are integral operators with $K^\infty$ kernels. Then at each regular value $\lambda\in\Pi(T)$ the Fredholm resolvent $T_\lambda$ of $T$ is an integral operator whose kernel $\boldsymbol{T}_\lambda$ is a $K^\infty$ kernel. If, in addition, $\Lambda=H^{\frac12}(I_{L^2}+S)H^{\frac12}$ is a diagonal operator with diagonal entries $\lambda_1,\lambda_2,\lambda_3,\dots$, then there are functions $\psi_n$, $\varphi_n$ ($n\in\mathbb{N}$) in $L^2$ for which the following formulae hold: $T\psi_n=\lambda_n\psi_n$, $T^*\varphi_m=\bar\lambda_m\varphi_m$, $\langle\psi_n,\varphi_m\rangle_{L^2}
=\lambda_n\delta_{nm}$ for all $n$, $m\in\mathbb{N}$, and $$\begin{gathered}
\left(D_2^jD_1^i\boldsymbol{T}_\lambda\right)(s,t)=
\sum_n\frac1{1-\lambda\lambda_n}\left[\psi_n\right]^{(i)}(s)
\overline{\left[\varphi_n\right]^{(j)}(t)}, \label{meijT14}
\\
\begin{split} \label{meijkt24}
\boldsymbol{t}_\lambda^{(i)}(s)&=\sum_n
\frac1{1-\overline{\lambda}\overline{\lambda_n}}
\overline{\left[\psi_n\right]^{(i)}(s)}\varphi_n,
\\
\left(\boldsymbol{t}^{\boldsymbol{\prime}}_\lambda\right)^{(j)}(t)&=
\sum_n\frac1{1-\lambda\lambda_n}
\overline{\left[\varphi_n\right]^{(j)}(t)}\psi_n,
\end{split}
\\
\left[T_\lambda f\right]^{(i)}(s)=
\sum_n\frac1{1-\lambda\lambda_n}\left\langle f,\varphi_n\right\rangle_{L^2}
\left[\psi_n\right]^{(i)}(s)\label{meijkTf34},\end{gathered}$$ for all $\lambda\in\Pi(T)$, all $i$, $j$, and all $s$, $t\in\mathbb{R}$, and all $f\in L^2$, where the series of converges $\mathbb{C}$-absolutely in $C\left(\mathbb{R}^2,\mathbb{C}\right)$, the two series of both converge in $C\left(\mathbb{R},L^2\right)$, and the series of converges $\mathbb{C}$-absolutely in $C(\mathbb{R},\mathbb{C})$.
Put $W=H^{\frac12}$, $V=(I_{L^2}+S^*)H^{\frac12}$, and let the orthonormal basis $\{u_n\}$ for $L^2$ allow the diagonal operator $\Lambda=H^{\frac12}(I_{L^2}+S)H^{\frac12}$ to be written as $$\label{LAMB}
\Lambda=\sum_n \lambda_n\left\langle\cdot,u_n\right\rangle_{L^2} u_n.$$ Since, by assumption, both operators $WW^*=H$ and $VV^*=(I_{L^2}+S^*)H(I_{L^2}+S)$ are integral and are defined by $K^\infty$ kernels, Corollary \[mfactor3\] can be applied with respect to factorization $T=H(I_{L^2}+S)=WV^*$ and basis $\{u_n\}$ to conclude that the Fredholm resolvent $T_\lambda$ of $T$ is an integral operator, with a kernel $\boldsymbol{T}_\lambda$ that is a $K^\infty$ kernel for which the expansion formulae - hold. By the change of notation: $\psi_n=Wu_n$, $\varphi_n=Vu_n$ ($n\in\mathbb{N}$), these formulae can be rewritten in forms -, respectively, because then $T\psi_n=H(I_{L^2}+S)H^{\frac12}u_n=H^{\frac12}\Lambda u_n=\lambda_n \psi_n$ by , and therefore $R_\lambda(T)\psi_n=\frac1{1-\lambda\lambda_n}\psi_n$. Moreover, $T^*\varphi_n=(I_{L^2}+S^*)HVu_n=(I_{L^2}+S^*)H(I_{L^2}+S^*)H^{\frac12}u_n
=(I_{L^2}+S^*)H^{\frac12}\Lambda^*u_n=\bar\lambda_n \varphi_n$ and $\left\langle\psi_n,\varphi_m\right\rangle_{L^2}=
\left\langle H^{\frac12}u_n,(I_{L^2}+S^*)H^{\frac12}u_m\right\rangle_{L^2}
=\left\langle\Lambda u_n,u_m\right\rangle_{L^2}
=\lambda_n\left\langle u_n,u_m\right\rangle_{L^2}=\lambda_n\delta_{nm}$ for all $m$, $n\in\mathbb{N}$. The corollary is proved.
For the next corollary, we recall that a family $\{\varphi_n\}$ of functions in $L^2$ is a *Riesz basis* (see [@Gohberg:Krejn]) for $L^2$ if there exist an invertible operator $A\in\mathfrak{R}(L^2)$ and an orthonormal basis $\{u_n\}$ for $L^2$ such that $\varphi_n=Au_n$ for all $n\in\mathbb{N}$.
\[Riesz\] Let $T\in\mathfrak{R}\left(L^2\right)$ be an integral operator induced by a $K^\infty$ kernel of Mercer type $\boldsymbol{T}_0$ and suppose that $T$ is representable as $$\label{merexp}
T=\sum_n \lambda_n\left\langle\cdot,\varphi_n\right\rangle_{L^2}\psi_n,\quad
\{\varphi_n\}\subset \{\varphi^\prime_n\},
\quad \{\psi_n\}\subset \{\psi^\prime_n\},$$ where $\{\varphi^\prime_n\}$ and $\{\psi^\prime_n\}$ are Riesz bases for $L^2$ and $\{\lambda_n\}$ is a family of complex numbers. Then $$\begin{gathered}
\left(D_2^jD_1^i\boldsymbol{T}_0\right)(s,t)=
\sum_n\lambda_n\left[\psi_n\right]^{(i)}(s)
\overline{\left[\varphi_n\right]^{(j)}(t)}, \label{meijT15}
\\
\boldsymbol{t}_0^{(i)}(s)=
\sum_n\overline{\lambda_n\left[\psi_n\right]^{(i)}(s)}\varphi_n,
\quad
\left(\boldsymbol{t}^{\boldsymbol{\prime}}_0\right)^{(j)}(t)=\sum_n\lambda_n
\overline{\left[\varphi_n\right]^{(j)}(t)}\psi_n,\label{meijkt25}
\\
\left[Tf\right]^{(i)}(s)=\sum_n\lambda_n\left\langle f,\varphi_n\right\rangle_{L^2}
\left[\psi_n\right]^{(i)}(s)\label{meijkTf35},\end{gathered}$$ for all $i$, $j$, and all $s$, $t\in\mathbb{R}$, and all $f\in L^2$, where the series of converges $\mathbb{C}$-absolutely in $C\left(\mathbb{R}^2,\mathbb{C}\right)$, the two series of both converge in $C\left(\mathbb{R},L^2\right)$, and the series of converges $\mathbb{C}$-absolutely in $C(\mathbb{R},\mathbb{C})$.
Assume, with no loss of generality, that $\lambda_n=\left|\lambda_n\right|e^{\imath\theta_n}\not=0$ for all $n\in\mathbb{N}$. Since both $\left\{\varphi^\prime_n\right\}$ and $\left\{\psi^\prime_n\right\}$ are Riesz bases, it follows that there exist two invertible operators $A$, $B\in\mathfrak{R}(L^2)$ and an orthonormal basis $\{u_n\}$ for $L^2$ such that $T=A\Lambda B$, where $$\label{ALB}
\Lambda=\sum_n \lambda_n\left\langle\cdot,u_{k_n}\right\rangle_{L^2}u_{m_n},
\quad u_{m_n}=A^{-1}\psi_n,\quad u_{k_n}=\left(B^*\right)^{-1}\varphi_n
\ (n\in\mathbb{N})$$ for some subsequences $\{m_n\}$, $\{k_n\}$ of the sequence $\{n\}_{n=1}^\infty$. Introduce operators $U$, $\left|\Lambda\right|
\in\mathfrak{R}\left(L^2\right)$, defined by $$U=\sum_n e^{\imath\theta_n}\left\langle\cdot,u_{k_n}\right\rangle_{L^2} u_{m_n},\quad
\left|\Lambda\right|=\sum_n \left|\lambda_n\right|\left\langle\cdot,u_{k_n}\right\rangle_{L^2} u_{k_n},$$ and it is clear that $\Lambda=U\left|\Lambda\right|$ and $\left|\Lambda\right|\ge0$. If $W=AU\left|\Lambda\right|^{\frac1{2}}$, $V=B^*\left|\Lambda\right|^{\frac1{2}}$, then $WW^*=AU\left|\Lambda\right| U^*A^*=TB^{-1}U^*A^*\in\mathcal{M}^{+}(T)$, $VV^*=B^*\left|\Lambda\right| B=B^*U^*A^{-1}T\in\mathcal{M}^{+}(T)$, and $T=A\Lambda B=AU\left|\Lambda\right|^{\frac1{2}}\left|\Lambda\right|^{\frac1{2}}B=WV^*$, that is to say $T=WV^*$ is an $\mathcal{M}$ factorization for $T$. In addition to this, for each $r\in\mathbb{N}$, $Wu_r=\delta_{rk_n}\left|\lambda_n\right|^{\frac12}AUu_{k_n}
=\delta_{rk_n}e^{\imath\theta_n}\left|\lambda_n\right|^{\frac12}Au_{m_n}
=\delta_{rk_n}e^{\imath\theta_n}\left|\lambda_n\right|^{\frac12}\psi_n$, and $Vu_r=B^*\left|\Lambda\right|^{\frac12}u_r
=\delta_{rk_n}\left|\lambda_n\right|^{\frac12}B^*u_{k_n}
=\delta_{rk_n}\left|\lambda_n\right|^{\frac12}\varphi_n$, by . Then $\left[Wu_r\right](s)\overline{\left[Vu_r\right](t)}
=\delta_{rk_n}\lambda_n[\psi_n](s)\overline{[\varphi_n](t)}$ for all $s$, $t\in\mathbb{R}$, so that formulae - stated in the theorem are already implied by corresponding formulae - in Theorem \[mfactor\](b) when the latter is applied, with $\lambda=0$, to the above $\mathcal{M}$ factorization $T=WV^*$ and basis $\{u_n\}$. The corollary is proved.
Since, at each fixed regular value $\lambda\in\Pi(T)$, the set $\left\{R_\lambda(T)\psi^\prime_n\right\}$ remains a Riesz basis for $L^2$ and, by Theorem \[mfactor\](a), the resolvent kernel $\boldsymbol{T}_\lambda$ for $\boldsymbol{T}_0$ is a $K^\infty$ kernel of Mercer type, formulae - continue to hold with $0$ and $\psi_n$ replaced respectively by $\lambda$ and $R_\lambda(T)\psi_n$, due to the same Corollary \[Riesz\].
When the Riesz bases $\{\varphi^\prime_n\}$ and $\{\psi^\prime_n\}$ are both orthonormal and when $\lambda_n\to0$ as $n\to\infty$, the assumed representation for $T$ is strongly reminiscent of Schmidt’s decomposition for compact operators (see [@Gohberg:Krejn], and below). For any $T\in\mathfrak{R}(\mathcal{H})$, meanwhile, there is a generalized Schmidt’s decomposition due to I. Fenyö [@Fenyo]:
\[fenyo\] For every operator $T\in \mathfrak{R}(\mathcal{H})$ there exist an orthonormal basis $\{x_n\}\subset\mathrm{Ran\,}T$ for $\overline{\mathrm{Ran\,}T}$, an orthonormal basis $\{y_n\}\subset \mathrm{Ran\,}T^*$ for $\overline{\mathrm{Ran\,}T^*}$, and bounded number sequences $\{\varkappa_n\}$, $\{\mu_n\}$ in $\mathbb{C}$, such that, for each $f\in\mathcal{H}$, $$\label{meT1}
Tf=\sum_n\alpha_n\langle f,v_n\rangle_{\mathcal{H}} x_n,\quad
Tf=\sum_{n}\beta_n\langle f,y_n\rangle_{\mathcal{H}} w_n$$ in the sense of convergence in $\mathcal{H}$, where $$\begin{gathered}
\alpha_n=\sqrt{|\varkappa_n|^2+|\mu_{n-1}|^2},\quad
v_n=\frac1{\alpha_n}\left(\overline \varkappa_ny_n+\overline \mu_{n-1} y_{n-1}\right)
\ (\mu_0=0),
\\
\beta_n=\sqrt{|\varkappa_n|^2+|\mu_n|^2},\quad
w_n=\frac1{\beta_n}\left(\varkappa_nx_n+\mu_nx_{n+1}\right).\end{gathered}$$
The elements $x_n$, $y_n\in\mathcal{H}$ ($n\in\mathbb{N}$) and the numbers $\varkappa_n$, $\mu_n\in\mathbb{C}$ ($n\in\mathbb{N}$), whose existence the proposition guarantees, can be determined simultaneously in a recursive way, as follows. Let $\varkappa_0=0$, and let $x_0=y_0=0$. If, for some positive integer $n$, the subset $\{x_j\}_{j=0}^{n-1}$ of $\mathrm{Ran\,}T$, the element $y_{n-1}\in\mathcal{H}$, and the number $\varkappa_{n-1}\in\mathbb{C}$, are already defined, and if $$\label{mespanker}
\mathrm{Span}\left(\{x_j\}_{j=0}^{n-1}\right)\not=\overline{\mathrm{Ran\,}T},$$ then let the number $\mu_{n-1}$ and the element $x_n$ of $\mathrm{Ran\,}T$ be chosen to satisfy $$\label{merules1}
\mu_{n-1}x_n=Ty_{n-1}-\varkappa_{n-1}x_{n-1}$$ subject to the restrictions $\sum_{j=0}^{n-1}|\langle x_n,x_j\rangle_{\mathcal{H}}|^2=0$, $\|x_n\|_{\mathcal{H}}=1$. Next, having made the proper choice of $\mu_{n-1}$ and $x_n$, let $\varkappa_n\in\mathbb{C}$ and $y_n\in\mathcal{H}$ be defined from $
\overline \varkappa_ny_n=T^*x_n-\overline \mu_{n-1}y_{n-1}
$ provided that $\|y_n\|_{\mathcal{H}}=1$. Either this process of determining $x_n$, $y_n$, $\mu_{n-1}$, and $\varkappa_n$, from the previously defined $\{x_j\}_{j=0}^{n-1}$, $y_{n-1}$, and $\varkappa_{n-1}$, is repeated indefinitely with unending growth of $n$, or it is terminated whenever inequality fails to hold. In any event, the process may generate different sequences $\{x_n\}$, $\{y_n\}$, $\{\varkappa_n\}$, and $\{\mu_n\}$, because of a freedom in choosing $x_n$ when the right side of happens to be zero. *We do not yet know whether, given a bi-integral operator $T$ on $\mathcal{H}=L^2$, there always exist operators $U_1$, $U_2\in\mathfrak{R}(L^2)$ such that $U_1v_n=x_n$ and $U_2w_n=y_n$ for all $n\in\mathbb{N}$, with the notation of .* If so, then an application of Theorem \[mfactor\](b) might yield the following result.
\[mefenyo\] If $T\in\mathfrak{R}(L^2)$ is an integral operator with a $K^\infty$ kernel $\boldsymbol{T}_0$ of Mercer type, then, with the notations of Proposition \[fenyo\], $$\begin{gathered}
\begin{split}\label{meijT16}
\left(D_2^jD_1^i\boldsymbol{T}_0\right)(s,t)=
\sum_n\alpha_n\left[x_n\right]^{(i)}(s)
\overline{\left[v_n\right]^{(j)}(t)},
\\
\left(D_2^jD_1^i\boldsymbol{T}_0\right)(s,t)=
\sum_n\beta_n\left[w_n\right]^{(i)}(s)
\overline{\left[y_n\right]^{(j)}(t)},
\end{split}
\\
\begin{split}\label{meijkt26}
\boldsymbol{t}_0^{(i)}(s)=
\sum_n\overline{\alpha_n\left[x_n\right]^{(i)}(s)}v_n,
\quad
\left(\boldsymbol{t}^{\boldsymbol{\prime}}_0\right)^{(j)}(t)=\sum_n\alpha_n
\overline{\left[v_n\right]^{(j)}(t)}x_n,
\\
\boldsymbol{t}_0^{(i)}(s)=
\sum_n\overline{\beta_n\left[w_n\right]^{(i)}(s)}y_n,
\quad
\left(\boldsymbol{t}^{\boldsymbol{\prime}}_0\right)^{(j)}(t)=\sum_n\beta_n
\overline{\left[y_n\right]^{(j)}(t)}w_n,
\end{split}
\\
\begin{split}\label{meijkTf36}
\left[Tf\right]^{(i)}(s)=\sum_n\alpha_n\left\langle f,v_n\right\rangle_{L^2}
\left[x_n\right]^{(i)}(s),
\\
\left[Tf\right]^{(i)}(s)=\sum_n\beta_n\left\langle f,y_n\right\rangle_{L^2}
\left[w_n\right]^{(i)}(s),
\end{split}\end{gathered}$$ for all $i$, $j$, all $s$, $t\in\mathbb{R}$, and all $f\in L^2$, where the series of converge $\mathbb{C}$-absolutely in $C\left(\mathbb{R}^2,\mathbb{C}\right)$, the series of converge in $C\left(\mathbb{R},L^2\right)$, and the series of converge $\mathbb{C}$-absolutely in $C(\mathbb{R},\mathbb{C})$.
Proof of Theorem \[infsmooth\] {#intrepr}
==============================
The proof is broken up into three steps. The first step is to specify a pair of orthonormal bases, $\left\{f_n\right\}$ for $\mathcal{H}$ and $\left\{u_n\right\}$ for $L^2$. The second step is to define a unitary operator from $\mathcal{H}$ onto $L_2$ by sending in a suitable manner the basis $\{f_n\}$ onto the basis $\{u_n\}$. This operator is suggested as $U$ in the theorem, and the third step of the proof is a straightforward verification that it is indeed as desired.
*Step 1.* Let $\mathcal{S}=\left\{S_\gamma\right\}_{\gamma\in\mathcal{G}}
\subset\mathfrak{R}(\mathcal{H})$ be a family satisfying for some orthonormal sequence $\left\{e_n\right\}_{n=1}^\infty$ in $\mathcal{H}$. Let us suppose that we have a pair of orthonormal bases: $\left\{f_n\right\}$ for $\mathcal{H}$ and $\left\{u_n\right\}$ for $L^2$, where the latter has the property that, for each $i$, the $i$-th derivative, $[u_n]^{(i)}$, of $[u_n]$ belongs to $C(\mathbb{R},\mathbb{C})$: $$\label{1}
\left[u_n\right]^{(i)}\in C(\mathbb{R},\mathbb{C})\quad(n\in\mathbb{N}).$$ Let us suppose further that each of these bases can be subdivided into two infinite subsequences: $\left\{f_n\right\}$ into $\{x_k\}_{k=1}^\infty$ and $\{y_k\}_{k=1}^\infty$, while $\left\{u_n\right\}$ into $\{g_k\}_{k=1}^\infty$ and $\{h_k\}_{k=1}^\infty$, such that $$\label{sumdxk}
\sum_k d(x_k)\le 2$$ and, for each $i$, $$\begin{gathered}
\label{hki}
\sum_k \left\|[h_k]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})}<\infty,
\\ \label{zndn}
\sum_k d(x_k)\left\|[g_k]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})}<\infty,\end{gathered}$$ where $$\label{dxk}
d(x_k):=2\left(\sup_{\gamma\in\mathcal{G}}\left\|S_\gamma x_k\right\|_{\mathcal{H}}^{1/4}+
\sup_{\gamma\in\mathcal{G}}\left\|\left(S_\gamma\right)^*x_k\right\|_{\mathcal{H}}^{1/4}\right)
\quad(k\in\mathbb{N}).$$ The proof will make use of the orthonormal bases $\left\{f_n\right\}$, $\left\{u_n\right\}$ just described to define the action of the desired unitary operator $U\colon\mathcal{H}\to L^2$ in the theorem. By using the assumption and some facts from the theory of wavelets, the following example affirmatively answers the existence question for such a pair of orthonormal bases.
Let $\mathbb{N}$ be decomposed into two infinite subsequences $\left\{l(k)\right\}_{k=1}^\infty$ and $\left\{m(k)\right\}_{k=1}^\infty$, and let $\{u_n\}$ be an orthonormal basis for $L^2$ that has property , and such that, for each $i$, $$\label{2}
\left\|\left[u_n\right]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})}\le N_nD_i
\quad(n\in\mathbb{N}),$$ where $\{N_n\}_{n=1}^\infty$, $\{D_i\}_{i=0}^\infty$ are two sequences of positive reals, the first of which is subject to the restriction that $$\label{3}
\sum_kN_{l(k)}<\infty.$$
Then the orthonormal basis $\{u_n\}$ can be paired (in the sense above) with an orthonormal basis $\{f_n\}$ for $\mathcal{H}$. Indeed, from conditions , it follows that the requirement may be satisfied for all $i$ by taking $h_k=u_{l(k)}$ ($k\in\mathbb{N}$). At the same time, on account of and of , the sequence $\left\{e_k\right\}_{k=1}^\infty$ does always have an infinite subsequence $\left\{x_k\right\}_{k=1}^\infty$ that satisfies both the requirement and the requirement for each $i$ with $g_k=u_{m(k)}$ ($k\in\mathbb{N}$). Once such a subsequence $\left\{x_k\right\}_{k=1}^\infty$ of $\left\{e_k\right\}_{k=1}^\infty$ has been fixed, the remaining task is simply to complete the $x_k$’s to an orthonormal basis, and to let $y_k$ ($k\in\mathbb{N}$) denote the new elements of that basis. Now $\{f_n\}=\{x_k\}_{k=1}^\infty\cup\{y_k\}_{k=1}^\infty$ and $\{u_n\}=\{g_k\}_{k=1}^\infty\cup\{h_k\}_{k=1}^\infty$ constitute a pair of orthonormal bases satisfying -.
In turn, an explicit example of a basis $\{u_n\}$ that obeys the above three conditions , , and , can be adopted from the wavelet theory, as follows. Let $\psi$ be the Lemarié-Meyer wavelet, $$\label{wave}
[\psi](s)=\dfrac1{2\pi}\int_{\mathbb{R}}e^{\imath\xi(\frac12+s)}
\mathrm{sgn\,}\xi b(\left|\xi\right|)\,d\xi\quad (s\in\mathbb{R})$$ with the bell function $b$ being infinitely smooth and compactly supported on $[0,+\infty)$ (see [@Her:Wei Example D, p. 62] for details). Then $[\psi]$ is of the Schwartz class $\mathcal{S}(\mathbb{R})$, so its every derivative $\left[\psi\right]^{(i)}$ is in $C(\mathbb{R},\mathbb{C})$. In addition, the “mother wavelet” $\psi$ generates an orthonormal basis $\left\{\psi_{\alpha\beta}\right\}_{\alpha,\,\beta\in\mathbb{Z}}$ for $L^2$ by $$\label{ujk}
\psi_{\alpha\beta}=2^{\frac \alpha2}\psi(2^\alpha\cdot-\beta)\quad (\alpha,\,\beta\in\mathbb{Z}).$$
In any manner, rearrange the two-indexed set $\{\psi_{\alpha\beta}\}_{\alpha,\,\beta\in\mathbb{Z}}$ into a simple sequence, so that it looks like $\left\{u_n\right\}_{n=1}^\infty$; clearly each term here, $u_n$, has property for each $i$. Besides, it is easily verified that a norm estimate like holds for each $[u_n]^{(i)}$: the factors in the right-hand side of may be taken as $$N_n=\begin{cases}
2^{\alpha_n^2}&\text{if $\alpha_n>0$,}\\
2^{\alpha_n/2}&\text{if $\alpha_n\le0$,}
\end{cases}
\qquad D_i=2^{\left(i+1/2\right)^2}\left\|[\psi]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})},$$ with the convention that $u_n=\psi_{\alpha_n\beta_n}$ ($n\in\mathbb{N}$) in conformity with that rearrangement. This choice of $N_n$ also gives $\sum_kN_{l(k)}<\infty$ (cf. ) whenever $\left\{l(k)\right\}_{k=1}^\infty$ is a subsequence of $\left\{l\right\}_{l=1}^\infty$ satisfying $\alpha_{l(k)}\to -\infty$ as $k\to\infty$.
*Step 2*. In this step our intention is to construct a candidate for the desired unitary operator $U\colon\mathcal{H}\to L^2$ in the theorem. Recalling that $$\begin{gathered}\label{SET}
\{f_1,f_2,f_3,\dots\}=\{x_1,x_2,x_3,\dots\}\cup\left\{y_1,y_2,y_3,\dots\right\},
\\
\{u_1,u_2,u_3,\dots\}=\{g_1,g_2,g_3,\dots\}\cup\left\{h_1,h_2,h_3,\dots\right\},
\end{gathered}$$ where the unions are disjoint, define the unitary operator $U$ from $\mathcal{H}$ onto $L^2$ by setting $$\label{2.12}
Ux_k=g_k,\quad Uy_k=h_k
\quad\text{for all $k\in \mathbb{N}$,}$$ in the harmless assumption that $Uf_n=u_n$ for all $n\in\mathbb{N}$.
*Step 3.* This step of the proof is to show that, in fact, if the unitary operator $U\colon \mathcal{H}\to L^2$ is defined as in , then the operators $US_{\gamma}U^{-1}\colon L^2\to L^2$ ($\gamma\in\mathcal{G}$) are all simultaneously bi-Carleman operators with $K^\infty$ kernels of Mercer type.
Fix, to begin with, an arbitrary index $\gamma\in\mathcal{G}$, abbreviate $S_\gamma$ to $S$, and let $T=USU^{-1}$. It is to be proved that $T$ is an integral operator with a $K^\infty$ kernel of Mercer type. The idea is first to conveniently split the operators $S$, $S^*$, each into two parts as follows. If $E$ is the orthogonal projection of $\mathcal{H}$ onto $\mathrm{Span}\left(\left\{x_k\right\}_{k=1}^\infty\right)$, write $$\label{2.7}
S=(I_\mathcal{H}-E)S+ES,\quad S^*=(I_\mathcal{H}-E)S^*+ES^*.$$ It is an immediate consequence of and that the operators $J=S E$ and $\widetilde{J}=S^* E$ are Hilbert-Schmidt. Write down their Schmidt’s decompositions $$\label{JSchdec}
J=\sum_n s_{n}\left\langle\cdot,p_{n}\right\rangle_{\mathcal{H}}q_{n},
\quad\widetilde{J}=\sum_n \widetilde{s}_{n}\left\langle\cdot,\widetilde{p}_{n}
\right\rangle_{\mathcal{H}}\widetilde{q}_{n},$$ and introduce compact operators $B$ and $\widetilde{B}$, which are defined by $$\label{aux}
B=\sum_n s_{n}^{\frac1{4}}
\left\langle\cdot,p_{n}\right\rangle_{\mathcal{H}} q_{n},
\quad
\widetilde{B}=\sum_n \widetilde{s}_{n}^{\frac1{4}}\left\langle\cdot,
\widetilde{p}_{n}\right\rangle_{\mathcal{H}}\widetilde{q}_{n};$$ here the $s_{n}$ are the singular values of $J$ (eigenvalues of $\left(J J^*\right)^{\frac1{2}}$), $\left\{p_n\right\}$, $\left\{q_n\right\}$ are orthonormal sets in $\mathcal{H}$ (the $p_{n}$ are eigenvectors for $J^*J$ and $q_{n}$ are eigenvectors for $JJ^*$). The explanation of the notation for $\widetilde{J}$ is similar.
For each $f\in\mathcal{H}$, let $$\label{2.8n}
c(f):=
\left\|Bf\right\|_{\mathcal{H}}
+
\left\|B^*f\right\|_{\mathcal{H}}
+
\left\|\widetilde{B}f\right\|_{\mathcal{H}}
+
\left\|\left(\widetilde{B}\right)^*f\right\|_{\mathcal{H}}.$$ Applying Schwarz’s inequality yields $$\label{2.8}
\begin{split}
c(x_k)&=
\sqrt{\sum_ns_n^{\frac12}
\left|\left\langle x_k,p_n\right\rangle_{\mathcal{H}}\right|^2}
+
\sqrt{\sum_ns_n^{\frac12}
\left|\left\langle x_k,q_n\right\rangle_{\mathcal{H}}\right|^2}
\\&\qquad
+
\sqrt{\sum_n\widetilde{s}_n^{\frac12}
\left|\left\langle x_k,\widetilde{p}_n\right\rangle_{\mathcal{H}}\right|^2}
+
\sqrt{\sum_n\widetilde{s}_n^{\frac12}
\left|\left\langle x_k,\widetilde{q}_n\right\rangle_{\mathcal{H}}\right|^2}
\\&
=
\left\|\left(J^*J\right)^{\frac18}x_k\right\|_{\mathcal{H}}
+
\left\|\left(JJ^*\right)^{\frac18}x_k\right\|_{\mathcal{H}}
\\&\qquad
+
\left\|\left(\left(\widetilde{J}\right)^*\widetilde{J}\right)^{\frac18}x_k\right\|_{\mathcal{H}}
+
\left\|\left(\widetilde{J}\left(\widetilde{J}\right)^*\right)^{\frac18}x_k\right\|_{\mathcal{H}}
\\&
\leq
\left\|J x_k\right\|_{\mathcal{H}}^{\frac14}
+
\left\|J^*x_k\right\|_{\mathcal{H}}^{\frac14}
+
\left\|\widetilde{J}x_k\right\|_{\mathcal{H}}^{\frac14}
+
\left\|\left(\widetilde{J}\right)^*x_k\right\|_{\mathcal{H}}^{\frac14}
\\&\qquad
\le 2\left(\left\|Sx_k\right\|^{\frac14}+
\left\|S^*x_k\right\|^{\frac14}\right)\le d(x_k)\quad(k\in\mathbb{N}),
\end{split}$$ whence it follows again from and that $B$ and $\widetilde{B}$ are both Hilbert-Schmidt operators, implying $$\label{2.9}
\sum_ns_n^{\frac1{2}}<\infty,\quad
\sum_n\widetilde{s}_n^{\frac1{2}}<\infty.$$
Now define $Q=(I_\mathcal{H}-E)S^*$, $\widetilde{Q}=(I_\mathcal{H}-E)S$. Since $\{y_k\}$ is, by construction, an orthonormal basis for the subspace $(I_\mathcal{H}-E)\mathcal{H}$, it follows that $$\begin{gathered}\label{2.10}
Qf=\sum_k\left\langle Qf,y_k\right\rangle_{\mathcal{H}}y_k
=
\sum_k\left\langle f,Sy_k\right\rangle_{\mathcal{H}}y_k,\\
\widetilde{Q}f=\sum_k\left\langle\widetilde{Q}f,y_k\right\rangle_{\mathcal{H}}y_k
=
\sum_k\left\langle f,S^* y_k\right\rangle_{\mathcal{H}}y_k
\end{gathered}$$ for all $f\in\mathcal{H}$, with the series converging in $\mathcal{H}$.
Having considered the splittings , which now look like $$\label{2.7new}
S=\widetilde{Q}+\left(\widetilde{J}\right)^*,\quad S^*=Q+J^*,$$ our next task is to verify that each of the four operators $UQU^{-1}$, $U\widetilde QU^{-1}$, $UJ^*U^{-1}$, and $U\left(\widetilde J\right)^*U^{-1}$, is a Carleman operator with a kernel satisfying conditions (i), (ii) in Definition \[Kmkernel\]. The checking is straightforward, and goes by representing all pertinent kernels and Carleman functions as infinitely smooth sums of termwise differentiable series of infinitely smooth functions, without any use of the method of Section \[prmfactor\]. Theorems on termwise differentiation of series will however be used repeatedly (and usually tacitly) in the subsequent analysis.
If $P=UQU^{-1}$, $\widetilde{P}=U\widetilde{Q}U^{-1}$, then from and it follows that, for each $f\in L^2$, $$\label{P}
Pf=\sum_k \left\langle f,Th_k\right\rangle_{L^2} h_k,
\quad
\widetilde{P}f=\sum_k\left\langle f,T^*h_k\right\rangle_{L^2}h_k,$$ in the sense of convergence in $L^2$. Represent the equivalence classes $Th_k$, $T^*h_k\in L^2$ ($k\in\mathbb{N}$) by the Fourier expansions with respect to the orthonormal basis $\{u_n\}$: $$Th_k=\sum_n \left\langle y_k,S^*f_n\right\rangle_{\mathcal{H}}u_n,
\quad
T^*h_k=\sum_n \left\langle y_k,Sf_n\right\rangle_{\mathcal{H}} u_n,$$ where the convergence is in the $L^2$ norm. But somewhat more than that can be said about convergence, namely that, for each fixed $i$, the series $$\label{2.15}
\sum_n \left\langle y_k,S^*f_n\right\rangle_{\mathcal{H}}[u_n]^{(i)}(s),
\quad
\sum_n \left\langle y_k,Sf_n\right\rangle_{\mathcal{H}}[u_n]^{(i)}(s)\quad(k\in\mathbb{N})$$ converge in the norm of $C(\mathbb{R},\mathbb{C})$. Indeed, all the series are dominated everywhere on $\mathbb{R}$ by one series $$\sum_n
\left(\left\|S^*f_n\right\|_{\mathcal{H}}+
\left\|Sf_n\right\|_{\mathcal{H}}\right)\left|[u_n]^{(i)}(s)\right|,$$ which is uniformly convergent on $\mathbb{R}$ for the following reason: its component subseries (cf. ) $$\begin{gathered}
\sum_k\left(\left\|Sx_k\right\|_{\mathcal{H}}+\left\|S^*x_k\right\|_{\mathcal{H}}\right)
\left|[g_k]^{(i)}(s)\right|,\quad
\sum_k\left(\left\|Sy_k\right\|_{\mathcal{H}}+\left\|S^*y_k\right\|_{\mathcal{H}}\right)
\left|[h_k]^{(i)}(s)\right|\end{gathered}$$ are uniformly convergent on $\mathbb{R}$ because they are in turn dominated by the convergent series $$\label{2.16}
\sum_k d(x_k)\left\|[g_k]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})},
\quad
\sum_k 2\|S\|\left\|[h_k]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})},$$ respectively (see , , , and ).
It is now evident that the pointwise sums of the series of define functions that belong to $C(\mathbb{R},\mathbb{C})$ and that are none other than $\left[Th_k\right]^{(i)}$, $\left[T^*h_k\right]^{(i)}$ ($k\in\mathbb{N}$), respectively. Moreover, the above arguments prove that given any $i$, there exists a positive constant $C_i$ such that $$\left\|\left[Th_k\right]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})}<C_i,
\quad
\left\|\left[T^*h_k\right]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})}<C_i,$$ for all $k$. Hence, by , it is possible to infer that, for all $i$, $j$, the series $$\sum_k \left[h_k\right]^{(i)}(s)
\overline{\left[Th_k\right]^{(j)}(t)},\quad
\sum_k \left[h_k\right]^{(i)}(s)
\overline{\left[T^*h_k\right]^{(j)}(t)}$$ converge and even absolutely in the norm of $C\left(\mathbb{R}^2,\mathbb{C}\right)$. This makes it clear that functions $\boldsymbol{P}$, $\boldsymbol{\widetilde{P}}\colon\mathbb{R}^2\to\mathbb{C}$, defined by $$\label{2.14}
\boldsymbol{P}(s,t)=\sum_k \left[h_k\right](s)\overline
{\left[Th_k\right](t)},\quad
\boldsymbol{\widetilde{P}}(s,t)=\sum_k\left[h_k\right](s)
\overline{\left[T^*h_k\right](t)},$$ satisfy condition (i) in Definition \[Kmkernel\].
Now we prove that (Carleman) functions $\boldsymbol{p}$, $\boldsymbol{\widetilde{p}}\colon\mathbb{R}\to L^2$, defined by $$\label{pp}
\boldsymbol{p}(s)=\overline{\boldsymbol{P}(s,\cdot)}=\sum_k
\overline{\left[h_k\right](s)}Th_k,\quad
\boldsymbol{\widetilde{p}}(s)=\overline{\boldsymbol{\widetilde{P}}(s,\cdot)}=
\sum_k\overline{\left[h_k\right](s)}T^*h_k,$$ are both subject to requirement (ii) in Definition \[Kmkernel\]. Indeed, the series displayed converge absolutely in the $C\left(\mathbb{R},L^2\right)$ norm, because those two series whose terms are respectively $\left|\left[h_k\right](s)\right|\|Th_k\|_{L^2}$ ($k\in\mathbb{N}$) and $\left|\left[h_k\right](s)\right|\left\|T^*h_k\right\|_{L^2}$ ($k\in\mathbb{N}$) are dominated by the second series of with $i=0$. For the remaining $i$, a similar reasoning implies the same convergence behavior of the series $\sum_k\overline{\left[h_k\right]^{(i)}(s)}Th_k$, $\sum_k\overline{\left[h_k\right]^{(i)}(s)}T^*h_k.$ The asserted property of both Carleman functions $\boldsymbol{p}$ and $\boldsymbol{\widetilde{p}}$ to satisfy (ii) then follows by the corresponding theorem on termwise differentiation of series.
From it follows that the series of , viewed as series in $C(\mathbb{R},\mathbb{C})$, converge (and even absolutely) in $C(\mathbb{R},\mathbb{C})$ norm, and therefore that their pointwise sums are none other than $\left[Pf\right]$ and $\left[\widetilde{P}f\right]$, respectively. On the other hand, the established properties of the series of and of make it possible to write, for each temporarily fixed $s\in\mathbb{R}$, the following chains of relations $$\begin{split}
\sum_k&\left\langle f,Th_{k}\right\rangle_{L^2} \left[h_{k}\right](s)
=\left\langle f,\sum\limits_k\overline{\left[h_{k}\right](s)}Th_{k}\right\rangle_{L^2}
\\&
=\int_\mathbb{R} \left(\sum\limits_k\left[h_{k}\right](s)
\overline{\left[Th_{k}\right](t)}\right)f(t)\,dt
=\int_{\mathbb{R}}\boldsymbol{P}(s,t)f(t)\,dt,
\end{split}$$ $$\begin{split}
\sum_k& \left\langle f,T^*h_{k}\right\rangle_{L^2} \left[h_{k}\right](s)
=\left\langle f,\sum\limits_k\overline{\left[h_{k}\right](s)}T^*h_{k}\right\rangle_{L^2}
\\&
=\int_\mathbb{R} \left(\sum\limits_k\left[h_{k}\right](s)
\overline{\left[T^*h_{k}\right](t)}\right)f(t)\,dt=\int_{\mathbb{R}}\boldsymbol{\widetilde{P}}(s,t)f(t)\,dt
\end{split}$$ whenever $f$ is in $L^2$. These imply that $P$ and $\widetilde{P}$ are Carleman integral operators, the kernels of which are $\boldsymbol{P}$ and $\boldsymbol{\widetilde{P}}$ respectively, both are subject to requirements (i), (ii) in Definition \[Kmkernel\] by what precedes.
Now we consider the Hilbert-Schmidt integral operators $F=UJ^*U^{-1}$ and $\widetilde{F}=U\left(\widetilde{J}\right)^*U^{-1}$ on $L^2$, and prove that the kernels of these operators are $K^\infty$ kernels. Associate with Schmidt’s decompositions for $J$, $\widetilde{J}$ two functions $\boldsymbol{F}$, $\boldsymbol{\widetilde{F}}\colon\mathbb{R}^2\to\mathbb{C}$, defined by $$\label{2.17}
\begin{split}
\boldsymbol{F}(s,t)
=\sum_n s_n^{\frac1{2}}\left[U B^*q_n\right] (s)
&\overline{\left[U Bp_n\right](t)}
\\&\left(=\sum_n s_n \left[U p_n\right](s)\overline{\left[U q_n\right](t)}\right),
\\
\boldsymbol{\widetilde{F}}(s,t)
=\sum_n\widetilde{s}_n^{\frac1{2}}\left[U\left(\widetilde{B}\right)^*\widetilde{q}_n
\right](s)&\overline{\left[U\widetilde{B}\widetilde{p}_n\right](t)}
\\&\left(=\sum_n\widetilde{s}_n\left[U\widetilde{p}_n\right](s)
\overline{\left[U\widetilde{q}_n\right](t)}\right),
\end{split}$$ whenever $s$, $t$ are in $\mathbb{R}$; for the auxiliary operators $B$, $\widetilde{B}$ here used, see . It is to be noted that without the square brackets the bilinear series just written do converge almost everywhere on $\mathbb{R}^2$ to Hilbert-Schmidt kernels that induce respectively $F$ and $\widetilde{F}$ (see ). Hence, and again because of , the conclusion that the above-defined functions $\boldsymbol{F}$ and $\boldsymbol{\widetilde{F}}$ are the kernels of $F$ and $\widetilde{F}$, respectively, and are subject to condition (i) of Definition \[Kmkernel\] can be inferred as soon as it is known that for each fixed $i$ the terms of the sequences $$\label{jsequences}
\left\{\left[U Bp_k\right]^{(i)}\right\},
\quad
\left\{\left[U B^*q_k\right]^{(i)}\right\},
\quad
\left\{\left[U\widetilde{B}\widetilde{p}_k\right]^{(i)}\right\},
\quad
\left\{\left[U\left(\widetilde{B}\right)^*\widetilde{q}_k\right]^{(i)}\right\}$$ make sense, are all in $C(\mathbb{R},\mathbb{C})$, and their $C(\mathbb{R},\mathbb{C})$ norms are bounded, regardless of $k$. To see that the conditions just listed are all fulfilled, it suffices to observe that, once $i$ is fixed, all the series $$\label{Fseries}
\begin{gathered}
\sum_n\left\langle p_k,B^*f_n\right\rangle_{\mathcal{H}}\left[u_n\right]^{(i)}(s),\quad
\sum_n\left\langle q_k,Bf_n\right\rangle_{\mathcal{H}} \left[u_n\right]^{(i)}(s),\quad
\\\sum_n\left\langle\widetilde{p}_k,\left(\widetilde{B}\right)^*f_n\right\rangle_{\mathcal{H}} \left[u_n\right]^{(i)}(s),\quad
\sum_n\left\langle\widetilde{q}_k,\widetilde{B}f_n\right\rangle_{\mathcal{H}}
\left[u_n\right]^{(i)}(s)
\quad(k\in\mathbb{N})
\end{gathered}$$ (which, with $i=0$, are merely the $L^2$-convergent Fourier expansions, with respect to the orthonormal basis $\{u_n\}$, for $UBp_k$, $UB^*q_k$, $U \widetilde{B}\widetilde{p}_k$, and $U(\widetilde{B})^*\widetilde{q}_k$, respectively) are dominated by one series $$\sum_nc(f_n)\left|\left[u_n\right]^{(i)}(s)\right|,$$ where $c(f)$ is defined in above. This last series is uniformly convergent on $\mathbb{R}$, because it is composed of the two subseries $$\sum_nc(x_k)\left|\left[g_k\right]^{(i)}(s)\right|,
\quad
\sum_nc(y_k)\left|\left[h_k\right]^{(i)}(s)\right|$$ that converge uniformly in $\mathbb{R}$, having as their dominant series the convergent series $$\sum_kd(x_k)\left\|[g_k]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})},
\quad
\sum_k2\left(\|B\|+\left\|\widetilde{B}\right\|\right)\left\|[h_k]^{(i)}\right\|_{C(\mathbb{R},\mathbb{C})}$$ (see , , ). Thus, for each $i$, all the series in converge (and even absolutely) in the $C(\mathbb{R},\mathbb{C})$ norm, and their sums are none other than, respectively, $[UBp_k]^{(i)}$, $\left[UB^*q_k\right]^{(i)}$, $\left[U\widetilde{B}\widetilde{p}_k\right]^{(i)}$, and $\left[U(\widetilde{B})^*\widetilde{q}_k\right]^{(i)}$, ($k\in\mathbb{N}$). Then, in virtue of and the above-established boundedness of the sequences of in $C(\mathbb{R},\mathbb{C})$, the series $$\begin{gathered}
\sum_ns_n^{\frac12}\left[UB^*q_n\right]^{(i)}(s)\overline{\left[UBp_n\right]^{(j)}(t)},
\quad
\sum_n\widetilde{s}_n^{\frac12}\left[U\left(\widetilde{B}\right)^*
\widetilde{q}_n\right]^{(i)}(s)\overline{\left[U\widetilde{B}\widetilde p_n\right]^{(j)}(t)}
\end{gathered}$$ converge (and even absolutely) in $C\left(\mathbb{R}^2,\mathbb{C}\right)$, for all $i$, $j$. This in conjunction with is sufficient to conclude that both the functions $\boldsymbol{F}$ and $\boldsymbol{\widetilde{F}}$ satisfy condition (i) of Definition \[Kmkernel\], and are the Hilbert-Schmidt kernels of $F$ and of $\widetilde{F}$, respectively.
Again by the properties of the sequences of and by , the series $$\begin{gathered}
\label{}
\sum_n s_n^{\frac1{2}}\overline{\left[U B^*q_n\right]^{(i)}(s)}U Bp_n,
\quad
\sum_n s_n^{\frac12}UB^*q_n\overline{\left[UBp_n\right]^{(i)}(s)},
\\
\sum_n\widetilde{s}_n^{\frac1{2}}\overline{\left[U\left(\widetilde{B}\right)^*\widetilde{q}_n
\right]^{(i)}(s)}U \widetilde{B}\widetilde{p}_n,
\quad
\sum_n \widetilde{s}_n^{\frac12}U\left(\widetilde{B}\right)^*\widetilde{q}_n
\overline{\left[U\widetilde{B}\widetilde{p}_n\right]^{(i)}(s)}\end{gathered}$$ converge and even absolutely in the $C\left(\mathbb{R},L^2\right)$ norm, for each $i$. Observe, via , that four of these series, namely those with $i=0$, represent the Carleman functions $\boldsymbol{f}(s)=\overline{\boldsymbol{F}(s,\cdot)}$, $\boldsymbol{f}^{\boldsymbol{\prime}}(s)=\boldsymbol{F}(\cdot,s)$, $\boldsymbol{\widetilde{f}}(s)=\overline{\boldsymbol{\widetilde{F}}(s,\cdot)}$, and $\boldsymbol{\widetilde{f}}^{\boldsymbol{\prime}}(s)=
\boldsymbol{\widetilde{F}}(\cdot,s)$, respectively, which therefore do satisfy conditions (ii), (iii) in Definition \[Kmkernel\]. This finally implies that the Hilbert-Schmidt kernels $\boldsymbol{F}$ and $\boldsymbol{\widetilde{F}}$ are $K^\infty$ kernels of $F$ and of $\widetilde{F}$, respectively.
In accordance with , the operator $T$ (which, recall, is the transform by $U$ of $S$) and its adjoint decompose as $T=\widetilde{P}+\widetilde{F}$, $T^*=P+F$ where all the terms are already known to be Carleman operators, with kernels satisfying (i), (ii). Hence, both $T$ and $T^*$ are Carleman operators, and their kernels, say $\boldsymbol{T}$ and $\boldsymbol{T}^{\boldsymbol\ast}$, defined as $$\label{2.18}
\boldsymbol{T}(s,t)=\boldsymbol{\widetilde{P}}(s,t)+\boldsymbol{\widetilde{F}}(s,t),
\quad
\boldsymbol{T}^{\boldsymbol{\ast}}(s,t)=\boldsymbol{P}(s,t)+\boldsymbol{F}(s,t)$$ for all $s$, $t\in\mathbb{R}$, inherit the smoothness properties (i), (ii) from their terms. But then, since the operator $T$ is bi-integral, $\boldsymbol{T}(s,t)=\overline{\boldsymbol{T}^{\boldsymbol{\ast}}(t,s)}$ for all $s$, $t\in\mathbb{R}$; hence $\boldsymbol{T}(\cdot,t)=
\overline{\boldsymbol{T}^{\boldsymbol{\ast}}(t,\cdot)}$ in the $L^2$ sense for each fixed $t\in\mathbb{R}$. That implies that the kernel $\boldsymbol{T}$ also satisfies condition (iii) in Definition \[Kmkernel\], and is thus a $K^\infty$ kernel that induces the bi-Carleman operator $T$.
This $K^\infty$ kernel $\boldsymbol{T}$ is moreover of Mercer type. A main ingredient in proving the claim is the following remark: if $d(x_k)$ are defined as in , if $U$ is defined as in , and if an operator $A\in\mathfrak{R}(\mathcal{H})$ fulfils $$2\left(\left\|Ax_k\right\|_{\mathcal{H}}^{\frac1{4}}
+\left\|A^*x_k\right\|_{\mathcal{H}}^{\frac1{4}}\right)\leq
d(x_k)\quad\text{for each $k\in\mathbb{N}$,}$$ then, like $T=USU^{-1}$, the operator $UAU^{-1}$ is a bi-Carleman operator with a $K^\infty$ kernel, no matter whether $A$ is in the initial family $\mathcal{S}$ or not. This additional feature of $U$ may be checked by directly applying the above verification procedure, leading from to , to the operator $A$ in place of $S$.
If now a nonzero operator $A$ is restricted to lie in the set $\mathcal{M}(S)$, then whatever its representation from among those in is, the scalar multiple $A_1=\left(16\max\left\{\|M\|,\|N\|\right\}\right)^{-1}\cdot A$ of $A$ obeys the following easily verifiable inequality valid for all $k\in\mathbb{N}$: $$2\left(\left\|A_1x_k\right\|_{\mathcal{H}}^{\frac1{4}}+
\left\|A_1^*x_k\right\|_{\mathcal{H}}^{\frac1{4}}\right)\leq
2\left(\left\|Sx_k\right\|_{\mathcal{H}}^{\frac1{4}}+
\left\|S^*x_k\right\|_{\mathcal{H}}^{\frac1{4}}\right)\leq
d(x_k).$$ By the above remark, this implies that the unitary operator $U$ defined in carries, besides $S$, every other member, $A$, of $\mathcal{M}(S)$ onto an integral operator, $UAU^{-1}$, with a $K^\infty$ kernel. Then, since $\mathcal{M}(T)=U\mathcal{M}(S)U^{-1}$, the $K^\infty$ kernel $\boldsymbol{T}$ of $T$, constructed in above, is of Mercer type by virtue of the definition. Moreover, since $T=US_\gamma U^{-1}$ where the index $\gamma$ was arbitrarily fixed, it follows that all the operators $T_\gamma=US_\gamma U^{-1}$ ($\gamma\in\mathcal{G}$) are bi-Carleman operators whose kernels are $K^\infty$ kernels of Mercer type.
The only thing that remains to be proved is that those $K^\infty$ kernels which induce finite linear combinations of $US_\gamma U^{-1}$ ($\gamma\in\mathcal{G}$) are also of Mercer type. Indeed, consider any finite linear combination $G=\sum z_\gamma S_\gamma$ with $\sum\left|z_\gamma\right|\leq 1$. It is seen easily that, for each $n$, $$\left\|\sum z_\gamma S_\gamma e_n\right\|_{\mathcal{H}}
\leq\sup_{\gamma\in\mathcal{G}}\left\|S_\gamma e_n\right\|_{\mathcal{H}},
\quad
\left\|\sum\overline{z}_\gamma \left(S_\gamma\right)^* e_n\right\|_{\mathcal{H}}
\leq\sup_{\gamma\in\mathcal{G}}\left\|\left(S_\gamma\right)^*e_n\right\|_{\mathcal{H}}.$$ There is, therefore, no barrier to assuming that $G$ was, from the start, in the initial family $\mathcal{S}$. The proof of the theorem is now complete.
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|
---
abstract: 'The internal magnetic field distribution in a mixed state of a cuprate superconductor, [Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$]{} ($T_{\rm c}\simeq28.5$ K, near the optimal doping), was measured by muon spin rotation ([$\mu$SR]{}) technique up to 60 kOe. The [$\mu$SR]{} linewidth $\Lambda(B)$ which exhibits excess broadening at higher fields ($B>5$ kOe) due to field-induced magnetism (FIM), is described by a relation, $\Lambda(B)\propto\sqrt{B}$. This suggests that the orbital current and associated quasiparticle excitation plays predominant roles in stabilizing the quasistatic correlation. Moreover, a slowing down of the vortex fluctuation sets in well above [$T_{\rm c}$]{} as inferred from the trace of FIM observed up to $\sim80$ K, and develops continuously without a singularity at [$T_{\rm c}$]{} as the temperature decreases.'
author:
- 'K. Ohishi'
- 'I. Yamada'
- 'A. Koda'
- 'S.R. Saha'
- 'R. Kadono'
- 'W. Higemoto'
- 'K.M. Kojima'
- 'M. Azuma'
- 'M. Takano'
title: |
Field-induced magnetism in the high-$T_c$ superconductor [Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$]{} \
with flat CuO$_2$ planes
---
The 1/8 anomaly, or the suppression of superconductivity at a specific hole carrier concentration, $p\simeq1/8$, has been understood in terms of the so-called “stripe" model, where a quasistatic charge and spin stripe order develops in place of superconductivity [@Tranquada:95]. While a similar situation is realized over a wide range of $p\simeq x$ in La$_{2-x-y}$Nd$_y$Sr$_x$CuO$_4$ (Ref.), a spatially modulated dynamical spin correlation is found in [La$_{2-x}$Sr$_x$CuO$_4$]{} (LSCO) that becomes static only near $p\simeq1/8$ (Ref.). It is speculated that the dynamical stripes tend to be static at $p\sim$ 1/8 by a synchronization of the periodicity between lattice and charge/spin modulations, as the buckling of CuO$_2$ planes in the tetragonal structure observed at low temperatures (LTT phase) seems to serve as a source of “pinning", leading to a stronger 1/8 anomaly.
Recent neutron scattering experiments in underdoped and optimally doped LSCO [@Katano:00; @Lake:01; @Lake:02] and underdoped La$_{2-x}$Ba$_x$CuO$_4$ (LBCO) [@Fujita:06] have revealed that the quasistatic stripe order is enhanced at around $p\simeq1/8$ by applying a moderate magnetic field of a few tesla parallel to the $c$ axis, as inferred from the enhanced intensity of incommensurate magnetic Bragg peaks. Moreover, such a field-induced static magnetic order has turned out to exhibit a three-dimensional correlation, which survives into the superconducting state, in LSCO ($p=0.10$), suggesting a predominant role of the orbital current in field-induced magnetism (FIM) [@Lake:05].
The occurrence of FIM in high-[$T_{\rm c}$]{} cuprates, which includes static and/or dynamical magnetic correlations, has also been suggested from muon spin rotation ([$\mu$SR]{}) measurements in underdoped and optimally doped LSCO [@Kadono:04; @Savici:05; @Ishida:07; @Chang:08], LBCO [@Savici:05], La$_{2-x-y}$Eu$_y$Sr$_x$CuO$_4$ (LESCO) [@Savici:05], and YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) [@Sonier:08]. There is a common feature that the muon spin relaxation is observed well above [$T_{\rm c}$]{} in these cuprates, suggesting that the FIM develops well above [$T_{\rm c}$]{}. Recently, the FIM has been observed even in overdoped LSCO [@Sonier:08; @MacDougall:06; @MacDougall]. The field-induced static magnetic order observed by neutron scattering experiments, on the other hand, occurs only below $T\sim$ [$T_{\rm c}$]{} in underdoped and optimally doped samples of LSCO. Here, the field-induced static magnetic order is presumed to originate from quasistatic moments around the vortices, and such a magnetic order is not observed for overdoped samples. A lack of experimental evidence from neutrons for static magnetic order persisting above [$T_{\rm c}$]{} led to the argument that the quasistatic FIM observed above [$T_{\rm c}$]{} by [$\mu$SR]{} could be due to uncorrelated static spins.
Regardless of such a correlation with [$T_{\rm c}$]{}, the field-induced static magnetic order has a common feature that it is observed in those high-[$T_{\rm c}$]{} cuprates that exhibit a buckling of the CuO$_2$ planes. From this view point, [Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$]{} (Na-CCOC) is a good candidate for studying the relation between the FIM and the buckling of CuO$_2$ planes, because Na-CCOC has flat CuO$_2$ planes without buckling, and therefore it would serve as a stage for testing the FIM as an intrinsic character of CuO$_2$ planes.
In this paper, we demonstrate the occurrence of FIM in Na-CCOC probed by [$\mu$SR]{}. While the crystal structure of Na-CCOC is isostructural with that of LSCO, it consists of flat CuO$_2$ planes, owing to the substitution of apical oxygen with chlorine. While the [$\mu$SR]{} linewidth $\Lambda$ in the mixed state decreases with increasing field at low magnetic induction, as expected for a normal flux line lattice (FLL), it exhibits a turnover at around $B\simeq5$ kOe, and increases in proportion to $\sqrt{B}$, strongly suggesting that the depolarization is due to quasistatic magnetism associated with magnetic vortices whose quasiparticle density is proportional to $\sqrt{B}$ in $d$-wave superconductors [@Volovik:93; @Nakai:04]. It is inferred from the muon Knight shift measurement that local spins of 0.15$\mu_{\rm B}$/Cu are responsible for the observed effect. Moreover, an enhancement of $\Lambda$ is observed up to 80 K under a field of 60 kOe, which is well above [$T_{\rm c}$]{} (=28.5 K). This reveals the possibility that the fluctuation of random vortices persists over a wide range of temperature far exceeding [$T_{\rm c}$]{}, which is in intriguing accord with the Nernst effect [@Xu:00; @Wang:01].
The single-crystalline specimen of Na-CCOC used in this study had been grown by a flux method under high pressure to yield slab samples with the $c$ axis normal to their plane [@Kohsaka:02; @Azuma:03]. The superconducting transition temperature determined from susceptibility measurement is 28.5 K (see the inset of Fig. \[KS\]), which corresponds to that of optimally doped compounds according to the phase diagram [@Hiroi:96; @Ohishi:05]. The samples were encapsulated in a polymide tape in a glove box to prevent the depletion of sodium and a subsequent deterioration. Transverse field (TF) [$\mu$SR]{} measurements were performed on the M15 beamline of TRIUMF, Vancouver, Canada. Muons with a momentum of 29 MeV/c were injected into the samples with their polarization rotated perpendicular to the beam momentum, so that the external field $H$ might be applied along the incoming beam axis (to minimize the disturbance to muon trajectory), which was parallel to the crystalline $c$ axis. An experimental setup with a high time resolution was employed to measure TF-[$\mu$SR]{} time spectra up to 60 kOe. The corresponding dc-susceptibility at 60 kOe was measured by a SQUID magnetometer.
Figure \[FFT\] shows the fast Fourier transform (FFT) of TF-[$\mu$SR]{} time spectra, which represent the density distribution of internal magnetic field $B$ at the muon site (with additional broadening due to random local fields from nuclear magnetic moments and that coming from a finite time window of FFT). In the normal state above [$T_{\rm c}$]{} (solid curve), the spectrum has a peak at $B=H=0.5$ kOe, which undergoes a shift to a lower frequency upon cooling to 2 K with an increased linewidth (dashed curve). This behavior is perfectly in line with the FLL formation in the superconducting state. Meanwhile, the spectrum exhibits further broadening with increasing field to 60 kOe (dotted curve), which is opposite to the predicted tendency of decreasing linewidth with increasing field in the conventional type II superconductors [@Brandt:88]. The data are analyzed by curve fits in the time domain using a phenomenological stretched exponential function, $$A\hat{P}(t) = A\exp\left[-\left(\Lambda t\right)^\beta\right]\exp(i\omega_\mu t+\phi),\label{stretch}$$ where $A$ is the positron decay asymmetry, $\Lambda$ is the depolarization rate (= linewidth in the frequency domain), $\beta$ is the power of the exponent, $\omega_\mu =\gamma_\mu B$ with $\gamma_\mu$ being the muon gyromagnetic ratio ($=2\pi\times$ 13.553 MHz/kOe), and $\phi$ is the initial phase of precession. The muon Knight shift, $K$, is then defined as, $$K=\frac{\omega_\mu-\omega_0}{\omega_0}=K_\mu+K_{\rm v}+K_{\rm dem},
\label{knightshift}$$ where $\omega_0=\gamma_\mu H$, $K_\mu$ is the shift due to the local spin susceptibility $\chi_{\rm spin}$, $K_{\rm v}$ is that due to the orbital current in the FLL state, and $K_{\rm dem}$ is the correction term consisting of demagnetization and Lorentz field \[$=4\pi(1/3-N)\rho_{\rm mol}\chi_{\rm mol}$, where $N\simeq 1$ is the demagnetization factor, $\rho_{\rm mol}=0.01480$ mol/cm$^3$ is the molar density, and $\chi_{\rm mol}$ is the molar susceptibility\].
Figure \[Hdep\] shows the field dependence of the relaxation rate at 40 K and 2 K, and that of the power of the exponent at 2 K. The relaxation rate, $\sigma$ ($=\Lambda_{\beta=2}$), at 40 K exhibits the $H$-linear behavior, consistent with previous results in LSCO [@Savici:05; @Sonier:08; @MacDougall:06], LBCO [@Savici:05], LESCO [@Savici:05], and YBCO [@Sonier:08]. Note that this linear behavior of $\sigma$ can be understood by assuming the existence of slowly fluctuating (staggered) random magnetic moments induced inside the vortices. As mentioned later, it is described that these fluctuating moments appear with the fluctuation of vortices, because the number of vortices corresponding to the fluctuating moments is proportional to the applied field. Therefore, this $H$-linear behavior of $\sigma$ is considered to be slowly fluctuating random magnetic moments. On the other hand, while $\Lambda$ at 2 K decreases with increasing field at lower fields, it exhibits a turnover around $H=5$ kOe and an increase represented by the relation $\Lambda\propto\sqrt{H}$ (where $H\simeq B$). The lineshape shows a change from that of the Gaussian ($\beta=2$) to single exponential decay ($\beta=1$) with increasing field, as shown Fig. \[Hdep\](b). The contribution of FLL at low fields, $\sigma_{\rm v}$, is extracted by subtracting $\sigma_{\rm n}$ from the total linewidth, $\sigma$, in quadrature, $\sigma^2_{\rm v}=\sigma^2-\sigma^2_{\rm n}$, with $\sigma_{\rm n}$ being the depolarization rate in the normal state (see below). The behavior observed at lower fields is consistent with the predicted $H$-dependence [@Brandt:88], $$\sqrt{2}\sigma_{\rm v}\simeq0.0274\frac{\gamma_\mu\Phi_0}{\lambda_{ab}^2(T,h)}(1-h)\sqrt{1+3.9(1-h)^2},
\label{lmdh}$$ where $\Phi_0$ is the magnetic flux quantum, $\lambda_{ab}(T,h)$ is the effective inplane London penetration depth, which can be expressed as $\lambda_{ab}(T,0)\cdot(1-\eta h)$ with $\eta>0$ for line nodes [@Kadono:07], and $h$ is the field normalized by the upper critical field ($h=H/H_{\rm c2}$). Meanwhile, the value of $\Lambda$ above $\sim$5 kOe far exceeds that observed at lower fields. It is unlikely that such an enhancement at a higher field is induced by flux pinning or other extrinsic artifacts, and thus can be uniquely attributed to FIM that is spatially inhomogeneous (as inferred from the single exponential-like lineshape). More interestingly, $\Lambda$ is excellently reproduced by the relation $\Lambda(B)\propto\sqrt{B}$ over the relevant field range. The inset of Fig. \[Hdep\](b) shows $\Lambda$, which is plotted against $\sqrt{B}$ to see the linearity. Note that this behavior is not observed in other high-[$T_{\rm c}$]{} cuprates that do not have flat CuO$_2$ planes [@Savici:05; @Ishida:07; @Chang:08; @Sonier:08; @MacDougall:06; @MacDougall]. It is established that quasiparticle excitation in the mixed state of $d$-wave superconductors is extended along the $(\pi,\pi)$ directions, leading to a non-linear field dependence of the quasiparticle density that is well approximated by such $\sqrt{B}$-dependence [@Volovik:93; @Nakai:04]. Here, we would like to consider the relation between the muon spin relaxation and the quasiparticle excitations in $d$-wave superconductors. In the case of the [*usual*]{} type II superconductors with $d$-wave symmetry, which do not show any FIM, muon spin relaxation rate decreases with increasing $H$. This is because the induced quasiparticle excitations around the nodal region due to pair breaking makes $\lambda$ increase. According to a relation of Eq. (\[lmdh\]), $\sigma_{\rm v}$ decreases with increasing $\lambda$. On the other hand, in the case of superconductors, which show FIM, muon spin relaxation rate increases because relaxation due to magnetic moments is dominant compared with that due to FLL (increase of $\lambda$) [@Kadono:04; @Savici:05; @Ishida:07; @Chang:08; @Sonier:08; @MacDougall:06; @MacDougall]. It is suggested that the induced quasiparticle excitations have quasistatic magnetic moments to describe the increase of $\Lambda(B)$. Consequently, we can assume that the observed field dependence of $\Lambda$ is one type of evidence for the $d$-wave superconductor, because quasiparticle excitations in $d$-wave superconductors shows a $\sqrt{B}$-dependence.
As shown in Fig. \[rlx-T\](a), while the linewidth, $\sigma$ ($=\Lambda_{\beta=2}$ observed at 0.5 kOe), is mostly independent of temperature above [$T_{\rm c}$]{} with a mean value of $\sigma_{\rm n}=0.144(3)$ $\mu$s$^{-1}$, it increases with decreasing temperature below [$T_{\rm c}$]{} as FLL is formed. The value of $\sigma_{\rm n}$ is in perfect agreement with that of random local fields from nuclear moments estimated from a previous [$\mu$SR]{} experiment [@Ohishi:05]. Fits by a power law, $\sigma_{\rm v}(T)=\sigma_{\rm v}(0)\left[1-\left(T/T_{\rm c}\right)^n\right]$, with [$T_{\rm c}$]{} as a free parameter yields $\sigma_{\rm v}(0)=0.519(5)$ $\mu$s$^{-1}$, $T_{\rm c}=27.7(2)$ K, and $n=3.2(1)$ \[Fig. \[rlx-T\](a), inset\]. Using Eq. (\[lmdh\]) for $h\ll1$, the magnetic penetration depth, $\lambda_{ab}(0,0)$, extrapolated to $T=0$ K is evaluated to be 382(4) nm. These results are quantitatively consistent with earlier literature on Na-CCOC (with the sodium concentration $x=0.18$, and $H=2$ kOe), except that no anomalous increase of $\sigma$ was observed below $\sim$5 K [@Khasanov:07].
The temperature dependence of $\Lambda$ observed at $H=60$ kOe is shown in Fig. \[rlx-T\](b). It increases below 80 K ($\gg T_c$), and continues to increase down to the lowest temperature without any saturation. This behavior is similar to those observed in LSCO at a high field [@Savici:05; @Ishida:07; @Chang:08; @Sonier:08; @MacDougall:06]. On the other hand, the muon Knight shift, $K$, decreases with decreasing temperature below [$T_{\rm c}$]{} \[Fig. \[rlx-T\](c)\], indicating that both the FIM and superconductivity coexist below [$T_{\rm c}$]{}. The behavior of $K$ is in line with that of the magnetic susceptibility \[$\chi$, measured after cooling under a field of 60 kOe, shown in Fig. \[KS\](a)\], except that below $\sim$10 K (see below), although the contributions of spin and orbital parts are not separated in these quantities. The spin part of the muon Knight shift, $K_\mu$, is extracted by Eq. (\[knightshift\]) for $T>T_{\rm c}$ (where $K_{\rm v}=0$) and mapped into the $K_\mu$-$\chi$ plot \[Fig. \[KS\](b)\]. All of the data points fall on a straight line without any sign of kink for those obtained below $\sim$80 K (smaller values of $K_\mu$-$\chi$), indicating that FIM is indeed carried by the paramagnetic Cu spins. The muon hyperfine parameter, $A_\mu$, is deduced as a gradient, $dK_\mu/d\chi$, in the normal state according to, $$K_\mu=K_0+\frac1{N_{\rm A}\mu_{\rm B}}A_\mu\chi,$$ which yields $A_\mu=696(2)$ Oe/$\mu_{\rm B}$, where $K_0$ ($\simeq0$) is a $T$-independent Fermi contact coupling with the conduction electrons, $N_{\rm A}$ is the Avogadro’s number and $\mu_{\rm B}$ is the Bohr magneton. The mean value $|\overline{\mu}_{\rm Cu}|$ for the moment size of the local copper spins is then obtained by comparing $A_\mu$ with that determined by the $z$-component of the dipole tensor, $$\frac{|\overline{\mu}_{\rm Cu}|}{\mu_{\rm B}}\sum_k a_k A^{zz}_{k}=
\frac{|\overline{\mu}_{\rm Cu}|}{\mu_{\rm B}}\sum_k a_k\sum_i\frac{1}{r^3_{ik}}\left[\frac{3z^2_{ik}}{r^2_{ik}}-1\right]\simeq A_\mu,
\label{dtensor}$$ where $a_k$ is the relative population of the $k$-th muon site (there are three of them in Na-CCOC [@Ohishi:05], see below), $r_{ik}$ is the distance between muon at the $k$-th site and the $i$-th Cu ions (with $z_{ik}$ being their $z$-component). Combining $A_\mu$ with other information on the local magnetic fields at the muon sites previously obtained for antiferromagnetically ordered phase of the parent compounds [@Ohishi:05], we reexamined the muon sites to reproduce all of the quantities concerning the local fields. The result is summarized in Table \[t1\], where the $\mu_1$ site turns out to be slightly off the previously assigned position as inferred from the sign of $A_\mu$. Accordingly, we have $\overline{A}^{zz}_{\rm dip}=\sum_k a_k A^{zz}_k=4788$ Oe$/\mu_{\rm B}$, and the effective moment size $|\overline{\mu}_{\rm Cu}|=0.1454(4)\mu_{\rm B}$ from Eq. (\[dtensor\]).
[c|cc|cc]{} $\mu_k$ site & $a_k$ & $r_{\rm Cu}$ \[nm\] & Coordinates & $A^{zz}_k$ \[Oe/$\mu_{\rm B}$\]\
$\mu_1$ & 0.68(2)$^a$ & 0.138$^a$ & (0,0,0.0917) & 6678\
$\mu_2$ & 0.08(1)$^a$ & 0.254$^a$ & (0,0.310,0.146) & 813\
$\mu_3$ & 0.23(1)$^a$ & 0.343$^a$ & (0,0.5,0.186) & 637\
\
It is noteworthy in Fig. \[KS\](a) that $\chi$ exhibits an upturn below $\sim$10 K, which might hint the occurrence of magnetic order. However, no such behavior is observed in the muon Knight shift $K$ shown in Fig. \[rlx-T\](c); it must be noted that $K_\mu$ (the spin part) is deduced using Eq. (\[knightshift\]) where $\chi$ comes in via the demagnetization term ($K_{\rm dem}\propto\chi$), so that the temperature dependence of $K_\mu$ is mostly parallel with that of $\chi$. We attribute this upturn in $\chi$ to unidentified impurities at this stage, considering the absence of a corresponding anomaly in $A_\mu$ anticipated for quasistatic magnetic order. The ambiguity on the absolute values of $\chi$ makes it difficult to discuss whether or not the observed change of sign in $K_\mu$ below $\sim$20 K is meaningful. However, such a behavior is readily explained by considering the contribution of $K_{\rm v}$ ($<0$) which is not discernible from other contributions in the present experiment.
Since Na-CCOC does not exhibit a quasistatic stripe correlation under a zero external field over the relevant hole concentration [@Ohishi:05], the occurrence of FIM can not be explained by a residual effect of the 1/8 anomaly, as it might have been in LSCO [@Savici:02]. A recent revelation of a weak 1/8 anomaly observed in Na-CCOC upon Zn substitution for Cu [@Satoh] strongly suggests that the situation is similar to that observed in YBCO [@Akoshima:00] and Bi$_2$Sr$_2$Ca$_{1-x}$Y$_x$Cu$_2$O$_{8+\delta}$ [@Watanabe:00]. This, together with the present result, supports the presumption that the instability of dynamical stripe correlation against local suppression of superconducting order parameter (i.e., by the Zn impurity or flux lines) is an intrinsic property of CuO$_2$ planes, common at least in Na-CCOC, LSCO and LBCO regardless of the buckling of the CuO$_2$ planes.
According to the reported result on the Nernst effect [@Xu:00; @Wang:01], it is suggested that strong fluctuation of superconductivity and associated flux lines persists well above [$T_{\rm c}$]{}. This naturally leads to a possibility that the FIM observed above [$T_{\rm c}$]{} may originate from the quasistatic stripe correlation segregated around the fluctuating vortices. Considering the result of field dependence of $\Lambda$ at 40 K, which behaves $H$-linear, this scenario is apparently against the expected field dependence in the FLL state [@Brandt:88]. However, it may be argued that the $\sqrt{B}$-dependence is expected only when the quasistatic FLL is established [@Volovik:93]. Moreover, this scenario is also supported by thermal conductivity measurements in LSCO, where the suppression of thermal conductivity observed above [$T_{\rm c}$]{} is attributed to the development of the quasistatic stripe order associated with fluctuating vortices [@Kudo:04]. Thus, the appearance of FIM above [$T_{\rm c}$]{} can be explained in terms of the instability of dynamical stripes upon the local suppression of the order parameter.
Finally, it would be worth mentioning that the suppression of the FIM near $p\sim 1/8$ observed in YBCO [@Sonier:08] has little or no direct relevance to our argument. While the relaxation rate is suppressed near $p\sim 1/8$ compared to other hole concentrations, they indeed observe that the FIM sets in at around $210$ K at $p\sim 1/8$, which is much higher than [$T_{\rm c}$]{}[@Sonier:08]. Thus, YBCO seems to share the feature of FIM that develops well above [$T_{\rm c}$]{} (while the onset temperature for the FIM might exhibit the $p$-dependence similar to [$T_{\rm c}$]{} characterized by a dip near $p\sim 1/8$).
In conclusion, our [$\mu$SR]{} measurements on optimally doped Na-CCOC demonstrate the occurrence of FIM above $\sim$5 kOe, which is highly inhomogeneous and coexists microscopically with superconductivity. Muon Knight shift measurements indicate that the magnetism is carried by local copper spins with an average moment size of 0.15$\mu_{\rm B}$. The field-dependence of the muon spin depolarization rate suggests that the magnetism comes from the quasistatic stripe correlation around the vortex cores (with $d$-wave paring). This is also in line with the interpretation that the observed FIM up to 80 K (well above [$T_{\rm c}$]{}) comes from the strong fluctuation of vortices suggested by the Nernst effect.
We would like to thank the staff of TRIUMF for their technical support during the experiments, and acknowledge helpful discussions with Y. Koike. This work was partially supported by a Grant-in-Aid for Creative Scientific Research (Grant No. 13NP0201) and a Grant-in-Aid for Scientific Research by the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grants No. 17105002, 19052008 and 19340098).
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abstract: 'The four-fermion contact-interaction searches in the process $e^+e^-\to\mu^+\mu^-$ at a future $e^+e^-$ Linear Collider with c.m. energy $\sqrt{s}=0.5$ TeV and with both beams longitudinally polarized are studied. We evaluate the corresponding model-independent constraints on the coupling constants, emphasizing the role of beam polarization, and make a comparison with the case of Bhabha scattering.'
---
[**Search for New Physics Effects at a Linear Collider with Polarized Beams [^1]** ]{}
[A. Babich and A. Pankov]{}\
The Pavel Sukhoi Gomel State Technical University, 246746 Belarus
Introduction
============
Contact interaction Lagrangians (CI) generally represent an effective description of the ‘low energy’ manifestations of some non-standard dynamics acting at new, intrinsic, mass scales much higher than the energies reachable at current particle accelerators. As such, they can be studied through deviations of the experimental observables from the Standard Model (SM) expectation, that reflect the additional presence of the above-mentioned new interaction. Typical examples are the composite models and the exchanges of extremely heavy neutral gauge bosons and leptoquarks [@Barger:1998nf; @Altarelli:1997ce].
Clearly, such deviations are expected to be extremely small, as they would be suppressed for dimensional reasons by essentially some power of the ratio between the available energy and the large mass scales. Accordingly, very high energy reactions in experiments with high luminosity are one of the natural tools to investigate signatures of contact interaction couplings. In general, these constants are considered as [*a priori*]{} free parameters, and one can quantitatively derive an assessment of the attainable reach and of the corresponding upper limits, essentially by numerically comparing the deviations with the expected experimental statistical and systematical uncertainties on the cross sections.
Here, we consider the muon pair production process $$e^++e^-\to \mu^+\mu^-, \label{proc}$$ at a Linear Collider (LC) with c.m. energy $\sqrt s=0.5\hskip 2pt{\rm TeV}$ and polarized electron and positron beams. We discuss the sensitivity of this reaction to the general, $SU(3)\times SU(2)\times U(1)$ symmetric $eeff$ contact-interaction effective Lagrangian, with helicity-conserving and flavor-diagonal fermion currents [@Eichten:1983hw]: $${\cal L}_{\rm CI}
=\sum_{\alpha\beta}g^2_{\rm eff}\hskip 2pt\epsilon_{\alpha\beta}
\left(\bar e_{\alpha}\gamma_\mu e_{\alpha}\right)
\left(\bar \mu_{\beta}\gamma^\mu \mu_{\beta}\right).
\label{lagra}$$ In Eq. (\[lagra\]): $\alpha,\beta={\rm L,R}$ denote left- or right-handed helicities, generation and color indices have been suppressed, and the CI coupling constants are parameterized in terms of corresponding mass scales as $\epsilon_{\alpha\beta}={\eta_{\alpha\beta}}/{{\Lambda^2_{\alpha\beta}}}$ with $\eta_{\alpha\beta}=\pm 1,0$ depending on the chiral structure of the individual interactions. Also, conventionally the value of $g^2_{\rm eff}$ is fixed at $g^2_{\rm eff}=4\pi$, as a reminder that, in the case of compositeness, the new interaction would become strong at $\sqrt s$ of the order of $\Lambda_{\alpha\beta}$. Obviously, in this parameterization, exclusion ranges or upper limits on the CI couplings can be equivalently expressed as exclusion ranges or lower bounds on the corresponding mass scales $\Lambda_{\alpha\beta}$.
For a given final lepton flavour $\mu$, ${\cal L}_{\rm CI}$ in Eq. (\[lagra\]) envisages the existence of eight individual, and independent, CI models corresponding to the combinations of the four chiralities $\alpha,\beta$ with the $\pm$ signs of the $\eta$’s, with [a priori]{} free, and nonvanishing, coefficients. Correspondingly, the most general (and model-independent) analysis of the process (\[proc\]) must account for the complicated situation where all four-fermion effective couplings defined in Eq. (\[lagra\]) are simultaneously allowed in the expression for the cross section, and in principle can interfere and weaken the bounds in case of accidental cancellations.
Of course, the different helicity amplitudes, as such, do not interfere. However, [*the deviations from the SM*]{} could be positive for one helicity amplitude, and negative for another. Thus, cancellations might occur.
The simplest attitude is to assume non-zero values for only one of the couplings (or one specific combination of them) at a time, with all others zero, this leads to tests of the specific models mentioned above. But, in principle, constraints obtained by simultaneously including couplings of different chiralities might become considerably weaker. Therefore, it should be higly desirable to apply a more general (and model-independent) approach to the analysis of experimental data, that simultaneously includes all terms of Eq. (\[lagra\]) as independent free parameters, and can also allow the derivation of separate constraints (or exclusion regions) on the values of the coupling constants.
To this aim, in the case of the process (\[proc\]) at the LC considered here, a possibility is offered by initial beam polarization, that enables us to extract from the data the individual helicity cross sections through the definition of particular, and optimal, polarized integrated cross sections and, consequently, to disentangle the constraints on the corresponding CI constants [@Pankov:1998ad]–[@Babich:2001ik]. In this note, we wish to to present a model-independent analysis of the CI that complements that of Refs. [@Pankov:1998ad]–[@Babich:2001ik], and is based on the measurements of the observables such as the total cross section, the forward-backward asymmetry $A_{\rm
FB}$, the left-right asymmetry $A_{\rm LR}$, and left-right forward-backward asymmetry $A_{\rm LR,FB}$.
Observables
===========
For the process Eq. (\[proc\]) we can neglect fermion masses with respect to $\sqrt s$, and express the amplitude in the Born approximation including the $\gamma$ and $Z$ $s$-channel exchanges plus the contact-interaction term of Eq. (\[lagra\]). With $P_e$ and $P_{\bar e}$ the longitudinal polarizations of the electron and positron beams, respectively, and $\theta$ the angle between the incoming electron and the outgoing fermion in the c.m. frame, the differential cross section can be expressed as [@Schrempp:1988zy]: $$\frac{{\rm d}\sigma}{{\rm d}\cos\theta}
=\frac{3}{8}
\left[(1+\cos\theta)^2 {\sigma}_+
+(1-\cos\theta)^2 {\sigma}_-\right].
\label{cross}$$ In terms of the helicity cross sections $\sigma_{\alpha\beta}$ (with $\alpha,\beta={\rm L,R}$), directly related to the individual CI couplings $\epsilon_{\alpha\beta}$: $$\begin{aligned}
{\sigma}_{+}&=&\frac{1}{4}\,
\left[(1-P_e)(1+P_{\bar{e}})\,\sigma_{\rm LL}
+(1+P_e)(1- P_{\bar{e}})\,\sigma_{\rm RR}\right]\nonumber \\
&=&\frac{D}{4}\,\left[(1-P_{\rm eff})\,\sigma_{\rm LL}
+(1+P_{\rm eff})\,\sigma_{\rm RR}\right],
\label{s+} \\
{\sigma}_{-}&=&\frac{1}{4}\,
\left[(1-P_e)(1+ P_{\bar{e}})\,\sigma_{\rm LR}
+(1+P_e)(1-P_{\bar{e}})\,\sigma_{\rm RL}\right] \nonumber \\
&=&
\frac{D}{4}\,\left[(1-P_{\rm eff})\,\sigma_{\rm LR}
+(1+P_{\rm eff})\,\sigma_{\rm RL}\right], \label{s-}\end{aligned}$$ where $$P_{\rm eff}=\frac{P_e-P_{\bar{e}}}{1-P_eP_{\bar{e}}}
\label{pg}$$ is the effective polarization [@Flottmann:1995ga], $\vert P_{\rm eff}\vert\leq 1$, and $D=1-P_eP_{\bar{e}}$. For unpolarized positrons $P_{\rm eff}\rightarrow P_e$ and $D\rightarrow 1$, but with $P_{\bar{e}}\ne0$, $\vert P_{\rm eff}\vert$ can be larger than $|P_e|$. Moreover, in Eqs. (\[s+\]) and (\[s-\]): $$\sigma_{\alpha\beta}=\sigma_{\rm pt}
\vert{\cal M}_{\alpha\beta}\vert^2,
\label{helcross}$$ where $\sigma_{\rm pt}\equiv\sigma(e^+e^-\to\gamma^\ast\to l^+l^-)
=(4\pi\alpha^2)/(3s)$. The helicity amplitudes ${\cal M}_{\alpha\beta}$ can be written as $${\cal M}_{\alpha\beta}=Q_e Q_f+g_\alpha^e\,g_\beta^{\mu}\,\chi_Z+
\frac{s}{\alpha}\,\epsilon_{\alpha\beta}
\label{amplit}$$ where $\chi_Z=s/(s-M^2_Z+iM_Z\Gamma_Z)$ represents the $Z$ propagator, $g_{\rm L}^{\mu}=(-1/2+ s_W^2)/s_W c_W$ and $g_{\rm R}^{\mu}=s_W^2/s_W c_W$ are the SM left- and right-handed fermion couplings of the $Z$ with $s_W^2=1-c_W^2\equiv \sin^2\theta_W$.
We now define, with $\epsilon$ the experimental efficiency for detecting the final state under consideration, the four, directly measurable, integrated event rates: $$N_{\rm L,F},\quad N_{\rm R,F},\quad N_{\rm L,B},\quad N_{\rm R,B},
\label{obsn}$$ where ($\alpha={\rm L,R}$) $$N_{\alpha,{\rm F}}=\frac{1}{2}{\cal L}_{\rm int}\,\epsilon
\int_{0}^{1}({\rm d}\sigma_\alpha/{\rm d}\cos\theta){\rm d}\cos\theta,
\label{nf}$$ $$N_{\alpha,{\rm B}}=\frac{1}{2}{\cal L}_{\rm int}\,\epsilon
\int_{-1}^{0}({\rm d}\sigma_\alpha/{\rm d}\cos\theta){\rm d}\cos\theta,
\label{nb}$$ and subscripts R and L correspond to two sets of beam polarizations, $P_e=+P_1$, $P_{\bar e}=-P_2$ ($P_{1,2}>0$) and $P_e=-P_1$, $P_{\bar e}=+P_2$, respectively, or, alternatively, $P_{\rm eff}=\pm P$ with $D$ fixed. In Eqs. (\[nf\]) and (\[nb\]), ${\cal L}_{\rm int}$ is the time-integrated luminosity, we assume it to be equally distributed over the two combinations of beam polarizations, L and R.
The set of ‘conventional’ observables we consider here for the discussion of bounds on the CI parameters are the unpolarized cross section: $$\label{unpol}
\sigma_{\rm unpol}
=\frac{1}{4}\left[\sigma_{\rm LL}+\sigma_{\rm LR}
+\sigma_{\rm RR}+\sigma_{\rm RL}\right];$$ the (unpolarized) forward-backward asymmetry: $$A_{\rm FB}=
\frac{3}{4}\,
\frac{\sigma_{\rm LL}-\sigma_{\rm LR}+\sigma_{\rm RR}-\sigma_{\rm RL}}
{\sigma_{\rm LL}+\sigma_{\rm LR}+\sigma_{\rm RR}+\sigma_{\rm RL}};
\label{afbth}$$ the left-right and the left-right forward-backward asymmetries (which both require polarization), that can be written as, respectively: $$A_{\rm LR}=
\frac{\sigma_{\rm LL}+\sigma_{\rm LR}-\sigma_{\rm RR}-\sigma_{\rm RL}}
{\sigma_{\rm LL}+\sigma_{\rm LR}+\sigma_{\rm RR}+\sigma_{\rm RL}},
\label{alrth}$$ and $$A_{\rm LR,FB}=
\frac{3}{4}\,
\frac{\sigma_{\rm LL}-\sigma_{\rm RR}+\sigma_{\rm RL}-\sigma_{\rm LR}}
{\sigma_{\rm LL}+\sigma_{\rm RR}+\sigma_{\rm RL}+\sigma_{\rm LR}}.
\label{alrfbth}$$ Using Eqs. (\[helcross\]) and (\[amplit\]), one can easily express the deviations of these observables from the SM predictions in terms of the SM couplings and the CI couplings $\epsilon_{\alpha\beta}$ of Eq. (\[lagra\]).
The above observables are connected to the measured ones through the integrated event rates $N_{\rm L,R;F,B}$, see Eq. (\[obsn\]), as follows: $$D\,\sigma_{\rm unpol}=\frac{N_{\rm tot }^{\rm exp}}{{\cal L}_{\rm
int}\,\epsilon}, \label{ntotth}$$ where $$N_{\rm tot}^{\rm exp}=N_{\rm L,F}+N_{\rm R,F}+N_{\rm L,B}+N_{\rm R,B}
\label{ntot}$$ is the total number of events observed with polarized beams (for the four measurements). Eq. (\[ntotth\]) expresses the well-known fact that, when both the electron and positron beams are polarized, the total annihilation cross section into fermion-antifermion pairs will be increased by the factor $D$, with $1\leq D\leq 2$.
For the experimental forward-backward asymmetry: $$A_{\rm FB}=A_{\rm FB}^{\rm exp}\equiv
\frac{N_{\rm L,F}+N_{\rm R,F}-N_{\rm L,B}-N_{\rm R,B}}
{N_{\rm L,F}+N_{\rm R,F}+N_{\rm L,B}+N_{\rm R,B}},
\label{afbexp}$$ Finally, for the experimental left-right and left-right forward-backward asymmetries the relations are $$P_{\rm eff}A_{\rm LR}=A_{\rm LR}^{\rm exp}\equiv
\frac{N_{\rm L,F}+N_{\rm L,B}-N_{\rm R,F}-N_{\rm R,B}}
{N_{\rm L,F}+N_{\rm L,B}+N_{\rm R,F}+N_{\rm R,B}},
\label{alrexp}$$ and $$P_{\rm eff}A_{\rm LR,FB}=A_{\rm LR,FB}^{\rm exp}\equiv
\frac{(N_{\rm L,F}-N_{\rm R,F})-(N_{\rm L,B}-N_{\rm R,B})}
{N_{\rm L,F}+N_{\rm R,F} + N_{\rm L,B}+N_{\rm R,B}}.
\label{alrfbexp}$$
In the following analysis, cross sections will be evaluated including initial- and final-state radiation by means of the program ZFITTER [@zfitter], which has to be used along with ZEFIT, adapted to the present discussion, with $m_{\rm top}=175$ GeV and $m_H=120$ GeV. One-loop SM electroweak corrections are accounted for by improved Born amplitudes [@Consoli:1989pc; @Altarelli:1990dt], such that the forms of the previous formulae remain the same. Concerning initial-state radiation, a cut on the energy of the emitted photon $\Delta=E_\gamma/E_{\rm beam}=0.9$ is applied for $\sqrt s=0.5\ {\rm TeV}$ in order to avoid the radiative return to the $Z$ peak, and increase the signal originating from the contact interaction contribution [@Djouadi:1992sx].
As numerical inputs, we shall assume the commonly used reference values of the identification efficiencies [@Damerell]: $\epsilon=95\%$ for $\mu^+\mu^-$. Concerning the statistical uncertainty, to study the relative roles of statistical and systematic uncertainties we shall vary ${\cal L}_{\rm int}$ from $50$ to $500\ \mbox{fb}^{-1}$ (half for each polarization orientation) with uncertainty $\delta{\cal L}_{\rm int}/{\cal L}_{\rm int}=0.5\%$, and a fiducial experimental angular range $|\cos\theta|\le 0.99$. Also, regarding electron and positron degrees of polarization, we shall consider the values: $\vert
P_e\vert=0.8$; $\vert P_{\bar e}\vert=0.0,\ 0.4\ {\rm and}\ 0.6$, with $\delta P_e/P_e=\delta P_{\bar e}/P_{\bar e}=0.5\ \%$.
Model independent constraints
=============================
The current bounds on $\Lambda_{\alpha\beta}$ cited in Sect. 1, of the order of several TeV, are such that for the LC c.m. energy $\sqrt{s}=0.5$ TeV the characteristic suppression factor $s/\Lambda^2$ in Eq. (\[amplit\]) is rather strong. Accordingly, we can safely assume a linear dependence of the cross sections on the parameters $\epsilon_{\alpha\beta}$. In this regard, indirect manifestations of the CI interaction (\[lagra\]) can be looked for, [*via*]{} deviations of the measured observables from the SM predictions, caused by the new interaction. The reach on the CI couplings, and the corresponding constraints on their allowed values in the case of no effect observed, can be estimated by comparing the expression of the mentioned deviations with the expected experimental (statistical and systematic) uncertainties.
To this purpose, assuming the data to be well described by the SM ($\epsilon_{\alpha\beta}=0$) predictions, i.e., that no deviation is observed within the foreseen experimental uncertainty, and in the linear approximation in $\epsilon_{\alpha\beta}$ of the observables (\[unpol\])–(\[alrfbth\]), we apply the method based on the covariance matrix: $$\begin{aligned}
\label{covarv}
&&V_{kl}=\langle({\cal O}_k-\bar{\cal O}_k)
({\cal O}_l-\bar{\cal O}_l)\rangle \nonumber \\
&&=
\sum_{i=1}^{4}\left(\delta N_i\right)^2
\left(\frac{\partial{\cal O}_k}{\partial N_i}\right)
\left(\frac{\partial{\cal O}_l}{\partial N_i}\right)
+\left(\delta {\cal L}_{\rm int} \right)^2
\left(\frac{\partial{\cal O}_k}{\partial{\cal L}_{\rm int}}\right)
\left(\frac{\partial{\cal O}_l}{\partial{\cal L}_{\rm int}}\right)
\nonumber \\
&&+
\left(\delta P_e\right)^2
\left(\frac{\partial{\cal O}_k}{\partial P_e}\right)
\left(\frac{\partial{\cal O}_l}{\partial P_e}\right)+
\left(\delta P_{\bar e}\right)^2
\left(\frac{\partial{\cal O}_k}{\partial P_{\bar e}}\right)
\left(\frac{\partial{\cal O}_l}{\partial P_{\bar e}}\right).\end{aligned}$$ Here, the $N_i$ are given by Eq. (\[obsn\]), so that the statistical error appearing on the right-hand-side is given by $$\delta N_i=\sqrt{N_i}, \label{deltani}$$ and the ${\cal O}_l=(\sigma_{\rm unpol}$, $A_{\rm FB}$, $A_{\rm LR}$, $A_{\rm LR,FB})$ are the four observables. The second, third and fourth terms of the right-hand-side of Eq. (\[covarv\]) represent the systematic errors on the integrated luminosity ${\cal L}_{\rm int}$, polarizations $
P_e$ and $P_{\bar e}$, respectively, for which we assume the numerical values reported in the previous Section. From the explicit expression of the matrix elements $V_{kl}$, one can easily notice that, apart from $\sigma_{\rm unpol}$ and $A_{\rm FB}$ that are uncorrelated ($V_{12}=0$), all other pairs of observables show a correlation. Indeed, the non-zero diagonal entries are given by: $$\begin{aligned}
\label{Eq:V-diag}
&&V_{11}= \frac{\sigma_{\rm unpol}^2}{N_{\rm tot}^{\rm exp}}
+ \sigma_{\rm unpol}^2 \,
\left[ \frac{P^2_e P^2_{\bar e}}{D^2}(\epsilon^2_e +\epsilon^2_{\bar e})
+ \epsilon^2_{\cal L} \right]; \nonumber \\
&&V_{22}=\frac{1-A^2_{\rm FB}}{N_{\rm tot}^{\rm exp}}, \hskip 4pt
V_{33}=\frac{1-A^2_{\rm LR} P^2_{\rm eff}}{P^2_{\rm eff} N_{\rm tot}^{\rm
exp}}
+ A^2_{\rm LR} \, \Delta^2_2; \nonumber \\
&&V_{44}=\frac{1-A^2_{\rm LR,FB} P^2_{\rm eff}}
{P^2_{\rm eff} N_{\rm tot}^{\rm exp}} + A^2_{\rm LR,FB}\, \Delta^2_2,
\label{diagv}\end{aligned}$$ and, for the non-diagonal ones we have: $$\begin{aligned}
&&V_{13}= \sigma_{\rm unpol} \, A_{\rm LR} \, \Delta^2_1; \qquad
V_{14}= \sigma_{\rm unpol}\,
A_{\rm LR, FB} \, \Delta^2_1; \nonumber \\
&&V_{23}=\frac{A_{\rm LR,FB}-A_{\rm FB} A_{\rm LR}}{N_{\rm tot}^{\exp}},
\hskip 4pt
V_{24}=\frac{A_{\rm LR}-A_{\rm FB} A_{\rm LR,FB}}{N_{\rm tot}^{\rm exp}};
\nonumber \\
&&V_{34}=\frac{A_{\rm FB}-A_{\rm LR} A_{\rm LR,FB}
P^2_{\rm eff}}{P^2_{\rm eff} N_{\rm tot}^{\rm exp}} + A_{\rm LR}\,
A_{\rm LR,FB}\, \Delta^2_2.
\label{nondiagv}\end{aligned}$$ Here: $$\begin{aligned}
&&\Delta^2_1=\frac{P_e \, P_{\bar e}}{P_{\rm eff} D^3}
\left[-(1-P^2_{\bar e})\, P_e \, \epsilon^2_e + (1-P^2_e)\,
P_{\bar e} \, \epsilon^2_{\bar e} \right], \nonumber \\
&&\Delta^2_2=\frac{(1-P^2_{\bar e})^2 \, P^2_e \, \epsilon^2_e
+ (1-P^2_e)^2 \, P^2_{\bar e} \, \epsilon^2_{\bar e}}{P^2_{\rm eff}D^4},\end{aligned}$$ and $\epsilon_e=\delta P_e/P_e$, $\epsilon_{\bar e}=\delta P_{\bar e}/P_{\bar e}$ and $\epsilon_{\cal L}=\delta {\cal L}_{\rm int}/{\cal L}_{\rm int}$ are the relative systematic uncertainties.
One can notice, from Eq. (\[Eq:V-diag\]), that systematic uncertainties in $\sigma_{\rm unpol}$ are induced by $\epsilon_e$, $\epsilon_{\bar e}$ [*and*]{} $\epsilon_{\cal L}$, while those in $A_{\rm LR}$ and $A_{\rm LR,FB}$ arise from $\epsilon_e$ and $\epsilon_{\bar e}$ only, and [*not*]{} from $\epsilon_{\cal L}$. Finally, $A_{\rm FB}$ is free from such systematic uncertainties.
Defining the inverse covariance matrix $W^{-1}$ as $$\left(W^{-1}\right)_{ij}=\sum_{k,l=1}^4 \left(V^{-1}\right)_{kl}
\left(\frac{\partial{\cal O}_k}{\partial\epsilon_i}\right)
\left(\frac{\partial{\cal O}_l}{\partial\epsilon_j}\right),$$ with $\epsilon_i=(\epsilon_{LL},\ \epsilon_{LR},\ \epsilon_{RL},\
\epsilon_{RR})$, model-independent allowed domains in the four-dimensional CI parameter space to 95% confidence level are obtained from the error contours determined by the quadratic form in $\epsilon_{\alpha\beta}$ [@eadie; @Cuypers:1996it]: $$\label{chi2}
\left(\epsilon_{LL}\ \epsilon_{LR}\ \epsilon_{RL}\ \epsilon_{RR}\right)
W^{-1}
\left( \begin{array}{c}
\epsilon_{LL}\\
\epsilon_{LR}\\
\epsilon_{RL}\\
\epsilon_{RR}
\end{array}\right)=9.49.$$ The value 9.49 on the right-hand side of Eq. (\[chi2\]) corresponds to a fit with four free parameters [@Groom:2000in; @James:1975dr].
The quadratic form (\[chi2\]) defines a four-dimensional ellipsoid in the $\left(\epsilon_{LL},\ \epsilon_{LR},\ \epsilon_{RL},\
\epsilon_{RR}\right)$ parameter space. The matrix $W$ has the property that the square roots of the individual diagonal matrix elements, $\sqrt{W_{ii}}$, determine the projection of the ellipsoid onto the corresponding $i$-parameter axis in the four-dimensional space, and has the meaning of the bound at 95% C.L. on that parameter regardless of the values assumed for the others. Conversely, $1/\sqrt{\left(W^{-1}\right)_{ii}}$ determines the value of the intersection of the ellipsoid with the corresponding $i$-parameter axis, and represents the 95% C.L. bound on that parameter assuming all the others to be exactly known. Accordingly, the ellipsoidal surface constrains, at the 95% C.L. and model-independently, the range of values of the CI couplings $\epsilon_{\alpha\beta}$ allowed by the foreseen experimental uncertainties.
For the chosen input values for integrated luminosity, initial beam polarization, and corresponding systematic uncertainties, such model-independent limits are listed as lower bounds on the mass scales $\Lambda_{\alpha\beta}$ in Table 1. All the numerical results exhibited in Table 1 can be represented graphically. In Fig. 1 we show the planar ellipses that are obtained by projecting onto the six planes ($\epsilon_{LL},\epsilon_{LR}$), ($\epsilon_{LL},\epsilon_{RL}$), ($\epsilon_{LL},\epsilon_{RR}$), ($\epsilon_{RR},\epsilon_{LR}$), ($\epsilon_{RR},\epsilon_{RL}$), ($\epsilon_{LR},\epsilon_{RL}$) the 95% C.L. allowed four-dimensional ellipsoid resulting from Eq. (\[chi2\]). In these figures, the inner and outer ellipses correspond to positron polarizations $\vert P_{\bar e}\vert=0.6$ and $\vert P_{\bar e}\vert=0.0$, respectively.
\[tab:table-1\]
\[Fig:m\]
(11.5,18) (-1.0,8.0) (-1.0,-1.0)
To appreciate the significant role of initial beam polarization we should consider that, in the unpolarized case the only available observables would be $\sigma\propto (\sigma_{\rm LL}+\sigma_{\rm RR})
+(\sigma_{\rm LR}+\sigma_{\rm RL})$ and $\sigma\cdot A_{\rm FB}\propto (\sigma_{\rm LL}+\sigma_{\rm RR})
-(\sigma_{\rm LR}+\sigma_{\rm RL})$, see Eqs. (\[unpol\]) and (\[afbth\]). Therefore, by themselves, the pair of experimental observables $\sigma_{\rm unpol}$ and $A_{\rm FB}$ are not able to limit separately the CI couplings within finite ranges, but could only provide a constraint among the [*linear combinations*]{} of parameters $(\epsilon_{\rm LL}+\epsilon_{\rm RR})$ and $(\epsilon_{\rm LR}+\epsilon_{\rm RL})$. In some planes, specifically in the ($\epsilon_{\rm LL},\epsilon_{\rm RR})$ and $(\epsilon_{\rm LR},\epsilon_{\rm RL})$ planes, this constraint has the form of (unlimited) bands of allowed values, or correlations, such as those limited by the straight lines in Fig. 1. With initial beam polarization, two more physical observables become available, [*i.e.*]{}, $A_{\rm LR}$ and $A_{\rm LR,FB}$, and this enables us to close the bands into the ellipses in Fig. 1. The allowed bounds obtained from the observables $\sigma_{\rm unpol}$ and $A_{\rm FB}$ are not affected by electron polarization (for unpolarized positrons). Therefore, the bounds in the form of straight lines are tangential to the outer ellipses referring to $P_{\bar e}=0$, and in this case the role of $P_e\ne 0$ is just to close the corresponding band to a finite area.
The crosses in Fig. 1 represent the constraints obtainable by taking only one non-zero parameter at a time, instead of all four simultaneously non-zero, and independent, as in the analysis discussed here. Similar to the inner and outer ellipses, the shorter and longer arms of the crosses refer to positron polarization $\vert P_{\bar e}\vert=0.6$ and 0.0, respectively. Such ‘one-parameter’ results are derived from a $\chi^2$ procedure applied to the combination of the four physical observables (\[unpol\])–(\[alrfbth\]), also taking the above-mentioned correlations among observables into account. This procedure leads to results numerically consistent with those presented from essentially the same set of observables in Ref. [@Riemann:2001bb], if applied to the same experimental inputs used there.
For comparison, we also show in Table 1 the corresponding limits obtained in the case of polarized Bhabha scattering [@pp2001]. The table shows that for $\Lambda_{\rm LL}$ and $\Lambda_{\rm RR}$ the restrictions from $e^+e^-\to\mu^+\mu^-$ and $e^+e^-\to e^+e^-$ are qualitatively comparable. Instead, the sensitivity to $\Lambda_{\rm LR}$, and the corresponding lower bound, is dramatically higher in the case of Bhabha scattering. In this regard, this is the consequence of the initial beams longitudinal polarization that allows, by measuring suitable combinations of polarized cross sections, to directly disentangle the coupling $\epsilon_{\rm LR}$. Indeed, without polarization, in general only correlations among couplings, rather that finite allowed regions, could be derived.
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to thank Prof. P. Osland and Prof. N. Paver for the fruitful and enjoyable collaboration on the topics covered here.
[99]{} V. Barger, K. Cheung, K. Hagiwara and D. Zeppenfeld, Phys. Rev. D [**57**]{} (1998) 391; D. Zeppenfeld and K. Cheung. G. Altarelli, J. Ellis, G. F. Giudice, S. Lola and M. L. Mangano, Nucl. Phys. B [**506**]{} (1997) 3; R. Casalbuoni, S. De Curtis, D. Dominici and R. Gatto, Phys. Lett. B [**460**]{} (1999) 135. E. Eichten, K. Lane and M. E. Peskin, Phys. Rev. Lett. [**50**]{} (1983) 811; R. Rückl, Phys. Lett. B [**129**]{} (1983) 363. A. A. Pankov and N. Paver, Phys. Lett. B [**432**]{} (1998) 159. A. A. Babich, P. Osland, A. A. Pankov and N. Paver, Phys. Lett. B [**476**]{} (2000) 95. A. A. Babich, P. Osland, A. A. Pankov and N. Paver, Phys. Lett. B [**481**]{} (2000) 263. A. A. Babich, P. Osland, A. A. Pankov and N. Paver, Phys. Lett. B [**518**]{} (2001) 128. A. A. Babich, P. Osland, A. A. Pankov and N. Paver, LC Note LC-TH-2001-021 (2001), hep-ph/0101150. B. Schrempp, F. Schrempp, N. Wermes and D. Zeppenfeld, Nucl. Phys. B [**296**]{} (1988) 1. K. Flottmann, DESY-95-064; K. Fujii and T. Omori, KEK-PREPRINT-95-127. S. Riemann, FORTRAN program ZEFIT Version 4.2; D. Bardin et al., Comput. Phys. Commun. [**133**]{} (2001) 229. M. Consoli, W. Hollik and F. Jegerlehner, CERN-TH-5527-89 [*Presented at Workshop on Z Physics at LEP*]{}. G. Altarelli, R. Casalbuoni, D. Dominici, F. Feruglio and R. Gatto, Nucl. Phys. B [**342**]{} (1990) 15. A. Djouadi, A. Leike, T. Riemann, D. Schaile and C. Verzegnassi, Z. Phys. C [**56**]{} (1992) 289. C. J. S. Damerell, D.J. Jackson, in [*Proceedings of the 1996 DPF/DPB Summer Study on New Directions for High Energy Physics*]{} (Snowmass 96), Edited by D.G. Cassel, L. Trindle Gennari, R.H. Siemann (SLAC, 1997) p. 442. W.T. Eadie, D. Drijard, F.E. James, M. Roos, B. Sadoulet, [*Statistical methods in experimental physics*]{} (American Elsevier, 1971). F. Cuypers and P. Gambino, Phys. Lett. B [**388**]{} (1996) 211. D. E. Groom [*et al.*]{} \[Particle Data Group Collaboration\], Eur. Phys. J. C [**15**]{} (2000) 1. F. James and M. Roos, Comput. Phys. Commun. [**10**]{} (1975) 343. S. Riemann, LC Note LC-TH-2001-007 (2001). A. A. Pankov and N. Paver, preprint IC/2001/125.
[^1]: Talk given at the International School-Seminar “The Actual Problems of Particle Physics”, Gomel, August 7 – 16, 2001
|
---
abstract: 'A novel class of Approximate Policy Iteration (API) algorithms have recently demonstrated impressive practical performance (*e.g.*, ExIt [@anthony2017thinking], AlphaGo-Zero [@silver2017mastering]). This new family of algorithms maintains, and alternately optimizes, two policies: a fast, reactive policy (*e.g.*, a deep neural network) deployed at test time, and a slow, non-reactive policy (e.g., Tree Search), that can plan multiple steps ahead. The reactive policy is updated under supervision from the non-reactive policy, while the non-reactive policy is improved via guidance from the reactive policy. In this work we study this class of Dual Policy Iteration (DPI) strategy in an *alternating optimization framework* and provide a convergence analysis that extends existing API theory. We also develop a special instance of this framework which reduces the update of non-reactive policies to model-based optimal control using learned local models, and provides a theoretically sound way of unifying model-free and model-based RL approaches with unknown dynamics. We demonstrate the efficacy of our approach on various continuous control Markov Decision Processes.'
author:
- |
Wen Sun$^1$, Geoffrey J. Gordon$^1$, Byron Boots$^{2}$, and J. Andrew Bagnell$^3$\
\
$^1$School of Computer Science, Carnegie Mellon University, USA\
$^{2}$College of Computing, Georgia Institute of Technology, USA\
$^{3}$Aurora Innovation, USA\
`{wensun, ggordon, dbagnell}@cs.cmu.edu`, `[email protected]`\
bibliography:
- 'reference.bib'
title: Dual Policy Iteration
---
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank Sergey Levine for valuable discussion. WS is supported in part by Office of Naval Research contract N000141512365
|
---
abstract: 'The active target defense differential game is addressed in this paper. In this differential game an Attacker missile pursues a Target aircraft. The aircraft is however aided by a Defender missile launched by, say, the wingman, to intercept the Attacker before it reaches the Target aircraft. Thus, a team is formed by the Target and the Defender which cooperate to maximize the separation between the Target aircraft and the point where the Attacker missile is intercepted by the Defender missile, while the Attacker simultaneously tries to minimize said distance. This paper focuses on characterizing the set of coordinates such that if the Target’s initial position belong to this set then its survival is guaranteed if both the Target and the Defender follow their optimal strategies. Such optimal strategies are presented in this paper as well.'
author:
- 'Eloy Garcia[^1], David W. Casbeer, and Meir Pachter'
---
INTRODUCTION {#sec:intro}
============
In multi-agent pursuit-evasion problems one or more pursuers try to maneuver and reach a relatively small distance with respect to one or more evaders, which strive to escape the pursuers. This problem is usually posed as a dynamic game [@Ganebny12], [@Huang11], [@Pham10]. Thus, a dynamic Voronoi diagram has been used in problems with several pursuers in order to capture an evader within a bounded domain [@Huang11], [@Bakolas10]. On the other hand, [@Sprinkle04] presented a receding-horizon approach that provides evasive maneuvers for an Unmanned Autonomous Vehicle (UAV) assuming a known model of the pursuer’s input, state, and constraints. In [@EarlDandrea07], a multi-agent scenario is considered where a number of pursuers are assigned to intercept a group of evaders and where the goals of the evaders are assumed to be known. Cooperation between two agents with the goal of evading a single pursuer has been addressed in [@Fuchs10] and [@Scott13].
In this paper we consider a zero-sum three-agent pursuit-evasion differential game. A two-agent team is formed which consists of a Target ($T$) and a Defender ($D$) who cooperate; the Attacker ($A$) is the opposition. The goal of the Attacker is to capture the Target while the Target tries to evade the Attacker and avoid capture. The Target cooperates with the Defender which pursues and tries to intercept the Attacker before the latter captures the Target. Cooperation between the Target and the Defender is such that the Defender will capture the Attacker before the latter reaches the Target. Such a scenario of active target defense has been analyzed in the context of cooperative optimal control in [@Boyell76], [@Boyell80]. Indeed, sensing capabilities of missiles and aircraft allow for implementation of complex pursuit and evasion strategies [@Zarchan97], [@Siouris04], and more recent work has investigated different guidance laws for the agents $A$ and $D$. Thus, in [@ratnoo11] the authors addressed the case where the Defender implements Command to the Line of Sight (CLOS) guidance to pursue the Attacker which requires the Defender to have at least the same speed as the Attacker. In [@rubinsky13] the end-game for the TAD scenario was analyzed based on the minimization/maximization of the Attacker/Target miss distance for a *non-cooperative* Target/Defender pair. The authors develop linearization-based Attacker maneuvers in order to evade the Defender and continue pursuing the Target. A different guidance law for the Target-Attacker-Defender (TAD) scenario was given by Yamasaki *et.al.* [@Yamasaki10], [@Yamasaki13]. These authors investigated an interception method called Triangle Guidance (TG), where the objective is to command the defending missile to be on the line-of-sight between the attacking missile and the aircraft for all time, while the Target aircraft follows some predetermined trajectory. The authors show, through simulations, that TG provides better performance in terms of Defender control effort than a number of variants of Proportional Navigation (PN) guidance laws, that is, when the Defender uses PN to pursue the Attacker instead of TG.
The previous approaches constrain and limit the level of cooperation between the Target and the Defender by implementing Defender guidance laws without regard to the Target’s trajectory.
Different types of cooperation have been recently proposed in [@Perelman11], [@Rusnak05], [@Rusnak11], [@Ratnoo12], [@Shima11], [@Shaferman10], [@Prokopov13] for the TAD scenario. In [@Rusnak11] optimal policies (lateral acceleration for each agent including the Attacker) were provided for the case of an aggressive Defender, that is, the Defender has a definite maneuverability advantage. A linear quadratic optimization problem was posed where the Defender’s control effort weight is driven to zero to increase its aggressiveness. The work [@Ratnoo12] provided a game theoretical analysis of the TAD problem using different guidance laws for both the Attacker and the Defender. The cooperative strategies in [@Shima11] allow for a maneuverability disadvantage for the Defender with respect to the Attacker and the results show that the optimal Target maneuver is either constant or arbitrary. Shaferman and Shima [@Shaferman10] implemented a Multiple Model Adaptive Estimator (MMAE) to identify the guidance law and parameters of the incoming missile and optimize a Defender strategy to minimize its control effort. In the recent paper [@Prokopov13] the authors analyze different types of cooperation assuming the Attacker is oblivious of the Defender and its guidance law is known. Two different one-way cooperation strategies were discussed: when the Defender acts independently, the Target knows its future behavior and cooperates with the Defender, and vice versa. Two-way cooperation where both Target and Defender communicate continuously to exchange their states and controls is also addressed, and it is shown to have a better performance than the other types of cooperation - as expected.
Our preliminary work [@Garcia14], [@Garcia15] considered the cases when the Attacker implements typical guidance laws of Pure Pursuit (PP) and PN, respectively. In these papers, the Target-Defender team solves an *optimal control* problem that returns the optimal strategy for the $T-D$ team so that $D$ intercepts the Attacker and at the same time the separation between Target and Attacker at the instant of interception of $A$ by $D$ is maximized. The cooperative optimal guidance approach was extended ([@Pachter14Allerton], [@Garcia15ACC], [@Garcia15JGCD]) to consider a differential game where also the Attacker missile solves an optimal control problem in order to minimize the final separation between itself and the Target. In this paper, we focus on characterizing the region of the reduced state space formed by the agents initial positions for which survival of the Target is guaranteed when both the Target and the Defender employ their optimal strategies. The optimal strategies for each one of the three agents participating in the active target defense differential game are provided in this paper as well.
The paper is organized as follows. Section \[sec:Problem\] describes the engagement scenario. Section \[sec:analysis\] presents optimal strategies for each one of the three participants in order to solve the differential game discussed in the paper. The Target escape region is characterized in Section \[sec:escape\]. Examples are given in Section \[sec:Example\] and concluding remarks are made in Section \[sec:concl\].
PROBLEM STATEMENT {#sec:Problem}
=================
The active target defense engagement in the realistic plane $(x,y)$ is illustrated in Figure \[fig:problem description\]. The speeds of the Target, Attacker, and Defender are denoted by $V_T$, $V_A$, and $V_D$, respectively, and are assumed to be constant. The simple-motion dynamics of the three vehicles in the realistic plane are given by: $$\begin{aligned}
\dot{x}_T&=V_T\cos\hat{\phi}, \ \ \ \ \ \ \ \ \ \ \ \dot{y}_T=V_T\sin\hat{\phi} \label{eq:fixedT} \\
\dot{x}_A&=V_A\cos\hat{\chi}, \ \ \ \ \ \ \ \ \ \ \ \dot{y}_A=V_A\sin\hat{\chi} \label{eq:fixedADG} \\
\dot{x}_D&=V_D\cos\hat{\psi}, \ \ \ \ \ \ \ \ \ \ \: \dot{y}_D=V_D\sin\hat{\psi} \label{eq:fixedD}\end{aligned}$$ where the headings of $T$, $D$, and $A$ are, respectively, $\hat{\phi}$, $\hat{\psi}$, and $\hat{\chi}$.
In this game the Attacker pursues the Target and tries to capture it. The Target and the Defender cooperate in order for the Defender interpose himself between the Attacker and the Target and to intercept the Attacker before the latter captures the Target. Thus, the Target-Defender team searches for a cooperative optimal strategy, optimal headings $\hat{\phi}^*$ and $\hat{\psi}^*$, to maximize the separation between the Target and the Attacker at the time instant of the Defender-Attacker collision. The Attacker will devise its corresponding optimal strategy, optimal heading $\hat{\chi}^*$, in order to minimize the terminal $A-T$ separation/miss distance. Define the speed ratio problem parameter $\alpha=V_T/V_A$. We assume that the Attacker missile is faster than the Target aircraft, so that $\alpha <1$. In this work we also assume the Attacker and Defender missiles are somewhat similar, so $V_D=V_A$.
In the following sections this problem is transformed to an aimpoint problem where each agent finds is optimal aimpoint. Furthermore, it is shown that the solution of the differential game involving three variables (the aimpoint of each one of the three agents) is equivalent to the solution of an optimization problem in only one variable.
DIFFERENTIAL GAME {#sec:analysis}
=================
We now undertake the analysis of the active target defense differential game. The Target ($T$), the Attacker ($A$), and the Defender ($D$) have “simple motion" $\grave{\text{a}}$ la Isaacs [@Isaacs65]. We also emphasize that $T$, $A$, and $D$ have constant speeds of $V_T$, $V_A$, and $V_D$, respectively. We assume that $V_A=V_D$ and the speed ratio $\alpha=\frac{V_T}{V_A}<1$. We confine our attention to point capture, that is, the $D-A$ separation has to become zero in order for the Defender to intercept the Attacker. $T$ and $D$ form a team to defend from $A$. Thus, $A$ strives to close in on $T$ while $T$ and $D$ maneuver such that $D$ intercepts $A$ before the latter reaches $T$ and the distance at interception time is maximized, while $A$ strives to minimize the separation between $T$ and $A$ at the instant of interception. Since the cost is a function only of the final time (the interception time instant) and the agents have simple motion dynamics, the optimal trajectories of each agent are straight lines.
In Figure \[fig:oNE\] the points $A$ and $D$ represent the initial positions of the Attacker and the Defender in the reduced state space, respectively. A Cartesian frame is attached to the points $A$ and $D$ in such a way that the extension to infinity of the segment $\overline{AD}$ in both directions represents the $X$-axis and the orthogonal bisector of $\overline{AD}$ represents the $Y$-axis. The state variables are $x_A$, $x_T$, and $y_T$. Notice that all points in the Left-Half-Plane (LHP) can be reached by the Defender before the Attacker does; similarly, all points in the Right-Half-Plane (RHP) can be reached by the Attacker before the Defender does.
In this paper we focus on the case where the Target is initially closer to the Attacker than to the Defender; in other words, assume that $x_T>0$.
With respect to Figure \[fig:oNE\] we note that the Defender will intercept the Attacker at point $I$ on the orthogonal bisector of $\overline{AD}$ at which time the Target will have reached point $T'$. The Attacker aims at minimizing the distance between the Target at the time instant when the Defender intercepts the Attacker, that is, the distance between point $T'$ and point $I$ on the orthogonal bisector of $\overline{AD}$ where the Defender intercepts the Attacker; the points $T$ and $T'$ represent the initial and terminal positions of the Target, respectively.
Cost/Payoff Function {#subsec:xTg0cp}
--------------------
When $x_T>0$ the Attacker and the Target are faced with a maxmin optimization problem: the Target chooses point $v$ and the Attacker chooses point $u$ on the Y-axis, see Figure \[fig:MaxminEj\]. Additionally, the Defender tries to intercept the Attacker by choosing his aimpoint at point $w$ on the Y-axis. Thus, the optimization problem is $$\begin{aligned}
\max_{v,w} \ \min_{u} \ J(u,v,w). \end{aligned}$$ where the function $J(u,v,w)$ represents the distance between the Target terminal position and the point where the Attacker is intercepted by the Defender. The target tries to cross the orthogonal bisector of $\overline{AD}$ into the LHP where the Defender will be able to allow it to escape by intercepting the Attacker at the point $(0,u)$ on the orthogonal bisector of $\overline{AD}$. Therefore, the Defender’s optimal policy is $w^*(u,v)=u$ in order to guarantee interception of the Attacker. The optimality of this choice by the Defender will be shown in Proposition \[prop:SPxtg0\].
Since the Defender’s optimal policy is $w^*=u$, the decision variables $u$ and $v$ jointly determine the distance $S$ between the Target terminal position $T'$ and the point $(0,u)$ where the Attacker is intercepted by the Defender. This distance is a function of the decision variables $u$ and $v$. Thus, the Attacker and the Target solve the following optimization problem $$\begin{aligned}
\max_{v} \ \min_{u} \ J(u,v). \end{aligned}$$ Now, let us analyze the possible strategies. If the Target chooses $v$, the Attacker will respond and choose $u$. If $u\neq v$ the Target would correct his decision and choose some $\bar{v}$ such that $\bar{S}>S$, as shown in Figure \[fig:MaxminEj\] for the case where $u>v$ and in Figure \[fig:MaxminEj2\] for the case where $u<v$. In general, choosing $u\neq v$ is detrimental to the Attacker since his cost will increase. Thus, the Attacker should aim at the point $v$ which is chosen by the Target, that is, $u^*(v)=\arg \min_u J(u,v)=v$.
Given the cost/payoff function $J(u,v)$, the solution $u^*$ and $v^*$ of the optimization problem $ \max_{v} \ \min_{u} J(u,v)$ is such that $$\begin{aligned}
u^*=v^*. \end{aligned}$$ Moreover, when $x_T>0$, the Attacker strategy is $u^*(v)=\arg \min_u J(u,v)=v$ so that it suffices to solve the optimization problem $\max_y J(y)$ where $$\begin{aligned}
J(y)=\alpha\sqrt{x_A^2+y^2}-\sqrt{(y-y_T)^2+x_T^2}. \label{eq:Costy} \end{aligned}$$
Critical Speed Ratio for Target Survival
----------------------------------------
We assume that the Attacker is faster than the Target, for otherwise the Target could always escape without the help of the Defender. Thus, we assume that the speed ratio $0<\alpha<1$. Also, we assume that $x_T>0$. The Target needs to be able to break into the LHP before being intercepted by the Attacker for the Defender to be able to assist the Target to escape, by intercepting the Attacker who is on route to the Target. Thus, a solution to the active target defense differential game exists if and only if the Apollonius circle, which is based on the segment $\overline{AT}$ and the speed ratio $\alpha$, intersects the orthogonal bisector of $\overline{AD}$. This imposes a lower limit $\bar{\alpha}$ on the speed ratio, that is, we need $\bar{\alpha}<\alpha<1$. The critical speed ratio $\bar{\alpha}$ corresponds to the case where the Apollonius circle is tangent to the orthogonal bisector of $\overline{AD}$. And if the speed ratio $\alpha\geq 1$ the Target always escapes and there is no need for a Defender missile, that is, no target defense differential game is played out. The optimal strategies for the case $x_T<0$ can be obtained in a similar way as shown in this paper and the critical speed ratio is $\bar{\alpha}=0$.
Assume that $x_T>0$. Then, the critical speed ratio $\bar{\alpha}$ is a function of the positions of the Target and the Attacker and is given by $$\begin{aligned}
\bar{\alpha}=\frac{\sqrt{(x_A+x_T)^2+y_T^2}-\sqrt{(x_A-x_T)^2+y_T^2}}{2x_A}. \label{eq:alphasol} \end{aligned}$$
*Proof*. The Attacker’s initial position, the Target’s initial position, and the center $O$ of the Apollonius circle are collinear and lie on the dotted straight line in Figure \[fig:oNE\] whose equation is $$\begin{aligned}
y=-\frac{y_T}{x_A-x_T}x + \frac{x_Ay_T}{x_A-x_T}. \nonumber\end{aligned}$$ The geometry of the Apollonius circle is as follows: The center of the circle, denoted by $O$, is at a distance of $\frac{\alpha^2}{1-\alpha^2}d$ from $T$ and its radius is $\frac{\alpha}{1-\alpha^2}d$, where $d$ is the distance between $A$ and $T$ and is given by $$\begin{aligned}
d=\sqrt{(x_A-x_T)^2+y_T^2}. \end{aligned}$$ Hence, the following holds $$\begin{aligned}
\Big(\frac{x_Ty_T}{x_A-x_T}-\frac{y_T}{x_A-x_T}x_0\Big)^2 + (x_0-x_T)^2
=\frac{\alpha^4}{(1-\alpha^2)^2}[(x_A-x_T)^2+y_T^2] %\label{eq:circleeq}\end{aligned}$$ and we calculate the coordinates of the center of the Apollonius circle $$\begin{aligned}
\left.
\begin{array}{l l}
x_O=\frac{1}{1-\alpha^2}x_T-\frac{\alpha^2}{1-\alpha^2}x_A, \ \
y_O=\frac{1}{1-\alpha^2}y_T.
\end{array} \label{eq:circCenter} \right.\end{aligned}$$ Consequently, the critical speed ratio $\bar{\alpha}$ is the positive solution of the quadratic equation $$\begin{aligned}
x_T-\alpha^2x_A=\alpha\sqrt{(x_A-x_T)^2+y_T^2} \label{eq:alphaeq} \end{aligned}$$ which is given by .
$\square$
In general, it can be seen from Figure \[fig:oNE\] that if $x_T<0$ then $\bar{\alpha}=0$ as well. We will assume $\bar{\alpha}<\alpha<1$, so that a solution to the active target defense differential game exists; otherwise, if $\alpha\leq\bar{\alpha}$, the Defender will not be able to help the Target by intercepting the Attacker before the latter inevitably captures the Target; and if $\alpha\geq 1$ then the Target can always evade the Attacker and there is no need for a Defender.
Optimal Strategies {#subsec:optimal}
------------------
When the Target is on the side of the Attacker, the Target chooses its aimpoint, denoted by $I$, on the orthogonal bisector of $\overline{AD}$ in order to maximize its payoff function , the final separation between Target and Attacker, and where $y$ represents the coordinate of the aimpoint $I$ on the orthogonal bisector of $\overline{AD}$. This is so because the Attacker will aim at the point $I$. In order to minimize the optimal strategy of the Attacker is to choose the same aimpoint $I$ on the orthogonal bisector of $\overline{AD}$, where it will be intercepted by the Defender.
In order to find the maximum of we differentiate eq. in $y$ and set the resulting derivative equal to zero $$\begin{aligned}
\frac{dJ(y)}{dy}=\frac{\alpha y}{\sqrt{x_A^2+y^2}}-\frac{y-y_T}{\sqrt{(y-y_T)^2+x_T^2}}=0. \label{eq:FderCosty} \end{aligned}$$ The following quartic equation in $y\geq 0$ is obtained $$\begin{aligned}
(1-\alpha^2)y^4 - 2(1-\alpha^2)y_Ty^3
+ \big((1\!-\!\alpha^2)y_T^2\!+\!x_A^2\!-\!\alpha^2x_T^2\big)y^2
- 2x_A^2y_Ty+x_A^2y_T^2=0. \label{eq:Quartic}\end{aligned}$$
In the sequel we focus on the case $0<\alpha<1$. In addition and without loss of generality assume that $y_T>0$. Let us divide both sides of eq. by $y_T^4$ and set $x_A=\frac{x_A}{y_T}$, $x_T=\frac{x_T}{y_T}$, and $y=\frac{y}{y_T}$, whereupon the quartic equation assumes the canonical form $$\begin{aligned}
(1-\alpha^2)y^4 - 2(1-\alpha^2)y^3
+ \big(1\!-\!\alpha^2 + x_A^2\!-\!\alpha^2x_T^2\big)y^2
- 2x_A^2y+x_A^2=0.
\label{eq:QuarticCan} \end{aligned}$$ We are interested in the real and positive solutions $y>0$ of the canonical quartic equation . Eq. has two real solutions, $$\begin{aligned}
0<y_{R_1}<1 \ \ \textsl{and} \ \ y_{R_2}>1. \nonumber\end{aligned}$$ When $x_T=0$, has two repeated solutions at $y=1$ and two complex solutions $y=\pm i\frac{1}{\sqrt{1-\alpha^2}}x_A$.
*Remark*. Writing the quartic equation as $f(y)=0$ we see that $f(0)=x_A^2y_T^2>0$, $f(y_T)=-\alpha^2 x_T^2y_T^2<0$, and $f(\infty)=+\infty$. Therefore, equation has two real solutions. Equation has a real solution $0<y<y_T$ and an additional real solution $y_T<y$, provided that $x_T\neq 0$. Note that the quartic equation is parameterized by $x_T^2$, so whether $x_T>0$ or $x_T<0$ makes no difference as far as the solutions to the quartic equation are concerned. However, if $x_T<0$ the applicable real solution is $y<y_T$, whereas if $x_T>0$ the applicable real solution is $y>y_T$.
When $x_T>0$, by choosing his heading, the Target (and the Defender) thus choose the coordinate $y$ to maximize $J(y)$; that is, $y$ is the Target’s (and Defender’s) choice. Then the payoff is given by eq. and the expression for $\frac{dJ(y)}{dy}$ was shown in . The second derivative of the payoff function $$\begin{aligned}
\frac{d^2J(y)}{dy^2}=\frac{\alpha x_A^2}{(x_A^2+y^2)^{3/2}}-\frac{x_T^2}{\big((y-y_T)^2+x_T^2\big)^{3/2}}. \label{eq:SderCosty} \end{aligned}$$ The Target is choosing $y$ to maximize the cost $J(y)$. Now, the Attacker reacts by heading towards the point $I$ on the orthogonal bisector of $\overline{AD}$ where, invariably, he will be intercepted by the Defender. Both the Target and the Attacker know that the three points $T,I,T'$ must be collinear. The defender will not allow the Attacker to cross the orthogonal bisector because then the Attacker will start to close in on the Target.
The optimal coordinate $y^*$ is the solution of the quartic equation such that the second-order condition for a maximum holds on $\frac{d^2J(y)}{dy^2}<0$. In view of we know that $$\begin{aligned}
\frac{1}{\sqrt{(y-y_T)^2+x_T^2}} = \alpha\frac{y}{y-y_T} \frac{1}{\sqrt{x_A^2+y^2}}. \label{eq:opteq} \end{aligned}$$ and inserting into yields $$\begin{aligned}
\frac{d^2J(y)}{dy^2}=\frac{\alpha}{(x_A^2\!+\!y^2)^{3/2}} \Big(x_A^2 - \alpha^2\Big(\frac{y}{y\!-\!y_T}\Big)^3x_T^2\Big). \label{eq:SderNeg} \end{aligned}$$ We have that $\frac{d^2J(y)}{dy^2}<0$ if and only if $$\begin{aligned}
\frac{1}{\alpha^2}\Big(\frac{x_A}{x_T}\Big)^2 < \Big(\frac{y}{y\!-\!y_T}\Big)^3. \label{eq:SderCond} \end{aligned}$$ Hence, the first real solution $y_1<y_T$ of the quartic equation does not fulfill the role of yielding a maximum and the second real solution $y_2>y_T$ of is the candidate solution. It is the Target who chooses $y^*$ to maximize the payoff $J(y)$. Note that $y_1=y_{R_1}y_T$ and $y_2=y_{R_2}y_T$, where $y_{R_1}$ and $y_{R_2}$ are the real solutions of .
Inserting eq. into eq. yields the Target and Defender payoff $$\begin{aligned}
\left.
\begin{array}{l l}
J^*(y) &= \alpha\sqrt{x_A^2+y^2} - \frac{1}{\alpha}\frac{y-y_T}{y}\sqrt{x_A^2+y^2} \\
&=\frac{1}{\alpha}\sqrt{x_A^2+y^2}\big(\frac{y_T}{y} - (1-\alpha^2)\big)
\end{array} \label{eq:payoff} \right.\end{aligned}$$ and using $y=y_2$ $$\begin{aligned}
\left.
\begin{array}{l l}
J^*(y)= \frac{1}{\alpha}\sqrt{x_A^2+y_2^2}\big(\frac{y_T}{y_2} - (1-\alpha^2)\big).
\end{array} \label{eq:payoff2} \right.\end{aligned}$$ When $\alpha>\bar{\alpha}$ we have that $J^*(y)>0$. Hence, the solution $y_2>y_T$ of the quartic equation must satisfy $$\begin{aligned}
\left.
\begin{array}{l l}
y_T<y_2<\frac{1}{1-\alpha^2} y_T. \label{eq:OptCondy}
\end{array} \right.\end{aligned}$$ This situation is illustrated in Figure \[fig:OpSol\] where the three points $T$, $I$, and $T'$ are collinear. Concerning expression , we also need the solution of the quartic equation to satisfy $$\begin{aligned}
y_2 < \frac{1}{1-\alpha^{2/3}(\frac{x_T}{x_A})^{2/3}} y_T. \label{eq:OptCondy2} \end{aligned}$$ The second real solution $y_2$ of the quartic equation must satisfy
$$\begin{aligned}
y_T< y_2 <\min\left\{\frac{1}{1-\alpha^2} y_T, \frac{1}{1-\alpha^{2/3}(\frac{x_T}{x_A})^{2/3}} y_T\right\}. \label{eq:OptCondymin} \end{aligned}$$
The points of intersection of the Apollonius circle with the $y$-axis (the orthogonal bisector) are $(0,\underline{y})$ and $(0,\overline{y})$, where $(0,\underline{y})$ and $(0,\overline{y})$ are the solutions of the quadratic equation $$\begin{aligned}
x_O^2+(y-y_O)^2=\frac{\alpha^2}{(1-\alpha^2)^2} d^2 \label{eq:yyQuad} \end{aligned}$$ where the distance $d=\sqrt{(x_A-x_T)^2+y_T^2}$ and the Apollonius circle’s center coordinates are given by . We have that $$\begin{aligned}
(y-y_O)^2=\frac{1}{(1-\alpha^2)^2}\big(\alpha^2d^2 - (x_T-\alpha^2x_A)^2\big) \nonumber \end{aligned}$$ where $$\begin{aligned}
\alpha^2d^2 - (x_T-\alpha^2x_A)^2>0 \nonumber\end{aligned}$$ because $\alpha>\bar{\alpha}$ and, from , we have that $\bar{\alpha}d = x_T-\bar{\alpha}^2x_A$. Hence, $$\begin{aligned}
y=y_O \pm \frac{1}{1-\alpha^2}\sqrt{\alpha^2d^2 - (x_T-\alpha^2x_A)^2} \nonumber \end{aligned}$$ which results in $$\begin{aligned}
\underline{y}= \frac{1}{1-\alpha^2}\Big(y_T - \sqrt{\alpha^2y_T^2 + (1-\alpha^2)(\alpha^2x_A^2-x_T^2)}\Big) \\
\overline{y}= \frac{1}{1-\alpha^2}\Big(y_T + \sqrt{\alpha^2y_T^2 + (1-\alpha^2)(\alpha^2x_A^2-x_T^2)}\Big). \end{aligned}$$ The Target’s choice of the optimal $y^*$, namely, the solution $y_2$ of the quartic equation must satisfy the inequalities $$\begin{gathered}
\frac{1}{1-\alpha^2}\Big(y_T \!-\! \sqrt{\alpha^2y_T^2 + (1\!-\!\alpha^2)(\alpha^2x_A^2-x_T^2)}\Big)
\leq y_2 \\
\leq \frac{1}{1-\alpha^2}\Big(y_T \!+\! \sqrt{\alpha^2y_T^2 + (1\!-\!\alpha^2)(\alpha^2x_A^2-x_T^2)}\Big). \end{gathered}$$
\[prop:SPxtg0\] (Saddle point equilibrium). Consider the case $x_T>0$. The strategy $y^*$ of the Target, where $y^*$ is the real solution of the quartic equation which maximizes , and the strategy of the Defender of heading to the point $(0,y^*)$, together with the strategy of the Attacker of aiming at the point $(0,y^*)$, constitute a strategic saddle point, that is $$\begin{aligned}
\left.
\begin{array}{l l}
&\left\{J(u^*,v^*,w),J(u^*,v,w^*),J(u^*,v,w)\right\}
<J(u^*,v^*,w^*)<J(u,v^*,w^*) . \nonumber
\end{array} \right. \end{aligned}$$
ESCAPE REGION {#sec:escape}
=============
In this section we analyze the Target’s escape region for given Target and Attacker speeds, $V_T$ and $V_A$, respectively. In other words, for given speed ratio $\alpha=V_T/V_A$. Consider the active target defense differential game where the Attacker and the Defender missiles have the same speeds. When $x_T>0$, the critical value of the speed ratio parameter $\bar{\alpha}$ can be obtained as a function of the Attacker’s and the Target’s coordinates $x_A$, $x_T$, and $y_T$ such that the Target is guaranteed to escape since the Defender will be able to intercept the Attacker before the latter reaches the Target.
Now, for a given Target’s speed ratio, $\alpha$, and for given Attacker’s initial position, $x_A$, we wish to characterize the region of the reduced state space for which the Target is guaranteed to escape. In other words, we want to separate the reduced state space into two regions: $R_e$ and $R_{e_o}$. The region $R_e$ is defined as the set of all coordinate pairs $(x,y)$ such that if the Target’s initial position $(x_T,y_T)$ is inside this region, then, it is guaranteed to escape the Attacker if both the Target and the Defender implement their corresponding optimal strategies. The region $R_{e_o}$, represents all other coordinate pairs $(x,y)$ in the reduced state space where the Target’s escape is not guaranteed.
For given speed ratio $\alpha$ and for given Attacker’s initial position $x_A$ in the reduced state space, the curve that divides the reduced state space into the two regions $R_e$ and $R_{e_o}$ is characterized by the right branch of the following hyperbola (that is, $x>0$) $$\begin{aligned}
\frac{x^2}{\alpha^2x_A^2} -\frac{y^2}{(1-\alpha)^2x_A^2} = 1. \label{eq:hb}\end{aligned}$$
*Proof*. The requirement for the Target to escape being captured by the Attacker is that the Apollonius circle intersects the Y-axis. The radius of the Apollonius circle is $$\begin{aligned}
r=\frac{\alpha}{1-\alpha^2}\sqrt{(x_A-x_T)^2+y_T^2} \label{eq:r}\end{aligned}$$ and the X-coordinate of its center is $$\begin{aligned}
x_O=\frac{1}{1-\alpha^2}(x_T-\alpha^2x_A) \label{eq:xo}\end{aligned}$$ If $x_T>\alpha^2x_A$, we need $r>x_O$ for the Defender to be of any help to the Target. Thus, $x_A>0$, $y_T\geq 0$, and $x_T>0$ must satisfy the condition $$\begin{aligned}
\frac{1}{1-\alpha^2}(x_T-\alpha^2x_A) < \frac{\alpha}{1-\alpha^2}\sqrt{(x_A-x_T)^2+y_T^2} \label{eq:cond}\end{aligned}$$ equivalently, $$\begin{aligned}
x_T-\alpha^2x_A < \alpha\sqrt{(x_A-x_T)^2+y_T^2} \label{eq:cond2}\end{aligned}$$ which is also equivalent to $$\begin{aligned}
\frac{x_A^2}{(\frac{x_T}{\alpha})^2} +\frac{y_T^2}{\big(\frac{\sqrt{1-\alpha^2}}{\alpha}x_T\big)^2} > 1. \label{eq:cond3}\end{aligned}$$ When the ‘greater than’ sign in inequality is changed to ‘equal’ sign, the resulting equation defines the curve that divides the reduced state space into regions $R_e$ and $R_{e_o}$. Additionally, since the symmetric case $y_T<0$ can be treated in a similar way as the case $y_T>0$, we do not need to restrict $y_T$ to be greater than or equal to zero. Thus, the coordinate pairs $(x,y)$ such that $$\begin{aligned}
\frac{x_A^2}{(\frac{x}{\alpha})^2} +\frac{y^2}{\big(\frac{\sqrt{1-\alpha^2}}{\alpha}x\big)^2} = 1 \nonumber\end{aligned}$$ can be written in the hyperbola canonical form shown in .
$\square$
*Remark*. Note that for a given speed ratio $\alpha$, the family of hyperbolas characterized by different values of $x_A>0$ shares the same center which is located at $C=(0,0)$, and the same asymptotes which are given by the lines $y=\frac{\sqrt{1-\alpha^2}}{\alpha}x$ and $y=-\frac{\sqrt{1-\alpha^2}}{\alpha}x$. One can also see that, for $0<\alpha<1$, the slope of the asymptotes increases as $\alpha$ decreases and viceversa. This behavior is expected since a relatively faster Target will be able to escape the Attacker when starting at the same position as a relatively slower Target.
It is important to emphasize that if $(x_T,y_T)\in R_{e_o}$ then capture of the Target by the Attacker is guaranteed if the Attacker employs its optimal strategy. In this case the optimal strategies described in Section \[subsec:optimal\] will result in $J(y^*)<0$. $J(y^*)$ being negative makes sense in terms of the differential game formulated in this paper (recall that the Attacker tries to minimize $J(y)$). However, the cost/payoff function $J(y)$ represents a distance and it does not make sense for it to be negative in the real scenario where the Attacker tries to capture the Target, i.e. the terminal separation $\overline{AT}$ should be zero instead of negative. Based on the solution of the differential game presented in Section \[subsec:optimal\], the Attacker is able to redefine its strategy and capture the Target, that is, to obtain $\overline{AT}=0$. The new strategy is as follows. The Attacker, by solving the differential game and obtaining the optimal cost/payoff, realizes that $J(y^*)<0$, then, it simply redefines its optimal strategy to be $u'^*(v')=v'$.
The details when the optimal strategies of Section \[subsec:optimal\] result in $J(y^*)<0$ are as follows. $T$ chooses his aimpoint to be $v'$ that lies on the Apollonius circle. $A$ realizes that $J(y^*)<0$ (equivalently, the Apollonius circle does not intersect the Y-axis) and chooses his aimpoint $u'$ also on the Apollonius circle - see Figure \[fig:Jneg\]. Given $T$’s choice of $v'$, the soonest $A$ can make $\overline{AT}=0$ is by capturing $T$ on the Apollonius circle (otherwise the Target will exit the Apollonius circle and the Defender may be able to assist the Target). Thus, $u'^*(v')=v'$.
Similarly, $T$ solves the differential game and obtains $J(y^*)<0$. This information is useful to $T$ and it realizes that $D$ is unable to intercept $A$. Thus, $T$ will be prepared to apply passive countermeasures such as releasing chaff and flares. $T$ can also change its objective and find some $u'^*$ in order to optimize a different criterion such as to maximize capture time; however, this topic falls outside the scope of this paper.
EXAMPLES {#sec:Example}
========
*Example 1*. Consider the speed ratio $\alpha=0.5$ and the Attacker’s initial position $x_A=6$. The right branch hyperbola shown in Figure \[fig:Reg1-1\] divides the Target escape/capture regions.
Simulation: Let the Target’s initial coordinates be $x_T=3$ and $y_T=2$. Note that $(x_T,y_T)\in R_e$. The Y-coordinate of the optimal interception point is given by $y^*=2.6108$. Figure \[fig:Ex1Sim\] shows the results of the simulation. The optimal cost/payoff is $J(y^*)=0.2102$ and the Target escapes being captured by the Attacker.
*Example 2*. For a given speed ratio $\alpha$, we can plot a family of right hand hyperbolas on the same plane for different values of $x_A$. Consider $\alpha=0.7$. Figure \[fig:RegM\] shows several hyperbolas for values of $x_A=1,2,...,8$.
CONCLUSIONS {#sec:concl}
===========
A cooperative missile problem involving three agents, the Target, the Attacker, and the Defender was studied in this paper. A differential game was analyzed where the Target and the Defender team up against the Attacker. The Attacker tries to pursue and capture the Target. The Target tries to evade the Attacker and the Defender helps the Target to evade by intercepting the Attacker before the latter reaches the Target. This paper provided optimal strategies for each one of the agents and also provided a further analysis of the Target escape regions for a given Target/Attacker speed ratio.
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[^1]: Corresponding Author: .\
Eloy Garcia is a contractor (Infoscitex Corp.) with the Control Science Center of Excellence, Air Force Research Laboratory, Wright-Patterson AFB, OH 45433.\
David Casbeer is with the Control Science Center of Excellence, Air Force Research Laboratory, Wright-Patterson AFB, OH 45433.\
Meir Pachter is with the Department of Electrical Engineering, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433.
|
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abstract: |
We revisit the problem of selecting an item from $n$ choices that appear before us in random sequential order so as to minimize the expected rank of the item selected. In particular, we examine the stopping rule where we reject the first $k$ items and then select the first subsequent item that ranks lower than the $l$-th lowest-ranked item among the first $k$. We prove that the optimal rule has $k \sim n/{\mathrm e}$, as in the classical secretary problem where our sole objective is to select the item of lowest rank; however, with the optimally chosen $l$, here we can get the expected rank of the item selected to be less than any positive power of $n$ (as $n$ approaches infinity). We also introduce a common generalization where our goal is to minimize the expected rank of the item selected, but this rank must be within the lowest $d$.
[*Key words:*]{} Secretary problem, relative ranks, stopping rule, optimization.
[*2010 AMS codes:*]{} Primary: 62L99; Secondary: 60C05.
author:
- |
Béla Bajnok\
[Department of Mathematics, Gettysburg College]{}\
[300 N. Washington Street, Gettysburg, PA 17325-1486 USA]{}\
[E-mail: [email protected]]{}\
\[.2in\] and\
Svetoslav Semov\
[Department of Economics, Boston University]{}\
[270 Bay State Rd, Boston, MA 02215 USA]{}\
[E-mail: [email protected]]{}\
\[.4in\]
date: 'March 12, 2013'
title: 'The “Thirty-seven Percent Rule” and the Secretary Problem with Relative Ranks\'
---
Introduction
============
Suppose that we have a job opening for which we have $n$ applicants of whom we must hire exactly one. The applicants appear before us in random order one at a time; all $n!$ permutations are equally likely. We assume that we do not know anything about the applicants until we interview them but that we have a clear preference between any two whom we have already met with. After interviewing a candidate, we can either accept them, in which case the search is over, or we reject them and move on to the next candidate. Our hiring decision is constrained by the rule that rejected candidates cannot be recalled; therefore, terminating the process early and waiting too long both come with risks. (In particular, if we reject each of the first $n-1$ applicants then we are forced to hire the last one.) What shall we do?
The answer, of course, depends on what our goal is exactly. In the classical “secretary problem,” we aim to maximize the probability that we hire the best candidate. It is easy to see that the best strategy is to reject the first $k$ candidates for some $k$ and then hire the applicant whom we prefer over all the ones that we have seen thus far; it was first proved by Gilbert and Mosteller in 1966 [@GilMos:1966a] — and now well known — that this stopping rule ${\cal R}_n(k)$ is optimal at $k \sim n/{\mathrm e}$. The general appeal of the secretary problem can be partly attributed to this surprisingly attractive answer, known popularly as the “thirty-seven percent rule.” There is a vast literature and many variations of the secretary problem; see, for example, Freeman [@Fre:1983a], Ferguson [@Fer:1989a], Pfeifer [@Pfe:1989a], Bruss and Ferguson [@BruFer:1993a], Assad and Samuel-Cahn [@AssSam:1996a], Quine and Law [@QuiLaw:1996a], Krieger and Samuel-Cahn [@KriSam:2009a], and their references.
Rather than focusing solely on the best applicant, one might instead aim to minimize the expected rank of the applicant hired. As we can see from Theorem \[En(k,l)\] below, the stopping rule ${\cal R}_n(k)$ just described yields a rather high expected rank, approximately $n/2\mathrm{e}$. Instead, as proved by Bearden [@Bea:2006a] in 2006, with the optimally chosen $k \sim \sqrt{n}$, the expected rank of the chosen candidate is considerably less, only about $\sqrt{n}$. A markedly better result can be achieved if we allow an adaptive strategy where our stopping rule is not given in advance but is dependent on the relative ranks of the applicants we have already seen. Such an optimal strategy was found by Lindley [@Lin:1961a] in 1961; a few years later Chow et al. [@Cho:1964a] proved that, as $n$ approaches infinity, the expected rank of the hired candidate using Lindley’s stopping rule tends to the constant value $$\prod_{k=1}^{\infty} \left( 1+ \frac{2}{k} \right)^{1/(k+1)} \approx 3.87.$$ This is an amazing result, though the strategy that achieves it can only be stated implicitly and employed via dynamic programming.
In Section 2 of this paper, we discuss an explicit and [*a priori*]{} stopping rule that generalizes Bearden’s result. Namely, we investigate the stopping rule ${\cal R}_n(k,l)$ that rejects the first $k$ candidates and then hires the first one after that that ranks lower than the $l$-th best candidate among the first $k$. As we shall see, for fixed $l$, the optimal value of $k$ is about $$k \sim n^{\frac{l}{l+1}}.$$ (Note that for $l=1$ we get Bearden’s result.) As it turns out, with this $k$, the value of $l$ for which the stopping rule ${\cal R}_n(k,l)$ minimizes the expected rank of the applicant hired is about $\log n -1$. This is a particularly pleasing result as $$n^{\frac{\log n -1}{\log n}}=n/\mathrm{e};$$ in other words, we are rejecting the first $\sim n/\mathrm{e}$ applicants, just like in the classical secretary problem, but now the expected value of the rank of the applicant hired is substantially lower, about $\mathrm{e} \log n /2$.
In Section 3 of our paper we introduce a common generalization of the classical secretary problem (where we are only interested in the best applicant) and the variation just discussed (where we have no absolute requirements on the applicant hired). Namely, we investigate what happens when we still want to minimize the rank of the applicant hired, but we insist on hiring one of the best $d$ applicants ($1 \leq d \leq n$). The case $d=1$ then yields the classical secretary problem, and the case $d=n$ corresponds to situation we analyzed in Section 2; we also provide a full analysis of the case $d=2$.
[**Acknowledgments.**]{} We thank Art Benjamin, Darren Glass, Benjamin Kennedy, Chuck Wessell, and Rebecca Zabel for valuable discussions of various parts of this paper.
The expected rank resulting from strategy ${\cal R}_n(k,l)$
===========================================================
Let $n$, $k$, and $l$ be fixed positive integers with $l \leq k \leq n-1$. By the stopping rule ${\cal R}_n(k,l)$ we mean the strategy that selects an element $s$ (for [*selected value*]{}) of a permutation $a_1,a_2,\dots,a_n$ of $\{1,2,\dots,n\}$ as follows. We let $t$ (for [*test value*]{}) denote the $l$-th lowest value among $a_1,\dots,a_k$ and set $$I=\{i \in \mathbb{N} \; | \; i \leq n-k \; \mbox{and} \; a_{k+i} < t\}.$$ We then let $$j = \left\{
\begin{array}{cl}
\min I & \mbox{if $I \neq \emptyset,$}\\
n-k & \mbox{if $I = \emptyset;$}\\
\end{array}\right.$$ the rule then selects $s=a_{k+j}$. In other words, the first $k$ candidates are rejected, after which the first value, if there is one, is selected that ranks lower than the $l$-th best candidate among the first $k$, and the last value is selected otherwise. We can determine the exact expected value of $s$ as follows.
\[En(k,l)\]
The expected value $E_n(k,l)$ of the rank of the candidate hired following strategy ${\cal R}_n(k,l)$ is $$E_n(k,l)=\frac{n+1}{2} \cdot \left( \frac{l}{k+1} + \frac{{n-l \choose k-l}}{{n \choose k}} \right).$$
[*Proof.*]{} First we introduce some terminology. We say that the sequence $a_1, a_2, \dots, a_n$ is [*successful*]{} if at least one of $a_{k+1}, a_{k+2}, \dots, a_n$ is less than $t$ (that is, the set $I$ above is nonempty); otherwise, if each of $a_{k+1}, a_{k+2}, \dots, a_n$ is more than $t$, we say that the sequence is [*unsuccessful*]{}. We will consider the cases of successful and unsuccessful sequences separately.
Suppose first that our search is successful and we select the value $s=a_{k+j}$ for some $j \in \mathbb{N}$. Note that $$1 \leq s \leq t-1;$$ furthermore, since exactly $l$ of the values $a_1,a_2,\dots,a_{k+j}$ are less than $t$ and $k-l+j-1$ are more than $t$, we must have $l+1 \leq t \leq n-k+l-j+1$. Consequently, $$1 \leq j \leq n-k+l-t+1$$ and $$l+1 \leq t \leq n-k+l$$ must hold.
We need to determine, for fixed values of $t$, $j$, and $s$, the number of sequences $a_1, a_2, \dots, a_n$ for which
- $t$ is the $l$-th lowest value among $a_1,a_2,\dots,a_k$;
- $a_{k+1},a_{k+2},\dots,a_{k+j-1}$ are each more than $t$; and
- $s=a_{k+j}$ is less than $t$.
We do this as follows.
First, we choose the sequence $a_1,a_2,\dots,a_k$. Since $l-1$ of the $k$ elements are less than $t$ but unequal to $s$ and $k-l$ of them are more than $t$, we have $${t-2 \choose l-1} \cdot {n-t \choose k-l}$$ ways to select the values; we then have $k!$ ways to arrange them into a sequence.
Next, we choose the sequence $a_{k+1}, a_{k+2}, \dots, a_{k+j-1}$. Since each of these terms must be more than $t$ and distinct from those $k-l$ values among $a_1,a_2,\dots,a_k$ that are also more than $t$, we have $${n-k+l-t \choose j-1}$$ ways to select the values; we then have $(j-1)!$ ways to arrange them into a sequence.
Finally, we have $(n-k-j)!$ ways to arrange the remaining $n-k-j$ elements of $\{1,2,\dots,n\}$ into the sequence $a_{k+j+1}, a_{k+j+2}, \dots, a_n$.
In summary, there are $${t-2 \choose l-1} \cdot {n-t \choose k-l} \cdot {n-k+l-t \choose j-1} \cdot k! \cdot (j-1)! \cdot (n-k-j)!$$ sequences $a_1, a_2, \dots, a_n$ satisfying the three conditions above, which we may rewrite as $${t-1 \choose l-1} \cdot {n-t \choose k-l} \cdot {n-k-j \choose t-l-1} \cdot \frac{(t-l) \cdot k! \cdot (n-k+l-t)!}{t-1}.$$ (Note that, in the case of a successful sequence, $t \neq 1$.) Therefore, for a given pair of integers $t$ and $s$, with $1 \leq s \leq t-1$, the probability $P^{\surd}_n(k,l;t,s)$ that the stopping rule ${\cal R}_n(k,l)$ results in the (successful) selection of $s$ is $$P^{\surd}_n(k,l;t,s)= \sum_{j=1}^{n-k+l-t+1} {t-1 \choose l-1} \cdot {n-t \choose k-l} \cdot {n-k-j \choose t-l-1} \cdot \frac{(t-l) \cdot k! \cdot (n-k+l-t)!}{(t-1) \cdot n!}.$$
We can simplify this expression by observing that ${n-k-j \choose t-l-1}$ enumerates the $(t-l)$-subsets of the set $\{1,2,\dots,n-k\}$ whose smallest element is $j$. Letting $j$ vary from 1 to $n-k+l-t+1$ includes all possibilities (when $j=n-k+l-t+1$, our set consists of the greatest $t-l$ elements of the set $\{1,2,\dots,n-k\}$). Therefore, $$\sum_{j=1}^{n-k+l-t+1} {n-k-j \choose t-l-1} = {n-k \choose t-l}=\frac{(n-k)!}{(n-k+l-t)! \cdot (t-l)!},$$ yielding $$\begin{aligned}
\label{Psurd}
P^{\surd}_n(k,l;t,s)= \frac{1}{{n \choose k}} \cdot {t-1 \choose l-1} \cdot {n-t \choose k-l} \cdot \frac{1}{t-1}.\end{aligned}$$
Let us turn now to the case of unsuccessful sequences. The sequence $a_1, a_2, \dots, a_n$ leads to an unsuccessful search when each of $a_{k+1}, a_{k+2}, \dots, a_n$ is more than the $l$-th smallest element among $a_1, a_2, \dots, a_k$. In other words, a sequence is unsuccessful exactly when each of $1,2,\dots,l$ is among the first $k$ terms of the sequence. Therefore, for a given integer $s$, with $l+1 \leq s \leq n$, the probability $P^{X}_n(k,l;t,s)$ that the stopping rule ${\cal R}_n(k,l)$ results in the (unsuccessful) selection of $s$ is $$\begin{aligned}
\label{PX}
P^{X}_n(k,l;s)= \frac{k!}{(k-l)!} \cdot \frac{(n-l-1)!}{n!}=\frac{1}{{n \choose k}} \cdot {n-l \choose k-l} \cdot \frac{1}{n-l}.\end{aligned}$$
We can then exhibit the expected value of $s$ as $$E_n(k,l) = \sum_{t=l+1}^{n-k+l} \sum_{s=1}^{t-1} P^{\surd}_n(k,l;t,s) \cdot s +
\sum_{s=l+1}^{n} P^{X}_n(k,l;s) \cdot s,$$ where $$\begin{aligned}
\sum_{t=l+1}^{n-k+l} \sum_{s=1}^{t-1} P^{\surd}_n(k,l;t,s) \cdot s & = & \sum_{t=l+1}^{n-k+l} \frac{1}{{n \choose k}} \cdot {t-1 \choose l-1} \cdot {n-t \choose k-l} \cdot \frac{1}{t-1} \cdot \frac{t^2-t}{2} \\
& = & \frac{l }{2 {n \choose k}} \cdot \sum_{t=l+1}^{n-k+l} {n-t \choose k-l} \cdot {t \choose l}\end{aligned}$$ and $$\begin{aligned}
\sum_{s=l+1}^{n} P^{X}_n(k,l;s) \cdot s & = & \frac{1}{{n \choose k}} \cdot {n-l \choose k-l} \cdot \frac{1}{n-l} \cdot \left(\frac{n^2+n}{2}-\frac{l^2+l}{2} \right) \\
& = & \frac{n+l+1 }{2 {n \choose k}} \cdot {n-l \choose k-l}.\end{aligned}$$
Next, observe that $$\sum_{t=l}^{n-k+l} {n-t \choose k-l} \cdot {t \choose l}={n+1 \choose k+1},$$ since both sides of the equation count the number of $(k+1)$-subsets of the set $\{1,2,\dots,n+1\}$. Indeed, the number of $(k+1)$-subsets of $\{1,2,\dots,n+1\}$ whose $(l+1)$-st smallest element equals $t+1$ is $${ n-t \choose k-l}\cdot {t \choose l};$$ as $t$ ranges from $l$ to $n-k+l$, we get all $(k+1)$-subsets of $\{1,2,\dots,n+1\}$. Therefore, $$\sum_{t=l+1}^{n-k+l} {n-t \choose k-l} \cdot {t \choose l}={n+1 \choose k+1}-{n-l \choose k-l},$$ and thus $$\begin{aligned}
E_n(k,l) & = & \frac{l }{2 {n \choose k}} \cdot \left( {n+1 \choose k+1}-{n-l \choose k-l} \right) + \frac{n+l+1 }{2 {n \choose k}} \cdot {n-l \choose k-l} \\
& = & \frac{n+1}{2} \cdot \left( \frac{l}{k+1} + \frac{{n-l \choose k-l}}{{n \choose k}} \right), \end{aligned}$$ as claimed. $\Box$
We can now use Theorem \[En(k,l)\] to find the stopping rule ${\cal R}_n(k,l)$ that minimizes the expected rank $E_n(k,l)$. For $l=1$ we have $$E_n(k,1)=\frac{n+1}{2} \cdot \left( \frac{1}{k+1} + \frac{k}{n} \right);$$ this quantity attains its minimum value when $k =\sqrt{n}-1$, and thus the stopping rule ${\cal R}_n(k,1)$ is optimal when $k=\lfloor \sqrt{n}-1 \rfloor$ or $k=\lceil \sqrt{n}-1 \rceil$, confirming Proposition 1 in [@Bea:2006a] (although the result there was under slightly different assumptions).
Minimizing for $l=2$ exactly is more complicated. We have $$E_n(k,2)=\frac{n+1}{2} \cdot \left( \frac{2}{k+1} + \frac{k^2-k}{n^2-n} \right),$$ and this quantity attains its minimum value when $$k =\frac{1}{2} \cdot \left( \left(\sqrt{(2n-1)^4-1}+(2n-1)^2 \right)^{1/3} -1 + \left(\sqrt{(2n-1)^4-1}+(2n-1)^2 \right)^{-1/3} \right);$$ for large $n$ this can be approximated as $k \sim n^{2/3}$.
More generally, for fixed $l$ and for large $k$ and $n$, we have $$E_n(k,l) = \frac{n+1}{2} \cdot \left( \frac{l}{k+1} + \frac{k \cdot (k-1) \cdot \cdots \cdot (k-l+1)}{n \cdot (n-1) \cdot \cdots \cdot (n-l+1)}\right) \sim \frac{n}{2} \cdot \left( \frac{l}{k} + \left( \frac{k}{n}\right)^l \right),$$ which attains its minimum at $k=n^{l/(l+1)}$. For this value of $k$, we have $$E_n(k,l) \sim \frac{n}{2} \cdot \left( \frac{l}{k} + \left( \frac{k}{n}\right)^l \right) = \frac{n}{2} \cdot \left( \frac{l}{n^{l/(l+1)}} + \left( \frac{n^{l/(l+1)}}{n}\right)^l \right) = \frac{l+1}{2} \cdot n^{1/(l+1)},$$ and this is minimal when $l=\log n -1$, in which case we have $$k=n^{l/(l+1)}= \frac{n}{\mathrm{e}}$$ and $$E_n(k,l) \sim \frac{l+1}{2} \cdot n^{1/(l+1)} = \frac{\mathrm{e}}{2} \log n .$$
We can thus see that whether our goal is to maximize the probability of hiring the top candidate, as in the classical secretary problem, or to minimize the expected rank of the applicant hired, we should let the first $k \sim n/\mathrm{e}$ applicants pass; but by hiring the $l$-th best candidate after the first $k$ with $l \sim \log n -1$ yields an expected rank of approximately $\mathrm{e} \log n /2$, a substantial improvement compared to the classical rule that aims to select the best candidate ($l=1$).
A generalization
================
In the classical secretary problem, one’s goal is to select the absolute lowest ranked applicant, while in the problem we just considered, we make no demands on the rank of the applicant selected. As a common generalization, here we consider the case when we still want to minimize the expected rank of the applicant hired, but we are only interested when this rank is within the top $d$ (for a given $1 \leq d \leq n$). We may reformulate the problem in terms of rewards: If an applicant of absolute rank $s$ is hired, our reward is $v_{n,d}(s)$ where $$v_{n,d}(s) = \left\{
\begin{array}{cl}
n+1-s & \mbox{if $s \leq d,$}\\
0 & \mbox{otherwise.}\\
\end{array}\right.$$ The classical secretary problem then corresponds to the case of $d=1$ (when the reward is $n$ if we hire the best applicant but 0 otherwise), and the no-demand problem of the previous section corresponds to $d=n$ (where the reward is $n$ for the best applicant, $n-1$ for the second best, and so on).
We can use our previous expressions (\[Psurd\]) and (\[PX\]) to compute the expected reward $V_{n,d}(k,l)$ following our stopping rule ${\cal R}_n(k,l)$ as $$\begin{aligned}
\label{Vformula}
V_{n,d}(k,l)=\sum_{t=l+1}^{n-k+l} \sum_{s=1}^{\min\{t-1,d\}} P^{\surd}_n(k,l;t,s) \cdot (n+1-s) +
\sum_{s=l+1}^{d} P^{X}_n(k,l;s) \cdot (n+1-s).\end{aligned}$$
We can then easily see that $$V_{n,n}(k,l) = (n+1) - E_n(k,l).$$ Indeed, we have $$\begin{aligned}
E_n(k,l) + V_{n,n}(k,l) & = & \sum_{t=l+1}^{n-k+l} \sum_{s=1}^{t-1} P^{\surd}_n(k,l;t,s) \cdot s +
\sum_{s=l+1}^{n} P^{X}_n(k,l;s) \cdot s \\
& & + \sum_{t=l+1}^{n-k+l} \sum_{s=1}^{t-1} P^{\surd}_n(k,l;t,s) \cdot (n+1-s) +
\sum_{s=l+1}^{n} P^{X}_n(k,l;s) \cdot (n+1-s) \\ \\
& = & (n+1) \cdot \left( \sum_{t=l+1}^{n-k+l} \sum_{s=1}^{t-1} P^{\surd}_n(k,l;t,s) +
\sum_{s=l+1}^{n} P^{X}_n(k,l;s) \right) \\ \\
& = & n+1,\end{aligned}$$ since $$\sum_{t=l+1}^{n-k+l} \sum_{s=1}^{t-1} P^{\surd}_n(k,l;t,s) +
\sum_{s=l+1}^{n} P^{X}_n(k,l;s)$$ is the sum of all probabilities and thus equals 1.
As we have seen in the previous section, with $k$ and $l$ approximately $n/\mathrm{e}$ and $\log n -1$, respectively, we have $$E_n(k,l) \sim \frac{\mathrm{e}}{2} \log n,$$ and therefore $$\lim_{n \to \infty} \frac{\max\{V_{n,n}(k,l) \mid k, l \}}{n} = \lim_{n \to \infty} \frac{\min\{n+1-E_{n}(k,l) \mid k, l \}}{n} =1.$$ Thus we can rest assured: with a large pool of applicants, when following a simple strategy but not deeming any applicant unacceptable, we are poised to end up with one of the best secretaries anyway.
In general, we are interested in finding (or estimating) for every (fixed) $d$, the value of $$c_d= \lim_{n \to \infty} \frac{\max\{V_{n,d}(k,l) \mid k, l \}}{n}.$$
Computational data suggest that $$c_1 \approx 0.37, \; c_2 \approx 0.51, \; c_3 \approx 0.63, \; c_4 \approx 0.71, \; c_5 \approx 0.77, \; c_6 \approx 0.81, \; c_7 \approx 0.84, \; c_8 \approx 0.87, \dots .$$ According to the classical secretary problem, and as we confirm below, we have $c_1=1/\mathrm{e};$ we will also find the exact value of $c_2$.
First, we establish the following explicit results.
\[Vnd\]
Let $H_n(k)$ denote the harmonic number $\sum_{i=1}^{n-k} 1/(n-i)$. The expected rewards $ V_{n,d}(k,l)$ for the stopping rule ${\cal R}_n(k,l)$ in the cases of $d=1$ and $d=2$ are as follows. $$\begin{aligned}
V_{n,1}(k,l) & = & \left\{
\begin{array}{ll}
k \cdot (H_n-H_k) & \mbox{if $l=1$}\\ \\
\frac{n}{l-1} \cdot \left( \frac{k}{n} - \frac{{n-l \choose k-l}}{{n \choose k}} \right) & \mbox{if $l \geq 2$}\\
\end{array}\right. \\ \nonumber \\ \nonumber \\
V_{n,2}(k,l) & = & \left\{
\begin{array}{ll}
\frac{2n-1}{n} \cdot k \cdot (H_n-H_k) - \frac{k}{n} \cdot (n-k-1) & \mbox{if $l=1$}\\ \\
\frac{2n-1}{l-1} \cdot \left( \frac{k}{n} - \frac{{n-l \choose k-l}}{{n \choose k}} \right) & \mbox{if $l \geq 2$}\\
\end{array}\right.\end{aligned}$$
[*Proof.*]{} Assume first that $l \geq d$. Then (\[Vformula\]) becomes $$V_{n,d}(k,l)=\sum_{t=l+1}^{n-k+l} \sum_{s=1}^{d} P^{\surd}_n(k,l;t,s) \cdot (n+1-s);$$ substituting (\[Psurd\]) and simplifying yields $$\begin{aligned}
\label{Vl>=d}
V_{n,d}(k,l)=\frac{d \cdot (2n+1-d)}{2 {n \choose k}} \cdot \sum_{t=l+1}^{n-k+l}{t-1 \choose l-1} \cdot {n-t \choose k-l} \cdot \frac{1}{t-1}.\end{aligned}$$ With $l=d=1$, (\[Vl>=d\]) becomes $$\begin{aligned}
\label{Vl=d=1}
V_{n,1}(k,1)=\frac{n}{{n \choose k}} \cdot \sum_{t=2}^{n-k+1} {n-t \choose k-1} \cdot \frac{1}{t-1}.\end{aligned}$$ Our result for $V_{n,1}(k,1)$ then follows from the identity $$\begin{aligned}
\label{Stirlingl=1}
\sum_{t=2}^{n-k+1} {n-t \choose k-1} \cdot \frac{1}{t-1} =
{n-1 \choose k-1} \cdot (H_n-H_k),\end{aligned}$$ which we prove next.
Note that (\[Stirlingl=1\]) is equivalent to $$\begin{aligned}
\label{Stirlingl=1'}
\sum_{t=2}^{n-k+1} {n-t \choose k-1} \cdot (k-1)! \cdot {n-k \choose n-t+1-k} \cdot (n-t+1-k)! \cdot (t-2)! +(n-1)! \cdot H_k=
(n-1)! \cdot H_n.\end{aligned}$$ Here we recognize that the right-hand side of (\[Stirlingl=1’\]) is Stirling’s cycle number $\left[{n \atop 2}\right]$, which counts the number of arrangements of the first $n$ positive integers into two disjoint nonempty cycles. (The order of the two cycles does not matter; as customary, we assume that the one containing 1 appears first. Furthermore, we assume that the cycles are both listed so that their smallest element appears first.) Indeed, if the first cycle has length $n-t$, then we have $$\frac{(n-1)!}{t!}$$ choices for it, leaving $(t-1)!$ choices for the second cycle. As $t$ ranges from 1 to $n-1$, we get all possibilities, and thus $$\left[{n \atop 2}\right]=\sum_{t=1}^{n-1} \frac{(n-1)!}{t!} \cdot (t-1)!=\sum_{t=1}^{n-1} \frac{(n-1)!}{t}=(n-1)! \cdot H_n.$$
We prove (\[Stirlingl=1’\]) by showing that $$\begin{aligned}
\label{Stirlingl=1'Left}
\sum_{t=2}^{n-k+1} {n-t \choose k-1} \cdot (k-1)! \cdot {n-k \choose n-t+1-k} \cdot (n-t+1-k)! \cdot (t-2)! \end{aligned}$$ counts those cycle-decompositions that contain each of $1,2,\dots,k$ in the same (first) cycle, while $$(n-1)! \cdot H_k$$ counts all others.
To verify the first claim, imagine that the first cycle (the one containing $1,2,\dots,k$) has length $n-t+1$; letting $t$ vary from 2 to $n-k+1$ assures that we have considered all possibilities where neither cycle is empty. To count the number of choices for the first cycle, recall that it starts with 1; we then have $${n-t \choose k-1}$$ ways to choose places for $2,3,\dots,k$, and $(k-1)!$ ways to arrange them. We also have $${n-k \choose n-t+1-k}$$ ways to choose the remaining elements in the first cycle, with $(n-t+1-k)!$ ways to arrange them. Our second cycle has length $t-1$, so we have $(t-2)!$ choices with the smallest element appearing first there too. Thus, the number of arrangements of $1,2,\dots, n$ into two disjoint nonempty cycles with each of $1,2,\dots,k$ in the first cycle is given by (\[Stirlingl=1’Left\]).
To enumerate the remaining cycle-decompositions, start by arranging $1,2,\dots,k$ into two disjoint nonempty cycles: there are $\left[{k \atop 2}\right]$ ways to do this. Then, place the elements $k+1,k+2,\dots,n$ into one of the two cycles one at a time; each can be put to the right of an already-placed element. (The two leading terms in the cycles must remain.) The number of choices to place these elements is $$\frac{(n-1)!}{(k-1)!},$$ proving our second claim as $$\left[{k \atop 2}\right] \cdot \frac{(n-1)!}{(k-1)!} = (n-1)! \cdot H_k.$$ This completes the proof of the formula for $V_{n,1}(k,1)$.
Next, we prove that for $l \geq 2$ we have $$\begin{aligned}
\label{Stirlingl>=2}
\sum_{t=l+1}^{n-k+l}{t-1 \choose l-1} \cdot {n-t \choose k-l} \cdot \frac{1}{t-1} =
\frac{1}{l-1} \cdot \left( {n-1 \choose k-1} - {n-l \choose k-l} \right) ,\end{aligned}$$ which then, with (\[Vl>=d\]), establishes our results for $V_{n,1}(k,l)$ and $V_{n,2}(k,l)$ when $l \geq 2$.
We start the proof of (\[Stirlingl>=2\]) by rewriting it as $$\begin{aligned}
\sum_{t=l}^{n-k+l}{t-1 \choose l-1} \cdot {n-t \choose k-l} \cdot \frac{1}{t-1} =
\frac{1}{l-1} \cdot {n-1 \choose k-1} ,\end{aligned}$$ and then as $$\begin{aligned}
\label{Stirlingl>=2'}
\sum_{t=l}^{n-k+l}{t-2 \choose l-2} \cdot (l-2)! \cdot {n-k \choose t-l} \cdot (t-l)! \cdot {n-t \choose k-l}\cdot (k-l)! \cdot (n-t-k+l)! =
\frac{(n-1)! \cdot (k-l)! \cdot (l-2)!}{(k-1)!}.\end{aligned}$$
In a manner similar to the one above, we establish this identity by showing that both sides count the number of arrangements of $1,2,\dots,n$ into two disjoint cycles with the added conditions that each of $1,2,\dots,l-1$ appears in the first cycle and each of $l,l+1,\dots,k$ appears in the second cycle. (As usual, we assume that the cycles are arranged with their smallest elements first.) Indeed, if the first cycle has length $t-1$, then we have $${t-2 \choose l-2} \cdot (l-2)!$$ ways to position $2,3,\dots,l-1$; $${n-k \choose t-l} \cdot (t-l)!$$ ways to choose and position the remaining elements in the first cycle; $${n-t \choose k-l}\cdot (k-l)!$$ ways to position $l+1,l+2,\dots,k$ in the second cycle; and $$(n-t-k+l)!$$ ways to arrange the remaining elements in the second cycle. As $t$ ranges from $l$ to $n-k+l$, we account for all desired cycle-decompositions, verifying the left-hand side of (\[Stirlingl>=2’\]).
To arrive at the right-hand side, we start by arranging the elements $2,3,\dots,l-1$ in the first cycle (all behind 1) and arranging $l+1,l+2,\dots,k$ in the second cycle (all behind $l$); there are $$(l-2)! \cdot (k-l)!$$ ways to do this. The elements $k+1,k+2,\dots,n$ can then be inserted, one at a time, to the right of previously-placed elements; this can be accomplished in $$\frac{(n-1)!}{(k-1)!}$$ ways. The result now follows.
Finally, we turn to the case of $l=1$ and $d=2$; from (\[Vformula\]) we have $$\begin{aligned}
V_{n,2}(k,1) & = & \sum_{t=2}^{n-k+1} \sum_{s=1}^{\min\{t-1,2\}} P^{\surd}_n(k,1;t,s) \cdot (n+1-s) +
\sum_{s=2}^{2} P^{X}_n(k,1;s) \cdot (n+1-s) \\ \\
& = & \sum_{t=2}^{n-k+1} \sum_{s=1}^{2} P^{\surd}_n(k,1;t,s) \cdot (n+1-s) - P^{\surd}_n(k,1;2,2) \cdot (n-1) +
P^{X}_n(k,1;2) \cdot (n-1),\end{aligned}$$ which, using (\[Psurd\]) and (\[PX\]) and simplifying, becomes $$\begin{aligned}
V_{n,2}(k,1)=\frac{2n-1}{{n \choose k}} \cdot \sum_{t=2}^{n-k+1} {n-t \choose k-1} \cdot \frac{1}{t-1} - \frac{k}{n} \cdot (n-k-1). \end{aligned}$$ Our claim for $V_{n,2}(k,1)$ now follows from (\[Stirlingl=1\]). $\Box$
For a fixed value of $d$, let $$c_d= \lim_{n \to \infty} \frac{\max\{V_{n,d}(k,l) \mid k, l \}}{n}.$$ Then $c_1=1/\mathrm{e} \approx 0.36788,$ and $c_2=2x-x^2 \approx 0.51239$ where $x$ is the smaller root of the equation $$2x - 2\log x = 3.$$
[*Proof.*]{} We will use Theorem \[Vnd\] to find the values of $k$ and $l$ for which $V_{n,d}(k,l)$ is maximal for $d=1$ and $d=2$.
Consider first $d=1$, in which case for $l=1$ we have $$V_{n,1}(k,1)=k \cdot (H_n-H_k)=k \cdot (n-k) \cdot \mathrm{Average}\left\{\frac{1}{n-i} \; | \; i=1,2,\dots,n-k\right\}$$ and for $l \geq 2$ we have $$\begin{aligned}
V_{n,1}(k,l) & = & \frac{n}{(l-1) \cdot {n \choose k}} \cdot \left({n-1 \choose k-1} - {n-l \choose k-l} \right) \\
& = & \frac{n}{(l-1) \cdot {n \choose k}} \cdot \left({n-2 \choose k-1} + {n-3 \choose k-2 } + \cdots + {n-l \choose k-l+1} \right) \\
& = & \frac{n}{(l-1) \cdot {n \choose k}} \cdot \left({n-2 \choose n-k-1} + {n-3 \choose n-k-1 } + \cdots + {n-l \choose n-k-1} \right) \\
& = & \frac{n}{ {n \choose k}} \cdot \mathrm{Average}\left\{ {n-j \choose n-k-1} \; | \; j=2,3,\dots,l \right\}\end{aligned}$$ where $\mathrm{Average} \; S$ is the arithmetic average of the finite set of real numbers $S$. Note that $\frac{1}{n-i}$ is an increasing function of $i$ but ${n-j \choose n-k-1}$ is a decreasing function of $j$, thus we have $$V_{n,1}(k,1) \geq k \cdot (n-k) \cdot \frac{1}{n-1}$$ and $$V_{n,1}(k,l) \leq \frac{n}{ {n \choose k}} \cdot {n-2 \choose n-k-1} = k \cdot (n-k) \cdot \frac{1}{n-1}$$ as $l \geq 2$. Therefore, for fixed $n$ and $k$, $V_{n,1}(k,l)$ is maximal for $l=1$.
To find the maximum of $V_{n,1}(k,1)$, we let $x$ denote the limit of $k/n$ as $n$ approaches infinity; we then have $$V_{n,1}(k,1)= k \cdot (H_n-H_k) \sim - n \cdot x \cdot \log x,$$ and this attains its maximum when $x =1/ \mathrm{e}$. Therefore, for $d=1$, $V_{n,1}(k,l)$ attains its maximum when $l=1$ and $k \sim n/\mathrm{e}$, in which case we have $V_{n,1}(k,1) \sim n/ \mathrm{e}.$
The situation is somewhat more complicated if $d=2$. We still find that for $l \geq 2$, $$V_{n,2}(k,l)=\frac{2n-1}{ {n \choose k}} \cdot \mathrm{Average} \left\{ {n-j \choose n-k-1} \; | \; j=2,3,\dots,l \right\}$$ which is a decreasing function of $l$, but comparing $V_{n,2}(k,1)$ and $V_{n,2}(k,2)$ is not possible. We will, however, show that, as $n$ approaches infinity, the maximum value of $V_{n,2}(k,1)$ is more than the maximum value of $V_{n,2}(k,2)$, as follows.
Clearly, $$V_{n,2}(k,2)=\frac{2n-1}{ {n \choose k}} \cdot {n-2 \choose n-k-1}=\frac{2n-1}{n^2-n} \cdot k \cdot (n-k)$$ is maximal when $k=n/2$; in which case we have $V_{n,2}(k,2) \sim n/2$.
To find the maximum of $V_{n,2}(k,1)$, we let $x$ denote the limit of $k/n$ as $n$ approaches infinity, as before; we then have $$V_{n,2}(k,1)= \frac{2n-1}{n} \cdot k \cdot (H_n-H_k) - \frac{k}{n} \cdot n \cdot \left(1-\frac{k}{n} - \frac{1}{n} \right) \sim - 2n \cdot x \cdot \log x - n \cdot x \cdot (1-x).$$ This attains its maximum when $2x - 2\log x = 3$ or $x \approx 0.30171$, in which case $$V_{n,2}(k,1) \sim n \cdot x \cdot (x-2\log x -1)=n \cdot x \cdot (2-x)$$ or about $0.51239 n $. Since this is larger than $n/2$, we have established our claim. $\Box$
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abstract: 'It is well known that we can use structural proof theory to refine, or generalize, existing paradigmatic computational primitives, or to discover new ones. Under such a point of view we keep developing a programme whose goal is establishing a correspondence between proof-search of a logical system and computations in a process algebra. We give a purely logical account of a process algebra operation which strictly includes the behavior of restriction on actions we find in Milner $\CCS$. This is possible inside a logical system in the Calculus of Structures of Deep Inference endowed with a self-dual quantifier. Using proof-search of proofs of such a logical system we show how to solve reachability problems in a process algebra that subsumes a significant fragment of Milner $\CCS$.'
bibliography:
- 'Roversi-SBVQ-computational-interpretation.bib'
title: |
[Communication, and concurrency with logic-based restriction inside a calculus of structures]{}\
[Luca Roversi]{}\
[Università di Torino — Dipartimento di Informatica[^1]]{}
---
[^1]: [*E-mail*]{}: [email protected]
|
---
abstract: 'We report the EXITE2 hard X-ray imaging of the sky around 3C273. A 2h observation on May 8, 1997, shows a $\sim$260 mCrab source detected at $\sim$4$\sigma$ in each of two bands (50-70 and 70-93 keV) and located $\sim$30’ from 3C273 and consistent in position with the SIGMA source GRS1227+025. The EXITE2 spectrum is consistent with a power law with photon index 3 and large low energy absorption, as indicated by the GRANAT/SIGMA results. No source was detected in more sensitive followup EXITE2 observations in 2000 and 2001 with 3$\sigma$ upper limits of 190 and 65 mCrab, respectively. Comparison with the flux detected by SIGMA shows the source to be highly variable, suggesting it may be non-thermal and beamed and thus the first example of a “type 2” (absorbed) Blazar. Alternatively it might be (an unprecedented) very highly absorbed binary system undergoing accretion disk instability outbursts, possibly either a magnetic CV, or a black hole X-ray nova.'
author:
- 'J.E. Grindlay'
- 'Y. Chou'
- 'P.F. Bloser'
- 'T. Narita'
title: EXITE2 Observation of the SIGMA Source GRS 1227+025
---
Introduction
============
GRS1227+025 was discovered serendipitously with SIGMA [@Bassani91; @Jourdain92] in the 40-80 keV energy band during an observation of the QSO 3C273. Separated from 3C273 by only $\sim$15’, GRS1227+025 (hereafter GRS1227) was detected at 5.5$\sigma$ whereas 3C273 was not detected. The source has a significantly steeper spectrum (photon index 3$^{+1.3}_{-0.9}$, [@Bassani91]) compared to 3C273, which was detected in subsequent observations, with its usual flatter photon index of $\sim$1.5. Upper limits (15-30 keV) with ART-P (simultaneous with SIGMA) showed that the spectrum must be heavily absorbed: N$_H \sim$1.5$\times$10$^{25}$ cm$^{-2}$ for a power law spectrum with photon index 3 [@Bassani93]. Followup observations done with SIGMA and OSSE (which could only measure the combined flux of 3C273 and GRS1227) showed that GRS1227 is probably variable by a factor 2-5 on $\sim$1 day time scales [@Bassani93; @Johnson95]. No obvious soft X-ray counterpart was found in the SIGMA error circle (5 arcmin, 1$\sigma$) with ROSAT (e.g. [@Leach96]), although an Einstein source (1E 1227+0224) in the SIGMA error circle (1$\sigma$) was identified with a QSO at z=0.57 that would have to be of “type 2” (heavily self-absorbed) if it were indeed the counterpart [@Grindlay93].
EXITE2 Observation and Data Analysis
====================================
The second-generation Energetic X-ray Imaging Telescope Experiment (EXITE2) was a phoswich (NaI(Tl)/CsI(Na)) imaging telescope designed to detect and image cosmic X-ray sources in the broad hard X-ray energy band (20 - 600 keV) from a high-altitude balloon. The details of the payload are described in [@Lum94; @Manandhar95; @Chou98; @Chou01; @Chou03]. EXITE2 flew three times (from Ft. Sumner, New Mexico) in 1997, 2000 and 2001 for scientific observations, which included the Crab Nebula, the X-ray binaries Cyg X-1 and 4U 0614+09, the microquasar GRS 1915+105, the Seyfert galaxy NGC 4151 and QSO 3C273/GRS1227. The data analysis methods and the observation results of the Crab and Cyg X-1 are reported in separate papers [@Bloser02a; @Chou03]. Other results from the 2000 and 2001 flights, including separate experiments for the development of imaging Cd-Zn-Te detectors (Bloser et al 2002b, Jenkins et al 2003) are reported elsewhere.
For the 1997 flight, during UT03:20 to UT05:32 on May 8 (local time 21:20 to 23:32 on May 7), the telescope was pointing to within 5 arcmin of the QSO 3C273 at (night) flight altitude $\sim$115,000 feet. The data were analyzed by the EXITE2 standard analysis methods described in Chou (2001), Bloser et al. (2002a) and Chou et al. (2003). The sky images showed that no significant ($>$3$\sigma$) detection of 3C273 was obtained during the $\sim$2 hour observation. However, a source $\sim$30’ away from 3C273 was detected in the EXITE2 energy band 5 (50-70 keV) and band 6 (70-93 keV) at 3.9$\sigma$ and 4.1$\sigma$, respectively. We estimated aspect errors as 5’ from agreement of the Crab and Cyg X-1 images with their source positions. The image for band 6 is shown as Figure \[3c273img\].
The source detected by EXITE2 in 1997 near 3C273 was located at $\alpha$(2000)= 12$^h$31$^m$13$^s$, $\delta$(2000)= +2$^\circ$18’, with positional uncertainty $\pm \, \sim$7 arcmin (1$\sigma$). Although $\sim$18’ away from the GRS1227 location reported by Jourdain et al (1992) and plotted in Figure \[grspos\], the 2$\sigma$ error circles for SIGMA vs. EXITE2 overlap. The detected energy bands and the combined 5$\sigma$ significance of the EXITE2 detection are both comparable to the original SIGMA detection, further supporting the reality of a highly variable source at this location and distinct from 3C273. The fluxes measured during the 2 hours observation were (6.6$\pm$1.7)$\times$10$^{-4}$ photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$ and (3.5$\pm$0.85)$\times$10$^{-4}$ photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$, for band 5 and 6 respectively, a factor of $\sim$10 brighter than SIGMA 1990 observations. The combined band 5-6 flux was (4.94$\pm$0.91)$\times$10$^{-4}$ photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$ ($\sim$260 mCrab). No source was significantly detected in band 4 (37-50 keV) or band 7 (93-127 keV) with 3$\sigma$ upper limits 1.8$\times$10$^{-3}$ photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$ and 1.2$\times$10$^{-4}$ respectively. We found that a spectrum of power law index $\sim$3 as reported by Bassani et al. 1991 for GRS1227 is consistent with the EXITE2 1997 observation results (see Figure \[grsspec\]). Alternatively, the source could still have a flatter spectrum (e.g. photon index $\sim$1.7 as for typical AGN) with high energy cutoff at $\sim$100 keV.
Additional observations of the field containing 3C273/GRS1227 were made in the EXITE2 flights on September 19 2000 (UT20:45 to UT21:30) and May 23 2001 (UT03:09 to UT06:17 on May 24). No source was detected during the $\sim$45 minute observation in the 2000 flight. The 3$\sigma$ upper limit for the combined bands 5-6 (50 to 93 keV) was 3.6$\times$10$^{-4}$ photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$ ($\sim$190 mCrab), which is a factor 1.5 below the 1997 value. For the 2001 observation, the telescope was pointed to the combined position of SIGMA/EXITE2, the intersection of the 2$\sigma$ error circles for EXITE2 and SIGMA at RA=12$^h$30$^m$14$^s$, DEC= +2$^\circ$14’53” (J2000), as shown in Figure \[grspos\]. No source was detected during a $\sim$3 hour observation. The 3$\sigma$ upper limit for the combined bands 5-6 (50 to 93 keV) was 1.25$\times$10$^{-4}$ photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$, or $\sim$65 mCrab, which is comparable to the original SIGMA detection. The upper limits for both EXITE2 observations are also plotted in Figure \[grsspec\]. The combined EXITE2-SIGMA observations indicate that the source is highly variable (by a factor of $\sim$10) at hard X-ray energies.
Discussion
==========
The GRS1227 source presents a major puzzle. Both EXITE2 and SIGMA detect a source at approximately the same location (within 2$\sigma$) and with similar spectral properties which are very distinct from 3C273. Although the EXITE2 detection is marginal, as was the original SIGMA detection, the combined results are more likely to be real. GRS1227 must be highly self-absorbed and yet highly variable. The SIGMA variations are a factor of $\sim$2 and SIGMA-EXITE2 variation is nearly a factor of $\sim$20. This is a combination not seen before, and suggests a new class of object: a type 2 (i.e. heavily absorbed) Blazar, since Blazars (BL Lacs) are usually the only AGN which show such extreme variability. However, it might also be a Narrow Line Seyfert 1 (NLSy) galaxy (e.g. RX J2217.9–5941, see [@Grupe01]) but must still be highly self-absorbed. It may be similar to the NLSys with significant internal absorption and significant variability described by [@Fabian99]. Alternatively it might be (an unprecedented) very highly absorbed binary system, possibly a low luminosity magnetic CV undergoing outburst, since intermediate polars at high inclination can also show significant internal absorption.
From the EXITE2 vs. SIGMA source error circles (2$\sigma$) shown in Figure \[grspos\], the combined source error box is $\sim$15’$\times$6’, centered on the EXITE2-SIGMA combined position RA=12$^h$30$^m$14$^s
(\pm$4’), DEC= +2$^\circ$14’53”($\pm$4’) (J2000). At this relatively uncrowded high latitude position, even a $\sim$2’ position could enable identification with an optical or radio variable. However the need for a 1’ position is obvious: if this is, as suspected, a highly absorbed non-thermal source (BL Lac), it may not be at all conspicuous in the optical and appear as a simple elliptical galaxy. The radio galaxies numbered (2) and (4) in Figure \[grspos\], or \[SRK80\]122803+023451 and 87GB\[BWE91\]1227+0230, are thus possible candidates. However, in the event that the radio emission is confined to the core, GRS1227 may not even be bright at radio wavelengths since it is likely Compton-thick. Given the space density of galaxies cataloged (SIMBAD and NED) within the combined EXITE2/SIGMA error circles (cf. Figure \[grspos\]), a 1’ source position is obviously needed for a unique galaxy identification.
Source variability measurements can partly constrain the underlying source nature. The generalized condition for maximum variability of an isotropically emitting source powered by accretion at rate [$\dot{{m}}$ ]{}and radiating luminosity L$_x$ is\
$\Delta$L$_x$ 2 $\times$ 10$^{41}$ $\eta_{0.1}$ $\Delta$t ergs/s,\
where $\eta_{0.1}$ = L$_x$/[$\dot{{m}}$ ]{}c$^2$ 0.1 is the likely maximum efficiency factor (cf. [@Guilbert83]). This can impose direct constraints on the source luminosity (and thus distance and nature) for an observed luminosity variation $\Delta$L$_x$ over time interval $\Delta$t. Therefore, a factor of 2 increase on 1d timescales (already suspected from SIGMA) would imply the source is beamed and thus a Type 2 Blazar if it were identified with an AGN at z 0.1 (like 3C273) with L$_x$ 10$^{45}$ erg/s.
GRS1227 exhibits an unusual spectral shape. The simultaneous detection by SIGMA (40 keV) and upper limit at 15 keV by the ART-P X-ray imager on GRANAT (Bassani 1991) indicated that the source is highly self-absorbed (N${_H} \sim$10$^{24-25}$ cm$^{-2}$) and thus a Compton-thick absorber (see [@Matt99] and references therein). If the spectrum above $\sim$40-50 keV is indeed a power law with photon index $\sim$3, it is consistent with a Blazar or NLSy. Although NLSys are also highly variable, these have not been reported as Type 2 objects. As mentioned above, the EXITE2 spectral shape (Figure 3) is equally consistent with a flatter spectrum with photon index 1.7 but high energy cutoff at $\sim$100 keV, as expected for a highly absorbed Seyfert 2 or Type 2 QSO. However the remarkable variability required argues against the Seyfert or Type 2 QSO interpretation and in favor of a Blazar, with highly variable non-thermal emission. In this case, large amplitude optical variability might also be expected, since [@Liller75] found a $\sim$5 magnitude brightening for the BL Lac PKS 1510-089 on the Harvard plates. Given the offset from 3C273, it is possible that significant optical variables could have been missed, and a search using the Harvard plates is in progress.
A magnetic CV interpretation is also possible even if the spectrum is indeed a (moderately steep) power law: the magnetic CV AE Aqr can be fit with a similar power law spectrum [@Eracleous96], and has been reported as a high energy gamma-ray source (cf. Meintjes and de Jager 2000 and references therein). A magnetic CV origin for GRS1227 might also explain the extreme flare variations: the object could be a dwarf nova (like GK Per) undergoing outbursts. Although the X-ray emission in classical dwarf novae outbursts is usually soft, an outburst in a magnetic CV may contain significant non-thermal emission. Historical optical measurements of the could again provide an important test, since the optical flux (from the disk) would be expected to increase by $\sim$5 mag. If the EXITE2 1997 detection was the peak of an outburst like that of GK Per, the expected X-ray optical flux ratio F$_x$/F$_v$$\sim$10 would imply an outburst magnitude V$\sim$10, and thus quiescence magnitude V$\sim$15-16, which is probably too bright to have been missed as a blue object in the Palomar Green survey (Green, Schmidt and Liebert 1986), which revealed no such objects near 3C273.
A high latitude black hole X-ray binary, an X-ray nova (XN) like XTE J1118+480. which showed evidence for magnetic flares and non-thermal optical emission (Merloni, DiMatteo and Fabian 2000) is conceivable. Here the power law hard X-ray spectrum is natural; this is seen in the very high state of BH novae (e.g. Nova Muscae, [@Esin97]) and is expected for the jet emission probable in a system like XTE J1118+480. It too would have to be highly self-absorbed, but - as with the CV hypothesis - is plausible if the system were at inclination $\sim$90$^\circ$, and self-absorbed at the required column density of N$_H\sim$10$^{25}$ cm$^{-2}$. Such a highly absorbed XN would be missed by the ASM on RXTE, and indeed no evidence for an ASM source is found at the time of the 1997 EXITE2 observation (or the 2000 and 2001 EXITE2 observations; cf. Figure 4). However, an optical outburst is again expected and should again be searched on historical monitoring (e.g. plate) data. Thus the galactic binary/transient scenarios might be distinguished from the Type 2 BLazar or NLSeyfert interpretation by historical optical or radio variations, which are probably larger in the binary case given the relatively low luminosity secondary star expected. The lack of any (large amplitude) optical variable would support the Type 2 Blazar or NLSeyfert interpretation.
Definitive confirmation of the source existence, and of course a refined hard X-ray position and spectrum, are obviously needed. Long after this paper was initially submitted, we have carried out an observation with INTEGRAL (IBIS telescope) to attempt to further measure the source position, spectrum and variability. Data have just been received and will be reported separately. An initial INTEGRAL/IBIS observation of 3C273 has been reported recently by Courvoisier et al (2003) which also provided a sensitive new upper limit ($\sim$10 mCrab) for emission from the position of GRS1227 – a factor of $\sim$6 below the SIGMA detection and 2001 EXITE2 upper limit and $\sim$25 below the 1997 EXITE2 detection. Thus the source variability at 40 keV must be at least this factor. Wide-field XMM observations of the 3C273 field should also be examined for evidence for a highly variable, highly absorbed (10 keV) X-ray source which would enable identification.
The authors thank T Gauron, J. Gomes, V. Kuosmanen, F. Licata, G. Nystrom, A. Roy, R. Scovel (SAO) and J. Apple (MSFC) for for the EXITE2 detector and gondola development, and NSBF personnel for their excellent support of the balloon launches and flight operations. This work was partially supported by NASA grants NAGW-5103 and NAG5-5279. PFB is a National Research Council Research Associate at NASA/Goddard Space Flight Center. TN acknowledges support from the College of the Holy Cross.
Bassani L. et al. 1999, Proc. of 22nd Int. Cosmic Ray Conf., 1, 173 Bassani L. et al. 1993, . 97, 89 Bloser, P. F., Chou, Y., Grindlay, J. E., Narita, T., & Monnelly, G., 2002a, Astroparticle Physics, 17, 393 Bloser, P., Narita, T., Jenkins, J., Perrin, M., Murray, R. and Grindlay, J. 2002b, Proc. SPIE, 4497, 88 Chou, Y., et al. 1998, in Conf. Record of the 1998 IEEE Nucl. Sci. Symposium (Piscataway, New Jersey: IEEE), 210 Chou, Y., 2001, Ph.D thesis, Harvard University Chou, Y., Bloser, P. F., Grindlay, J. E., Jenkins, J. A., & Narita, T., 2003, submitted to ApJ Courvoisier, T., Beckmann, V., Bourban, G., Chenevez, J., Chernyakova, M., Deluit, S., Favre, P., Grindlay, J. et al. 2003, , in press (astro-ph/0308212) Eracleous, M. and Horne, K. 1996, , 471,427 Esin, A. A., McClintock, J. E., & Narayan, R., 1997, , 486, 865 Fabian, A. C. and Iwasawa, K., 1999, , 303, L34 Green, R. F., Schmidt, M., and Liebert, J., 1986, Ap.J. Suppl., 61, 305 Grindlay, J. E. 1993, A&AS, 97, 113 Grupe, D., Thomas, H. -C. and Leighly, K. M., 2001, , 369, 450 Guilbert, P., Fabian, A., & Rees, M. 1983, , 205, 593 Johnson, W. N. et al. 1995, , 445, 182 Jourdain, E. et al. 1992, , 395, L69 Leach, C. M. & McHardy I. M., 1996, , 278, 465 Liller , M. H. and Liller, W., 1975, , 199, L133 Lum, K. S. K. et al. 1994, IEEE Trans. on Nucl. Sci., NS-41, 1354 Manandhar, R. P. 1995, Ph.D. Dissertation, Harvard University Matt, G., Pompilio, F. & La Franca, F., 1999 New Astro. 4, 91 Meintjes, P.J. and de Jager, O.C. 2000, , 311, 611 Merloni, A., DiMatteo, T. and Fabian, A. 2000, , 318, L15
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abstract: |
A current challenge for disease modeling and public health is understanding pathogen dynamics across scales since their ecology and evolution ultimately operate on several coupled scales. This is particularly true for vector-borne diseases, where within-vector, within-host, and between vector-host populations all play crucial roles in diversity and distribution of the pathogen. Despite recent modeling efforts to determine the effect of within-host virus-immune response dynamics on between-host transmission, the role of within-vector viral dynamics on disease spread is overlooked. Here we formulate an age-since-infection structured epidemic model coupled to nonlinear ordinary differential equations describing within-host immune-virus dynamics and within-vector viral kinetics, with feedbacks across these scales. We first define the *within-host viral-immune response and within-vector viral kinetics dependent* basic reproduction number $\mathcal R_0.$ Then we prove that whenever $\mathcal R_0<1,$ the disease free equilibrium is locally asymptotically stable, and under certain biologically interpretable conditions, globally asymptotically stable. Otherwise if $\mathcal R_0>1,$ it is unstable and the system has a unique positive endemic equilibrium. In the special case of constant vector to host inoculum size, we show the positive equilibrium is locally asymptotically stable and the disease is weakly uniformly persistent. Furthermore numerical results suggest that within-vector-viral kinetics and dynamic inoculum size may play a substantial role in epidemics. Finally, we address how the model can be utilized to better predict the success of control strategies such as vaccination and drug treatment.\
[Keywords: Vector-host model, multi-scale modeling, stability analysis, Lyapunov function, vector to host inoculum size, within-vector viral kinetics, reproduction number]{}
[AMS Subject Classification: 92D30, 92D40]{}
address: 'Mathematics Department, University of Louisiana at Lafayette, Lafayette, LA '
author:
- 'Hayriye Gulbudak$^*$'
bibliography:
- 'NSF\_References.bib'
title: 'An Immuno-Epidemiological Vector-Host Model with Within-Vector Viral Kinetics '
---
[^1]
Introduction
============
The ecology and evolution of infectious diseases operate on several interdependent scales. This is particularly true for vector-borne diseases, where coupled within-vector, within-host, and between vector-host population dynamics together determine the diversity and distribution of the pathogen. One of the major mechanisms in determining disease abundance and virus evolution is within-vector viral kinetics [@forrester2014arboviral; @sim2014mosquito; @tabachnick2013nature]. Yet, from the mathematical point of view, the within-vector viral dynamics has been overlooked, and rarely studied [@reiner2013systematic; @rock2015age; @wang2017global]. There is also a serious need for an integrated modeling approach that links all scales [@handel2015crossing]. However, traditional modeling approaches treat within-host, within-vector, and between-host pathogen dynamics as separate systems. Multi-scale mathematical models can be both an avenue for understanding the complex features displayed by vector-borne diseases and tools for targeting interventions.
More than $17\%$ of all infectious diseases are accounted to be vector-borne diseases, causing more than $700, 000$ deaths annually world-wide according to World Health Organization (WHO). In particular, a vast majority of vector-borne vertebrate infecting viruses (arboviruses) are responsible for a number of severe diseases in humans (yellow fever (YFV), dengue (DENV), various encephalitides, etc.) and livestock (West Nile encephalomyelitis (WNV), Rift Valley fever (RVF), vesicular stomatitis, etc.). Mathematical modeling can help us understanding the impact of mechanisms behind the establishment and transmission of vector-borne viruses, which are crucial for developing effective intervention strategies. Mosquito-borne diseases are spread when mosquitoes bite hosts and release microscopic parasites, which live in the salivary glands of the mosquitoes, into the hosts’ bloodstream. In a recent work, Churcher et al. [@churcher2017probability] show that the number of parasites each mosquito carries influences the chance of successful malaria infection. In particular, it demontrates that the more parasites present in a mosquito’s salivary glands, the more likely it was to be infectious, and also the faster any infection would develop, highlighting the importance of within-mosquito viral kinetics. So far disease control authorities, including WHO (World Health Organization), has relied on the average number of potentially infectious mosquito bites per person per year. However, not every infectious mosquito bite will result in disease, and not all equally infectious. Therefore, there is an urgent need for unified models to understand how the within-vector viral kinetics can be scaled up to the host population level disease transmission.
A large number of previous studies formulate and analyze coupled nonlinear ODE models, describing population level vector-host disease spread. In a study that extensively reviewed mosquito-borne pathogen models, Reiner et al.[@reiner2013systematic] suggests that “moving forward in mosquito-borne disease modeling and addressing public health challenges will require modeling efforts on the heterogeneities such as variation in individual hosts and mosquitoes and their consequences for heterogeneous biting." Some of the previous vector-borne disease modeling studies focus on the impact of mechanisms such as temperature and rainfall on the vector population [@eikenberry2018mathematical; @taghikhani2018mathematics], or preventative measures, such as mosquito reduction strategies or personal protection [@prosper2014optimal]. Due to the extrinsic and intrinsic incubation periods and vector maturation process, time delayed vector-borne disease models have been deployed in [@cai2017global; @martcheva2013unstable; @nah2014malaria; @fan2010impact]. Dynamical properties of vector-borne diseases are also studied in age-since-*host* infection structured PDE models with direct transmission, time delay or reinfection [@wei2008epidemic; @lashari2011global; @cai2013dynamical; @xu2016hopf]. Furthermore, in within-host scale, arbovirus-immune response dynamics such as within-host DENV transmission is modeled and analyzed in [@ciupe2017host]. Immunology is also coupled with between vector-host disease dynamic models in prior works [@gulbudak2017vector; @tuncer2016structural]. Gulbudak et al. [@gulbudak2017vector] construct a multi-scale ODE-PDE hybrid model (similar to [@gilchrist2002modeling]), coupling within-host viral-immune response and between vector-host disease transmission on population scale and study coevolution of both vector-borne pathogen and host. In a companion paper, Tuncer et. al.[@tuncer2016structural] fit RVF multi-scale data to the multi-scale model in [@gulbudak2017vector], and utilize identifiability analysis for the model parameters. Furthermore, in a recent work, the effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple strain pathogens is also investigated in a two strain immuno-epidemiological DENV model structured by dynamic antibody size [@DENVimmunoepi2019].
Within-vector viral kinetics are more often overlooked in disease dynamic models, despite its role on disease transmission and virus evolution [@tabachnick2013nature; @reiner2013systematic]. Many prior modeling studies do not take into account how infectious each of those bites and each bite is considered equally infectious. In a recent article, Rock et al.[@rock2015age] develop and numerically study an age-since-vector infection and -bite model for vector-borne diseases and illustrate the distinct dynamics induced by complex interaction between feeding and life-expectancy, vector-infection-age, and bite distributions; yet the heterogeneity among host infectivity is ignored. Similarly, in another study, Wang et al. [@wang2017global] formulate and analyze a hybrid hyperbolic ODE-PDE model, where vector and host are stratified by infection ages. Nevertheless, to our best knowledge, none of the modeling studies exclusively coupled between vector-host disease dynamics, within-vector viral dynamics and within-host virus-immune response, that are crucial competences in disease spread.
Our modeling framework, introduced here, allows for variable vector to host inoculum size and infectivity of mosquitoes, both dependent on within-vector viral kinetics, and both inducing heterogeneity among immune-pathogen dynamics and the infected host population. The multi-scale framework also enables direct incorporation of within-vector viral data, which can be utilized to assess the role of within-vector viral dynamics on the disease spread. As opposed to some other multi-scale models [@agusto2019transmission; @garira2019coupled; @kitagawa2019mathematical], our PDE-ODE system does not reduce to a simpler set of ODEs because both the within-host and within-vector components operate on a different timescale with dynamics stratifying the structured PDE epidemic model. Interestingly, analytical and numerical results show that the within-vector-viral dynamics can be a *driving mechanism* in large epidemics. The model here can be applied to many arbovirus diseases including WNV, DENV, RVFV, and may help us understanding the role of mosquitoes, environmental factors, and host immune response on disease dynamics and the impact of disease control strategies, such as vaccination, drug treatment, and Wolbachia biocontrol strategy. It scales within-vector viral kinetics and within-host viral-immune response process up to the host population level disease dynamics.
Here, we organize the paper as follows: in section , we present an immuno-epidemiological model, incorporating within-vector viral kinetics, described with a modified logistic model with Allee effect. Distinct from the existent studies, the modeling formulation that we introduced here not only tracks the *infection-immune states of host population*, but also the *heterogeneity among infectivity of vectors*, depending on *their within-viral kinetics*. In section , we define the within-host immune-response and within-vector-viral kinetics dependent reproduction number $\mathcal R_0$ and investigate the threshold properties of the multi-scale system such as local and global stability of the equilibria in addition to the disease persistence for a special case. In section , we introduce a feasible way to incorporate “epidemiological feedback” between epidemiological and immunological scales. By doing so, we numerically assess the impact of both mechanisms *within-vector viral kinetics* and *host infectivity* on the population scale disease transmission. Finally, we summarize the results in Conclusion section .
[> lX]{} Variable/Parameter & Meaning\
\[0.5ex\]\
$P_0^{v\rightarrow h}=\hat{c}V(s,V_0)$ & Vector to host inoculum size (*the amount of pathogen that an infectious mosquito, that has infection age $s$, injects to host upon giving a bite*)\
$P(\tau,s)$ & Pathogen concentration at $\tau$ days post host infection *with initial condition $(P_0=P_0^{v\rightarrow h}, M_0, G_0),$*\
$M(\tau,s)$ & The concentration of IgM immune response antibodies at host infection age $\tau,$ *with initial condition $(P_0^{v\rightarrow h}, M_0, G_0),$*\
$G(\tau,s)$ & The concentration of IgG immune memory antibodies at host infection age $\tau,$ *with initial condition $(P_0^{v\rightarrow h}, M_0, G_0),$*\
$r_h$ & Within-host parasite growth rate,\
$K_h$ & Within-host parasite carrying cappacity,\
$\epsilon$ & The efficiency of the IgM immune response at killing the parasite,\
$\delta$ & The efficiency of the IgG immune response at killing the parasite,\
$a$ & The IgM immune response activation rate,\
$q$ & The per-capita rate at which the IgM immune response antibody production switches to the IgG immune response antibody production,\
$b$ & The IgG activation rate upon coming into contact with the pathogen,\
$c$ & IgM immune response antibody decay rate,\
[> lX]{} Variable/Parameter & Meaning\
\[0.5ex\]\
$V_0^{h\rightarrow v}$ & Host to vector inoculum size (*the amount of pathogen acquired from the blood meal of an infectious host by a susceptible vector upon giving a bite*)\
$V(s)=V(s,V_0^{h\rightarrow v})$ & Pathogen concentration within-an infectious vector at $s$ days post infection\
$r_v$ & Within-vector pathogen growth rate,\
$U_v$ & Allee threshold,\
$K_v$ & Within-vector pathogen carrying cappacity.\
Coupling disease dynamics across the scales {#acc_scales}
===========================================
****** Consider a modified logistic model with *Allee effect*, describing the within-vector viral dynamics:
where the variable $V(s)$ represents the pathogen concentration within an infected mosquito at infection age $s;$ i.e. $s$ days passed after vector infection. The parameters $r_v, K_v$ represent the intrinsic pathogen growth rate within a vector, and carrying capacity, respectively. If the initial pathogen density, denoted by $V_0^{h\rightarrow v}(=V(0))$ (*the amount of pathogen acquired in the blood meal of an infectious host by a susceptible vector upon a bite*), is less than $U_v,$ then the virus within a vector (mosquito) eventually clears. Otherwise if $V_0^{h\rightarrow v}>U_v,$ then the viruses persist and asymptotically converge to the carrying capacity $K_v.$ The model parameters are fitted to within-mosquito WNV viral data given in [@fortuna2015experimental] in Fig.2.
The Allee effect in the model refers that when host to vector inoculum size $V_0^{h \rightarrow v}$ is below $U_v$ (which is assume to be a small threshold size), the virus clears faster due to loss of viruses during transportation of them from foregut to mosquito midgut and mosquito immune response [@sim2014mosquito]. However when the inoculum size $V_0^{h \rightarrow v}>U_v,$ despite the loss during transportation, due to larger initial density, it replicates faster and reaches to carrying capacity $K_v$ [@franz2015tissue].
Here, in particular, we focus on mosquito-borne viruses such as WNV (Cx. Pipiens), DENV ( Aedes aegypti), and RVFV (most commonly transmited by the Aedes and Culex mosquitoes). Arbovirus infection of mosquitoes is generally asymptomatic and persists for the life of the vector, but the virus appears to be continually targeted by innate immune response [@sim2014mosquito]. Here for simplicity, we do not consider mosquito immune response.
****** To capture the short-term and long-term host-immune response to pathogen ($P$) introduction, we model two particular antibodies IgM ($M$) and IgG ($G$), released by B-cell lymphocytes. The within-host infection model takes the following form ([@gulbudak2017vector; @tuncer2016structural]): [$$\begin{aligned}
\label{PB}
P_{\tau}&=(\tilde{f}(P)-\epsilon M-\delta G)P, \quad M_{\tau}=\left(aP-(q+c)\right)M,
\quad G_{\tau} =qM + bGP, \\ \quad P(0,s)&=h(V(s, V_0)), M(0,s)=M_0, G(0,s)=G_0.\nonumber \end{aligned}$$]{}
Note that $s$ in in the initial condition $(P(0,s), M(0,s), G(0,s)),$ means that the solutions to system depend on the vector to host inoculum size, $P_0^{v\rightarrow h}.$ In particular, $P_0 =P_0^{v\rightarrow h}=h(V(s, V_0)).$ The solutions are denoted by $P(\tau,s)$, $M(\tau,s)$ and $G(\tau,s),$ where the time variable $\tau$ refers time-since-infection within a host, and $s$ denotes vector infection age. Note that upon *viral progression within an infected mosquito midgut*, the amount of pathogen in an *infected mosquito saliva* dynamically changes [@salas2015viral], determining the vector to host inoculum size $P_0^{v\rightarrow h}$ which is *the amount of pathogen that is injected to a susceptible host by an infectious vector.* If the mosquito has $V(s, V_0)$ amount of pathogen within at the time giving a bite to a host, then we assume that the vector to host inoculum size, $P_0^{v\rightarrow h},$ is a function of the amount of the pathogen within-vector, $V(s, V_0),$ i.e. $P_0^{v\rightarrow h}=h(V(s, V_0))$. Here we consider $h(V(s, V_0))=\hat{c} V(s, V_0),$ with a constant $\hat{c} \in (0,1)$. The intuitive assumption is that the more pathogen an infectious mosquito has within-its midgut, the more pathogen it can inject to its hosts. Furthermore, Fortuna et al. [@fortuna2015experimental] displays multiple experimental data suggesting that there is a linear relationship between the amount of pathogen within a vector and the amount of pathogen in the saliva of this mosquito.
We assume that the parasite replicates with a logistic growth rate $\tilde{f}(P)=r_h(1-P/ K_h),$ where the parameters $r_h$ and $K_h$ represent net viral growth rate, and the carrying capacity within a host, respectively. Upon exposure to virus, the IgM immune response activates at a rate $a,$ and decays at a rate $c.$ The IgM immune response antibodies are responsible for *rapid destruction of virus* and kills the pathogen at a rate $\epsilon.$ Furthermore, B-cells switch production of IgM antibodies to IgG antibodies with a per-capita rate $q$ [@honjo2002molecular]. The pathogen also stimulates the IgG antibody immune response that activates at a rate $b$ and kills the pathogen at a rate $\delta.$ Yet, the IgG immune response antibodies mainly responsible for *life-long immunity*.
All parameters and dependent variables of this within-host model and their definitions are given in Table \[table:immunological parameters\].
\[withhost\] If $M_0>0 (\text{ or } G_0>0$), then the pathogen (within-the host) eventually clears ($\lim_{\tau \rightarrow \infty}P(\tau,s)=0$), the IgM immune response antibodies decays to zero after viral clearance, and subsequently the IgG immune memory antibodies reach a steady-state; i.e. $\ \lim_{\tau \rightarrow \infty}M(\tau,s)=0 \ \text{and} \ \lim_{\tau \rightarrow \infty}G(\tau,s)=G^+,$ where $G^+>0$ depends on the initial condition; i.e. $G^+=z(P_0^{v\rightarrow h},M_0,G_0).$
Let $P_0^{v \rightarrow h}(=P(0,s))>0.$ By the first equation in (\[PB\]), we obtain $$\begin{aligned}
\label{Plim}
P(\tau,s)\leq P_0^{v \rightarrow h}e^{\int_0^\tau{\left[r_h-\delta G(\sigma,s)\right]}d\sigma},
\end{aligned}$$ where $s$ is fixed. Without loss of generality, assume that $M_0>0.$ Then $\exists \tilde{\tau}:$ $\frac{\partial}{\partial\tau}G(\tau,s) \geq 0$ for all $\tau \geq \tilde{\tau}.$ Therefore $G(\tau,s)$ increases for all $\tau \geq \tilde{\tau}$.\
Case (i) Assume $G(\tilde{\tau},s)>\dfrac{r_h}{\delta}$. Then, we obtain $G(\tau,s)>\dfrac{r_h}{\delta},$ for all $\tau > \tilde{\tau}.$ Therefore as $\tau \rightarrow \infty$, the RHS of the inequality (\[Plim\]) goes to zero. Then by comparison principle, we obtain $\lim_{\tau \rightarrow \infty}P(\tau,s)=0.$ Thus $\lim_{\tau \rightarrow \infty}M(\tau,s)=0$. Then $G(\tau,s)$ saturates as $\tau \rightarrow \infty;$ i.e $\lim_{\tau \rightarrow \infty}G(\tau,s)=\bar{G},$ for some $\bar{G}>0,$ depending on the initial condition $(P_0^{v \rightarrow h}, M_0, G_0).$\
Case (ii)Now suppose that $G(\tilde{\tau},s)<\dfrac{r_h}{\delta}.$ Assume that there exists $\hat{\tau}$: $P(\hat{\tau},s)=0. $ Then we obtain that $P(\tau,s)=0,$ for all $\tau> \hat{\tau}$ and $\lim_{\tau \rightarrow \infty}M(\tau,s)=0$ and $\lim_{\tau \rightarrow \infty}G(\tau,s)=\bar{G}$, for some $\bar{G}>0.$ Now assume that $P(\tau,s)>0,$ for all $\tau>0.$ Then $\frac{\partial}{\partial\tau}G(\tau,s)>0,$ for all $\tau>0.$ Hence there exists $\tau^{+}>0: \ G(\tau^{+})>\dfrac{r}{\delta}$. The rest of the proof follows the argument in case (i), completing the proof.
[> lX]{} Variable/Parameter & Meaning\
\[0.5ex\]\
$S_v(t)$ & The number of susceptible vectors at time $t$,\
$i_v(t,s)$ & The density of infected vectors with infection age $s$ at time $t$,\
$S_H(t)$ & The number of susceptible hosts at time $t,$\
$ i_H(t,\tau,s)$ & The density of the infected hosts (whom infected with a vector with infection age $s$) with host infection age $\tau$ at time $t,$\
$ R_H(t)$ & The number of recovered hosts at time $t,$\
$ \Lambda$ & Susceptible host recruitment rate\
$ \eta$ & Susceptible vector recruitment rate\
$\beta_v(s)$ & Infected vector transmission rate at $s$ days post infection\
$\beta_H(\tau,s)$ & Infected host transmission rate (whom infected with a vector with infection age $s$) at $\tau$ days post infection\
$\nu_H(\tau,s)$ & Additional host mortality rate (whom infected with a vector with infection age $s$) due to disease at $\tau$ days post infection\
$\gamma_H (\tau,s)$ & Per capita host recovery rate (whom infected with a vector with infection age $s$) at $\tau$ days post infection\
$d$ & Host natural death rate\
$\mu$ & Vector natural death rate\
\[table:variables\]
[> lX]{} Variable/Parameter & Meaning\
\[0.5ex\]\
$b_0$ & the parasite cost coefficient,\
$a_0$ & the transmission efficiency of the parasitic infection,\
$b_1$ & the immune response cost coefficient,\
$a_1$ & half-saturation constant in transmission rate,\
$c_0$ & saturation constant in recovery rate,\
$\epsilon_0$ & half-saturation constant in recovery rate,\
$d_1$ & half-saturation constant of vector transmission rate,\
$d_0$ & saturation constant of vector transmission rate,\
(main)[Vector Population]{}; (sv) at (\[xshift=2cm\]main.west); (svtext) at (\[xshift=-0.7cm,yshift=-0.2cm\]sv.north) [$S_v(t)$]{}; (d1) at (\[yshift =-0.3cm,xshift=-0.5cm\]svtext.south) ; (d2) ; (d3) ; (d4) ; (Iv) ; (Ivtext) at (\[xshift=-0.7cm,yshift=-0.2cm\]Iv.north) [$i_v(t,s)$]{}; (dr1) at (\[yshift =-0.3cm,xshift=-0.5cm\]Ivtext.south) ; (dr2) ; (dr3) ; (dr4) ;
(host)[Host Population]{}; (sh) at (\[xshift=2cm,yshift=-0.35cm\]host.west); (shtext) at (\[xshift=-0.7cm,yshift=-0.2cm\]sh.north) [$S_H(t)$]{}; (c1) at (\[yshift =-0.3cm,xshift=-0.5cm\]shtext.south) ; (c2) ; (c3) ; (c4) ; (Ih) ; (Ihtext) at (\[xshift=-0.7cm,yshift=-0.2cm\]Ih.north) [$i_H(t,\tau,s)$]{}; (cr1) at (\[yshift =-0.3cm,xshift=-0.5cm\]Ihtext.south) ; (cr2) ; (cr3) ; (cr4) ; (rh) ; (rhtext) at (\[xshift=-0.7cm,yshift=-0.2cm\]rh.north) [$R_H(t)$]{}; (cb1) at (\[yshift =-0.3cm,xshift=-0.5cm\]rhtext.south) ; (cb2) ; (cb3) ; (cb4) ;
(whd); (pathogen) at (\[yshift =-0.4cm\]whd.north); (pt) [Pathogen level in host ($P(\tau,s)$)]{}; (antibody) []{}; (a1t) [IgM level in host ($M(\tau,s)$)]{}; (antibody2) []{}; (a2t) [IgG level in host ($G(\tau,s)$)]{}; ;
(\[xshift =0.3cm\]pathogen.east) – (pt); (antibody) – (a1t); (antibody2) – (a2t); (tangent cs:node=cr3,point=[(whd.west)]{},solution=2) – (tangent cs:node=whd,point=[(cr3.west)]{}); (tangent cs:node=cr3,point=[(whd.east)]{},solution=1) – (tangent cs:node=whd,point=[(cr3.east)]{},solution=2);
(sv.east) – node\[name=empty,yshift=0.2cm,text=black\][Transmission]{} (Iv.west); (Ih.north) – (empty);
(svout); (svin); (Ivout); (\[yshift =-0.5cm\]sv.west)– node\[xshift=-0.2cm\] [Deaths]{}(svout); (svin)– node\[xshift=-0.2cm\] [Births]{}(\[yshift =0.5cm \]sv.west); (Iv.east)– node\[xshift=0.2cm\][Deaths]{}(Ivout);
(shout); (shin); (Ihout); (rhout); (\[yshift =-0.5cm\]sh.west)– node\[xshift=-0.2cm\] [Deaths]{}(shout); (shin)– node\[xshift=-0.2cm\] [Births]{}(\[yshift =0.5cm \]sh.west); (\[yshift=0.5cm\]Ih.east)– node\[xshift=0.2cm\][Deaths]{}(Ihout); (rh.east)– node\[xshift=0.2cm\][Deaths]{}(rhout); (sh.east) – node\[name=empty2,yshift=0.2cm,text=black\][Transmission]{} (Ih.west); (\[yshift=-0.2cm\]Iv.west) – (empty2); (Ih.east) – node[Recovery]{} (rh.west);
\[nestedframe\]
****** To incorporate heterogeneity among vector to host inoculum size (across the vectors with different infection age), we formulate the infected host compartment as follows:
where $i_H(t,\tau,s)$ represents *the density of hosts infected at time $t-\tau$ by a vector with infection age $s.$* In other words, $i_H(t,\tau,s)$ represents the density of hosts infected at time $t-\tau,$ with an infectious mosquito with vector to host inoculum size $P_0^{v\rightarrow h}$ (which is a function of viral density within-the infectious vectors; i.e. $P_0^{v\rightarrow h}=h(V(s, V_0)).$)
Furthermore the rates of change in the dynamics of susceptible ($S_H(t)$ ), and recovered ($R_H(t)$ ) host population size are described as follows:
where parameters for the host population include: $f(N(t))$ the host recruitment rate with total host population size $N_H(t)=S_H(t)+\int_0^\infty\int_0^\infty{i_H(t,\tau,s)}d\tau ds+R_H(t)$, $d$ the natural death rate of host, $\beta_v$ the transmission rate of infection from infected vectors to hosts, $\gamma_H$ host recovery rate, and $\nu_H$ host disease-induced death rate. For simplicity, we consider the recruitment rate to be constant:$f(N(t))= \Lambda.$\
Vectors are the only mechanism transmitting the disease to susceptible hosts. The *age-since-infection structured vector model* is given by:
where $i_v(t,s)$ represents the density of infected vectors at time $t$ with infection age $s.$ The parameters related to vector dynamics are: $\eta$ the birth/recruitment rate of vectors, $\mu$ the natural death rate of vectors, $\beta_H$ the transmission rate of infection from infected hosts to vectors. In the system -, a portion of the susceptible hosts move to the infected compartment with a rate $\int_0^{\infty} {\beta_v(s)i_v(t,s)ds}$ through bites by infected vectors $i_v(t,s)$ with infection age $s$.
****** The epidemiological parameters $\beta_H(\tau,s)$, $\gamma_H(\tau,s)$, and $\nu_H(\tau,s)$ are formulated similar to previous studies [@gulbudak2017vector] (confirmed by data [@handel2015crossing; @fraser2014virulence]) as follows:
The data [@handel2015crossing; @fraser2014virulence] suggests that the transmission rate $\beta_H(\tau,s)$ is a Holling type II function with respect to the within-host pathogen load $P(\tau,s),$ where within-host pathogen load depends on host and vector infection age: $P(\tau,s)=P(\tau,V(s,V_0)).$ The parameters $a_0$ and $a_1$ are transmission and half saturation constants, respectively. In addition, similar to previous study [@gulbudak2017vector], we formulate the recovery rate as a function of immune response $G(\tau,s)$ and inversely related to the viral load as shown in , where $c_0$ is transmission constant and $\epsilon_0>0$ is proportionality constant and a small number. This formulation of recovery translates into low pathogen load with sufficient IgG memory antibodies to prevent subsequent rise in pathogen load. Thus recovery rate is a decreasing function of pathogen ($P$) and increasing function of IgG immune response ($G$). Furthermore, disease induced death rate $\nu$ is simply formulated as a linear function of $P$ (death due to pathogen resource use), and a function of $M$ (death due to aggressive immune response).
In addition, we formulate the transmission rate from an infectious vector to susceptible host ($\beta_v(s)$) as follows: $ \beta_v(s)=\dfrac{d_0 V(s)}{d_1+V(s)},$ where the parameters $d_0, d_1$ are saturation and half saturation constants, respectively.\
Note that the within-vector viral kinetics $V(s)$ affects both the vector to host transmission $\beta_v(s)$ and vector to host inoculum size $P_0^{v\rightarrow h}$, and the latter alters the within-host virus-immune dynamics, and in turn the host to vector transmission $\beta_H(\tau,s)$. This chain of across-scale interactions effectively introduces feedback from both scales. In contrast to previous attempts of incorporating feedback between scales [@gandolfi2015epidemic], our approach is amenable to analysis and biologically relevant for vector-borne diseases. Although the host to vector inoculum size $V_0^{h \rightarrow v}$ (amount of pathogen in blood meal) can affect within-vector kinetics, the within-host dynamics are more sensitive to vector to host inoculum size. Thus we reserve the significant mathematical complexity of a two-way “infinite-dimensional” feedback for potential future modeling work, and introduce a “friendly” formulation in next section to vary the host to vector inoculum size $V_0^{h \rightarrow v}.$
Analytical results {#analytical}
==================
Basic properties of the system
------------------------------
We assume that all parameters of the model are non-negative. In addition to that, we also consider the immune model initial conditions to be nonnegative: $P_0=P_0^{v\rightarrow h}, \ M_0, \ G_0\geq0$. Through this article, it satisfies that $$\begin{aligned}
\beta_H(.,.),\nu_H(.,.), \gamma_H(.,.)\in L^\infty(0,\infty)^2, \quad \beta_v(.) \in L^\infty(0,\infty)\end{aligned}$$ We introduce $$\pi_H(\tau,s) =e^{-{\displaystyle}\int_0^{\tau}{\left( \nu_H(u,s)+\gamma_H(u,s)+d \right)du}}, \text{ with } \tau, s \geq 0.$$ Integrating the second equation of the system - along the characteristic lines, we obtain $$\label{charac_equ}
i_H(t,\tau,s)=
\begin{cases}
\pi_H(\tau,s)i_H(t-\tau,0,s), & \text{ if } t > \tau \geq 0,\\
\dfrac{\pi_H(\tau,s)}{\pi_H(\tau-t,s)}i_H(0,\tau-t,s), & \text{ if } \tau>t\geq 0,\\
\end{cases}$$ where $\pi_H(\tau,s)$ can be interpreted as the probability of host (whom is bitten with a vector at infection age $s$) still being in the infected class at host infection age $\tau.$
First note that by the equation , we have $\lim_{\tau \rightarrow \infty}i_H(t,\tau,s)=0, \ \forall \ t \in [0,\infty).$ Then by integrating both side of the equation with respect to both independent variables $s$ and $\tau,$ we obtain the following equation: $$\label{IH}
I_H^{\prime}(t)=S_H(t)\int_0^\infty{\beta_v(s)i_v(t,s)ds}-\int_0^\infty{\int_0^\infty{\left( \nu_H(\tau,s)+\gamma_H(\tau,s)+d \right)i_H(t,\tau,s)d\tau}ds}$$ where $I_H(t)=\int_0^\infty \int_0^\infty{i_H(t,\tau,s)d\tau}ds.$ Similarly, integrating the last equation of the system - along the characteristic lines, we obtain $$\label{charac_equ_vector}
i_v(t,s)=
\begin{cases}
\pi_v(s)i_v(t-s,0), & \text{ if } t > s \geq 0,\\
\dfrac{\pi_v(s)}{\pi_H(s-t)}i_v(0,s-t), & \text{ if } s>t\geq 0,\\
\end{cases}$$ where $\pi_v(s)$ can be interpreted as the probability of vector still being in the infected class at infection age $s.$ Then by similar argument above, we also obtain $$\label{Iv}
I_v^{\prime}(t)=S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau, s)i_H(t,\tau,s) ds d\tau - \mu I_v(t)$$ Upon plugging in the boundary and initial conditions of and to and , respectively, we obtain a system of integro-differential equations. Utilizing contraction mapping arguments, similar to methods in [@gulbudak2014modeling; @webb], the existence of unique solutions to the coupled system - can be shown, which remain non-negative for any time $t.$ After adding all equations in the system -, we obtain $$\begin{aligned}
N_H^{\prime}\leq \Lambda -d N_H, \text{ and } N^{\prime}_v =\eta -\mu N_v, \text{ implying }
\limsup\limits_{t\rightarrow \infty} N_H(t)\leq \dfrac{\Lambda}{d} \text{ and } \lim_{t\rightarrow \infty} N_v(t)=\dfrac{\eta}{\mu}.\end{aligned}$$ Since solutions $\phi(t):=(S_H(t),i_H(t,\tau,s),R_H(t), S_v(t), i_v(t,s))$ remain non-negative, we have $$\begin{aligned}
0\leq \limsup\limits\limits\limits_{t\rightarrow \infty} S_H(t), I_H(t), R_H(t) \leq \limsup\limits_{t\rightarrow \infty} N_H(t)\leq \dfrac{\Lambda}{d} \text{ and }0\leq \limsup\limits_{t\rightarrow \infty} S_v(t), I_v(t) \leq lim_{t\rightarrow \infty} N_v(t)=\dfrac{\eta}{\mu}.\end{aligned}$$ Thus solutions remain bounded for all time $t$ and are attracted to a bounded set as $t\rightarrow\infty$. Furthermore, solutions to the system - form a $C^0$-semigroup, denoted $\Psi(t)$, in the state space $X:=\mathbb R_+\times L_+^1(0,\infty)^2 \times \mathbb R_+\times L_+^1(0,\infty)$ [@thieme]. In particular, for $x\in X$, where $\phi(t)=\Psi(t)x$ denotes solution with initial condition $x$ (written above), the following holds: $$\begin{aligned}
\Psi(t+s)x=\Psi(t)(\Psi(s)x). \label{semi}\end{aligned}$$\
The long term behavior of the solutions is determined in part by the equilibria that are time-independent solutions of the system -. The system - has a DFE $\mathcal E_0=(S_H^0, 0, 0, S^0_v,0),$ where $S^0_H=\Lambda / d, S_v^0=\eta/ \mu.$\
Define the reproduction number as follows:
The *basic reproduction number* $\mathcal R_0$ keeps track of the number of secondary infectious hosts produced by one infected host during its infectious time period in an entirely susceptible host population. The term $ S_v^0 {\displaystyle}\int_0^\infty \beta_H(\tau,s) e^{-{\displaystyle}\int_0^\tau ( \nu_H(u,s)+\gamma_H(u,s)+ d)du} d\tau$ is the average number of secondary infectious vectors produced one infectious host (whom is bitten by an infectious vector at infection age $s$) during its lifespan in a wholly susceptible vector population. In addition, $e^{-d\tau}$ is the probability of host having survived to infection age $\tau.$
The DFE $\mathcal E_0$ is locally asymptotically stable if $\mathcal R_0<1$ and unstable if $\mathcal R_0>1.$
To study the behavior of the solutions nearby an equilibrium, we first linearize the vector-host model ($\ref{SIRhost_n})-(\ref{SIvector_n}$) about the equilibrium $(S_H^+, \ \bar{i}_H(\tau,s),\ R_H^+,\ S_V^+, \ \bar{i}_v(s))$ by taking $S_H(t)= S_H^+ + x_H(t), i_H(t,\tau,s)= \bar{i}_H(\tau,s)+ y_H(t,\tau,s), R_H(t)= R^+_H + z_H(t)$, $S_v(t)=S_v^++ x_v(t)$ and $i_v(t,s)=\bar{i}_v(s) + y_v(t,s).$ We look for eigenvalues of the linear operator - that is we look for solutions of the form $x_H(t)= \overline{x}_He^{\lambda t}, y_H(t,\tau,s)= \overline{y}_H(\tau,s)e^{\lambda t},z_H(t)= \overline{z}_He^{\lambda t}$, $x_v(t)=\overline{x}_v e^{\lambda t},$ and $y_v(t,s)= \overline{y}_v(s)e^{\lambda t}$, where $\overline{x}_H,\ \overline{y}_H(\tau), \ \overline{z}_H, \ \overline{x}_v$ , $\overline{y}_v$ are arbitrary non-zero constants (a function of $\tau$ or $s$ in the case of $y_i$ for $i \in \{ H,v\} $). This process results in the following system (the bars have been omitted):
-0.09in Solutions of the system (\[linearized system\]) give the eigenvectors and eigenvalues $\lambda$ of the differential operator. As in [@martcheva2003progression], it can be shown that knowing the distribution of the eigenvalues is sufficient to determine the stability of a given equilibrium for PDEs operators. In other words, as similar to ODEs, if all eigenvalues have negative real parts, the corresponding equilibrium is locally stable; if there is an eigenvalue with a positive real part, then the equilibrium is unstable. Because of that, we will concentrate on investigating eigenvalues.
Equilibrium of interest is DFE $\mathcal E_0=(S^0_H,0,0,S^0_v,0)$. Hence the system (\[linearized system\]) simplifies to the following system:
$$\label{linearized system at DFE}
\begin{cases}
\lambda x_H&\hspace{-4mm}= -S^0_H{\displaystyle}\int_0^{\infty}{\beta_v(s)y_v(s)ds}+\omega z_H-dx_H, \vspace{1.0mm}\\
{\displaystyle}\frac{d y_H(\tau,s)}{d \tau} +\lambda y_H(\tau,s)
&\hspace{-4mm}=
- \left( \nu_H(\tau,s)+\gamma_H(\tau,s)+d \right)y_H(\tau,s), \vspace{1.0mm}\\
y_H(0,s) &\hspace{-4mm} = S^0_H \beta_v(s)y_v(s) \vspace{1.0mm},\\ \\
\lambda z_H &\hspace{-4mm} ={\displaystyle}\int_0^{\infty}{\displaystyle}\int_0^{\infty}{\gamma_H(\tau,s)y_H(\tau,s)ds d\tau}-(d+\omega)z_H \vspace{1.0mm},\\
\lambda x_v &\hspace{-4mm}= -S_v^0 {\displaystyle}\int_0^\infty {\displaystyle}\int_0^{\infty}{\beta_H(\tau,s)y_H(\tau,s)ds d\tau} -\mu x_v \vspace{1.0mm},\\
{\displaystyle}\frac{d y_v(s)}{ds}+\lambda y_v(s) &\hspace{-4mm}=-\mu y_v(s) \vspace{1.0mm}.\\
y_v(0) &\hspace{-4mm} = S_v^0 {\displaystyle}\int_0^\infty {\displaystyle}\int_0^{\infty}{\beta_H(\tau,s)y_H(\tau,s)ds d\tau} \vspace{1.0mm},\\ \\
\end{cases}$$
-0.09in Solving the differential equations in the linearized system , we obtain $$\label{y_H}
y_H(\tau,s) = y_H(0,s)e^{-\lambda \tau}\pi_H(\tau,s), \textit{ where } \pi_H(\tau,s) =e^{-\int_0^{\tau}{\left( \nu_H(u,s)+\gamma_H(u,s)+d \right)du}},$$ and $$\label{y_v}
y_v(s)=y_v(0)e^{-\lambda s}\pi_v(s), \textit{ where } \pi_v(s)=e^{-\mu s}.$$ Also by the boundary conditions in , we have $$\label{y_0}
y_H(0,s)=S^0_H\beta_v(s)y_v(s), \ y_v(0)=S_v^0 {\displaystyle}\int_0^\infty {\displaystyle}\int_0^{\infty}{\beta_H(\tau,s)y_H(\tau,s)ds d\tau}.$$ Substituting (\[y\_H\]) and into (\[y\_v\]) and canceling $y_v(0)$, we get the following characteristic equation for $\lambda:$ $$\label{chrac_eqn}
1=S_H^0S_v^0 {\displaystyle}\int_0^\infty {\displaystyle}{ \beta_v(s)\pi_v(s) \int_0^\infty{\beta_H(\tau,s) \pi_H(\tau,s) e^{-\lambda(\tau+s)}d\tau}ds}.$$ The equation is a trancedental equation; i.e. it involves trancedental functions. The above equation may have many solutions. To show stability of the DFE, we need to show that all solutions $\lambda$ of the above equation have negative real parts. If there is a solution $\lambda$ with positive real part, then the DFE is unstable. To investigate this, we denote by $$\begin{aligned}
G (\lambda) = S_H^0S_v^0 {\displaystyle}\int_0^\infty {\displaystyle}{ \beta_v(s)\pi_v(s) \int_0^\infty{\beta_H(\tau,s) \pi_H(\tau,s) e^{-\lambda(\tau+s)}d\tau}ds.}\end{aligned}$$ Notice that $G(0)=\mathcal R_0.$ If $\mathcal R_0>1$ and $\beta_H(\tau,s)$ is strictly positive on a positive interval, then the function $G(\lambda)$ is a decreasing function of $\lambda.$ Since $G(0)>1$ and $\lim_{\lambda \rightarrow \infty} G(\lambda)=0,$ then there exists $\lambda^+>0:\ G(\lambda^+)=1.$ Hence the DFE is unstable.\
Now let $\mathcal R_0<1.$ Then for all $\lambda=a+ib$ with $a \geq 0,$ we have
$$\begin{aligned}
\left| G(\lambda) \right| & \leq S_H^0S_v^0 {\displaystyle}\int_0^{\infty} {\displaystyle}\beta_v(s)\pi_v(s) \int_0^{\infty} \beta_H(\tau,s) \pi_H(\tau,s) \left| e^{-\lambda(\tau+s)} \right| d\tau ds \\
&\leq S_H^0S_v^0 {\displaystyle}\int_0^{\infty} {\displaystyle}\beta_v(s)\pi_v(s) \int_0^{\infty} \beta_H(\tau,s) \pi_H(\tau,s) e^{-a(\tau+s)}d\tau ds \leq \mathcal R_0 <1. \end{aligned}$$
Then $\lambda$’s whose real part is non-negative can not satisfy the equation $G(\lambda)=1.$ Therefore the DFE is locally asymptotically stable in this case.
Under certain conditions, this result can be extended to global stability of $\mathcal E_0$ by means of Lyapunov functions.
Suppose that $$\mathcal R_v:=S_H^0\int_0^{\infty}{\beta_v(s)\pi_v(s)ds}\leq 1, \quad \textit{and} \quad
\mathcal R_H:=S_v^0 \left( \max\limits_{s \in [0,\infty)}\int{\beta_H(\tau,s)\pi_H(\tau,s)d\tau}\right)\leq 1.$$ Then the DFE $\mathcal E_0$ is globally asymptotically stable.
First define the positive functions $$\begin{aligned}
\alpha(s):=\int_s^{\infty}{\beta_v(l) e^{-\int_s^l{\mu dz}}dl,} \quad \omega(\tau,s):=\int_{\tau}^{\infty}{\beta_H(l,s) e^{-\int_\tau^l{\tilde{\delta}(z,s)dz}}dl},\end{aligned}$$ where $\tilde{\delta}(\tau,s):= \nu_H(\tau,s)+\gamma_H(\tau,s)+ d$. Then by Leibiniz rule, the derivatives of $\alpha(s)$ with respect to $s$ and $ \omega(\tau,s)$ with respect to $\tau$ satisfy $$\begin{aligned}
\label{derivatives}
\alpha^\prime (s)=\dfrac{d \alpha(s)}{ds}=\mu \alpha(s)-\beta_v(s) \quad \textit{and} \quad
\dfrac{\partial \omega(\tau,s)}{\partial \tau}=\tilde{\delta}(\tau,s)\omega(\tau,s)-\beta_H(\tau,s).\end{aligned}$$ Let us consider any solution $(S_H(t),i_H(t,\tau,s), R_H(t),S_v(t), i_v(t,s))$ of the model - with the non-negative initial data. We define a function $W(t)$ as follows: $$\begin{aligned}
W=S_H^0g(\dfrac{S_H}{S_H^0})+S_v^0g(\dfrac{S_v}{S_v^0})+\dfrac{1}{\alpha(0)}\int_0^\infty{\alpha(s)i_v(t,s)ds}+\int_0^\infty{\dfrac{1}{\omega(0,s)} \int_0^\infty{\omega(\tau,s) i_H(t,\tau,s)ds}d\tau}\end{aligned}$$ where $g(x)= x-\ln(x)-1.$ First note that $$\begin{aligned}
&\dfrac{d}{dt}\left(\dfrac{1}{\alpha(0)}\int_0^\infty{\alpha(s)i_v(t,s)ds} \right)\\
&\qquad=\dfrac{1}{\alpha(0)}\left(\int_0^\infty{\alpha(s)\left( -\dfrac{\partial i_v(t,s)}{\partial s} -\mu i_v(t,s) \right)ds}\right)\\
&\qquad=\dfrac{1}{\alpha(0)}\left( -\int_0^\infty{\alpha(s)\mu i_v(t,s)ds}-\int_0^\infty{\alpha(s)\dfrac{\partial i_v(t,s)}{\partial s}ds}\right) \\
&\qquad\qquad\qquad \textit{(by integration by parts)}\\
&\qquad=\dfrac{1}{\alpha(0)}\left(
-\int_0^\infty{\alpha(s)\mu i_v(t,s)ds}-\alpha(s) i_v(t,s)|_{s=0}^{\infty} +\int_0^\infty{\alpha^\prime(s) i_v(t,s)ds}\right)\\
&\qquad\qquad\qquad \textit{(Note that $\alpha(s) i_v(t,s)|_{s=0}^{\infty}=- \alpha(0) i_v(t,0)$)}\\
&\qquad=\dfrac{1}{\alpha(0)}\left(
-\int_0^\infty{\alpha(s)\mu i_v(t,s)ds}+\alpha(0) S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau,s)i_H(t,\tau,s) ds d\tau
\right.\\
&\qquad\qquad\qquad \left.+\int_0^\infty{\alpha^\prime(s) i_v(t,s)ds}\right)\\
&\qquad\qquad\qquad \textit{(Recall that $\alpha^\prime (s)=\mu \alpha(s)-\beta_v(s)$ by \eqref{derivatives})}\\
&\qquad=\dfrac{1}{\alpha(0)}\left(
\alpha(0) S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau,s)i_H(t,\tau,s) ds d\tau-\int_0^\infty{\beta_v(s) i_v(t,s)ds}\right)\\
&\qquad=S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau,s)i_H(t,\tau,s) ds d\tau-\dfrac{1}{\alpha(0)}\int_0^\infty{\beta_v(s) i_v(t,s)ds},\end{aligned}$$ $$\begin{aligned}
&\dfrac{d}{dt}\left(\int_0^\infty{\dfrac{1}{\omega(0,s)}\int_0^\infty{\omega(\tau,s)i_H(t,\tau,s)d\tau}ds} \right)\\
&\qquad=\int_0^\infty{\dfrac{1}{\omega(0,s)}\int_0^\infty{ \omega(\tau,s)\dfrac{\partial i_H(t,\tau,s)}{\partial t} d\tau}ds} \\
&\qquad=-\int_0^\infty{\dfrac{1}{\omega(0,s)}\int_0^\infty{ \omega(\tau,s) \left( \dfrac{\partial i_H(t,\tau,s)}{\partial \tau}+\tilde{\delta}(\tau,s)i_H(t,\tau,s) \right) d\tau}ds}\\
&\qquad=-\int_0^\infty{\dfrac{1}{\omega(0,s)} \int_0^\infty{ \omega(\tau,s) \dfrac{\partial i_H(t,\tau,s)}{\partial \tau}d\tau} ds}
-\int_0^\infty{\dfrac{1}{\omega(0,s)} \int_0^\infty{ \omega(\tau,s)\left(\tilde{\delta}(\tau,s)i_H(t,\tau,s) \right) d\tau}ds}\\
&\qquad=-\int_0^\infty{\dfrac{1}{\omega(0,s)} \left(\omega(\tau,s)i_H(t,\tau,s)|_{\tau=0}^{\infty}-\int_0^\infty{\dfrac{\partial \omega(\tau,s)}{\partial \tau} i_H(t,\tau,s)d\tau}\right)ds} \\
&\qquad\qquad\qquad -\int_0^\infty{\dfrac{1}{\omega(0,s)} \int_0^\infty{ \omega(\tau,s)\left(\tilde{\delta}(\tau,s)i_H(t,\tau,s) \right) d\tau}ds}\\
&\qquad\qquad\qquad \textit{(Recall that $\dfrac{\partial \omega(\tau,s)}{\partial \tau}=\tilde{\delta}(\tau,s)\omega(\tau,s)-\beta_H(\tau,s)$ by \eqref{derivatives})}\\
&\qquad=-\int_0^\infty{\dfrac{1}{\omega(0,s)} \left(-\omega(0,s)i_H(t,0,s)-\int_0^\infty{\left(\tilde{\delta}(\tau,s)\omega(\tau,s)-\beta_H(\tau,s) \right) i_H(t,\tau,s)d\tau}\right)ds} \\
&\qquad\qquad\qquad -\int_0^\infty{\dfrac{1}{\omega(0,s)} \int_0^\infty{ \omega(\tau,s)\left(\tilde{\delta}(\tau,s)i_H(t,\tau,s) \right) d\tau}ds}\\
&\qquad=\int_0^\infty{i_H(t,0,s)}ds+\int_0^\infty{\dfrac{1}{\omega(0,s)}\left( \int_0^\infty{\left(\tilde{\delta}(\tau,s)\omega(\tau,s)-\beta_H(\tau,s) \right) i_H(t,\tau,s)d\tau}\right)ds} \\
&\qquad\qquad\qquad -\int_0^\infty{\dfrac{1}{\omega(0,s)} \int_0^\infty{ \omega(\tau,s)\left(\tilde{\delta}(\tau,s)i_H(t,\tau,s) \right) d\tau}ds}\\
&\qquad=\int_0^\infty{i_H(t,0,s)}ds-\int_0^\infty{\dfrac{1}{\omega(0,s)}\left( \int_0^\infty{\beta_H(\tau,s) i_H(t,\tau,s)d\tau}\right)ds} \\
&\qquad=\int_0^\infty{S_H(t)\beta_v(s)i_v(t,s)}ds-\int_0^\infty{\dfrac{1}{\omega(0,s)}\left( \int_0^\infty{\beta_H(\tau,s) i_H(t,\tau,s)d\tau}\right)ds}.\end{aligned}$$ Therefore the derivative of $W$ along solutions is: $$\begin{aligned}
\dot{W}=&\dfrac{d}{dt}\left(S_H^0g(\dfrac{S_H}{S_H^0})+S_v^0g(\dfrac{S_v}{S_v^0})\right)+\dfrac{d}{dt}\left(\dfrac{1}{\alpha(0)}\int_0^\infty{\alpha(s)i_v(t,s)ds}\right)\\
&\qquad\qquad+\dfrac{d}{dt}\left(\int_0^\infty{\dfrac{1}{\omega(0,s)} \int_0^\infty{\omega(\tau,s) i_H(t,\tau,s)ds}d\tau}\right)\\
&= \left(1-\dfrac{S_H^0}{S_H} \right)(S_H^\prime)+ \left(1-\dfrac{S_v^0}{S_v} \right)(S_v^\prime)+\dfrac{d}{dt}\left(\dfrac{1}{\alpha(0)}\int_0^\infty{\alpha(s)i_v(t,s)ds}\right)\\
&\qquad\qquad+\dfrac{d}{dt}\left(\int_0^\infty{\dfrac{1}{\omega(0,s)} \int_0^\infty{\omega(\tau,s) i_H(t,\tau,s)ds}d\tau}\right)\\
&= \left(1-\dfrac{S_H^0}{S_H} \right)\left(\Lambda -S_H(t)\int_0^{\infty} {\beta_v(s)i_v(t,s)ds}-d S_H(t)\right)\\
&\qquad\qquad+ \left(1-\dfrac{S_v^0}{S_v} \right)\left(\eta - S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau,s)i_H(t,\tau,s) ds d\tau-\mu S_v(t)\right)\\
&\qquad\qquad +S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau,s)i_H(t,\tau,s) ds d\tau-\dfrac{1}{\alpha(0)}\int_0^\infty{\beta_v(s) i_v(t,s)ds}\\
&\qquad\qquad+\int_0^\infty{S_H(t)\beta_v(s)i_v(t,s)}ds-\int_0^\infty{\dfrac{1}{\omega(0,s)}\left( \int_0^\infty{\beta_H(\tau,s) i_H(t,\tau,s)d\tau}\right)ds}\\
&=\left(\Lambda -S_H(t)\int_0^{\infty} {\beta_v(s)i_v(t,s)ds}-d S_H(t)\right)-\dfrac{S_H^0}{S_H} \left(\Lambda -S_H(t)\int_0^{\infty} {\beta_v(s)i_v(t,s)ds}-d S_H(t)\right)\\
&\qquad\qquad+ \left(1-\dfrac{S_v^0}{S_v} \right)\left(\eta - S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau,s)i_H(t,\tau,s) ds d\tau-\mu S_v(t)\right)\\
&\qquad\qquad +S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau,s)i_H(t,\tau,s) ds d\tau-\dfrac{1}{\int_0^{\infty}{\beta_v(l) \pi_v(l)dl}}\int_0^\infty{\beta_v(s) i_v(t,s)ds}\\
&\qquad\qquad+\int_0^\infty{S_H(t)\beta_v(s)i_v(t,s)}ds-\int_0^\infty{\dfrac{1}{\int_{0}^{\infty}{\beta_H(l,s) \pi_H(l,s)}dl}\left( \int_0^\infty{\beta_H(\tau,s) i_H(t,\tau,s)d\tau}\right)ds}\\
&=\left(\Lambda -d S_H(t)\right)-\dfrac{S_H^0}{S_H} \left(\Lambda -S_H(t)\int_0^{\infty} {\beta_v(s)i_v(t,s)ds}-d S_H(t)\right)\\
&\qquad\qquad+ \left(\eta -\mu S_v(t)\right)-\dfrac{S_v^0}{S_v} \left(\eta - S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau,s)i_H(t,\tau,s) ds d\tau-\mu S_v(t)\right)\\
&\qquad\qquad -\dfrac{1}{\int_0^{\infty}{\beta_v(l) \pi_v(l)dl}}\int_0^\infty{\beta_v(s) i_v(t,s)ds}\\
&\qquad\qquad-\int_0^\infty{\dfrac{1}{\int_{0}^{\infty}{\beta_H(l,s) \pi_H(l,s)}dl}\left( \int_0^\infty{\beta_H(\tau,s) i_H(t,\tau,s)d\tau}\right)ds}\\
&=-\dfrac{d}{S_H(t)}(S_H^0 -S_H(t))^2 - \dfrac{\mu}{S_v(t)}(S_v^0-S_v(t))^2\\
&\qquad\qquad +(1-\dfrac{1}{S_H^0\int_0^{\infty}{\beta_v(l) \pi_v(l)dl}})S_H^0\int_0^\infty{\beta_v(s) i_v(t,s)ds}\\
&\qquad\qquad+\int_0^\infty{(1-\dfrac{1}{S_v^0\int_{0}^{\infty}{\beta_H(l,s) \pi_H(l,s)}dl})\left(S_v^0 \int_0^\infty{\beta_H(\tau,s) i_H(t,\tau,s)d\tau}\right)ds}
\\
&\leq-\dfrac{d}{S_H(t)}(S_H^0 -S_H(t))^2 - \dfrac{\mu}{S_v(t)}(S_v^0-S_v(t))^2+S_H^0\left(1-\dfrac{1}{\mathcal R_v}\right)\int_0^\infty{\beta_v(s) i_v(t,s)ds}\\
&\qquad\qquad+S_v^0\left(1-\dfrac{1}{\mathcal R_H}\right)\int_0^\infty\int_0^\infty \beta_H(\tau,s) i_H(t,\tau,s)d\tau ds
\end{aligned}$$ Therefore $\mathcal R_v\leq 1$ and $\mathcal R_H\leq 1$ ensures that $\dot W\leq 0$ holds. Note that the within-host infection eventually clears (Theorem \[withhost\]), implying that there is a finite maximum age of host-vector transmission. Also the within-vector viral load converges to equilibrium, implying that transmission rate $\beta_v(s)$ and inoculum $h(V(s,V_0^{h \rightarrow v}))$ are eventually constant, allowing us essentially to separate solutions into a part with variable rates dependent on $s$ for $s<s_M$ and constant rates for $s>s_M$. These two features ensure a finite maximum age (so that all forward paths have compact closure), and allow us to apply Lyapunov-Lasalle Invariance Principle for functional differential equations [@hale1963stability]. Thus solutions tend to the largest invariant set, $\mathcal A$, where $\dot W=0$. Equality requires that $S_H(t)=S_H^0$ and $S_v(t)=S_v^0$. If $\mathcal R_v<1$ or $\mathcal R_H< 1$, then either $i_H(t,\tau,s)\equiv 0$ or $i_v(t,s)\equiv 0$ on account of characteristic solutions - and invariance of $\mathcal A$. This readily implies $i_H(t,\tau,s)\equiv 0$ and $i_v(t,s)\equiv 0$. If $\mathcal R_v= 1$ and $\mathcal R_H=1$, from $S_H'$ and $S_v'$ equations, we still obtain that $\int_0^\infty{\beta_v(s) i_v(t,s)ds}=0$ and $\int_0^\infty\int_0^\infty \beta_H(\tau,s) i_H(t,\tau,s)d\tau ds=0$. So the same argument implies $i_H(t,\tau,s)\equiv 0$ and $i_v(t,s)\equiv 0$. Therefore the DFE $\mathcal E_0$ is globally asymptotically stable.
We remark that $\mathcal R_0\leq \mathcal R_v \mathcal R_H$ by Hölder’s inequality. The interpretation of $\mathcal R_v$ is the average secondary host transmissions due to an infected vector, and for $\mathcal R_H$, the expected secondary vector transmissions by an infected host maximized over all possible vector to host inoculum sizes, $V(s,V_0^{h \rightarrow v})$. The result guarantees when both are less than unity, the disease eradicates.
Assume that $V(s,V_0^{h \rightarrow v})=V_0^{h \rightarrow v}.$ Then if $\mathcal R_0<1,$ the DFE $\mathcal E_0$ is globally asymptotically stable.
Recall the solutions of the system - obtained along the characteristic lines: $$\label{SIRsol}
i_H(t,\tau,s) =
\begin{cases}
i_H(t-\tau,0,s) \pi_H(\tau,s) &, t>\tau\\
i_H^0(\tau-t,s) \displaystyle\frac{\pi_H(\tau,s)}{\pi_H(\tau-t,s)} &, t<\tau
\end{cases}$$ Substituting (\[SIRsol\]) in the first equation in (\[SIvector\_n\]), and noting that $S_V(t)\le S_v^0$ and $S_H(t)\le S_H^0,$ we obtain $$\label{ineq1}
\begin{array}{l}
\displaystyle \frac{d I_V}{d t} = S_V(t){\displaystyle}\int_0^\infty {\displaystyle}\int_0^t \beta_H(\tau,s) i_H(t-\tau,0,s)
\pi_H(\tau,s) \, d\tau ds \\
\qquad\qquad\qquad\qquad+ S_V(t) {\displaystyle}\int_0^{\infty} {\displaystyle}\int_t^{\infty} \beta_H(\tau,s)
i_H^0(\tau-t,s)\displaystyle\frac{\pi
_H(\tau,s)}{\pi_H(\tau-t,s)}\,d\tau ds -\mu I_V(t)\\
\qquad \le\displaystyle S_H^0 S_v^0{\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty \beta_v(s)i_v(t,s) \beta_H(\tau,s)\pi_H(\tau,s) d\tau ds +\underbrace{S_v^0 \bar{\beta}e^{-d t}
\int_0^\infty \int_0^\infty i_H^0(\tau,s) d\tau ds}_{\mathcal{O}{(e^{-dt})}} -\mu I_V(t)\\
\quad \textit{(Note that $V(s,V_0^{h \rightarrow v})=V_0^{h \rightarrow v} \Rightarrow \beta_v(s)=\tilde{\beta}_v$ for some constant $\tilde{\beta}_v=\dfrac{d_0 V_0^{h \rightarrow v}}{d_1+V_0^{h \rightarrow v}}>0$} \\
\quad \textit{ and $\beta_H(\tau,s)=\tilde{\beta}_H(\tau), \pi_H(\tau,s)=\tilde{\pi}_H(\tau) \ \forall s\in [0, \infty)$ )}\\
\qquad=S_H^0 S_v^0 \tilde{\beta}_v{\displaystyle}\int_0^\infty i_v(t,s)ds {\displaystyle}\int_0^\infty\tilde{\beta}_H(\tau)\tilde{\pi}_H(\tau) d\tau +\underbrace{S_v^0 \bar{\beta}e^{-d t}
\int_0^\infty \int_0^\infty i_H^0(\tau,s) d\tau ds}_{\mathcal{O}{(e^{-dt})}} -\mu I_v(t)\\
\qquad=S_H^0 S_v^0 \tilde{\beta}_v I_v(t) {\displaystyle}\int_0^\infty\tilde{\beta}_H(\tau)\tilde{\pi}_H(\tau) d\tau +\underbrace{S_v^0 \bar{\beta}e^{-d t}
\int_0^\infty \int_0^\infty i_H^0(\tau,s) d\tau ds}_{\mathcal{O}{(e^{-dt})}} -\mu I_V(t)\\
\qquad \le\displaystyle
\mu \mathcal R_0 I_v(t)+\mathcal{O}{(e^{-dt})} -\mu I_v(t)=\mu I_v(t)(\mathcal R_0-1)+\mathcal{O}{(e^{-dt})}
\\ \\
\end{array}$$ Now, define $ \limsup\limits_t I_v :=I_v^\infty $. Then by Fluctuation Lemma \cite{}, $\exists \{ t_n\}: I_v'(t_n)\to 0$ and $I_v(t_n) \rightarrow I_v^\infty $ as $t \rightarrow \infty,$ implying that $0\leq (\mathcal R_0-1)\mu I_v^\infty.$ Since $\mathcal R_0<1$, this implies that $ I_v ^\infty=0$. Furthermore, since $S_H(t)$ is bounded, $\lim \limits_{t \to \infty} i_H(t,0,s) =0$. Similar inequalities as the ones in (\[SIRsol\]) imply that $\lim \limits_{t \to \infty} I_H(t) =0$. Therefore, by the differential equation in the system -, $\lim \limits_{t \to \infty} R_H(t) =0.$ Since $\lim \limits_{t \to \infty} N = S^0_H$, that implies that $\lim \limits_{t \to \infty} S_H(t)
=S^0_H$. Similar reasoning applies to the vector population, resulted in $\lim \limits_{t \to \infty} S_v(t) = S^0_v.$ This completes the proof.
If $\mathcal R_0>1,$ there exists a unique positive endemic equilibrium $$\mathcal E^+ =\left(S^+_H,\ I^+_H={\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty { \bar{i}_H(\tau,s)ds d\tau},\ R^+_H,\ S^+_v, \ I^+_v={\displaystyle}\int_0^\infty {\bar{i}_v(s)ds}\right),$$ with components $$\begin{aligned}
\label{eq_comp}
S^+_H &=\dfrac{S^0_H (\mathcal R_0-1)}{\mu S^0_v\mathcal R_0}
\left[1/ \left(\dfrac{1}{\eta}+ \dfrac{\mathcal R_0}{S^0_H S^0_v}+\dfrac{1}{d}(1+ {\displaystyle}\int_0^\infty \beta_v(s) \pi_v(s) {\displaystyle}\int_0^\infty \nu_H(\tau,s)\pi_H(\tau,s) d\tau ds ))\right)\right]
+ \dfrac{S^0_H}{\mathcal R_0} , \vspace{1mm}\\ \vspace{1mm}
I^+_H&=S^+_H \bar{i}_v(0){\displaystyle}\int_0^\infty \beta_v(s)\pi_v(s){\displaystyle}\int_0^\infty \pi_H(\tau,s)d\tau ds, \quad R^+_H =\dfrac{S^+_H \bar{i}_v(0)}{d} {\displaystyle}\int_0^\infty \beta_v(s)\pi_v(s) {\displaystyle}\int_0^\infty \gamma_H(\tau,s) \pi_H(\tau,s)d\tau ds, \vspace{1mm}\\
S^+_v &= \frac{S^0_HS^0_v}{S^+_H \mathcal R_0},\quad I^+_v= \bar{i}_v(0){\displaystyle}\int_0^\infty \pi_v(s) ds,\vspace{1mm}\\
\end{aligned}$$ and $$\begin{aligned}
\bar{i}_v(0)= \left( \dfrac{ S^0_H}{S^+_H}-\dfrac{S^0_H}{S^+_H\mathcal R_0}\right) / \left[ \dfrac{1}{\eta}+ {\displaystyle}\int_0^\infty \beta_v(s) \pi_v(s) {\displaystyle}\int_0^\infty \pi_H(\tau,s) d\tau ds \right. \\
\left. \qquad\qquad\qquad\qquad +\dfrac{1}{d}(1+{\displaystyle}\int_0^\infty \beta_v(s) \pi_v(s) {\displaystyle}\int_0^\infty \nu_H(\tau,s)\pi_H(\tau,s) d\tau ds) \right].\end{aligned}$$
To find the endemic equilibria, we look for time-independent solutions with at least one non-zero infected compartment, which satisfy the system - with the time derivatives equal to zero: $$\label{vectorhostequ.}
\begin{cases}
0 &\hspace{-4mm}= \Lambda - S^+_H {\displaystyle}\int_0^\infty \beta_v(s)\bar{i}_v(s)ds -d S^+_H, \ {\displaystyle}\frac{\partial \bar{i}_H(\tau,s)}{\partial \tau} =
-(\nu_H (\tau,s)+ \gamma_H(\tau,s)+ d)\bar{i}_H(\tau,s) \vspace{1.0mm},\\
\bar{i}_H(0,s) &\hspace{-4mm} = S^+_H \beta_v(s)\bar{i}_v(s) , \ 0 ={\displaystyle}\int_0^\infty {\displaystyle}\int _0^\infty {\gamma_H(\tau,s)\bar{i}_H(\tau,s)ds d\tau}- d R^+_H \vspace{1.0mm},\\
0 &\hspace{-4mm}= \eta - S^+_v {\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty {\beta_H(\tau,s) \bar{i}_H(\tau,s)ds d \tau}-\mu S^+_v , \ {\displaystyle}\frac{d \bar{i}_v(s)}{ds} =
-\mu(s) \bar{i}_v(s) \vspace{1.0mm},\\
\bar{i}_v(0) &\hspace{-4mm} = S^+_v{\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty \beta_H(\tau,s) \bar{i}_H(\tau,s) ds d\tau \vspace{1.0mm},\\
\end{cases}$$ -0.09in An endemic equilibrium will be given by a non-trivial solution $(S_H^+, \ \bar{i}_H(\tau,s), \ R_H^+, \ S_v^+,\bar{i}_v(s)).$ We first solve the differential equations in the system and obtain the following implicit solutions: $$\begin{aligned}
\label{iheq}
\bar{i}_H(\tau,s) = \bar{i}_H(0,s)\pi_H(\tau,s)= S^+_H \beta_v(s)\bar{i}_v(s) \pi_H(\tau,s), \ \bar{i}_v(s)= \bar{i}_v(0)\pi_v(s)\end{aligned}$$ Substituting this expression into the vector boundary condition in (\[vectorhostequ.\]) and canceling $\bar{i}_v(0)$, we obtain $1= S^+_H S^+_v {\displaystyle}\int_0^\infty \beta_v(s)\pi_v(s) {\displaystyle}\int_0^\infty \beta_H(\tau,s)\pi_H(\tau,s)d\tau ds.$ Then the susceptible host equilibrium is: $$\begin{aligned}
\label{bwoc1}
S^+_H= 1\ / \left(S^+_v {\displaystyle}\int_0^\infty \beta_v(s)\pi_v(s) {\displaystyle}\int_0^\infty \beta_H(\tau)\pi_H(\tau)d\tau ds \right).\end{aligned}$$ From the third equation in (\[vectorhostequ.\]), we can express $R^+_H$ in the terms of $\bar{i}_v(0):$ $$\label{R_H}
R^+_H=\frac{S^+_H \bar{i}_v(0)}{d} \Gamma, \text{ where } \Gamma= {\displaystyle}\int_0^\infty \beta_v(s)\pi_v(s) {\displaystyle}\int_0^\infty \gamma_H(\tau,s) \pi_H(\tau,s)d\tau ds.$$ Integrating the second differential equation in (\[vectorhostequ.\]), we obtain $$\begin{aligned}
\label{I_H'}
0= S^+_H\beta_v(s)\bar{i}_v(s) - {\displaystyle}\int_0^\infty{\left( \nu_H(\tau,s)+\gamma_H(\tau,s) \right)\bar{i}_H(\tau,s)d\tau ds}-d{\displaystyle}\int_0^\infty{\bar{i}_H(\tau,s)}d\tau,\end{aligned}$$ where $\lim_{\tau \rightarrow \infty} \bar{i}_H(\tau,s)=0.$ Adding this equation to the first and the fourth equations in , we obtain the population size for the host at equilibrium as follows: $$\begin{aligned}
\label{N_H}
N^+_H =\left(\Lambda-{\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty \nu_H(\tau,s) \bar{i}_H(\tau,s)d\tau ds\right) \ /d.\end{aligned}$$ Next, we substitute in the equilibrium for the total population size $N^+_H = S^+_H+I^+_H +R^+_H,$ and obtain $$\begin{aligned}
\label{sm}
\left(\Lambda-{\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty \nu_H(\tau,s) \bar{i}_H(\tau,s)d\tau ds\right)\ /d &= 1\ / \left(S^+_v {\displaystyle}\int_0^\infty \beta_v(s)\pi_v(s) {\displaystyle}\int_0^\infty \beta_H(\tau,s)\pi_H(\tau,s)d\tau ds\right) \\
&+ {\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty \bar{i}_H(0,s)\pi_H(\tau,s)d\tau ds+ \frac{S^+_H \bar{i}_v(0)}{d} \Gamma. \nonumber\\ \nonumber\end{aligned}$$ Next we will solve the equation for $\bar{i}_v(0).$ Notice that by having an explicit expression for $\bar{i}_v(0),$ we can obtain $\bar{i}_H(\tau,s).$ From the equation , we have $$\begin{aligned}
\label{i_H}
S^0_H\left( 1-d \ / (\Lambda S^+_v {\displaystyle}\int_0^\infty \beta_v(s)\pi_v(s) {\displaystyle}\int_0^\infty \beta_H(\tau,s)\pi_H(\tau,s)d\tau ds ) \right) = & {\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty \bar{i}_H(0,s)\pi_H(\tau,s)d\tau ds \\
&\hspace{-20mm} + \left(S^+_H \bar{i}_v(0)+{\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty \nu_H(\tau,s) \bar{i}_H(\tau,s)d\tau ds \right) / d \nonumber \\ \nonumber\end{aligned}$$ By the fourth equation in (\[vectorhostequ.\]), we have the susceptible vector equilibrium as follows: $$\begin{aligned}
\label{S_V}
S^+_v=\eta \ / \left( {\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty {\beta_H(\tau,s) \bar{i}_H(\tau,s)ds d \tau}+\mu \right)\end{aligned}$$ Substituting (\[S\_V\]) into the equation (\[i\_H\]) and rearranging it, we obtain $$\begin{aligned}
\label{i_H111}
\dfrac{S^0_H} {S^+_H \bar{i}_v(0)}(1-\frac{1}{\mathcal R_0}) &= \dfrac{1}{\eta}
+ {\displaystyle}\int_0^\infty \beta_v(s) \pi_v(s) {\displaystyle}\int_0^\infty \pi_H(\tau,s) d\tau ds \\
&\qquad\qquad + \dfrac{1}{d}(1+{\displaystyle}\int_0^\infty \beta_v(s) \pi_v(s) {\displaystyle}\int_0^\infty \nu_H(\tau,s)\pi_H(\tau,s) d\tau ds) .\nonumber\end{aligned}$$ Rearranging the equation , we get $$\begin{aligned}
\bar{i}_v(0)= \left( \dfrac{ S^0_H}{S^+_H}-\dfrac{S^0_H}{S^+_H\mathcal R_0}\right) / \left[ \dfrac{1}{\eta}+ {\displaystyle}\int_0^\infty \beta_v(s) \pi_v(s) {\displaystyle}\int_0^\infty \pi_H(\tau,s) d\tau ds \right. \\
\left. \qquad\qquad\qquad\qquad +\dfrac{1}{d}(1+{\displaystyle}\int_0^\infty \beta_v(s) \pi_v(s) {\displaystyle}\int_0^\infty \nu_H(\tau,s)\pi_H(\tau,s) d\tau ds) \right]. \nonumber\end{aligned}$$ Therefore whenever $\mathcal R_0>1,$ $\bar{i}_v(0)$ is positive, establishing the result.
Assume that within-vector viral load is constant through vector infection period; i.e. $V(s,V_0^{h \rightarrow v})=V_0^{h \rightarrow v}.$ Then if $\mathcal R_0>1,$ then $\mathcal E^+$ is locally asymptotically stable whenever it exists.
Consider the linearized system (\[linearized system\]), where the equilibrium of interest is the endemic equilibrium. We eliminate $x_H(t), \ y_H(t,\tau,s), \ x_v(t), \ y_v(t,s)$ so that an equation in $\lambda$ is obtained. The first equation in (\[linearized system\]) is an equality in the terms of $x_H(t), \ y_v(t,s)$ and $\lambda.$ Also by the second equality and boundary condition in the system (\[linearized system\]), we have $y_H(t,\tau,s)= y_H(0,s,t)e^{-\lambda \tau}\pi_H(\tau,s),$ where $y_H(0,t,s) = x_H(t)\beta_v(s)\bar{i}_v(s) + S^+_H \beta_v(s)y_v(t,s).$ Substituting this equality in the last equation in (\[linearized system\]), we obtain an equality in the terms of $x_H, \ y_v,\ x_v$ and $\lambda.$ By the third equation in the system (\[linearized system\]), we have $$x_v=-S_v^+ {\displaystyle}\int_0^\infty {\displaystyle}\int_0^{\infty}{\beta_H(\tau,s)y_H(t,\tau,s)ds d\tau} / \left(\lambda+ {\displaystyle}\int_0^\infty{\beta_H(\tau,s)\bar{i}_H(\tau,s)d\tau ds}+\mu \right),$$ which is in terms of $y_H(\tau,s),$ and $\lambda.$ Then substituting this equality also in the last equation in (\[linearized system\]), we obtain the last equality in the terms of $x_H, \ y_v(0,t)$ and $\lambda.$ We obtain the following system: $$\label{linerized system1}
\footnotesize{\begin{cases}
0&\hspace{-150mm}=\lambda x_H + y_v(t,0)S_H^+{\displaystyle}\int_0^\infty{\beta_v(s) e^{-\lambda s}\pi_v(s)ds}+ x_H {\displaystyle}\int_0^\infty{\beta_v(s)\bar{i}_v(s)ds} +dx_H \vspace{1.0mm},\\
0&\hspace{-150mm}=y_v(t,0) \left( \lambda +\mu + {\displaystyle}\int_0^\infty{\displaystyle}\int_0^\infty{\beta_H(\tau,s)\bar{i}_H(\tau,s)dsd\tau}-(\lambda+\mu) S_v^+ S_H^+ {\displaystyle}\int_0^\infty{\displaystyle}\int_0^\infty{\beta_H(\tau,s)\beta_v(s)\pi_H(\tau,s)\pi_v(s)e^{-\lambda (\tau+s)}dsd\tau} \right) \\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad - x_H(t)(\lambda+\mu)S_v^+{\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty {\beta_H(\tau,s)\beta_v(s)\bar{i}_v(s)\pi_H(\tau,s)e^{-\lambda \tau} dsd\tau} \vspace{1.0mm}.\\
\end{cases}}$$ -0.09in Since $x_H(t)\neq 0$ and $y_v(0,t)\neq 0$, we set the determinant below equal to zero $$\begin{aligned}
\left|\footnotesize{
\begin{array}{ll}
\dfrac{\lambda}{(\lambda+\mu)} + {\displaystyle}\int_0^\infty{\beta_v(s)\bar{i}_v(s)ds} +d
&\dfrac{S_H^+}{(\lambda+\mu)}{\displaystyle}\int_0^\infty{\beta_v(s) e^{-\lambda s}\pi_v(s)ds}\\
\\
-S_v^+{\displaystyle}\int_0^\infty {\displaystyle}\int_0^\infty {\beta_H(\tau,s)\beta_v(s)\bar{i}_v(s)\pi_H(\tau,s)e^{-\lambda \tau} dsd\tau}&
1 + \tilde{T}- S_v^+ S_H^+ {\displaystyle}\int_0^\infty{\displaystyle}\int_0^\infty{\beta_H(\tau,s)\beta_v(s)\pi_H(\tau,s)\pi_v(s)e^{-\lambda (\tau+s)}dsd\tau}\\
\end{array}}
\right|,\end{aligned}$$ where $\tilde{T}=\dfrac{T}{(\lambda +\mu)}$, with $T={\displaystyle}\int_0^\infty{\displaystyle}\int_0^\infty{\beta_H(\tau,s)\bar{i}_H(\tau,s)dsd\tau}.$ Notice that $T \geq 0.$ We obtain the following characteristic equation:
Now assuming $V(s,V_0^{h \rightarrow v})=V_0^{h \rightarrow v},$ we have $\beta_v(s)=\tilde{\beta}_v$ for some constant $\tilde{\beta}_v=\dfrac{d_0 V_0^{h \rightarrow v}}{d_1+V_0^{h \rightarrow v}}>0.$ Then after cancelling some terms and rearranging , we obtain the following characteristic equation: where $\beta_H(\tau)=\beta_H(\tau,V_0^{h \rightarrow v}),\pi_H(\tau)=\pi_H(\tau,V_0^{h \rightarrow v}).$ Now by the way of contradiction, suppose that the characteristic equation can have a solution $\lambda$ with positive real part. Let $\lambda=a+bi$ and assume $a\geq 0.$ Taking the absolute value of both side of the equality above, we get Notice that For $\lambda$ with nonnegative real part, the LHS of the inequality remains strictly greater than one, while the RHS is strictly smaller than one. Thus, such $\lambda$’s cannot satisfy the characteristic equation (\[characteristic\_equ\_scl\]). Hence the endemic equilibrium is locally asymptotically stable whenever it exists.
However the stability of $\mathcal E^+$ when $\mathcal R_0>1,$ for general case, is unknown. The interesting question is: *Is it possible that the heterogeneity among the vector infectivity destabilizes the endemic equilibrium, leading to oscillatory dynamics via *Hopf bifurcation*?* From dynamical systems view, it is not uncommon that while structured PDE models can present oscillatory dynamics, but not a special case of it; for instance the ODE version of the system, where the model parameters are constant [@browne2016immune]. The characteristic equation is too complicated for analysis to infer existence of a Hopf bifurcation, but future work will explore the possibility.
In the presence of a disease, one also would like to understand under what conditions the disease will remain endemic for large time. We say the disease is *uniformly weakly endemic* if there exists some $\tilde{\varepsilon}>0$ independent of the initial conditions such that $$\begin{aligned}
\limsup\limits_{t \rightarrow \infty} I(t)>\tilde{\varepsilon}, \text{ whenever } I(0)>0,\end{aligned}$$ for all solutions of the model. However the disease is *uniformly strongly endemic* if there exists some $\tilde{\varepsilon}>0$ independent of the initial conditions such that $$\begin{aligned}
\liminf\limits_{t \rightarrow \infty} I(t)>\tilde{\varepsilon}, \text{ whenever } I(0)>0,\end{aligned}$$ for all solutions of the model. In the following results, we identify the conditions that result in the prevalence being bounded away from zero.
Assume that within-vector viral load is constant through vector infection period; i.e. $V(s,V_0^{h \rightarrow v})=V_0^{h \rightarrow v}.$ Then if $\mathcal R_0>1,$ then disease is uniformly weakly endemic.
By the way of contradiction, assume that there exists a solution $I_H(t),$ with $I_H(0)>0,$ such that $\lim\limits_{t\rightarrow \infty} I_H(t)=0.$ Let $\varepsilon_1>0$ be given. Then $\exists t_0>0: I_H(t) \varepsilon_1, \ \forall t \geq t_0.$ Consequently, the semigroup properties of a solution imply that without loss of generality we can assume the above inequality valid for all $t\geq 0.$\
Next note that $$\int_0^\infty \int_0^\infty \beta_H (\tau,s)i_H(t,\tau,s)d\tau ds \leq K,$$ for some positive real number $K,$ since $\beta_H(\tau)\leq a_0 (:=\hat{\beta}_H)$ and $\limsup\limits_{t\rightarrow \infty} N_H(t) \leq \dfrac{\Lambda}{d}.$\
Then by the third equation of the system , $$R^{\prime}_H \leq \hat{\gamma}\varepsilon_1-d R_H.$$ Then $\limsup\limits_{t\rightarrow \infty} R_H(t) \leq \dfrac{\hat{\gamma}\varepsilon_1}{d}.$ Hence by the inequality above, we have $R_H(t)\leq \dfrac{\hat{\gamma}\varepsilon_1}{d}+\delta_0,$ for given $\delta_0>0$ and $\forall \ t\geq 0,$ by semigroup property. By similar argument above, we also obtain $\limsup\limits_{t\rightarrow \infty} I_v(t)\leq S^0_v\dfrac{\hat{\beta}_H\varepsilon_1}{\mu}.$ Then $I_H(t), R_H(t),I_v(t) \leq \varepsilon_2,$ where $\varepsilon_2=\dfrac{\hat{\gamma}\varepsilon_1}{d}+\delta_0+S^0_v\dfrac{\hat{\beta}_H\varepsilon_1}{\mu}+\varepsilon_1.$ Next by the first and fourth equation in the system -, we obtain $$\begin{aligned}
S^{\prime}_H & = \Lambda -S_H(t)\int_0^{\infty} {\beta_v(s)i_v(t,s)ds}-d S_H(t) \geq \Lambda-S_H(t)\hat{\beta}_v \varepsilon_2-dS_H(t).\\
S^{\prime}_v&=\eta - S_v(t) \int_0^{\infty}\int_0^{\infty}\beta_H(\tau,s)i_H(t,\tau,s) ds d\tau-\mu S_v(t) \geq \eta -S_v(t)(\hat{\beta}_H\varepsilon_1+\mu)\end{aligned}$$ Then $$\begin{aligned}
\liminf \limits_{t} S_H(t) \geq \dfrac{\Lambda}{\hat{\beta}_v \varepsilon_2+d}:=S_H^0(\varepsilon_2), \
\liminf \limits_{t} S_v(t) \geq \dfrac{\eta}{\hat{\beta}_H \varepsilon_1+\mu}=S_v^0(\varepsilon_1).\end{aligned}$$ Since the functions defined above are continuous and $\lim\limits_{\varepsilon_j\rightarrow 0}S^0_i(\varepsilon_j)=S^0_i$, $i \in \{v,H\}, j \in \{1,2\}$, it follows that for given $\varepsilon_3>0, \exists t_1: S_i(t)\geq S^0_i- \varepsilon_3$ for $i \in \{v,H\},$ and $\forall t\geq t_1.$ Again by semigroup property, w.l.o.g. the inequality above is valid for all $t>0.$ Now assuming $V(s,V_0^{h \rightarrow v})=V_0^{h \rightarrow v},$ we have $\beta_v(s)=\tilde{\beta}_v$ for some constant $\tilde{\beta}_v=d_0 V_0^{h \rightarrow v}/ (d_1+V_0^{h \rightarrow v})>0.$
Next note that by the equation , we have $$\begin{aligned}
\label{comp_prin}
I^{\prime}_v(t)&\geq (S_v^0-\varepsilon_3)(S_H^0-\varepsilon_3)\tilde{\beta}_v I_v(t)\int_0^\infty{\beta_H(\tau)\pi_H(\tau)d\tau}-\mu I_v(t),\end{aligned}$$ where $\beta_H(\tau)=\beta_H(\tau,V_0^{h \rightarrow v}),\pi_H(\tau)=\pi_H(\tau,V_0^{h \rightarrow v}).$ Thus we can write in the following form: $$\begin{aligned}
I^{\prime}_v(t)&\geq \mu I_v(t)\left( \mathcal R_0(\varepsilon_3)-1\right), \label{Ivcomp}\end{aligned}$$ where $\mathcal R_0(\varepsilon_3)\searrow \mathcal R_0 $ as $\varepsilon_3\searrow 0$. Therefore, since $\mathcal R_0>1$, for sufficiently small $\epsilon_3$, and by comparison principle applied to , $I_H(t), I_v(t) $ goes to infinity, as $t \rightarrow \infty.$ This is a contradiction to boundedness of solutions.
Here we conjecture that, for general case, $\mathcal R_0>1$ implies *uniform persistence* of disease, and preserve it as future work. We turn our attention to crucial extensions of this modeling framework, and epidemiological implications.\
\[Within\_vectorfitting\]
{width="41.00000%"}
\[Within\_vector\_pathogen\_vs\_R0\]
(a){width="44.00000%"} b){width="45.00000%"}
\[fig\_R0\]
{width="90.00000%"}
The signatures of within-vector viral kinetics on disease dynamics {#implications}
==================================================================
An interesting question is: *Is it possible that infectiousness of mosquitoes can be a good predictor of disease outbreaks?* In a recent study, Churcher et al.[@churcher2017probability] found that the amount of parasites in a mosquito’s salivary glands not only is a good indicator for how much the mosquito bite can be infectious, and also how faster infection would develop within-host upon receiving the bite. Therefore it is crucial to understand *how the within-vector viral kinetics can be scaled up to the disease dynamics among host population for prediction and disease intervention.*
In previous section, we show that the basic reproduction number, $\mathcal R_0,$ is a threshold quantity, providing whether a disease can persist or eventually die out. To numerically calculate the initial transmission risk $\mathcal R_0$, we first compute the probability function $\pi_H(\tau,s),$ depending on the within-host model variables, with initial condition, vector to host inoculum size $P_0^{v\rightarrow h}=h(V(s_i,{\mbox{\boldmath $ p $}}_v)),$ where $s_i$ is the vector-infection-age, and ${\mbox{\boldmath $ p $}}_v$ represent the within-vector model parameters. Notice that the epidemiological parameters $\beta_H(\tau,s), \nu_H(\tau,s), \gamma_H(\tau,s)$ are functions of within-host immunological variables $P(\tau,s), M(\tau,s), G(\tau,s),$ and the vector to host inoculum size, $P_0^{v\rightarrow h},$ governed by within-vector variable $V(s,{\mbox{\boldmath $ p $}}_v)$. Let $V(s,{\mbox{\boldmath $ p $}}_v)$ be the solution of the system with initial pathogen concentration $V_0^{h \rightarrow v}$. Here, we consider the host to vector inoculum size, $V_0^{h\rightarrow v},$ to be constant, representing the mean. Then by implementing trapezoidal rule multiple times with chosen fixed time step size $\Delta \tau =0.0005,$ we estimated $\mathcal R_0$ and the steady state disease abundance $ \bar{\mathcal I}_H$ (host) and $\bar{\mathcal I}_v$ (vector population) (see Fig.3).
For numerical simulations, we obtain the value of within-host virus-immune response model and epidemiological parameters from the literature, presented in Tables \[table:fitparamPB\], \[table:fixparam2\], and \[table:fitparamepi\], respectively. To estimate the within-vector parameters, we extracted within-mosquito WNV viral data, given in [@fortuna2015experimental], by using MATLAB code *grabit.m*, and numerically fit these data by using the *least square error*. Fig.2 displays the fitted model solution and the within-mosquito viral data (blue dots) given in [@fortuna2015experimental]. The fitted parameter values are: $r_v= 0.3258,\ K_v= 1.2303\times 10^3, \ U_v= 0.9933.$
Prior field studies suggest that environmental factors can manipulate the mosquito’s competence [@tabachnick2013nature; @anderson2010effects]. For example, it has been shown that as temperature increases, virus replication increases in a mosquito’s tissues; therefore increasing viral replication within-mosquito and viral transmission to host. Some of the observed effects of temperature, it is suggested, are due to increased viral replication at higher temperatures that often results in a shortening of the EIP (the time it takes for a mosquito to become infectious once it has taken a viremic blood meal). In addition, it is revealed that the host to vector inoculum size and the length of the exposed period influence the effect of WNV transmission by Cx. nigripalpus [@anderson2010effects]. However the impact of these factors on disease dynamics is still unclear.
In Fig.3, we vary the value of within-vector viral growth rate as $r_v=0.1$ (orange line), $r_v=0.5$ (green line), $r_v=1$ (blue line), and plot the corresponding values of basic reproduction number for distinct values of within-host viral growth rate $r_h \in [0.01, 7]$, and ask: *How does the effect of external factors such as temperature can be scaled up to host population level disease transmission?* The inserted subfigures in Fig.3 displays the corresponding within-vector viral dynamics with respect to varying values of the vector parameter $r_v$(right), and the within-host viral-immune response antibody dynamics for $r_h=7.$ and $P_0^{v \rightarrow h}=0.01.$ We observe that increasing value of $r_v$ shortens virus incubation period for vectors, mimicking field studies, mentioned above [@tabachnick2013nature]. Our numerical results suggest that in return, at host population scale, these mechanisms may lead to significant increase in initial transmission risks, $\mathcal R_0.$ The biological insight is that increasing viral replication rates inside the mosquito decreases the time needed for a blood-fed mosquito to be able to pass on the virus to another host [@dohm2002effect; @soverow2009infectious; @reisen2014effects]. For example, in Fig.3, an increase in within-vector parasite growth rate $r_v$ from $0.5$ to $1,$ shortens the incubation period for $4$ days (see inserted left figure), and increases the epidemic (disease persistence) parameter range of $r_h$ ($\mathcal R_0>1$) from $[0.15 \ 3.9]$ to $[0.14 \ 4.68].$ Therefore these results demonstrate that in an environment where disease may not persist due low susceptibility of host population (lower $r_h$), an increase in within-vector viral replication rate $r_v$ via external factors may lead prolonged epidemic (disease persistence). For instance, in an environment, where a resident mosquito population has a mean value of $r_v=0.5,$ the disease only persists when within-host viral growth rate $r_h$ is in the parameter range $ [0.15 \ 3.9].$ However when within-vector parasite growth rate is increased to $r_v=1$, this range increases to $[0.14 \ 4.68],$ implying that the infectivity of vector population can make host population more susceptible to epidemics. Therefore within-vector viral kinetics (whether it is manipulated by external factors or not) can change the fate of the disease outcomes, and might be a good predictor for disease outbreaks.
In Fig.4, we also assess how the initial transmission risk, $\mathcal R_0,$ changes across the distinct values of $V_0^{h \rightarrow v}$ and the within-host immune response parameter for $a.$ The numerical results suggest that when the immune activation parameter $a$ is small; i.e. when host population does not have strong immunity or protection against infection, the disease dynamics is very sensitive to host population infectivity. This implies that the impact of host to vector inoculum size $V_0^{h \rightarrow v}$ on the disease dynamics among the host population is more magnified in a host population with low immune profile. When there is no sufficient host immune response, larger $V_0^{h \rightarrow v}$ increases the probability of the parasite transmission from host to vector, in return increases the transmission risk among the host population significantly. These findings could have significant implications for public health, magnifying importance of control strategies such as drug treatment or vaccination, which can be utilized to slow down the viral production within-host scale. Another crucial motivation for considering within-vector viral dynamics explicitly in a tractable system is to assess the impact of vector to host inoculum size on the efficacy of control strategies. For example, the recent evidence suggests that the vaccine was less effective when mice or humans were bitten by mosquitoes carrying a greater number of parasites. Assuming that vaccination mainly works by increasing the within-host immune response activation rates $a,$ (or $b$), our results (mentioned above) mimic the observations from the field studies as follows: larger host to vector inoculum size $V_0^{h \rightarrow v}$ results in shorter vector exposed period (see left inserted subfigure in Fig.4), and subsequently generates infectious vector distribution with larger inoculum size $P_0^{v \rightarrow h}.$ Among host population with low immune state, this resulting increase in $P_0^{v \rightarrow h}$ causes larger vector to host virus transmission, ultimately leading an increase in the number of secondary cases ($\mathcal R_0$). The biological insight behind of these findings from field studies is that because the vaccine can only kill a certain proportion of the parasites, it is overwhelmed when the parasite population is too large, suggesting that “it will become epidemiologically important to know how infected a mosquito is for disease elimination”. One drawback of our model (\[SIRhost\_n\]-\[SIvector\_n\]) is that the host to vector inoculum size $V_0^{h \rightarrow v}$ is chosen to be constant, representing the mean. Indeed it should also depend on the within-host viral load $P(\tau,s).$ Next we argue the motivation and challenges in overcoming this strain, and develop a feasible way to vary the host to vector inoculum size $V_0^{h \rightarrow v},$ depending on host infectiousness. Recall that in our model, vector to host inoculum size depends on within-vector pathogen dynamics; i.e. $P^{v \rightarrow h}_0=h(V(s, V_0^{v\rightarrow h}))$, but host to vector inoculum size is assumed to be constant, $V_0^{v\rightarrow h}=P_0.$
\[Infection\_cycle\] {width="70.00000%"}
\[fig\_R01upd2\]
(a)![a)The distribution of host to vector inoculum size $V_0^{h \rightarrow v}$ with respect to varying within-vector viral growth rate $r_v.$ b) $\mathcal R_0$ versus the within-vector viral growth rate $r_v$ with highly infectious (solid line, $V_0^{h \rightarrow v} \in [0,2]$), medium infectious (dashed line, $V_0^{h \rightarrow v} \in [0,1.2]$), and less infectious host population (dotted line, $V_0^{h \rightarrow v} \in [0,1]$), respectively. ](distribution_p01.pdf "fig:"){width="35.00000%"} (b)![a)The distribution of host to vector inoculum size $V_0^{h \rightarrow v}$ with respect to varying within-vector viral growth rate $r_v.$ b) $\mathcal R_0$ versus the within-vector viral growth rate $r_v$ with highly infectious (solid line, $V_0^{h \rightarrow v} \in [0,2]$), medium infectious (dashed line, $V_0^{h \rightarrow v} \in [0,1.2]$), and less infectious host population (dotted line, $V_0^{h \rightarrow v} \in [0,1]$), respectively. ](R0_varying_infectiousness.pdf "fig:"){width="40.00000%"}
Incorporating two-way feedback between epidemiological and immunological scales {#two_way_feedback}
-------------------------------------------------------------------------------
Mosquito vectors are often exposed to hosts that individually vary in pathogen loads, which can result in variation in the proportion of the mosquito population that becomes infectious. Although recent immuno-epidemiological models [@gulbudak2017vector; @tuncer2016structural] have been successful in measuring host to vector transmission as a function of within-host viral load, in these models the varying within-host pathogen loads only implicitly affect the number of secondary infectious cases, where impact of host to vector inoculum size cannot be studied explicitly. Here we introduce a feasible way to incorporate host to vector inoculum size $V_0^{h \rightarrow v},$ by defining a distribution. It affects within-vector viral kinetics, subsequently vector-to-host disease transmission chain. Finally, we numerically and analytically explored how the infectivity of host population (measured by $V_0^{h \rightarrow v}$) impacts arbovirus disease dynamics.
Notice that host to vector inoculum size, $V_0^{h \rightarrow v}$, which is a function of within-host viral load $P(\tau,s)$, affect the within-vector dynamics, described by $V^\prime(s)=g(V(s),V_0^{h \rightarrow v}).$ It is in return affect the distribution of the vector to host inoculum size $P_0^{v\rightarrow h}=h(V(s,V_0^{h \rightarrow v})),$ governing the within-host dynamics and ultimately between-host disease dynamics. This chain of across-scale interactions effectively introduces feedback from both scales (See Fig.5). Yet, due to the significant mathematical complexity of a two-way “infinite-dimensional” feedback, closing the loop is not feasible. Next we describe a feasible way to improve this limitation. In contrast to previous attempts of incorporating feedback between scales [@gandolfi2015epidemic], our approach is amenable to analysis and biologically relevant for vector-borne diseases.
To define the distribution, we assume that the probability distribution of $V_0^{h \rightarrow v}$ depend on two factors: (i) the amount of pathogen that a susceptible vector might get upon biting an infected host, depending on *the host infectiousness* $\beta_H(P(\tau,s), P_0^{v\rightarrow h})$ and (ii) the *infected host density* $\bar{i}_H(\tau, s, P_0^{v\rightarrow h}).$ Therefore, we consider a distributed $V_0^{h \rightarrow v},$ depending on within-host dynamics, given by , and the corresponding steady state infected host density, $\bar{i}_H(\tau, s,P_0^{v\rightarrow h}),$ given by , as follows:
$$\label{PDF}
p(V_0^{h \rightarrow v})=\dfrac{\int_0^{\infty}\int_0^{\infty}{\beta_h(P(\tau, P_0^{v\rightarrow h}=h(V(s, V_0))))\bar{i}_H(\tau, s,P_0^{v\rightarrow h}=h(V(s, V_0)))d\tau}ds}{\int_{V^0_{lower}}^{V^0_{upper}}\int_0^{\infty} \int_0^{\infty}{\beta_h(P(\tau, P_0^{v\rightarrow h}=h(V(s, V_0))))\bar{i}_H(\tau, s, P_0^{v\rightarrow h}=h(V(s, V_0)))ds d\tau}d V_0}.$$
Then the system has the basic reproduction number as follows: $$\int {p(V_0^{h \rightarrow v})\mathcal R_0(V_0^{h \rightarrow v}, a) dV_0^{h \rightarrow v}}.$$
The Fig. 6(a) displays the distribution of host to vector inoculum size with respect to varying within-vector viral growth rate $r_v$. The Fig. 6(b) displays how $\mathcal R_0$ changes w.r.t. varying $r_v,$ given how infectious the host population is, which is defined by the range of $V_0^{h \rightarrow v}$. Numerical results suggest that in a highly infectious host population (larger inoculum size range: $V_0^{h \rightarrow v} \in [0.01\ 2] $), increasing within-vector viral growth rate $r_v$ increases initial transmission risk, $\mathcal R_0.$ However, an increase in $r_v$ among a less infectious host population (inoculum size $V_0^{h \rightarrow v} \in [0.01\ 1] $) decreases $\mathcal R_0,$ suggesting that when the host population is less infectious due to resulting distribution $p(V_0^{h \rightarrow v}),$ vectors as intermediate carries only dilute the effect of disease transmission due to Allee effect.
These findings could have significant implications for disease control. Our numerical results highlight that: (i) a significant reduction in host to vector inoculum size, which can be accomplish utilizing a drug treatment or vaccination (to hamper virus transmission from host to vector) can reduce $\mathcal R_0,$ significantly, in which case the disease can be ultimately eradicated (see Fig.6), (ii) when host population is very infectious, vector infectivity magnifies the disease outcomes.
A more rigorous approach in assessing the impact of vector, or host parameters on disease dynamics requires investigating the sensitivity of within-vector viral kinetics, within-host immune response and epidemic parameters to $\mathcal R_0,$ and $ \mathcal E^+$ which we reserve as a future work.
Conclusion {#conc}
==========
Within-vector viral dynamics can be a driving mechanism in disease dynamics and vector-borne pathogen evolution. Assessing the impact of within-vector viral kinetics on disease dynamics at population scale requires tractable models to measure this impact. In this study, we develop a multi-scale vector-borne disease model, connecting all scales from within-vector viral kinetics to between vector-host disease spread. By doing so, we investigate the impact of within-vector viral kinetics on disease dynamics and, in particular, address:
- How host and vector infectivity, measured by inoculum size, might affect the landscape of the initial transmission risks, $\mathcal R_0.$
- And how within-host immune response combined with within-vector viral kinetics might affect the success of a control strategy such as vaccination, and drug treatment.
There are several reasons for explicitly modeling the heterogeneity in the within-vector dynamics in the novel manner of this work. First the overall aim is to construct a multi-scale model so variations in parameters associated with within-vector viral kinetics can be extrapolated to the overall epidemic dynamics. These variations may come from climate or environmental factors, or bio-control strategies such as *Wolbachia*, and the within-vector viral model can be validated directly from experiment. Tracking dynamics within vectors, as opposed to using average quantities for vector epidemiological parameters (as in ODE models), allows for important biological and mathematical features to be captured. For instance, the delay between infecting bite and viral growth to infectious levels within the mosquito, known as *extrinsic incubation period (EIP)*, can be a very sensitive quantity for determining disease spread [@tjaden2013extrinsic]. While a delay differential equation (DDE) can be used and is also a special case of the PDE, a DDE will still not capture all of the heterogeneity levels of infectiousness that this model. Furthermore, our multi-scale framework provides a natural way to have within-vector parameters shape the EIP, and, in turn, the overall epidemic, as exemplified in Fig. $3$ where within-vector viral growth rate can largely affect $\mathcal R_0$.
In addition, a goal of this work is to model how vector-to-host inoculum size affects the dynamics. In previous work [@gulbudak2017vector], we showed that this inoculum size, as a parameter, has a very large effect on virulence evolution for vector-borne diseases. Here by also incorporating vector viral infection kinetics, we can directly model variable inoculum size based on the within-vector dynamics determining both infectiousness and the initial condition in our within-host model which has a large impact on the host infection, as observed in Fig. $4$.
For analytical results, upon defining the basic reproduction number, $\mathcal R_0,$ (depending on host and vector infectious status), we prove that if $\mathcal R_0<1,$ the disease-free equilibrium $\mathcal E_0$ is locally (via linearization) and globally asymptotically stable (via comparision principle). Otherwise if $\mathcal R_0>1,$ the system has a unique endemic equilibrium, $\mathcal E^\dagger,$ and it is locally asymptotically stable when the vector to host inoculum size, $P_0^{v \rightarrow h},$ is constant during vector infectious time period. However, for general case, the stability of $\mathcal E^\dagger,$ might not be guaranteed via standard linearization method. We provide a condition that if holds, the system might present a Hopf bifurcation, leading oscillatory dynamics. Given the constant vector to host inoculum size, we also show that whenever $\mathcal R_0>1,$ the disease is uniformly weakly persistent.
Our numerical results suggest that when immune response is very low among host population, or when vaccination do not provide sufficiently large immunity, the disease transmission between and within-host are very sensitive to vector to host inoculum size $P_0^{v \rightarrow h}$, mimicking field studies. Indeed, recent field studies suggest that when mosquitoes were very infectious (large vector to host inoculum size), the vaccine was less effective when mice or humans were bitten by mosquitoes carrying a greater number of Malaria parasites, due to “overwhelmed” immune response. Therefore the within-vector viral kinetics, providing how infectious mosquito population, can be crucial determining the transmission risk, and it can impact the outcomes of disease control strategies such as vaccination. In addition, we extend this model by incorporating a distribution of host to vector inoculum size $V_0^{h \rightarrow v}.$ Our results suggest that in a highly infectious host population, the disease outbreaks are highly sensitive to vector competence, magnifying the importance of disease control strategies such as drug treatment, and vaccination, which can slow down the viral progression within-hosts.
In conclusion, in this paper, we develop an immuno-epidemiological model, coupling within-vector viral kinetics, within-host virus-immune response and between host -vector disease transmission. As one of the crucial applications, we show that the developed multi-scale model can be utilized to assess the impact of disease control strategies, when the infectiousness of host population and vector population vary across scales. We also investigate how environmental factors such as temperature can magnify the role of vectors in disease outcomes. Field studies also suggest that nutrition and competition during the larval stage may also influence the transmission capability of arboviruses for the resulting adult females. In addition, environmental factors such as exposure to insecticides in the adult or larval stages has been shown to influence mosquito competence for arboviruses [@muturi2011larval; @yadav2005effect; @muturi2011larval]. Future work will investigate the role of these mechanisms on disease dynamics. In addition in host scale, we only consider adaptive, but not innate immune response, which might affect the disease outcomes. Future work will include these complexities. The multi-scale modelling framework, introduced here, can be utilized to assess the role of vectors on disease dynamics, given many external factors affecting the vector competence and host immune response. In addition, it can be used for assessing the impact of Wolbachia-based biocontrol strategy, mainly utilized to interfere within-vector viral growth to slow down disease transmission among host population, which will be the future work. In summary, the modeling work contained in this paper can help to understand the effect of within-vector viral kinetics on arbovirus disease dynamics and help to guide policies on strategies for disease control.
Acknowledgment
==============
This project may have benefited from discussions with Gabriela Blohm (University of Florida, College of Public Health and Health Professions). In addition, the author also thanks two anonymous reviewers for their helpful comments and feedback on the manuscript, and James M. Hyman (Tulane University), Carrie Manore (Los Alamos National Laboratory), and Gerardo Chowell (Georgia State University) for their suggestions and comments during *New Orleans workshop on Modeling the Spread of Infectious Diseases* at Tulane University. This work was supported by a grant from the Simons Foundation/SFARI($638193$, HG).
[> lXXl]{} Parameter & Estimate &Units & Reference\
\[0.5ex\]\
$r_v$ & $0.3258$ &$ (\mbox{TCID}_{50}\times \mbox{ days})^{-1}$ & See Section \[acc\_scales\]\
\[0.5 ex\] $K_v$ & $1.2303\times 10^3$ & TCID$_{50}$ & See Section \[acc\_scales\]\
$U_v$ & $0.9933$& TCID$_{50}$ & See Section \[acc\_scales\]\
\[0.5 ex\]
\[table:fitparam\]
[> lXX]{} Parameter & Estimate &Units\
\[0.5ex\]\
$r$ & $7.21859433$ &$ (\mbox{TCID}_{50}\times \mbox{ days})^{-1}$\
\[0.5 ex\] $K$ & $5.828521156794433\times 10^7$ & TCID$_{50}$\
$a$ & $1.1\times 10^{-7}$& $(\mbox{ELISA PP} \times \mbox{days})^{-1}$\
\[0.5 ex\] $q$ & $ 0.48442451$ &$\mbox{days}^{-1}$\
\[0.5 ex\] $w$ & $0.40599756$ &$\mbox{days}^{-1}$\
\[0.5 ex\] $b$ & $5\times 10^{-8}$ & $(\mbox{ELISA PP} \times \mbox{days})^{-1}$\
\[0.5 ex\]
\[table:fitparamPB\]
\[table:fixparam2\]
\[table:fitparamepi\]
[^1]: $^*$author for correspondence
|
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abstract: 'An informative sampling design leads to unit inclusion probabilities that are correlated with the response variable of interest. However, multistage sampling designs may also induce higher order dependencies, which are typically ignored in the literature when establishing consistency of estimators for survey data under a condition requiring asymptotic independence among the unit inclusion probabilities. We refine and relax this condition of asymptotic independence or asymptotic factorization and demonstrate that consistency is still achieved in the presence of residual sampling dependence. A popular approach for conducting inference on a population based on a survey sample is the use of a pseudo-posterior, which uses sampling weights based on first order inclusion probabilities to exponentiate the likelihood. We show that the pseudo-posterior is consistent not only for survey designs which have asymptotic factorization, but also for designs with residual or unattenuated dependence. Using the complex sampling design of the National Survey on Drug Use and Health, we explore the impact of multistage designs and order based sampling. The use of the survey-weighted pseudo-posterior together with our relaxed requirements for the survey design establish a broad class of analysis models that can be applied to a wide variety of survey data sets.'
author:
- |
Matthew R. Williams\
Substance Abuse and Mental Health Services Administration\
and\
Terrance D. Savitsky\
U.S. Bureau of Labor Statistics
bibliography:
- 'refs\_june2018.bib'
title: '**Bayesian Estimation Under Informative Sampling with Unattenuated Dependence**'
---
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[1]{}
[**Bayesian Estimation Under Informative Sampling with Unattenuated Dependence**]{}
[*Keywords:*]{} Cluster sampling, Stratification, Survey sampling, Sampling weights, Markov Chain Monte Carlo.
Introduction
============
Bayesian formulations are increasingly popular for modeling hypothesized distributions with complicated dependence structures. The primary interest of the data analyst is to perform inference about a finite population generated from an unknown model, $P_{0}$. The observed data are often collected from a sample taken from that finite population under a complex sampling design distribution, $P_{\nu}$, resulting in probabilities of inclusion that are associated with the variable of interest. This association could result in an observed data set consisting of units that are not independent and identically distributed. This association induces a correlation between the response variable of interest and the inclusion probabilities. Sampling designs that induce this correlation are termed, “informative", and the balance of information in the sample is different from that in the population. Failure to account for this dependence caused by the sampling design could bias estimation of parameters that index the joint distribution hypothesized to have generated the population [@holt:1980]. While emphasis is often placed on the first order inclusions probabilities (individual probabilities of selection), an “informative” design may also have other features such as clustering and stratification that use population information and impact higher order joint inclusion probabilities. The impact of these higher order terms are more subtle but we will demonstrate that they can also impact bias and consistency.
@2015arXiv150707050S proposed an automated approach that formulates a sampling-weighted pseudo-posterior density by exponentiating each likelihood contribution by a sampling weight constructed to be inversely proportional to its marginal inclusion probability, $\pi_{i} = P\left(\delta_{i} = 1\right)$, for units, $i = 1,\ldots,n$, where $n$ denotes the number of units in the observed sample. The inclusion of unit, $i$, from the population, $U$, in the sample is indexed by $\delta_{i} \in \{0,1\}$. They restrict the class of sampling designs to those where the pairwise dependencies among units attenuate to $0$ in the limit of the population size, $N$, (at order $N$) to guarantee posterior consistency of the pseudo-posterior distribution estimated on the sample data, at $P_{0}$ (in $L_{1}$). While some sampling designs will meet this criterion, many won’t; for example, a two-stage clustered sampling design where the number of clusters increases with $N$, but the number of units in each cluster remain relatively fixed such that the dependence induced at the second stage of sampling never attenuates to $0$. A common example are designs which select households as clusters. Despite the lack of theoretical results demonstrating consistency of estimators under this scenario, use of first order sampling weights performs well in practice. This work provides new theoretical conditions for when consistency can be achieved and provides some examples for when these conditions are violated and consistency may not be achieved.
Motivating Example: The National Survey on Drug Use and Health
--------------------------------------------------------------
Our motivating survey design is the National Survey on Drug Use and Health (NSDUH), sponsored by the Substance Abuse and Mental Health Services Administration (SAMHSA). NSDUH is the primary source for statistical information on illicit drug use, alcohol use, substance use disorders (SUDs), mental health issues, and their co-occurrence for the civilian, non institutionalized population of the United States. The NSDUH employs a multistage state-based design [@MRB:Sampling:2014], with the earlier stages defined by geography within each state in order to select households (and group quarters) nested within these geographically-defined primary sampling units (PSUs). The sampling frame was stratified implicitly by sorting the first-stage sampling units by a core-based statistical area (CBSA) and socioeconomic status indicator and by the percentage of the population that is non-Hispanic and white. First stage units (census tracts) were then selected with probability proportionate to a composite size measure based on age groups. This selection was performed ‘systematically’ along the sort order gradient. Second and third stage units (census block groups and census blocks) were sorted geographically and selected with probability proportionate to size (PPS) sequentially along the sort order. Fourth stage dwelling units (DU) were selected systematically with equal probability, selecting every $k^{th}$ DU after a random starting point. Within households, 0, 1 or 2 individuals were selected with unequal probabilities depending on age with youth (age 12- 17) and young adults (age 18-25) over-sampled.
This paper provides conditions for asymptotic consistency for designs like the NSDUH, which are characterized by:
- Cluster sampling, such as selecting only one unit per cluster, or selecting multiple individuals from a dwelling unit.
- Population information used to sort sampling units along gradients.
Both features are common, in practice, and create pairwise sampling dependencies that do not attenuate even if the population grows. The consistency of estimators under these sampling designs are not addressed in the literature. For example, we will examine the relationship between depression and smoking. Cigarette use and depression vary by age, metropolitan vs. non-metropolitan status, education level, and other demographics [@DT:2014; @MHDT:2014]. Both smoking and depression have the potential to cluster geographically and within dwelling units, since these related demographics may cluster. Yet the current literature, such as in @2015arXiv150707050S, is silent on the issue of non-ignorable clustering that may be informative (i.e. related to the response of interest). The results presented in this work establish conditions for a wide variety of survey designs and provide a theoretical justification that this relationship can be estimated consistently even under a complex multistage design such as the NSDUH.
Review of Methods to Account for Dependent Sampling
---------------------------------------------------
For consistency results, assumptions of approximate or asymptotic independence of sample selection (or factorization of joint inclusion probabilities into a product of individual inclusion probabilities) are ubiquitous. For example, @Isaki82 assume asymptotic factorization to demonstrate the consistency of the Horvitz-Thompson estimator and related regression estimators. More recently, @zbMATH06017974 used a similar assumption to demonstrate consistency of survey-weighted regression trees and @2015arXiv150707050S used it to show consistency of a survey-weighted pseudo-posterior.
@chambers2003analysis [Ch.2] review the construction of a sample likelihood using a Bayes rule expression for the population likelihood defined on the sampled units, $f_{s}(y) = f_{U}(y|I = 1)$ (similar to @pkr:1998). They explicitly state the assumption that the “sample inclusion for any particular population unit is independent of that for any other unit (and is determined by the outcome of a zero-one random variable, $I$)”. The further assumption of independence of the population units stated in @pkr:1998 means that weighting each likelihood contribution multiplied together in the sample is an approximation of the likelihood for the $N$ population units.
@pkr:1998 maintains the assumption of unconditional independence of the population units, but defines two classes of sampling designs: (1) The first class is independent, with replacement sampling, so the sample inclusions are all independent. (2) The second class is some selected with replacement designs that are asymptotically independent. @chambers2003analysis discuss the pseudo-likelihood (and cite @Kish74, @Binder83 and @Godambe86) for estimation via a weighted score function. They assume that the correlation between inclusion indicators has an *expected value* of 0, where the expectation is with respect to the population generating distribution. We note that they do not assume this correlation to be exactly equal to 0. However this condition still appears to be more restrictive than that of asymptotic factorization in which deviations from factorization shrink to 0 at a rate inverse to the population size $N$: $\order{N^{-1}}$.
The assumptions above are relied on to show consistency. However in practice, approximate sampling independence is only assumed for the first stage or primary sampling units (PSUs), with dependence between secondary units within these clusters commonly assumed. This setup is the defacto approach for design-based variance estimation (for example, see @heeringa2010applied [Ch.3] and [@Rao92]) and is used in all the major software packages for analyzing survey data. One goal of the current work is to reconile this discrepancy by extending the class of designs for which consistency results are available to cover designs seen in practice such as those for which design-based variance estimation strategies already exist.
We focus on extending the results of the survey-weighted pseudo-posterior method of @2015arXiv150707050S which provides for flexible modeling of a very wide class of population generating models. By refining and relaxing the conditions on factorization, we expand results to include many common sampling designs. These conditions for the sampling designs can be applied to generalize many of the other consistency results mentioned above. There are some population models of interest for which marginal inclusion probabilities may not be sufficient and pairwise inclusion probabilities and composite likelihoods can be used to achieve consistent results [@2016yi; @2017pair]. However, @2017pair demonstrate that both a very specific population model (for example conditional behavior of spouse-spouse pairs within households) and specific sample design (differential selection of pairs of individuals within a household related to outcome) are needed for marginal weights to lead to bias. In the usual setting of inference on a population of individuals (rather than on a population of joint relationships *within* households), pairwise weights and marginal weights are numerically similar, converging to one another for moderate sample sizes. The theory presented in the current work also clarifies why both approaches lead to consistent results. Furthermore, the current work also applies when individual units are mutually exclusive; for example, only selecting one individual from a household to the exclusion of all others. Such designs are not covered by the composite likelihood with pairwise weights approach, which require non-zero joint inclusion probabilities.
The remainder of this work proceeds as follows: In section \[sec:pseudop\] we briefly review the pseudo-posterior approach to account for informative sampling via the exponentiaion of the likelihood with sampling weights. Our main result, presented in section \[results\], provides the formal conditions controlling for sampling dependence. In section \[sec:sims\], we provide two simulations. We first demonstrate consistency for a multistage survey design analogous to the NSDUH. We next create a pathological design based on sorting. The design violates our assumptions for sampling dependence and estimates fail to converge. However, we show that this design will lead to consistency if embedded within stratified or clustered designs. Lastly, we revisit the NSDUH with a simple example (section \[sec:NSDUH\]) and provide some conclusions (section \[sec:conc\]).
Pseudo-Posterior Estimator to Account for Informative Sampling {#sec:pseudop}
==============================================================
We briefly review the pseudo-likelihood and associated pseudo-posterior as constructed in @2015arXiv150707050S and revisited by @2017pair.
Suppose there exists a Lebesgue measurable population-generating density, $\pi\left(y\vert\bm{\lambda}\right)$, indexed by parameters, $\bm{\lambda} \in \Lambda$. Let $\delta_{i} \in \{0,1\}$ denote the sample inclusion indicator for units $i = 1,\ldots,N$ from the population. The density for the observed sample is denoted by, $\pi\left(y_{o}\vert\bm{\lambda}\right) = \pi\left(y\vert \delta = 1,\bm{\lambda}\right)$, where “$o$" indicates “observed".
The plug-in estimator for posterior density under the analyst-specified model for $\bm{\lambda} \in \Lambda$ is $$\hat{\pi}\left(\bm{\lambda}\vert \mathbf{y}_{o},\tilde{\mathbf{w}}\right) \propto \left[\mathop{\prod}_{i = 1}^{n}p\left(y_{o,i}\vert \bm{\lambda}\right)^{\tilde{w}_{i}}\right]\pi\left(\bm{\lambda}\right), \label{pseudolike}$$ where $\mathop{\prod}_{i=1}^{n}p\left(y_{o,i}\vert \bm{\lambda}\right)^{\tilde{w}_{i}}$ denotes the pseudo-likelihood for observed sample responses, $\mathbf{y}_{o}$. The joint prior density on model space assigned by the analyst is denoted by $\pi\left(\bm{\lambda}\right)$. The sampling weights, $\{\tilde{w}_{i} \propto 1/\pi_{i}\}$, are inversely proportional to unit inclusion probabilities and normalized to sum to the sample size, $n$. Let $\hat{\pi}$ denote the noisy approximation to posterior distribution, $\pi$, based on the data, $\mathbf{y}_{o}$, and sampling weights, $\{\tilde{\mathbf{w}}\}$, confined to those units *included* in the sample, $S$.
Consistency of the Pseudo Posterior Estimator {#results}
=============================================
A sampling design is defined by placing a *known* distribution on a vector of inclusion indicators, $\bm{\delta}_{\nu} = \left(\delta_{\nu 1},\ldots,\delta_{\nu N_{\nu}}\right)$, linked to the units comprising the population, $U_{\nu}$. The sampling distribution is subsequently used to take an *observed* random sample of size $n_{\nu} \leq N_{\nu}$. Our conditions needed for the main result employ known marginal unit inclusion probabilities, $\pi_{\nu i} = \mbox{Pr}\{\delta_{\nu i} = 1\}$ for all $i \in U_{\nu}$ and the second order pairwise probabilities, $\pi_{\nu ij} = \mbox{Pr}\{\delta_{\nu i} = 1 \cap \delta_{\nu j} = 1\}$ for $i,j \in U_{\nu}$, which are obtained from the joint distribution over $\left(\delta_{\nu 1},\ldots,\delta_{\nu N_{\nu}}\right)$. We denote the sampling distribution by $P_{\nu}$.
Under informative sampling, the inclusion probabilities are formulated to depend on the finite population data values, $\mathbf{X}_{N_{\nu}} = \left({\mathbf{X}}_{1},\ldots,{\mathbf{X}}_{N_{\nu}}\right)$. Information from the population is used to determine size measures for unequal selection $\pi_{\nu i}$ and used to establish clustering and stratification which determine joint inclusions probabilities $\pi_{\nu ij}$. Since the balance of information is different between the population and a resulting sample, a posterior distribution for $\left({\mathbf{X}}_{1}\delta_{\nu 1},\ldots,{\mathbf{X}}_{N_{\nu}}\delta_{\nu N_{\nu}}\right)$ that ignores the distribution for $\bm{\delta}_{\nu}$ will not lead to consistent estimation.
Our task is to perform inference about the population generating distribution, $P_{0}$, using the observed data taken under an informative sampling design. We account for informative sampling by “undoing" the sampling design with the weighted estimator, $$p^{\pi}\left({\mathbf{X}}_{i}\delta_{\nu i}\right) := p\left({\mathbf{X}}_{i}\right)^{\delta_{\nu i}/\pi_{\nu i}},~i \in U_{\nu},$$ which weights each density contribution, $p({\mathbf{X}}_{i})$, by the inverse of its marginal inclusion probability. This approximation for the population likelihood produces the associated pseudo-posterior, $$\label{inform_post}
\Pi^{\pi}\left(B\vert {\mathbf{X}}_{1}\delta_{\nu 1},\ldots,{\mathbf{X}}_{N_{\nu}}\delta_{\nu N_{\nu}}\right) = \frac{\mathop{\int}_{P \in B}\mathop{\prod}_{i=1}^{N_{\nu}}\frac{p^{\pi}}{p_{0}^{\pi}}({\mathbf{X}}_{i}\delta_{\nu i})d\Pi(P)}{\mathop{\int}_{P \in \mathcal{P}}\mathop{\prod}_{i=1}^{N_{\nu}}\frac{p^{\pi}}{p_{0}^{\pi}}({\mathbf{X}}_{i}\delta_{\nu i})d\Pi(P)},$$ that we use to achieve our required conditions for the rate of contraction of the pseudo-posterior distribution on $P_{0}$. We note that both $P$ and $\bm{\delta}_{\nu}$ are random variables defined on the space of measures ($\mathcal{P}$ and $ B \subseteq \mathcal{P}$) and the distribution, $P_{\nu}$, governing all possible samples, respectively. An important condition on $P_{\nu}$ formulated in @2015arXiv150707050S that guarantees contraction of the pseudo-posterior on $P_{0}$ restricts pairwise inclusion dependencies to asymptotically attenuate to $0$. This restriction narrows the class of sampling designs for which consistency of a pseudo-posterior based on marginal inclusion probabilities may be achieved. We will replace their condition that requires marginal factorization of all pairwise inclusion probabilities with a less restrictive condition allowing for non-factorization for a small partition of pairwise inclusion probabilities. This expands the allowable class of sampling designs under which frequentist consistency may be guaranteed. We assume measurability for the sets on which we compute prior, posterior and pseudo-posterior probabilities on the joint product space, $\mathcal{X}\times\mathcal{P}$. For brevity, we use the superscript, $\pi$, to denote the dependence on the known sampling probabilities, $\{\pi_{\nu ij}\}_{i,j \in U_{\nu}}$; for example, $$\displaystyle\Pi^{\pi}\left(B\middle\vert {\mathbf{X}}_{1}\delta_{\nu 1},\ldots,{\mathbf{X}}_{N_{\nu}}\delta_{\nu N_{\nu}}\right) := \Pi\left(B\middle\vert \left({\mathbf{X}}_{1}\delta_{\nu 1},\ldots,{\mathbf{X}}_{N_{\nu}}\delta_{\nu N_{\nu}}\right),
\{\pi_{\nu ij}: i,j \in U_{\nu} \} \right).$$
Our main result is achieved in the limit as $\nu\uparrow\infty$, under the countable set of successively larger-sized populations, $\{U_{\nu}\}_{\nu \in \mathbb{Z}^{+}}$. We define the associated rate of convergence notation, $a_{\nu} = \order{b_{\nu}}$, to denote $ |a_{\nu}| \le M |b_{\nu}|$ for a constant $M > 0$.
Empirical process functionals {#empirical}
-----------------------------
We employ the empirical distribution approximation for the joint distribution over population generation and the draw of an informative sample that produces our observed data to formulate our results. Our empirical distribution construction follows @breslow:2007 and incorporates inverse inclusion probability weights, $\{1/\pi_{\nu i}\}_{i=1,\ldots,N_{\nu}}$, to account for the informative sampling design, $$\mathbb{P}^{\pi}_{N_{\nu}} = \frac{1}{N_{v}}\mathop{\sum}_{i=1}^{N_{\nu}}\frac{\delta_{\nu i}}{\pi_{\nu i}}\delta\left({\mathbf{X}}_{i}\right),$$ where $\delta\left({\mathbf{X}}_{i}\right)$ denotes the Dirac delta function, with probability mass $1$ on ${\mathbf{X}}_{i}$ and we recall that $N_{\nu} = \vert U_{\nu} \vert$ denotes the size of of the finite population. This construction contrasts with the usual empirical distribution, $\mathbb{P}_{N_{\nu}} = \frac{1}{N_{v}}\mathop{\sum}_{i=1}^{N_{\nu}}\delta\left({\mathbf{X}}_{i}\right)$, used to approximate $P \in \mathcal{P}$, the distribution hypothesized to generate the finite population, $U_{\nu}$.
We follow the notational convention of @Ghosal00convergencerates and define the associated expectation functionals with respect to these empirical distributions by $\mathbb{P}^{\pi}_{N_{\nu}}f = \frac{1}{N_{\nu}}\mathop{\sum}_{i=1}^{N_{\nu}}\frac{\delta_{\nu i}}{\pi_{\nu i}}f\left({\mathbf{X}}_{i}\right)$. Similarly, $\mathbb{P}_{N_{\nu}}f = \frac{1}{N_{\nu}}\mathop{\sum}_{i=1}^{N_{\nu}}f\left({\mathbf{X}}_{i}\right)$. Lastly, we use the associated centered empirical processes, $\mathbb{G}^{\pi}_{N_{\nu}} = \sqrt{N_{\nu}}\left(\mathbb{P}^{\pi}_{N_{\nu}}-P_{0}\right)$ and $\mathbb{G}_{N_{\nu}} = \sqrt{N_{\nu}}\left(\mathbb{P}_{N_{\nu}}-P_{0}\right)$.
The sampling-weighted, (average) pseudo-Hellinger distance between distributions, $P_{1}, P_{2} \in \mathcal{P}$, $d^{\pi,2}_{N_{\nu}}\left(p_{1},p_{2}\right) = \frac{1}{N_{\nu}}\mathop{\sum}_{i=1}^{N_{\nu}}\frac{\delta_{\nu i}}{\pi_{\nu i}}d^{2}\left(p_{1}(\mathbf{X}_{i}),p_{2}(\mathbf{X}_{i})\right)$, where $d\left(p_{1},p_{2}\right) = \left[\mathop{\int}\left(\sqrt{p_{1}}-\sqrt{p_{2}}\right)^{2}d\mu\right]^{\frac{1}{2}}$ (for dominating measure, $\mu$). We need this empirical average distance metric because the observed (sample) data drawn from the finite population under $P_{\nu}$ are no longer independent. The associated non-sampling Hellinger distance is specified with, $d^{2}_{N_{\nu}}\left(p_{1},p_{2}\right) = \frac{1}{N_{\nu}}\mathop{\sum}_{i=1}^{N_{\nu}}d^{2}\left(p_{1}(\mathbf{X}_{i}),p_{2}(\mathbf{X}_{i})\right)$.
Main result {#main:results}
-----------
We proceed to construct associated conditions and a theorem that contain our main result on the consistency of the pseudo-posterior distribution under a broader class of informative sampling designs at the true generating distribution, $P_{0}$. This approach follows the main in-probability convergence result of @2015arXiv150707050S which extends @ghosal2007 by adding new conditions that restrict the distribution of the informative sampling design. Instead of the standard asymptotic factorization condition, we provide two alternative conditions which allow for residual dependence between sampling units:
Suppose we have a sequence, $\xi_{N_{\nu}} \downarrow 0$ and $N_{\nu}\xi^{2}_{N_{\nu}}\uparrow\infty$ and $n_{\nu}\xi^{2}_{N_{\nu}}\uparrow\infty$ as $\nu\in\mathbb{Z}^{+}~\uparrow\infty$ and any constant, $C >0$,
(A1)\[existtests\]
: (Local entropy condition - Size of model) $$\mathop{\sup}_{\xi > \xi_{N_{\nu}}}\log N\left(\xi/36,\{P\in\mathcal{P}_{N_{\nu}}: d_{N_{\nu}}\left(P,P_{0}\right) < \xi\},d_{N_{\nu}}\right) \leq N_{\nu} \xi_{N_{\nu}}^{2},$$
(A2)\[sizespace\]
: (Size of space) $$\displaystyle\Pi\left(\mathcal{P}\backslash\mathcal{P}_{N_{\nu}}\right) \leq \exp\left(-N_{\nu}\xi^{2}_{N_{\nu}}\left(2(1+2C)\right)\right)$$
(A3)\[priortruth\]
: (Prior mass covering the truth) $$\displaystyle\Pi\left(P: -P_{0}\log\frac{p}{p_{0}}\leq \xi^{2}_{N_{\nu}}\cap P_{0}\left[\log\frac{p}{p_{0}}\right]^{2}\leq \xi^{2}_{N_{\nu}} \right) \geq \exp\left(-N_{\nu}\xi^{2}_{N_{\nu}}C\right)$$
(A4)\[bounded\]
: (Non-zero Inclusion Probabilities) $$\displaystyle\mathop{\sup}_{\nu}\left[\frac{1}{\displaystyle\mathop{\min}_{i \in U_{\nu}}\vert\pi_{\nu i}\vert}\right] \leq \gamma, \text{ with $P_{0}-$probability $1$.}$$
(A5.1)\[deprestrict\]
: (Growth of dependence is restricted)\
For every $U_{\nu}$ there exists a binary partition $\{S_{\nu 1}, S_{\nu 2}\}$ of the set of all pairs $S_{\nu}= \{\{i,j\}: i\ne j \in U_{\nu}\}$ such that $$\displaystyle\mathop{\limsup}_{\nu\uparrow\infty} \left\vert S_{\nu 1} \right\vert = \order{N_{\nu}},$$ and $$\displaystyle\mathop{\limsup}_{\nu\uparrow\infty} \mathop{\max}_{i,j \in S_{\nu 2}}\left\vert\frac{\pi_{\nu ij}}{\pi_{\nu i}\pi_{\nu j}} - 1\right\vert = \order{N_{\nu}^{-1}}, \text{ with $P_{0}-$probability $1$}$$ such that for some constants, $C_{4},C_{5} > 0$ and for $N_{\nu}$ sufficiently large, $$\left\vert S_{\nu 1} \right\vert \le C_{4} N_{\nu},$$ and $$\displaystyle N_{\nu}\mathop{\sup}_{\nu}\mathop{\max}_{i,j \in S_{\nu 2}}\left\vert\frac{\pi_{\nu ij}}{\pi_{\nu i}\pi_{\nu j}} - 1\right\vert \leq C_{5},$$
(A5.2)\[depblock\]
: (Dependence restricted to countable blocks of bounded size)\
For every $U_{\nu}$ there exists a partition $\{B_1,\dots, B_{D_{\nu}}\}$ of $U_{\nu}$ with $D_{\nu} \le N_{\nu}$, $\mathop{\lim}_{\nu\uparrow\infty} D_{\nu} = \order{N_{\nu}}$, and the maximum size of each subset is bounded: $$1 \le \displaystyle\mathop{\sup}_{\nu} \displaystyle\mathop{\max}_{d \in 1,\dots, D_{\nu}}\left\vert B_d\right\vert \leq C_{4},$$ Such that the set of all pairs $S_{\nu}= \{\{i,j\}: i\ne j \in U_{\nu}\}$ can be partitioned into $S_{\nu 1} = \left\{\{i,j\}: i\ne j \in B_{d}, d \in \{1, \ldots, D_{\nu} \}\right\}$ and\
$S_{\nu 2} = \left\{\{i,j\}: i \in B_d \cap j \notin B_{d}, d \in \{1, \ldots, D_{\nu}\}\right\}$ with $$\displaystyle\mathop{\limsup}_{\nu\uparrow\infty} \mathop{\max}_{i,j \in S_{\nu_2}}\left\vert\frac{\pi_{\nu ij}}{\pi_{\nu i}\pi_{\nu j}} - 1\right\vert = \order{N_{\nu}^{-1}}, \text{ with $P_{0}-$probability $1$}$$ such that for some constant, $C_{5} > 0$, $$\displaystyle N_{\nu}\mathop{\sup}_{\nu}\mathop{\max}_{i,j \in S_{\nu_2}}\left\vert\frac{\pi_{\nu ij}}{\pi_{\nu i}\pi_{\nu j}} - 1\right\vert \leq C_{5}, \text{ for $N_{\nu}$ sufficiently large.}$$
(A6)\[fraction\]
: (Constant Sampling fraction) For some constant, $f \in(0,1)$, that we term the “sampling fraction", $$\mathop{\limsup}_{\nu}\displaystyle\biggl\vert\frac{n_{\nu}}{N_{\nu}} - f\biggl\vert = \order{1}, \text{ with $P_{0}-$probability $1$.}$$
The first three conditions are the same as @ghosal2007. They restrict the growth rate of the model space (e.g., of parameters) and require prior mass to be placed on an interval containing the true value. Condition requires the sampling design to assign a positive probability for inclusion of every unit in the population because the restriction bounds the sampling inclusion probabilities away from 0. Condition ensures that the observed sample size, $n_{\nu}$, limits to $\infty$ along with the size of the partially-observed finite population, $N_{\nu}$, such that the variation of information about the population expressed in realized samples is controlled.
@2015arXiv150707050S rely on asymptotic factorization for all pairwise inclusion probabilities. Their (A.5) condition is a conservative approach to establish a finite upper bound for the un-normalized posterior mass assigned to those models, $P$, at some minimum distance from the truth, $P_0$. They require all terms, a set of size $\order{N_{\nu}^{2}}$, to factorize with the maximum deviation term shrinking at a rate of $\order{N_{\nu}^{-1}}$, since there are $N^2$ terms divided by $N$ (inherited from an empirical process).
Although their condition guarantees the $L_1$ contraction result, it defines an overly narrow class of sampling designs under which this guaranteed result holds. As discussed in the introduction, multistage household survey designs are not members of this allowed class because the within household dependency does not attenuate for a set of pairs of size $\order{N_{\nu}}$. We replace their (A5) with , which allows up to $\order{N_{\nu}}$ pairwise terms to not factor, such that there remains a residual dependence. We show in the Appendix that the contraction result may, nevertheless, be guaranteed under this condition as each of the non-factoring terms has an $\order{1}$ bound. The implication of our condition is that we have constructed a wider class of sampling designs that includes those from @2015arXiv150707050S, in addition to the multistage cluster designs for fixed cluster sizes.
Our condition is a special case of specified for cluster designs where the number of units per cluster is bounded by a constant, which encompasses the multistage NSDUH household design from which we draw our application data set. We walk from to by constructing $S_{\nu 1}$ through a collection of clusters $(B_{\nu 1},…,B_{\nu D_{\nu}})$, where the size $|B_{\nu d}|$ is bounded from above. Sampling dependence within each cluster $B_{\nu d}$ is unrestricted, while dependence across clusters must asymptotically factor.
\[main\] Suppose conditions - hold. Then for sets $\mathcal{P}_{N_{\nu}}\subset\mathcal{P}$, constants, $K >0$, and $M$ sufficiently large, $$\begin{aligned}
\label{limit}
&\mathbb{E}_{P_{0},P_{\nu}}\Pi^{\pi}\left(P:d^{\pi}_{N_{\nu}}\left(P,P_{0}\right) \geq M\xi_{N_{\nu}} \vert {\mathbf{X}}_{1}\delta_{\nu 1},\ldots,{\mathbf{X}}_{N_{\nu}}\delta_{\nu N_{\nu}}\right) \leq\nonumber\\
&\frac{16\gamma^{2}\left[\gamma \mathbf{C_{2}} +C_{3}\right]}{\left(Kf + 1 - 2\gamma\right)^{2}N_{\nu}\xi_{N_{\nu}}^{2}} + 5\gamma\exp\left(-\frac{K n_{\nu}\xi_{N_{\nu}}^{2}}{2\gamma}\right),\end{aligned}$$ which tends to $0$ as $\left(n_{\nu}, N_{\nu}\right)\uparrow\infty$.
The proof follows exactly that in @2015arXiv150707050S where we bound the numerator (from above) and the denominator (from below) of the expectation with respect to the joint distribution of population generation and the taking of a sample of the pseudo-posterior mass placed on the set of models, $P$, at some minimum pseudo-Hellinger distance from $P_{0}$. We reformulate one of the enabling lemmas of @2015arXiv150707050S, which we present in an Appendix, where the reliance on (their) condition (A5) requiring asymptotic factoring of pairwise unit inclusion probabilities is here replaced by condition that allows for non-factorization of a subset of pairwise inclusion probabilities.
As noted in @2015arXiv150707050S, the rate of convergence is decreased for a sampling distribution, $P_{\nu}$, that expresses a large variance in unit pairwise inclusion probabilities such that $\gamma$ will be relatively larger. Samples drawn under a design that expresses a large variability in the first order sampling weights will express more dispersion in their information relative to a simple random sample of the underlying finite population. We construct $C_3 = C_5 + 1$ and $C_2 = C_4 + 1$. Under the more restrictive condition (A5) of @2015arXiv150707050S, our constant $C_4 = 0$ and thus $C_2 = 1$.
Simulation Examples {#sec:sims}
===================
We construct a population model to address our inferential interest of a binary outcome $y$ with a linear predictor $\mu$.
$$\label{pop_like}
y_{i} \mid \mu_{i} {\stackrel{\mathrm{ind}}{\sim}}Bern \left(F^{-1}_l(\mu_{i}) \right),~ i = 1,\ldots,N$$
where $F^{-1}_l$ is the quantile function (inverse cumulative function) for the logistic distribution. We let $\mu$ depend on two predictors $x_1$ and $x_2$. The variable $x_1$ represents the observed information available for analysis, whereas $x_2$ represents information available for sampling, which is either ignored or not available for analysis. The $x_1$ and $x_2$ distributions are $\mathcal{N}(0,1)$ and $\mathcal{E}(r =1/5)$ with rate $r$, where $\mathcal{N}(\cdot)$ and $\mathcal{E}(\cdot)$ represent normal and exponential distributions, respectively. The size measure used for sample selection is $\tilde{\bm{x}}_{2} = \bm{x}_{2} - \min (\bm{x}_{2}) + 1$. $$\bm{\mu} = -1.88 + 1.0 \bm{x}_{1}+ 0.5 \bm{x}_{2}$$ where the intercept was chosen such that the median of $\mu$ is approximately 0, therefore the median of $F^{-1}_l(\bm{\mu})$ is approximately 0.5.
Even though the population response $y$ was simulated with $\mu = f(x_1,x_2)$, we estimate the marginal models at the population level for $\mu = f(x_1)$. This exclusion of $x_2$ is analogous to the situation in which an analyst does not have access to all the sample design information and ensures that our sampling design instantiates informativeness (where $y$ is correlated with the selection variable, $x_{2}$, that defines inclusion probabilities). In particular, we estimate the models under each of several sample design scenarios and compare the population fitted models, $\mu = f(x_1)$, to those from the samples.
We formulate the logarithm of the sampling-weighted pseudo-likelihood for estimating $(\bm{\mu},\lambda)$ from our observed data for the $ n\leq N$ sampled units, $$\begin{aligned}
\label{pseudo_like}
\log\left[\mathop{\prod}_{i=1}^{n} p\left(y_{i}\mid \mu_{i}\right)^{w^{\ast}_{i}}\right] &= \mathop{\sum}_{i=1}^{n}w^{\ast}_{i}\log p \left( y_{i}\mid \mu_{i}\right) \nonumber\\
&= \mathop{\sum}_{i=1}^{n} w^{\ast}_{i} y_{i} \log(\theta_i) + w^{\ast}_{i} (1- y_{i}) \log(1-\theta_i),\end{aligned}$$ where $\theta_i = F^{-1}_l(\mu_{i})$ and the sampling weights, $w^{\ast}_{i}$ are normalized such that the sum of the weights equals the sample size $\mathop{\sum}_{i=1}^{n}w^{\ast}_{i} = n$.
Finally, we estimate the joint posterior distribution using Equation \[pseudo\_like\], coupled with our prior distributions assignments, using the NUTS Hamiltonian Monte Carlo algorithm implemented in Stan [@stan:2015].
Multistage Cluster Designs
--------------------------
We begin by abstracting the five-stage, geographically-indexed NSDUH sampling design [@MRB:Sampling:2014] to a simpler, three stage design of {area segment, household, individual} that we use to draw samples from a synthetic population in a manner that still generalizes to the NSDUH (and similar multistage sampling designs where the number of last stage units does not grow with overall population size). We simulate a population of N = 6000, with 200 primary sampling units (PSUs) each containing 10 households (HHs) which each contain 3 individuals with independent responses $y_i$.
For the simulation, the number of selected PSUs was varied $K \in \{10, 20, 40, 80, 160\}$, the number of HHs within each PSU was fixed at 5, and the number of selected individuals within each HH was 1. Each setting was repeated $M = 200$ times. Details for the selection at each stage follows:
1. For each PSU indexed by $k$, an aggregate size measure $X_{2,k} = \sum_{ij} x_{2,ij|k}$ was created summing over all individuals $i$ and HHs $j$ in PSU $k$. PSUs are then selected proportional to this size measure based on Brewer’s PPS algorithm [@BrewerPPS].
2. Once PSUs are selected, for each HH within the selected PSUs indexed by $j$, an aggregate size measure $X_{2,j|k} = \sum_{i} x_{2,i|jk}$ was created summing over all individuals $i$ within each HH in the selected PSUs. HHs are selected independently across PSUs. Within each PSU, HHs are selected systematically with equal probability by first sorting on $X_{2,j|k}$ and then selecting a random starting point.
3. Within each selected HH, a single person is selected with probability proportional to size $x_{2,i|jk}$.
The nested structure of the sampling induces asymptotic independence between PSU’s. Within PSUs, the systematic sampling of HHs creates a block of non-attenuating dependence between households. Likewise, the sampling of only one person within each HH creates a joint dependence $\pi_{ii'|jk} = 0$ between individuals within the same HH. Therefore, non-factorization of the second order inclusions remains within each PSU (see Figure \[fig:3stagefactor\]). Figure \[fig:bin2pred\] compares the bias and mean square error (MSE) for estimation with equal weights (black) and inverse probability weights (blue). As expected, the sampling weights remove bias and lead to convergence, since the non-factoring pairwise inclusion probablities are of $\order{N}$.
Dependent Sampling of First Stage Units
---------------------------------------
We now use the same population response model and distributions for $y$, $x_1$, and $x_2$ but consider the case of single stage sampling designs where the sample size is half the population (i.e. a partition of size $N/2$). In particular, we construct a design with second order dependence that grows $\order{N^{2}}$ and demonstrate that estimates for this design fail to converge. However, with slight modifications, the design can be altered into $\order{N}$ dependence and does demonstrate convergence, as predicted by the theory.
One simple way to create an informative design is to use the size measure $\tilde{\bm{x}}_{2}$ to sort the population. Partition the population $U$ into a “high” ($U_1$) group with the top $N/2$ and a “low” ($U_2$) group with the bottom $N/2$. This partition rule leads to an outcome space with only two possible samples of size $N/2$: $U_1$ and $U_2$. For simplicity, assume an equal probability of selection of $1/2$. Then it follows that $\pi_{i} = 1/2$, for all $i \in 1,\ldots, N$, and $\pi_{ij} = 1/2$ if $i \ne j \in U_k$, for $k = 1,2$ and 0 otherwise. In fact, all joint inclusions, from orders 2 to $N/2$, are 1/2 if all members indexed are in the same partition and 0 otherwise. These second and higher order inclusion probabilities do not factor with increasing population size $N$. Thus, the number of pairwise inclusions probabilities that do not factor ($\pi_{ij} \ne 1/4$) grows at rate $\order{N^{2}}$, violating condition .
Alternatively, we could embed the partitioning procedures within strata, where the strata are created according to rank order, have a fixed size, and the number of strata grow with population size $N$. For example grouping every 50 units into a strata, then partitioning within each. Such a modification is relatively minor, but leads to factorization for all but $\order{N}$ pairwise inclusion probabilities. This can be visualized as the diagonal blocks in the full pairwise inclusion matrix (see Figure \[fig:factorstrata\]).
For each $N \in \{100, 200, 400, 800, 1600\}$, we generate a single population and compare the relative convergence of the original dyadic partitions and the stratified versions. Figure \[fig:rankpar\] compares the bias and mean square error (MSE) of the two partitions (red and blue) compared to the average of 100 samples from the stratified version (black). It’s clear that as the population size (and sample size) grows, the bias of the two partitions does not go away (the variability is due to a single realization of the population at each size), while the overall bias and MSE of the stratified version clearly decreases with increasing N, consistent with the theory.
Application to the NSDUH {#sec:NSDUH}
========================
A simple logistic model of current (past month) smoking status by past year major depressive episode (MDE) was fit via the survey weighted psuedo-posterior as described in section \[sec:sims\] using both equal and probability-based analysis weights for adults from the 2014 NSDUH public use data set (Figure \[fig:NSDUH\]). It is reasonable to assume that equal weights lead to higher estimates of smoking, as young adults are more likely to smoke and are over-sampled. Based on the theoretical results and the simulation study presented in this paper, we have justification that the probability-based weights have removed this bias and provide consistent estimation. The large number of strata and the asymptotically independent first stage of selection creates factorization for all but $\order{N}$ pairwise inclusion probabilities, even though the clustering and the sorting of units before selection may be informative.
Conclusions {#sec:conc}
===========
This work is motivated by the discrepancy between the theory available to justify consistent estimation for survey sample designs and the practice of estimation for complex, multistage cluster designs such as the NSDUH. Previous requirements for approximate or asymptotic factorization of joint sampling probabilities exclude such designs, leaving the practitioner unable to fully justify their use. We have presented an alternative requirement that allows for unrestricted sampling dependence to persist asymptotically rather than to attenuate. For example, dependence between units within a cluster is unrestricted provided that the cluster size is bounded and dependence between clusters attenuates. This dependence can be positive (joint selection) or negative (mutual exclusion). Results are further demonstrated via a simulation study of a simplified NSDUH design. Additional simulations expand our understanding of the impact of sorting. While the direct application of these methods can lead to dependence among all units (effectively one cluster of infinite size), embedding these features within stratified or clustered designs can be justified (for subsequent estimation using marginal sampling weights) by our main results and performs well in simulation and in practice. For example, geographic units sorted along a gradient can now be fully justified for the NSDUH, because the sampling along this gradient occurs independently across a large number of strata.
With this work, the use of the sample weighted pseudo-posterior [@2015arXiv150707050S] is now available to a much wider variety of survey programs. We note that while establishing consistency is essential, understanding other properties of pseudo-posteriors such as posterior intervals, still requires more research. Furthermore, so called “Fully Bayesian” methods, which avoid a plug-in estimator for the sampling weights by jointly modelling the outcome and the sample selection process, are also being researched [@2017arXiv171000019N]. The theory uses the stricter conditions for asymptotic factorization of the sample design and could be generalized by using conditions for the sample design that are similar to those presented in this work.
Enabling Lemmas
===============
\[numerator\] Suppose conditions and hold. Then for every $\xi > \xi_{N_{\nu}}$, a constant, $K>0$, and any constant, $\delta > 0$, $$\begin{aligned}
\mathbb{E}_{P_{0},P_{\nu}}\left[\mathop{\int}_{P\in\mathcal{P}\backslash\mathcal{P}_{N_{\nu}}}\mathop{\prod}_{i=1}^{N_{\nu}}
\frac{p^{\pi}}{p_{0}^{\pi}}\left({\mathbf{X}}_{i}\delta_{\nu i}\right)d\Pi\left(P\right)\left(1-\phi_{n_{\nu}}\right)\right] \leq \Pi\left(\mathcal{P}\backslash\mathcal{P}_{N_{\nu}}\right)& \label{outside}\\
\mathbb{E}_{P_{0},P_{\nu}}\left[\mathop{\int}_{P\in\mathcal{P}_{N_{\nu}}:d^{\pi}_{N_{\nu}}\left(P,P_{0}\right)> \delta\xi}\mathop{\prod}_{i=1}^{N_{\nu}}
\frac{p^{\pi}}{p_{0}^{\pi}}\left({\mathbf{X}}_{i}\delta_{\nu i}\right)d\Pi\left(P\right)\left(1-\phi_{n_{\nu}}\right)\right] &\leq \nonumber \\
2\gamma\exp\left(\frac{-K n_{\nu}\delta^{2}\xi^{2}}{\gamma}\right).&\label{inside}\end{aligned}$$
See @2015arXiv150707050S, @2016arXiv160607488S, and @2017pair for details and modifications.
\[denominator\] For every $\xi > 0$ and measure $\Pi$ on the set, $$B = \left\{P:-P_{0}\log\left(\frac{p}{p_{0}}\right) \leq \xi^2, P_{0}\left(\log\frac{p}{p_{0}}\right)^{2} \leq \xi^{2}\right\}$$ under the conditions , , , , we have for every $C > 0 $, $C_{2} = C_{4}+1$, $C_{3} = C_{5}+1$, and $N_{\nu}$ sufficiently large, $$\label{denomresult}
\mbox{Pr}\left\{\mathop{\int}_{P\in\mathcal{P}}\displaystyle\mathop{\prod}_{i=1}^{N_{\nu}}\frac{p^{\pi}}{p_{0}^{\pi}}
\left({\mathbf{X}}_{i}\delta_{\nu i}\right)d\Pi\left(P\right)\leq \exp\left[-(1+C)N_{\nu}\xi^{2}\right]\right\}
\leq \frac{\gamma \mathbf{C_{2}}+C_{3}}{C^{2} N_{\nu}\xi^{2}},$$ where the above probability is taken with the respect to $P_{0}$ and the sampling generating distribution, $P_{\nu}$, jointly.
\[AppDenominator\] The proof follows that of @2015arXiv150707050S by bounding the probability expression on left-hand size of Equation \[denomresult\] with, $$\begin{aligned}
&\mbox{Pr}\left\{\mathbb{G}^{\pi}_{N_{\nu}}\mathop{\int}_{P\in\mathcal{P}}\log\frac{p}{p_{0}}
d\Pi\left(P\right)\leq -\sqrt{N_{\nu}}\xi^{2}C\right\}\nonumber\\
&\leq\frac{\displaystyle\mathop{\int}_{P\in\mathcal{P}}\left[\mathbb{E}_{P_{0},P_{\nu}}
\left(\mathbb{G}^{\pi}_{N_{\nu}}\log\frac{p}{p_{0}}\right)^{2}\right]d\Pi\left(P\right)}
{N_{\nu}\xi^{4}C^{2}}\label{chebyshev:e2},\end{aligned}$$ where we have used Chebyshev to achieve the right-hand bound of Equation \[chebyshev:e2\]. We now proceed to further bound the numerator in the right-hand side of Equation \[chebyshev:e2\]. @2015arXiv150707050S and @2017pair establish the following: $$\label{gbound}
\mathbb{E}_{P_{0},P_{\nu}}\left[\mathbb{G}^{\pi}_{N_{\nu}}\log\frac{p}{p_{0}}\right]^{2} \leq N_{\nu}\mathbb{E}_{P_{0},P_{\nu}}\left[\left(\mathbb{P}^{\pi}_{N_{\nu}} - \mathbb{P}_{N_{\nu}}\right)\log\frac{p}{p_{0}}\right]^{2} +~ \xi^{2}$$ We proceed to further simplify the bound in the first term on the right in Equation \[gbound\]:
$$\begin{aligned}
&N_{\nu}\mathbb{E}_{P_{0},P_{\nu}}\left[\left(\mathbb{P}^{\pi}_{N_{\nu}} - \mathbb{P}_{N_{\nu}}\right)\log\frac{p}{p_{0}}\right]^{2}\nonumber\\
&= \displaystyle\frac{1}{N_{\nu}}\mathop{\sum}_{i=j\in U_{\nu}}\mathbb{E}_{P_{0}}\left[\left(\frac{1}
{\pi_{\nu i}} - 1 \right)\left(\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\right)^{2}\right]\nonumber\\
&+ \frac{1}{N_{\nu}}\mathop{\sum}_{i=j \in U_{\nu}}\mathbb{E}_{P_{0}}\left[\left(\frac{\pi_{\nu ij}}{\pi_{\nu i}\pi_{\nu j}}-1\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{j}\right)\right]\label{firstbro}\\
&\le \displaystyle\frac{1}{N_{\nu}}\mathop{\sum}_{i=j\in U_{\nu}}\mathbb{E}_{P_{0}}\left[\left(\frac{1}
{\pi_{\nu i}} - 1 \right)\left(\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\right)^{2}\right]\nonumber\\
&+ \frac{1}{N_{\nu}}\mathop{\sum}_{i=j \in U_{\nu}} \left|\mathbb{E}_{P_{0}}\left[\left(\frac{\pi_{\nu ij}}{\pi_{\nu i}\pi_{\nu j}}-1\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{j}\right)\right]\right|\label{secondbro}\\
&= \displaystyle\frac{1}{N_{\nu}}\mathop{\sum}_{i=j\in U_{\nu}}\mathbb{E}_{P_{0}}\left[\left(\frac{1}
{\pi_{\nu i}} - 1 \right)\left(\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\right)^{2}\right]\nonumber\\
&+ \frac{1}{N_{\nu}}\mathop{\sum}_{i\ne j\in S_{\nu 1}} \left|\mathbb{E}_{P_{0}}\left[\left(\frac{\pi_{\nu ij}}{\pi_{\nu i}\pi_{\nu j}}-1\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{j}\right)\right]\right|\nonumber\\
&+ \frac{1}{N_{\nu}}\mathop{\sum}_{i \ne j\in S_{\nu 2}}\left|\mathbb{E}_{P_{0}}\left[\left(\frac{\pi_{\nu ij}}{\pi_{\nu i}\pi_{\nu j}}-1\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{j}\right)\right]\right|\label{thirdbro}\\
&\leq \left(\gamma - 1 \right)\displaystyle\frac{1}{N_{\nu}}\mathop{\sum}_{i=j\in U_{\nu}}\mathbb{E}_{P_{0}}\left[\left(\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\right)^{2}\right]\nonumber\\
&+ \max\{1,\gamma - 1\}\frac{1}{N_{\nu}}\mathop{\sum}_{S_{\nu 1}}\left|\mathbb{E}_{P_{0}}\left[\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{j}\right)\right]\right|\nonumber\\
&+ C_{5} N_{\nu}^{-1}\frac{1}{N_{\nu}}\mathop{\sum}_{S_{\nu 2}}\left|\mathbb{E}_{P_{0}}\left[\log\frac{p}{p_{0}}\left({\mathbf{X}}_{i}\right)\log\frac{p}{p_{0}}\left({\mathbf{X}}_{j}\right)\right]\right|\label{fourthbro}\\
&\leq\left(\gamma (1 +C_{4}) + C_{5}\right)\xi^{2}\label{fifthbro},\end{aligned}$$
for sufficiently large $N_{\nu}$. The first equality (\[firstbro\]) is derived from the quadratic expansion and subsequent expectation of the inclusion indicators $\delta_{\nu i},\delta_{\nu j}$ with respect to the conditional distribution of $P_{\nu}$ given and follows @2015arXiv150707050S and @2017pair. The next inequality (\[secondbro\]) is needed because the pairwise terms could be negative, so the sum is bounded by the sum of the absolute value. The two pairwise terms are equivalently partitioned by $S_{\nu 1}$ and $S_{\nu 2}$ in the next equality (\[thirdbro\]). Condition implies the following bounds: $$-1 \le \left(\frac{\pi_{\nu ij}}{\pi_{\nu i}\pi_{\nu j}}-1\right) \le
\left (\frac{1}{\pi_{\nu i}} - 1 \right) \le (\gamma - 1)$$ since $0 \le \pi_{\nu ij} \le \min\{\pi_{\nu i},\pi_{\nu j}\}$, which are used in \[fourthbro\]. The size bounds from condition and the definition of the space $B$ provide the remaining bounds (in \[fifthbro\]).
We may now bound the expectation on the right-hand size of Equation \[chebyshev:e2\], $$\begin{gathered}
\mathbb{E}_{P_{0},P_{\nu}}\left[\mathbb{G}^{\pi}_{N_{\nu}}\log\frac{p}{p_{0}}\right]^{2} \leq \left(\gamma (1 +C_{4}) + C_{5}\right)\xi^{2}+ \xi^2 \\ \leq \left(\gamma (1 +C_{4}) + C_{5} + 1\right)\xi^{2} = (\gamma C_{2} + C_{3})\xi^2,\end{gathered}$$ for $N_{\nu}$ sufficiently large, where we set $C_{2} := C_{4}+1$ and $C_{3} := C_{5}+1$. This concludes the proof.
|
---
abstract: 'We discuss historical attempts to formulate a physical hypothesis from which Turing’s thesis may be derived, and also discuss some related attempts to establish the computability of mathematical models in physics. We show that these attempts are all related to a single, unified hypothesis.'
author:
- 'Matthew P. Szudzik'
bibliography:
- 'IsTuringsThesis.bib'
date: 2 April 2012
title: 'Is Turing’s Thesis the Consequence of a More General Physical Principle?'
---
Introduction {#s:intro}
============
Alan Turing [@aT36] proposed the concept of a computer—that is, the concept of a mechanical device that can be programmed to perform any conceivable calculation—after studying the processes that humans use to perform calculations. In particular, he claimed that any function of non-negative integers which can be effectively calculated by humans is a function that can be calculated by a Turing machine. This claim, known as *Turing’s thesis*, is an empirical principle that has withstood many tests to its validity.[^1] But why does Turing’s thesis seem to be true? Can it be derived, for example, from a principle of contemporary sociology, from a principle of human biology, or perhaps from a principle of fundamental physics?
We will be concerned with the last of these questions, namely the question of whether Turing’s thesis is the consequence of a physical principle. But before discussing some of the historically important attempts to answer this question, let us introduce the following formalism. Given a physical system, define a *deterministic physical model* for the system to be
1. a set $S$ of *states*
together with
2. a set $A$ of functions from $S$ to the real numbers, and
3. for each non-negative real number $t$, a function $\varPhi_t$ from $S$ to $S$.
If the system begins in state $s$, then we understand $\varPhi_t\left(s\right)$ to be the state of the system after $t$ units of time. Furthermore, we identify each member $\alpha$ of $A$ with an *observable quantity* of the system,[^2] and we consider $\alpha\left(s\right)$ to be the *value predicted* for the observable quantity when the system is in state $s$. For example, the following is a deterministic physical model for a particle that is moving uniformly along a straight line with a velocity of $3$ meters per second.
\[m:detline\] Let $S$ be the set of all real numbers and define $\varPhi_t\left(x\right)=x+3t$ for all states $x$ in $S$, where $t$ is the time measured in seconds. The position of the particle on the line, measured in meters, is given by the function $\alpha\left(x\right)=x$.
We say that a deterministic physical model is *faithful* if and only if there is a state $s_0$ such that, for each time $t$, the actual values of the observable quantities at that time agree with the values that are predicted when the system is in state $\varPhi_t\left(s_0\right)$. Deterministic physical models are commonly studied in the theory of dynamical systems.
Now, the first notable attempt to derive Turing’s thesis from a principle of physics appears to have been made by Robert Rosen [@rR62]. Rosen hypothesized[^3] that every physical system has a faithful deterministic physical model where
1. the set $S$ is the set of all non-negative integers,
2. \[i:observable\] each $\alpha$ in $A$ is a total recursive function,[^4] and
3. when restricted to non-negative integer times $t$, $\varPhi_t\left(s\right)$ is a total recursive function of $s$ and $t$.
Note that since recursive functions are functions from non-negative integers to non-negative integers, Condition \[i:observable\] implies that the values of all observable quantities are non-negative integers. This hypothesis is justified by the fact that actual physical measurements have only finitely many digits of precision. For example, a distance measured with a meterstick is a non-negative integer multiple $m$ of the length of the smallest division on the meterstick. Therefore, without loss of generality, we can consider $m$ to be the value of the measurement. Nevertheless, Condition \[i:observable\] does not forbid time from being measured with a non-negative real number, since time is not regarded as an observable quantity in deterministic physical models. For example, it is consistent with Rosen’s hypotheses to specify that $$\varPhi_t\left(s\right)=\varPhi_{\left\lfloor t\right\rfloor}\left(s\right)$$ for all non-negative real numbers $t$ and for all states $s$, where $\left\lfloor t\right\rfloor$ denotes the largest integer less than or equal to $t$.
Now, Turing’s thesis can be derived from Rosen’s hypotheses as follows. First, note that by definition, if a function $\psi$ is *effectively calculable* then there is a physical system that can reliably be used to calculate $\psi\left(x\right)$ for every non-negative integer $x$. This means that there must be a system where the input can be observed, where the output can be observed, and where it is possible to observe that the system is finished with the calculation. In the context of deterministic physical models this means that the system has observable quantities $\alpha$, $\beta$, and $\gamma$, respectively, such that for each non-negative integer $x$
1. there exists a state $s$ and a time $u$ such that $\alpha\left(s\right)=x$ and $\gamma\left(\varPhi_t\left(s\right)\right)=1$ for all $t\geq u$, and
2. if $\alpha\left(s\right)=x$ for any state $s$ then $\beta\left(\varPhi_t\left(s\right)\right)=\psi\left(x\right)$ where $t$ is any time such that $\gamma\left(\varPhi_t\left(s\right)\right)=1$.
It then immediately follows from Rosen’s hypotheses that every effectively calculable function is recursive, whether it is calculated by a human being or by any other physical system. That is, Turing’s thesis is a consequence of Rosen’s hypotheses.
Informally, Rosen’s hypotheses can be understood as stating that the universe is discrete, deterministic, and computable. It is important then to ask whether we are living in a discrete, deterministic, computable universe. Unfortunately, according to the currently-understood laws of physics, the answer appears to be “No.” The universe, as we currently understand it, does not seem to satisfy Rosen’s hypotheses. Nevertheless, it may be possible to reformulate the currently-understood physical laws so that Rosen’s hypotheses are satisfied. This sort of reformulation was first attempted by Konrad Zuse [@kZ67; @kZ69] in the 1960’s. Since then, increasingly sophisticated attempts have been made by Edward Fredkin [@eF90] and Stephen Wolfram [@sW02]. In contrast, Roger Penrose [@rP89; @rP94] has speculated that the universe might not be computable, but efforts to find experimental evidence for this assertion have not succeeded. There is, in fact, very little that can currently be said. The question of whether Turing’s thesis is the consequence of a valid physical principle is too difficult to be answered conclusively at this time.
Physical Models
===============
Although deterministic physical models are widely used in the study of dynamical systems and have important real-world applications, they are not the only sort of model that is useful in physics. The sorts of models used in quantum electrodynamics [@rF85], for example, are necessarily non-deterministic. And it is difficult to reconcile the way that time is modeled in general relativity [@kG49] with the way that it is treated as a linear quantity external to the states in a deterministic physical model. Anyone who has attempted to reformulate the laws of physics so that Rosen’s hypotheses are satisfied has had to grapple with these obstacles, and no one has had complete success.
For this reason, we have proposed [@mS12] a more general sort of model which we simply call a *physical model*. A physical model for a system is a set $S$ of states together with a set $A$ of functions from $S$ to the real numbers. Each member $\alpha$ of $A$ is identified with an observable quantity of the system,[^5] and $\alpha\left(s\right)$ is the value predicted for that observable quantity when the system is in state $s$. Deterministic physical models are special sorts of physical models. For example, the deterministic physical model that was described in the introduction (Model \[m:detline\]) can be expressed as the following physical model.
\[m:line\] Let $S$ be the set of all triples $\left(x,x_0,t\right)$ of real numbers such that $t>0$ and $x=x_0+3t$. The position of the particle on the line, measured in meters, is given by the function $\alpha\left(x,x_0,t\right)=x$. The initial position of the particle (for example, recorded in the observer’s notebook) is given by the function $\beta\left(x,x_0,t\right)=x_0$. The time, measured in seconds, is given by the function $\gamma\left(x,x_0,t\right)=t$.
In contrast to Model \[m:detline\], time is treated as an observable quantity in this model, as is the initial position. The inclusion of these observable quantities in the model can be justified physically by noting that if a researcher were to test Model \[m:detline\] in a laboratory experiment, he would be required to measure the initial position $x_0$ of the particle and the position $x$ of the particle at some later time $t$. That is, besides the position $x$, observations of both the time and the initial position are fundamental to the system.
Physical models can also be used to describe non-deterministic systems. For example, suppose that one atom of the radioactive isotope nitrogen-$13$ is placed inside a detector at time $t=0$. We say that the detector has status $1$ at time $t$ if the decay of the isotope was detected at any earlier time, and we say that the detector has status $0$ otherwise. We use a non-negative integer to represent the history of the detector. In particular, if $b_i$ is the status of the detector at time $t=i$, then the history of the detector at time $t=n$ is $h=\sum_{i=1}^nb_i2^{n-i}$. Note that when $h$ is written as an $n$-bit binary number, the $i$th bit (counting from left to right) is $b_i$. For example, if the isotope decays sometime between $t=2$ and $t=3$, then the history of the detector at time $t=5$ is $7$ because the binary representation of $7$ is $\left(00111\right)_2$. Now, if the history of the detector is displayed on a computer screen, then the following is a physical model for the history of the detector as observed by a researcher looking at the screen.
\[m:radio\] Let $S$ be the set of all triples $\left(t,2^d-1,k\right)$ where $d$, $t$, and $k$ are non-negative integers such that $d\leq t$, $t\ne0$, and $2k\leq2^d-1$. The history of the detector is given by the function $\alpha\left(t,h,k\right)=h$, and the time, measured in units of the half-life of the isotope, is given by the function $\beta\left(t,h,k\right)=t$.
In a deterministic model such as Model \[m:line\], the state of the system at a particular time is uniquely determined from its initial state. But this is not the case for Model \[m:radio\], which is a non-deterministic physical model. For example, if we are given as an initial condition that the detector had status $0$ at time $t=1$, then there are four different states at time $t=3$ which are consistent with that initial condition.
$\left(3,\left(000\right)_2,0\right)$$\left(3,\left(001\right)_2,0\right)$$\left(3,\left(011\right)_2,0\right)$$\left(3,\left(011\right)_2,1\right)$
In addition, Model \[m:radio\] has the property that if these states are considered to be equally likely, then the corresponding probability of observing a given history at time $t=3$ matches the probability predicted by the conventional theory of radioactive decay. For example, since the history $\left(011\right)_2$ is observed in half of the four states, there is a $\frac{1}{2}$ probability that the detector’s history will be $\left(011\right)_2$ at time $t=3$ if the detector had status $0$ at time $t=1$. See [@mS12] for a more complex example of a non-deterministic physical model that involves incompatible quantum measurements.
Computable Physical Models
==========================
Define a *computable physical model* to be a physical model where $S$ is a recursive set of non-negative integers and where each $\alpha$ in $A$ is a total recursive function.[^6] Of course, deterministic physical models that satisfy Rosen’s hypotheses can all be expressed as computable physical models. Non-deterministic models, such as the model for radioactive decay (Model \[m:radio\]), can also be expressed as computable physical models. Now, we assert the following hypothesis.
The laws of physics can be expressed as a computable physical model.
To show that Turing’s thesis is a consequence of the computable universe hypothesis, first recall that if a function $\psi$ is *effectively calculable* then there is a physical system that can reliably be used to calculate $\psi\left(x\right)$ for every non-negative integer $x$. This means that in some state of the universe it must be possible to observe a record of the the function’s input, to observe the function’s output, and to observe that the calculation has finished. In the context of a physical model for the universe, this means that there are observable quantities $\alpha$, $\beta$, and $\gamma$, respectively, such that for each non-negative integer $x$
1. there exists a state $s$ such that $\alpha\left(s\right)=x$ and $\gamma\left(s\right)=1$, and
2. if $\alpha\left(s\right)=x$ and $\gamma\left(s\right)=1$ for any state $s$, then $\beta\left(s\right)=\psi\left(x\right)$.
It then immediately follows from the computable universe hypothesis that every effectively calculable function is recursive, whether it is calculated by a human being or by any other system that is governed by physical law. Hence, Turing’s thesis is a consequence of the computable universe hypothesis.
Of course, as was the case with Rosen’s hypotheses, it is not known whether the computable universe hypothesis is true. But since the computable universe hypothesis allows for non-determinism and for more complex temporal relationships, it may be somewhat easier to reformulate the currently-understood physical laws so as to satisfy this less-restrictive hypothesis.
We conclude this section by considering the following example that helps to clarify certain features of our definition of effective calculability in the context of non-deterministic computable physical models. First, let $\left\langle x,y\right\rangle$ denote a non-negative integer that encodes the pair $\left(x,y\right)$ of non-negative integers. For example, using Cantor’s pairing function, we could define $$\left\langle x,y\right\rangle=\frac{1}{2}\left(x^2+2x y+y^2+3x+y\right)\enspace.$$ Triples $\left(x,y,z\right)$ of non-negative integers can then be encoded as $\left\langle\left\langle x,y\right\rangle,z\right\rangle$. We will use $\left\langle x,y,z\right\rangle$ as an abbreviation for $\left\langle\left\langle x,y\right\rangle,z\right\rangle$. Also define $x\bmod2$ to be the rightmost bit in the binary representation of the non-negative integer $x$. Now consider the following computable physical model for the history of a ‘noisy’ detector.
\[m:noise\] Let $S$ be the set of all pairs $\left\langle t,h\right\rangle$ where $t$ and $h$ are non-negative integers such that $t\ne0$ and $h\leq2^t-1$. The time is given by the function $\alpha\left\langle t,h\right\rangle=t$, the current status of the detector is given by the function $\beta\left\langle t,h\right\rangle=h\bmod2$, a trivial observable quantity is given by the function $\gamma\left\langle t,h\right\rangle=1$, and the history of the detector is given by the function $\delta\left\langle t,h\right\rangle=h$.
Although superficially similar to Model \[m:radio\], all histories are possible in Model \[m:noise\]. That is, there are no restrictions on the status of the detector. At any point in time and regardless of the detector’s past history, the status of the detector can be $1$ or $0$. Now imagine forming a tree by taking each state $\left\langle t,h\right\rangle$ of Model \[m:noise\] as a node of the tree, and by taking the children of this node to be those states of the form $\left\langle t+1,h^\prime\right\rangle$ where the first $t$ bits of $h^\prime$ agree with the first $t$ bits of $h$. We call this the *tree of histories* for Model \[m:noise\], and every branch on this tree corresponds to an alternate sequence of histories for the detector. Moreover, we can think of the status of the detector as defining a function along each branch of the tree. For each state $s$ on the branch, the input of the function is $\alpha\left(s\right)$, the corresponding output is $\beta\left(s\right)$, and the fact that the calculation has finished is indicated by $\gamma\left(s\right)$. But since all histories are possible for this detector, every possible function $\psi$ from the positive integers to the set $\left\{0,1\right\}$ is calculated by the detector along some branch, including functions $\psi$ which are not recursive (that is, not computable by a Turing machine). Therefore, Model \[m:noise\] is an example of a computable physical model where non-recursive functions may be calculated along certain branches of the tree of histories. It is important to note, though, that according to our definition of effective calculability, these non-recursive functions are not effectively calculable, since for each state $s$ of Model \[m:noise\] there is another state $s^\prime$ such that $\alpha\left(s\right)=\alpha\left(s^\prime\right)$ and $\gamma\left(s\right)=\gamma\left(s^\prime\right)$, but $\beta\left(s\right)\ne\beta\left(s^\prime\right)$. In other words, the detector cannot reliably be used to calculate $\psi$ because the detector is behaving non-deterministically.
Continuous Models
=================
Although Turing’s thesis holds in all discrete, deterministic, computable universes, Turing himself did not believe that the universe is discrete. In particular, Turing [@aT50] stated that
> digital computers … may be classified amongst the ‘discrete state machines’. These are the machines which move by sudden jumps or clicks from one quite definite state to another. These states are sufficiently different for the possibility of confusion between them to be ignored. Strictly speaking there are no such machines. Everything really moves continuously.
But discreteness is not a prerequisite for computability. In fact, Georg Kreisel [@gK74] has hypothesized that the universe may be continuous and computable.
Before we turn to a discussion of Kreisel’s hypothesis, define a *functional physical model* for a system to be a set $D$ of finitely many real-valued functions, each of which takes $k$ real numbers as input, for some non-negative integer $k$. We identify each member $\delta$ of $D$ with an observable quantity of the system, and also identify each of the $k$ inputs with an observable quantity. The observable quantities that are identified with the inputs are said to be the *given quantities* for the model,[^7] and the observable quantities that are identified with the members of $D$ are said to be the *predicted quantities* of the model. We say that a functional physical model is *faithful* if and only if the values of the predicted quantities in the model match the values that are actually measured whenever the values of the given quantities in the model match the values that are actually measured.[^8]
For example, the deterministic physical model that was described in the introduction (Model \[m:detline\]) can be expressed as the following functional physical model.
\[m:func\] Let the position of the particle on the line, measured in meters, be given by the function $\delta\left(x_0,t\right)=x_0+3t$. The initial position of the particle is $x_0$. The time, measured in seconds, is $t$.
In this case, the position of the particle is the only predicted quantity. The particle’s initial position and the time are the given quantities.
Now, note that every integer $i$ can be encoded as a non-negative integer $\zeta\left(i\right)$, where $\zeta\left(i\right)=2i$ if $i\geq0$ and where $\zeta\left(i\right)=-2i-1$ if $i<0$. Each rational number $\frac{a}{b}$ in lowest-terms with $b>0$ can be encoded as a non-negative integer $\rho\left(\frac{a}{b}\right)$ where $$\rho\left(\frac{a}{b}\right)=\zeta\left(\left(\mathrm{sgn}\:a\right)2^{\zeta\left(a_1-b_1\right)} 3^{\zeta\left(a_2-b_2\right)} 5^{\zeta\left(a_3-b_3\right)} 7^{\zeta\left(a_4-b_4\right)} 11^{\zeta\left(a_5-b_5\right)}\cdots\right)$$ and where $$a=\left(\mathrm{sgn}\:a\right)2^{a_1}3^{a_2}5^{a_3}7^{a_4}11^{a_5}\cdots$$ and $$b=2^{b_1}3^{b_2}5^{b_3}7^{b_4}11^{b_5}\cdots$$ are the prime factorizations of $a$ and $b$, respectively. For each pair of rational numbers $q$ and $r$, define $\left(q\,;r\right)$ to be the non-negative integer $\left\langle\rho\left(q\right),\rho\left(r\right)\right\rangle$. We will use $\left(q\,;r\right)$ to represent the open interval with endpoints $q$ and $r$.
Next, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{R}$ denote the set of real numbers. Note that every real number $x$ can be represented as a nested sequence of open intervals whose intersection is $x$. We say that a function $\phi:\mathbb{N}\to\mathbb{N}$ is a *nested oracle* for $x\in\mathbb{R}$ if and only if $\phi\left(0\right)=\left(a_0\,;b_0\right)$, $\phi\left(1\right)=\left(a_1\,;b_1\right)$, $\phi\left(2\right)=\left(a_2\,;b_2\right)$, … is a sequence of nested intervals whose intersection is $x$. A function $\delta:\mathbb{R}\to\mathbb{R}$ is said to be *computable* (in the sense defined by Lacombe [@dL55a; @dL55b; @dL55c]) if and only if there is a total recursive function $\xi$ such that if $\phi$ is a nested oracle for $x\in\mathbb{R}$, then $\lambda m\left[\,\xi\left(\phi\left(m\right)\right)\right]$ is a nested oracle for $\delta(x)$. This definition can naturally be extended to functions from $\mathbb{R}^k$ to $\mathbb{R}$, for any $k\in\mathbb{N}$. Finally, we say that a functional physical model is *computable* if and only if every member of the set $D$ is computable in the sense that we have just described.
Let us now return to a discussion of Kreisel’s hypothesis. Kreisel hypothesized that every faithful functional physical model is computable. For example, Model \[m:func\] is computable because in that model the function $\delta:\mathbb{R}^2\to\mathbb{R}$ is computed by the total recursive function $\xi$ that is defined so that $$\xi\left(\left(a\,;b\right),\left(c\,;d\right)\right)=\left(a+3c\,;b+3d\right)$$ for all rational numbers $a$, $b$, $c$, and $d$. Moreover, every computable functional physical model can be expressed as a computable physical model. For example, Model \[m:func\] can be expressed as the following computable physical model.
\[m:contline\] Let $S$ be the set of all triples $\left\langle \left(a\,;b\right),\left(c\,;d\right),\left(a+3c\,;b+3d\right)\right\rangle$ where $a$, $b$, $c$, and $d$ are rational numbers such that $a<b$ and $c<d$. The initial position of the particle on the line, represented as a range of positions measured in meters, is given by the function $\alpha\left\langle \left(a\,;b\right),\left(c\,;d\right),\left(e\,;f\right)\right\rangle=\left(a\,;b\right)$. The time interval, measured in seconds, is given by the function $\beta\left\langle \left(a\,;b\right),\left(c\,;d\right),\left(e\,;f\right)\right\rangle=\left(c\,;d\right)$. The position of the particle, represented as a range of positions measured in meters, is given by the function $\gamma\left\langle \left(a\,;b\right),\left(c\,;d\right),\left(e\,;f\right)\right\rangle=\left(e\,;f\right)$.
Note that the states of Model \[m:contline\] are all of the form $$\left\langle\left(a\,;b\right),\left(c\,;d\right),\xi\left(\left(a\,;b\right),\left(c\,;d\right)\right)\right\rangle\enspace,$$ and this guarantees that if $\phi$ is a nested oracle for some initial position $x_0$ and if $\psi$ is a nested oracle for some time $t$, then there is a unique sequence of states $s_0$, $s_1$, $s_2$, … in $S$ such that $\alpha\left(s_m\right)=\phi\left(m\right)$ and $\beta\left(s_m\right)=\psi\left(m\right)$ for all $m\in\mathbb{N}$. Therefore, $$\gamma\left(s_m\right)=\xi\left(\phi\left(m\right),\psi\left(m\right)\right)$$ for all $m\in\mathbb{N}$, and $\lambda m\left[\gamma\left(s_m\right)\right]$ is a nested oracle for the position $\delta\left(x_0,t\right)$ that is predicted by Model \[m:func\]. Thus, there is a direct correspondence between the functional physical model (Model \[m:func\]) and the computable physical model (Model \[m:contline\]). See [@mS12] for more information regarding this correspondence.
In summary, the computable physical models comprise a very general class of models, capable of expressing discrete deterministic models such as those studied by Rosen, non-deterministic models such as the model for radioactive decay, and continuous models such as those studied by Kreisel. Furthermore, Turing’s thesis is a consequence of the hypothesis that the laws of physics can be expressed as a computable physical model. It is very tempting, therefore, to wonder whether this hypothesis might be true.
[^1]: Indeed, every time a software developer attempts to write a computer program to implement an explicitly-described procedure, Turing’s thesis is tested.
[^2]: To ensure that the predictions of the model are unambiguous, it is common practice in physics to define each observable quantity operationally [@pB27].
[^3]: Rosen did not fully formalize his hypotheses. The hypotheses given here are reasonable formal interpretations of Rosen’s “Hypothesis I” [@rR62].
[^4]: We will use terminology from the theory of recursive functions [@hR67] throughout this paper. It follows from Turing’s work [@aT37] that the set of all functions from non-negative integers to non-negative integers that are computable by Turing machines is identical to the set of recursive functions.
[^5]: As before, we require that each observable quantity be defined operationally.
[^6]: See the isomorphism theorems in [@mS12] for other, equivalent definitions of a computable physical model.
[^7]: To ensure that each function $\delta$ in $D$ is defined for all real number inputs, the scale of each given quantity should be adjusted so that it ranges over all real numbers. For example, if a given quantity is a temperature and if $\delta$ is undefined for temperatures below $0$ degrees Kelvin, then a logarithmic temperature scale should be used instead of the Kelvin scale.
[^8]: In practice, of course, measurement errors will prevent us from knowing these values exactly.
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The environment bias $b_\text{e}$ expresses the matter overdensity in the surroundings of massive halos. Observational campaigns have ascertained the halo bias and its mass dependence[@joh+al07; @ser+al15_bias; @dvo+al17] but efforts to detect enhancing mechanisms or secondary biases for massive halos have been inconclusive. Contamination by foreground or background groups hampers the analysis in stacked cluster subsamples[@zu+al17; @bu+wh17]. Here, we measure for the first time the environment bias of a single massive cluster by detection of the weak lensing (WL) signal. WL distorts the shape of the background galaxies. The correlated matter around the halo imprints a peculiar feature in the shear profile[@og+ta11]. No proxy is needed. Mass and concentration of the halo and environment bias can be determined by fitting the shear profile up to very large distances. Even though the measurement of the shear around a single cluster is very challenging due to high noise, the interpretation is much more direct than for stacked samples, where the noise is reduced at the price of averaging over heterogeneous or contaminated samples.
Sunyaev-Zel’dovich (SZ) selected galaxy clusters appear to be trustful tracers of the massive end of the cosmological halo mass function[@ros+al16]. We studied the environment bias in the PSZ2LenS sample[@ser+al17_psz2lens], which consists of the 35 galaxy clusters detected by the [*Planck*]{} mission[@planck_2015_XXVII] in the sky portion covered by the lensing surveys CFHTLenS (Canada France Hawaii Telescope Lensing Survey)[@hey+al12] and RCSLenS (Red Cluster Sequence Lensing Survey) [@hil+al16]. PSZ2LenS is a statistically complete and homogeneous subsample of the PSZ2[@planck_2015_XXVII] catalogue. It is approximately mass limited and the main halo properties are in excellent agreement with the $\Lambda$CDM (Cold Dark Matter with a cosmological constant $\Lambda$) scenario of structure formation [@ser+al17_psz2lens].
The WL quality data up to very large radii enables us to investigate the environment bias. Our main target is PSZ2 G099.86+58.45, the highest redshift cluster, $z_\text{cl}=0.616$, of the PSZ2LenS sample, see Fig. \[fig\_PSZ0478\_map\]. PSZ2 G099.86+58.45 is a hot cluster. The temperature of the intra-cluster medium is $T_\text{X}=8.9^{+2.8}_{-1.1}$ keV, as derived from the spectroscopic analysis of [*XMM*]{}-Newton data. The galaxy velocity dispersion is $\sigma_\text{v}=680^{+160}_{-130}\text{km~s}^{-1}$.
The cluster is located at the centre of the CFHT-Wide 3 field. The angular diameter distance to the cluster is very significant, $D_\text{d}\simeq 0.98~\text{Gpc}~h^{-1}$, where $h=H_0/(100~\text{km}~\text{s}^{-1}~\text{Mpc}^{-1})$, which is $\sim 80$% of the maximum angular diameter distance reachable in our universe. The shear can then be investigated up to very large proper projected distances from the cluster centre, which we identify with the dominant brightest cluster galaxy (BCG). However, the lens redshift is still such that we can measure the shape distortion of a significant number of faint sources in the background. This makes PSZ2 G099.86+58.45 an ideal target.
We recovered the mass distribution around the cluster with the WL analysis of the differential surface density $\Delta\Sigma_+$, see Fig. \[fig\_PSZ0478\_Delta\_Sigma\]. The shear signal was collected from more than 150000 galaxies up to $\sim 25.1~ \text{Mpc}~h^{-1}$ ($\sim 1.46$ degrees).
Shape distortions of galaxies were measured in CFHT wide-field images in the $i$ optical band[@erb+al13; @mil+al13]; photometric redshifts were estimated exploiting observations in the $u$, $g$, $r$, $i$ and $z$ bands[@hil+al12; @ben+al13]. We selected background galaxies using their colours or their photometric redshifts. The effective redshift of the selected source galaxies is $z_\text{s} \sim 0.96$. The overall level of shear systematics due to calibration errors, fitting procedure, contamination by foreground or cluster member galaxies, photometric redshift uncertainties and intrinsic alignment is at the $\sim5\%$ level.
The mass distribution as inferred from WL can be compared with the light density of the galaxies at the cluster redshift, selected if their photometric redshift is within $|z - z_\text{cl}|\le 0.06\times(1+z_\text{cl})$. Notwithstanding the poor angular resolution of the WL analysis, the map comparison tentatively suggests that matter peaks coincide with galaxies overdensities, see Fig. \[fig\_PSZ0478\_map\].
We measured the WL signal in circular annuli, see Fig. \[fig\_PSZ0478\_Delta\_Sigma\]. All the matter along the line of sight contributes to the lensing phenomenon. We can identify three main agents: (i) the main lens, i.e. the collapsed and nearly virialized cluster, which is dominant at radii $\lesssim 3~\text{Mpc}$; (ii) the correlated matter in the surroundings ($\gtrsim 3~\text{Mpc}$), comprising the satellite halos, the filamentary structure and the smoothly accreting matter[@col+al05; @eck+al15]; (iii) the uncorrelated matter of the large scale structure (LSS) which fills the line-of-sight.
The measured signal in the range $10<R~[\text{Mpc}~h^{-1}]<25.1$, where the correlated matter is the dominant term, is $\Delta \Sigma_{+,\text{obs}}=32.1\pm4.5(\text{stat.})\pm8.1(\text{LSS})\pm1.6(\text{sys.})~M_\odot~h~\text{pc}^{-2}$. The signal-to-noise ratio is $\text{SNR}\simeq 3.4$. This provides a clear and model independent detection of the cluster surroundings.
The $\Lambda$CDM paradigm makes strong predictions on clusters and surroundings in terms of mass and redshift of the main halo. The cluster can be modelled with a cuspy density profile[@nfw96] whose mass and concentration are correlated[@men+al14]. The mass profile is truncated at the splash-back radius beyond which the matter is still infalling [@die+al17]. The correlated matter is dominant beyond the splash-back radius. It can be expressed as a 2-halo term[@og+ta11], where the halo bias is a function of the peak height[@tin+al10]. The LSS contribution acts as a noise whose amplitude is fixed by the effective projected power spectrum[@sch+al98b]. It is significant on very large scales.
We reconstructed the matter distribution and compared it with $\Lambda$CDM predictions. We used two parametric modellings. Firstly, we fitted the shear profile only in the region more sensitive to the main halo ($0.1<R<3~ \text{Mpc}~h^{-1}$) with an informative prior on the mass-concentration relation[@men+al14] and environment bias $b_\text{e}$ modelled as a 2-halo term[@tin+al10]. This is the $\Lambda$CDM model, which gives mass $M_{200} \sim (8.3\pm3.2)\times 10^{14} M_\odot h^{-1}$, concentration $c_{200} =3.4.\pm 0.9$, and $b_\text{e,$\Lambda$CDM}=11.1\pm2.5$. The uncertainty on the theoretical fitting function of the halo bias is $\sim6\%$, as estimated from the simulation to simulation scatter [@tin+al10], even though simulations poorly cover the mass and redshift range around PSZ2 G099.86+58.45 and some extrapolation is needed.
The measured WL mass $M_{500} \sim (5.5\pm2.0)\times 10^{14} M_\odot h^{-1}$ is in good agreement with expectations based on multi-probe proxies: $M_{\text{X},500} \sim (5.1 \pm1.8)\times 10^{14} M_\odot h^{-1}$ based on $T_\text{X}$; $M_{\text{SZ},500} \sim (6.1 \pm0.8)\times 10^{14} M_\odot h^{-1}$ based on the integrated Compton parameter[@planck_2015_XXVII]; $M_{\sigma,500} \sim (1.2 \pm0.8)\times 10^{14} M_\odot h^{-1}$ based on the galaxy velocity dispersion. Secondly, we fitted the full profile ($0.1<R<25.1~ \text{Mpc}~h^{-1}$) with the bias as a free parameter. Now, $ b_\text{e}$ quantifies how much the total matter around the halo is overdense. We found $b_\text{e}=78\pm11$. Priors on mass and concentration of the main halo affects very marginally this bias estimate. The multi-probe analysis confirms that the mass measurement of the cluster is solid and that the bias excess cannot be explained in terms of an under-estimated halo mass.
The measured signal in the range $10<R~[\text{Mpc}~h^{-1}]<25$ is much larger than the average $\Lambda$CDM prediction, $\Delta \Sigma_{+,\Lambda\text{CDM}}=5.6\pm1.5~M_\odot~h~\text{pc}^{-2}$, hinting to two possible, not exclusive causes: very overdense correlated matter boosted by formation mechanisms or projection effects from uncorrelated matter.
To quantify the degree of discrepancy, we performed numerical simulations exploiting the Lagrangian perturbation theory, where the hierarchical formation of dark matter halos is realised from an initial density perturbation field[@mon+al13]. This method is very effective in covering the mass and redshift range we are interested in, which can be challenging for standard $N$-body simulations.
We derived the shear around 128 simulated halos in the redshift range $0.54\lesssim z \lesssim 0.71$ with average mass and redshift as PSZ2 G099.86+58.45. The signal is consistent with the analytical $\Lambda$CDM model, see Fig. \[fig\_PSZ0478\_Delta\_Sigma\]. The probability that an overdense line of sight boosts the shear at the measured level is of $\lesssim 0.5\%$, see Fig. \[fig\_PSZ0478\_histo\_DeltaSigma\_sim\]. This is also confirmed by the analysis of the shear distortions at random locations in the CFHTLS-W3 field, where the measured signal is greater than the excess $\Delta \Sigma_{+,\text{obs}} - \Delta \Sigma_{+,\Lambda\text{CDM}}$ with a probability of only $\sim0.9\%$ (Fig. \[fig\_PSZ0478\_histo\_DeltaSigma\_sim\]).
The largest simulated shears are associated to overdense regions at the lens redshift. Whereas the typical simulated system shows multiple peaks of uncorrelated matter along the line of sight, the simulation with the highest signal, $\Delta \Sigma_{+,\text{sim}}\simeq31.0~M_\odot~h~\text{pc}^{-2}$, shows a single prominent peak at the lens redshift. Simulations strongly disfavour uncorrelated noise as the only source of signal excess, which is related to the cluster surroundings.
The shear excess can be also investigated for the full PSZ2LenS sample, see Fig. \[fig\_PSZ2LenS\_bias\_PDF\]. We first stacked the shear measurements of the PSZ2LenS clusters, which lie either in the CFHTLenS or in the RCSLenS, and we then fitted the combined differential density profile[@ser+al17_psz2lens]. The effective lensing weighted redshift of PSZ2LenS is $z_\text{stack}=0.20$. As for PSZ2 G099.86+58.45, we fitted the signal with the $\Lambda$CDM modelling in the radial range $0.1<R~[\text{Mpc}~h^{-1}]<3.1$ or with $b_\text{e}$ as a free parameter for $0.1<R~[\text{Mpc}~h^{-1}]<25.1$.
The stacked analysis probes the bias at an unprecedented halo mass of $(4.3\pm0.5)\times10^{14}M_\odot h^{-1}$. Even though the stacked analysis cannot probe any assembly bias, since we combined the signal with no regard to the secondary halo properties, we can still look for bias excess in the favourable case of high SNR ($\sim3.9$) and low LSS noise, which is now greatly reduced by averaging over different sky regions. Even though the noise affecting the stacked sample is much smaller than for PSZ2 G099.86+58.45 ($1.1$ vs $9.3~M_\odot~h~\text{pc}^{-2}$), SNRs are comparable, which further stresses the extremely high signal produced by the surroundings of PSZ2 G099.86+58.45.
The $\Lambda$CDM expectation for the stacked PSZ2LenS sample ($b_\text{e,$\Lambda$CDM}=5.4\pm0.3$) is compatible with the measured environment bias of the stacked clusters ($b_\text{e}=8.1\pm2.2$) at the 11.9% level, see Fig. \[fig\_PSZ2LenS\_bias\_PDF\], which is statistically significant. The result is not driven by PSZ2 G099.86+58.45, which has a low lensing weight given the high redshift and the relatively small number of background galaxies. In fact, the results do not change significantly if we exclude PSZ2 G099.86+58.45 from the stacking or we consider the subsample at low redshift.
The SZ selection is unique in sampling the massive end of the halo mass function and unveiling new cluster properties. The environment bias for the PSZ2LenS sample is statistically consistent with $\Lambda$CDM predictions and the correlated matter around PSZ2 G099.86+58.45 lies in the extreme value tail. This is an extremely rare case. Clustering around cluster-sized halos can be amplified for low halo concentrations or high spins or a significant number of subhalos with a large average distance, even tough it is still uncertain why and if these different proxies of halo assembly history can exhibit different trends[@mao+al17]. According to the statistics of peaks, the extreme environment bias of PSZ2 G099.86+58.45 is associated to a peak of the primordial Gaussian density field with a very low value of the curvature $s=|d\langle\delta\rangle/d\log M |$, where $\delta$ is the density fluctuation and $M$ is the mass[@dal+al08]. These are the locations of larger background density and enhanced clustering for very massive halos. Formation and evolution mechanisms can be very effective in boosting the environment density. Next generation of galaxy surveys will routinely perform the lensing analysis of single halos out to very large radii that we have presented here for the first time.
Methods {#methods .unnumbered}
=======
**GL signal.** Our analysis exploited the public CFHTLenS and RCSLenS shear catalogs. WL data were processed with `THELI`[@erb+al13] and shear measurements were obtained with `lensfit`[@mil+al13]. We computed the differential projected surface density $\Delta\Sigma_+$ in circular annuli as $$\label{eq_Delta_Sigma_3}
\Delta \Sigma_+ (R) = \frac{\sum_i (w_i \Sigma_{\text{cr},i}^{-2}) \epsilon_{\text{+},i} \Sigma_{\text{cr},i}} {\sum_i ( w_i \Sigma_{\text{cr},i}^{-2}) },$$ where $\epsilon_{\text{+},i}$ is the tangential component of the ellipticity of the $i$-th source galaxy after bias correction and $w_i$ is the weight assigned to the source ellipticity. The sum runs over the galaxies included in the annulus at transverse proper distance $R$ from the centre, i.e. the position of the brightest galaxy cluster (BCG). $\Sigma_\text{cr}$ is the critical density for lensing, $$\label{eq_Delta_Sigma_2}
\Sigma_\text{cr}=\frac{c^2}{4\pi G} \frac{D_\text{s}}{D_\text{d} D_\text{ds}},$$ where $c$ is the speed of light in vacuum, $G$ is the gravitational constant, and $D_\text{d}$, $D_\text{s}$ and $D_\text{ds}$ are the angular diameter distances to the lens, to the source, and from the lens to the source, respectively. As reference cosmological framework, we assumed the concordance flat $\Lambda$CDM model with total matter density parameter $\Omega_\text{M}=0.3$, baryonic parameter $\Omega_\text{B}=0.05$, Hubble constant $H_0=70~\text{km~s}^{-1}\text{Mpc}^{-1}$, power spectrum amplitude $\sigma_8=0.8$ and initial index $n_\text{s}=1$. When $H_0$ is not specified, $h$ is the Hubble constant in units of $100~\text{km~s}^{-1}\text{Mpc}^{-1}$.
The differential surface density $\Delta\Sigma_{+}$ was measured between 0.1 and $\sim 25.12~\text{Mpc}~h^{-1}$ from the cluster centre in 24 radial circular annuli equally distanced in logarithmic space. The binning is such that there are 10 bins per decade, i.e. 10 bins between 0.1 and $1~\text{Mpc}~h^{-1}$.
The raw ellipticity components of the sources, $e_{\text{m}, 1}$ and $e_{\text{m}, 2}$, were calibrated and corrected by applying a multiplicative and an additive correction, $$\label{eq_calibration}
e_{i} = \frac{e_{\text{m}, i} - c_i}{1 + {\bar m}} \, \hspace{1cm} (i=1,2) \, .$$ Each selected source galaxy measurement was individually corrected for the estimated additive bias. The multiplicative bias $m$ mostly depends on the shape measurement technique and was identified from the simulated images[@hey+al12; @mil+al13]. In each annulus, we considered the average $\bar{m}$, which was evaluated taking into account the weight of the associated shear measurement[@vio+al15], $$\label{eq_Delta_Sigma_5}
\bar{m}(R) = \frac{\sum_i w_i \Sigma_{\text{cr},i}^{-2} m_i}{\sum_i w_i \Sigma_{\text{cr},i}^{-2}}.$$
We identified the population of background galaxies either with a colour-colour selection $g-r-i$[@ogu+al12; @cov+al14], $$\label{eq_col_1}
(g-r < 0.3) \\
\, \text{OR} \\
\, (r - i > 1.3) \\
\, \text{OR} \\
\, (r - i > g-r)
\, \text{AND} \\
\, z_\text{s} > z_\text{lens} +0.05,$$ or with criteria based on the photometric redshifts, $$\label{eq_zphot_1}
\texttt{ODDS}>0.8 \,\, \text{AND} \,\, z_\text{2.3\%} > z_\text{lens} +0.05 \,\, \text{AND} \,\, z_\text{max} < z_\text{s} < 1.2,$$ where the parameter $\texttt{ODDS}$ quantifies the relative importance of the most likely redshift[@hil+al12] and $z_\text{2.3\%}$ is the lower bound of the region including the 95.4% of the probability density distribution. $z_\text{min}=0.2$ $(0.4)$ for the CFHTLenS (RCSLenS) fields.
The SNR of the WL detection was defined in terms of the weighted differential density $\langle \Delta \Sigma_{+} \rangle_{R_\text{min} <R< R_\text{max}}$ in the corresponding radial range, $$\label{eq_SNR_1}
\text{SNR}=\frac{\langle \Delta \Sigma_+ \rangle_{R_\text{min} <R< R_\text{max}}}{\delta_+},$$ where $\delta_+$ includes statistical uncertainty and cosmic noise added in quadrature. For our analysis of the correlated matter, we considered the signal in the range $10<R~[\text{Mpc}~h^{-1}]<25.1$.
**Lens modelling.** The profile of the differential projected surface density of the lens was modelled as $$\Delta \Sigma_\text{tot}=\Delta \Sigma_\text{h}+\Delta \Sigma_\text{e} \pm \Delta \Sigma_\text{LSS} \pm \Delta \Sigma_\text{Stat}.$$ The main halo responsible for $\Delta \Sigma_\text{h}$ is a smoothly truncated cuspy density profile[@bal+al09], $$\rho_\text{BMO} = \frac{\rho_\text{s}}{(r/r_\text{s}) (1 + r/r_\text{s})^2} \left(\frac{r_\text{t}^2}{r^2 + r_\text{t}^2} \right)^2,$$ where $r_\text{s}$ is the inner scale length, $\rho_\text{s}$ is the characteristic density and $r_\text{t}$ is the truncation radius. For our analysis, we set $r_\text{t} = 3\,r_{200} $[@og+ha11; @cov+al14], and we expressed $r_\text{s}$ and $\rho_\text{s}$ in terms of mass $M_{200}$ and concentration $c_{200}$. The suffix $200$ ($500$) refers to the region wherein the main halo density is $200$ ($500$) times the cosmological critical density at the cluster redshift. At $R\gtrsim10~\text{Mpc}$, the shear fitting analysis is independent of details about main halo truncation and outskirts.
The contribution of the local environment surrounding the halo is [@og+ta11; @og+ha11] $$\Delta \Sigma_\text{e} (\theta; M, z) = b_\text{e} \frac { \bar{\rho}_\text{M}(z)}{ (1+z)^3 D_\text{d}^2} \int \frac{l d l}{2 \pi} J_2(l \theta)P_\text{m}(k_l; z),
\label{eq:gamma_t2}$$ where $\bar{\rho}_\text{M}$ is the mean cosmological matter density at the cluster redshift, $ \theta$ is the angular radius, $J_n$ is the Bessel function of $n$-th order, and $k_l \equiv l / [ (1+z) D_\text{d}(z) ]$. $ b_\text{e}$ is the environment bias with respect to the linear dark matter power spectrum $P_\text{m}(k_l; z)$ [@sh+to99; @tin+al10; @bha+al13]. We computed $P_\text{m}$ with semi-analytical approximations[@ei+hu99].
Large scale structure induces a correlated noise. The cross-correlation between two angular bins $\Delta \theta_i$ and $\Delta \theta_j$ is[@sch+al98b; @hoe03] $$\label{eq_lss_1}
\langle \Delta \Sigma_\text{LSS}(\Delta \theta_i) \Delta \Sigma_\text{LSS}(\Delta \theta_j) \rangle = 2 \pi \Sigma_\text{cr}^2 \int_0^{\infty} P_k(l)g(l, \Delta \theta_i) g(l, \Delta \theta_j) \ dl \ ,$$ where $P_k(l)$ is the effective projected power spectrum of lensing. The effects of non-linear evolution on the power spectrum were accounted for with standard methods[@smi+al03]. The function $g$ is the filter. In an angular bin $\theta_1 < \Delta \theta< \theta_2$, $$g=\frac{1}{\pi(\theta_1^2 -\theta_2^2)l} \left[ \frac{2}{l} \left( J_0(l \theta_2) -J_0(l \theta_1) \right) +\theta_2 J_1(l \theta_2) -\theta_1 J_1(l \theta_1) \right].$$
**Inference.** In a predetermined cosmological model, the lensing system is characterized by three parameters, $\textbf{p} = (M_{200},c_{200},b_\text{e})$, which we measured with a standard Bayesian analysis[@ser+al15_cM]. The posterior probability density function of the parameters given the data $\{{\Delta\Sigma_{+}} \}$ is $$p(\textbf{p}| \{{\Delta\Sigma_{+}} \}) \propto {\cal L}(\textbf{p}) p_\text{prior}(\textbf{p}),$$ where $\cal L$ is the likelihood and $p_\text{prior}$ represents a prior.
The likelihood can be expressed as ${\cal L}\propto \exp (-\chi^2)$, where the $\chi^2$ is written as $$\chi^2 = \sum_{i,j} \left[ \Delta \Sigma_{+,i} - \Delta \Sigma_i (\textbf{p}) \right]^{-1} C_{ij}^{-1} \left[ \Delta \Sigma_{+,j} - \Delta \Sigma_j (\textbf{p}) \right];$$ the sum extends over the radial annuli and the effective radius $R_i$ of the $i$-th bin is estimated as a shear-weighted radius[@ser+al17_psz2lens]; $\Delta\Sigma_{+}(R_i)$ is the measured differential surface density and $ \Delta \Sigma (\textbf{p}) = \Delta \Sigma_\text{h}+\Delta \Sigma_\text{e}$.
$\Delta \Sigma_\text{LSS}$ and $\Delta \Sigma_\text{Stat}$ are treated as uncertainties. The total uncertainty covariance matrix is $$\label{eq_inf_2}
C= C^\text{stat}+ C^\text{LSS},$$ where $C^\text{stat}$ accounts for the uncorrelated statistical uncertainties in the measured shear whereas $C^\text{LSS}_{i,j} = \langle \Delta \Sigma_\text{LSS}(\Delta \theta_i) \Delta \Sigma_\text{LSS}(\Delta \theta_j) \rangle$ is due to LSS, see equation (\[eq\_lss\_1\]).
As mass prior, we considered a uniform probability distribution in the ranges $0.05 \le M_{200}/(10^{14}h^{-1}M_\odot) \le 100$, with the distributions being null otherwise.
For the concentration, we considered a lognormal distribution in the range $1<c_{200}<10$, with median value[@men+al14], $$c_{200}= A \left( \frac{1.34}{1+z}\right)^B \left( \frac{M_{200}}{8\times 10^{14}h^{-1}M_\odot}\right)^C,$$ where $A=3.757$, $B=0.288$, and $C=-0.058$. The scatter of the mass-concentration relation is $0.25$ in natural logarithms.
The prior on the bias is either a Dirac $\delta$ function of the peak height $\nu$, $b_\text{e}=b_\text{h}[\nu(M_{200},z)]$[@tin+al10] for the $\Lambda$CDM model or an uniform distribution in the range $0.02<b_\text{e} <200$.
**WL stacking.** We stacked the lensing measurements of the PSZ2LenS clusters following a standardized approach[@joh+al07; @man+al08; @cov+al14; @ser+al17_psz2lens; @ser+al15_bias]. The lensing signals of multiple clusters were combined in physical proper length units. The weight factor is mass-independent and the effective mass and concentration of the stacked clusters is unbiased [@oka+al13; @ume+al14]. Clusters were centred on the respective BCGs. We fitted a single profile to the stacked signal to determine the ensemble properties[@ser+al17_psz2lens].
**WL systematics.** Systematic uncertainties on the shear signal are listed in table \[tab\_systematics\_psz2lens\]. Errors not accounted for in equation (\[eq\_inf\_2\]) can be quantified by an analysis of the full PSZ2LenS sample[@ser+al17_psz2lens]. The main contributors to the error budget are the calibration uncertainty of the multiplicative shear bias, the photo-$z$ accuracy and precision, and the selection of the source galaxies.
The multiplicative bias is well controlled, but a calibration uncertainty in the shape measurements can persist at the level of a few per cents. By detailed comparison of separate shape catalogues[@jar+al16], the systematic uncertainty can be estimated in $\delta m \sim 0.03$.
Cluster members or foreground galaxies can dilute the lensing signal. Our conservative selection criteria based on either photometric redshifts or colour-colour cuts suffer by a $\lesssim 2\%$ contamination.
Miscentring can underestimate the shear signal at small scales and affect the concentration measurement[@joh+al07]. The effect is however negligible at large scales $\gtrsim 10~\text{Mpc}$.
Photometric redshift systematics can impact weak lensing analyses by biasing the estimation of the surface critical density. As source redshifts, we considered the peak of the probability density, as applicable to well behaved and single peaked distributions. The systematic error associated to either a bias or a scatter in the photo-$z$ estimates was estimated for PSZ2LenS clusters with simulations[@ser+al17_psz2lens].
Improper halo modelling can affect the mass and concentration estimate at a few percents[@ser+al16_einasto]. However, as far as the truncation of the main halo is modelled, the effect on $b_\text{e}$ is negligible.
The role of cluster projection is marginal too. Two clusters that fall along the same line of sight may be blended by the SZ cluster finder into a single, larger cluster. Whereas the Compton parameters add approximately linearly, the lensing amplitude $\Delta \Sigma_+$ is a differential measurement and the estimated $b_\text{e}$ of the blended system can be well below the sum of the two the aligned halos. However, the chance to have two or more [*Planck*]{} clusters aligned is $\sim 5\%$. At $z=0.616$, the systematic error is then negligible ($< 1\%$).
Intrinsic galaxy alignment of physically nearby galaxies (II) can contaminate the signal. Furthermore, background galaxies experience a shear caused by the foreground tidal gravitational field. If the foreground galaxy has an intrinsic ellipticity that is linearly correlated with this field, shape and shear are correlated (GI). In the intrinsic alignment model[@br+ki07; @hey+al13], the power spectra of intrinsic alignment II and GI are proportional to the matter power spectrum, $$P_\text{II} = F_z^2 P_\delta \ , \ \ P_\text{GI} = F_z^2 P_\delta \ ,$$ with $$F_z = - C_1 \rho_\text{Cr}(z=0) \frac{\Omega_\text{M}}{D(z)} ,$$ where $D(z)$ is the linear growth factor normalized to unity today and $C_1 = 5 \times 10^{-14} h^{-2}M_\odot^{-1}\text{Mpc}^3$. At $z=0.616$, the combined systematic error from II and GI is then negligible ($< 1\%$).
The total level of systematic uncertainty is $\sim5\%$.
[l r@[$\,\pm\,$]{}l r r@[$\,\pm\,$]{}l ]{} & & SNR &\
reference & 32. & 9. & 3.5 & 78. & 11.\
\
WL pass & 24. & 10. & 2.4 & 58. & 14.\
$c_{2}=0$ & 32. & 9. & 3.5 & 78. & 11.\
\
NW & 55. & 12. & 4.5 & 117. & 22.\
NE & 3. & 12. & 0.3 & 20. & 14.\
SE & 36. & 12. & 3.1 & 80. & 20.\
SW & 33. & 12. & 2.7 & 87. & 22.\
\
$gri$ & 31. & 9. & 3.3 & 73. & 11.\
$z_\text{phot}$ & 31. & 12. & 2.5 & 71. & 22.\
\
bias & 32. & 11. & 2.9 & 77. & 15.\
scatter & 38. & 11. & 3.4 & 82. & 16.\
\
SZ centred & 32. & 9. & 3.4 & 81. & 11.\
$R>0.5~\text{Mpc}~h^{-1}$ & 32. & 9. & 3.5 & 77. & 11.\
\
WMAP9 & 32. & 9. & 3.5 & 73. & 10.\
NL-$P_\delta$ & 32. & 9. & 3.5 & 71. & 9.\
**WL stress tests.** We checked for potential residual sources of erros in the WL analysis of PSZ2 G099.86+58.45. Not properly corrected systematics can affect the cross component of the shear signal. We verified that it is consistent with zero as expected, see Fig. \[fig\_PSZ0478\_Delta\_Sigma\_X\]. The $p$-value of the null hypothesis is 0.08.
We then repeated the analysis of the tangential component under a series of assumptions to check whether the systematic level is sufficiently smaller than the statistical noise, see table \[tab\_systematics\_0478\].
The cluster catalogue and the shape measurements in our analysis are taken from completely different data sets. The distribution of lenses is then uncorrelated with residual systematics in the shape measurements[@miy+al15]. However, significant residual errors can hamper the shape measurements if the PSF (point spread function) is very anisotropic. We then considered only fields that passed the CFHTLenS weak lensing selection (WL pass), i.e. fields with a low level of PSF anisotropy contamination as estimated from the analysis of the star-galaxy cross correlation[@hey+al12; @erb+al13]. The result is in very good agreement with the reference case, as also confirmed by the analysis performed disregarding the additive bias correction, i.e. by putting $c_2=0$.
To check if the excess signal can be associated to a single structure nearby or in projection, we measured the shear signal in sectors. The signal in the North-West (NW), North-East (NE), South-East (SE) and South-West (SW) quadrants is compatible with the signal in the full field of view, whereas the bias exceeds the average $\Lambda$CDM prediction in each sector. This confirms that the excess is not related to foreground or background massive halos.
The selection of background galaxies was checked by comparing results obtained using either the color-color, see equation (\[eq\_col\_1\]), or the photo-$z$ method, see equation (\[eq\_zphot\_1\]).
The effects of centering or cluster member dilution were checked by considering the SZ centroid as lens center or excising the inner region at $R <0.5~\text{Mpc}~h^{-1}$.
The extent of errors affecting the photo-$z$ estimates was checked either adding a positive bias $+0.01(1+z)$ to the peak of the redshift distributions of the galaxies in the field or randomly scattering the peaks with a Gaussian distribution with standard deviation $\sigma_z=0.04 (1+z)$.
Variations in the shear signal due to the cosmological model are negligible too, as checked by considering the cosmological parameters from the nine-year Wilkinson Microwave Anisotropy Probe (WMAP9) observations[@hin+al13].
Halo bias is usually defined with respect to the liner power spectrum. However, even considering non-linearities, the estimated bias is still much larger than the average $\Lambda$CDM prediction (NL-$P_\delta$).
**Random pointings.** General features of the large scale structure can be studied by extracting the signal around random points with the same procedure used for the cluster analysis. We measured the differential density associated to 1000 random positions in the CFHT-W3 field at redshift $z=0.616$ and we stacked the signals. Both the tangential and the cross-component of the shear are consistent with a null signal, see Fig. \[fig\_PSZ0478\_Delta\_Sigma\_random\]. The $p$-value of the null hypothesis is 0.30 (0.12) for the tangential (cross) component. This further confirms that the main systematics have been eliminated and that the signal excess around PSZ2 G099.86+58.45 is significant, see Fig. \[fig\_PSZ0478\_histo\_DeltaSigma\_sim\].
Residual systematics can affect the stacking analysis due to incomplete annuli for clusters near the border of the field of view or to partially overlapping regions for nearby clusters. We computed the stacked signal for 100 PSZ2LenS-like catalogs of 35 random sources reproducing the input redshift distribution and the field locations. Both the tangential and the cross component of the stacked shear are very well consistent with a null signal, see Fig. \[fig\_PSZ2LenS\_Delta\_Sigma\_random\]. The $p$-value of the null hypothesis is 0.49 (0.25) for the tangential (cross) component.
**Numerical simulations of the WL signal.** We generated the WL signal of cluster and large scale structure using a sample of self-consistent halo model simulations. We produced a set of halo catalogues within past-light cone simulations up to $z=1$ using `Pinocchio`[@mon+al13]. `Pinocchio` is a fast code to generate catalogues of cosmological dark matter halos starting from an initial power spectrum and perturbing it using the Lagrangian perturbation theory (LPT) model. For this work we performed the `Pinocchio` simulations using the 3LPT approximation[@mun+al17]. The large scale matter density distribution in halos by `Pinocchio` accurately reproduces the results of $N$-body simulations.
Out of 512 light-cone realisations, we extracted a sample of 128 halos with mass and redshift similar to PSZ2 G099.86+58.45. We constructed the effective convergence map of the full light-cone of the main cluster plus correlated and uncorrelated systems up to redshift $z=1$ (Fig. \[fig\_map\_sim\]), using the `MOKA`[@gio+al12a] and the `WL-MOKA`[@gio+al17] tools. In particular, we constructed the effective convergence map of the cluster using `MOKA`, which generates triaxial systems populated with dark matter substructures mimicking halos from numerical simulations. We located a BCG, which was modelled using a Jaffe profile, in the centre of the cluster. The halo dark matter distribution is adiabatically contracted as consequence. Correlated matter and uncorrelated LSS are modelled as isolated NFW[Navarro, Frenk & White, @nfw96] halos with a mass-concentration relation consistent with field halos[@men+al14]. The aperture of our field of view is 3 degrees by side; by construction our light-cones are pyramids where the observer is located at the vertex and the base is at a fixed source redshift, $z_\text{s}=1$. We computed the shear field from the convergence maps using fast Fourier methods.
We finally measured the reduced shear profile around the cluster centre assuming a source density of 32 galaxies per arcminute$^2$. The very large background density of sources make it sure that the signal measured in the simulated systems is due to real features in the matter distribution. It is not due to measurement uncertainties.
Alternatively to the shear signal, the cluster environment was studied by analyzing the mass distribution as a function of redshift. We measured the total matter (except for the central cluster) collapsed in halos above the minimum threshold of $7\times10^{11}M_\odot/h$. The redshift slices were $\Delta z= 0.05$ thick and we considered only halos in limited angular radial apertures ranging from 0.15 to 1.5 degree. In Fig. \[fig\_PSZ0478\_Mz\_sim\], we plot the result for an aperture of 1 degree. The mass was rescaled by the lensing distance kernel $D_\text{lens}=D_\text{ds}/D_{s}$.
**X-ray analysis.** The X-ray analysis was performed on archival [*XMM*]{}-Newton data observed on November 8th, 2013. We applied the standard calibration in order to obtain the event lists for the EPIC detector, using the `cifbuild`, `odfingest`, and `emchain` packages [@sno+al08]. Background sources were excluded with the `cheese` tool. We preliminarily applied a standard filtering with the `mos-filter` and `pn-filter` package for the MOS and PN detector, respectively, in order to check the contamination by soft-proton background. The high number of CCDs (three) in the anomalous mode for the MOS1 detector led us to consider only the MOS2 and the PN detectors for our analysis. The particle background model for our starting images and spectra was produced with the `mos_back` and `pn_back` packages.
We then selected the time intervals less contaminated by soft-proton background. Using images in the soft and hard [*XMM*]{} bands, we identified and removed extra X-ray sources located in the region of interest. Finally, we obtained the spectral files for the source, the background, and the instrumental responses.
The spectral analysis was performed with `XSPEC`[@anr96]. We considered an absorbed `APEC` thermal model for the cluster component, with metal abundance fixed at $Z=0.3 Z_{\odot}$.
We took into account different background sources: i) an unabsorbed thermal component representing the local hot bubble[@ku+sn08]; ii) an absorbed thermal component which models the intergalactic medium and the cool halo[@ku+sn08]; and iii) an absorbed power-law with spectral index $\alpha = 1.46$ representing the unresolved background of cosmological sources[@tak+al11]. In addition, we included emission lines rising from the solar wind charge exchange at 0.56 and 0.65 keV. We finally included three Gaussian models in order to consider bright fluorescent lines at 1.49, 1.75 and 8 keV, due to the K$\alpha$ of the Al, Si and Cu, respectively.
The spectra were finally fitted in the range \[0.4–7.2\] keV. The X-ray temperature was converted in mass exploiting calibrated scaling relations[@vik+al09].
**Optical spectroscopy.** The spectroscopic redshift analysis was performed under an International Time Project (ITP13-08) from August 2012 to July 2013 [@planck_int_XXXVI]. We preliminarily calculated the photometric redshift of the cluster using archival Sloan Digital Sky Survey (SDSS) DR12 data [@ala+al15]. We identified likely cluster members showing coherent colours in agreement with $z_\text{phot}=0.63\pm 0.03$.
In order to confirm the cluster and obtain an estimate of the galaxy velocity dispersion, we performed spectroscopic observations using the OSIRIS spectrograph of the 10.4m GTC telescope, at Roque de los Muchachos Observatory (ORM) in Canary Island, during March 2014. We obtained spectroscopic redshift for 8 galaxy members by setting the long-slit in two position angles. The exposure time was $3~\text{ks}$ for each position. The full wavelength range, 4000-9000 Å, was covered with a resolution $R\sim500$.
The spectroscopic data reduction was performed using standard `IRAF` tasks[@tod86]; radial velocities were obtained using `XCSAO`, i.e. the cross-correlation technique [@ton79] implemented in the `IRAF` task `RVSAO`, with six spectrum templates of different galaxy morphologies: E, S0, Sa, Sb, Sc, and Irr [@ken92]. We measured radial velocities for 8 galaxy cluster members, including the BCG at $\text{RA}=213.696611\deg$, $\text{DEC}=54.784321\deg$ (J2000) and $z_\text{BCG}=0.6139\pm0.0002$. In addition, we also considered 4 spectroscopic redshifts from SDSS DR12. All galaxy members are placed within $2.5~\text{Mpc}$ from the cluster centre and show velocities within $\pm 2500~\text{km~s}^{-1}$ from the BCG. The full spectroscopic dataset reveals that PSZ2 G099.86+58.45 is at $z_\text{spec}=0.616\pm0.001$. The galaxy velocity dispersion $\sigma_\text{v}=680^{+160}_{-130}~\text{km~s}^{-1}$ was estimated using the gapper scale estimator [@bee+al90]. This estimate can be used as a mass proxy through a calibrated scaling relation [@evr+al08].
**Data availability.** The data that support the findings of this study are available at <http://pico.oabo.inaf.it/~sereno/> or from the corresponding author upon reasonable request. The WL data were obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/IRFU, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. The CFHTLenS and RCSLenS catalogues including photometry and lensing shape information are publicly available at <http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/community/CFHTLens/query.html> and <http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/community/rcslens/query.html>, respectively.
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Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank José Alberto Rubiño Martín for coordinating the spectroscopic campaign and Luca D’Avino for suggestions on the rendering of Fig. \[fig\_PSZ0478\_map\]. SE and MS acknowledge financial support from the contracts ASI-INAF I/009/10/0, NARO15 ASI-INAF I/037/12/0, ASI 2015-046-R.0, and ASI-INAF n.2017-14-H.0. CG acknowledges support from the Italian Ministry for Education, University, and Research (MIUR) through the SIR individual grant SIMCODE, project number RBSI14P4IH, and the Italian Ministry of Foreign affairs and International Cooperation, Directorate General for Country Promotion for Country Promotion. LI acknowledges support from the Spanish research project AYA 2014-58381-P. LM thanks the support from the grants ASI n.I/023/12/0 and PRIN MIUR 2015. AF, AS, RB acknowledge financial support from the Spanish Ministry of Economy and Competitiveness (MINECO) under the AYA2014-60438-P, the ESP2013-48362-C2-1-P and the 2011 Severo Ochoa Program MINECO SEV-2011-0187 projects. This article includes observations made with the Gran Telescopio Canarias (GTC) operated by Instituto de Astrofísica de Canarias (IAC) with telescope time awarded by the CCI International Time Programme at the Canary Islands observatories (program ITP13-8). The simulations were run on the Marconi supercomputer at Cineca thanks to the projects IsC10\_MOKAlen3 and IsC49\_ClBra01.
Author contributions {#author-contributions .unnumbered}
====================
All authors contributed to the interpretation and presentation of the results. MS: lead author; project concept, planning, and design; writing; lensing, statistical, and cosmological analyses. CG: numerical simulations. LI: X-ray analysis. FM, AV: cosmological analysis. SE, LM: planning and interpretation. GC: cluster sample selection. AF, AS, RB: galaxy kinematics.
Additional information {#additional-information .unnumbered}
======================
[**Correspondence and requests for materials**]{} should be addressed to MS at <[email protected]>.\
Competing interests {#competing-interests .unnumbered}
===================
The authors declare no competing financial interests.\
|
---
abstract: 'There are two formulas for charged lepton mass relation: One is a formula (formula A) which was proposed based on a U(3) family model on 1982. The formula A will be satisfied only masses switched off all interactions except for U(3) family interactions. Other one (formula B) is an empirical formula which we have recognized after a report of the precise measurement of tau lepton mass, 1992. The formula B is excellently satisfied by pole masses of the charged leptons. However, this excellent agreement may be an accidental coincidence. Nevertheless, 2009, Sumino has paid attention to the formula B. He has proposed a family gauge boson model and thereby he has tried to understand why the formula B is so well satisfied with pole masses. In this talk, the following views are given: (i) What direction of flavor physics research is suggested by the formula A; (ii) How the Sumino model is misunderstood by people and what we should learn from his model; (iii) What is strategy of my recent work, U(3)$\times$U(3)$''$ model.'
---
[**Sumino Model and My Personal View**]{}[^1].
[**Yoshio Koide**]{}
[*Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan*]{}\
[*E-mail address: [email protected]*]{}
[**1 Two formulas for charged lepton masses**]{}
Prior to discussing the Sumino model [@Sumino_PLB09], let us review a charged lepton mass relation, We know two formulas for the charged lepton masses. One is a formula (formula A) which was proposed based on a U(3) family model on 1982 [@K-mass_82]: $$K(m_{ei} ) \equiv \frac{m_e + m_\mu +m_\tau}{\left(\sqrt{m_e}+
\sqrt{m_\tau} +\sqrt{m_\tau}\right)^2} = \frac{2}{3} .
\eqno(1)$$ The formula A will be satisfied only masses which are given in the world switched off all interactions except for the U(3) family interactions. Other one (formula B) is an empirical formula which we have recognized since precise observation of tau lepton mass [@tau-mass_92], 1992: $$K(m_{ei}^{pole}) = \frac{2}{3}
\times(0.999989 \pm 0.000014) .
\eqno(2)$$ The formula B is excellently satisfied with pole masses of the charged leptons. However, this excellent agreement may be an accidental coincidence.
I regret that some people simple-honestly tried to search mathematical quantities which leads to the form $K=2/3$. They did not understand that $m_{ei}$ in the formula B are pole masses, and besides, that the mass spectra cannot discuss independently of the flavor mixing. Most of their attempts could not left any physical result. Their attempts are nothing but playing of a mathematical puzzle, not physics.
Independently of whether the formula A can give well numerical agreement or not, if we accept the formula A, we will also accept the following points of view:
\(i) We know the quark mixing and neutrino mixing. Therefore, the formula A holds only in the charged lepton sector, so that a similar relation never hold in other sectors (up-quark, down-quark and neutrino sectors). In other words, we should discuss flavor physics on the diagonal bases of the charged lepton mass matrix $M_{e}$.
\(ii) Masses and mixings should be investigated based on $M_e^{1/2}$, not $M_e$.
\(iii) The observed hierarchical mass spectra in quarks and leptons suggest that those cannot be understood from a conventional symmetry approach (symmetry + a small breaking), and it has to be understood form vacuum expectation values (VEVs) of scalars with Higgs-like mechanism [@YK_MPLA90; @Yukawaon].
\(iv) If we put $m_e=0$ in the formula A, we obtain unwelcome ratio of $m_\mu/m_\tau=1/(2+\sqrt{3})$. This suggests that we have to seek for mass matrix model in which the electron mass should be given by a non-zero value from the beginning even if it is very small. The mass spectrum $(m_e, m_\mu, m_\tau)$ has to be understood simultaneously, that is, without considering a mass generation model with two or three steps.
[**2 Impact of the Sumino model**]{}
However, against such my personal view given in the previous section, 2009, Sumino [@Sumino_PLB09] has paid attention to the formula B. He has proposed a family gauge boson model and thereby, he has tried to understand why the formula B is so well satisfied with pole masses.
The formula A is invariant under a transformation $$m_{ei} \ \rightarrow \ m_{ei}(1 + \varepsilon_0 ) ,
\eqno(3)$$ where $\varepsilon_0$ is a constant which is independent of the family-number. The deviation from the formula A due to QED correction comes from $\log m_{ei}$: $$\delta m_{ei} = m_{ei} \left( 1 + c_1^{QED} \log m_{ei} +
c_0^{QED} \right) ,
\eqno(4)$$ at the level of the one-loop correction [@Arason_PRD92]. Therefore, Sumino has assumed an existence of family gauge bosons (FGBs) $A_i^{ j}$ with the masses $M_{ij}^2 =k (m_{ei} + m_{ej})$ and thereby, he has proposed a cancellation mechanism between $\log m_{ei} $ in the QED correction and $\log M_{ii}$ in the FGB one-loop correction: $$\delta m_{ei} = m_{e_i} \left( c_1^{QED} \log m_{ei} +
c_1^{FGB} \log M_{ii} + const \right) = m_{ei} \times const.
\eqno(5)$$
His model is based on U(3)$\times$O(3) symmetry. In his model, in order to obtain a minus sign for the cancellation, the quarks and leptons $f$ are assigned to $(f_L, f_R)=({\bf 3}, {\bf 3}^*)$ of U(3), so that the model is not anomaly free. Besides, effective interactions with $\Delta N_{fam} = 2$ ($N_{fam}$ is a family number) appears. Therefore, in order to remove those shortcomings, an extended Sumino model has been proposed based on U(3)$\times$U(3)$'$ symmetry and with an inverted mass hierarchy of FGBs [@K-Y_PLB12]. However, the purpose of this talk is not to review those details.
The big objection is that there are many diagrams which we should take into consideration, so that the Sumino cancellation mechanism cannot work effectively. However, it is misunderstanding for the Sumino mechanism. Sumino has already taken those effects into consideration. The Sumino cancellation mechanism does not mean complete cancellation, but it means practical cancellation at a level of the present experimental accuracy. In fact, Sumino says that if accuracy in the present tau lepton mass measurement can be improved to one order, the deviation from the formula B will be observed. Also, he has said that the upper bound in which the cancellation mechanism is effective is $10^3$-$10^4$ TeV. We hope that the soon coming tau mass measurement will verify Sumino’s conjecture.
In his model, the masses $M_{ij}$ are related to the charged lepton masses $m_{ei}$, and the family gauge coupling constant $g_F$ is related to QED gauge coupling constant $e$. Therefore, the FGB model has highly predictability.
The most notable point of Sumino FGB model is that there is a upper limit of the FGB mass scale, which comes from applicability of the Sumino mechanism. Therefore, when Sumino FGBs cannot be discovered at the expected scale, we cannot excuse the undiscovered fact by extending the scale to one order. In such a case, we have to abandon the Sumino model.
Even apart from the Sumino cancellation mechanism, his FGB model has many notable characteristics. In his model, the FGB masses $M_{ij}$ and the charged lepton masses $m_{ei}$ are generated by the same scalar $\Phi =({\bf 3}, {\bf 3})$ of U(3)$\times$O(3), so that, when the charged lepton mass matrix $M_e$ is diagonal, the FGB mass matrix is also diagonal. Therefore, family-number violation does not occur in the lepton sector. Family-number violation appears only in the quark sector only via quark mixing. Therefore, in the limit of zero quark mixing, family-number violation disappears in the quark sector, too. Thus, the Sumino family FGB model offers us FGBs with considerably low scale without constraining the conventional view from the observed $K^0$-$\bar{K}^0$ mixing and so on [@YK_PLB14]. Now we may expect observations of FGBs in terrestrial experiments. We will have fruitful new physics related to Sumino FGB model.
[**3 Strategy of the U(3)$\times$U(3)$'$ model**]{}
Stimulating by the Sumino model, I have recently investigated a unified description of quarks and leptons based on U(3)$\times$U(3)$'$ symmetry [@K-N_PRD15R; @K-N_MPLA16]. Here, quarks and leptons are assigned to $({\bf 3},{\bf 1})$ of U(3)$\times$U(3)$'$. Nevertheless, we need additional symmetry U(3)$'$ with considerably higher scale. Why? The reason will see an example in the following mass matrix model (although our interest is not only in masses and mixing).
In order to give a review of the U(3)$\times$U(3)$'$ model concretely, let us take a mass matrix model based on the U(3)$\times$U(3)$'$ symmetry. In this model, heavy fermions $F_\alpha$ ($\alpha=1,2,3$) are introduced in addition to quarks and leptons $f_i$ ($i=1,2,3$). $F_\alpha$ and $f_i$ belong to $({\bf 1}, {\bf 3})$ and $( {\bf 3}, {\bf 1})$ of U(3)$\times$U(3)$'$, respectively. We consider a seesaw-like mass matrix: $$(\bar{f}_L^i \ \ \bar{F}_L^\alpha )
\left(
\begin{array}{cc}
(0)_i^{\ j} & (\Phi_f)_i^{\ \beta} \\
(\bar{\Phi}_{f})_\alpha^{\ j} & -(S_f)_\alpha^{\ \beta}
\end{array} \right)
\left(
\begin{array}{c}
f_{Rj} \\
F_{R\beta}
\end{array}
\right) .
\eqno(6)$$ Since we consider that U(3)$'$ is broken into a discrete symmetry S$_3$, a VEV form of $\hat{S}_f$, in general, takes a form (unit matrix + democratic matrix): $$\langle \hat{S}_f \rangle = v_{S} ({\bf 1} + b_f X_3) ,
\eqno(7)$$ where ${\bf 1}$ and $X_3$ are defined as $${\bf 1} = \left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) , \ \ \ \ \
X_3 = \frac{1}{3} \left(
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array} \right) ,
\eqno(8)$$ and $b_f$ are complex parameters. On the other hand, we consider that U(3) is broken by VEVs of $\Phi_f$ with $({\bf 3}, {\bf 3}^*)$ of U(3)$\times$U(3)$'$, not by $({\bf R}, {\bf 1})$ of U(3)$\times$U(3)$'$. Here, the VEV forms are diagonal and given by $$\langle \Phi_f \rangle = v_\Phi\, {\rm diag}( z_1 e^{i\phi^f_1},
z_2 e^{i\phi^f_2}, z_3 e^{i\phi^f_3}) .
\eqno(9)$$ Since we consider $|\langle\hat{S}_f\rangle | \gg |\langle \Phi_f
\rangle |$, we obtain a seesaw-like Dirac mass matrix for $f$ [@K-F_ZPC96] $$(\hat{M}_f)_i^{\ j} = \langle\Phi_f\rangle_i^{\ \alpha}
\langle \hat{S}_f^{-1} \rangle_\alpha^{\ \beta}
\langle \bar{\Phi}_f\rangle_\beta^{\ j} .
\eqno(10)$$ Since our model gives $b_e=0$ for the charged lepton sector, so that the charged lepton mass matrix is diagonal, the parameters $z_i$ given in Eq.(9) can be expressed as $$z_i = \frac{ \sqrt{m_{ei}} }{\ \sqrt{
m_e + m_\mu + m_\tau} } .
\eqno(11)$$ As a result, masses and mixings of quarks and neutrinos are only the family-number independent parameters $b_f$. (We will take the phase factors $\phi^f_i$ as $\phi^f_i=0$ except for $f=u$.) Even for the family-number dependent parameters $\phi^u_i$, we can express those by the parameters $(z_1,z_2, z_3)$ and two family-number independent parameters [@K-N_PRD15].) Thus, masses and mixings of quarks and leptons are governed by rules in U(3)$\times$U(3)$'$, not in U(3).
Note that we have used the observed values of charged lepton masses for the parameters $z_i$ given in Eq.(11). We never ask any origin of the charged lepton mass spectrum $(m_e, m_\mu, m_\tau)$. Our strategy is as follows: our aim is to describe quarks and neutrino masses and mixings only by using the observed values $(m_e, m_\mu, m_\tau)$ and without using any family-number dependent parameters. We consider that it is too early to investigate the origin of $(m_e, m_\mu, m_\tau)$, that is, U(3) symmetry breaking mechanism. It is a future task to us.
However, there are still many remaining tasks in the U(3)$\times$U(3)$'$ model. We have to improved this model into more simple and reliable model. (For a recent work in the U(3)$\times$U(3)$'$ model, for example, see Ref.[@K-N_17].)
[99]{} Y. Sumino, Phys. Lett. [**B 671**]{} (2009) 477; JHEP 0905 (2009). Y. Koide, Lett. Nuovo Cim. [**34**]{} (1982) 201; Phys. Lett. B [**120**]{}(1983) 161; Phys. Rev. D [**28**]{} (1983) 252. H. Albrecht [*et al.*]{} ARGUS collab., Phys.=Lett. [**B292**]{}, 221 (1992); J. Z. Bai [*el al.*]{} BES collab., Phys. Rev. Lett. [**69**]{}, 3021 (1982); M. Daoudi [*et al.*]{} CLEO collab., a talk given at the XXVI int. Conf. on High Energy Physics, Dallas, 1992. Y. Koide, Mod. Phys. Lett. A [**28**]{}, 2319 (1990). Y. Koide, Phys. Rev. [**D 79**]{} (2009), 033009; Phys. Lett. [**B 680**]{} (2009) 76. H. Arason, [*et al*]{}., Phys. Rev. [**D 46**]{}, 3945 (1992). Y. Koide and T. Yamashita, Phys. Lett. B [**711**]{}, 384 (2012). Y. Koide, Phys. Lett. B [**736**]{}, 499 (2014). Y. Koide and H. Nishiura, Phys.Rev. [**D 92**]{} (2015) 111301(R). Y. Koide and H. Nishiura, Mod.Phys.Lett. A [**31**]{} (2016) 1650125. Y. Koide and H. Fusaoka, Z. Phys. [**C 71**]{} (1996) 459. Y. Koide and H. Nishiura, Phys. Rev. [**D 91**]{} (2015) 116002. Y. Koide and H. Nishiura, in preparation.
[^1]: A talk given at a mini-workshop on “quarks, leptons and family gauge bosons", Osaka University, Osaka, Japan, December 26-27, 2016.
|
---
abstract: 'The work of formation of a critical nucleus is sometimes written as $W = n\Delta\mu + \gamma A$. The first term $W_\mathit{vol} = n\Delta\mu$ is called the volume term and the second term $\gamma A$ the surface term with $\gamma$ being the interfacial tension and $A$ the area of the nucleus. Nishioka and Kusaka \[J. Chem. Phys. **96** (1992) 5370\] derived $W_\mathit{vol} = n\Delta\mu$ with $n = V_\beta/v_\beta$ and $\Delta\mu = \mu_\beta(T,p_\alpha) - \mu_\alpha(T,p_\alpha)$ by rewriting $W_\mathit{vol} = -(p_\beta-p_\alpha)V_\beta$ by integrating the isothermal Gibbs-Duhem relation for an incompressible $\beta$ phase, where $\alpha$ and $\beta$ represent the parent and nucleating phases, $V_\beta$ is the volume of the nucleus, $v_\beta$, which is constant, the molecular volume of the $\beta$ phase, $\mu$, $T$, and $p$ denote the chemical potential, the temperature, and the pressure, respectively. We note here that $\Delta\mu = \mu_\beta(T,p_\alpha) - \mu_\alpha(T,p_\alpha)$ is, in general, not a directly measurable quantity. In this paper, we have rewritten $W_\mathit{vol} = -(p_\beta-p_\alpha)V_\beta$ in terms of $\mu_\mathit{re} - \mu_\mathit{eq}$, where $\mu_\mathit{re}$ and $\mu_\mathit{eq}$ are the chemical potential of the reservoir (equaling that of the real system, common to the $\alpha$ and $\beta$ phases) and that at equilibrium. Here, the quantity $\mu_\mathit{re} - \mu_\mathit{eq}$ is the directly measurable supersaturation. The obtained form is similar to but slightly different from $W_\mathit{vol} = n\Delta\mu$.'
address: 'Institute of Technology and Science, The University of Tokushima, 2-1 Minamijosanjima, Tokushima 770-8506, Japan'
author:
- 'Atsushi Mori[^1]'
title: Volume term of work of critical nucleus formation in terms of chemical potential difference relative to equilibrium one
---
A1 Critical nucleus formation work; A1 volume term A1 Gibbs formula; B1 $n\Delta\mu$\
PACS numbers: 82.60.Nh, 64.60.Q-
Introduction \[sec:intro\]
==========================
To calculate the reversible work of formation of a critical nucleus is one of the purposes of the theory of nucleation, because one can predict the steady state nucleation rate $J_{s} = J_0 \exp(-W^*/k_BT)$ through the work of formation of the critical nucleus $W$, where $k_BT$ is the temperature multiplied by Boltzmann’s constant. Here, $W^* \equiv W(R^*)$ is the height of the nucleation barrier with $R^*$ being the radius of the critical nucleus. We can refer a theory not including molecular level quantities to as a classical nucleation theory. We often encounter the following formula \[Eq. (\[eq:common\])\] or equivalent one in the classical nucleation theory: $$\label{eq:common}
W = n\Delta\mu + \gamma A,$$ with $\gamma$ begin the interfacial tension, $A \equiv 4\pi R^2$ the area of the interface (rigorously speaking, $R$ is the radius of the surface of tension) in textbooks such as [@wunderlich; @chernov; @mutaftschiev; @saito; @kashchiev; @markov] as well as research papers such as [@auer2001; @gasser2001; @kashchiev2004; @merikanto2007; @wedekind2008; @kashchiev2010; @kawasaki2010]. Nishioka and Kusaka [@nishioka1992] called this formula \`\`a commonly used formula.“ In this formula, one regards $\Delta\mu$ the chemical potential difference between the parent phase (the $\alpha$ phase) and the nucleating phase (the $\beta$ phase), i.e., $\Delta\mu \equiv \mu_\beta - \mu_\alpha (<0)$ \[in the field of crystal growth, one sometimes defines the supersaturation as $\Delta\mu \equiv \mu_\alpha - \mu_\beta (>0)$ and the negative sign arises as $-n\Delta\mu$\]. We note here that the term \`\`supersaturation” is sometimes used to express the such thermodynamic driving force, apart form the literal meaning. In this paper, we will adopt this terminology. One can understand $n$ as the numbers of molecules undergone the phase transition from the $\alpha$ phase to the $\beta$ phase. Apart form its implication, meaning of $\Delta\mu$ in this expression is immediately ambiguous; for the critical nucleus, the chemical potential of the $\beta$ phase is equal to that of the $\alpha$ phase. In this paper, we will give correct formulas for $W$ in terms of the chemical potential difference relative to equilibrium one.
The exactly correct form for $W$ given by Gibbs [@gibbs] is $$\label{eq:exact}
W = -(p_\beta - p_\alpha)V_\beta + \gamma A,$$ where $p$ denotes the pressure and $V_\beta \equiv 4\pi R^3/3$ is the volume of the nucleus. Rigorously speaking, $p_\beta$ is the pressure of the hypothetical cluster defined such as possessing the bulk property and filling the inside of the surface of tension, and thus $V_\beta$ the volume inside the surface of tension. A transparent explanation for the volume term $W_\mathit{vol} = -(p_\beta - p_\alpha)V_\beta$ can be given through a grand potential formalism — the grand potential is defined as $\Omega = -pV$; the grand potential formalism for interfaces was given in, for example, a textbook by Landau and Lifshitz [@landau] and exactness of $W_\mathit{vol} = -(p_\beta - p_\alpha)V_\beta$ is shown by a textbook by Vehkamäki [@vehkamaki] (see also [@yang1983]). The author wish to introduce a heuristic paper [@mori2013] for readers’ convenience. That is, one can readily understand the form of $W_\mathit{vol}$ on the basis of the fact that the reversible work of formation of the critical nucleus is the grand potential difference. In the following sense, Eq. (\[eq:exact\]) is entirely exact. We divide the process of nucleus formation into two. One is the formation of a hypothetical cluster of radius $R$, within which the bulk $\beta$ phase fulfills. The other is the formation of interfacial structure on the mathematical boundary of radius $R$. The former is given by the first term in Eq. (\[eq:exact\]) and the latter is expressed by the second term. The problem of classical theories is that the interfacial tension $\gamma$ is treated as constant (in particular, in a capillary approximation $\gamma$ for the flat interface is employed), whereas the curvature dependence is unnecessarily negligible [@tolman1949; @koenig1950; @buff1950; @bogdan1997; @koga1998; @bartell2001; @blokhuis2006; @troster2012].
In Sec. \[sec:common\] we will give the derivation of $W_\mathit{vol} = n [\mu_\beta(T,p_\alpha)-\mu_\alpha(T,p_\alpha)]$ due to Nishioka and Kusaka [@nishioka1992]. Here, we will give a brief review on this form. To express the pressures in terms of the chemical potentials using the Gibbs-Duhem relation was seen in Oxtoby and Kashchiev’s paper on the nucleation theorem [@oxtoby1994]. In their paper, however, $W_\mathit{vol} $ was not calculated. Laaksonen *et al.* [@laaksonen1999] pointed out that $W_\mathit{vol}$ could be calculated by extending the Oxtoby and Kashchiev’s line. This is completely the same as Nishioka and Kusaka’s [@nishioka1992]. Also, the same procedure was followed by Debenedetti and Reiss [@debenedetti1998]. In order to avoid defocusing of the point of the present paper, we will not give a general review on the nucleation theorem.
Differentiating Eq. (\[eq:exact\]) with respect to $R$ and equating zero and then solving for $R$, we have formula for the size of the critical nucleus as $$\label{eq:criticalR}
R^* = \frac{2\gamma}{p_\beta-p_\alpha}.$$ By substituting Eq. (\[eq:criticalR\]) for $R$ in Eq. (\[eq:exact\]) the hight of the nucleation barrier is obtained as $$\label{eq:criticalW}
W^* = \frac{1}{2}(p_\beta-p_\alpha)V_\beta^* = \frac{1}{3}\gamma A^* = \frac{16\pi\gamma^3}{3(p_\beta-p_\alpha)^2},$$ with $V_\beta^* \equiv 4\pi (R^*)^3/3$ and $A^* \equiv 4\pi (R^*)^2$. Corresponding to Eq. (\[eq:common\]), one has $$W^* = \frac{16\pi v^2 \gamma^3}{3(\Delta\mu)^2},$$ instead of the last expression of Eq. (\[eq:criticalW\]). Here, $v$ represent \`\`the molecular volume"; that is, $n$ (or $n^*$) is expressed as $V_\beta/v$ (or $V_\beta^*/v$). We note here that the definition of $v$ is immediately ambiguous; whose phase is not specified, or $v$ may be common to the $\alpha$ and $\beta$ phases. Also, as mentioned above, the definition of $\Delta\mu$ immediately unclear. For a rarefied $\alpha$ phase, one uses very often $$\label{eq:lnS}
\Delta\mu = k_BT \ln S,$$ where $S$ is the supersaturation ratio $p_\mathit{re}/p_\mathit{eq}$ with $p_\mathit{re}$ being the pressure of the reservoir, which equals $p_\alpha$, and $p_\mathit{eq}$ the equilibrium pressure. In other words, $\Delta\mu$ is defined as $$\label{eq:incorrect}
\Delta\mu = \mu_\alpha(T, p_\mathit{re}) - \mu_\alpha(T, p_\mathit{eq}),$$ where $T$ is the temperature, which is assumed to be uniform throughout the system. Hereafter, $T$ will be omitted for brevity. As shall be shown below, this definition is, however, incorrect. Nevertheless, expression $W^*$ in terms of this $\Delta\mu$ is strongly desired because this quantity is directly measurable. Indeed, Eq. (\[eq:incorrect\]) is widely used such as in [@mcgraw1981; @wilemski1991; @yasuoka1998; @holten2005; @noguera2006; @wasai2007; @uwaha2010].
Derivation of commonly used formula \[sec:common\]
==================================================
Nishioka and Kusaka [@nishioka1992] found out that in case that the $\beta$ phase is incompressible, the volume term $W_\mathit{vol} = -(p_\beta - p_\alpha)V_\beta$ can be rewritten in the form $n(\mu_\beta-\mu_\alpha)$. That is, they integrated $$\label{eq:dmudp}
\left(\frac{\partial\mu}{\partial p}\right)_T = v,$$ which is nothing other than Gibbs-Duhem relation for the isothermal case, for the $\beta$ phase. Unfortunately, they concluded incorrectly that the form (\[eq:common\]) was valid only for the case of the incompressible $\beta$ phase such as the nucleation of an incompressible liquid phase in a vapor phase. For example, this condition is, in a mathematical form, valid for a bubble nucleation in an incompressible liquid phase. In this paper, however, we limit ourselves to the incompressible $\beta$ phase to avoid the confusion in argument. Note that the form of Eq. (\[eq:common\]) cannot be derived in general — we can derive in some approximations.
For the case of the incompressible $\beta$ phase, $W_\mathit{vol}$ is given by [@nishioka1992] $$\label{eq:wvolbeta}
W_\mathit{vol} =\frac{V_\beta}{v_\beta} [\mu_\beta(p_\alpha) - \mu_\alpha(p_\alpha)].$$ To reach to this expression we have used the fact that the chemical potential of the nucleation phase $\mu_\beta(p_\beta)$ is equal to that of the parent phase $\mu_\alpha(p_\alpha)$, i.e., $$\mu_\beta(p_\beta) = \mu_\alpha(p_\alpha),$$ (that is, the chemical potential is uniform throughout the system) as indicated by a horizontal dashed lines in Fig. \[fig:mudrop\]. The vertical solid lines in Fig. \[fig:mudrop\] depicts $\mu_\beta(p_\alpha) - \mu_\alpha(p_\alpha)$ \[Eq. (\[eq:wvolbeta\])\]. Figure \[fig:mudrop\] (a) is for a normal case such as a liquid droplet in a vapor phase. On the other hand, Fig. \[fig:mudrop\] (b) is for an abnormal case such as formation a nucleus of less denser, incompressible $\beta$ phase in a denser $\alpha$ phase. In both cases, $\mu_\beta(p_\alpha) - \mu_\alpha(p_\alpha)$ does not coincide to the supersaturation, which is the chemical potential difference relative to the chemical potential at the $\alpha$-$\beta$ phase equilibrium (sometimes with a negative sign as note in Sec. \[sec:intro\]).
Volume term in general \[sec:general\]
======================================
In these ways, one knows two cases where the form of eq. (\[eq:common\]) is valid. Let us develop a general consideration; we consider the case that $v$ in eq. (\[eq:dmudp\]) is a function of $p$ according to the mean value theorem. Following Nishioka and Kusaka [@nishioka1992] we integrate the Gibbs-Duhem relation of the form of eq. (\[eq:dmudp\]) from $p_1$ to $p_2$. $$\label{eq:tmv}
\mu(p_2) - \mu(p_1) = v(\tilde{p})(p_2-p_1),$$ where $\tilde{p}$ is a certain value lying in the interval ($p_2$,$p_1$). Let us define $p_\mathit{eq}$ as the equilibrium pressure of the $\alpha$-$\beta$ phase equilibrium (the saturation pressure of the $\alpha$ phase with respect to the $\beta$ phase), which is the solution to $$\mu_\beta(p)=\mu_\alpha(p).$$ For latter convenience, let us define an abbreviation $\mu_\mathit{eq}$ for $\mu_\beta(p_\mathit{eq}) = \mu_\alpha(p_\mathit{eq})$. Also for latter convenience, we define $\mu_\mathit{re} \equiv \mu_\beta(p_\beta) = \mu_\alpha(p_\alpha)$, which is the chemical potential of the reservoir in $\mu VT$ ensemble. Applying eq. (\[eq:tmv\]) for the $\beta$ and $\alpha$ phase, we have $\mu_\beta(p_\beta) - \mu_\beta(p_\mathit{eq}) = v_\beta(p_\mathit{eq}+\theta_\beta(p_\beta-p_\mathit{eq}))(p_\beta-p_\mathit{eq})$ and $\mu_\alpha(p_\alpha) - \mu_\alpha(p_\mathit{eq}) = v_\alpha(p_\mathit{eq}+\theta_\alpha(p_\alpha-p_\mathit{eq}))(p_\alpha-p_\mathit{eq})$ with $\theta_\beta$ and $\theta_\alpha$ being certain values laying in an interval (0,1), i.e., $$\begin{aligned}
\label{eq:tmvbeta}
\mu_\mathit{re} - \mu_\mathit{eq} &=& v_\beta(\tilde{p_\beta})(p_\beta-p_\mathit{eq}), \\
\label{eq:tmvalpha}
\mu_\mathit{re} - \mu_\mathit{eq} &=& v_\alpha(\tilde{p_\alpha})(p_\alpha-p_\mathit{eq}),\end{aligned}$$ where $v_\beta$ and $v_\alpha$ are molecular volumes of respective phase, $\tilde{p_\beta}$ and $\tilde{p_\alpha}$ certain values respectively lying in intervals ($p_\mathit{eq}$,$p_\beta$) and ($p_\mathit{eq}$,$p_\alpha$) for cases of $p_\beta > p_\alpha > p_\mathit{eq}$ as shown in Fig. \[fig:mugeneral\] (a) \[for cases of $p_\alpha < p_\beta < p_\mathit{eq}$ as shown in Fig. \[fig:mugeneral\] (b) the intervals are replaced with ($p_\beta$, $p_\mathit{eq}$) and ($p_\alpha$, $p_\mathit{eq}$), respectively\]. By dividing Eq. (\[eq:tmvbeta\]) by $v_\beta(\tilde{p_\beta})$ and Eq. (\[eq:tmvalpha\]) by $v_\alpha(\tilde{p_\alpha})$ and subtracting the latter from the former, we eliminate $p_\mathit{eq}$ to have $$\label{eq:diffp}
\left[\frac{1}{v_\beta(\tilde{p_\beta})}-\frac{1}{v_\alpha(\tilde{p_\alpha})}\right](\mu_\mathit{re}-\mu_\mathit{eq}) = p_\beta-p_\alpha.$$ We can rewrite the first term in Eq. (\[eq:exact\]) by substituting $p_\beta-p_\alpha$ by Eq. (\[eq:diffp\]). $$\begin{aligned}
\nonumber
W_\mathit{vol} &=& -\left[\frac{1}{v_\beta(\tilde{p_\beta})}-\frac{1}{v_\alpha(\tilde{p_\alpha})}\right](\mu_\mathit{re}-\mu_\mathit{eq}) V_\beta \\
\label{eq:wvoltmv}
&=& -(n_\beta-n_\alpha)(\mu_\mathit{re}-\mu_\mathit{eq}),\end{aligned}$$ where $n_\beta \equiv V_\beta/v_\beta(\tilde{p_\beta})$ and $n_\alpha \equiv V_\beta/v_\alpha(\tilde{p_\alpha})$. This form is slightly different from the first term in Eq. (\[eq:common\]). In a case that the $\alpha$ phase is a rarefied vapor phase, the quantity $n_\alpha$ tends to vanish and the form of the first term in eq. (\[eq:common\]) is obtained. This case can be included in the case of Fig. \[fig:mudrop\] (a) (nucleation of an incompressible liquid droplet in an infinitely rarefied vapor). In this case the curve of $\mu_\alpha$ tends to a vertical line. In this limiting case the horizontal location of the intersection of horizontal dashed line with the $\mu_\alpha$ curve locates on the intersection of two $\mu$ curves; that is, $\mu_\beta(p_\alpha)-\mu_\alpha(p_\alpha)$ in eq. (\[eq:wvolbeta\]) tends to coincide to $\mu_\mathit{re}-\mu_\mathit{eq}$. Here, we note that $-(\mu_\mathit{re}-\mu_\mathit{eq})$ is the true supersaturation for the case of $p_\beta > p_\alpha > p_\mathit{eq}$ \[Fig. \[fig:mugeneral\] (a)\] and the true undersaturation for the case of $p_\beta < p_\alpha < p_\mathit{eq}$ \[Fig. \[fig:mugeneral\] (b)\]. In this way, we have successfully express the work term in terms of the supersaturation and revealed under what case the commonly used formula holds.
We have obtained a form of Eq. (\[eq:common\]) with $\Delta\mu = \mu_\mathit{re} - \mu_\mathit{eq}$. The result of Eq. (\[eq:wvoltmv\]) includes an issue concerning physicochemical problems, although it is of a mathematically beautiful form; in definitions $n_\beta \equiv V_\beta/v_\beta(\tilde{p_\beta})$ and $n_\alpha \equiv V_\beta/v_\alpha(\tilde{p_\alpha})$, the denominators $v_\beta(\tilde{p_\beta})$ and $v_\alpha(\tilde{p_\alpha})$ are not fixed constant values. We will give a prompt solution here. Assuming a smallness of $\mu_\mathit{re} - \mu_\mathit{eq}$ \[rigorously speaking the smallness should be described in terms of a dimensionless quantity — that is, the present statement reads $|(\mu_\mathit{re} - \mu_\mathit{eq})/\mu_\mathit{eq}| \ll 1$ or $|\mu_\mathit{re} - \mu_\mathit{eq}|/k_BT \ll 1$\], let us make an second-order expansion instead of Eqs. (\[eq:tmvbeta\]) and (\[eq:tmvalpha\]). Instead of Eq. (\[eq:dmudp\]), it is more convenient to start with $$\label{eq:dpdmu}
\left(\frac{\partial p}{\partial\mu}\right)_T = \rho,$$ with $\rho=1/v$ being the number density. The second-order expansions are $$\begin{aligned}
\nonumber
p_\beta-p_\mathit{eq} &=&
\rho_\beta(p_\mathit{eq}) (\mu_\mathit{re} - \mu_\mathit{eq})
+ \frac{1}{2} \rho_\beta^2(p_\mathit{eq}) \kappa_\beta(p_\mathit{eq}) (\mu_\mathit{re} - \mu_\mathit{eq})^2 \\
\label{eq:tpbetapeq}
&&
+ O((\mu_\mathit{re} - \mu_\mathit{eq})^3), \\
\nonumber
p_\alpha-p_\mathit{eq} &=&
\rho_\alpha(p_\mathit{eq}) (\mu_\mathit{re} - \mu_\mathit{eq})
+ \frac{1}{2} \rho_\alpha^2(p_\mathit{eq}) \kappa_\alpha(p_\mathit{eq}) (\mu_\mathit{re} - \mu_\mathit{eq})^2 \\
\label{eq:palphaeq}
&&
+ O((\mu_\mathit{re} - \mu_\mathit{eq})^3),\end{aligned}$$ where $\kappa$ denotes the isothermal compressibility. Subtraction Eq. (\[eq:palphaeq\]) from Eq. (\[eq:tpbetapeq\]), we have $$\begin{aligned}
\nonumber
p_\beta-p_\alpha &=&
[\rho_\beta(p_\mathit{eq})-\rho_\alpha(p_\mathit{eq})] (\mu_\mathit{re} - \mu_\mathit{eq}) \\
\nonumber
&& + \frac{1}{2} [\rho_\beta^2(p_\mathit{eq}) \kappa_\beta(p_\mathit{eq}) - \rho_\alpha^2(p_\mathit{eq}) \kappa_\alpha(p_\mathit{eq})]
(\mu_\mathit{re} - \mu_\mathit{eq})^2 \\
\label{eq:pbetapalba2nd}
&&
+ O((\mu_\mathit{re} - \mu_\mathit{eq})^3).\end{aligned}$$ Neglecting the second and higher order terms in Eq. (\[eq:pbetapalba2nd\]) and inserting in the first term of Eq. (\[eq:exact\]), we have the form of Eq. (\[eq:wvoltmv\]) with fixed $n_\beta$ and $n_\alpha$ as an approximation. Taking into account the second term, we can improve the approximation.
Coincidence between the present result and the commonly used formula with Eq. (\[eq:incorrect\]) can be understood as follows. For nucleations of incompressible $\beta$ phase in a rarefied gas $\mu_\beta(p_\alpha) - \mu_\alpha(p_\alpha)$ can be rewritten as $$\begin{aligned}
\nonumber
\mu_\beta(p_\alpha) - \mu_\alpha(p_\alpha) &=& [\mu_\beta(p_\alpha)-\mu_\mathit{eq}] - [\mu_\alpha(p_\alpha)-\mu_\mathit{eq}] \\
\nonumber
&=& [\mu_\beta(p_\alpha)-\mu_\beta(p_\mathit{eq})] - [\mu_\alpha(p_\alpha)-\mu_\alpha(p_\mathit{eq})] \\
\label{eq:kblnS}
&=& v_\beta (p_\alpha-p_\mathit{eq}) - k_BT \ln \frac{p_\alpha}{p_\mathit{eq}}.\end{aligned}$$ The last term can be expanded in $p_\alpha-p_\mathit{eq}$ as $$\begin{aligned}
\nonumber
k_BT \ln \frac{p_\alpha}{p_\mathit{eq}} &=& k_BT \ln \left[ 1 + \frac{p_\alpha - p_\mathit{eq}}{p_\mathit{eq}} \right] \\
\nonumber
&\cong& k_BT \frac{p_\alpha - p_\mathit{eq}}{p_\mathit{eq}} \\
&=& v_\alpha(p_\mathit{eq}) (p_\alpha - p_\mathit{eq}).\end{aligned}$$ To reach to the last line, the equation of state for the ideal gas has been used. Because the $\alpha$ phase is a rarefied gas, the inequality $v_\alpha \gg v_\beta$ holds and then we find that the first term in Eq. (\[eq:kblnS\]) can be neglected.
Discussion \[sec:discussion\]
=============================
In 1984, Wilemski [@wilemski1984] divided the number of molecule of species $i$ included in a nucleus for the binary system as $$\label{eq:nbns}
n_i = n_i^b + n_i^s,$$ with the superscripts $b$ and $s$ denoting bulk and surface and wrote down the condition of the critical nucleus as $$\begin{aligned}
\nonumber
0 &=& (\Delta\mu + \gamma(\partial A/n_j)_{n_i})(dn_j)_{n_i} \\
\nonumber
&& + n_1^b d\mu_1^l + n_2^b d\mu_2^l \\
\label{eq:wilemski}
&&+ n_1^s d\mu_1^l + n_2^s d\mu_2^l + Ad\gamma \mbox{\hspace{1em} (const. $T$,$P$)}.\end{aligned}$$ In 1999, Laaksonen *et al.* [@laaksonen1999] revealed the work of formation of the nucleus underlaying Eq. (\[eq:wilemski\]) as $$\label{eq:gdeltamu}
\Delta G = \sum_i (\mu_\mathit{li}(P_v) - \mu_\mathit{vi}(P_v)) g_i + A\gamma,$$ with $g_i \equiv n_\mathit{li} - n_\mathit{vi} + n_\mathit{si}$, where the subscripts l and v denote the liquid and vapor phases (in Wilemski’s paper, the starting equation is $\Delta G = n_1 \Delta\mu_1 + n_2 \Delta\mu_2 + A\gamma$). The so-called surface excess number of molecules, $n_s$, is identical to the superficial number of molecules in the Gibbs interfacial thermodynamics [@gibbs] and we find that $n_s$ is proportional to $A$.
As mentioned in Sec. \[sec:intro\], in the Gibbs interfacial thermodynamics the work of nucleus formation is divided into the formation of the hypothetical cluster and that of the interfacial structure. A quantity proportional to a superficial quantity is categorized as the latter. In this respect, in Eq. (\[eq:gdeltamu\]) the volume term is regarded as $$\label{eq:wvollaaknonen}
W_\mathit{vol} = \sum_i (\mu_\mathit{li}(P_\mathit{v}) - \mu_\mathit{vi}(P_\mathit{v})) (n_\mathit{li} - n_\mathit{vi})$$ While the factors regarded as $\Delta\mu$ are different with each other in Eqs. (\[eq:wvoltmv\]) and (\[eq:wvollaaknonen\]), the coefficients to \`\`$\Delta\mu$" coincide with each other. The difference is that in Eq. (\[eq:gdeltamu\]) the quantity $\mu_\beta(p_\alpha)-\mu_\alpha(p_\alpha)$ appears, not the quantity $\mu_\mathit{re}-\mu_\mathit{eq}$.
Concluding remarks \[sec:summary\]
==================================
We have successfully rewritten the volume term of the work of formation of a critical nucleus in terms of the supersaturation. The result is similar to the form of $W_\mathit{vol} = n\Delta\mu$ but slightly different; $n$ in this form has been replaced with $n_\beta-n_\alpha$. The form $W_\mathit{vol} = n\Delta\mu$ with $\Delta\mu$ being the supersaturation (the chemical potential difference relative to equilibrium) is recovered in a limiting case that the parent phase is a rarefied gas. This is a finding that requires a concerning note in textbooks — as mentioned in Sec. \[sec:intro\], some textbooks lead readers to understanding that $W_\mathit{vol} = n\Delta\mu$ is exact, and in some literatures this form is valid only for an incompressible nucleating phase.
We wish to postpone comparisons with experimental studies after formulation in a form of the nucleation theorem. That is, $W^*$ should be plotted against experimentally determined $\Delta\mu$’s to evaluate the deferential coefficient. In relation with experiments, a crucial comment arises: the mathematical formula itself (without expansion made in the latter part of Sec. \[sec:general\]) makes a sense — if one forcibly write the work of formation of a critical nucleus in the form like Eq. (\[eq:common\]), then uncertainties necessarily accompany in interpreting the experimental results.
acknowledgment {#acknowledgment .unnumbered}
==============
The author gratefully acknowledges discussions with Dr. Y. Suzuki. Also he thanks Prof. E. Yokoyama for reading the manuscript.
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[^1]: Corresponding author: E-mail [email protected], Tel +81-88-656-9417, Fax +81-88-656-9435
|
---
abstract: 'Learning the dynamics of robots from data can help achieve more accurate tracking controllers, or aid their navigation algorithms. However, when the actual dynamics of the robots change due to external conditions, on-line adaptation of their models is required to maintain high fidelity performance. In this work, a framework for on-line learning of robot dynamics is developed to adapt to such changes. The proposed framework employs an incremental support vector regression method to learn the model sequentially from data streams. In combination with the incremental learning, strategies for including and forgetting data are developed to obtain better generalization over the whole state space. The framework is tested in simulation and real experimental scenarios demonstrating its adaptation capabilities to changes in the robot’s dynamics.'
author:
- 'Bilal Wehbe, Marc Hildebrandt and Frank Kirchner[^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'library.bib'
title: '**A Framework for On-line Learning of Underwater Vehicles Dynamic Models** '
---
INTRODUCTION
============
Accurate modelling of robot dynamics is a critical aspect for most of control algorithms, navigation, path planing and robot simulation [@albiez2015flatfish; @hildebrandt2008computerised]. At its core, a model describes the relation between the robot’s states of motion, the actuation input and the dynamic forces and torques in play. Typically, models are manually engineered and fined-tuned to fit a specific robot design or application. This not just the case in underwater systems but also for complex robots like in [@lemburg2011AILA]. In future applications we will be able to combine this approach with techniques as presented in [@eich2010semantic] and drive underwater vehicles to perform very complex tasks.
Constructing a model for an underwater system involves computing the inertia of the robot, Coriolis and centripetal forces, and external forces such as damping and gravity. Well tuned models would serve the control purpose given that the robotic system is mechanically stationary and no external disturbances are to be expected. Nevertheless, a well-engineered but fixed model will naturally have limited usability when the mechanical structure of the system changes, or when the operating environment is non-static. This situation arises commonly in marine robotic applications, where changing a sensor or payload on the robot will result in a completely new hydrodynamic behaviour, not to mention the environmental disturbances such as bio-fouling or density fluctuations that can influence the model’s integrity.
Model learning is a technique that could avoid the manual crafting of the dynamic models, where the dynamic relations between the robot’s actions and states can be directly inferred from the data collected during operation [@nguyen2011model]. In the context of non-stationary dynamics, on-line adaptation is a beneficial perk for any control or navigation algorithm, since the changes in the system’s dynamics can be captured [@sigaud2011line].
This paper addresses the problem of on-line learning of robot dynamic models, where we use an autonomous underwater vehicle (AUV) named “Dagon” as our test subject. This vehicle was developed in the CUSLAM project [@hildebrandt2010design] and has been extensively modified and used in a number of subsequent research activities [@hildebrandt2012two; @wehbe2017experimental]. This included the addition of a hydrodynamic hull around the pressure compartments, as well as the implementation of multiple additional sensor systems. This lead to a number of severe changes in the vehicle’s hydrodynamic properties, some of which are shown in Fig. \[fig:dagon\]. Where, the top figures (\[fig:dagon1a\] & \[fig:dagon1b\]) show the current default configuration of Dagon by which it can be used for regular survey missions. The middle figures (\[fig:dagon2a\] & \[fig:dagon2b\]) show Dagon when equipped with an imaging sonar as payload, whereas the bottom figures (\[fig:dagon3a\] & \[fig:dagon3b\]) show a third configuration that represents an accidental partial loss of the vehicle’s hydrodynamic hull during operation.
Building upon our previous findings in [@wehbe2017learning; @wehbe2017online], we develop a general framework for on-line learning of AUV dynamics. Specifically, we use an incremental support vector regression (IncSVR) algorithm to model the fully coupled non-linear dynamics of the robot. To support the on-line adaptation, a forgetting strategy is implemented using density estimation technique to regulate pruning of the weights associated to data samples from different model contexts. Additionally, a node for including new data samples is used to prevent the accumulation of redundant data, and an outlier rejection node which is responsible for pruning faulty measurements. The proposed framework is validated on a dataset collected from a simulation, as well as a real dataset collected from experimentation with Dagon given the configurations listed above.
Related Work
------------
Robotic model learning has been an active topic of research during the last decade. There has been an extensive work in the domain of humanoid and manipulator arms dynamics, but not excluding any other domain of robotics. A general survey on model learning for robot control can be found in [@nguyen2011model], as well a survey on on-line regression methods for robot model learning is found in [@sigaud2011line]. An infinite mixture of linear experts approach was used in [@jamone2014incremental] to model the dynamic contexts of a manipulator arm carrying different weights. One drawback of this method is that all training samples need to be stored in the memory. Since linear experts model the dynamics only locally, the number of experts increases quickly when modelling a more complex system. The authors of [@jamone2014incremental] show that 60 experts where needed to model only two contexts of the dynamics. A mixture of Gaussian process (GP) experts with a support vector machine (SVM) classifier was used to learn different contact models of a manipulator arm in [@calandra2015learning]. This method nevertheless did not show any on-line adaptation rather the models were trained offline. In [@mckinnon2017learning], mixture of GPs was used to model different dynamics of a wheeled robot loaded with different weights. Benefiting from the Bayesian properties of GPs, a Dirichlet process (DP) was used instead as an unsupervised classifier. Two drawbacks of this method can be pointed out. Although GPs are commonly used for modelling robot dynamics, their cubic computational complexity $O(n^3)$ is a major issue when used for on-line learning [@rueckert2017learning]. Secondly, the Dirichlet process classifier is dependent on the density distribution of the data, i.e. two datasets from the same model but having different density distributions will be classified as two different models. The authors of [@nguyen2009sparse; @nguyen2011incremental] proposed an online sparsification method by using an independency measure to control the sample pruning. This method nevertheless suffers from high computational demand and memory overhead since for a dictionary of saved data points of size $n$, a matrix of size $(n-1)$ has to be saved for every sample.
In the field of marine robotics, [@fagogenis2014improving] used locally weighted projection regression to compensate the mismatch between the physics based model and the sensors reading of the AUV Nessie. Auto-regressive networks augmented with a genetic algorithm as a gating network were used to identify the model of a simulated AUV with variable mass. In a previous work [@wehbe2017online], an on-line adaptation method was proposed to model the change in the damping forces resulting from a structural change of an AUVs mechanical structure. The algorithm showed good adaptation capability but was only limited to modelling the damping effect of an AUV model. In this work we build upon our the results of [@wehbe2017learning; @wehbe2017online] to provide a general framework for on-line learning of AUV fully coupled nonlinear dynamics, and validating the proposed approach on simulated data as well as real robot data.
Online learning of AUV Dynamics {#sec:online}
===============================
In this section, we introduce the main framework for on-line learning of an AUV dynamic model. First, the dynamics of an AUV are briefly introduced. Next, we introduce the incremental support vector regression as the core learning algorithm, followed by the strategies for including, forgetting samples and outlier rejection.
AUV Dynamics
------------
The standard dynamics of submersed vehicles are expressed as the combination of Newtonian rigid-body dynamics and radiation-induced forces and moments [@fossen2002marine]. Radiation-induced forces and moments are expressed as three components: (1) added mass due to the inertia of the surrounding fluid, (2) potential damping, and (3) restoring forces due to Archimedes. In addition to potential damping, friction due to fluid viscosity and votrex shedding define the total hydrodynamic damping [@fossen2002marine]. Given $\eta$ as the pose of the body in a fixed reference coordinate frame, and $\nu$ as the velocity in body-fixed coordinates, the full dynamics of the body can be expressed as $$\bm{\dot{\nu}}= M^{-1}\left(\tau + \zeta(\bm{\eta},\bm{\nu},\bm{\tau}) - C(\bm{\nu})\bm{\nu}-d(\bm{\nu})-g(\bm{\eta})\right)\, ,
\label{dynamic}$$ where $M$, $C(\bm{\nu})\bm{\nu}$, $d(\bm{\nu})$, $g(\bm{\eta})$, $\tau$ and $ \zeta(\bm{\eta},\bm{\nu},\bm{\tau})$ are respectively the combined rigid-body and added mass inertia matrix, the Coriolis and centripetal effect, the hydrodynamic damping effect, the restoring forces and moments, the external forces applied to the body and the unmodelled effects. A compact form of the dynamic model can be written as $$\label{model}
\bm{\dot{\nu}} = \mathcal{F}(\bm{\eta},\bm{\nu},\bm{\tau})\,.\\$$ In this work, we model the thruster dynamics directly with the vehicle model, thus we eq. (2) can be written as $$\label{model2}
\bm{\dot{\nu}} = \mathcal{F^*}(\bm{\eta},\bm{\nu},\bm{n})\,.\\$$ where $\bm{n}$ is a vector representing the rotational velocities of each thruster.
Model Learning with Incremental SVR
-----------------------------------
The goal of model learning is to estimate the function $\mathcal{F^*}$ in equation (\[model2\]), by having access to its inputs and outputs. For such purpose, we use a method know as Support vector regression (SVR) [@smola2004tutorial]. One of the advantages of this method is that the model is represented by a smaller subset of the training data known as the support vectors $SV$, which we will make use of in the including and forgetting strategies explained in details later. Another advantage is the lighter computational cost $O(n^2)$ of this method, as compared to Gaussian process regression.
SVR is a supervised learning method which takes in input-output pairs as training data and learns the relation between the input and the output. The goal is thus to fit a function $f(x)=\Phi(x)^T\bm{w}$ onto a training data set $\mathcal{D}=\{(x_i,y_i)|i=1,...,n \}$, where $\Phi(x)$ is a mapping from the input space onto a higher dimension, and $\bm{w}$ is an associated weight vector. The weight vector can be expressed as a linear combination of the input features $\bm{w}=\sum^{i=1}_{n}\omega_i\Phi(x_i)$, thus the regression function can be written as $$f(x) = \sum^{i=1}_{n}\omega_i\langle \Phi(x_i), \Phi(x)\rangle = \sum^{i=1}_{n}\omega_i \kappa(x_i,x)\, ,$$ where $\kappa$ is a kernel function and $\omega_i $ are the linear weights to be estimated. SVR uses the $\epsilon$-insensitive function defined in [@smola2004tutorial] as a loss function, which penalizes the residual of the predicted output $f(x)$ and its training value $y$ only beyond a margin $\epsilon$. As opposed to GPR which estimates the weights $\omega_i$ by matrix inversion, SVR solves the problem by using the Lagrangian multipliers optimization method. This transforms the optimization into a convex problem as follows (a more detailed explanation can be found in [@smola2004tutorial]) :
$$\label{dual}
\begin{aligned}
\min_{\alpha,\beta} & \left\{ \begin{aligned}
& \frac{1}{2}\sum_{i,j=1}^n (\alpha_i-\beta_i)(\alpha_j-\beta_j)\kappa(x_i,x_j) \\
& + \epsilon \sum_{i=1}^n (\alpha_i+\beta_i) -\sum_{i=1}^n y_i(\alpha_i-\beta_i)
\end{aligned} \right. \, ,\\
s.t. & \quad 0 \leq \alpha_i,\beta_i \leq C \qquad \forall i : 1 \leq i \leq n \, , \\
& \quad \sum_{i=1}^n (\alpha_i-\beta_i)=0.
\end{aligned}$$
Where $\alpha_i,\,\beta$ are the Lagrangian multipliers, $C$ is a constant that compromises between having a more generalizing model with low weights or having too large deviations, and $\kappa(x,z)$ the kernel function. To solve (\[dual\]), the sequential minimal optimization method, as implemented in [@CC01a] was used, resulting in the final regression function $$f(x)=\sum_{i=1}^m(\alpha_i-\beta_i)\kappa(x_i,x)\, .$$ As kernel we use the squared exponential defined as $$\kappa(x,x') = \text{exp}\left(-(x-x')^T S^{-1} (x-x')\right)$$ where $S=\Sigma/\gamma$ is a matrix proportional to the covariance of the training data.
One advantage of SVRs is the $\epsilon$-insensitive loss function where all samples with a residual below the threshold $\epsilon$ are assigned a zero weight; thus, the set of samples left are used for producing predictions. These samples are called the set of support vectors $SV$. Another beneficial aspect of solving for the weights as an optimization problem is the quadratic computational complexity $O(n^2)$ in contrast to matrix inversion which is of cubic complexity $O(n^3)$.
However, for on-line learning continuous updates of the estimated weights is required as the data arrives sequentially. Starting from a set of support vectors with their corresponding weights, as a new sample arrives it is added on top of the existing samples and thus the new set is passed to the optimization algorithm. The weights of the already existing samples are kept as a starting point for the next optimization step which results in a much faster conversion since we start from a more optimal solution. Nevertheless, having a stream of continuously arriving data will lead to an unlimited growth of the set of support samples causing the memory to increase boundlessly. To obey the memory and computational constraints, we limit the set of support samples to a fixed buffer. Therefore as new samples get in, older samples has to be removed to keep a fixed size of the buffer. The straight-forward way is to first remove the oldest samples in the buffer, or what is commonly known as first-in-first-out (FIFO), since assumingly they are the most outdated samples. However, this approach could lead to a dangerous situation where the model will lose information about certain regions of the model space, if for a certain reason the new coming samples get concentrated in a local region. For example, if the mission requirements demands the robot to operate in low speeds for a prolonged time, the information about the dynamics in higher speeds can be lost over time. To cope with such shortcoming we propose the following strategies to control the adding and removal of data samples form the SVR buffer.
### Forgetting strategy
Given a limited buffer size, the main idea of the forgetting strategy is to keep a balanced global distribution of support vectors over the model’s sample space , as well as keeping the support samples inside the buffer up-to-date. However, there must be a trade-off between the density distribution of a sample inside the buffer and its queue time. The higher the density and the older a sample is, the more likely this sample will be removed. We define thus the following metric $$\label{forget}
\phi = \frac{d}{\sqrt{t_s} +k}\, ,$$ where $d$, $t_s$ and $k$ are respectively the density of a sample, the time stamp when the sample was recorded, and a constant weighting the importance of the age of the sample over its density. We use the square root of the timestamp to avoid very high values which would lead the forgetting score $\phi$ to approach zero. Eventually, the samples with the highest scores will be removed from the buffer until the maximum allowed size is reached.
To estimate the density of the samples we use multivariate kernel density estimation (KDE), where we average the correlation of a sample with respect to its neighbouring observations. The kernel function a measure of the correlation between two samples, therefore a higher weight is given if the samples are closer to each other whereas the weight vanishes as the samples get more distant. The general equation of a KDE can be written as $$\label{density}
d(x) = \frac{1}{n} \sum_{i=1}^{n} \frac{1}{det(H)} \kappa\left( H^{-1} (x-x_i)\right) \, ,$$ where $H$ is a nonsingular bandwidth matrix which need to be tuned carefully. A cross-validation optimization is quite costly with more variables in the input features vector. To avoid computational power, we can reduce the cross-validation optimization to tuning only one parameter by choosing a bandwidth matrix that is proportional to the covariance of the input data [@hardle2012nonparametric].
### Including strategy
As the stream of data is fed into the SVR learner, every sample is passed first through a filtering gate to determine if it will be used by the learner or discarded from the training step. The main motivation behind this approach is to save memory and computational power and as well prevent redundancy in the set of support vectors learned by the model. In practice, new samples that fall in a very close proximity of already existing support samples, or the residual of the target and the prediction is less than the threshold $\epsilon$ will not have any significant influence on the regression function. Additionally we make benefit from the kernel function to measure the proximity of a newly arriving sample with respect to the support vectors. Thus, a new sample $(x_i,y_i)$ is discarded from training if for any support vector $(x_{sv},y_{sv})$ one of the following is true $$\label{eq:inc_1}
\kappa(x_i,x_{sv}) > \xi \quad \text{and} \quad |f(x_i)-y_i|<\epsilon$$ $$\label{eq:inc_2}
\kappa(x_i,x_{sv}) > a \quad \text{and} \quad |y_i - y_{sv}|<b$$ where $\xi$, $a$ and $b$ are constant thresholds that can be selected through cross validation.
![Flow diagram of the on-line training framework.[]{data-label="fig:framework"}](framework.pdf){width="48.00000%"}
### Outlier rejection
The last node of the framework is to deal with outliers in the training data due to faulty sensor measurements. Although most outliers will be filtered out in the including node due to equation (\[eq:inc\_2\]), a sample that is not lying in the vicinity of any support sample would still be considered as a novelty and admitted in for training. Therefore, the outlier filter is applied on the set of support samples after each training step. To satisfy on-line training restraints, we keep a computationally low-cost approach for filtering outliers. We compute the residue of each support sample and its corresponding prediction $$\text{res}_{sv}=f(x_{sv})-y_{sv}.$$ The interquarile range (IQR) of the residuals are then computed and all samples with their corresponding residuals falling outside side of the range of $$\label{IQR}
\text{IQR} = [q1-1.5(q3-q1),q3+1.5(q3-q1)]$$ are flagged as outliers. Here, $q1$ and $q3$ represent the limits of the first and last quartile [@rousseeuw1993alternatives].
Finally, the overall framework can be viewed as five main nodes as shown in Fig. \[fig:framework\]. Data samples will be received from the stream into the including node first, then passed into the sample collector where they are merged with the processed buffer of support samples from the previous step. The stack of new training samples and older support samples are then fed into the training node. The resulting new set of support vectors are then processed by the outlier rejection and forgetting nodes and made ready for the next iteration. A pseudocode of the overall framework can be found in Algorithm \[alg1\]. We note here that although our framework uses SVR, which unlike GPR, does not estimate a true confidence interval on prediction, using the kernel density estimation can provide a measure of uncertainty of a prediction.
new data sample $(x_i,y_i)$ delete $(x_i,y_i)$ concatenate $(x_i,y_i)$ and $SV$ solve eq. (\[dual\]): compute $\alpha_i$ and $\beta_i$ compute $ \text{res}_{sv}=f(x_{sv})-y_{sv}$ calculate IQR using eq. (\[IQR\]) delete $(x_{sv},y_{sv})$ with the corresponding res is outside IQR compute $\phi$ using eq. (\[forget\]) and (\[density\]) delete $(x_{sv},y_{sv})$ with the highest $\phi$
Evaluation and results
======================
In this section, we present the evaluation procedure of our approach, where we test its performance on two datasets, one acquired from simulation and one from an actual experimental trial with the robot. Dagon is a hovering type AUV equipped with two vertical thrusters for depth and pitch stabilization (roll is passively stable), and three lateral thrusters that are used differentially to control the vehicle in the surge, sway and yaw directions. Stabilizing the vehicle in depth and pitch will help us to reduce the dimensionality of the problem, where the regression problem is then formulated as a mapping between an input feature vector composed of the vehicle’s surge, sway and yaw velocities and the lateral thrusters’ rotational speed, and an output target vector defined as the the vehicle’s acceleration in surge sway and yaw.
Data Acquisition
----------------
### Simulation Setup
As a first evaluation step of our on-line learning framework, we design a simulation environment for Dagon that allows us to easily induce changes into the dynamics. For the dynamics simulation we use the underwater dynamics plugin[^3] [@britto2017improvements] for the Gazebo simulator [@koenig2004design]. A dataset is collected for three different configurations of the vehicle’s dynamics, where the first simulates the default configuration of the robot, the second represents a damage in one of its thrusters, and the third represents a change in the damping effect due to a mechanical change in the vehicle’s structure. Actuation inputs are given to the later thrusters in the form of a sinusoidal signal of varying frequencies in order to cover a bigger range of the model’s state space. Every thruster is given a different sine wave with a period that changes randomly between 20 and 70 seconds. We sample the simulation at a frequency of 1 Hz, generating a dataset of 30,000 samples in total, divided equally between the three mentioned configurations.
### Experimental Setup
A set of experiments were carried out with Dagon in a salty water basin with a static water volume and no induced currents. The vehicle was controlled in a similar fashion as in the simulation, where we stabilize pitch and depth and let it run freely in a horizontal plane by using its lateral thruster. Dagon is equipped with a number of navigation sensors from which we will use a doppler velocity log (DVL) to measure the linear velocities and a fiber-optics-gyroscope to measure the angular velocities. All thrusters are also equipped with Hall effect sensors to measure their rotational speeds. Similar to the simulation runs, the thrusters receive a separate sinusoidal command with randomly shifting periods. A separate trial was conducted for each of the configurations shown in Fig. \[fig:dagon\], resulting in a dataset with 31300 samples, split as 11567, 10050 and 9683 samples respectively for configurations 1, 2 and 3.
Offline Learning
----------------
![Validation results of the offline trained models. Every model is trained with data from one configuration at a time, and then tested against the validation data of all three configurations. The validation results of each model shows a high score when trained and tested on data from a single configuration whereas a lower score is observed when testing with other configurations.[]{data-label="fig:offline"}](offline_both1.pdf){width="48.00000%"}
For each of the two test scenarios; simulation and real data; 80$\%$ of the data is used for training and 20$\%$ for validation. The splits are done in a stratified manner, which means for each different configuration, 20$\%$ of the data is left out for validation and the rest is used for training. As a goodness-of-fit scoring metric, we use the coefficient of determination: $$R^2 = 1- \frac{\sum_{i=1}^{n} (y^{predicted}_i-y^{true}_i)^2}{\sum_{i=1}^{n} (\bar{y}-y^{true}_i)^2},\\$$ where $\bar{y}$ is the mean of $y^{true}$. For each of the two datasets, we train three separate models as baselines for our evaluation. This means, a separate SVR is trained with the data corresponding to each configuration. Thus, the notation *“Baseline 1 - Real”* means a model is trained with the data from configuration 1 of the real dataset, *“Baseline 2 - Sim”* would indicate a model is trained with the second configuration of the simulation data, and so forth. Each baseline model is a supervised SVR by itself that has been cross-validated with the validation data to ensure the highest performance that could achieved. In Fig. \[fig:offline\] we report the validation scores of all six models tested with the validation sets of every configuration separately. The blue, orange and green bars represent the scores of the baselines trained with configurations 1, 2 and 3 of the simulation data respectively, whereas the red, purple and brown bars correspond to the baselines of the real data. It can be observed in both scenarios, that every model achieves a high validation score when it is trained and tested on data from the same configuration, whereas lower scores are reported when tested on data from other configurations. These results are not very surprising as they demonstrate that any static model trained with one configuration of dynamics cannot describe accurately other configurations, which emphasizes the need of on-line adaptation.
{width="99.00000%"}
On-line Learning with Changing Dynamics {#sec:training}
---------------------------------------
In this section we demonstrate the capability of the proposed algorithm of on-line learning and adapting to the changes in dynamics. We use the same splits of the data as of the offline scenario, but this time the data is provided sequentially to the learner. The evaluation scheme used is described as the following. (1) We continuously evaluate the performance by testing on the validation data after every training step, as the training data stream is fed in. (2) We evaluate always on the validation data that corresponds to the training data seen by the learner, i.e., if training data from the first configuration is observed by the learner, then evaluation is done on the validation data from the first configuration as well.
Hyperparameter Surge Sway Yaw
---------------- ----------- ----------- -----------
$\epsilon$ 0.1 0.001 0.1
$C$ 10 10 10
$\gamma$ 100 40 20
buffer size 900 900 900
$k$ 10 10 1
$a$ 0.99 0.99 0.99
$b$ 10$^{-2}$ 10$^{-2}$ 10$^{-2}$
$\xi$ 0.99 0.99 0.99
: Results of hyperparameter optimization[]{data-label="tab:hyperparam"}
First, the hyper-parameters for the SVR as well as the additional parameters due to the including and forgetting nodes were optimized using only the data from the first configuration, and fixed for the rest. The values of the hyper-parameters chosen via cross-validation are reported in TABLE \[tab:hyperparam\].
In Fig. \[fig:validation\], the propagation of the $R^2$ score of the real data is plotted, as the training takes place. This score results from testing against the validation data after every training step. The training is started from scratch without any prior knowledge of the model. We start by feeding in the training data from the first configuration of the robot, at this point the validation data correspond to the first configuration as well. At the early stages, a very low performance can be observed since the model has not gained enough information about the full dynamics. As time passes by, the learner experiences more training data, which shows a gradual increase in the performance until it reaches a comparable value to the offline baseline (*“Baseline 1 - Real”*). At this point, a general description of the dynamics of the first configuration is learned. At around 3000 seconds, a transition to the second configuration of the robot starts. From this point on, the validation is switched to the second configuration as well, where a drop in the performance can be noticed since the memory of the learner is still populated with information about the old configuration. As time progresses, the evaluation score increases again as the model adapts to the new configuration. A similar behaviour can also be observed from 5700 seconds on, where the data from the third configuration flows in. It can be noticed that the proposed on-line method adapts well to new configurations of the dynamic, as well as being able to reach, in every configuration, a performance comparable to the corresponding offline baselines. After every training set is finished, the prediction results over the corresponding test set are shown in Fig. \[fig:prediction\].
![Prediction results computed at the end of every training set just before switching to a new configuration. Note that all test sets are separate data that the online model did not see at training time.[]{data-label="fig:prediction"}](prediction_subplots.pdf){width="50.00000%"}
Discussion
----------
In this section we demonstrate the necessity of the including and forgetting strategies as we compare the performance of the proposed framework with the FIFO approach as well as the sparse-online-SVR (SOSVR) presented in [@nguyen2009sparse]. For the FIFO model, we use the same incremental SVR approach with a fixed buffer, but with removing the oldest samples first as new samples are included. A grid search is used to optimize the hyperparameters of the SOSVR. Fig. \[fig:comparison\] shows a comparison between all three methods for the simulation and the real test scenarios. The scores of the method presented in this paper are showed in blue whereas the FIFO method is plotted in orange and the SOSVR in green. In the early stages of training, it can be observed that all methods report similar validation scores. As the training continues, our approach shows a consistent performance in both cases, where the performance converges to a stable and accurate state. On the other hand, as the buffer of the FIFO method gets fully occupied, removing only the oldest samples in the buffer results in an unstable and jittery performance. The fluctuation of the validation score of the FIFO method can be observed throughout the whole training process. This behaviour can be interpreted by the uncontrolled pruning of data, which results in loss of important information in some regions of the model’s state space. Alternatively, incorporating the density of the support samples into the pruning procedure, helps keeping a balanced distribution, yet up-to-date samples in the learning buffer. On the other hand, the SOSVR method shows consistent adaptation with changing dynamics but a significant lower accuracy compared to the method proposed in this work.
CONCLUSIONS
===========
In this work we presented a framework for learning on-line the model of a robot and adapting to changes in its dynamics. An including and forgetting strategies were developed to control the pruning of old data data, without losing information about the global state space of the model. The proposed framework was validated in two test scenarios, a simulation and real experimental data from an AUV with different configurations, which showed its adaptation capabilities to the new dynamics.
As future work, we aim extend this method to include a database where the learned models can be stored and reused by the robot if a similar situation is encountered again.
ACKNOWLEDGMENT {#acknowledgment .unnumbered}
==============
This work was supported by the Mare-IT (grant No. O1lS17029A) and EurEx-SiLaNa (grant No. 50NA1704) projects which are funded by the German Federal Ministry of Economics and Technology (BMWi).
[^1]: All authors are with DFKI - Robotic Innovation Center, and the Department of Mathematics and Informatics, University of Bremen, Germany.
[^2]: [{name}.{lastname}@dfki.de]{}
[^3]: github.com$/$rock-gazebo$/$simulation-gazebo$\_$underwater
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